JOURNAL OF APPLIED FUNCTIONAL ANALYSIS · Approximation Theory,Nonlinear Operators. 28) Vassilis Papanicolaou Department of Mathematics National Technical University of Athens Zografou

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Bounds in spaces of Morrey under Cordes type conditions

A. Canale∗

∗ Universita degli Studi di Salerno

Facolta di Scienze MM.FF.NN.

Dipartimento di Ingegneria dell’Informazione e Matematica Applicata

Via S.Allende, 84081 Baronissi (Salerno), Italy.

[email protected]

Abstract In the study of boundary value problems for linear elliptic equations in

nondivergence form with discontinuous coefficients we consider the class of discontinuity

of Cordes type.

In particular we state some local and non local a priori bounds for solutions of Dirichlet

problem in unbounded domains.

The coefficients of lower terms in the differential operator belong to Morrey spaces

and the principal coefficients are ‘near’ to functions satisfying a condition of Cordes type.

Our results are based on embedding theorems which allow us to require a summability

lower than n for the coefficients of the operator L.

11

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS,VOL.3,NO.1,11-32,COPYRIGHT 2008 EUDOXUS PRESS, LLC

We introduce a modulus of continuity of the functions in Morrey spaces to obtain the

dependence of the constants in the estimates. We state also a result about the multiplica-

tion operator from W 1(Ω) in L2(Ω).

Mathematics Subject Classifications: 35J25, 46E35

Key words: elliptic equations, embedding theorems, a priori bounds, Morrey spaces,

Cordes condition.

1. Introduction

Boundary value problems for linear elliptic equations in nondivergence form with

discontinuous coefficients have been widely studied in bounded open sets. The paper of

Miranda [24] represent a point of reference for many authors in the study of Dirichlet

problem when coefficients have derivatives in the Ln spaces. Subsequent results were

stated, for example, in [21, 23, 28].

Other results can be found in [2, 13, 15, 16] in wider classes of spaces while different

classes of discontinuous operators were studied in [17, 18, 19, 20, 25].

When Ω is an unbounded open set, the problem was studied in more general spaces

than Ln spaces in [26], in spaces of Morrey type in [7, 9, 10, 11] and in weighted spaces in

[3, 4, 5, 6, 8, 12].

Basic tools for proving existence and, sometimes, uniqueness of solution of elliptic

boundary value problems in Sobolev spaces are a priori bounds.

CANALE

12

In this paper we state some a priori bounds for solutions of the problemLu = f , f ∈ L2(Ω) ,

u ∈W 2(Ω) ∩W 10 (Ω) ,

(1.1)

where L is the operator

Lu = −n∑

i,j=1

aij uxixj+

n∑i=1

ai uxi+ a u . (1.2)

The coefficients ai and a of the operator L belong to the class of Morrey type spaces

Mp,λ introduced in [27] which are larger than Ln spaces. We observe that, when Ω is a

bounded open set, the spaces Mp,λ(Ω) are reduced to the classical Morrey space Lp,λ(Ω)

(see [13], [14]) while, if Ω = Rn, include Lp,λ(Rn).

We require a lower summability for the coefficients of the operator L when we work

with Morrey spaces with respect the other spaces. The reason is that our embedding

theorems use some results stated by C.Fefferman [22], so we do not need to achieve n.

In this paper we consider the following class of discontinuity: the so-called Cordes

condition introduced by H.O.Cordes in the study of Holder continuity of solutions of elliptic

equations. The requirement is that the eigenvalues of the matrix of the coefficients of the

operator L do not scatter too much.

The interest of this type of conditions in the study of a priori bounds is due to

the fact we get local estimates without the introduction of functions more regular close to

coefficients aij and without further assumptions. The reason why we can apply embedding

theorems also for |x| large ‘enough’ is the kind of functions eij which approximate aij .

Derivatives of such a functions are equal to zero and, so, we do not need further hypotheses

on derivatives to use embedding results in the local a priori bounds.

BOUNDS IN SPACES OF MORREY...

13

A priori bounds (see Theorem 6.1 and Corollary 6.1 in Section 6) are obtained using

embedding theorems (see Section 3) and local a priori bounds stated in Section 5.

In particular we prove that

‖u‖W 2(Ω) ≤ c(‖Lu+ λβu‖L2(Ω) + ‖u‖L2(Ωo)

),

where λ ≥ 0, β : Ω → R+ and Ωo is an bounded open subset of Ω.

We study also the dependence of the constants. This dependence turns out to be

crucial to achieve some existence results.

To this aim it is necessary to introduce a kind of modulus of continuity of a function

g ∈ Mp,λ(Ω) (see Section 2 for definitions) and to study the multiplication operator

u −→ gu

from W 1(Ω) in L2(Ω) (see Lemma 3.1 and 3.2).

A recent paper [7] deals with problem (1.1) under conditions of Chicco type.

We remark that the two types of discontinuity require different hypotheses in the

study of local bounds.

2. Notations and function spaces

Let E be a Lebesgue measurable subset of Rn and Σ(E) the σ-algebra of Lebesgue

measurable subsets of E.

We denote by D(A) the class of restrictions to A, A ∈ Σ(E), of functions φ ∈ C∞o (Rn)

such that supp φ ∩ A ⊂ A and by Lploc(A) the class of functions f : A → C such that

CANALE

14

φf ∈ Lp(A) for any φ ∈ D(A). We set

|f |p,A = ‖f‖Lp(A) , 1 ≤ p ≤ +∞ .

Let B(x, r), x ∈ Rn, r ∈ R+, be the open ball with center in x and radius r.

For r ∈ R+, we set Br = B(0, r) and denote by ζr a function of class C∞o (Rn) such

that

suppζr ⊂ B2r, 0 ≤ ζr ≤ 1, ζr|Br=1, (ζr)x ≤ 2r.

Let Ω be an open subset of Rn. We set

Ω(x, r) = Ω ∩B(x, r) ∀x ∈ Ω , ∀ r ∈ R+ .

Let us consider the spaces Mp,λ(Ω), Mp,λ(Ω), Mp,λo (Ω) defined in [27] (we refer also to

[10] where we can find many properties of these spaces).

Let us define, for 1 ≤ p < +∞ and 0 ≤ λ < n, n ≥ 2,

Mp,λ(Ω) is the space of functions g ∈ Lploc(Ω) such that

‖g‖Mp,λ(Ω) = supx∈Ω

0<τ≤1

τ−λ/p‖g‖Lp(Ω∩B(x,τ)) < +∞ , (2.1)

equipped with the norm defined in (2.1);

Mp,λ(Ω) is the closure of L∞(Ω) in Mp,λ(Ω);

Mp,λo (Ω) is the closure of C∞

o (Ω) in Mp,λ(Ω).

From the results in [27] we have the following characterizations of the spaces Mp,λ(Ω)

and Mp,λo (Ω):


15

Mp,λ(Ω) is the subspace of Mp,λ(Ω) of the functions g ∈Mp,λ(Ω) such that:

∀ε ∈ R+ ∃δε ∈ R+ s.t. (E ∈ Σ(Ω), supx∈Ω

|E ∩B(x, 1)| ≤ δε ⇒ ‖gχE‖Mp,λ(Ω) ≤ ε) , (2.2)

Mp,λo (Ω) is the subspace of Mp,λ(Ω) of the functions g ∈Mp,λ(Ω) such that:

∀ε ∈ R+ ∃hε, kε ∈ R+ s.t. (E ∈ Σ(Ω), |E ∩B(0, kε)| ≤ hε ⇒ ‖gχE‖Mp,λ(Ω) ≤ ε) . (2.3)

Let us set:

Mp(Ω) = Mp,0(Ω) , Mp(Ω) = Mp,0(Ω) , Mpo (Ω) = Mp,0

o (Ω).

The spaces Mp(Ω) and Mpo (Ω) have been introduced and studied in [26].

It is useful to recall some results about Morrey type spaces introduced above.

We have the embedding:

Mpo,λ0(Ω) → Mp,λ(Ω) , p ≤ po ,λ− n

p≤ λ0 − n

po

which implies in particular that:

L∞(Ω) → Mp,λ(Ω) .

The following inclusions hold:

Ln(Ω) ⊂Mn,0(Ω) ⊂M s,n−s(Ω) , s ∈]2, n[ . (2.4)

For example the constant functions belong to Mn,0(Ω) but do not belong to Ln(Ω). Fur-

thermore the function f(x) = 11+|x|α ∈ Mp,0(Ω) if α > 0 while belongs to Lp(Ω) if

α ∈ [0, np [.

CANALE

16

Remark 2.1 - We remark that if g ∈ Lploc(Ω), 1 ≤ p < +∞, and φ ∈ D(Ω), then

φ g ∈Mp,λ0 (Ω) and as a consequence to the space Mp,λ(Ω) (see [7, Lemma 2.1 and Remark

2.1]).

3. Embedding results

Embedding results due to C.Fefferman [22] (see also [14]) allow us to state the following

lemma (see [27]).

Lemma 3.1 - If Ω has the cone property and g ∈M s,n−s(Ω), s ∈]2, n], then for any

u ∈W 1(Ω) we get gu ∈ L2(Ω) and

|g u|2,Ω ≤ H ‖g‖Ms,n−s(Ω) ‖u‖W 1(Ω) , (3.1)

where the constant H, independent of g and u, depends on n and s.

Let us define the modulus of continuity of a function g ∈ Mp,λ(Ω) (see also [9]).

If p ∈ [1,+∞[, λ ∈ [0, n[ and g ∈ Mp,λ(Ω) , we set

τpλ [g](t) = sup

E∈Σ(Ω)supx |E∩B(x,1)|≤t

‖g χE‖Mp,λ(Ω) , t ∈ R+ ,

where χE is the characteristic function of E. From (2.2) it follows that that g ∈ Mp,λ(Ω)

if and only if g ∈Mp,λ(Ω) and

limt→0

τpλ [g](t) = 0 .

We define the modulus of continuity of g ∈ Mp,λ(Ω) as a function τ [g] : R+ → R+

satisfying

τpλ [g](t) ≤ τ [g](t) , ∀t ∈ R+ , lim

t→0τ [g](t) = 0 .


17

In the case g : Ω → R, we put

Ar(g) = x ∈ Ω : |g(x)| ≥ r , r ∈ R+ .

If g ∈ Lploc(Ω), p ∈ [1,+∞[, we get

limr→+∞ |Ar(g) ∩B(x, 1)| = 0 .

Let us denote, for all k ∈ R+, by rk = rk(g) a real number such that

|Ark(g) ∩B(x, 1)| ≤ 1

k + 1(3.2)

and by r[g] the function

r[g] : k ∈ R+ → r[g](k) = rk ∈ R+ . (3.3)

Now we state the following lemma which we will use later. In [27] and in [10] we can find

a similar inequality, but in this paper we emphasize the dependence of the constant in the

final bound.

Lemma 3.2 - In the same hypotheses of Lemma 3.1 and if g ∈ M s,n−s(Ω), s ∈]2, n],

then for any k ∈ R+ we have

|g u|2,Ω ≤ H τ [g](

1k + 1

)‖u‖W 1(Ω) + r[g](k) ‖u‖L2(Ω) ∀u ∈W 1(Ω) ,

where H is the constant in (3.1), τ [g] is the modulus of continuity of g in M s,n−s(Ω) and

r[g] is the function defined by (3.3).

Proof. Let

gk = (1 − χArk) g .

CANALE

18

The function gk so defined belongs to the space L∞(Ω). From Lemma 3.1 we get

|g u|2,Ω ≤ |(g − gk) u|2,Ω + |gk u|2,Ω ≤

≤ H‖g − gk‖Ms,n−s(Ω) ‖u‖W 1(Ω) + |gk u|2,Ω =

= H‖gχArk‖Ms,n−s(Ω) ‖u‖W 1(Ω) + r[g](k)|u|2,Ω .

Taking in mind (3.2) and modulus of continuity we deduce the result.

4. Hypotheses

Let us set

B+ = x ∈ B1 : xn > 0 , Bo = x ∈ B1 : xn = 0 ,

and suppose that

h1) there are a d ∈ R+, an open cover Uii∈I of ∂Ω and, for any i ∈ I, a

C2-diffeomorphism ψi : U i → B1 such that:

• ψi(Ui ∩ Ω) = B+ , ψi(Ui ∩ ∂Ω) = Bo ;

• the components of ψi and ψ−1i and of their first and second derivatives are

bounded by a constant independent of i;

• for any x ∈ Ωd there exists an i ∈ I such that B(x, d) ⊂ Ui and, for any

x ∈ Ω \ Ωd, we get B(x, d) ⊂ Ω, where Ωd = x ∈ Ω : dist(x, ∂Ω) < d.

Remark 4.1 - It is easy to prove that h1) holds when Ω has the uniform C2 -

regularity property defined in [1].


19

Remark 4.2 - The condition h1) implies that there exists a number ρ ∈ R+ such

that, for any x ∈ Rn, B(x, ρ) ∩ ∂Ω = ∅ or B(x, ρ) ∩ ∂Ω = ∅ and B(x, ρ) ⊂ Ui for some

i ∈ I.

Let us consider in Ω the second order linear differential operator

Lu = −n∑

i,j=1

aij uxixj+

n∑i=1

ai uxi+ a u (4.1)

with the following conditions on the coefficients:

h2) aij = aji ∈ L∞(Ω) , i, j = 1, . . . , n ,

h3) ai ∈ M s,n−s(Ω) , i = 1, . . . , n , a ∈ M t(Ω) ,

where

s ∈]2, n] , t = 2 if n = 3 , t > 2 if n = 4 , t =n

2if n > 4 .

h4) (Cordes type condition)

ess infΩ

(n∑

i=1

aii

)2 n∑

i,j=1

a2ij

−1

> n− 1 .

Previous condition can be written in the following equivalent form

ess supΩ

n∑i,j=1

(δij − gaij

)2

< 1 , (4.2)

where g =∑

n

i,j=1δij aij∑n

i,j=1a2

ij

.

CANALE

20

Let us set

ux =( n∑

i=1

u2xi

)1/2

, uxx =( n∑

i,j=1

u2xixj

)1/2

.

We consider a function β : Ω → R+ such that the following hypothesis holds:

h5) β ∈ M t(Ω) and ∃ δ ∈ M s,n−s(Ω) such that βx ≤ β δ .

For example, some functions which satisfy the hypothesis h5) are given by β = 1 or

β(x) = 1(1+|x|2)τ , x ∈ Ω , τ > 0 .

Remark 4.3 - Let us note that hypothesis h4) implies that operator L defined in

(4.1) is uniformly elliptic in Ω.

Remark 4.4 - One can show that under hypotheses h1)−h3) and h5) it follows that

for any s, λ ∈ R the operator

u ∈W 2(Ω) → Lu+ λβ u ∈ L2(Ω)

is bounded.

5. Local a priori bounds

Let us set

Lou = −n∑

i,j=1

aij uxixj ,

and let us fix a bounded open subset V of Rn such that

V ⊂ Ω or V ∩ ∂Ω = ∅ and V ⊂ Ui for some i ∈ I .


21

We can prove the following Lemma using as tool Lemma 3.2. We remark that Cordes

conditions are sufficient to get estimates for |x| large enough.

Lemma 5.1 − If the conditions h1) − h5), hold and λ1 is a real number, then there

exists a constant c ∈ R+ such that for any λ ∈ [λ1,+∞[ and for any function v satisfying

v ∈W 2(Ω) ∩W 10 (Ω) , supp v ⊂ V .

we get

|vxx|2,Ω ≤ c(|Lv + λ g−1 β v|2,Ω + |vx|2,Ω + |v|2,Ω

), (5.1)

where c is a positive constant depending on n, s, t, ‖aij‖∞, τ [δ], τ [β], τ [ai], τ [a], r[δ],

r[β], r[ai], r[a].

Proof. We start proving the inequality

|vxx|22,Ω ≤∣∣∣∣∣∣−

n∑i,j=1

δijvxixj+ λβ v

∣∣∣∣∣∣2

2,Ω

+ |η vx|22,Ω , (5.2)

where η =∑n

i,j=1 δij δ and λ ≥ 0.

In fact we have for λ ≥ 0

∫Ω

(−

n∑i,j=1

δijvxixj+ λβ v

)2

dx ≥∫

Ω

(−

n∑i,j=1

δijvxixj

)2

dx+ λ2

∫Ω

β2v2 dx+

+ 2λ∫

Ω

β v2x dx− 2λ

∫Ω

β η |v| vx dx .

(5.3)

Using the inequality

∫Ω

β η |v| vx dx ≤ λ

2

∫Ω

β2v2 dx+12λ

∫Ω

|η vx|2 dx ,

from (5.3) we get (5.2).

CANALE

22

We proceed using hypothesis of Cordes type to get the result. Indeed if we set

h = ess supΩ

n∑

i,j=1

|δij − gaij|2

1/2

,

from inequality (5.2) we get

|vxx|2,Ω ≤∣∣∣∣∣−

n∑i,j=1

(δij − g aij)vxixj+ g Lo v + λβ v

∣∣∣∣∣2,Ω

+ |η vx|2,Ω ≤

≤ h |vxx|2,Ω + ‖g‖∞∣∣Lov + λ g−1β v

∣∣2,Ω

+ |η vx|2,Ω ,

from which we deduce the inequality

|vxx|2,Ω ≤ c1

(|Lov + λ g−1β v|2,Ω + |η vx|2,Ω

), (5.4)

since 1 − h > 0 from (4.2).

The function η ∈ M s,n−s(Ω), then we can use Lemma 3.2 to estimate the last term

in (5.4) to get

|vxx|2,Ω ≤ c2

(|Lo v + λ g−1 β v|2,Ω + |vx|2,Ω + |v|2,Ω +

+H τ [η](

1k + 1

)|vxx|2,Ω

).

(5.5)

By definition of modulus of continuity given in Section 3 it follows that there exists k0 ∈ R+

such that from (5.5)

|vxx|2,Ω ≤ c3

(|Lo v + λ g−1β v|2,Ω + |vx|2,Ω + |v|2,Ω

). (5.6)

If λ1 < 0, we fix λ ∈ [λ1, 0[.

Using h5) and applying to β Lemma 3.2 we get the bound

|λ g−1 β v|2,Ω ≤ c5|λ1| (ess inf g)−1(|vx|2,Ω + |v|2,Ω

). (5.7)


23

Now if we consider the inequality (5.6) with λ = 0, from (5.7) we easly deduce (5.1) with

Lo instead of L.

Finally, applying Lemma 3.2 to the functions ai and a verifying hypothesis h3) we

obtain the result.

Remark 5.1 - Lemma 5.1 can be proved in more general hypotheses, that is under

Chicco type conditions (see [7] and, in weighted spaces, [3], [4]). In such a case the function

η depends also on derivatives of functions which approximate aij and we can apply Lemma

3.2 for |x| large enough introducing further assumptions. So the two types of discontinuity

require different hypotheses in the study of local bounds.

6. A priori bounds

We assume that the following further hypotheses hold:

h6) there exists a function γ : R+ → R+ such that

ess supΩ\Bk

n∑i,j=1

|cij − gaij| ≤ γ(k) , ∀k ∈ R+ , limk→+∞

γ(k) = 0 ,

where cij , for i, j = 1, ..., n, are constant functions satisfying

cij = cji , i, j = 1, . . . , n ,n∑

i,j=1

cij ξi ξj ≥ ν |ξ|2 ∀ ξ ∈ Rn , a.e. in Ω ,

with ν positive constant independent of x and ξ ;

h7)

ai ∈M s,n−s0 (Ω) , i = 1, ..., n , ess inf

Ωa > 0 .

CANALE

24

Local a priori bound stated in Lemma 5.1 allows us to prove the following result.

Theorem 6.1 − If the hypotheses h1)−h7) hold, then there exist a constant c ∈ R+

and a bounded open set Ωo ⊂⊂ Ω such that

‖u‖W 2(Ω) ≤ c

(|Lu+ λ g−1 β u|2,Ω + |u|2,Ωo

)(6.1)

∀u ∈W 2(Ω) ∩W 10 (Ω) , ∀λ ≥ 0 ,

where c is a positive constant depending on Ω, ν, n, s, t, ai, ‖aij‖∞ , cij , τ [δ], τ [β],

τ [a], r[δ], r[β], r[a].

Proof. • Step 1 (Estimates at infinity).

If the principal coefficients of L are suitable constants, we can use Corollary 5.2 in

[10] to get the bound (6.2). Therefore if

Lo = −n∑

i,j=1

cij∂2

∂xi∂xj,

and g =∑n

i,j=1δij aij∑

n

i,j=1a2

ij

, we have that

‖(1 − ζk) u‖W 2(Ω) ≤ c1

∣∣∣∣Lo((1 − ζk) u) + (ga+ λβ)(1 − ζk) u∣∣∣∣2,Ω

, (6.2)

from which

‖(1−ζk) u‖W 2(Ω) ≤ c1

(∣∣∣∣−n∑

i,j=1

(cij − gaij

)((1 − ζk) u

)xixj

+

− g

n∑i,j=1

aij

((1 − ζk) u

)xixj

+ (ga+ λβ)(1 − ζk) u∣∣∣∣2,Ω

)≤

≤ c1

(‖g‖∞|Lo

((1 − ζk) u

)+ (a+ λ g−1 β)(1 − ζk) u|2,Ω+

+ γ(k)|((1− ζk) u)xx|2,Ω

).

(6.3)


25

Taking in mind h6), by a suitable choice k = k0 ∈ R+ we get from (6.3)

‖(1 − ζk0) u‖W 2(Ω) ≤ c2

∣∣∣∣Lo

((1 − ζk0) u

)+ (a+ λ g−1 β)(1 − ζk0) u

∣∣∣∣2,Ω

. (6.4)

• Step 2 (Estimates on bounded sets).

Let us consider a function ϕ ∈ C∞0 (Rn) such that:

ϕ|B 12

= 1, supp ϕ ⊂ B1 , supRn

|∂αϕ| ≤ cα ∀α ∈ Nno .

Let us define for x ∈ Ω,

Φ = Φx : y ∈ Rn → ϕ

(x− y

τ

).

We have

Φ|B(x, τ2 ) = 1, supp Φ ⊂ B(x,τ) , sup

Rn

|∂αΦ| ≤ c′α ∀α ∈ Nn0 ,

where c′α = cατ−|α|.

So, if u ∈ W 2(Ω) ∩W 10 (Ω), then the function v = Φu ∈ W 2(Ω) ∩W 1

0 (Ω) and either

supp v ⊂ Ω or supp v ∩ ∂Ω = ∅ and supp v ⊂ Ui for some i ∈ N .

Let us fix k ∈ R+ and set w = ζku. Then, we can apply Lemma 5.1 with v = Φw and

L = Lo + a to get

|(Φw)xx|2,Ω ≤ c3

(|Lo(Φw) + (a+ λg−1β)Φw|2,Ω + |(Φw)x|2,Ω + |Φw|2,Ω

). (6.5)

The first term of the right hand side in (6.5) can be bounded as it follows:

|Lo(Φw) + (a+ λg−1β)Φw|2,Ω ≤ |Φ(Low + (a+ λg−1β)w)|2,Ω+

+ 2 supi,j

‖aij‖L∞(Ω)|Φxwx|2,Ω + supi,j

‖aij‖L∞(Ω)|Φxxw|2,Ω ≤

≤ c4

(|Low + (a+ λg−1β)w|2,Ω(x,τ) + |wx|2,Ω(x,τ) + |w|2,Ω(x,τ)

).

(6.6)

CANALE

26

Hence from (6.5) and (6.6) we deduce the inequality

|wxx|2,Ω(x, τ2 ) ≤ c5

(|Low + (a+ λg−1β)w|2,Ω(x,τ) + |wx|2,Ω(x,τ) + |w|2,Ω(x,τ)

).

Therefore, applying Lemma 1.1 in [10], we obtain

|(ζku)xx|2,Ω ≤ c6

(∣∣(Lo(ψku)+(a+ λg−1β)ψku∣∣2,Ω

+ |(ζk)x u|2,Ω+

+ |ζk ux|2,Ω + |ζku|2,Ω

).

(6.7)

Using the well known inequality (see [1])

|ux|2,suppζk≤ K(ε|uxx|2,suppζk

+ ε−1|u|2,suppζk) , (6.8)

where K = K(n,Ω) and 0 < ε < ε0, ε0 > 0, by (6.7)

‖ζku‖W 2(Ω) ≤ c7

(∣∣(Lo(ζku) + (a+ λg−1β)ζku∣∣2,Ω

+ |ζku|2,Ω

). (6.9)

By (6.4) and (6.9) with k = k0 and using again (6.8) we get

‖u‖W 2(Ω) ≤ c8(|Lo u+ (a+ λ g−1 β) u|2,Ω + |u|2,Ω′

o

), (6.10)

with Ω′o = supp ζk0 .

Moreover from Lemma 3.4 in [10] we have that for any ε ∈ R+ there exist c(ε) ∈ R+

and an open set Ωε ⊂⊂ Ω such that

n∑i=1

‖aiuxi‖L2

s(Ω) ≤ ε ‖u‖W 2s (Ω) + c(ε) |u|2,Ωε

. (6.11)

From (6.10) and (6.11) we deduce the assertion with Ωo = Ω′o ∪ Ωε .

Remark 6.1 − We observe that in Theorem 6.1 we can suppose in place of the

condition ess infΩ a > 0 in h7)

a = a′ + a′′ , a′ ∈M t0(Ω) , ess inf

Ωa′′ > 0 .


27

Remark 6.2 − A different assumption in Theorem 6.1 could be the convergence

of aij to more regular functions αij at infinity. For example functions such that (αij)xh

belong to the space M s,n−s0 (Ω). Then we can modify the proof in Step 1 setting

Lo = −n∑

i,j=

αij∂2

∂xi∂xj

and using h6) with αij in place of cij .

From Theorem 6.1 it follows the following

Corollary 6.2 − In the same hypotheses of Theorem 6.1 and if

β−1 ∈ L∞loc(Ω) (6.12)

then for any s ∈ R there exist c, λ0 ∈ R+ such that

‖u‖W 2(Ω) ≤ c∣∣Lu+ λ g−1 β u

∣∣2,Ω

(6.13)

∀u ∈W 2(Ω) ∩W 10 (Ω) , ∀λ ≥ λ0 ,

where c has the same dependence of the constant in Theorem 6.1.

Proof. Using hypotheses (6.12) and taking in mind Remark 4.4 and Theorem 6.1

it follows that

λ|u|2,Ωo≤ c1|λβg−1u|2,Ωo

≤ c2

(|Lu+ λβg−1u|2,Ω + ‖u‖W 2(Ω)

)≤

≤ c3

(|Lu+ λβg−1u|2,Ω + |u|2,Ωo

) (6.14)

for any u ∈ W 2(Ω) ∩ W 10 (Ω) and for any λ ∈ R+, where Ωo is the open set in

Theorem 6.1. For λ large enough we deduce the result by (6.1) and (6.14).

CANALE

28

Remark 6.3 − Inequality (6.13) can be obtained under different assumptions if we

suppose coefficients of the operator L more regular. We refer to the paper [10] where we

can find some results. We remark that Theorem 6.1 allows us to obtain the result stated in

Corollary 6.2 under hypotheses considerably weakened with respect to previous papers.

References

[1] R.A.Adams, Sobolev spaces, Academic Press, 1975.

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first derivatives weakly-Ln, Ann. Mat. Pura Appl. 138 (1984), 331-340.

[3] A. Canale, A priori bounds in weighted spaces, J. Math. Anal. Appl. Vol. 287 (4)

(2003), 101-117.

[4] A.Canale, Existence and uniqueness results in weighted spaces, J. Concr. Appl. Math.

Vol. 1 (4) (2003), 307-323.

[5] A. Canale, On some results in weighted spaces under Cordes conditions, preprint,

2005.

[6] A. Canale, On some results in weighted spaces under Chicco conditions, preprint,

2005.

[7] A. Canale, Bounds in spaces of Morrey under Chicco type conditions, preprint, 2005.

[8] A. Canale - L. Caso - M. Transirico, Second order elliptic equations with disconti-

nuous coefficients in irregular domains Rend. Accad. Naz. Sci. XL Mem. Mat., 24

(2000), 63-79.


29

[9] A.Canale - P.Di Gironimo - A.Vitolo, Functions with derivatives in spaces of Morrey

type and elliptic equations in unbounded domains, Studia Math. (3) 128 (1998),

199-218.

[10] A.Canale - M.Longobardi - G.Manzo, Second order elliptic equations with discontin-

uous coefficients in unbounded domains, Rend. Accad. Naz. Sci. XL Mem. Mat. 18

(1994), 41-56.

[11] A.Canale - M.Longobardi - G.Manzo, Existence and uniqueness results for second

order elliptic equations in unbounded domains, Rend. Accad. Naz. Sci. XL Mem.

Mat. 18 (1994), 171-187.

[12] L.Caso - M.Transirico, The Dirichlet problem for second order elliptic equations with

singular data, Acta Math. Hungar. 76 (1997), 1-16.

[13] F.Chiarenza - M.Franciosi, A generalization of a theorem by C.Miranda, Ann. Mat.

Pura Appl. 161 (1992), 285-297.

[14] F.Chiarenza - M.Frasca, A remark on a paper by C.Miranda, Proc. Amer. Math.

Soc. 108 (1990), 407-409.

[15] F.Chiarenza - M.Frasca - P.Longo, Interior W 2,p estimates for non divergence elliptic

equations with discontinuous coefficients, Ricerche Mat. 40 (1991), 149-168.

[16] F.Chiarenza - M.Frasca - P.Longo, W 2,p-solvability of the Dirichlet problem for non-

divergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336

(1993), 841-853.

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[17] M.Chicco, Equazioni ellittiche del secondo ordine di tipo Cordes con termini di ordine

inferiore, Ann. Mat. Pura Appl. (4) 85 (1970), 347-356.

[18] M.Chicco, Dirichlet problem for a class of linear second order elliptic partial differential

equations with discontinuous coefficients, Ann. Mat. Pura Appl. 92 (1972), 13-22.

[19] M.Chicco, Principio di massimo per soluzioni di equazioni ellittiche del secondo ordine

di tipo Cordes, Ann. Mat. Pura Appl. 100 (1974), 239-258.

[20] M.Chicco, Osservazioni sulla risolubilita del problema di Dirichlet per una classe

di equazioni ellittiche a coefficienti discontinui, Rend. Sem. Mat. Univ. Padova 66

(1982), 137-141.

[21] M.Chicco, Su un classe di equazioni ellittiche del secondo ordine in forma non varia-

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form equations with rough coefficients, Ricerche Mat., 38 (1989), 93-106.

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31

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divergence form on unbounded domains, Boll. Unione Mat. Ital. 9 (1995), 153-174.

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Mat. 4 (1984), 617-632.

CANALE

32

Applications of Randomly Pseudo-monotoneOperators with Randomly Upper semicontinuity inGeneralized Random Quasivariational Inequalities

M.K. Ahmad?, A.H. Siddiqi?? and Salahuddin?

?Department of MathematicsAligarh Muslim University, Aligarh-202 002 (U.P.), Indiaahmad [email protected]; [email protected]

??Department of Mathematical SciencesKing Fahd University of Petroleum & Minerals,KFUPM 1745, Dhahran - 31261, Saudi Arabia

[email protected]

Abstract: Let (;) be a measurable space, E a topological vector space and X a nonemptysubset of E. Let S : X ! 2X and T : X ! E be two random mappings. Thenthe generalized random quasi-variational inequality (GRQVI) is to nd for a measurable mapy : ! X such that y(!) 2 S(!; y(!)), w 2 T (!; y(!)) and

Rehw; y(!) x(!)i 0; 8 x(!) 2 S(!; y(!)):

We use Chowdhury and Tan's [6] generalized version of Ky Fan's minimax inequality as a tool

to obtain some general theorems on random solutions of the GRQVI on a paracompact set X

in a Hausdor locally convex space. The random multivalued operator T is either randomly

strong psudo-monotone or randomly pseudo-monotone and is randomly upper semicontinuous

from Co(A) to the weak-topology on E for each nonempty nite subset A of X.

Key words: Measurable space, generalized quasi-variational inequality, locally convexspace, partition of unity, paracompact set, randomly lower semicontinuous, randomly up-per semicontinuous, randomly strong pseudo-monotone, -algebra.

AMS Subject Classications: 47H04, 47H05, 47H10, 49J35, 54C60

1. Introduction

The theory of variational inequalities provides a natural and elegant framework forthe study of many seemingly unrelated free boundary value problems arising in variousbranches of engineering and mathematical sciences. Variational inequalities have manynice results related to nonlinear partial dierential equations. Complementarity problem,

33


which is closely related to variational inequality problem, plays an important role ingeneral equilibrium theory, economics, management sciences and operations research.

An important and useful generalization of variational inequality is the quasi-variationalinequality introduced and considered by Bensousson and Lions [2]. For further details,we refer to Baiocchi and Capelo [1]. On the other hand, Pang [17] has considered thequasi-complementarity problem. Karamardian [13] showed that if the set involved in avariational inequality problem is a convex cone, then variational inequality and comple-mentarity problems have the same solution set. Pang [17] proved that the same relationis true for the quasi-complementarity problem and quasi-variational inequality problem.

The fundamental theory of random operators is an important branch of stochasticanalysis and its development is required for the study of several classes of random oper-ator equations. Almost half a century ago, the systematic study of random xed pointwas initiated by the Prague school of probabilists. However, it received the attention itdeserved only after the appearance of the survey paper by Bharucha-Reid [3] in 1976.Since then this discipline has been developed further in which many profound conceptsand results were established with considerable generality, see for instance, the work ofShahzad [16], Xu [24], Itoh [12], Liu [15], Papageorgiou [18], Tan and Yuan [23], Yuan[25], Salahuddin [20], Khan and Salahuddin [11] etc.

The aim of this paper is to make further investigations in the same direction. Weshall use Chowdhury and Tan's results [7,8] and Ky Fan's minimax inequality [10] astools to obtain some general theorems on solutions of the GRQVI on a paracompact setX in a locally convex Hausdor topological vector space, where the multivalued randomoperator T is randomly strong pseudo-monotone or randomly pseudo-monotone and isupper semicontinuous from Co(A) to the weak-topology on E for each A 2 F(X).

We shall use our following multivalued generalization of the classical random pseudo-monotone operators. The classical denition of a pseudo-monotone operator was intro-duced by Brezis, Nirenberg and Stampacchia in [4].

Let X be a set, 2X the family of all nonempty subsets of X and F(X) the family of allnonempty nite subsets of X. Let E be a topological vector space and E its continuousdual, hw; xi the pairing between E and E for w 2 E and x 2 E, and Rehw; xi the realpart of hw; xi. If X E, S : X ! 2X and T : X ! E, the quasi-variational inequality(QVI) is to nd a point y 2 S(y) such that

RehT (y); y xi 0; for all x 2 S(y);

which is introduced by Bensousson and Lions in 1973, see [2]. Again, we consider amultivalued mapping T : X ! 2E

, then the generalized quasi-variational inequality(GQVI) is to nd a point y 2 S(y) and a point w 2 T (y) such that

Rehw; y xi 0; for all x 2 S(y);

which is introduced and studied by Chan and Pang [5] in 1982.

AHMAD ET AL

34

A measurable space (;) is a pair, where is a set and a -algebra of subsets of. If X is a set, A X and D is nonempty family of subsets of X, we shall denote byD\A the family fD\A : D 2 Dg and by X(D) the smallest -algebra onX generated byD. If X is a topological space with topology X , we shall use B(X) to denote X(X), theBorel -algebra on X if there is no ambiguity on the topology X . Let X be a topologicalspace and F : (;) ! 2X be a correspondence, then F is said to be measurable (resp.weakly measurable) if F1(B) = f! 2 : F (!) \ B 6= g 2 E for each closed (resp.open) subset B of X. The mapping F is said to have a measurable graph if

GrafF = f(!; y) 2 X : y 2 F (!)g 2 B(X):

A function F : ! X is a measurable selection of F if f is a measurable functionsuch that f(!) 2 F (!), for all ! 2 .

Denition 1.1. Let (;) be a measurable space, E a topological vector space, X anonempty subset of E and T : X ! 2E

. If h : X ! R, then T is said to be

(i) randomly h-pseudo-monotone if for each xed ! 2 , y(!) 2 X and every randomnet fy(!)g2 in X converging to y(!) with

lim sup

[ infu2T (!;y(!))

Rehu; y(!) y(!)i+ h(!; y(!)) h(!; y(!))] 0;

we have

lim inf[ infu2T (!;y(!))

Rehu; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]

infw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!)); for all x(!) 2 X;

(ii) randomly pseudo-monotone if T is randomly h-pseudo-monotone with h 0.

2. Generalized Random Quasi-variational Inequalities for Randomly Strong

pseudo-monotone Operators

In this section, we shall introduce the notion of randomly pseudo-monotone operatorsand obtain some general theorem on solution of the GRQVI on paracompact sets in locallyconvex Hausdor topological vector spaces.

We shall begin with the following:

Denition 2.1. Let (;) be a measurable space, E a topological vector space, X anonempty subset of E, and T : X ! 2E

. If h : X ! R, then T is said to be

(i) randomly strong h-pseudo-monotone if for each continuous function : X ![0; 1], for xed ! 2 , y(!) 2 X and every random net fy(!)g2 in X convergingto y(!) with

lim sup

[(!; y(!))f infu2T (!;y(!))

Rehu; y(!)y(!)i+h(!; y(!))h(!; y(!))g] 0;

RANDOMLY PSEUDO-MONOTONE OPERATORS...

35

we have

lim sup

[(!; y(!))f infu2T (!;y(!))

Rehu; y(!)x(!)i+h(!; y(!))h(!; x(!))g]

[(!; y(!))f infw2T (!;y(!))

Rehw; y(!)x(!)i+h(!; y(!))h(!; x(!))g];

for all x(!) 2 X,

(ii) randomly strong pseudo-monotone if random operator T is randomly strong h-pseudo-monotone with h 0.

Remark 2.1. Every randomly strong pseudo-monotone operator is also a randomlypseudo-monotone operator.

Proposition 2.1. Let (;) be measurable space, X a nonempty subset of a topologicalvector space E. If T : X ! E is randomly monotone and continuous from therelative weak topology on X to the weak-topology on E, then random operator T israndomly strong pseudo-monotone.

Proof. Let : X ! [0; 1] be any arbitrary continuous random functional. Supposefy(!)g2 is a random net in X and for each ! 2 , y(!) 2 X with y(!)! y(!) (and

lim sup

[(!; y(!))fRehT (!; y(!)); y(!) y(!)ig] 0):

For any x(!) 2 X, ! 2 and > 0, there are 1; 2 2 with

j (!; y(!))RehT (!; y(!)); y(!) y(!)i j<

2; for all 1

and

j (!; y(!))RehT (!; y(!)) T (!; y(!)); y(!) x(!)i j<

2; for 2:

Choose 0 2 with 0 1; 2. Thus

(!; y(!))RehT (!; y(!)); y(!)x(!)i

= (!; y(!))hT (!; y(!)); y(!)y(!)i+(!; y(!))RehT (!; y(!)); y(!)x(!)i

(!; y(!))RehT (!; y(!)); y(!)y(!)i+(!; y(!))RehT (!; y(!)); y(!)x(!)i

= (!; y(!))RehT (!; y(!)); y(!)y(!)i+(!; y(!))RehT (!; y(!)T (!; y(!)); y(!)x(!)i

+ (!; y(!))RehT (!; y(!)); y(!)x(!)i

>

2

2+ (!; y(!))RehT (!; y(!)); y(!) x(!)i; for all 0;

so that

inf0

(!; y(!))RehT (!; y(!)); y(!) x(!)i

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+ inf0

(!; y(!))RehT (!; y(!)); y(!) x(!)i:

It follows that

lim sup

(!; y(!))RehT (!; y(!)); y(!) x(!)i

lim inf(!; y(!))RehT (!; y(!)); y(!) x(!)i

+ (!; y(!))RehT (!; y(!)); y(!) x(!)i:

As > 0 is arbitrary,

lim sup

(!; y(!))RehT (!; y(!)); y(!)x(!)i (!; y(!))RehT (!; y(!)); y(!)x(!)i:

Hence random operator T is randomly pseudo-monotone.

Theorem 2.1. Let (;) be a measurable space, E a locally convex Hausdor topologicalvector space, X a nonempty paracompact convex subset of E and h : E ! R beconvex. Let S : X ! 2X be randomly upper semicontinuous such that for each xed! 2 , each S(!; x(!)) is compact convex and T : X ! 2E

a randomly strong h-pseudo-monotone and randomly upper semicontinuous from Co(A) to the weak-topologyon E, for each A 2 F(X) and for each xed ! 2 such that T (!; x(!)) is weak-compactconvex. Suppose that the set

A = f! 2 ; y(!) 2 X : supx(!)2S(!;y(!))

[ infw2T (!;y(!))

Rehw; y(!) x(!)i

+h(!; y(!)) h(!; x(!))] > 0g

is open in X. Suppose further that there exists a nonempty compact subset K of X anda point x0(!) 2 X for xed ! 2 , such that x0(!) 2 K \ S(!; y(!)) and

infw2T (!;y(!))

Rehw; y(!); x0(!)i+ h(!; y(!)) h(!; x0(!)) > 0;

for all y 2 XnK for ! 2 .

Then there exists a measurable map y : ! K such that

(i) y 2 S(!; y(!)) and

(ii) there exists ! 2 T (!; y(!)) with

Rehw; y(!) x(!)i h(!; x(!)) h(!; y(!)); for all ! 2 ; x(!) 2 S(!; y(!)):


37

Proof. We divide the proof into two steps:

Step 1. There exists a measurable map y : ! X such that y(!) 2 S(!; y(!)) and

supx(!)2S(!;y(!))

[ infw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))] 0:

Suppose the contrary, for each xed ! 2 and y(!) 2 X, either y(!) 62 S(!; y(!))or there exists x(!) 2 S(!; y(!)) such that

infw2T (!;y(!))

Rehw; y(!) x(!))i+ h(!; y(!)) h(!; x(!)) > 0;

i.e., y(!) 62 S(!; y(!)) or y(!) 2 A, for each xed ! 2 . If for each ! 2 , y(!) 62S(!; y(!)), then by Hahn-Banach separation theorem, there exists p 2 E such that

Rehp; y(!)i supx(!)2S(!;y(!))

hp; x(!)i > 0:

For each ! 2 , y(!) 2 X, set

(y(!)) = supx(!)2S(!;y(!))

[ infw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]:

Let V0 = f! 2 ; y(!) 2 X j (y(!)) > 0g = A and for each p 2 E, set

Vp = f! 2 ; y(!) 2 X : Rehp; y(!)i supx(!)2S(!;y(!))

hp; x(!)i 0g:

Then X = V0 [S

p2E

Vp. Since each Vp is open in X, by Lemma 1 in [21], and V0

is open in X by hypothesis, then fV0; Vp : p 2 Eg is an open covering of X. SinceX is paracompact, there is a continuous partition of unity f0; p : p 2 Eg for X

subordinated to the open cover fV0; Vp : p 2 Eg (see Theorem VIII. 4.2.[9]), i.e., for eachp 2 E; p : X ! [0; 1] and 0 : X ! [0; 1] are continuous functions such that for eachp 2 E; p(y) = 0; for all y 2 XnVp and 0(y) = 0, for all y 2 XnVp and fsupport 0,support p : p 2 Eg is locally nite and 0(y(!))+

Pp2E

p(y(!)) = 1, for each y(!) 2 X.

Note that for each A 2 F(X), h is randomly continuous on Co(A), see [19, page 83].Dene : X X ! R by

(!; x(!); y(!)) = 0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i;

for each x(!); y(!) 2 X, for xed ! 2 .

Then we have the following.

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(1) Since E is Hausdor, for each A 2 F(X) and for each xed x(!) 2 Co(A), therandom map

y(!)! minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))

is randomly lower semicontinuous on Co(A), by Lemma 3 in [6]; and the fact that h israndomly continuous on Co(A), the random map

y(!)! 0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]

is randomly lower semicontinuous on Co(A), by Lemma 3 in [22]. For ! 2 and for eachxed x(!) 2 X

y(!)!X

p2E

p(!; y(!))Rehp; y(!) x(!)i

is randomly continuous on X. Hence for each A 2 F(X) and for each ! 2 , x(!) 2Co(A), the random map y(!) ! (!; x(!); y(!)) is randomly lower semicontinuous onCo(A).

(2) For xed ! 2 and for each A 2 F(X), y(!) 2 Co(A),

minx(!)2A

(!; x(!); y(!)) 0:

Indeed, if this were false, then for some A = fx1(!); ; xn(!)g 2 F(X), for xed

! 2 and some y(!) 2 Co(A), (say y(!) =nPi=1

ixi(!), where 1; ; n 0 with

nPi=1

i = 1), we have

min1in

(!; xi(!); y(!)) > 0:

Then for each i = 1; 2; ; n,

0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!)xi(!)i+h(!; y(!))h(!; xi(!))]

+X

p2E

p(!; y(!))Rehp; y(!) xi(!)i > 0;

so that

0 = (!; y(!); y(!)) = 0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!)nX

i=1

ixi(!)i


39

+h(!; y(!)) h(!;nX

i=1

ixi(!))] +X

p2E

p(!; y(!))Rehp; y(!)nX

i=1

ixi(!)i

nX

i=1

i(0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!) xi(!)i+ h(!; y(!)) h(!; xi(!))]

+X

p2E

p(!; y(!))Rehp; y(!) xi(!)i) > 0;

which is a contradiction.

(3) For xed ! 2 , suppose A 2 F(X), x(!); y(!) 2 Co(A) and fy(!)g2 is arandom net in X converging to y(!) with (!; tx(!) + (1 t)y(!); y(!)) 0, forall 2 and t 2 [0; 1]. Then for t = 0, we have

(!; y(!); y(!)) 0; for all 2 ; i.e.;

0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!) y(!)i+ h(!; y(!)) h(!; y(!))]

+X

p2E

p(!; y(!))Rehp; y(!) y(!)i 0; for all 2 :

Hence

lim sup

[0(!; y(!))( minw2T (!;y(!))

Rehw; y(!) y(!)i+ h(!; y(!)) h(!; y(!)))]

+ lim inf(X

p2E

p(!; y(!))Rehp; y(!) y(!)i)

lim sup

[0(!; y(!))( minw2T (!;y(!))

Rehw; y(!) y(!)i+ h(!; y(!)) h(!; y(!))

+X

p2E

p(!; y(!))Rehp; y(!) y(!)i)] 0:

Therefore

lim sup

[0(!; y(!))( minw2T (!;y(!))

Rehw; y(!) y(!)i+ h(!; y(!)) h(!; y(!)))] 0:

Since random operator T is randomly strong h-pseudo-monotone, we have

lim sup

[0(!; y(!))( minw2T (!;y(!))

Rehw; y(!)x(!)i+h(!; y(!))h(!; x(!)))]

0(!; y(!))( minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))):

Thus

lim sup

[0(!; y(!))( minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!)))]

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40

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i

0(!; y(!))( minw2T (!;y(!))

Rehw; y(!)x(!)i+h(!; y(!))h(!; x(!)))

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i: (2:1)

For t = 1, we have

(!; x(!); y(!)) 0; for all 2 ; i.e.;

0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i 0; for all 2 :

Therefore

lim sup

[0(!; y(!))( minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!)))]

+ lim inf[X

p2E

p(!; y(!)Rehp; y(!) x(!)i]

lim sup

[0(!; y(!))( minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i)] 0:

Thus

lim sup

[0(!; y(!))( minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!)))]

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i 0: (2:2)

Hence by (2.1) and (2.2), we have

(!; x(!); y(!)) 0:

(4) By hypothesis, there exists a nonempty compact subset K of X and a point x0(!) 2X such that x0(!) 2 K \ S(!; y(!)) and

infw2T (!;y(!))

Rehw; y(!)x0(!)i+h(!; y(!))h(!; x0(!)) > 0; for each xed ! 2 ; y 2 XnK:

Thus for each y 2 XnK,

0(!; y(!))[ infw2T (!;y(!))

Rehw; y(!) x0(!)i+ h(!; y(!)) h(!; x0(!))] > 0;


41

whenever 0(!; y(!)) > 0 and Rehp; y(!) x0(!)i > 0, for p 2 E: Consequently,

(!; x0(!); y(!)) = 0(!; y(!))[ infw2T (!;y(!))

Rehw; y(!)x0(!)i+h(!; y(!))h(!; x0(!))]

+X

p2E

p(!; y(!))Rehp; y(!) x0(!)i > 0;

for all y 2 XnK; for xed ! 2 .

Therefore satises all hypothesis of Theorem 2 in [6]. Hence by Theorem 2 in [6],there exists a measurable map y : ! K such that

(!; x(!); y(!)) 0; for all x(!) 2 X and for each ! 2 ; i.e.;

0(!; y(!))[ infw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i 0; (2:3)

for each xed ! 2 .

If (y(!)) 0, choose any x(!) 2 S(!; y(!)), such that

infw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!)) (y(!))

2> 0:

If 0(!; y(!)) > 0, then y(!) 2 V0 2 A, so that (y(!)) > 0. It follows that

0(!; y(!))[ infw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))] > 0:

If p(!; y(!)) > 0, for some p 2 E, then y(!) 2 Vp and hence

Rehp; y(!)i > supx(!)2S(!;y(!))

Rehp; x(!)i Rehp; x(!)i;

so thatRehp; y(!) x(!)i > 0:

Therefore, p(!; y(!))Rehp; y(!) x(!)i > 0, whenever p(!; y(!)) > 0, for p 2 E.

Since 0(!; y(!)) > 0 or p(!; y(!)) > 0 for some p 2 E, it follows that

(!; x(!); y(!)) = 0(!; y(!))[ infw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i > 0;

which contradicts (2.3). This contradiction proves step 1.

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42

Step 2. There exists a measurable map y : ! X such that w 2 T (!; y(!)) and

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!)) 0;

for all x(!) 2 S(!; y(!)) and for xed ! 2 .

Note that for each xed x(!) 2 S(!; y(!)),

w ! Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))

is convex and randomly continuous on T (!; y(!)) and for each xed w 2 T (!; y(!))

x(!)! Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))

is concave on S(!; y(!)). Then by Kneser's Minimax Theorem in [14], we have

minw2T (!;y(!))

maxx(!)2S(!;y(!))

[Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]

= maxx(!)2S(!;y(!))

minw2T (!;y(!))

[Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]:

Hence

minw2T (!;y(!))

maxx(!)2S(!;y(!))

[Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))] 0;

by step 1. Since T (!; y(!)) is compact, there exists a measurable map y : ! X withw 2 T (!; y(!)) such that

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!)) 0; for all x(!) 2 S(!; y(!))

and for each xed ! 2 . This completes the proof.

3. Generalized Random Quasi-Variational Inequalities for Randomly

Pseudo-monotone Operators

In this section, we shall obtain some existence theorems of generalized random quasi-variational inequalities for randomly pseudo-monotone operators on para compact convexsets.

Theorem 3.1. Let (;) a measurable space, E a locally convex Hausdor topolog-ical vector space, X a nonempty paracompact convex and bounded subset of E andh : E ! R be convex such that h(X) is bounded. Let S : X ! 2X be randomlyupper semicontinuous such that each S(!; x(!)), for each xed ! 2 , is compact convexand T : X ! 2E

the randomly h-pseudo-monotone and randomly upper semicon-tinuous from Co(A) to the weak-topology on E, for each A 2 F(X) such that for xed! 2 , each T (!; x(!)) a weak-compact convex on T (X) is randomly bounded. Supposethat the set

A = f! 2 ; y(!) 2 X : supx(!)2S(!;y(!))

[ infw2T (!;y(!))

Rehw; y(!) x(!)i


43

+h(!; y(!)) h(!; x(!))] > 0g

is open in X. Suppose further that there exists a nonempty compact subset K of X anda random point x0(!) 2 X such that x0(!) 2 K \ S(!; y(!)) and

infw2T (!;y(!))

Rehw; y(!) x0(!)i+ h(!; y(!)) h(!; x0(!)) > 0; for all y 2 XnK;

for each xed ! 2 . Then there exists a measurable map y : ! K such that

(i) y(!) 2 S(!; y(!)) and

(ii) there exists w 2 T (!; y(!)) with

Rehw; y(!) x(!)i h(!; x(!)) h(!; y(!)); for all x 2 S(!; y(!))

and for each xed ! 2 .

Proof. We divide the proof into two steps:

Step 1. There exists a measurable map y : ! X such that y(!) 2 S(!; y(!)) and foreach xed ! 2 ,

supx(!)2S(!;y(!))

[ infw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))] 0:

Suppose the contrary, for each xed ! 2 and y(!) 2 X, either y(!) 62 S(!; y(!))or there exists x(!) 2 S(!; y(!)) such that

infw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!)) > 0;

i.e., y(!) 62 S(!; y(!)) or y(!) 2 A. If y(!) 62 S(!; y(!)), then by Hahn-Banach separa-tion theorem, there exists p 2 E such that

Rehp; y(!)i supx(!)2S(!;y(!))

Rehp; x(!)i > 0:

For each xed ! 2 , y(!) 2 X, set

(y(!)) = supx(!)2S(!;y(!))

[ infw2T (!;y(!))

Rehw; y(!)x(!)i+h(!; y(!))h(!; x(!))]:

Let V0 = f! 2 ; y(!) 2 X j (y(!)) > 0g = A and for each p 2 E, set

Vp = f! 2 ; y(!) 2 X : Rehp; y(!)i supx(!)2S(!;y(!))

hp; x(!)i 0g:

Then X = V0 [S

p2E

Vp. Since each Vp is open in X by Lemma 1 in [21] and V0 is

open in X by hypothesis, then fV0; Vp : p 2 Eg is an open covering of X. Since X is

AHMAD ET AL

44

paracompact, there is a continuous partition of unity f0; p : p 2 Eg for X subordinatedto the open cover fV0; Vp : p 2 Eg. For each A 2 F(X), h is a randomly continuous onCo(A), see [19, p.83].

Dene : X X ! R by

(!; x(!); y(!)) = 0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i;

for each x(!); y(!) 2 X and for xed ! 2 .

Then we have

(1) The same argument in proving (1) in the proof of Theorem 2.1, shows that foreach A 2 F(X) and for each xed x(!) 2 Co(A), the random mapping y(!) !(!; x(!); y(!)), for xed ! 2 , is lower semicontinuous on Co(A).

(2) The same argument in proving (2) in the proof of Theorem 2.1, shows that for xed! 2 and for each A 2 F(X), y(!) 2 Co(A),

minx(!)2A

(!; x(!); y(!)) 0:

(3) Suppose A 2 F(X), for xed ! 2 , x(!); y(!) 2 Co(A), fy(!)g2 is a randomnet in X converging to y(!) with

(!; tx(!) + (1 t)y(!); y(!)) 0 for all 2 and t 2 [0; 1]:

Case 1. 0(!; y(!)) = 0. Note that 0(!; y(!)) 0 for each 2 and 0(!; y(!))!0. Since T (X) is randomly strong bounded and fy(!)g2 a randomly bounded net, itfollows that

lim sup

[0(!; y(!))f minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))g] = 0:

(3:1)Also

0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))] = 0:

Thus

lim sup

[0(!; y(!))f minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i


45

=X

p2E

p(!; y(!))Rehp; y(!) x(!)i (by (3:1))

= 0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i: (3:2)

For t = 1, we have (!; x(!); y(!)) 0, for all 2 ; i.e.,

0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i 0; for all 2 : (3:3)

Therefore

lim sup

[0(!; y(!))f minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))g]

+ lim inf[X

p2E

p(!; y(!))Rehp; y(!) x(!)i]

lim sup

[0(!; y(!))f minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))g

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i] 0 by (3:3):

Thus

lim sup

[0(!; y(!))( minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) + h(!; x(!)))]

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i 0: (3:4)

Hence by (3.2) and (3.4), we have

(!; x(!); y(!)) 0:

Case 2. 0(!; y(!)) > 0. Since 0(!; y(!))! 0(!; y(!)), there exists 2 such that0(!; y(!)) > 0 for all and for each xed ! 2 . Then for t = 0, we have

(!; y(!); y(!)) 0; for all 2 ; i.e.;

0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!) y(!)i+ h(!; y(!)) h(!; y(!))]

+X

p2E

p(!; y(!))Rehp; y(!) y(!)i 0

AHMAD ET AL

46

for all 2 and for each xed ! 2 . Thus

lim sup

[0(!; y(!))f minw2T (!;y(!))

Rehw; y(!) y(!)i+ h(!; y(!)) h(!; y(!))

+X

p2E

p(!; y(!))Rehp; y(!) y(!)i] 0: (3:5)

Hence

lim sup

[0(!; y(!))f minw2T (!;y(!))

Rehw; y(!) y(!)i+ h(!; y(!)) h(!; y(!))g]

+ lim inf[X

p2E

p(!; y(!))Rehp; y(!) y(!)i]

lim sup

[0(!; y(!))f minw2T (!;y(!))

Rehw; y(!) y(!)i+ h(!; y(!)) h(!; y(!))

+X

p2E

p(!; y(!))Rehp; y(!) y(!)i] 0; by (3:5):

Sincelim inf

[X

p2E

p(!; y(!))Rehp; y(!) y(!)i] = 0;

we have

lim sup

[0(!; y(!))f minw2T (!;y(!))

Rehw; y(!) y(!)i+ h(!; y(!)) h(!; y(!))g] 0:

(3:6)Since 0(!; y(!)) > 0, for all . It follows that

0(!; y(!)) lim sup

[ minw2T (!;y(!))

Rehw; y(!)y(!)i+h(!; y(!))h(!; y(!))]

= lim sup

[0(!; y(!))f minw2T (!;y(!))

Rehw; y(!)y(!)i+h(!; y(!))h(!; y(!))g]: (3:7)

Since 0(!; y(!)) > 0, by (3.6) and (3.7), we have

lim sup

[ minw2T (!;y(!))

Rehw; y(!)y(!)i+h(!; y(!))h(!; y(!))] 0:

Since T is randomly h-pseudomonotone, we have

lim inf[ minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]

minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!)):

Since 0(!; y(!)) > 0, we have

0(!; y(!))[lim inff minw2T (!;y(!))

Rehw; y(!)x(!)i+h(!; y(!))h(!; x(!))]


47

0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]:

Thus,

0(!; y(!))[lim inff minw2T (!;y(!))

Rehw; y(!)x(!)i+h(!; y(!))h(!; x(!))g]

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i

0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i: (3:8)

For t = 0, we also have

(!; x(!); y(!)) 0; for all 2 ; and ! 2 ; i.e.;

0(!; y(!))[ minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))]

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i 0;

for all 2 , and for each xed ! 2 . Therefore

0 lim inf[0(!; y(!))f min

w2T (!;y(!))Rehw; y(!)x(!)i+h(!; y(!))h(!; x(!))g

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i]

lim inf[0(!; y(!))f min

w2T (!;y(!))Rehw; y(!)x(!)i+h(!; y(!))h(!; x(!))g]

+ lim inf[X

p2E

p(!; y(!))Rehp; y(!) x(!)i]

= 0(!; y(!))[lim inff minw2T (!;y(!))

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!))g]

+X

p2E

p(!; y(!))Rehp; y(!) x(!)i: (3:9)

Consequently, by (3.8) and (3.9), we have

(!; x(!); y(!)) 0:

Now, the remaining part of the proof of step 1 is similar to the proof in step 1 ofTheorem 2.1.

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48

Step 2. There exists w 2 T (!; y(!)) such that for each ! 2 ,

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!)) 0; for all x(!) 2 S(!; y(!)):

Also the same proof in step 2 of Theorem 2.1, shows that there exists w 2 T (!; y(!))such that

Rehw; y(!) x(!)i+ h(!; y(!)) h(!; x(!)) 0;

for all x(!) 2 S(!; y(!)), and for each xed ! 2 . Hence result is completed.

References

[1] C. Baiocchi and A. Capelo, Variational and Quasi-Variational Inequalities, Appli-cations to Free Boundary Problems, J. Wiley, New York 1984.

[2] A. Bensousson and J.L. Lions, Applications des inequations variationelles en controlstochastique, Dunod, Paris, 1978.

[3] A. T. Bharucha-Reid, Fixed point theorem in probabilistic analysis, Bull. Amer.Math. Soc., 82(1976), 641-657.

[4] H. Brezis, L. Nirenberg and G. Stampacchia, A remark on Ky Fan's minimax prin-ciple, Bollettino U.M.I., 6(4)(1972), 293-300.

[5] D. Chan and J.S. Pang, The generalized quasi-variational inequality problem, Math.Oper. Res., 7(1982), 211-222.

[6] M.S.R. Chowdhury and K.K. Tan, Generalization of Ky Fan's minimax inequalitywith applications to generalized variational inequalities for pseudomonotone opera-tors and xed point theorems, J. Math. Anal. Appl., 204(1996), 910-929.

[7] M.S.R. Chowdhury and K.K. Tan, Generalized quasi-variational inequalities for up-per semicontinuous operators on non compact sets, Nonlinear Analysis, Proceedingsof the Second World Congress of Nonlinear Analysis, 30(8)(1997), 5389-5394.

[8] M.S.R. Chowdhury and K.K. Tan, Applications of pseudomonotone operators withsome kind of upper semicontinuity in generalized quasivariational inequalities onnon-compact sets, Proceedings American Mathematical Society, 126(10)(1998), 2957-2968.

[9] J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.

[10] K. Fan, A minimax inequality and applications, In: \Inequalities", Vol. III, Pro-ceedings Third Symposium on Inequalities" (O. Shisha Ed.), Academic Press, NewYork, 103-113, 1992.


49

[11] M.F. Khan and Salahuddin, On generalized vector variational -like inequalities, Non-linear Analysis, 59(6)(2004), 879-889.

[12] S. Itoh, A random xed point theorem for a multivalued contraction mapping, PacicJ. Math., 68(1977), 85-90.

[13] S. Karamardian, Generalized complementarity problems, J. Optim. Theory Appl.,8(1971), 161.

[14] H. Kneser, Sur un theoreme fundamental de la theorie des jeux, C.R. Acad. Sci.Paris, 234(1952), 2418-2420.

[15] L.S. Liu, Some random approximations and random xed point theorem for 1-set-contractive random operators, Proc. Amer. Math. Soc., 125(1997), 515-521.

[16] N. Shahzad, Some general random coincidence point theorems, New Zealand J.Math., 33(2004), 95-103.

[17] J.S. Pang, The implicit complementarity problem in nonlinear programming (Man-gasarian, Meyer and Robinson Eds.) 487, Academic Press, London, 1981.

[18] N.S. Papageorgiou, Random xed point theorems for measurable multifunction inBanach spaces, Proc. Amer. Math. Soc. 97(1986), 507-514.

[19] R. T. Rockefeller, Convex Analysis, Princeton University Press, Princeton, 1970.

[20] Salahuddin, Some Aspects of Variational Inequalities, Ph.D. Thesis, A.M.U., Ali-garh, 2001.

[21] M. H. Shih and K. K. Tan, Generalized quasi-variational inequalities in locally con-vex topological vector spaces, J. Math. Anal. Appl., 108(1985), 333-343.

[22] W. Takahashi, Nonlinear variational inequalities and xed point theorems, J. Math.Soc. Japan, 28(1976), 166-181.

[23] K.K. Tan and X.Z. Yuan, Random xed point theorems and approximations in cones,J. Math. Anal. Appl., 185(1994), 378-390.

[24] H.K. Xu, Some random xed point theorems for condensing and nonexpansive op-erators, Proc. Amer. Math. Soc., 110(1990), 495-500.

[25] George X.Z. Yuan, The study of minimax inequalities and applications to economicsand variational inequalities, Mem. Amer. Math. Soc. 132 No. 625(1998), 1-140.

[26] George X.Z. Yuan, KKM Theorem and Applications in Nonlinear Analysis, MarcelDekker, New York, 1999.

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On Uniform Continuity and Lebesgue Property inIntuitionistic Fuzzy Metric Spaces

Cihangir Alaca1 and Hakan Efe21Department of Mathematics, Faculty of Science and Arts, Ondokuz

Mayis University, Kurupelit, 55139 Samsun, Turkey.e-mail: [email protected], [email protected]

2Department of Mathematics, Faculty of Science and Arts, GaziUniversity, Teknikokullar, 06500 Ankara, Turkey.

e-mail: [email protected]

In this paper, we introduce the concepts of uniform continuity, R-uniform continuity, equinormality and Lebesgue property in intuition-istic fuzzy metric spaces. We show that every continuous functionon a compact intuitionistic fuzzy metric space is uniformly continuous.Thereafter, we prove every real valued continuous function is uniformlycontinuous in intuitionistic fuzzy metric spaces.KeyWords. Intuitionistic fuzzy metric space, uniformly continuous

function, equinormal intuitionistic fuzzy metric, Lebesgue intuitionisticfuzzy metric.M.S.C. (2000). 54A40, 54E35, 54E40

1. INTRODUCTION

Since the introduction of the concept of fuzzy set by Zadeh [19] in1965, many authors have introduced the concept of fuzzy metric spacein di¤erent ways [3, 4, 6, 9, 10, 13, 14]. George and Veeramani [6, 8]modied the concept of fuzzy metric space introduced by Kramosil andMichalek [14] and dened a Hausdor¤ topology on this fuzzy metricspace. They also showed that every metric induces a fuzzy metric.Gregori et al. [11] gave with the help of appropriate fuzzy notionsof equinormality and Lebesgue property, several characterizations ofthose fuzzy metric spaces, in the sense of George and Veeramani [6, 8],for which every real valued continuous function is uniformly continuouswas obtained.Park [16] using the idea of intuitionistic fuzzy sets, dened the no-

tion of intuitionistic fuzzy metric spaces with the help of continuoust-norm and continuous t-conorm as a generalization of fuzzy metricspace due to George and Veeramani. Alaca et al. [1] dened the com-pletions of intuitionistic fuzzy metric spaces. A complete intuitionisticfuzzy metric space Y is said to be an intuitionistic fuzzy completionof a given intuitionistic fuzzy metric space X if X is isometric to adense subspace of Y . They gave an example of an intuitionistic fuzzymetric space that does not admit any intuitionistic fuzzy metric com-pletion. However, they proved that every standard intuitionistic fuzzymetric space has an (up to isometry) unique intuitionistic fuzzy metriccompletion. They also showed that for each intuitionistic fuzzy metric

51


space there is an (up to uniform isomorphism) unique complete intu-itionistic fuzzy metric space that contains a dense subspace uniformlyisomorphic to it. Many authors studied the concept of intuitionisticfuzzy metric space and its applications [12, 17 ].The purpose of this paper is to introduce the concepts of uniform

continuity, R-uniform continuity, equinormality and Lebesgue propertyin intuitionistic fuzzy metric spaces. We show that every continuousfunction on a compact intuitionistic fuzzy metric space is uniformlycontinuous. Thereafter, we prove every real valued continuous functionis uniformly continuous in intuitionistic fuzzy metric spaces. Also wegive some relationships between equinormality and Lebesgue propertyin intuitionistic fuzzy metric spaces.

2. ON INTUITIONISTIC FUZZY METRIC SPACES

Denition 1 ([18]). A binary operation : [0; 1] [0; 1] ! [0; 1] iscontinuous t-norm if is satisfying the following conditions: (i) iscommutative and associative; (ii) is continuous; (iii) a 1 = a forall a 2 [0; 1]; (iv) a b c d whenever a c and b d, anda; b; c; d 2 [0; 1].

Denition 2 ([18]). A binary operation : [0; 1] [0; 1] ! [0; 1] iscontinuous t-conorm if is satisfying the following conditions: (i) is commutative and associative; (ii) is continuous; (iii) a0 = afor all a 2 [0; 1]; (iv) ab cd whenever a c and b d, anda; b; c; d 2 [0; 1].

Denition 3 ([16]). A 5-tuple (X;M;N; ;) is said to be an intu-itionistic fuzzy metric space if X is an arbitrary set, is a continu-ous t-norm, is a continuous t-conorm and M; N are fuzzy sets onX2 (0;1) satisfying the following conditions: for all x; y; z 2 X, s;t > 0,

(IFM-1) M(x; y; t) +N(x; y; t) 1;(IFM-2) M(x; y; t) > 0;(IFM-3) M(x; y; t) = 1 if and only if x = y;(IFM-4) M(x; y; t) =M(y; x; t);(IFM-5) M(x; y; t) M(y; z; s) M(x; z; t+ s) ;(IFM-6) M(x; y; :) : (0;1)! [0; 1] is continuous;(IFM-7) N(x; y; t) 0;(IFM-8) N(x; y; t) = 0 if and only if x = y;(IFM-9) N(x; y; t) = N(y; x; t);(IFM-10) N(x; y; t)N(y; z; s) N(x; z; t+ s);(IFM-11) N(x; y; :) : (0;1)! [0; 1] is continuous.

Then (M;N) is called an intuitionistic fuzzy metric on X. ThefunctionsM(x; y; t) andN(x; y; t) denote the degree of nearness and thedegree of non-nearness between x and y with respect to t, respectively.

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52

Remark 1. Every fuzzy metric space (X;M; ) is an intuitionisticfuzzy metric space of the form (X;M; 1 M; ;) such that t-norm and t-conorm are associated [15], i.e., xy = 1 ((1x) (1y))for any x; y 2 [0; 1].

Remark 2. In intuitionistic fuzzy metric space X, M(x; y; :) is non-decreasing and N(x; y; :) is non-increasing for all x; y 2 X.

Example 1. Let (X; d) be a metric space. Denote a b = ab andab = minf1; a+ bg for all a; b 2 [0; 1] and let Md and Nd be fuzzy setson X2 (0;1) dened as follows:

Md(x; y; t) =htn

htn +md(x; y), Nd(x; y; t) =

md(x; y)

htn +md(x; y)

for all h;m; n 2 R+. Then (X;Md; Nd; ;) is an intuitionistic fuzzymetric space.

Remark 3. Note the above example holds even with the t-norm a b =minfa; bg and the t-conorm ab = maxfa; bg and hence (M;N) is anintuitionistic fuzzy metric with respect to any continuous t-norm andcontinuous t-conorm. In the above example by taking h = m = n = 1,we get

Md(x; y; t) =t

t+ d(x; y), Nd(x; y; t) =

d(x; y)

t+ d(x; y).

We call this intuitionistic fuzzy metric induced by a metric d the stan-dard intuitionistic fuzzy metric.

Example 2. Let X = Nnf0g. Dene a b = maxf0; a + b 1g andab = a+ b ab for all a; b 2 [0; 1] and let M and N be fuzzy sets onX2 (0;1) as follows:

M(x; y; t) =

xy

if x y,yxif y x, , N(x; y; t) =

yxy

if x y,xyx

if y x,

for all x; y 2 X and t > 0. Then (X;M;N; ;) is an intuitionisticfuzzy metric space.

Remark 4. Note that, in the above example, t-norm and t-conorm are not associated. And there exists no metric d on X satisfying

M(x; y; t) =t

t+ d(x; y), N(x; y; t) =

d(x; y)

t+ d(x; y)

where M(x; y; t) and N(x; y; t) are as dened in above example. Alsonote the above functions (M;N) is not an intuitionistic fuzzy metricwith the t-norm and t-conorm dened as a b = minfa; bg and ab =maxfa; bg.

ON UNIFORM CONTINUITY AND LEBESGUE PROPERTY...

53

Denition 4 ([6]). Let (X;M; ) be a fuzzy metric space and let r 2(0; 1); t > 0 and x 2 X. The set BM(x; r; t) = fy 2 X : M(x; y; t) >1 rg is called the open ball with center x and radius r with respect tot.

Denition 5 ([16]). Let (X;M;N; ;) be an intuitionistic fuzzy met-ric space and let r 2 (0; 1), t > 0 and x 2 X. The set B(M;N)(x; r; t) =fy 2 X :M(x; y; t) > 1 r, N(x; y; t) < rg is called the open ball withcenter x and radius r with respect to t.

Theorem 1 ([16]). Every open ball B(M;N)(x; r; t) is an open set.

Remark 5. Let (X;M;N; ;) be an intuitionistic fuzzy metric space.Dene (M;N) = fA X :for each x 2 A, there exist t > 0, r 2 (0; 1)such that B(M;N)(x; r; t) Ag. Then (M;N) is a topology on X.Lemma 1 ([17]). Let (X;M;N; ;) be an intuitionistic fuzzy metricspace. Then (X; (M;N)) is a metrizable topological space.

It was proved Lemma 1 for each n 2 N and

Un =

(x; y) 2 X X :M(x; y;

1

n) > 1 1

n, N(x; y;

1

n) <

1

n

,

fUn : n 2 Ng is a base for uniformity U(M;N) on X whose inducedtopology coincides with (M;N).Let us recall that a uniformity U on a set X has the Lebesgue prop-

erty provided that for each open cover G of X there is U 2 U such thatfU(x) : x 2 Xg renes G, and U is said to be equinormal if for eachpair of disjoint nonempty closed subsets A and B of X there is U 2 Usuch that U(A) \ B = ?. A metric d on X has the Lebesgue prop-erty provided that the uniformity Ud, induced by d, has the Lebesgueproperty and d is equinormal provided that Ud so is (see, for instance,[5]).In this paper R and N will denote the set of real numbers and positive

integer numbers, respectively.3. MAIN RESULTS

Gregori et al. [11] gave with the help of appropriate fuzzy notionsof equinormality and Lebesgue property, several characterizations ofthose fuzzy metric spaces, in the sense of George and Veeramani [6, 8],for which every real valued continuous function is uniformly continuouswere obtained. Now, we give fundamental denitions in intuitionisticfuzzy metric spaces as follows:

Denition 6. A mapping f from an intuitionistic fuzzy metric space(X;M;N; ;) to an intuitionistic fuzzy metric space (Y;M 0; N 0; 0;0)is called uniformly continuous if for each " 2 (0; 1) and each t > 0,there exist r 2 (0; 1) and s > 0 such that M 0(f(x); f(y); t) > 1 " andN 0(f(x); f(y); t) < " whenever M(x; y; s) > 1 r and N(x; y; s) < r.

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54

In this paper by a compact intuitionistic fuzzy metric space wemean an intuitionistic fuzzy metric space (X;M;N; ;) such that(X; (M;N)) is a compact topological space.

Remark 6. Every uniformly continuous mapping from the intuitionis-tic fuzzy metric space (X;M;N; ;) to the intuitionistic fuzzy metricspace (Y;M 0; N 0; 0;0) is continuous from (X; (M;N)) to (Y; (M 0;N 0)):

Now, we dene R-uniform continuity in intuitionistic fuzzy metricspaces.

Denition 7. A real valued function f on the intuitionistic fuzzy met-ric space (X;M;N; ;) is R-uniformly continuous provided that foreach " > 0 there exist r 2 (0; 1) and s > 0 such that jf(x) f(y)j < "whenever M(x; y; s) > 1 r and N(x; y; s) < r.Theorem 2. Let f be a continuous mapping from the compact intu-itionistic fuzzy metric space (X;M;N; ;) to the intuitionistic fuzzymetric space (Y;M 0; N 0; 0;0). Then f is uniformly continuous.Proof. We put " 2 (0; 1) and t > 0; then there exists > 0 such that(1 ) 0 (1 ) > (1 ") and 0 < "; by the continuity of 0and 0:So, for each x 2 X there exist rx; r0x 2 (0; 1) and sx > 0 such that

fB(M;N)(x; r

0x; sx)

B(M 0;N 0)(f(x); ; t=2)

and (1 rx) (1 rx) > (1 r0x) and rxrx < r0x: Now, there existsa nite subset A of X such that X = [x2AB(M;N)(x; rx; sx=2). Putr = minfrx : x 2 Ag and s = maxfsx=2 : x 2 Ag. It is routine toshow that M 0(f(x); f(y); t) > 1 " and N 0(f(x); f(y); t) < " wheneverM(x; y; s) > 1 r and N(x; y; s) < r. So f is uniformly continuous.This completes the proof. Now, we dene equinormality in intuitionistic fuzzy metric spaces.

Denition 8. An intuitionistic fuzzy metric (M;N) on a set X iscalled equinormal if for each pair of disjoint nonempty closed subsetsA and B of (X; (M;N)) there is s > 0 such that supfM(a; b; s) : a 2A; b 2 Bg < 1 and inffN(a; b; s) : a 2 A; b 2 Bg > 0.Now, we dene Lebesgue property in intuitionistic fuzzy metric spaces.

Denition 9. An intuitionistic fuzzy metric (M;N) on a set X hasthe Lebesgue property if for each open cover G of (X; (M;N)) there existr 2 (0; 1) and s > 0 such that fB(M;N)(x; r; s) : x 2 Xg renes G.Remark 7. Notice that if (X; d) is a metric space, then the intuition-istic fuzzy metric (Md; Nd) has the Lebesgue property (resp. is equinor-mal) if and only if d has the Lebesgue property (resp. is equinormal).

Theorem 3. Let (X;M;N; ;) and (Y;M 0; N 0; 0;0) are intuitionis-tic fuzzy metric spaces. Then, following are equivalent:


55

(i) For each intuitionistic fuzzy metric space (Y;M 0; N 0; 0;0)anycontinuous mapping from (X; (M;N)) to (Y; (M 0;N 0)) is uni-formly continuous as a mapping from (X;M;N; ;) to (Y;M 0; N 0; 0;0).

(ii) Every real valued continuous function on (X; (M;N)) isR-uniformlycontinuous on (X;M;N; ;).

(iii) Every real valued continuous function on (X; (M;N)) is uni-formly continuous on (X; U(M;N)).

(iv) (M;N) is an equinormal intuitionistic fuzzy metric on X.(v) U(M;N) is an equinormal uniformity on X.(vi) The uniformity U(M;N) has the Lebesgue property.(vii) The intuitionistic fuzzy metric (M;N) has the Lebesgue prop-

erty.

Proof. (i)) (ii). Let f be a real valued continuous function on (X; (M;N))and " > 0. We may assume without loss of generality that " 2 (0; 1).Choose n 2 N such that 1 " > 1

n. By assumption f is uniformly con-

tinuous as a mapping from (X;M;N; ;) to (R;Md; Nd; 0;0), where(Md; Nd) is the Euclidean intuitionistic fuzzy metric on R. Hence, thereexist r 2 (0; 1) and s > 0 such that

1n

1n+ jf(x) f(y)j

> 1 " and jf(x) f(y)j1n+ jf(x) f(y)j

< "

whenever M(x; y; s) > 1 r and N(x; y; s) < r. An easy computa-tion shows that jf(x) f(y)j < " whenever M(x; y; s) > 1 r andN(x; y; s) < r. We conclude that f is R-uniformly continuous on(X;M;N; ;).(ii) ) (iii) Let f a real valued continuous function on (X; (M;N))

and " > 0. By assumption, there exist r 2 (0; 1) and s > 0 such thatjf(x) f(y)j < " wheneverM(x; y; s) > 1r and N(x; y; s) < r. Taken 2 N such that 1

n minfr; sg. Then for all x; y 2 X such that (x; y) 2

Un, we obtain by Remark 2,

M(x; y; s) M(x; y; 1n) > 1 1

n 1 r

and

N(x; y; s) N(x; y; 1n) <

1

n r,

so jf(x) f(y)j < ". We conclude that f is uniformly continuous on(X; U(M;N)).(iii) ) (iv). Let A and B be two disjoint nonempty closed subsets

of (X; (M;N)). There exists a continuous function f : X ! [0; 1]such that f(A) f0g and f(B) f1g. Since by assumption f isuniformly continuous on (X; U(M;N)), for " = 1 there is n 2 N suchthat jf(x) f(y)j < 1 wheneverM(x; y; 1

n) > 1 1

nandN(x; y; 1

n) < 1

n.

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56

Hence

M(a; b;1

n) 1 1

nand N(a; b;

1

n) 1

nfor all a 2 A and b 2 B. We conclude that (M;N) is equinormalintuitionistic fuzzy metric on X.(iv)) (v). Let A and B be two disjoint nonempty closed subsets of

(X; (M;N)). By assumption, there exist r 2 (0; 1) and s > 0 such thatsupfM(a; b; s) : a 2 A; b 2 Bg = 1 r

andinffN(a; b; s) : a 2 A; b 2 Bg = r.

Put

U = f(x; y) 2 X X :M(x; y; s) > 1 r, N(x; y; s) < sg .Then U 2 U(M;N) and U(A) \ B = ?. Hence U(M;N) is an equinormaluniformity on X.(v) ) (vi). Its clear from [2] Theorem 2.3.1.(vi) ) (vii). Let G be an open cover of X: From our assumption

it follows that there is an n 2 N such that fB(M;N)(x; 1n ;1n) : x 2 Xg

renes G:Hence (M;N) is a Lebesgue intuitionistic fuzzy metric on X:(vii)) (i). Let (Y;M 0; N 0; 0;0) an intuitionistic fuzzy metric space

and f a continuous mapping from (X; (M;N)) to (Y; (M 0;N 0)). Fix" 2 (0; 1) and t > 0: There is > 0 such that (1 ) 0 (1 ) >1 " and 0 < ". Since f is continuous, for each x 2 X there isan open neighborhood Vx of x such that f(Vx) B(M 0;N 0)(f(x); ;

t2).

By assumption there exist r = r(t; ") 2 (0; 1) and s > 0 such thatfB(M;N)(x; r; s) : x 2 Xg renes fVx : x 2 Xg.Now ifM(x; y; s) > 1r andN(x; y; s) < s we have y 2 B(M;N)(x; r; s),

so x; y 2 Vz for some z 2 X:Hence, f(x) and f(y) are inB(M 0;N 0)(f(z); ;t2).

Thus,

M 0(f(x); f(y); t) M 0(f(x); f(z);t

2) 0M 0(f(z); f(y);

t

2) > 1 "

and

N 0(f(x); f(y); t) N 0(f(x); f(z);t

2)0N 0(f(z); f(y);

t

2) < ".

Then f is uniformly continuous from (X;M;N; ;) to (Y;M 0; N 0; 0;0).

It is well known (see, for instance, [2]) that a metrizable topologicalspace admits a metric with the Lebesgue property if and only if the setof nonisolated points is compact. From this result and the precedingtheorem we deduce the following corollary.

Corollary 1. A (intuitionistic fuzzy) metrizable topological space ad-mits an intuitionistic fuzzy metric with the Lebesgue property if andonly if the set of nonisolated points is compact.


57

Remark 8. Given a metrizable topological spaceX we denote by F ~N(M;N)the supremum of all uniformities U(M;N) induced by all compatible intu-itionistic fuzzy metrics for X: It is easy to see that F ~N(M;N) is exactlythe ne uniformity of X. Hence, the classical theorem that if topolog-ical space X admits a metric d with the Lebesgue property, then theuniformity Ud coincides with the ne uniformity of X, can be refor-mulated as follows: If a topological space admits an intuitionistic fuzzymetric (M;N) with the Lebesgue property, then the uniformity U(M;N)coincides with the uniformity F ~N(M;N).

We conclude the paper with an example which illustrates the ob-tained results.

Example 3. Let X be the et of natural numbers and let is a con-tinuous t-norm, is a continuous t-conorm dened by a b = ab andab = minf1; a + bg for all a; b 2 [0; 1]. For each x; y 2 X and t > 0let

M(x; y; t) =

1 if x = y,1xy

if x 6= y, and N(x; y; t) =

0 if x = y,xy1xy

if x 6= y,

It is easy to check that (X;M;N; ;) is an intuitionistic fuzzy met-ric space. Note that there is no metric d on X for which (M;N) isthe intuitionistic fuzzy metric induced by d. For each pair of disjointnonempty subsets of X, A and B, we have

supfM(a; b; s) : a 2 A; b 2 Bg 1

2and inffN(a; b; s) : a 2 A; b 2 Bg 1

2

for all t > 0. From this fact it follows that (M;N) is an equinormalintuitionistic fuzzy metric on X (so (X;M;N; ;) satises conditionsof our theorem), and that the (M;N) discrete topology on X since foreach x 2 X ,

supfM(a; b; s) : a 2 A; b 2 Bg 1

2and inffN(a; b; s) : a 2 A; b 2 Bg 1

2

and thus B(M;N)(x; 12 ;12) = fxg.

Acknowledgement. The authors would like to thank the refereesfor their help in the improvement of this paper.

References

[1] C. Alaca, H. Efe and C. Yildiz, On completion of intuitionistic fuzzy metricspaces, Chaos, Solitons & Fractals in press, doi : 10:1016=j:chaos:2006:01:039.

[2] G. Beer, Topologies on Closed and Convex Sets, Kluwer Acad., Dordrecth1993.

[3] Z. K. Deng, Fuzzy pseudo-metric spaces, J. Math. Anal. Appl. 86, 74 95(1982).

[4] M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69, 205230 (1979).

ALACA-EFE

58

[5] P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, NewYork, 1982.

[6] A. George and P. Veeramani, On some results in fuzzy metric spaces, FuzzySets and Systems 64, 395 399 (1994).

[7] A. George and P. Veeramani, Some theorems in fuzzy metric spaces, J. FuzzyMath. 3, 933 940 (1995).

[8] A. George and P. Veeramani, On some results of analysis for fuzzy metricspaces, Fuzzy Sets and Systems 90, 365 368 (1997).

[9] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27,385 389 (1988).

[10] V. Gregori and S. Romeguera, Some properties of fuzzy metric spaces, FuzzySets and Systems 115, 485 489 (2000).

[11] V. Gregori, S. Romeguera and A. Sapena, Uniform continuity in fuzzy metricspaces, Rend. Istit. Mat. Univ. Trieste 32, 81 88 (2001).

[12] V. Gregori, S. Romaguera and P. Veeramani, A note on intuitionistic fuzzymetric spaces, Chaos, Solitons & Fractals 28, 902 905 (2006).

[13] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems12, 225 229 (1984).

[14] O. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Ky-bernetica 11, 326 334 (1975).

[15] R. Lowen, Fuzzy Set Theory, Kluwer Academic Pub., Dordrecht 1996.[16] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons & Fractals 22,

1039 1046 (2004).[17] R. Saadati and J. H. Park, On the intuitionistic topological spaces, Chaos,

Solitons & Fractals 27, 331 344 (2006).[18] B. Schweizer and A. Sklar, Statistical metric spaces, Pacic J. Math. 10, 314

334 (1960).[19] L. A. Zadeh, Fuzzy sets, Inform and Control 8, 338 353 (1965).


59

60

Sharp estimates for solutions of parabolic equations

with a lower order term

Angelo Alvino - Roberta Volpicelli - Bruno Volzone

Dipartimento di Matematica e Applicazioni ”Renato Caccioppoli”

Universita degli studi di Napoli ”Federico II”

Complesso Monte S. Angelo - Via Cintia

80126 NAPOLI - ITALY

e-mails: [email protected]; [email protected];

[email protected]

Running head: Sharp estimates

Corresponding author: Angelo Alvino

61


Abstract

We give a comparison result for solutions of Cauchy-Dirichlet prob-

lems for parabolic equations by means of Schwarz symmetrization. The

result takes into account the influence of the zero order term, on which

any boundedness or sign assumption is assumed.

2000 Mathematics Subject Classification: 35B45, 35K15, 35K20.

Key words and phrases: Rearrangements, Comparison results, Schwarz

symmetrization, Parabolic equations.

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ALVINO ET AL

62

1 Introduction

Among the a priori estimates for solutions of elliptic boundary value prob-

lems, a particular role is assumed by those estimates, known as isoperimetric

inequalities, that show interesting properties of geometric type. Indeed, the

solution of the starting problem is compared with the solution of a suitable

symmetric problem which is, in some sense, the ”worst” one.

The pioneristic result in this direction is due to G.Talenti [22]. After-

wards, this result has been presented in several and different situations: for

instance, we refer to the papers [3], [4], [6], [8], [13], [23], in which a particu-

lar emphasis has been reserved to the influence of the lower order terms. We

restrict ourselves to describe briefly the case involving the zero order term.

If Ω is an open bounded subset of RN , let u ∈ H10 (Ω) be the weak solution

of the equation

−∆u + cu = f

where c is a nonnegative function. This condition guarantees that the solu-

tion u is nonnegative if f is nonnegative. Moreover, if Ω# is the ball of RN

centered at the origin having the same measure as Ω, let v ∈ H10 (Ω#) be the

weak solution of the symmetrized problem

−∆v + c#v = f#,

where c#, f# are the increasing spherical rearrangement of c and the de-

creasing spherical rearrangement of f . Roughly speaking, the functions c#

and f# have some nice properties like simmetry, monotonicity and preserve

3

SOLUTIONS OF PARABOLIC EQUATIONS...

63

the measures of the level sets of c and f respectively (we refer to section 2

for their precise definitions).

Then, it is known (see [3], [6], [8]) that u is dominated by v in the sense

of rearrangements, i.e.

∫ s

0u∗ ≤

∫ s

0v∗ ∀s ∈ [0, |Ω|] , (1.1)

where u∗, v∗ are the decreasing rearrangements of u and v. Inequality (1.1)

implies the following property (see [10], [8]):

∫Ω

F (|u|) dx ≤∫

Ω#

F (|v|) dx

for any, increasing convex function F on R+, such that F (0) = 0. So that

any Luxemburg norm of u can be estimated by the same norm of v (see

[10]).

In the meantime, an analogous theory has been developed for parabolic

operators (see for example [3], [7], [8], [14], [15], [20], [24], [25], [26]). Con-

sider the problem

ut −N∑

i,j=1

(aij (x, t) uxi)xj+ cu = f in Ω× (0, T )

u = 0 on ∂Ω× (0, T )

u (x, 0) = u0 (x) x ∈ Ω,

(1.2)

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ALVINO ET AL

64

where we assume that the operator is uniformly parabolic, i.e.

N∑i,j=1

aij (x, t) ξiξj ≥ |ξ|2 a.e. (x, t) ∈ Ω× (0, T ) , ∀ξ ∈ RN , (1.3)

and

c (x) ≥ 0 ;

then, if v is the solution of the ”symmetrized” problem

vt −∆v = f# in Ω# × (0, T )

v = 0 on ∂Ω# × (0, T )

v (x, 0) = u#0 (x) x ∈ Ω#,

(1.4)

for all t ∈ [0, T ] the following inequality holds

∫ s

0u∗ (σ, t) dσ ≤

∫ s

0v∗ (σ, t) dσ, ∀s ∈ [0, |Ω|] . (1.5)

In problem (1.4) and in (1.5), the spherical rearrangement f# and the de-

creasing rearrangements u∗, v∗ are meant to be calculated with respect to

x, for t fixed.

The main difficulty that appears in each of the above mentioned papers

is linked to the presence of the time derivative term. This last one can

be treated by two different methods. Following the approach contained

in a paper of C. Bandle (see [7]), the crucial part consists in proving a

delicate derivation formula with respect to the time variable for functions

5


65

defined by integrals. In [7], such a formula is proved under strong regularity

assumptions on the solutions. These hypotheses have been removed later

in a paper of Mossino-Rakotoson (see [20]), where the formula is proved for

fuctions u ∈ H1(0, T ;L2 (Ω)

)by using the notion of relative rearrangement.

Some generalizations of this result have been obtained in [5] or [16], where

a formula concerning the second derivatives is also given.

Another approach uses the implicit time discretization scheme. In this

way we replace the time derivative with a difference quotient, and by using

a partition of the time interval [0, T ] of the form 0 = t0 < t1 < . . . < tn = T,

we are reduced to apply the above quoted comparison result to a sequence

of elliptic problems with zero order term of the form

−(a

(k)ij (x) u

(k)xi

)xj

+(

c +1

tk − tk−1

)u(k) = f (k) +

u(k−1)

tk − tk−1

u(k) ∈ H10 (Ω) ,

for k = 1, . . . , n, where u(k) = u (x, tk) and f (k), a(k)ij are suitable discretiza-

tion of the functions f = f (x, t), aij = aij (x, t). Then we reach the aim by

passing to limit (see [3], [24], [25]). This method is described in more details

in remark 2.

In all the results for parabolic operators mentioned above, the influence

of the zero order term cu is always neglected, since this term is essentially

omitted in (1.4) by using the sign condition c (x) ≥ 0. Our aim is to find a

comparison result of the type (1.5), between the solution u of the problem

(1.2) and the solution v of a spherically symmetric problem which keeps in

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ALVINO ET AL

66

mind the zero order term. The candidate problem is the following

vt −∆v + c#v = f# in Ω# × (0, T )

v = 0 on ∂Ω# × (0, T )

v (x, 0) = u#0 (x) x ∈ Ω#,

(1.6)

where

c# =(c+)#−(c−)#

,

c+ and c− being the positive and the negative part of c and (c+)# , (c−)#

respectively their increasing and decreasing spherical rearrangements.

We consider weak solutions of the problem (1.2): namely we deal with

functions u ∈ L2(0, T ;H1

0 (Ω))∩ C

([0, T ] ; L2 (Ω)

)such that

∂u

∂t∈ L2 (Ω× (0, T ))

and

∫Ω

∂u

∂tϕdx +

∫Ωaijuxiϕxjdx +

∫Ωcuϕdx =

∫Ωfϕdx,

u (0) = u0,

(1.7)

for all ϕ ∈ H10 (Ω) and for a.e. t ∈ [0, T ] . The existence of such a solution is

guaranteed under suitable assumptions on the data.

The result is the following:

7


67

Theorem 1. Let Ω be a bounded open subset of RN , assume that the coef-

ficients aij ∈ L∞ (Ω× (0, T )) satisfy (1.3) and suppose

c ∈ Lr (Ω) with r > N/2 if N ≥ 2, r ≥ 1 if N = 1,

f ∈ L2 (Ω× (0, T )) and u0 ∈ L2 (Ω). Let u and v be the weak solutions of

problems (1.2) and (1.6) respectively, then for all t ∈ [0, T ] , (1.5) holds.

In section 2 we prove inequality (1.5) by assuming that c is bounded

from below. Obviously, this assumption allows us to reduce the study to

the case c ≥ 0. In fact, if c (x) ≥ λ for a.e. x ∈ Ω, the function eλtu is

the solution of a problem of type (1.2) in which the zero order coefficient is

(c− λ). This situation was already studied in [26]. We give a simpler proof

that avoids to proceed by means of the approximation used in [26].

In section 3 we deal with the more general case, in which c is not bounded

from below. The motivation of this study, besides its intrinsic interest, is

also connected to some recent results obtained by various authors (see [9],

[12], [17]), related to the existence of solutions of parabolic equations where

the coefficient c has a singularity of the type

c (x) = − λ

|x|2.

This situation can be classified as a limit case, in the sense that c (x) /∈

LN/2 (N ≥ 3), but belongs to the Lorentz space L (N/2,∞) . Moreover the

operator −∆u−(λ/ |x|2

)u is coercive if and only if λ < λN := (N − 2)2 /4,

where λN is the best constant in the classical Hardy inequality; it is positive

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ALVINO ET AL

68

when λ = λN : hence, the standard existence and regularity theories do not

apply in this case. We will discuss this limit case in a forthcoming paper.

2 Proof of Theorem 1: the case c (x) ≥ λ

Before going into a detailed proof of Theorem 1, we begin this section by re-

calling some definitions that are useful in the following. Let Ω be a bounded

open subset of RN and u be a real measurable function on Ω, we define the

distribution function µu of u as

µu (θ) = |x ∈ Ω : |u (x)| > θ| , θ ≥ 0,

the decreasing and the increasing rearrangement of u as

u∗ (s) = sup θ ≥ 0 : µu (θ) > s , s ∈ (0, |Ω|) ,

u∗ (s) = u∗ (|Ω| − s) , s ∈ (0, |Ω|) .

Furthermore, if ωN is the measure of the unit ball in RN and Ω# is the ball

of RN centered at the origin having the same measure as Ω, the functions

u# (x) = u∗(ωN |x|N ) , x ∈ Ω#,

u# (x) = u∗(ωN |x|N ) , x ∈ Ω#,

are respectively the decreasing spherical rearrangement and the increasing

spherical rearrangement of u. Here we just quote the well known Hardy-

9


69

Littlewood inequality (see [18]): if u, v are measurable functions on Ω, then

∫ |Ω|

0u∗ (s) v∗ (s) ds ≤

∫Ω|u (x) v (x)| dx ≤

∫ |Ω|

0u∗ (s) v∗ (s) ds . (2.1)

As we pointed out in the introduction, our aim is to obtain a comparison

result for problems of the type (1.2). For this reason, in the following we

will consider real functions u defined on the set Ω×(0, T ) , where T is a real

positive number, that are measurable with respect to the space variable x

and denote by µu (θ, t), u∗ (s, t) , u∗ (s, t) , u# (x, t) , u# (x, t) the distribution

function and the rearrangements of u (x, t), with respect to x for t fixed. In

other words, u# (x, t) is the Steiner symmetrization of u (x, t) with respect

to the line x = 0.

As in [7], [26], problem (1.2) can be dealt by using a classic method

introduced by Talenti in [22]. This method consists in choosing a suitable

test function in (1.7), and it leads to the study of derivation formulas with

respect to the variable t of integrals of the type

∫u(x,t)>u∗(s,t)

u (x, t) dx.

Such formula, obtained in [7] for regular functions (see also [5] and [16]),

allows us to say that

∫u(x,t)>u∗(s,t)

∂u

∂t(x, t) dx =

∫ s

0

∂u∗

∂t(σ, t) dσ. (2.2)

In the paper [20], (2.2) has been obtained for less regular functions. More

precisely, the following result is proved:

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ALVINO ET AL

70

Lemma 2. If u is a nonnegative function in H1(0, T ;L2 (Ω)

), then u∗

belongs to H1(0, T ;L2 (0, |Ω|)

)and (2.2) holds if |u (x, t) = u∗ (s, t)| = 0,

for a.e. s ∈ (0, |Ω|) .

Now we are able to prove Theorem 1 assuming that c is nonnegative.

In the first part of the proof we follow an approach similar to the one given

in [26] (see also [3], [8], [20]) and we report it for completeness. It consists in

deriving an integro-differential inequality for the decreasing rearrangement

of u. In the second part we get the result by means of a maximum princi-

ple. This maximum principle does not make use of the method contained in

[26], that consists in approximating the solution v of problem (1.6) with a

sequence of solutions of suitable perturbated problems.

Since we will need that ∂u∂t ∈ L2 (QT ), we assume, for instance, the

additional conditions (see [11])

∂aij

∂t∈ C0 (Ω× [0, T ]) , i, j = 1, . . . , N

u0 ∈ H10 (Ω) .

Fixed t ∈ [0, T ] , h > 0 and θ ≥ 0, we choose the function

ϕh (x) =

sign (u) if |u| > θ + h

|u| − θ

hsign (u) if θ < |u| ≤ θ + h

0 otherwise

11


71

as a test function in (1.7). Letting h go to 0 and using (1.3) we obtain

− ∂

∂θ

∫|u|>θ

|∇u|2 dx ≤∫|u|>θ

[f (x, t)− c (x) u− ∂u

∂t

]sign (u) dx. (2.3)

The left hand side of (2.3) can be estimated from below using the following

inequalities

N2ω2/NN µu (θ, t)2−(2/N) ≤

(−∂µu

∂θ

)(− ∂

∂θ

∫|u|>θ

|∇u|2 dx

)(2.4)

which are consequences of the isoperimetric inequality, the Fleming Rishel

formula and the Schwartz inequality (we refer to [22] for more details). As

regards to the term involving the derivative of u with respect to t, we notice

that, since u ∈ H10 (Ω) for a.e. t, it follows u∗ ∈ C (]0, |Ω|]) (see [20]),

therefore |u = θ| = 0 and u∗ (µu (θ, t)) = θ, for a.e. θ, hence by lemma 1

∫|u|>θ

∂u

∂tsign (u) dx =

∫ µu(θ,t)

0

∂u∗

∂tds for a.e. θ ≥ 0. (2.5)

The remaining terms of (2.3) can be treated by using (2.1) in the following

way:

−∫|u|>θ

c (x) u sign (u) dx ≤ −∫ µu(θ,t)

0c∗ (s) u∗ (s, t) ds ,∫

|u|>θf (x, t) sign (u) dx ≤

∫ µu(θ,t)

0f∗ (s, t) ds.

(2.6)

Collecting (2.4), (2.5) and (2.6), we have

N2ω2/NN µu (θ, t)2−(2/N) ≤

(−∂µu

∂θ

)(∫ µu(θ,t)

0

f∗ (s, t)− c∗ (s) u∗ (s, t)− ∂u∗

∂t

ds

).

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ALVINO ET AL

72

Making a change of variable (see [20]), we get

∫ s

0

∂u∗

∂t(σ, t) dσ − p (s)

∂u∗

∂s+∫ s

0c∗ (σ) u∗ (σ, t) dσ (2.7)

≤∫ s

0f∗ (σ, t) dσ for a.e. (s, t) ∈ Q∗

T := (0, |Ω|)× (0, T ) ,

where p (s) := N2ω2/NN s2−(2/N) . On the other hand, if we consider the

solution v of problem (1.6), all the inequalities we used to get (2.7) become

equalities and so we have

∫ s

0

∂v∗

∂t(σ, t) dσ − p (s)

∂v∗

∂s+∫ s

0c∗ (σ) v∗ (σ, t) dσ (2.8)

=∫ s

0f∗ (σ, t) dσ for a.e. (s, t) ∈ Q∗

T .

From (2.7)-(2.8) it follows that

∂

∂t

∫ s

0w (σ, t) dσ − p (s)

∂w

∂s+∫ s

0c∗ (σ) w (σ, t) dσ ≤ 0,

for a.e. (s, t) ∈ Q∗T , where w := u∗ − v∗. Then, if we set

χ (s, t) :=∫ s

0w (σ, t) dσ with (s, t) ∈ Q∗

T ,

13


73

by the boundary conditions of (1.2), (1.6), we have that χ satisfies

∂χ

∂t− p (s)

∂2χ

∂s2+∫ s

0c∗ (σ)

∂χ

∂σ(σ, t) dσ ≤ 0 a.e in Q∗

T

χ (0, t) =∂χ

∂s(|Ω| , t) = 0 ∀t ∈ [0, T ]

χ (s, 0) = 0 ∀s ∈ [0, |Ω|] .

(2.9)

Our aim is to show that

χ ≤ 0.

It can be easily proved that the function χ is continuous in Q∗T , so it has

a maximum in Q∗T . We will prove that this maximum has to be zero. We

argue by contradiction: let (s0, t0) be a maximum point of χ in Q∗T such

that χ (s0, t0) > 0 and first assume that (s0, t0) ∈ Q∗T . We begin observing

that the term∫ s0 c∗ (σ) ∂χ

∂σ (σ, t) dσ in the differential inequality of problem

(2.9) can be neglected in a suitable square neighbourhood of (s0, t0). Indeed,

integrating by parts we obtain

∫ s0

0c∗ (σ)

∂χ

∂σ(σ, t0) dσ =

∫ s0

0c∗ (σ) dχ (σ, t0) (2.10)

=∫ s0

0[χ (s0, t0)− χ (σ, t0)] dc∗ (σ) > 0,

then by continuity it is possible to find a suitable square neighbourhood

Qδ = Qδ (s0, t0) of (s0, t0) in which the function∫ s0 c∗ (σ) ∂χ

∂σ (σ, t) dσ is still

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ALVINO ET AL

74

positive. Therefore, provided to choose δ conveniently, from (2.9) we find

∂χ

∂t− p (s)

∂2χ

∂s2< 0 a.e. in Qδ. (2.11)

Hence we can reduce our study of problem (2.9) to the study of inequality

(2.11) in the neighbourhood Qδ. This allows us to proceed as in [20]. How-

ever, we cannot multiply directly both sides of (2.11) for χ+, and integrate

onto the interval (s0 − δ, s0 + δ), because we don’t know the values of χ on

the parabolic boundary Γδ of Qδ. So it is natural to consider, instead of χ+,

the function ϕ defined as

ϕ :=(

χ/Qδ−max

Γδ

χ

)+

.

For our purpose we can suppose, without loss of generality, that

χ (s, t) < χ (s0, t0) ∀ (s, t) ∈ Qδ\ (s0, t0) ,

therefore ϕ (s0,t0) > 0.Multiplying the inequality (2.11) by s(2/N)−2ϕ we

find

s(2/N)−2 ∂χ

∂tϕ ≤ N2ω

2/NN

∂2χ

∂s2ϕ a.e.in Qδ . (2.12)

We can prove as in [20] that

χ ∈ W 2,∞ (s0 − δ, s0 + δ) ,

ϕ ∈ H1 (s0 − δ, s0 + δ) ,

15


75

so the integration by parts leads to :

∫ s0+δ

s0−δ

∂2χ

∂s2ϕds =

[∂χ

∂sϕ

]s0+δ

s0−δ

−∫ s0+δ

s0−δ

(∂ϕ

∂s

)2

ds ≤ 0 .

Then, from (2.12) and by the definition of ϕ we have, for any t ∈ (t0 − δ, t0 + δ),

0 ≥ 2∫ t

t0−δ

∫ s0+δ

s0−δs(2/N)−2 ∂χ

∂tϕdsdτ =

∫ s0+δ

s0−δs(2/N−2)

[∫ t

t0−δ

∂

∂τ

(ϕ2 (s, τ)

)dτ

]ds

=∫ s0+δ

s0−δs(2/N)−2ϕ2 (s, t) ds,

therefore ϕ = 0 in Qδ = (s0 − δ, s0 + δ) × (t0 − δ, t0 + δ) but this is a con-

tradiction.

By similar arguments we come to the same conclusion if we suppose

that (s0, t0) belongs either to the segment line (s, T ) : s ∈ (0, |Ω|) or to

the segment line (|Ω| , t) : t ∈ (0, T ) .

Remark 1. Obviously the case in which c is unbounded from below could

also be treated by truncating the coefficient c and passing to the limit.

3 Proof of Theorem 1: the general case

In this section we conclude the proof of theorem 1 considering the general

case c ∈ Lr (Ω) , with r > N/2 if N ≥ 2, and r ≥ 1 if N = 1.

The first part of the proof is exactly the same as the one of the previous

case, the main difference consists in the proof of the maximum principle.

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ALVINO ET AL

76

As before, we have that u satisfies

∫ s

0

∂u∗

∂t(σ, t) dσ − p (s)

∂u∗

∂s+∫ s

0

[(c+)∗ −

(c−)∗]

u∗ (σ, t) dσ

≤∫ s

0f∗ (σ, t) dσ for a.e. (s, t) ∈ Q∗

T

and this inequality holds as equality replacing u with the solution v of the

symmetrized problem (1.6). However, if we set

χ (s, t) =∫ s

0(u∗ − v∗) dσ,

it is not possible to neglect the term∫ s0 χ (σ, t) d

dσ

[(c+)∗ − (c−)∗

]dσ by

means of the pointwise arguments we used to get (1.5), since c could not be

bounded from below. In order to treat this term we proceed by approxima-

tion.

Let vε = v + εv0 be the solution of the following perturbated problem

vεt −∆vε +[(c+)# − (c−)#

]vε = f# + εδ in Ω# × (0, T )

vε = 0 on ∂Ω# × (0, T )

vε (x, 0) = u#0 (x) + εv0 (x) x ∈ Ω#,

(3.1)

where ε > 0, δ is the Dirac measure concentrated at the origin and v0 is the

unique solution vanishing on ∂Ω# of the equation

−∆v0 +[(

c+)#−(c−)#]

v0 = δ in Ω#. (3.2)

17


77

A suitable definition of weak solution to equation (3.2), together with some

existence and regularity results, can be found in the monograph of G. Stam-

pacchia [21]. Suppose for the sake of simplicity N ≥ 3 : since c ∈ Lr (Ω) with

r > N/2, by a regularity result contained in [2] to have that the solution v0

of (3.2) is in the Lorentz space L (N/ (N − 2) ,∞) .

Proceeding as in the proof of the case c (x) ≥ 0, setting

χε (s, t) :=∫ s

0(u∗ − v∗ε ) dσ,

we get

∂χε

∂t− p (s)

∂2χε

∂s2+∫ s

0

[(c+)∗ − (c−)∗

] ∂χε

∂σ(σ, t) dσ

≤ −ε a.e in Q∗T

χε (0, t) =∂χε

∂s(|Ω| , t) = 0 ∀t ∈ [0, T ]

χε (s, 0) = −ε

∫ s

0v∗0 (σ) dσ.

(3.3)

We want to prove that χε ≤ 0 in Q∗T , so that

∫ s

0u∗ (σ, t) dσ ≤

∫ s

0v∗ε (σ, t) dσ , ∀ (s, t) ∈ Q∗

T ,

and this obviously gives the desired result by letting ε goes to zero.

Let us suppose that χε > 0 in a subset F of Q∗T and let t be the minimum

18

ALVINO ET AL

78

value of the projection of F on the t axis. We notice that

χε (s, t) =∫ s

0(u∗ − v∗) dσ − ε

∫ s

0v∗0dσ

and then

lims→0

(c−)∗ (s) χε (s, t) = 0.

Indeed, since v0 is in L (N/ (N − 2) ,∞) we get

∣∣∣∣(c−)∗ (s)∫ s

0v∗0 (σ) dσ

∣∣∣∣ ≤ K(c−)∗ (s) s

2N ≤ Ks

2N−1

∫ s

0

(c−)∗

dσ

≤ Ks2N− 1

r

∥∥c−∥∥Lr(Ω)

;

obviously a similar estimate holds for the term (c−)∗ (s)∫ s0 (u∗ − v∗) dσ,

since u∗, v∗ ∈ L2∗ (0, |Ω|) . Hence integrating by parts we have that

∫ s

0

[(c+)∗ −

(c−)∗] ∂χε

∂σ(σ, t) dσ =

[(c+)∗ −

(c−)∗]

χε (s, t)

−∫ s

0χε (σ, t) d

[(c+)∗ −

(c−)∗]

.

By continuity arguments, we can choose τ > 0 such that, for t < t + τ, it

results ∫ s

0χε (σ, t) d

[(c+)∗ −

(c−)∗]− ε ≤ 0 ∀s ∈ [0, |Ω|] ,

so that by (3.3) we have

∂χε

∂t− p (s)

∂2χε

∂s2+[(c+)∗ − (c−)∗

]χε (s, t) ≤ 0 ,

a.e. in (0, |Ω|)× (0, t + τ) .

(3.4)

19


79

A similar computation can be done in the cases N = 1 and N = 2. Indeed,

v0 is bounded if N = 1, while in the case N = 2 we find that v0 belongs to

the Zygmund space Lexp (Ω) (see [1]), namely

sups∈(0,|Ω|)

v∗∗0 (s)(1 + log |Ω|

s

) < ∞,

where

v∗∗0 (s) =1s

∫ s

0v∗0 (σ) dσ

is the maximal function of v∗0.

Dividing both sides of the inequality (3.4) by p (s) we can rewrite it as

p (s)−1 ∂χε

∂t− ∂2χε

∂s2+ p (s)−1 [(c+)∗ − (c−)∗

]χε ≤ 0 ,

a.e. in (0, |Ω|)× (0, t + τ) .

(3.5)

Multiplying both sides of (3.5) by χ+ε and integrating between 0 and |Ω| ,taking

into account the boundary conditions in (3.3), we get

γN

∫ |Ω|

0s(2/N)−2 ∂χε

∂tχ+

ε ds +∫ |Ω|

0

(∂χ+

ε

∂s

)2

ds

+γN

∫ |Ω|

0

[(c+)∗ − (c−)∗

]s(2/N)−2 (χ+

ε )2 ds ≤ 0,

(3.6)

where γN := N−2ω−2/NN . We want to prove that if we replace the function

χε with the function Uε := e−λtχε (where λ > 0 is a suitable constant),

the sum of those terms in (3.6) that don’t contain the time derivative is

non negative. This is essentially due to the fact that if c ∈ Lr (Ω) with

r > N/2, the operator −∆u + cu is coercive unless to multiply both sides

20

ALVINO ET AL

80

of the equation by e−λt. Indeed, by (3.6) we have that Uε satisfies

γN

∫ |Ω|

0s(2/N)−2 ∂Uε

∂tU+

ε ds +∫ |Ω|

0

(∂U+

ε

∂s

)2

ds

+γN

∫ |Ω|

0

[λ− (c−)∗

]s(2/N)−2 (U+

ε )2 ds ≤ 0.

(3.7)

Now, if Bλ := x : c− (x) > λ and B∗λ :=

s : (c−)∗ (s) > λ

= [0, |Bλ|), we

notice that

∫ |Ω|

0

(∂U+

ε

∂s

)2

ds + γN

∫ |Ω|

0

[λ−

(c−)∗]

s(2/N)−2(U+

ε

)2ds (3.8)

≥∫ |Ω|

0

(∂U+

ε

∂s

)2

ds− γN

∫B∗

λ

[(c−)∗ − λ

]s(2/N)−2

(U+

ε

)2ds.

If s ∈ B∗λ, it results

[(c−)∗ (s)− λ

]s2/N ≤ s(2/N)−1

∫ s

0

[(c−)∗ (σ)− λ

]dσ

≤(∫ s

0

[(c−)∗ (σ)− λ

]N/2dσ

)2/N

≤

(∫ |Bλ|

0

[(c−)∗ (σ)− λ

]N/2dσ

)2/N

=∥∥(c−)∗ − λ

∥∥LN/2(B∗

λ).

Using this last inequality and the one dimensional Hardy inequality, we

21


81

obtain

∫B∗

λ

[(c−)∗ − λ

]s(2/N)−2

(U+

ε

)2ds ≤

∫B∗

λ

[(c−)∗ (σ)− λ

]s2/N

(U+

ε

s

)2

ds

≤∥∥(c−)∗ − λ

∥∥LN/2(B∗

λ)

∫B∗

λ

(U+

ε

s

)2

ds

≤ 4∥∥(c−)∗ − λ

∥∥LN/2(B∗

λ)

∫ |Ω|

0

(∂U+

ε

∂s

)2

ds,

and by (3.8) we deduce

∫ |Ω|

0

(∂U+

ε

∂s

)2

ds + γN

∫ |Ω|

0

[λ−

(c−)∗]

s(2/N)−2(U+

ε

)2ds

≥[1− 4γN

∥∥(c−)∗ − λ∥∥

LN/2(B∗λ)

] ∫ |Ω|

0

(∂U+

ε

∂s

)2

ds.

Then, provided we choose λ sufficiently large, we get

αN :=[1− γN

∥∥(c−)∗ − λ∥∥

LN/2(B∗λ)

]> 0,

so we have

∫ |Ω|

0

(∂U+

ε

∂s

)2

ds + γN

∫ |Ω|

0

[λ− (c−)∗

]s(2/N)−2 (U+

ε )2 ds

≥ αN

∫ |Ω|

0

(∂U+

ε

∂s

)2

ds.

(3.9)

We notice that we can obtain a similar estimate also in the cases N = 1, 2 :

indeed, the case N = 1 is much simpler, while in the case N = 2 we find

∫B∗

λ

[(c−)∗ − λ

]s−1

(U+

ε

)2ds ≤ 4C

∥∥(c−)∗ − λ∥∥

Lr(B∗λ)

∫ |Ω|

0

(∂U+

ε

∂s

)2

ds,

22

ALVINO ET AL

82

for some r > 1 and a suitable constant C.

By (3.7) it follows

∫ |Ω|

0s(2/N)−2 ∂Uε

∂tU+

ε ds ≤ 0 (3.10)

for a.e. t ∈ (0, t + τ). Therefore, by integrating (3.10) between 0 and t with

t ∈ (0, t + τ), and using again the boundary conditions of (3.3) we get

0 ≥ 2∫ t

0dτ

∫ |Ω|

0s(2/N)−2 ∂Uε

∂τU+

ε ds =∫ |Ω|

0s(2/N)−2

(∫ t

0

∂

∂τ

(U+

ε

)2dτ

)ds

=∫ |Ω|

0s(2/N)−2

(U+

ε

)2 (s, t) ds

which implies U+ε = 0 in [0, |Ω|] for every t ∈ (0, t + τ). This means that

Uε ≤ 0 in [0, |Ω|]× [t, t + τ), and then also χε ≤ 0 in [0, |Ω|]× [t, t + τ), but

in the same rectangle the function χε is positive. Then χε ≤ 0 in Q∗T .

Remark 2. If N > 2, we could obtain the result of theorem 1 under the

weaker assumption c ∈ LN/2 (Ω), by using the implicit time discretization

scheme. We can take a partition of lenght τ = T/n (n ∈ N) of the interval

(0, T ) and we approximate the solutions u and v of problems (1.2)-(1.6) by

the sequences

un (x, t) := u(k) (x, t) x ∈ Ω, t ∈ [(k − 1) τ, kτ ]

vn (x, t) := v(k) (x, t) x ∈ Ω#, t ∈ [(k − 1) τ, kτ ]

23


83

where u(k) is the solution of the elliptic problem

u(k) − u(k−1)

τ−(a

(k)ij (x) u

(k)xi

)xj

+ cu(k) = f (k) in Ω

u(k) = 0 on ∂Ω,

(3.11)

with

a(k)ij (x) :=

1τ

∫ kτ

(k−1)τaij (x, t) dt,

f (k) (x) :=1τ

∫ kτ

(k−1)τf (x, t) dt

for any k = 1, . . . , n and u(0) := u0, while v(k) is the solution of the sym-

metrized problem

v(k) − v(k−1)

τ−∆v(k) + c#v(k) = f (k)# in Ω#

v(k) = 0 on ∂Ω#,

(3.12)

with v(0) := u#0 . Using the results of [4] (see theorem 3.4) we can prove by

induction that ∫ s

0u(k)∗ (σ) dσ ≤

∫ s

0v(k)∗ (σ) dσ (3.13)

for k = 1, . . . , n. Actually the results of [4] can be applied in the case

c (x)+ 1τ ∈ L∞ (Ω) , but they can be easily extended to the case c (x)+ 1

τ ∈

LN/2 (Ω) , since the operator

L(k)u(k) := −(a

(k)ij (x) u(k)

xi

)xj

+(

c (x) +1τ

)u(k)

24

ALVINO ET AL

84

is coercive (see [27]). Finally we pass to the limit and get (1.5).

References

[1] A. ALBERICO, V. FERONE: Regularity properties of solutions of el-

liptic equations in R2 in limit cases, Rend. Mat. Acc. Lincei, s. 9, 6

(1995), 237-250.

[2] A. ALVINO: Formule di maggiorazione e regolarizzazione per soluzioni

di equazioni ellittiche del secondo ordine in un caso limite, Atti Accad.

Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (8) 52 (1977), 335-340.

[3] A. ALVINO, P.L. LIONS and G. TROMBETTI: Comparison results

for elliptic and parabolic equations via Schwarz symmetrization, Ann.

Inst. Henri Poincare, (2) 7 (1990), 37-65.

[4] A. ALVINO, S. MATARASSO and G. TROMBETTI: Variational in-

equalities and rearrangements, Rend. Mat. Acc. Lincei, s. 9, 3 (1992),

271-285.

[5] A.ALVINO, J.I.DIAZ, P.L.LIONS, G.TROMBETTI: Elliptic Equa-

tions and Steiner Symmetrization, Comm. Pure Appl. Math., Vol.

XLIX, 217-236 (1996).

[6] A. ALVINO, S. MATARASSO and G. TROMBETTI: Elliptic bound-

ary value problems: comparison results via symmetrization, Ric. Mat.,

(2) 51 (2002), 341-356.

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[7] C. BANDLE: On symmetrizations in parabolic equations, J.Anal.

Math., 30 (1976), 98-112.

[8] C. BANDLE: Isoperimetric inequalities and applications, Monographs

and Studies in Math., No. 7, Pitman, London, 1980.

[9] P. BARAS and J. A. GOLDSTEIN: The heat equation with a singular

potential, Trans. Amer. Math. Soc., 284 (1984).

[10] C. BENNET, R. SHARPLEY: Interpolation of Operators, Pure and

Appl. Math. Vol. 129, Academic Press, 1988.

[11] A. BENSOUSSAN and J.L. LIONS: Applications des inequations

variationnelles en controle stochastique, Dunod, Collection ”Methodes

Mathematiques de l’Informatique”, 1978.

[12] X. CABRE and Y. MARTEL: Existence versus instantaneous blow-

up for linear heat equations with singular potentials, C. R. Acad. Sci.

Paris, Ser. I Math., (11) 329 (1999), 973-978.

[13] G. CHITI: Norme di Orlicz delle soluzioni di una classe di equazioni

ellittiche, Boll. U.M.I., (5) 16-A (1979), 178-185.

[14] J.I. DIAZ: Simetrizacion de problemas parabolicos no lineales: aplica-

cion a ecuaciones de reaccion-difusion, Mem. Real Academia De Cien-

cias Exactas, Fisicas Y Naturales, Serie de Ciencias Exactas, Tomo

XXVIII, 1991.

[15] J.I. DIAZ: Symmetrization of nonlinear elliptic and parabolic problems

and applications: a particular overview, Progress in partial differential

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equations: elliptic and parabolic problems (Pont-a-Mousson, 1991), 1–

16, Pitman Res. Notes Math. Ser., 266, Longman Sci. Tech., Harlow,

1992.

[16] V. FERONE and A. MERCALDO: A second order derivation formula

for functions defined by integrals, C.R. Acad. Sci. Paris, t.326, Serie I

(1998), 549-554.

[17] J. GARCIA AZORERO and I. PERAL ALONSO: Hardy Inequalities

and some critical elliptic and parabolic problems, J. Diff. Eq., (2) 144

(1998), 441-476.

[18] G. H. HARDY, J. E. LITTLEWOOD and G. POLYA: Inequalities,

Cambridge University Press, 1964.

[19] B. KAWHOL: Rearrangements and convexity of level sets in PDE,

Lecture Notes in Math., No. 1150, Springer-Verlag, Berlin-New York,

1985.

[20] J. MOSSINO and J.M. RAKOTOSON: Isoperimetric inequalities in

parabolic equations, Ann. Scuola Norm. Sup. Pisa, 13 (1986), 51-73.

[21] G. STAMPACCHIA: Le probleme de Dirichlet pour les equations el-

liptiques du second ordre a coefficients discontinus, Ann. Inst. Fourier

Groenoble, 15 (1965).

[22] G. TALENTI: Elliptic equations and rearrangements, Ann. Scuola

Norm. Sup. Pisa, (4) 3 (1976), 697-718.

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[23] G. TALENTI: Linear Elliptic P.D.E.’s: Level Sets, Rearrangements

and a priori Estimates of Solutions, Boll. U.M.I., (6) 4-B (1985), 917-

949.

[24] J.L. VAZQUEZ: Symmetrization for ut = ∆ϕ (u) and applications

(French), C.R. Acad. Sci. Paris, Serie I , (2) 295 (1982), 71-74.

[25] J.L. VAZQUEZ: Symmetrization and Mass Comparison for Degenerate

Nonlinear Parabolic and related Elliptic Equations, Advanced Nonlinear

Studies, 5 (2005), 87-131.

[26] R. VOLPICELLI: Comparison results for solutions of parabolic equa-

tions, Ric. Mat., (1) XLII (1993), 179-192.

[27] A. SZULKIN and M. WILLEM: Eigenvalue problems with indefinite

weight, Studia Math., (2) 135 (1999), 191–201.

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88

Integral and Differential Calculus in Riesz

spaces and applications

A. Boccuto - D. Candeloro

Dipartimento di Matematica e Informatica

via Vanvitelli, 1 - I-06123 Perugia (ITALY)

e-mail: [email protected], [email protected]


via Vanvitelli, 1 - I-06123 Perugia (ITALY)


contact author: Domenico Candeloro


Via Vanvitelli, 1 - 06123 Perugia (ITALY)

tel. +39 075 5852936 – fax: +39 075 5855024


1

89


2 A. Boccuto--D. Candeloro

Abstract

In this paper we outline a new theory about integral and differ-

ential calculus for Riesz space-valued mappings defined on suitable

Riesz spaces. In our abstract context, we prove some theorems

similar to the classical ones, like for example the Fundamental

Formula of Calculus and the theorem about exchanging order be-

tween limits and derivatives. As applications, we give some results

about power series, a fixed point theorem, and some models of dif-

ferential functional equations.

2000 AMS Mathematics Subject Classification: 28B15, 28B05, 28B10,

46G10.

Keywords: Riesz space, convergence, continuity, differentiability, Taylor

formula, series, differential functional equations.

90

Integral and Differential Calculus... 3

0 Introduction

In this paper a new theory is presented, concerning integral and differ-

ential calculus for functions defined in a suitable Riesz space and with

values in another Riesz space, linked together with a ”product” structure.

This approach is, in a certain sense, a generalization of the one given in

[1]. The concepts of uniform continuity, uniform differentiability, Rie-

mann integrability are introduced and investigated, and some theorems

like the corresponding classical ones are proved: among them we quote

the Fundamental Formula of Calculus. Moreover a version of the Taylor

formula is demonstrated: here we express the ”remainder term” by means

of our introduced abstract integral. One can find applications for exam-

ple in the Ito formula, proved in [2], however we shall not deal with it

here. Furthermore, a theory about exchanging order between limits and

derivatives, power series and analyticity in our abstract context is given:

a fixed point theorem, and some examples of differential and functional

equations are then deduced. These equations too might have interesting

formulations in the Stochastic Calculus, and some of them also in Theory

of Fractals, though we chose not to treat them here.

91


1 Basic definitions and assumptions

A Riesz space R is said to be Dedekind complete if every nonempty subset

A ⊂ R, bounded from above, has supremum in R.

From now on, we assume that R is a Dedekind complete Riesz space.

Given a bounded sequence (pn)n in R, we set:

lim supn

pn = infn∈IN

[supm≥n

pm]; lim infn

pn = supn∈IN

[ infm≥n

pm];

and we say that limn pn = l ∈ R if lim supn pn = lim infn pn = l.

This corresponds to the classical definition of order convergence or (o)-

convergence (see also [5], [6]).

Assumptions 1.1 Let R1, R2, R be three Dedekind complete Riesz

spaces. We say that (R1, R2, R) is a product triple if there exists a map

· : R1 ×R2 → R, which we will call product, such that

1.1.1) (r1 + s1) · r2 = r1 · r2 + s1 · r2, r1 · (r2 + s2) = r1 · r2 + r1 · s2,

1.1.2) [r1 ≥ s1, r2 ≥ 0] ⇒ [r1 · r2 ≥ s1 · r2], [r1 ≥ 0, r2 ≥ s2] ⇒ [r1 · r2 ≥

r1 · s2] for all rj, sj ∈ Rj, j = 1, 2;

1.1.3) if (aλ)λ∈Λ is any family in R1 with aλ ≥ 0 ∀λ and infλ aλ = 0, and

R2 3 b ≥ 0, then infλ (aλ · b) = 0; if (bλ)λ is any family in R2 with

bλ ≥ 0 ∀λ and infλ bλ = 0, and R1 3 a ≥ 0, then infλ (a · bλ) = 0.

92


A Dedekind complete Riesz space R is called an algebra if (R,R,R) is a

product triple.

2 A Riemann-type integral in Riesz spaces

Let (R1, R2, R) be a product triple of Riesz spaces.

Given two elements a, b ∈ R1, with a ≤ b, we denote by [a, b] and call

order interval (or in short interval) the set of all elements r ∈ R1, such

that a ≤ r ≤ b. Given an order interval [a, b] ⊂ R1, a division of [a, b] is

any finite set T = x0, x1, . . . , xn ⊂ [a, b], such that x0 = a, xn = b and

xi ≤ xi+1, xi 6= xi+1 for all i = 0, . . . , n− 1. The mesh of a division T is

the quantity η(T ) = supni=1 (xi − xi−1).

A decomposition of [a, b] is a set E = ([xi−1, xi], ξi) : i = 1, . . . , n,

where x0, x1, . . . , xn is a division T of [a, b] and ξi ∈ [xi−1, xi] ∀ i =

1, . . . , n. For such a decomposition E, we shall put |E| = η(T ).

We now introduce a Riemann-type integral in our setting, which will

be useful in the sequel in order to prove our version of the Taylor formula.

If f : [a, b] → R2 is a map and E is a decomposition of [a, b], E =

([xi−1, xi], ξi): i = 1, . . . , n, we denote by S(f, E) and call Riemann

sum associated with E the element of R given byn∑

i=1

(xi − xi−1) · f(ξi).

A function f : [a, b] → R2 is said to be Riemann integrable (in short,

93


integrable) in [a, b] if there exists an element Y ∈ R such that

infr∈R+

1

(sup|S(f, E)− Y | : |E| ≤ r) = 0,

where R+1 is the set of all elements r ∈ R1 such that r ≥ 0 and r 6= 0. In

this case we write

∫ b

a

f(t) dt = Y .

It is easy to see that such an element Y is uniquely determined.

The following results are easy to prove and will be useful in the sequel.

Proposition 2.1 If f1 and f2 are integrable in [a, b] and α1, α2 ∈ IR,

then α1f1 + α2f2 is integrable in [a, b] too, and in this case we have

∫ b

a

(α1f1 + α2f2)(t) dt = α1

∫ b

a

f1(t) dt+ α2

∫ b

a

f2(t) dt.

If f1 and f2 are integrable in [a, b] and f1 ≤ f2, then

∫ b

a

f1(t) dt ≤∫ b

a

f2(t) dt.

If f is integrable in an order interval [a, b], then f is also integrable in

any order interval I ⊂ [a, b].

Thus it follows that for any integrable function f : [a, b] → R2 the indef-

inite integral is defined as an additive R-valued interval function on the

family of all intervals in [a, b]. We shall denote it by F (I) =

∫I

f , with

the convention to define F ([b, a]) = −F ([a, b]) (∀ a, b ∈ R1, a ≤ b), so

that F ([a, a]) = 0 ∀ a ∈ R1, and we shall call integral function associated

94


with f the map defined (with abuse of notation) as follows:

F (x) ≡ F ([a, x]), x ∈ [a, b].

Like in the classical case, uniform continuity implies integrability.

Definition 2.2 We say that a function f : [a, b] → R2 is uniformly

continuous in [a, b] if

infr∈R+

1

[sup|f(v)− f(u)| : u, v ∈ [a, b], u ≤ v, v − u ≤ r] = 0.

Proposition 2.3 Every uniformly continuous function f : [a, b] → R2

is integrable in [a, b] too.

Proof. The proof is similar to the one in [4]. 2

3 An abstract derivative in Riesz spaces

Throughout this section, we always assume that R1 and R2 are two

Dedekind complete Riesz spaces and that (R1, R2, R2) is a product triple;

let now [a, b] ⊂ R1 be an interval. We begin with the following:

Definition 3.1 We say that a function f : [a, b] → R2 is uniformly

differentiable in [a, b] if there exist a bounded function f ′ : [a, b] → R2

and an increasing family (pr)r∈R+1

such that infr∈R+1pr = 0 and

|f(v)− f(u)− (v − u) f ′(x)| ≤ (v − u) pr (1)

95


for every r ∈ R+1 and whenever u, v, x ∈ [a, b], u ≤ x ≤ v, v − x ≤ r and

x − u ≤ r. In this case we say that f ′ is a uniform derivative of f or,

when no confusion can arise, that f ′ is a derivative of f .

We observe that, in general, f ′ is not unique. Indeed, let R1 and R2 be

the spaces of all bounded measurable real-valued functions, defined on

[0, 1], vanishing on [0, 1/2] and ]1/2, 1] respectively. For every ψ1 ∈ R1

and ψ2 ∈ R2, ψ1 · ψ2 is identically zero (here, · is the usual product

between functions): thus, it is not difficult to see that (R1, R2, 0) is a

product triple with respect to this product. Let [a, b] be any arbitrary

order interval of R1, and f : [a, b] → R2 be any constant function: then

clearly every function f1 : [a, b] → R2 is a derivative of f .

This fact will not affect our results, and it will be clear from the

context in which sense we deal with derivatives.

For instance, it is quite clear that every function f : [a, b] → R2,

uniformly differentiable in [a, b], is uniformly continuous in [a, b].

Usual differentiation rules hold in our setting, for example:

Proposition 3.2 Let (R1, R2, R2), (R1, S2, S2), (R2, S2, T1), (R1, T1, T1)

be four product triples, and [a, b] ⊂ R1 be an interval. If f : [a, b] → R2,

g : [a, b] → S2 are two uniformly differentiable functions with derivatives

f ′, g′ respectively, then the map h = f · g : [a, b] → T1 is uniformly

96


differentiable too, with derivative h′ given by h′(x) = f ′(x) · g(x) + f(x) ·

g′(x), x ∈ [a, b].

Therefore every ”polynomial” function (in a commutative algebra R) is

uniformly differentiable, and the usual differentiation rule is valid.

The following results are fundamental theorems of Integral Calcu-

lus, as in the classical case. We begin with the following version of the

Torricelli-Barrow theorem: the proof is easy.

Theorem 3.3 Let (R1, R2, R2) be a product triple, and f : [a, b] → R2

be a uniformly continuous function (in [a, b]). Then its integral function

F is uniformly differentiable in [a, b] and F ′(x) = f(x) ∀x ∈ [a, b].

We now turn to a version of the Fundamental Formula of Integral Cal-

culus in an abstract setting.

Theorem 3.4 Let (R1, R2, R2) be a product triple, [a, b] ⊂ R1 be an

interval and f : [a, b] → R2 be a uniformly differentiable function, with

derivative f ′. Then, f ′ is integrable, and

∫ b

a

f ′(t) dt = f(b)− f(a).

Proof. Choose arbitrarily r ∈ R+1 and take any decomposition E =

([xi−1, xi], ξi) : i = 1, . . . , n of [a, b], with |E| ≤ r. Let (pr)r∈R+1

be a

97


family as in the definition of uniform differentiability. We get:

0 ≤

∣∣∣∣∣n∑

i=1

(xi − xi−1) · f ′(ξi)− [f(b)− f(a)]

∣∣∣∣∣≤

n∑i=1

|f(xi)− f(xi−1)− (xi − xi−1) · f ′(ξi)|

≤

(n∑

i=1

(xi − xi−1)

)· pr = (b− a) pr.

Thus the assertion follows. 2

Remark 3.5 We can observe that Theorem 3.4 is true also if the end-

points a and b are not comparable, provided they are contained in a

larger interval [A,B] where f is uniformly differentiable, and f ′ is its

derivative. In fact, in case A ≤ a, b ≤ B, we can set h = b−a, and define

∫ b

a

f ′(t)dt =

∫ a+h

a

f ′(t)dt =

∫ a+h+

a

f ′(t)dt−∫ a+h+

a+h

f ′(t)dt : (2)

indeed, as B − a ≥ 0, from h = b − a ≤ B − a it follows h+ ≤ B − a,

and hence [a, a+ h+] ⊂ [A,B]; moreover, it follows also [a+ h, a+ h+] =

[b, a + h+] ⊂ [b, B]. Thus, applying 3.4 to the last member of (2), it

follows easily

∫ b

a

f ′(t)dt = f(a+ h+)− f(a) + f(a+ h)− f(a+ h+) = f(b)− f(a).

98


4 The Taylor formula

We shall prove a version of the Taylor formula in our context. Besides

the obvious applications in approximating functions, this formula has

applications in stochastic integration (see [2]).

Definition 4.1 If a function f : [a, b] → R2 is uniformly differentiable

and if its derivative f ′ is uniformly differentiable with derivative f ′′, we

will say that f ′′ is a uniform second derivative or, when no confusion can

arise, second derivative of f . By induction it is possible to introduce the

(uniform) derivatives of order n for every n ∈ IN . If f : [a, b] → R2 is

uniformly differentiable up to the order n, and if its n-th derivative f (n)

is uniformly continuous, we say that f is of class Cn([a, b]). Furthermore,

if S ⊂ R1 contains at least an order interval, we say that f : S → R2 is

of class Cn(S) if it is of class Cn([a, b]) for every order interval [a, b] ⊂ S,

and that f : S → R2 is of class C∞(S) if it is of class Cn(S) ∀n ∈ IN .

Theorem 4.2 Let R be an algebra, [a, b] ⊂ R be an interval, and f :

[a, b] → R have derivatives up to the order n+1: f ′, f ′′, . . . , f (n), f (n+1).

Fix arbitrarily x0 ∈ [a, b] and h ∈ R, such that x0 + h ∈ [a, b]. Then we

have:

f(x0 + h) = f(x0) +h f ′(x0)

1!+ . . .+

hn f (n)(x0)

n!+B(x0, h),

99


where |B(x0, h)| ≤|h|n+1

n!sup

x∈[a,b]

|f (n+1)(x)|.

Proof. Fix x0 and h as in the hypotheses, and define an auxiliary func-

tion F : [a, b] → R as follows:

F (t) = f(x0 + h)− f(t)− (x0 + h− t) f ′(t)

1!− . . .− (x0 + h− t)n f (n)(t)

n!.

By hypothesis, F is uniformly differentiable and we have, ∀ t ∈ [a, b]:

F ′(t) = −(x0 + h− t)n f (n+1)(t)

n!,

and F ′ is bounded. Put M = supx∈[a,b] |f (n+1)(x)|. By Theorem 3.4 and

Remark 3.5 we get:

F (x0) = −∫ x0+h

x0

F ′(t) dt =

∫ x0+h

x0

(x0 + h− t)n

n!f (n+1)(t) dt

=

∫ x0+h+

x0

(x0 + h− t)n

n!f (n+1)(t)dt−

∫ x0+h+

x0+h

(x0 + h− t)n

n!f (n+1)(t)dt,

and hence

|F (x0)| ≤ M

(∫ x0+h+

x0

|x0 + h− t|n

n!dt+

∫ x0+h+

x0+h

|x0 + h− t|n

n!dt

)

≤ Mh+|h|n + h−|h|n

n!= M

|h|n+1

n!,

since |x0 + h− t| ≤ |h|. Thus the assertion follows. 2

5 Sequences of differentiable functions

In this section we give some conditions, under which it is possible to

exchange the order between limits and derivatives. First of all we intro-

100


duce the concept of uniform convergence for sequences of functions. We

always suppose that [a, b] ⊂ R1 is an order interval.

Definition 5.1 A sequence (fn : [a, b] → R2)n is said to be uniformly

convergent to f : [a, b] → R2 if limn [supt∈[a, b] |fn(t)− f(t)| ] = 0.

We now give two fundamental properties of uniform convergence, which

will be useful in the sequel. The proofs are straightforward.

Theorem 5.2 Let (fn : [a, b] → R2)n be a sequence of integrable func-

tions, uniformly convergent to a map f : [a, b] → R2. Then f is integrable

and

limn

∫ b

a

fn(t) dt =

∫ b

a

f(t) dt.

Theorem 5.3 Let (fn)n be a sequence of uniformly continuous functions

fn : [a, b] → R2, uniformly convergent to a mapping f : [a, b] → R2. Then

f is uniformly continuous.

Thanks to Theorems 3.4, 5.2, 5.3 and 3.3, it is possible to use a classical

technique in order to prove the next result.

Theorem 5.4 Let (fn : [a, b] → R2)n be a sequence of uniformly dif-

ferentiable functions, with derivatives f ′n, n ∈ IN . Moreover, assume

that the sequence (f ′n)n is uniformly convergent in [a, b] and that there

101


exists limn fn(a) in R2. Then the sequence (fn)n is uniformly conver-

gent in [a, b] to a uniformly differentiable function f : [a, b] → R2, and

f ′ = limn f ′n in [a, b].

We recall that, analogously as in the classical case, it is possible to

give the concept of series of elements of any Riesz space R and the ones of

convergence and absolute convergence, and to deduce an analogue of the

Cauchy criterion, together with its usual consequences. Thus, Theorem

5.4 implies the analogue of the classical result concerning differentiation

term-by-term of a series of functions. We shall not write it down here,

however we shall use it later.

6 Power series and applications

In this section we deal with power series: this will be the main tool in

the subsequent applications.

Definition 6.1 Let R be any commutative algebra. We shall suppose

that there exists a multiplicative unit in R, which will be denoted by 1.

For every positive real number k, and for every positive element r ∈ R, we

denote by Sk(r) the following subset of R: Sk(r) = x ∈ R : |x| ≤ k · r;

moreover, for each positive real number t, we set Ut(r) =⋃

0<k<t Sk(r),

Rr =⋃

t>0 Ut(r). In case r = 1, we shall simply write Sk and Ut rather

102


than Sk(1) and Ut(1). A power series is a series of the type

∞∑n=0

anxn, (3)

with x ∈ R, an ∈ R ∀n, and with the convention x0 = 1 ∀x ∈ R.

Proposition 6.2 If the series (3) converges at some x ∈ R, then it

converges uniformly and absolutely in every set Sk(|x|) with 0 < k < 1.

Proof. From convergence of the series at x, it follows that the sequence

(|anxn|)n is bounded in R: let M denote any upper bound for that se-

quence. Now, for any real number k ∈]0, 1[ and every element r ∈ Sk(|x|),

we have |anrn| ≤ |an|kn|xn| ≤ Mkn for all positive integers n. This

clearly implies the assertion. 2

The following results have many consequences: for example they show

that some elements in R have an inverse, and give an expression for it.

Proposition 6.3 The geometric series∞∑

n=0

xn absolutely converges in

the set U1; moreover, for every element x ∈ U1 there exists the inverse of

1 − x in the algebra R: such inverse is the sum of the geometric series

above.

Proof. Let us fix x ∈ U1, and choose any real number α ∈]0, 1[ such that

|x| ≤ α1. Then we get |xn| ≤ αn1: this clearly implies convergence of

the geometric series at x. Moreover, for every positive integer n we have

103


(1− x)n∑

j=0

xj = 1− xn+1. From convergence of the series, we deduce

that limn xn = 0, and finally (1− x)

∞∑n=0

xn = 1. 2

Proposition 6.4 The exponential series∞∑

n=0

xn

n!converges everywhere

in the set R1.

Proof. First of all, we observe that the exponential series obviously

converges at the points ν1, for every positive integer ν. To conclude the

proof, it will suffice to apply Proposition 6.2. 2

As usual, the sum of the exponential series∞∑

n=0

xn

n!will be denoted

by exp(x), whenever it exists. Moreover, usual techniques show that

this function coincides with its derivative (see also Theorem 6.7 below),

and obeys the usual algebraic rules of the exponential function, therefore

exp(x) has always an inverse element, i.e. exp(−x).

Another consequence involves Taylor series:

Proposition 6.5 Let f : Ut → R be a function of class C∞(Ut). A

sufficient condition for convergence to f of its Mc-Laurin series

∞∑n=0

f (n)(0)

n!xn (4)

is that there exists a positive element M ∈ R such that |f (n)(x)| ≤ Mn1

holds, for every n ∈ IN and every x ∈ Ut.

104


Proof. The proof is an easy consequence of Theorem 4.2 and of conver-

gence of the exponential series. 2

Now, given the power series (3), it is possible to associate to it its ”deriva-

tive series”

∞∑n=1

n anxn−1, x ∈ R. (5)

We now prove the following:

Theorem 6.6 Fix t > 0. The series (3) converges at every x ∈ Ut if

and only if the series (5) converges at every x ∈ Ut.

Proof. We begin with the ”if” part. Fix arbitrarily x ∈ Ut, and let

k ∈]0, t[ be any real number such that x ∈ Sk. By hypotheses and by

Proposition 6.2, it follows that the series (5) absolutely converges at x.

Now, from |an xn| = |x||an x

n−1| ≤ |x| |nanxn−1| it follows that the series

(3) converges at x. Concerning the ”only if” part, assume that the series

(3) converges in Ut, and fix any element x ∈ Ut. Let k be any positive

real number, k < t, such that x ∈ Sk. Set k′ :=k + t

2, k′′ :=

k′ + k

2, so

that k < k′′ < k′ < t, and put x1 = k′1, x2 = k′′1. Clearly, x1 ∈ Ut and

therefore, thanks to Proposition 6.2, the series (3) converges absolutely

at x1; now, from |n an xn−12 | = n |an| (k′′)n−11 ≤ |an| (k′)n1 (which holds

at least definitely), we can deduce convergence of the series (5) at x2.

From Proposition 6.2 there follows convergence of (5) at x. 2

105


A consequence of Theorems 6.6 and 5.4 is the following:

Theorem 6.7 Fix any positive real number t. If the series (3) converges

∀x ∈ Ut and if f is its sum, that is

f(x) =∞∑

n=0

anxn ∀x ∈ Ut,

then we get

f ′(x) =∞∑

n=1

n anxn−1 ∀x ∈ Ut.

Proof. Fix arbitrarily x ∈ Ut and take an order interval [−k1, k1] con-

taining x and contained in Ut. Then, by Theorem 6.6, the series (5)

converges (absolutely) at every r ∈ Ut, and by 6.2 the series (3) and

(5) converge uniformly in [−k1, k1]. The assertion follows from this and

term-by-term differentiation. 2

Of course, Theorem 6.7 implies that the sum of a power series∞∑

n=0

anxn,

convergent in Ut, is of class C∞(Ut), and its Mc-Laurin series coincides

with the initial power series.

A first consequence of the previous results is a fixed point theorem,

of the type of Banach.

Theorem 6.8 Let f : [a, b] → [a, b] be any mapping, satisfying

|f(x2)− f(x1)| ≤ K |x2 − x1|

106


for a suitable positive element K ∈ U1, and all x1, x2 ∈ [a, b]. Then

there exists a unique fixed point s for f . Moreover, s is the limit of every

sequence (sn)n defined by choosing s0 arbitrarily in [a, b] and requiring

sn+1 = f(sn) ∀n ∈ IN ∪ 0.

Proof. Let us fix arbitrarily s0 ∈ [a, b], and define the sequence (sn)n as

described, by iterations of f . For all n, p ∈ IN we get

|sn+p − sn| ≤ |s1 − s0|Kn(1−K)−1

by means of usual techniques. As limnKn = 0, the sequence (sn)n is

clearly convergent, and its limit s satisfies f(s) = s thanks to continuity

of f . Uniqueness can be proved by similar techniques. 2

As a remark, we can observe that a mapping f : [a, b] → [a, b] satisfies

the contraction condition of 6.8 as soon as f is uniformly differentiable,

and its derivative satisfies |f ′(x)| ≤ K for all x ∈ [a, b]: this follows easily

from Theorem 4.2.

Further applications allow us to obtain solutions of suitable func-

tional equations, according with the following theorems. Though these

equations are nothing but examples, from them one could find wide gen-

eralizations and also formulations in different contexts, for example in

Stochastic Analysis, by specializing the underlying Riesz space.

107


Theorem 6.9 Assume that κ(u) =∞∑

n=1

γnun is a fixed convergent power

series, with u ∈ R and γn ∈ R ∀n ∈ IN . Then, for every element

s ∈ U1, there exist functions Y : R→ R such that Y (u)− κ(u) = Y (su),

for all u ∈ R.

Proof. We note that the given series converges absolutely at every u ∈ R

(see Proposition 6.2). Moreover, as s ∈ U1, also sn ∈ U1 for every positive

integer n, and (1 − sn)−1 exists, thanks to 6.3. Consider the following

power series:

Y (u) =∞∑

n=1

γn(1− sn)−1un.

From Proposition 6.3 we deduce:

|γn(1− sn)−1un| ≤ |γnun|

∞∑j=0

|s|j = (1− |s|)−1|γnun| ∀n ∈ IN.

By means of the Cauchy criterion, we then obtain (absolute) convergence

of Y (u) for every u ∈ R. Now,

Y (u)− Y (su) =∞∑

n=1

(1− sn)γnun(1− sn)−1 = κ(u)

for every u ∈ R, thus the equation is fulfilled. Clearly, if Y is any solution,

also Y + a is a solution, for every constant element a ∈ R. 2

A slightly different equation does not require analyticity of κ.

Existence and uniqueness of the solution could be deduced from some

modifications of Theorem 6.8, but we have chosen a more direct approach.

108


Theorem 6.10 Assume that κ : R → R is any bounded function, and

fix two elements in R, α, β, with α ∈ U1. Then there is a unique bounded

function Y : R→ R satisfying Y (u)− κ(u) = α Y (β u), for all u ∈ R.

Proof. Set Y (u) =∞∑

n=0

κ(βnu) αn. Clearly Y is well-defined, because κ is

bounded. Moreover, Y is bounded and satisfies α Y (βu) = Y (u)− κ(u),

as required. Now, assume that another bounded function Y1 : R → R

exists, satisfying the same equation. Then we must have κ(u) = Y1(u)−

αY1(βu) and therefore

Y (u) =∞∑

n=0

κ(βnu)αn =∞∑

n=0

(Y1(βnu)− αY1(β

n+1u))αn (6)

= Y1(u)−∞∑

n=1

αnY1(βnu) +

∞∑n=1

αnY1(βnu)

for all u ∈ R (absolute convergence of the series being ensured by bound-

edness of Y1). The conclusion is now obvious. 2

We remark that the function Y here obtained is a generalization of the

so-called Weierstrass functions, which are continuous but nowhere differ-

entiable, and have self-similarity features. We also notice that uniqueness

in the previous theorem is strictly related to boundedness of the function

Y : if we drop such condition, there may exist many different solutions.

For example, let us assume κ = 0, β = 2 1, α = 12

1; the equation then

reduces to Y (2u) = 2Y (u): the solution given by Theorem 6.10 is iden-

tically 0, but every function of the type Y1(u) = ru clearly is a solution

109


(though unbounded), for every constant r ∈ R.

Finally, we turn to some kind of differential functional equation. For

the sake of simplicity, we shall deal with a very particular type of equa-

tion, as described in the following theorem.

Theorem 6.11 Fix any positive element s ∈ R1. Then there exist non-

trivial differentiable functions Y : R1 → R satisfying the equation:

Y ′(u) = Y (s u) (7)

for all u ∈ R1.

Proof. First of all, let us observe that the space of solutions is a linear

one. Next, put

Y (u) =∞∑

n=0

s(n2−n)/2

n!un, u ∈ R1 :

it is not difficult to deduce convergence of the series in R1 and that the

function Y obtained in this way is non-trivial and satisfies (7). 2

References

[1] A. BOCCUTO, Differential and Integral Calculus in Riesz Spaces,

Tatra Mountains Math. Publ. 14 (1998), 293-323.

110


[2] A. BOCCUTO, D. CANDELORO & E. KUBINSKA, Kondurar

Theorem and Ito formula in Riesz spaces, (2004), to appear on J.

Concr. Appl. Math.

[3] D. CANDELORO, Riemann-Stieltjes Integration in Riesz Spaces,

Rend. Mat. (Roma) 16 (1996), 563-585.

[4] M. DUCHON & B. RIECAN, On the Kurzweil-Stieltjes integral in

ordered spaces, Tatra Mountains Math. Publ. 8 (1996), 133-142.

[5] W. A. J. LUXEMBURG & A. C. ZAANEN, Riesz Spaces, I, (1971),

North-Holland Publ. Co.

[6] B. Z. VULIKH, Introduction to the theory of partially ordered spaces,

(1967), Wolters - Noordhoff Sci. Publ.

[7] A. C. ZAANEN, Riesz spaces, II, (1983), North-Holland Publ. Co.

111

112

Random fixed point theorems for uniformly Lipschitzain

and asymptotically regular random operators∗

Poom Kumam† and Somyot Plubtieng‡

†Department of Mathematics, King Mongkut’s University of Technology Thonburi,

Bangkok 10140. THAILAND.

‡Department of Mathematics, Naresuan University, Pitsanulok 65000. THAILAND.

October 13, 2007

Abstract. Let (Ω, Σ) be a measurable space, X a Banach space, C a weakly

compact convex subset of X and T : Ω × C → C a random operator. We prove

the random version of a deterministic fixed point theorem when T is uniformly

Lipschitzian random operators and satisfies property P such that σ(T (ω, ·)) ≤√WCS(X) for all ω ∈ Ω, and T is asymptotically regular on C. Our results also

give stochastic version generalization of some results of Domınguez Benavides and

Xu [A new geometrical coefficient for Banach spaces and its applications in fixed

point theory, Nonlinear Anal. 25 No. 3 (1995), 311-325.].

∗

2000 Mathematics Subject Classification: 47H10, 47H09, 47H04.

Key words and phrases: random fixed point, uniformly Lipschitzian mapping, Lifschititz characteristic, ran-

dom asymptotically regular.∗Supported by The Thailand Research Fund under grant BRG49800018/2549.†Corresponding author.

Email addresses: [email protected] (P. Kumam) and [email protected] (S. Plubtieng)

1

113


2 Poom Kumam and Somyot Plubtieng

1 Introduction

The study of random fixed point theorems was initiated by the Prague school of probability

in the 1950s. Random operator theory is needed for the study of various classes of random

equations (see [9] and references therein). Random fixed point theory has received much

attention for the last two decades because of its importance in probabilistic functional analysis;

the reader is referred to Beg and Shahzad [2], Shahzad and Latif [11] and Tan and Yaun [12].

Generalizations of the random fixed point theorems for continuous selfmaps to the case of

non-selfmaps have been considered by many authors (see e.g. Beg et al. [2], and Shahzad

and Latif [11]). On the other hand, the first fixed point theorem for uniformly Lipschitzian

mapping in Banach spaces was given by Goebel and Kirk [8] who state a relationship between

the existence of fixed point for uniformly Lipschitzian mappings and clarkson modulus of

convexity.

In [3], Casini and Maluta prove the existence of fixed points of uniformly k-Lipschitzian

mapping T with k <√

N(X) in a space X with uniform normal structure. (N(X) is the normal

structure coefficient of X.) In 1995, Benavides and Xu [7] prove the existence of fixed point of

a uniformly Lipschitzian mapping T such that the Lipschitz’s constant σ(T ) <√

WCS(X)

and WCSX > 1. In 1996, Xu [14] gave the random version of Theorem 3.1 of Casini-Maluta

[3] for uniformly Lipschitzian mappings.

The main goal of this paper is to establish some random fixed point theorems for Uni-

formly Lipschitzian and asymptotically regular operators. We will prove the random fixed

point theorems for nonlinear uniformly Lipschitzian mappings in the frame work of a Banach

space with WCS(X) > 1.

2 Preliminaries and notations

Through this paper we will consider a measurable spaces (Ω,Σ) (where Σ is a σ−algebra of

subset of Ω) and (X, d) will be a metric spaces. We denote by CL(X)(resp.CB(X),KC(X))

the family of all nonempty closed (resp. closed bounded, compact) subset of X.

A set-valued operator T : Ω → 2X is call (Σ)− measurable if, for any open subset B of

X,

T−1(B) := ω ∈ Ω : T (ω) ∩B 6= ∅

114

Random Fixed Point Theorems 3

belongs to Σ. A mapping x : Ω → X is said to be a measurable selector of a measurable

set-valued operator T : Ω → 2X if x(·) is measurable and x(ω) ∈ T (ω) for all ω ∈ Ω. Let M

be a nonempty closed subset of X. An operator T : Ω×M → 2X is call a random operator if,

for each fixed x ∈ M , the operator T (·, x) : Ω → 2X is measurable. We will denote by F (ω)

the fixed point set of T (ω, ·), i.e.,

F (ω) := x ∈ M : x ∈ T (ω, x) .

Note that if we do not assume the existence of fixed point for the deterministic mapping

T (ω, ·) : M → 2X , F (ω) may be empty. A measurable operator x : Ω → M is said to be a

random fixed point of a operator T : Ω×M → 2X if x(ω) ∈ T (ω, x(ω)) for all ω ∈ Ω. Recall

that T : Ω ×M → 2X is continuous if, for each fixed ω ∈ Ω, the operator T : (Ω, ·) → 2X is

continuous.

Let C be a closed bounded convex subset of a Banach spaces X. A random operator

T : Ω × C → C is said to be nonexpansive if, for fixed ω ∈ Ω the map T : (ω, ·) → C

is nonexpansive. We will say that T is uniformly Lipschitzian if there exists a function

k : Ω → [1, +∞) such that

‖Tn(ω, x)− Tn(ω, y)‖ ≤ k(ω)‖x− y‖

for all x, y ∈ C and for each integer n ≥ 1. Here Tn(ω, x) is the valued at x of the nth iterate

of the map T (ω, ·). We will say that T is asymptotically nonexpansive if there exists a sequence

of function kn : Ω → [1, +∞) such that for each fixed ω ∈ Ω, limn→∞ kn(ω) = 1 and

‖Tn(ω, x)− Tn(ω, y)‖ ≤ kn(ω)‖x− y‖

for all x, y ∈ C and integer n ≥ 1. The nonexpansive random map T is called asymptotically

regular if for each x ∈ K,

limn→∞

‖Tn+1(ω, x)− Tn(ω, x)‖ = 0

for each ω ∈ Ω.

Now recall the weakly convergent sequence coefficient WCS(X) [7] of X is defined by

WCS(X) = inf

A(xn)infy∈coxn lim supn→∞ ‖xn − y‖ : xn is a weakly convergent sequence

which is not norm-convergent

,

where A(xn) = lim supn→∞‖xi−xj‖ : i, j ≥ n is the asymptotic diameter of xn. We will

use next relationship between the asymptotically center of a sequence and the characteristic

of convexity of the space. Let C be a nonempty bounded closed subset of Banach spaces X

115


and xn bounded sequence in X, we use r(C, xn) and A(C, xn) to denote the asymptotic

radius and the asymptotic center of xn in C, respectively, i.e.

r(C, xn) = inf r(x, xn) : x ∈ C , wherer(x, xn) = lim supn

‖xn − x‖,A(C, xn) = x ∈ C : r(x, xn) = r(C, xn) .

If D is a bounded subset of X, the Chebyshev radius of D relative to C is defined by

rC(D) := inf sup‖x− y‖ : y ∈ D : x ∈ C .

Definition 2.1. Let xn and C be a nonempty bounded closed subset of Banach spaces X.

Then xn is called regular with respect to C if r(C, xn) = r(C, xni) for all subsequences

xni of xn.

We are going to list several result related to the concept of measurability which will be

used repeatedly in next section.

Theorem 2.2. ( cf. Wagner [13]). Let (X, d) be a complete separable metric spaces and

F : Ω → CL(X) a measurable map. Then F has a measurable selector.

Theorem 2.3. ( cf. Tan and Yuan [12]). Let X be a separable metric spaces and Y a metric

spaces. If f : Ω×X → Y is a measurable in ω ∈ Ω and continuous in x ∈ X, and if x : Ω → X

is measurable, then f(·, x(·)) : Ω → Y is measurable.

Follows form the separability of C and form Theorem 1.2 of Bharucha-Reid’s book [?],

we can easily prove the following proposition.

Proposition 2.4. Let C be a closed convex separable subset of a Banach space X and (Ω,Σ)

be a measurable space. Suppose f : Ω → C is a function that is w-measurable, i.e., for

each x∗ ∈ X∗, the dual space of X, the numerically-valued function x∗f : Ω → (−∞,∞) is

measurable, then f is measurable.

Theorem 2.5. (Benavidel, Lopez and Xu cf.[5]). Suppose C is a weakly closed nonempty

separable subset of a Banach space X, F : Ω → 2X a measurable with weakly compact values,

f : Ω× C → Ris a measurable, continuous and weakly lower semicontinuous function. Then

the marginal function r : Ω → R defined by

r(ω) := infx∈F (x)

f(ω, x)

and the marginal map. R : Ω → X defined by

R(ω) := x ∈ F (x) : f(ω, x) = r(ω)

are measurable.

116


Proposition 2.6. ( Xu cf.[14]) Let M be a separable metric space and f : Ω × C → R be a

Caratheodory map, i.e., for every x ∈ M , then the map f(·, x) : Ω → R is measurable and

for every ω ∈ Ω, the map f(ω, ·) : M → R is continuous. Then for any s ∈ R, the map

Fs : Ω → M defined by

Fs(ω) = x ∈ M : f(ω, x) < s, ω ∈ Ω

is measurable.

Let M be a bounded convex sunset of a Banach space X. We recall that the Lifschitz

characteristic for asymptotically regular mappings, is defined;

(i) A number b ≥ 0 is said to have property (Pω) with respect to M if there exists some

a > 1 such that for all x, y ∈ M and r > 0 with ‖x−y‖ > r and each weakly convergent

sequence ξn ⊂ M for which lim sup ‖ξn − x‖ ≤ ar and lim sup ‖ξn − y‖ ≤ br, there

exists some z ∈ M such that lim inf ‖ξn − z‖ ≤ r;

(ii) κω(M) = supb > 0 : b has property (Pω) w.r.t. M;

(iii) κω(X) = infκω(m) : M as above.

If S is a mapping from a set C into itself, then we use the symbol |S| to denote the

Lipscitz constant of S, i.e.

|S| = sup‖Sx− Sy‖

‖x− y‖ : x, y ∈ C, x 6= y

.

For a mapping T on C, we set

σ(T ) = lim infn→∞

|Tn|.

A random operator T (ω, ·) on C has property (P ) if there exists subsequence Tnj (ω, ·)of Tn(ω, ·) converges uniformly to lim infn→∞ |Tn(ω, ·)|.

3 The Main results

In the framework of random nonexpansive operators Domınguez Benavides and Xu [7]. proved

the following result:

117


Theorem 3.1. Let C be a nonempty weakly compact convex separable subset of a Banach

space with WCS(X) > 1 and T : Ω × C → C be a uniformly Lipschitzian random operator

and T (ω, ·) has property (P ), such that σ(T (ω, ·)) ≤√

WCS(X) for all ω ∈ Ω. Suppose in

addition that T is asymptotically regular on C. Then T has a random fixed point.

Proof. It is easy to see that (cf. [7]),

WCS(X) = sup

M > 0 : M · lim supn→∞

‖xn − x∞‖ ≤ Dxn

,

where the supremum is taken over all weakly (not strongly) convergent sequence xn in X

and x∞ is the weak limit of xn and Dxn = lim supm→∞ lim supn→∞ ‖xn − xm‖.

Fixed x0 ∈ C, and consider the measurable function x0(ω) = x0. Define a map G1 : Ω →CB(C) by

G1(ω) := w − clTn((ω), x0), ω ∈ Ω.

(Here w− cl denote the closure under the weak topology of X.) Then G1 : Ω → CB(C) is w-

measurable. By Lemma 2.2, G1 has a w- measurable selector x1 : Ω → C. Since C is separable

x1 is actually measurable by Proposition 2.4. By the definition of G1, we note that x1(ω)

is a weak cluster point of Tn((ω), x0) for each ω ∈ Ω. Hence, for a fix ω ∈ Ω, there exists a

subsequence nk(1) of positive integer n such that Tnk(1)((ω), x0) converging weakly to

x1(ω) and satisfies

σ(T (ω, ·)) ≤ σ(T (ω, ·)) <√

WCS(X).

Next, define a map G2 : Ω → C by

G2(ω) := w − clTn((ω), x1), ω ∈ Ω.

Then G2 : Ω → C is w- measurable. By Lemma 2.2, G2 has a w- measurable selector

x2 : Ω → C. Since C is separable x2 is actually measurable by Proposition 2.4. Since for

each ω ∈ Ω, x2(ω) is a weak cluster point of Tn((ω), x1). In fact, by definition of G2, for

a fix ω ∈ Ω, we have a subsequence Tnk(2)((ω), x1) of Tn((ω), x1) converging weakly to

x2(ω).

By induction, for each m ≥ 1 we construct Gm(ω) := w − clTn((ω), xm−1), ω ∈ Ω.

Then Gm : Ω → C is w- measurable. It follows again from Lemma 2.2, that Gm has a w-

measurable selector xm which measurable by Proposition 2.4. By definition of Gm, for a fix

ω ∈ Ω, we have a subsequence Tnk(m)((ω), xm−1(ω)) xm(ω) for some nj of n. That

is we can construct a sequence xm of measurable function xm : Ω → C such that for each

118


ω ∈ Ω and integer m ≥ 1,

xm(ω) = w − limj→∞

Tnj (ω, xm−1(ω)),

where nj := nk(m), and for each ω ∈ Ω.

Note that the asymptotic regular of T (ω.·) we have

xm(ω) = w − limj→∞

Tnj+p(ω, xm−1(ω)), ∀p ≥ 0.

We now show that xm(ω) converges strongly to foxed point of T . Set for each integer

m ≥ 1,

Bm(ω) = lim supj

‖Tnj (ω, xm(ω))− xm+1(ω)‖, Lm(ω) = |Tnm(ω)|

for each ω ∈ Ω and set

α =[σ(T )]2

WCS(X).

Then α < 1 and by the above definition of WCS(X), for each ω ∈ Ω, we have

Bm(ω) ≤ 1WCS(X)

D(Tnj (ω, xm(ω)))

However, since T (ω, ·) have property P and from the w-lower semicontinuous of the norm of

X, it follows that

D(Tnj (ω, xm(ω))) = lim supj lim supi ‖Tni(ω, xm(ω))− Tnj (ω, xm(ω))‖= lim supj lim supi ‖Tni+nj (ω, xm(ω))− Tnj (ω, xm(ω))‖≤ lim supj |Tnj (ω) lim supi ‖Tni(ω, xm(ω))− xm(ω)‖= limnj Lj(ω) lim supi ‖Tni(ω, xm(ω))− xm(ω)‖≤ σ(T (ω, ·)) lim supi (lim infj ‖Tni(ω, xm(ω))− Tnj (ω, xm−1(ω))‖)≤ σ(T (ω, ·))(lim supi Li(ω)) lim supj ‖xm(ω)− Tnj (ω, xm−1(ω))‖= [σ(T (ω, ·))]2Bm−1(ω).

We, therefore conclude that

Bm(ω) ≤ [σ(T (ω, ·))]2WCS(X)

Bm−1(ω) ≤ αBm−1(ω).

Now using the w-lower semicontinuous of the norm of X again, we deduce that

‖xm(ω)− xm+1(ω)‖ ≤ lim supi ‖xm(ω)− Tni(ω, xm(ω))‖+ lim supi ‖Tni(ω, xm(ω))− xm+1(ω)‖

≤ lim supi lim supj ‖Tnj (ω, xm−1(ω))− Tni(ω, xm(ω))‖+ Bm(ω)

≤ lim supi |Tnj | lim supj ‖Tnj (ω, xm−1(ω))− xm(ω)‖+ Bm(ω)

= σ(T (ω, ·))Bm−1(ω) + Bm(ω).

119


This implies that xm(ω) is Cauchy sequence for each ω ∈ Ω. For any ω ∈ Ω, let x(ω) =

limm→∞ xm(ω). We will that x(ω) is a random fixed point of T. Indeed, for each j ≥ 1 we

have

‖x(ω)− Tnj (ω, x(ω))‖ ≤ ‖x(ω)− xm+1(ω))‖+ ‖xm+1(ω)− Tnj (ω, xm(ω))‖+‖Tnj (ω, xm(ω))− Tnj (ω, x(ω))‖

≤ ‖x(ω)− xm+1(ω))‖+ ‖xm+1(ω)− Tnj (ω, xm(ω))‖+|Tnj |‖xm(ω)− x(ω)‖.

Taking the upper limit as j →∞ yields

lim supj

‖x(ω)− Tnj (ω, x(ω))‖ ≤ ‖x(ω)− xm+1(ω)‖+ Bm(ω) + σ(T (ω, ·))‖xm(ω)− x(ω)‖

which implies Tnj (ω, x(ω))−x(ω) → 0 as m →∞. Since T (ω, ·) is continuous and asymptotic

regular, it follows that x(ω) = T (ω, x(ω)). Observe that x(ω) is the limit of measurable

mappings, so it is measurable. Hence x(ω) is a random fixed point of T . This completes the

proof.

Remark 3.2. Theorem 3.1 is stochastic version of the Theorem 3.2 of Domınguez Benavides

and Xu in [7]. for uniformly Lipschitzian and asymptotically regular mappings.

Acknowledgement. The authors would like to thanks The Thailand Research Fund for

financial support.

References

[1] A.T. Bharucha-Reid,“Fixed point theorem in proobabilistic analysis,” Bull. Amer. Math.

Soc. 82(1976) 641-645.

[2] I. Beg and N. Shahzad. Random approximations and random fixed point theorems, J.

Appl. Math. Stoch. Anal. 7, 2, 145–150 (1994).

[3] E. Csaini and E. Maluta, Fixed points of uniformly Lipschitzian mappings in spaces with

uniformly normal structure, Nonlinear Analysis 9, 103-108 (1985).

[4] T. Domınguez Benavides, “Fixed point theorems for uniformly Lipschitziane mappings

and asymptotically regular mappings”, Nonlinear Anal. 32 No. 1 (1998), 15-27.

[5] T. Domınguez Benavides G. Lopez Acedo and H.-K Xu,“Random fixed point of set-valued

operator”, Proc. Amer. Math. Soc.124 (1996), 838-838.

120

[6] T. Domınguez Benavides G. Lopez Acedo and H.-K Xu“Weak uniform normal structure

and iterative fixed points of nonexpansive mappings”,Coll. Math., (1995), LXVIII(I),17-

23.

[7] T. Domınguez Benavides and H.-K Xu“A new geometrical coefficient for Banach spaces

and its applications in fixed point theory”, Nonlinear Anal. 25 No. 3 (1995), 311-325.

[8] K. Goebel and W. A. Kirk, “Topic in metric fixed point theorem,” Cambridge University

Press, Cambridge (1990).

[9] S. Itoh, “Random fixed point theorem for a multivalued contraction mapping ”, Pacific

J. Math. 68 (1977), 85-90.

[10] P. Lorenzo Ramırez,“Random fixed point of uniformly Lipschitzian mappings”, Nonlinear

Anal. 57 (2004), 23-34.

[11] N. Shahzad and S. Latif,“Random fixed points for several classes of 1-ball-contractive

and 1-set-contractive random maps”, J. Math. Anal. Appl. 237 (1999), 83-92.

[12] K.-K. Tan and X.Z. Yuan, “Some random fixed point theorem ”, in: K.-K. Tan (Ed),

Fixed Point Theory and Applications, Wold Sciedtific, Singapro, 1992,334-345.

[13] D.-H. Wagner, “Survey of measurable selection theorems”, SIAM J. Control Optim. 15

(1977), 859-903.

[14] H. K. Xu,“Random fixed point theorems for nonlinear uniform Lipschitzian mappings ”,

Nonlinear Anal. 26(1996)No.7, 1301-1311.

[15] S. Reich,“Fixed point in locally convex spaces”, Math. Z. 125(1972), 17-31.

[16] X. Yuan and J. Yu,“Random fixed point theorems for nonself mappings ”, Nonlinear

Anal. 26(1996)No.6, 1097-1102.

9

RANDOM FIXED POINT THEOREMS

121

122

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TABLE OF CONTENTS, JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOLUME 3, NO.1, 2008 BOUNDS IN SPACES OF MORREY UNDER CORDES TYPE CONDITIONS, A.CANALE,………………………………………………………………………...11 APPLICATIONS OF RANDOMLY PSEUDO-MONOTONE OPERATORS WITH RANDOMLY UPPER SEMICONTINUITY IN GENERALIZED RANDOM QUASIVARIATIONAL INEQUALITIES,M.K.AHMAD,A.H.SIDDIQI AND SALAHUDDIN,………………………………………………………………….…33 ON UNIFORM CONTINUITY AND LEBESGUE PROPERTY IN INTUITIONISTIC FUZZY METRIC SPACES,C.ALACA AND H.EFE………………………………51 SHARP ESTIMATES FOR SOLUTIONS OF PARABOLIC EQUATIONS WITH A LOWER ORDER TERM, A.ALVINO,R.VOLPICELLI,B.VOLZONE,…………..61 INTEGRAL AND DIFFERENTIAL CALCULUS IN RIESZ SPACES AND APPLICATIONS,A.BOCCUTO,D.CANDELORO,………………………………..89 RANDOM FIXED POINT THEOREMS FOR UNIFORMLY LIPSCHITZIAN AND ASYMPTOTICALLY REGULAR RANDOM OPERATORS, P.KUMAM,S.PLUBTIENG,………………………………………………………..113

Volume 3,Number 2 April 2008 ISSN:1559-1948 (PRINT), 1559-1956 (ONLINE) EUDOXUS PRESS,LLC


SCOPE AND PRICES OF

















Multi-anisotropic Gevrey classes and ultradistributions

Daniela Calvo

Dipartimento di Matematica, Universita di Torino

Via Carlo Alberto 10, 10123 Torino, Italy

e-mail: [email protected],

Tel. 0039 011 6702836, Fax 0039 011 6702878

Alessandro Morando

Dipartimento di Matematica, Universita di Torino

Via Carlo Alberto 10, 10123 Torino, Italy

e-mail: [email protected],

Tel. 0039 011 6702803, Fax 0039 011 6702878

1

139JOURNAL OF APPLIED FUNCTIONAL ANALYSIS,VOL.3,NO.2,139-162,COPYRIGHT 2008 EUDOXUS PRESS, LLC

Abstract

We consider a relevant generalization of the standard Gevrey classes, the so-called multi-anisotropic

spaces, defined in terms of a given complete polyhedron. With respect to the previous literature on

the subject, we concentrate here in the study of the topology. It is defined as inductive and projec-

tive limit of Banach spaces, in two equivalent ways, based on the estimates on the derivatives and

on the Fourier transform, respectively. We consequently introduce the dual space, the class of the

multi-anisotropic ultradistributions, of which we give different characterizations, study topological

and algebraic properties and present some applications.

AMS Subject Classification (MSC 2000): 46F05, 46E10, 35A99.

Key Words: Generalized Gevrey functions, ultradistributions, inductive and projective limits of Ba-

nach spaces, complete polyhedra.

2

CALVO-MORANDO140

1 Introduction

The standard Gevrey classes play an important role in the study of partial differential equations as

intermediate setting between the C∞ and the analytic spaces. In particular, whenever the behavior of a

differential operator (for instance local solvability and well-posedness of the Cauchy problem) is different

in the C∞ and analytic frame, it is natural to study the problem in Gevrey classes. Here, we deal with

Roumieu type Gevrey classes, cf. [29, 30]; the classes of Beurling type, defined in [2] (cf. also [20]), have

a parallel study.

Let us recall the definition of the standard Gevrey classes, in order to pass then to their generalizations.

Let Ω be an open subset of Rn and s ∈ R, s ≥ 1. We say that a function f belongs to the Gevrey class

Gs(Ω) if f ∈ C∞(Ω) and for every compact subset K of Ω there is a constant C > 0 such that:

supx∈K

|Dαf(x)| ≤ C |α|+1|α|s|α|, ∀α = (α1, . . . , αn) ∈ Nn. (1)

Observe that G1(Ω) is the set of the analytic functions in Ω and that Gs(Ω) ( C∞(Ω) for any s ≥ 1. A

useful characterization of the Gevrey functions with compact support Gs0(Rn) = Gs(Rn) ∩ C∞

0 (Rn), for

s > 1, is expressed by a decay estimate of the Fourier transform; namely the following result holds (cf.

f.i. [28]).

A compactly supported distribution f ∈ E ′(Rn) belongs to Gs0(Rn) if and only if there are two constants

C, ε > 0 such that the Fourier transform of f , f(ξ) =∫

Rn e−ix·ξf(x) dx, satisfies

|f(ξ)| ≤ C exp(−ε〈ξ〉 1s ), ∀ξ ∈ Rn, (2)

where 〈ξ〉 :=√

1 + |ξ|2.The Gevrey classes Gs(Ω), Gs

0(Ω) can be endowed with natural locally convex topologies and the topo-

logical dual spaces, called ultradistributions, can be constructed, cf. for instance [20].

In literature, two kinds of generalizations of Gevrey classes were introduced, proceeding from the esti-

mates on the derivatives (1) or the decay of the Fourier transform (2). The first approach consists of

replacing |α|s|α| in (1) with a more general quantity Mα, depending on the multi-index α and obeying

suitable conditions related to the properties of the corresponding function spaces. This case, usually

referred to as ultradifferentiable classes, was analyzed by Roumieu in [30], under general hypotheses on

the sequence Mα. When Mα depends on α only through its length |α| := α1 + · · · + αn, the related

Roumieu spaces coincide with the classes EMp(Ω), DMp(Ω), widely studied by many authors, among

which Roumieu [29], Lions-Magenes [23], Mandelbrojt [24], Komatsu [20], Braun-Meise-Taylor [4], Rudin

[31] and Matsumoto [25]. The second approach was followed by Bjorck [3], Liess-Rodino [22] and Calvo-

Morando-Rodino [10], where the inhomogeneous Gevrey classes Gs,λ(Ω) are introduced by replacing in

(2) the weight 〈ξ〉 with a function λ(ξ) fulfilling suitable conditions.

The present paper is devoted to an important extension of the standard Gevrey classes and their dual

spaces: the multi-anisotropic Gevrey functions and ultradistributions, defined by a generalization of the

3

MULTI-ANISOTROPIC GEVREY CLASSES... 141

estimates (1) on the derivatives based on the notion of multi-quasi-elliptic or complete polyhedra in-

troduced by Friberg [13], Pini [27] and Barozzi [1] in relation with hypoellipticity. They include, for

particular choices of the polyhedron, the standard Gevrey classes and the anisotropic Gevrey classes

studied, for instance, in [11], [14], [36]. An equivalent characterization via Fourier transform shows that

the multianisotropic Gevrey functions and ultradistributions are a particular case of the inhomogeneous

case of [3], [10], [22], for a suitable choice of the weight function. Comparing with the more general

inhomogeneous Gevrey classes, our case allows a wider study, more direct proofs and more precise char-

acterizations, as we can proceed with estimates on the derivatives.

We observe that, in general, the sequence Mα related to the multi-anisotropic Gevrey spaces cannot be

written in terms of the length of α, and therefore they cannot be recovered by the theory of the classes

EMp(Ω),DMp(Ω); neither they fit in the frame of Roumieu [30]. In fact, the Roumieu ultradifferen-

tiable classes (providing the test functions) are assumed to be rings of multiplication (cf. [30], Theorem

4), that is not true in our case (cf. Proposition 3.4), through very restrictive conditions on the sequence

Mα. However, the multi-anisotropic spaces keep the main features displayed by the Roumieu ultradis-

tributions.

The multi-anisotropic Gevrey classes have many applications in the study of partial differential equations,

among which we mention the works of Corli [12], Gindikin-Volevich [14], Hakobyan-Markaryan [16] and

Kazharyan [18] regarding the hypoellipticity, Bouzar-Chaili [5, 6], Calvo-Hakobyan [9] and Zanghirati

[35] for the iterates, Calvo [7, 8] for hyperbolic problems. The dual spaces, the multi-anisotropic ultra-

distributions, also have applications in the study of partial differential equations. In fact, all the results

concerning hypoellipticity and iterates of a multi-quasi-elliptic operator ([1], [5], [6], [9], [12], [13], [14],

[16], [18], [27], [35], [36]) can be reformulated by replacing the class of the Schwartz distributions with

the space D′s,P(Ω) corresponding to the Newton polyhedron P of the operator; moreover, we study in

this frame the well-posedness of the Cauchy problem for weakly hyperbolic operators (cf. Theorem 4.3).

The paper is planned as follows.

In Section 2 we recall the notion of complete polyhedron in Rn and collect some related definitions and

properties.

In Section 3 we introduce the multi-anisotropic Gevrey classes Gs,P(Ω), s ≥ 1, and Gs,P0 (Ω), s > 1. In

analogy with the standard Gevrey case, we can endow these spaces with locally convex topologies, defined

as inductive and projective limits of Banach spaces. On the other hand, we can also characterize their

topology in an equivalent way, through the behavior of the Fourier transform, that coincides with the one

of the inhomogeneous Gevrey classes (cf. [10]). This allows also to see that the topology of the standard

Gevrey classes (cf. f.i. [20] and [28]) can be equivalently formulated in terms of the Fourier transform,

and therefore related to the topology of the inhomogeneous Gevrey classes. We then study the algebraic

and topological properties of the operations.

In Section ?? we construct the topological dual of Gs,P0 (Ω), the space of multi-anisotropic ultradistri-

butions D′s,P(Ω), and of Gs,P(Ω), the space of compactly supported multi-anisotropic ultradistributions

4

CALVO-MORANDO142

E ′s,P(Ω), of which we present different characterizations. In particular, the latter can be defined also by an

exponential-type estimate on the Fourier transform or on the Fourier-Laplace transform (cf. Theorems 4.1

and 4.2); the latter gives a version of the Paley-Wiener-Schwartz theorem for distributions in our frame.

We study the topological and algebraic properties of the multi-anisotropic ultradistributions, and show

under which conditions we can set in these spaces a theory of linear partial differential operators with

variable coefficients. As an application, we deal with the Cauchy problem, showing the well-posedness in

the set of multi-anisotropic ultradistributions for a class of weakly hyperbolic operators (a generalization

of the s-hyperbolic operators of Larsson [21]), called multi-quasi-hyperbolic and modeled in such a way

to have well-posedness in the multi-anisotropic Gevrey classes (cf. [7]).

2 Complete polyhedra

To introduce our study of multi-anisotropic Gevrey functions and ultradistributions, we start by de-

scribing complete polyhedra and some related properties. Let P be a convex polyhedron in Rn, then Pcan be obtained as convex hull of a finite set V(P) ⊂ Rn of convex-linearly-independent points, called

the vertices of P and uniquely determined by P. Moreover, if P has non-empty interior and the ori-

gin belongs to P, there is a finite set N (P) = N0(P) ∪ N1(P), with |ν| = 1 ∀ν ∈ N0(P), such that

P = z ∈ Rn|ν · z ≥ 0,∀ν ∈ N0(P), ν · z ≤ 1,∀ν ∈ N1(P) (N1(P) is the set of the normal vectors to the

faces of P).

Definition 2.1. A complete polyhedron is a convex polyhedron P ⊂ Rn+ such that

1. V(P) ⊂ Qn (i.e. all vertices have rational coordinates);

2. the origin (0, 0, . . . , 0) belongs to P;

3. N0(P) = e1, e2, . . . , en, with ej = (0, . . . , 0, 1j−th, 0, . . . , 0) ∈ Rn for j = 1, . . . , n;

4. every ν ∈ N1(P) has strictly positive components.

Remark 1. The condition 4 implies that for every x ∈ P the set Q(x) = y ∈ Rn|0 ≤ y ≤ x is included

in P and if s belongs to a face of P and r > s, then r 6∈ P (where for x, y ∈ Rn, y ≤ x means that

yi ≤ xi, i = 1, . . . , n; and y < x means y ≤ x, y 6= x).

Let us now summarize some notations related to a complete polyhedron P:

k(s,P) = inft > 0 : t−1s ∈ P = maxν∈N1(P) ν · s, ∀s ∈ Rn+;

µj(P) = maxν∈N1(P) ν−1j ;

µ = µ(P) = maxj=1,...,n µj the formal order of P;

µ(0) = µ(0)(P) = minγ∈V(P)\0 |γ| the minimum order of P;

µ(1) = µ(1)(P) = maxγ∈V(P) |γ| the maximum order of P.

5


Finally, we define the weight function associated to P:

|ξ|P :=( ∑

v∈V(P)

|ξv|) 1

µ

, ∀ξ ∈ Rn. (3)

It is a weight function according to the definition of Liess-Rodino [22].

The definition of the previous quantities is clarified by the following result (for the proof we refer to

[7, 16]).

Proposition 2.1. Let P be a complete polyhedron in Rn with vertices sl = (sl1, . . . , s

ln), for l =

1, . . . , n(P). Then

1. for every j = 1, 2, . . . , n, there is a vertex slj of P such that slj = sljj ej, s

ljj = maxs∈P sj =: mj(P);

2. the boundary of P has at least one vertex lying outside the coordinate axes if and only if µj >

mj , ∀j = 1, . . . n, that is equivalent to ask that the formal order µ(P) is greater than the maximum

order µ(1)(P);

3. if s belongs to P, then |ξs| ≤∑n(P)

l=1 |ξsl |, ∀ξ ∈ Rn, where ξs =∏n

j=1 ξsj

j and n(P) is the number of

vertices of P, including the origin.

Proposition 2.2. For any complete polyhedron P and any s ∈ Rn+, k(s,P) is well defined and bounded

as follows:|s|µ(1)

≤ k(s,P) ≤ |s|µ(0)

.

The associated weight function |ξ|P satisfies for some constants C1, C2 > 0 and all ξ ∈ Rn:

C1〈ξ〉µ(0)

µ ≤ |ξ|P ≤ C2〈ξ〉µ(1)

µ .

Considering a polynomial with complex coefficients, we can regard it as the symbol of a differential

operator and associate a polyhedron to it.

Definition 2.2. Let P (D) =∑

|α|≤m cαDα, cα ∈ C, be a differential operator in Rn with complex

coefficients and P (ξ) =∑

|α|≤m cαξα, ξ ∈ Rn, its symbol. The Newton polyhedron or characteristic

polyhedron associated to P (D) (or P (ξ)) is the convex hull of the set 0⋃α ∈ Zn

+ : cα 6= 0.

The Newton polyhedron of an hypoelliptic operator is complete (cf. Friberg [13]), but the converse is not

true in general (cf. Bouzar-Chaili [5, 6], Calvo-Hakobyan [9] and Zanghirati [35, 36]).

To clarify our treatment, we give now some examples of complete polyhedra (for more details cf. [7, 8]).

1. If P (D) is an elliptic operator of order m, then its Newton polyhedron is the complete polyhedron

of vertices 0,mej , j = 1, . . . , n. The set N1(P) is reduced to the point ν = m−1∑m

j=1 ej , and

mj(P) = µj(P) = µ(0)(P) = µ(1)(P) = µ(P) = m, for all j = 1, 2, . . . , n; the weight function |ξ|Passociated to P is equivalent to 〈ξ〉.

6

CALVO-MORANDO144

2. If P (D) is a quasi-elliptic operator of order m (cf. for instance [14], [28], [36]), its characteristic

polyhedron P is complete and has vertices 0,mjej , j = 1, . . . , n, where mj = mj(P) are fixed

integers (the anisotropic case). The set N1(P) is reduced to a point ν =∑n

j=1 m−1j ej ; then

µj(P) = mj , for all j = 1, . . . , n, µ(0)(P) = minj=1,...,n mj , µ(P) = µ(1)(P) = maxj=1,...,n mj = m.

The weight function associated to P is |ξ|P = (1 + |ξ1|m1 + · · ·+ |ξn|mn)1m .

3. If P ⊂ R2 is the polyhedron of vertices V(P) = (0, 0), (0, 3), (1, 2), (2, 0), then P is complete and

N1(P) =ν1 =

(13 , 1

3

), ν2 =

(12 , 1

4

). We have m1(P) = µ(0)(P) = 2, m2(P) = m(P) = µ(1)(P) =

3, µ(P) = 4. We observe that in this case the formal order µ(P) is bigger than the maximal order

m(P), as P has a vertex lying outside the coordinate axes (cf. Proposition 2.1). The weight function

associated to P is |ξ|P = (1 + |ξ1|2 + |ξ2|3 + |ξ1ξ22 |)

14 .

Following Hakobyan-Markaryan [16], we define the complementary polyhedron associated to a complete

polyhedron P and give an important property involving the corresponding weight function; the latter

will be closely related to the behavior of the pointwise product, cf. Proposition 3.4.

Definition 2.3. Let P be a complete polyhedron in Rn and let µj = µj(P), j = 1, . . . , n. The comple-

mentary polyhedron associated to P is P∗ = x ∈ Rn+ : x · λ0 ≤ 1, for λ0 =

(1

µ1, . . . , 1

µn

).

Remark 2. The formal orders of P and P∗ coincide by definition. The complementary polyhedron P∗

has only one face (besides the faces on the coordinate hyperplanes); P is included in P∗ and they coincide

only when P has only one face (the anisotropic and standard cases).

Proposition 2.3. Let P be a complete polyhedron and P∗ its complementary polyhedron as in Definition

2.3, then the associated weight functions satisfy for some constant C > 0:

|ξ + η|P ≤ C(|ξ|P + |η|P∗), ∀ξ, η ∈ Rn.

Moreover, for all polyhedra P ′ such that µ(P ′) = µ(P) and P∗ 6⊆ P ′, there exist two sequences ξkk∈N,

ηkk∈N in Rn such that

limk→∞

|ξk + ηk|P|ξk|P + |ηk|P′

= ∞.

It therefore follows that |ξ|P satisfies the ring condition (cf. [3], [22]): |ξ + η|P ≤ C(|ξ|P + |η|P) for some

C > 0 and all ξ, η ∈ Rn, if and only if P has only one face (the anisotropic and standard cases).

Remark 3. For every complete polyhedron P, P∗ is always included in the polyhedron of vertices

0, µej , j = 1, . . . , n defining the standard Gevrey classes, and therefore, |ξ + η|P ≤ C(|ξ|P + 〈η〉),∀ξ, η ∈ Rn.

Remark 4. When P has more than one face, the inclusion P ⊂ P∗ is strict; therefore, from Proposition

2.3, there are two sequences ξkk∈N, ηkk∈N in Rn such that

limk→∞

|ξk + ηk|P|ξk|P + |ηk|P

= ∞.

As an example, we can consider the polyhedron P ⊂ R2 with vertices V(P) = (0, 0), (0, 3), (2, 2), (3, 0),for which |(ξ1, ξ2)|P = (1 + |ξ1|3 + ξ2

1ξ22 + |ξ2|3)

16 , and take the sequences ξk = (k, 0), ηk = (0, k), k ∈ N.

7


3 Multi-anisotropic Gevrey classes

We now introduce the multi-anisotropic Gevrey classes associated to a complete polyhedron P. Firstly

we define these spaces by suitable estimates on the growth of the derivatives, that generalize the con-

dition (1) of the standard Gevrey classes. We then give some equivalent definitions; in particular the

characterization by means of their Fourier transform is very useful for the applications to partial differ-

ential equations (cf. f.i. [7, 8]); it also shows that the multi-anisotropic Gevrey classes can be seen as

a particular case of the inhomogeneous Gevrey classes Gs,λ(Ω) (cf. [10]), for λ(ξ) = |ξ|P . We finally

present their topological and algebraic properties.

Definition 3.1. Let P be a complete polyhedron in Rn, Ω an open set in Rn and s ∈ R, s ≥ 1. We

denote by Gs,P(Ω) the multi-anisotropic Gevrey class of order s associated to P, that is the set of all

f ∈ C∞(Ω) such that for any compact subset K of Ω there is a constant C > 0 such that

supx∈K

|Dαf(x)| ≤ C |α|+1(µk(α,P))sµk(α,P), ∀α ∈ Nn. (4)

For s > 1, the multi-anisotropic Gevrey classes with compact support are Gs,P0 (Ω) = Gs,P(Ω) ∩ C∞

0 (Ω).

Remark 5. If P is the Newton polyhedron of an elliptic operator (cf. Example 1 of Section 2), then

Gs,P(Ω) coincides with Gs(Ω), the set of the standard Gevrey functions in Ω. If P is the Newton polyhe-

dron of a quasi-elliptic operator (cf. Example 2 of Section 2), then Gs,P(Ω) = Gs,q(Ω) is the set of the

anisotropic Gevrey functions, cf. [14], [28], [36].

Remark 6. We have the following inclusions:

Gs(Ω) ⊆ Gs µ

µ(1) (Ω) ⊆ Gs,P(Ω) ⊆ Gs µ

µ(0) (Ω), (5)

that show that the multi-anisotropic Gevrey classes include the standard Gevrey classes of the same order.

For the characterization via Fourier transform of the multi-anisotropic Gevrey classes and, in the next

section, of the multi-anisotropic ultradistributions (cf. Theorem 4.1), it is useful to introduce some spaces

of multi-anisotropic Gevrey classes in which the Fourier transform is an automorphism. At this aim we

define the space Ss,P(Rn), in analogy with [3], [29], [30].

Definition 3.2. We say that a function f ∈ L1(Rn) belongs to Ss,P(Rn) if f, f ∈ C∞(Rn) and there is

a constant ε > 0 such that for all α ∈ Nn it holds:

pα,ε(f) = supx∈Rn

exp(ε|x|1s

P)|Dαf(x)| < ∞

πα,ε(f) = supξ∈Rn

exp(ε|ξ|1s

P)|Dαf(ξ)| < ∞.(6)

We easily see that Gs,P0 (Rn) ⊂ Ss,P(Rn) ⊂ Gs,P(Rn) and Ss,P(Rn) is included in the Schwartz space

S(Rn). In analogy with [3], from (6) we can see that the Fourier transform is an automorphism in

Ss,P(Rn).

¿From now on P is a complete polyhedron and the Gevrey order s is strictly bigger than 1.

8

CALVO-MORANDO146

Theorem 3.1. Let f ∈ E ′(Rn), then f belongs to Gs,P0 (Rn) if and only if there are two positive constants

C, ε such that |f(ξ)| ≤ C exp(−ε|ξ|1s

P), ∀ξ ∈ Rn.

Remark 7. Theorem 3.1 can be reformulated in the setting Ss,P(Rn) as follows.

Let f ∈ S ′(Rn), then f belongs to Ss,P(Rn) if and only if there are two positive constants C, ε such that

|f(ξ)| ≤ C exp(−ε|ξ|1s

P), ∀ξ ∈ Rn.

Then the following version of Paley-Wiener-Schwartz Theorem can be established for multi-anisotropic

Gevrey classes with compact support.

Theorem 3.2. Let K be a convex compact subset of Rn. If f is a function in Gs,P0 (Rn) with support

contained in K then there exist two positive constants C, ε such that the Fourier-Laplace transform F of

f satisfies:

|F (ζ)| ≤ C|K| exp(HK(=ζ)− ε|ζ|1s

P), ∀ζ ∈ Cn, (7)

where |K| is the Lebesgue measure of K, HK(t) := supx∈K t · x, ∀t ∈ Rn is the so-called support function

of K and the weight function | · |P is defined by naturally extending formula (3) to Cn. Conversely, every

entire analytic function F in Cn satisfying (7) is the Fourier-Laplace transform of a function in Gs,P0 (Rn)

with support contained in K.

For the proof of Theorem 3.1 we refer to Calvo [7], Theorem 3.2 can be proved analogously.

A natural topological structure is defined in the multi-anisotropic Gevrey spaces analogously to the

standard Gevrey and ultradifferentiable functions (cf. f.i. [20], [28], [30]).

Let K be a compact subset of Rn. We denote by C∞(K) the space of all the C∞-Whitney jets on K

(cf. [34]). We also write C∞0 (K) for the space of all the functions of class C∞ in Rn with support

(possibly empty) contained in K; and, in analogy with [20], we proceed to construct the topology of the

multi-anisotropic Gevrey classes.

Definition 3.3. For any compact subset K of Ω and any positive constant C, the space Gs,P(K, C) is

the set of all f ∈ C∞(K) such that:

‖f‖Gs,P(K,C) := supα∈Nn

C−|α|(µk(α,P))−sµk(α,P) supx∈K

|Dαf(x)| < ∞. (8)

For s > 1, we also set Gs,P0 (K, C) := Gs,P(K, C) ∩ C∞

0 (K).

The spaces Gs,P(K, C) and Gs,P0 (K, C) are Banach spaces with respect to the norm (8).

Let us begin to describe the topology of Gs,P0 (Ω). Since Gs,P

0 (Ω) =⋃

K⊂⊂Ω,C>0 Gs,P0 (K, C), then Gs,P

0 (Ω)

is endowed with the inductive limit topology of the Banach spaces Gs,P0 (K, C), as K ranges over the family

of all compact subsets of Ω and C over R+; therefore we write

Gs,P0 (Ω) = indlimK⊂⊂Ω,C>0G

s,P0 (K, C). (9)

For a detailed study of the inductive limit topology in an abstract topological setting, we refer to [19], [32].

In a similar way, for any compact subset K of Rn, we endow the space Gs,P0 (K) := Gs,P(Rn)∩C∞

0 (K) with

9


the inductive limit topology of the Banach spaces Gs,P0 (K, C) and write Gs,P

0 (K) = indlimC>0Gs,P0 (K, C).

Analogously to [20], we have the following result.

Proposition 3.1. For any 0 < C1 < C2 and any compact subset K of Ω, the inclusion mappings

Gs,P(K, C1) → Gs,P(K, C2), Gs,P0 (K, C1) → Gs,P

0 (K, C2)

are compact operators.

It turns out that for any compact subset K of Rn, Gs,P0 (K) and Gs,P

0 (Ω) are (DFS)-spaces according

to [19, 20]; in particular they are separable complete bornologic Montel and Schwartz spaces. By the

characterization of the Gevrey multi-anisotropic functions via Fourier transform (cf. Theorem 3.1), we

can construct an equivalent topology on Gs,P0 (Ω), that can be seen as a particular case of the topology

of the inhomogeneous Gevrey classes (cf. [10]).

Definition 3.4. Let K be a compact subset of Rn and ε a positive constant. We define the space

Gs,|·|P0 (K, ε) := f ∈ E ′(Rn) : supp f ⊆ K, ‖f‖

Gs,|·|P0 (K,ε)

:= supξ∈Rn

exp(ε|ξ|1s

P)|f(ξ)| < ∞.

We observe that we can also take f in E ′(Ω), considering its null extension in Rn as a distribution in

E ′(Rn). Gs,|·|P0 (K, ε) is a Banach space with norm ‖ · ‖

Gs,|.|P0 (K,ε)

. The following result holds (cf. [10]).

Lemma 3.1. Let K1 ⊆ K2 be two compact subsets of Ω and 0 < ε2 < ε1. Then the inclusion map

Gs,|·|P0 (K1, ε1) → G

s,|·|P0 (K2, ε2) is a compact operator.

¿From Theorem 3.1 and Definition 3.4, it follows that the space Gs,P0 (Ω) can be described as Gs,P

0 (Ω) =⋃K⊂⊂Ω,ε>0 G

s,|·|P0 (K, ε). Then Gs,P

0 (Ω) is endowed with the inductive limit topology of the Banach spaces

Gs,|·|P0 (K, ε), as K ranges over the family of the compact subsets of Ω and ε on R+. Analogously to (9),

we can write

Gs,P0 (Ω) = indlimK⊂⊂Ω,ε>0G

s,|·|P0 (K, ε). (10)

For any compact subset K of Rn we write also Gs,|·|P0 (K) := indlimε>0G

s,|·|P0 (K, ε).

Proposition 3.2. Let τ0 be the inductive limit topology on Gs,P0 (Ω) defined by (9) and τ1 the topology

defined in Gs,P0 (Ω) by (10), then τ0 and τ1 are equivalent.

Proof. In view of the Open Mapping Theorem for (LF)-spaces (cf. [26], Theorem 8.4.11), it suffices to

prove that the identity map id : (Gs,P0 (Ω), τ0) → (Gs,P

0 (Ω), τ1) is continuous. Therefore, we have to

prove that for any compact subset K of Ω and any constant C1 > 0 the restriction of the operator id to

Gs,P0 (K, C1) → (Gs,P

0 (Ω), τ1) is continuous. If f belongs to Gs,P0 (K, C1) then, analogously to the proof

of Theorem 3.1 (cf. [7]), its Fourier transform satisfies

|f(ξ)|1

µs ≤ ‖f‖1

µs

Gs,P(K,C1)C

1µs

2 (n(P))1

µs |K|1

µs(2n(P)C

1µ )

Ns eNN !

(|ξ|1s

P)N, ∀ξ ∈ Rn,

10

CALVO-MORANDO148

for N = 0, 1, 2, . . . , where n(P) is the number of vertices of P, C2 is a suitable positive constant in-

dependent of f and C = max(µµsCµ1 , C2). Summing over N = 0, 1, 2, . . . , we obtain for a suitable

ε > 0

|f(ξ)| ≤ C2‖f‖Gs,P(K,C1)n(P)|K| exp(−ε|ξ|1s

P), ∀ξ ∈ Rn.

Therefore, for any C1 > 0 there is an ε > 0 such that ‖f‖G

s,|·|P0 (K,ε)

≤ C2n(P)|K|‖f‖Gs,P(K,C1) for all

f ∈ Gs,P0 (K, C1); thus for a suitable positive ε the inclusion Gs,P

0 (K, C1) → Gs,|·|P0 (K, ε) is a continuous

embedding of Banach spaces. Since the inclusion map Gs,|·|P0 (K, ε) → (Gs,P

0 (Ω), τ1) is continuous, by the

definition of the inductive topology τ1, the result is proved.

We treat now the topology of the multi-anisotropic Gevrey functions with arbitrary support. First we

define for any compact subset K of Ω: Gs,P(K) :=⋃

C>0 Gs,P(K, C) and we endow Gs,P(K) with the

inductive limit topology of Banach spaces Gs,P(K, C), writing Gs,P(K) = indlimC>0Gs,P(K, C). ¿From

Proposition 3.1, it turns out that Gs,P(K) are (DFS)-spaces; in particular they are reflexive and complete

Hausdorff, Montel and Schwartz spaces. Let Kjj∈N be an exhaustive sequence of compact subsets of

Ω, i.e. for every j = 1, 2, . . . Kj−1 ⊂

Kj (with K0 := ∅) and⋃

j∈N Kj = Ω. For any pair of indices l, j

with l > j we denote by ρlj the restriction operator ρl

j : Gs,P(Kl) → Gs,P(Kj) defined by ρlj(f) = f|Kj

for all f ∈ Gs,P0 (Kl). Of course, the mappings ρl

j are linear and continuous and, for any j = 1, 2, . . . ,

they satisfy ρj+1j ρj+2

j+1 = ρj+2j . Therefore, following [32] (cf. also [19]), the projective limit of the sequence

Gs,P0 (Kj)j∈N with respect to the mappings ρl

jl>j is defined as the space of all sequences fjj∈N of

C∞-jets fj ∈ Gs,P(Kj) such that

ρj+1j (fj+1) = fj , j = 1, 2, . . . . (11)

This projective space is isomorphic to Gs,P(Ω), if we identify any sequence fjj∈N fulfilling (11) with

the function f in Gs,P(Ω) defined by setting f(x) = fj(x) for x ∈ Kj , j = 1, 2, . . . . Then Gs,P(Ω)

is endowed with the projective topology of the sequence Gs,P(Kj)j∈N with respect to the mappings

ρljl>j (cf. [15], [19]); this procedure does not depend on the sequence Kjj∈N and makes Gs,P(Ω) a

complete Schwartz space (cf. [15], [19]). Therefore we will write

Gs,P(Ω) = projlimK⊂⊂ΩGs,P(K) = projlimK⊂⊂Ω(indlimC→∞Gs,P(K, C)).

Concerning the topology in the spaces Ss,P , we just observe that the semi-norms pα,ε, πα,ε in (6) endow

Ss,P with a locally convex topology.

Remark 8. The inclusions (5) of Gevrey classes are continuous with respect to the previously defined

topology.

We now consider the algebraic properties of multi-anisotropic Gevrey classes. It is clear that if u, v ∈Gs,P(Ω), k ∈ C, then u + v ∈ Gs,P(Ω), ku ∈ Gs,P(Ω).

11


Proposition 3.3. If α ∈ Nn, then for any 0 < C < C ′ and any compact subset K of Ω, the derivative

Dα : Gs,P(K, C) → Gs,P(K, C ′) is continuous and it satisfies ‖Dαf‖Gs,P(K,C′) ≤ M‖f‖Gs,P(K,C) for

every f ∈ Gs,P(K, C) and some constant M > 0 depending only on α, C and C ′. Therefore, the

derivative Dα defines a continuous linear operator from each of the spaces Gs,P(Ω) and Gs,P0 (Ω) into

itself.

The proof is immediate by direct computation, or after observing that the sequence (µk(α,P))µsk(α,P),

α ∈ Nn, fulfills the differentiation property stated by Roumieu (cf. [30], inequality (7)). The problem of

the pointwise multiplication for multi-anisotropic Gevrey functions is solved by the following result.

Proposition 3.4. Let Ω be an open subset of Rn, P be a complete polyhedron and P∗ its complementary

polyhedron given by Definition 2.3; then for any s > 1 we have: Gs,P(Ω) · Gs,P∗(Ω) ⊆ Gs,P(Ω) and for

any complete polyhedron P ′ such that µ(P ′) = µ(P) and P∗ 6⊆ P ′: Gs,P(Ω) ·Gs,P′(Ω) 6⊆ Gs,P(Ω).

For the proof and counterexamples we refer to Calvo [8] and Hakobyan-Markaryan [16].

Remark 9. In view of the inclusion P ⊆ P∗, we have Gs,P∗(Ω) ⊆ Gs,P(Ω) and they coincide only in

the standard or the anisotropic case (cf. [8], Proposition 2.1). Moreover, from Proposition 3.4, Gs,P(Ω)

is not an algebra under the pointwise multiplication, except the standard and anisotropic cases.

Remark 10. As the inclusion Gs(Ω) ⊆ Gs,P∗(Ω) holds for any complete polyhedron P, then Gs(Ω) ·

Gs,P(Ω) ⊆ Gs,P(Ω).

¿From the topological point of view, the following results can be proved.

Proposition 3.5. Let P be a complete polyhedron, P∗ its complementary polyhedron, C1, C2 > 0 and K a

compact subset of Ω. Then ‖fg‖Gs,P(K,C1+C2) ≤ ‖f‖Gs,P(K,C1)‖g‖Gs,P∗ (K,C2), for every f ∈ Gs,P(K, C1)

and g ∈ Gs,P∗(K, C2).

Proposition 3.6. Let P be a complete polyhedron in Rn and P∗ its complementary polyhedron. Then,

for any open subset Ω of Rn, any compact subset K of Ω and s > 1:

1. Gs,P(Ω) is a topological Gs,P∗(Ω) -module;

2. Gs,P0 (Ω) is a Gs,P∗

(Ω) -module in which the pointwise multiplication is hypocontinuous.

By the properties of the multiplication of multi-anisotropic Gevrey functions (cf. Remark 10, Propo-

sition 3.5), we can give an equivalent topological description of the space Gs,P(Ω) in analogy with the

inhomogeneous Gevrey case.

Corollary 3.1. Let Ω be an open subset of Rn and Kjj∈N an exhaustive sequence of compact subsets of

Ω; chosen any χj ∈ Gs0(Kj) such that 0 ≤ χj ≤ 1 and χj |Kj−1 ≡ 1 then Gs,P(Ω) = projlimj→∞G

s,|·|P0 (Kj),

where the projective limit in the right-hand side is taken with respect to the mappings ϕj : Gs,|·|P0 (Kj+1) →

Gs,|·|P0 (Kj), f 7→ χjf .

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CALVO-MORANDO150

Corollary 3.2. If Ω is an open subset of Rn and K1,K2 are compact subsets of Ω such that K1 ⊆ K2,

then the inclusion maps: Gs,P0 (K1) → Gs,P(K2), Gs,P

0 (K1) → Gs,P(Ω) are topological homomorphisms.

The proof is analogous to that in [4], following from Propositions 3.5 and 3.6.

Combining Propositions 3.3, 3.4, 3.6, we get the following result.

Proposition 3.7. Let P (x,D) =∑

|α|≤m aα(x)Dα be a linear partial differential operator with coeffi-

cients aα ∈ Gs,P∗(Ω). Then the following are linear continuous operators:

P (x,D) : Gs,P(Ω) → Gs,P(Ω), P (x,D) : Gs,P0 (Ω) → Gs,P

0 (Ω).

Concerning the convolution product of multi-anisotropic Gevrey functions, the following result can be

established.

Proposition 3.8. Let f ∈ Gs,P(Rn) and g ∈ L1comp(Rn). Then the convolution product f ∗ g belongs

to Gs,P(Rn); more precisely for any compact subset K of Rn there is a constant C > 0 such that

‖f ∗ g‖Gs,P(K,C) ≤ ‖f‖Gs,P(K−H,C)‖g‖L1(H), where H = supp g. Analogously, if f ∈ Gs,P0 (Rn) and

g ∈ L1loc(Rn) then f ∗ g ∈ Gs,P(Rn); in particular for any compact subset K of Rn there exists a constant

C > 0 such that ‖f ∗ g‖Gs,P(K,C) ≤ ‖f‖Gs,P(H,C)‖g‖L1(K−H), where H = supp f .

Corollary 3.3. Let P be a complete polyhedron in Rn and set S(f, g) := f ∗ g; then the following are

hypocontinuous bilinear maps:

S : Gs,P(Rn)× L1comp(Rn) → Gs,P(Rn), S : Gs,P

0 (Rn)× L1loc(Rn) → Gs,P(Rn).

Remark 11. In view of Proposition 3.8, analogously to the standard Gevrey case we can prove that

Gs,P0 (Ω) is dense in Gs,P(Ω), Gs,P

0 (Ω) is dense in C∞0 (Ω) and Gs,P(Ω) is dense in C∞(Ω) with continuous

inclusions. The proof is based on the convolution with functions ϕε ∈ Gs(Ω) ⊆ Gs,P(Ω), ϕε → δ (cf. [4],

Lemma 3.8 and [30], Corollary 3).

4 Multi-anisotropic ultradistributions

Now we present the topological dual spaces of the multi-anisotropic Gevrey classes: the multi-anisotropic

ultradistributions. They admit equivalent characterizatons according to the topology defined in the

multi-anisotropic Gevrey spaces; in particular we study the properties of the Fourier transform and prove

a version of the Paley-Wiener-Schwartz Theorem for multi-anisotropic ultradistributions with compact

support (cf. Theorems 4.1, 4.2). We then examine the algebraic and topological properties and give an

application concerning the well posedness of the Cauchy problem for a certain class of operators with

constant coefficients.

Definition 4.1. For any s > 1 and any open set Ω of Rn, the space of multi-anisotropic ultradistributions

D′s,P(Ω) is the topological dual space of the multi-anisotropic Gevrey class Gs,P

0 (Ω), endowed with the

strong dual topology.

13


Proposition 4.1. The following conditions are equivalent:

1. u belongs to D′s,P(Ω);

2. u is a linear form on Gs,P0 (Ω) such that u(fj) → 0 for any sequence fjj∈N ⊂ Gs,P

0 (Ω) converging

to 0 in Gs,P0 (Ω);

3. for any compact subset K of Ω and any ε > 0, there is a constant Cε > 0 such that for all f ∈Gs,P

0 (Ω), with supp f ⊆ K, it holds |u(f)| ≤ Cε supα∈Nn ε|α|(µk(α,P))−sµk(α,P) supx∈K |Dαf(x)|.

The proof follows from standard topological arguments (cf. [19], [32], [33]).

Remark 12. For any complete polyhedron P in Rn, any open subset Ω of Rn and t > s > 1, the

inclusions D′(Ω) ⊂ D′t,P(Ω) ⊂ D′

s,P(Ω) hold continuously (as a consequence of the continuous inclusions

Gs,P0 (Ω) ⊂ Gt,P

0 (Ω) ⊂ C∞0 (Ω), cf. Remark 11). ¿From Remark 6, we derive also the following inclusions

of multi-anisotropic and standard ultradistributions: D′s µ

µ(0)(Ω) ⊆ D′

s,P(Ω) ⊆ D′s µ

µ(1)(Ω) ⊆ D′

s(Ω).

Let us mention some particular cases of multi-anisotropic ultradistributions.

1. If P is the Newton polyhedron of an elliptic operator, then D′s,P(Ω) coincides with D′

s(Ω), the space

of the ultradistributions associated to the standard Gevrey class Gs0(Ω);

2. If P is the Newton polyhedron of a quasi-elliptic operator, then D′s,P(Ω) = D′

s,q(Ω) is the set of the

anisotropic ultradistributions associated to the anisotropic Gevrey class Gs,q0 (Ω). From Proposition

4.1, D′s,q(Ω) is the set of the linear forms on Gs,q

0 (Ω) such that for any compact subset K of Ω

and any ε > 0 there is a constant Cε > 0 for which the inequality |u(f)| ≤ Cε supα∈Nn ε|α|(α ·q)−sα·q supx∈K |Dαf(x)| holds for all f ∈ Gs,q

0 (Ω) with supp f ⊆ K.

Definition 4.2. For every s > 1, we denote by E ′s,P(Ω) the topological dual space of the multi-anisotropic

Gevrey class Gs,P(Ω), endowed with the strong dual topology.

As in the space of Schwartz distributions, we can define the support of a multi-anisotropic ultradistribution

u ∈ D′s,P(Ω) as the intersection of all closed subsets of Ω in whose complement u vanishes; then, in analogy

with Proposition 4.1, we have the following result.

Proposition 4.2. The following conditions are equivalent:

1. u belongs to E ′s,P(Ω);

2. u is a linear form on Gs,P(Ω) such that u(fj) → 0 for any sequence fjj∈N ⊂ Gs,P(Ω) converging

to 0 in Gs,P(Ω);

3. u is a linear form on Gs,P(Ω) and there is a compact subset K of Ω such that for any ε > 0 there

is a constant Cε > 0 such that for all f ∈ Gs,P(Ω) it holds:

|u(f)| ≤ Cε supα∈N

ε|α|(µk(α,P))−sµk(α,P) supx∈K

|Dαf(x)|. (12)

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CALVO-MORANDO152

4. u belongs to D′s,P(Ω) and has compact support in Ω.

We can equivalently define the multi-anisotropic ultradistributions with compact support by means of

the Fourier transform and the Fourier-Laplace transform, in analogy with the standard ultradistributions

(cf. [20], [28]). It follows that the multi-anisotropic ultradistributions can be seen as a particular case of

the inhomogeneous ultradistributions treated in [10].

If u belongs to E ′s,P(Rn), then its Fourier transform can be defined by setting u(ξ) = ux(e−ix·ξ), ∀ξ ∈ Rn,

that is well defined as e−ix·ξ belongs to Gs(Rn) ⊂ Gs,P(Rn). The rules of derivation and multiplication

of the Fourier transform in E ′(Rn) are still satisfied in E ′s,P(Rn).

Theorem 4.1. If u belongs to E ′s,P(Rn), then for any ε > 0 there is a constant Cε > 0 such that

|u(ξ)| ≤ Cε exp(ε|ξ|1s

P), ∀ξ ∈ Rn. (13)

Proof. If u belongs to E ′s,P(Rn), then u(ξ) = ux(e−ix·ξ) is a function, since u has compact support. From

formula (12), taking f(x) = e−ix·ξ we get:

|u(ξ)| = |ux(e−ix·ξ)| ≤ Cε supα∈Nn

ε|α|(µk(α,P))−sµk(α,P) supx∈H

|Dαx e−ix·ξ|

≤ Cε supα∈Nn

ε|α|(µk(α,P))−sµk(α,P)|ξ|µk(α,P)P .

Applying the inequality td ≤ etdd for t = ε1s |ξ|

1s

P and d = µk(α,P) with an arbitrary ε1 > 0, we can

estimate

|u(ξ)| ≤ Cε supα∈Nn

ε|α|(µk(α,P))−sµk(α,P) exp(ε1|ξ|

1s

P

)(µk(α,P))sµk(α,P)

(s

ε1

)sµk(α,P)

≤ Cε supα∈Nn

ε|α|(

s

ε1

)s µµ0|α|

exp(ε1|ξ|

1s

P

).

Now taking ε =(

sε1

)−s µ

µ(0)we obtain |u(ξ)| ≤ Cε exp(ε1|ξ|

1s

P), that gives the result.

Definition 4.3. The space of multi-anisotropic ultradistribution S ′s,P(Rn) is the topological dual of

Ss,P(Rn).

We have obviously the inclusions E ′s,P(Rn) ⊂ S ′s,P(Rn) ⊂ D′s,P(Rn) and S ′(Rn) ⊂ S ′s,P(Rn), where

S ′(Rn) is the space of tempered distributions, dual space of S(Rn).

In analogy with the spaces S ′(Rn), it is possible to define the Fourier transform also for u ∈ S ′s,P(Rn),

by the use of Parseval’s formula: u(f) = u(f), for all f ∈ Ss,P(Rn) (as also f ∈ Ss,P(Rn)). For u ∈ E ′s,P ,

this definition obviously coincides with the previous one.

Proposition 4.3. The Fourier transform is an automorphism of S ′s,P(Rn).

Corollary 4.1. Any function U(ξ) satisfying for all ε > 0 the condition (13) in Rn belongs also to

S ′s,P(Rn), and therefore is the Fourier transform of an element u of S ′s,P(Rn).

15


It follows from the inclusion S ′s,P(Rn) ⊂ D′s,P(Rn) that if U(ξ) satisfies (13), then it is the Fourier

transform of an element of D′s,P(Rn) (as we will need in the proof of Theorem 4.3).

Finally, we prove a version of Paley-Wiener-Schwartz Theorem for multi-anisotropic ultradistributions.

Theorem 4.2. If u belongs to E ′s,P(Rn) and suppu ⊆ K, where K is a convex compact subset of Rn,

then for any ε > 0 there is a constant Cε > 0 such that the Fourier-Laplace transform U of u satisfies

|U(ζ)| ≤ Cε exp(HK(=ζ) + ε|ζ|

1s

P

), ∀ζ ∈ Cn, (14)

where HK(t) := supx∈K x · t, for t ∈ Rn. If an entire analytic function U(ζ) satisfies (14), then U is the

Fourier-Laplace transform of an ultradistribution u ∈ E ′s,P(Rn) with suppu ⊆ K.

Proof. If u belongs to E ′s,P(Rn) and suppu ⊆ K, then (14) holds in analogy with Theorem 4.1.

Conversely, let U(ζ) satisfy (14). Then, as in the standard Gevrey case (cf. [4]), the linear form:

u(ϕ) := (2π)−n

∫U(−ξ)ϕ(ξ)dξ, ∀ϕ ∈ Gs,P

0 (Rn)

defines a multi-anisotropic ultradistribution u ∈ D′s,P(Rn) with suppu ⊆ K, whose Fourier-Laplace

transform coincides with U . Therefore, in view of 4. of Proposition 4.2, u belongs to E ′s,P(Rn).

Remark 13. The inequality (14) is an estension to the complex space Cn of (13).

We now study the topological properties of the multi-anisotropic ultradistributions D′s,P(Ω) and E ′s,P(Ω).

In view of Definition 4.1 and according to [32], Theorem 4, the strong dual topology makes D′s,P(Ω) a (FS)-

space; in particular it is a complete bornologic Montel and Schwartz space. An explicit fundamental sys-

tem of continuous semi-norms is given by setting for any u ∈ D′s,P(Ω): pBj

m(u) := supf∈Bj

m|u(f)|, j, m =

1, 2, . . . , where, for any pair of positive integers (j, m), Bjm := f ∈ Gs,P

0 (Kj , Cj) : ‖f‖Gs,P(Kj ,Cj) ≤ m,Kjj∈N is an exhaustive sequence of compact subsets of Ω and Cjj∈N is an increasing sequence of

positive numbers diverging to +∞. Following the arguments of Komatsu ([20], Proposition 5.11 and

Theorem 5.12), E ′s,P(Ω) are complete bornologic Montel and Schwartz spaces.

In the space of multi-anisotropic ultradistributions, we can define the usual elementary operations.

Let P be a complete polyhedron and P∗ its complementary polyhedron. For any u ∈ D′s,P(Ω), f ∈

Gs,P∗(Ω) and α ∈ Nn the following operations are well defined:

- the product fu(ϕ) := u(fϕ), ∀ϕ ∈ Gs,P0 (Ω);

- the derivative Dαu(ϕ) := (−1)−|α|u(Dαϕ), ∀ϕ ∈ Gs,P0 (Ω).

¿From Propositions 3.3 and 3.4, the derivative is continuous inD′s,P(Ω) and E ′s,P(Ω) and setting M(f, u) :=

fu the following maps are bilinear and hypocontinuous:

M : Gs,P∗(Ω)×D′

s,P(Ω) → D′s,P(Ω),

M : Gs,P∗(Ω)× E ′s,P(Ω) → E ′s,P(Ω),

M : Gs,P∗

0 (Ω)×D′s,P(Ω) → E ′s,P(Ω).

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CALVO-MORANDO154

Corollary 4.2. Let P be a complete polyhedron and P∗ its complementary polyhedron. Let P (x, D) :=∑|α|≤m aα(x)Dα be a linear partial differential operator with coefficients aα ∈ Gs,P∗

(Ω). Then the

following are continuous linear operators:

P (x,D) : D′s,P(Ω) → D′

s,P(Ω), P (x,D) : E ′s,P(Ω) → E ′s,P(Ω).

If u ∈ D′s,P(Rn) and f ∈ Gs,P

0 (Rn), or u ∈ E ′s,P(Rn) and f ∈ Gs,P(Rn), the convolution product u ∗ f is

defined by (u ∗ f)(x) = uy(f(x− y)), where uy acts on f(x− y) as an ultradistribution in y.

Proposition 4.4. Let f ∈ Gs,P(Rn) and u ∈ E ′s,P(Rn) (or f ∈ Gs,P0 (Rn) and u ∈ D′

s,P(Rn)); then the

convolution product u ∗ f belongs to Gs,P(Rn).

Proof. We prove the proposition for f ∈ Gs,P(Rn) and u ∈ E ′s,P(Rn); the other case is similar and we

omit it. For all α ∈ Nn, Dαx (u ∗ f)(x) = uy(Dα

x f(x − y)) is well defined, since Dαx f(x − y) ∈ Gs,P(Rn

x)

for all y ∈ Rn and u is linear and continuous in Gs,P(Rn); therefore u ∗ f belongs to C∞(Rn); now we

prove that it is also in Gs,P(Rn). From the definitions of Gs,P(Rn) and E ′s,P(Rn) (cf. formulas (4) and

(12), respectively), for any compact subset K of Rn there is a C > 0 such that for any α ∈ Nn it holds:

supx∈K

|Dαx (u ∗ f)(x)| = sup

x∈K|uy(Dα

x f(x− y))|

≤ Cε supβ∈Nn

ε|β|(µk(β,P))−sµk(β,P) supx∈K

supy∈H

|Dαx Dβ

y f(x− y)|

≤ Cε supβ∈Nn

ε|β|(µk(β,P))−sµk(β,P) supz∈H−K

|Dα+βf(z)|

≤ Cε supβ∈Nn

ε|β|(µk(β,P))−sµk(β,P)C |α|+|β|+1(µk(α + β,P))sµk(α+β,P)

≤ Cε supβ∈Nn

ε|β|(µk(β,P))−sµk(β,P)C |α|+|β|+1(µk(α,P) + µk(β,P))s(µk(α,P)+µk(β,P)),

where in the last step we have applied

k(α + β,P) = maxν∈V(P)

(α + β) · ν ≤ maxν∈V(P)

α · ν + maxν∈V(P)

β · ν = k(α,P) + k(β,P).

¿From the elementary inequality td ≤ ddet−d we obtain:

supx∈K

|Dαx (u ∗ f)(x)| ≤ Cε sup

β∈Nn

C |α|+|β|+1ε|β|(µk(α,P))sµk(α,P)esµk(β,P)esµ|α|

≤ CCε(Cesµ)|α|(µk(α,P))sµk(α,P) supβ∈Nn

ε|β|esµk(β,P)C |β|.

So, taking ε = (Cesµ)−1 and C1 = Cesµ, C2 = CCε for the previous choice of ε, we get the estimate

supx∈K

|Dαx (u ∗ f)(x)| ≤ C2C

|α|1 (µk(α,P))sµk(α,P), ∀α ∈ Nn,

that implies that u ∗ f belongs to Gs,P(Rn).

The convolution product satisfies the usual properties (analogously to the case of Schwartz distributions);

in particular, for any f ∈ Gs,P(Rn), u ∈ E ′s,P(Rn) (or f ∈ Gs,P0 (Rn), u ∈ D′

s,P(Rn)) and g ∈ E ′s,P(Rn), it

17


holds (u ∗ f) ∗ g = u ∗ (f ∗ g) and supp(u ∗ f) ⊆ suppu + supp f . Therefore, we can define the product

by convolution of two ultradistributions u, v ∈ D′s,P(Rn), one of which has compact support, by setting

(u ∗ v) ∗ g = u ∗ (v ∗ g) for all g ∈ Gs,P0 (Rn); this is well defined in view of Proposition 4.4 and u ∗ v

belongs to D′s,P(Rn).

To conclude our treatment, we show some applications of the multi-anisotropic ultradistributions to the

study of partial differential equations.

We first point out that the multi-anisotropic ultradistributions are the suitable setting for the problems

(for instance hypoelliptcity, local solvability and iterates of operators) in which existence or regularity

results are proved in the frame of multi-anisotropic Gevrey classes, cf. for instance [1], [5], [6], [9], [12],

[13], [14], [16], [18], [27], [35], [36]. In fact, all these results can be reformulated by replacing the class of

Schwartz distributions with the corresponding multi-anisotropic ultadistributions.

As further application, we study the Cauchy problem in multi-anisotropic ultradistributions for weakly

hyperbolic operators. The well-posedness will be proved for the so-called multi-quasi-hyperbolic operators

introduced by Calvo [7] and generalizing the s-hyperbolic operators of Larsson [21]; they are shaped in

such a way that the well-posedness of the Cauchy problem holds for multi-anisotropic Gevrey classes.

Let us recall the definition; for more precise characterizations and properties we refer to [7].

Definition 4.4. We say that a differential operator with constant coefficients in Rt × Rnx

P (D) = P (Dt, Dx) = Dmt +

∑|ν|+j≤m,j 6=m

aνjDνxDj

t (15)

is multi-quasi-hyperbolic with respect to P of order s > 1 (for short (s,P)-hyperbolic) if there exists a

constant C > 0 such that for any (τ, ξ) ∈ C× Rn the symbol of P (D) satisfies the condition:

P (τ, ξ) = τm +∑

|ν|+j≤m,j 6=m

aνjξντ j = 0 =⇒ |=τ | ≤ C|ξ|

1s

P .

Obviously, if P (D) is (s,P)-hyperbolic then it is also weakly hyperbolic, i.e. all the roots τ of the

characteristic equation Pm(τ, ξ) = τm +∑

|ν|+j=m,j 6=m ξντ j = 0 are real. In the opposite direction, if

Pm(τ, ξ) satisfies the weakly hyperbolic assumption, in order to have the (s,P)-hyperbolicity we need to

ask some Levi-type conditions on the lower order terms, modeled on P, cf. [7]. We just point out the

following result, in order to give an idea of the multi-quasi-hyperbolicity conditions.

Proposition 4.5. Let P (D) in (15) be weakly hyperbolic and suppose that the multiplicity of the roots

of its principal symbol Pm(τ, ξ) is equal to M ≤ m and the lower order terms satisfy for some k < M

and a constant C > 0:

|aνjξν | ≤ C|ξ|kP〈ξ〉m−M−j for |ν|+ j ≤ m− 1. (16)

Then P (D) is (Mk ,P)-hyperbolic.

In [7] it was proved that, under the hypothesis of (s,P)-hyperbolicity, the Cauchy problem (17) admits

a unique solution u ∈ C∞(R, Gs,P(Rn)) for any data uk ∈ Gr,P(Rn) if r < s.

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CALVO-MORANDO156

Theorem 4.3. Let P (D) be an (s,P)-hyperbolic differential operator in Rt × Rnx. Let 1 < r < s and

uk ∈ D′r,P(Rn

x) (k = 0, 1, . . . ,m− 1). Then the Cauchy problemP (D)u = Dmt u +

∑|ν|+j≤m, j 6=m aνjD

νxDj

t u = 0

Dkt u(0, x) = uk(x), k = 0, 1, . . . ,m− 1

(17)

admits a unique solution u ∈ C∞(R,D′r,P(Rn

x)).

Proof. For the finite speed of propagation of weakly hyperbolic operators with constant coefficients, it is

not restrictive to take the data with compact support uk ∈ E ′s,P(Rn), k = 0, . . . ,m− 1. Therefore, after

performing the partial Fourier transform (with respect to the x variable and taking t as parameter) of

the Cauchy problem (17), the unique solution of the equivalent Cauchy problem (an ordinary Cauchy

problem in t, with ξ as parameter) is given by:

u(t, ξ) =m−1∑j=0

uj(ξ)Fj(t, ξ), (18)

where uj (j = 0, . . . ,m−1) are the Fourier transforms of the data of (17) and Fj are the unique solutions

of the Cauchy problems P (Dt, ξ)Fj = 0

Dkt Fj(0, ξ) = δjk, k = 0, . . . ,m− 1,

for δjk = 1 if j = k and 0 otherwise. Let us fix now an arbitrary T > 0. The Fj are estimated by

Lemma 12.7.7 of Hormander [17] thanks to the hypothesis of weak hyperbolicity of P (D), and therefore

for suitable constants c1, C1, c2, C2, C′ > 0 and all ξ ∈ Rn, t ∈ [−T, T ] we have:

|Fj(t, ξ)| ≤ (C1〈ξ〉)m+1c1 exp(C ′|t||ξ|1s

P) ≤ c2 exp(C2(1 + |t|)|ξ|1s

P).

Then we can estimate u(t, ξ) given by (18) as follows: for all ε > 0 there is a constant Cε > 0 such that

|u(t, ξ)| ≤m−1∑j=0

c2Cε exp(ε|ξ|1r

P) exp(C2(1 + |t|)|ξ|1s

P).

As r < s, for t ∈ [−T, T ] and for all ε′ > 0, taking ε sufficiently small (depending on ε′, r, s, T ), there is

a constant Cε′ > 0 such that it holds |u(t, ξ)| ≤ Cε′ exp(ε′|ξ|1r

P), ∀ξ ∈ Rn; this proves that for every fixed

t ∈ [−T, T ] we have u ∈ D′r,P(Rn). Similar estimates for ∂j

t u(t, ξ) (cf. [17], Lemma 12.7.7) show that

∂jt u ∈ D′

r,P(Rn) for all t ∈ [−T, T ]; since T is arbitrarily fixed, we conclude that u ∈ C∞(R,D′r,P(Rn).

We finally illustrate the notion of multi-quasi-hyperbolicity and the previous result by some examples.

1. Let us consider the case when P is the Newton polyhedron of an elliptic operator, i.e. |ξ|P = 〈ξ〉.From Proposition 4.5, any differential operator P (D) = Pm(D) + Q(D) in Rn+1, such that its

principal part Pm(D) is hyperbolic and Q(D) has order q < m, is mq -hyperbolic; therefore, for any

r < mq and any data uk ∈ D′

r(Rn), k = 0, . . . ,m− 1, there is a unique solution u ∈ C∞(R,D′r(Rn))

19


of the corresponding Cauchy problem.

If Pm(D) is hyperbolic, with multiplicity of the characteristics equal to M and q = m−M + k for

a given k, 0 < k < M , then P (D) is Mk -hyperbolic. When k = M − 1, (16) is always satisfied and

we obtain the well-known result of Gs well-posedness for s < MM−1 (cf. for instance [21]).

2. Let P be the complete polyhedron in R2 with vertices V(P) = (0, 0), (0, 2), (1, 0); then the

associated weight function is |(ξ, η)|P =(1 + |ξ|+ |η2|

) 12 . Consider the operator of order 3:

P (Dx, Dy, Dt) = P3(Dx, Dy, Dt) + P2(Dx, Dy, Dt) + P1(Dx, Dy, Dt) + c, (19)

such that the principal part P3(Dx, Dy, Dt) is hyperbolic, P2(Dx, Dy, Dt) = c1D2y and P1(Dx, Dy, Dt)

is any operator of order 1, with c1, c ∈ C. It is multi-quasi-hyperbolic of order 32 with respect to P

(cf. Proposition 4.5) and therefore, from Theorem 4.3, for any r < 32 and any data uk ∈ D′

r,P(Rn),

k = 0, 1, 2, there is a unique solution u ∈ C∞(R,D′r,P(Rn)) of the corresponding Cauchy problem.

3. Let P be the same polyhedron as in Example 2; then if we ask the conditions of Proposition

4.5 with M = 2, k = 1 to the operator (19), we have that P3(Dx, Dy, Dt) is hyperbolic with

multiplicity of the characteristics equal to 2 and the lower order terms must be of the following

kind: P2(Dx, Dy, Dt) = c1D2y + c2DxDy + c3DyDt and P1(Dx, Dy, Dt) is any operator of order

1, with c1, c2, c3 ∈ C. This implies that P (D) is multi-quasi-hyperbolic of order 2 with respect

to P and for any r < 2 and any data uk ∈ D′r,P(Rn), k = 0, 1, 2, there is a unique solution

u ∈ C∞(R,D′r,P(Rn)) of the corresponding Cauchy problem. Observe that the lower order terms

here are more general than in Example 2.

20

CALVO-MORANDO158

5 Acknowledgements

The authors are grateful to Prof. O. Liess, University of Bologna, and Prof. L. Rodino, University of

Torino, for useful ideas and valuable remarks during the writing of this paper.

21


References

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Un. Mat. It. 24, Pitagora, Bologna, 1983.

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Group, Dordrecht, 1992.

[15] A. Grothendieck, Sur les espaces (F) et (DF), Summa Brasil. Math. 3, 57-123 (1954).

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CALVO-MORANDO160

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Nats. Akad. Nauk Armenii Mat. 31 (2), 40-56 (1998); translation in J. Contemp. Math. Anal. 31

(2), 33-47 (1996).

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Berlin, 1983.

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(2001-2002).

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131, Amsterdam, 1987.

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23


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Math. Soc. 36, 63-89 (1934).

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Un. Mat. Ital. Suppl. 1, 177-195 (1980).

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Univ. Padova 67, 85-104 (1982).

24

CALVO-MORANDO162

Periodic Solutions for ppp-Laplacian Duffing Equations

with a Deviating Argument

Wing-Sum Cheung

Department of Mathematics, The University of Hong Kong

Pokfulam, Hong Kong

[email protected]

and

Jingli Ren

Department of Mathematics, Zhengzhou University

Zhengzhou, Henan 450052, P.R. China

ren [email protected]

Suggested Running Head: Periodic Solutions for p-Laplacian Duffing Eqautions

Correspondence Author:

Wing-Sum Cheung

Department of Mathematics

The University of Hong Kong

Pokfulam, Hong Kong

Tel: (852) 2859-1996

Fax: (852) 2559-2225

Email: [email protected]

1


Abstract

In this paper, by means of Mawhin’s continuation theorem, the existence of periodic solutions

for a p-Laplacian Duffing equation with deviating argument is obtained.

Keywords: periodic solution, Mawhin’s continuation theorem, deviating argument.

2000 AMS Subject Classification: 34K13, 34L30

2

CHEUNG-REN164

1. INTRODUCTION

Consider the p-Laplacian Duffing equation with a deviating argument

(ϕp(x′(t)))′ + g(x(t− τ(t))) = e(t), (1.1)

where p > 1 is a constant, ϕp : R→ R, ϕp(u) = |u|p−2u is a one-dimensional p-Laplacian, e, τ are

periodic functions with period T > 0, and g, e, τ ∈ C(R,R).

There has been a great deal of research works on such an equation which is used to describe

fluid mechanical and nonlinear elastic mechanical phenomena. For example, when p = 2 and

τ(t) ≡ 0, the existence of T−periodic solutions to Eq.(1.1) was extensively studied in [1-3]. In [4-6,

9, 13], by using the time maps and the phase plane analysis, the authors discussed the existence of

periodic solutions to Eq.(1.1) for p 6= 2 and τ(t) ≡ 0. On the other hand, for p = 2 and τ(t) 6≡ 0,

the existence of T−periodic solutions to several second order scalar differential equations were also

studied in [8, 10-12]. In [8], X. Huang and Z. Xiang studied the following type of Duffing equation

with a single constant deviating argument

x′′(t) + g(x(t− τ)) = p(t). (1.2)

Under a one-sided boundedness condition imposed on g(x) such as

|g(x)| < R0 for x > M, (1.3)

where M > 0, R0 > 0 are constants, and a signal condition xg(x) > 0 for |x| > M , the authors

obtained a periodic solution for Eq.(1.2). In [12], S. Ma, Z. Wang and J. Yu studied delay Duffing

equations of the type

x′′(t) + m2x(t) + g(x(t− τ)) = p(t). (1.4)

They established several criteria to guarantee the existence of periodic solutions of Eq.(1.4) by

assuming

supx∈R

|g(x)| < ∞. (1.5)

Recently, S. Lu and W. Ge in [10] discussed the existence of periodic solutions for the second order

differential equation with multiple deviating arguments

x′′(t) + f(x(t))x′(t) +n∑

j=1

βj(t)g(x(t− γj(t))) = p(t). (1.6)

In their work, some linear growth condition imposed on g(x) such as

lim|x|→+∞

|g(x)||x| = r ∈ [0, +∞). (1.7)

was needed.

The main technique of these works [8, 10-12] is to convert the problem into the abstract form

Lx = Nx, with L being a non-invertible linear operator. Thus the existence of solutions of the

3

P-LAPLACIAN DUFFING EQUATIONS 165

problem can be given by the Mawhin’s continuation theorem [7]. But as far as we are aware of,

the corresponding problem of Eq.(1.1) with p 6= 2 and τ(t) 6≡ 0 has never been studied. This is

mainly due to the fact that the Mawhin’s continuation theorem is not applicable directly since the

p-Laplacian ϕp(u) = |u|p−2u is not linear with respect to u except when p = 2.

In this paper, we translate equation (1.1) into a two-dimension system to ensure Mawhin’s

continuation theorem can be applied. This method can also be used to solve problems for other

equations with p-Laplacian. Moreover, the one-side growth condition we impose on g(x) in order

to obtain a priori bound of periodic solutions for Eq.(1.1) is weaker than the corresponding ones

in (1.3), (1.5) and (1.7).

2. MAIN RESULT

First, we recall Mawhin’s continuation theorem which our study is based upon.

Let X and Y be real Banach Spaces and let L : D(L) ⊂ X → Y be a Fredholm operator

with index zero, here D(L) denotes the domain of L. This means that Im L is closed in Y and

dim Ker L = dim(Y/Im L) < +∞. Consider the supplementary subspaces X1 and Y1 such that

X = Ker L ⊕ X1 and Y = Im L ⊕ Y1 and let P : X → Ker L and Q : Y → Y1 be the natural

projections. Clearly, Ker L∩(D(L)∩X1) = 0, thus the restriction LP := L|D(L)∩X1 is invertible.

Denote by K the inverse of LP .

Now, let Ω be an open bounded subset of X with D(L) ∩Ω 6= φ. A map N : Ω → Y is said to

be L− compact in Ω, if QN(Ω) is bounded and the operator K(I −Q)N : Ω → X is compact.

MAWHIN’S CONTINUATION THEOREM [7] Suppose that X and Y are two Banach

spaces, and L : D(L) ⊂ X → Y is a Fredholm operator with index zero. Furthermore, Ω ⊂ X is

an open bounded set and N : Ω → Y is L− compact on Ω. If

(1)Lx 6= λNx, ∀x ∈ ∂Ω ∩D(L), λ ∈ (0, 1);

(2)Nx /∈ Im L, ∀x ∈ ∂Ω ∩Ker L; and

(3)degJQN, Ω ∩Ker L, 0 6= 0, where J : Im Q → Ker L is an isomorphism,

then the equation Lx = Nx has a solution in Ω⋂

D(L).

In order to use Mawhin’s continuation theorem to study the existence of T−periodic solutions

for Eq.(1.1), we rewrite Eq.(1.1) in the following form

x′1(t) = ϕq(x2(t)) = |x2(t)|q−2x2(t)

x′2(t) = −g(x1(t− τ(t))) + e(t),(2.1)

where q > 1 is a constant with 1p + 1

q = 1. Clearly, if x(t) = (x1(t), x2(t))> is a T−periodic solution

to Eqs.(2.1), then x1(t) must be a T−periodic solution to Eq.(1.1). Thus, the problem of finding

a T−periodic solution for Eq. (1.1) reduces to finding one for Eq. (2.1).

Now, we set CT = φ ∈ C(R,R) : φ(t + T ) ≡ φ(t) with norm |φ|0 = maxt∈[0,T ] |φ(t)|,X = Y = x = (x1(·), x2(·)) ∈ C(R,R2) : x(t) ≡ x(t + T ) with norm ||x|| = max|x1|0, |x2|0.

4

CHEUNG-REN166

Clearly, X and Y are Banach spaces. Meanwhile, let

L : D(L) ⊂ X → Y, Lx = x′ =

x′1

x′2

N : X → Y, Nx =

ϕq(x2)

−g(x1(t− τ(t))) + e(t).

It is easy to see that Ker L = R2, Im L = y ∈ Y :∫ T

0y(s)ds = 0. So L is a Fredholm operator

with index zero. Let P : X → Ker L and Q : Y → Im Q ⊂ R2 be defined by

Px =1T

∫ T

0

x(s)ds; Qy =1T

∫ T

0

y(s)ds,

and let K denote the inverse of L|KerP∩D(L). Obviously, KerL = ImQ = R2 and

[Ky](t) =∫ T

0

G(t, s)y(s)ds. (2.2)

where

G(t, s) =

s

T, 0 ≤ s < t ≤ T.

s− T

T, 0 ≤ t ≤ s ≤ T.

From (2.2), one can easily see that N is L−compact on Ω, where Ω is an open, bounded subset of

X.

THEOREM 1. Suppose the following conditions are satisfied,

[A1]∫ T

0e(s)ds = 0 and e(t) 6≡ 0;

[A2] there exists a constant d > 0 such that ug(u) > 0 for |u| > d or ug(u) < 0 for |u| > d;

[A3] there is a constant r0 ≥ 0 such that limu→−∞

|g(u)||u|p−1 = r0,

then Eq.(1.1) has at least one T−periodic solutions if 2r0Tp < 1.

PROOF. Considering the following operator equation

Lx = λNx, λ ∈ (0, 1). (2.3)

Let Ω1 ∈ x : x ∈ X, Lx = λNx, λ ∈ (0, 1). If x(t) =

x1(t)

x2(t)

∈ Ω1, then from (2.3), we see

x′1(t) = λϕq(x2(t)) = λ|x2(t)|q−2x2(t)

x′2(t) = −λg(x1(t− τ(t))) + λe(t)(2.4)

From the first equation of (2.4), we have x2(t) = ϕp( 1λx′1(t)). Hence by the second equation of

(2.4),

[ϕp(1λ

x′1(t))]′ + λg(x1(t− τ(t))) = λe(t),

i.e.,

[ϕp(x′1(t))]′ + λpg(x1(t− τ(t))) = λpe(t). (2.5)

5


Integrating the two sides of (2.5) on [0, T ], we get∫ T

0

g(x1(t− τ(t)))dt = 0 (2.6)

So there is a constant ξ ∈ [0, T ] such that

g(x1(ξ − τ(ξ))) = 0.

From assumption [A2], we see that |x1(ξ − τ(ξ))| ≤ d. Write ξ − τ(ξ) = kT + t0, where k ∈ Z and

t0 ∈ [0, T ). Then

|x1(t0)| = |x1(ξ − τ(ξ))| ≤ d,

which implies

|x1|0 ≤ d +∫ T

0

|x′1(s)|ds. (2.7)

On the other hand, taking absolute values and integrating both sides of Eq.(2.5) on [0, T ], we

have ∫ T

0|[ϕp(x′1(t))]

′|dt ≤ λp[∫ T

0|g(x1(t− τ(t)))|dt +

∫ T

0|e(t)|dt]

<∫ T

0|g(x1(t− τ(t)))|dt +

∫ T

0|e(t)|dt.

(2.8)

In view of r0Tp < 1, it is easy to see that there is a constant ε > 0 (independent of λ) such that

(r0 + ε)T p < 1. (2.9)

For such a constant ε > 0, we have from assumption [A3] that there is a constant ρ > d (independent

of λ) such that

|g(u)| ≤ (r0 + ε)|u|p−1, for u < −ρ. (2.10)

Let

E1 = t ∈ [0, T ] : |x1(t− τ(t))| ≤ ρ,E2 = t ∈ [0, T ] : x1(t− τ(t)) > ρ,E2 = t ∈ [0, T ] : x1(t− τ(t)) < −ρ.

From (2.6), we know that

(∫

E1

+∫

E2

+∫

E3

)g(x1(t− τ(t)))dt = 0.

It follows from [A2] that∫

E2

|g(x1(t− τ(t)))|dt = |∫

E2

g(x1(t− τ(t)))dt| ≤∫

E1

|g(x1(t− τ(t)))|dt +∫

E3

|g(x1(t− τ(t)))|dt

and then by (2.10), we have

∫ T

0|g(x1(t− τ(t)))|dt = (

∫E1

+∫

E2+

∫E3

)|g(x1(t− τ(t)))|dt

≤ 2∫

E1|g(x1(t− τ(t)))|dt + 2

∫E3|g(x1(t− τ(t)))|dt

≤ 2gρT + 2(r0 + ε)T |x1|p−10 ,

(2.11)

6

CHEUNG-REN168

where gρ = max|u|≤ρ

|g(u)|. Substituting (2.11) into (2.8), we get

∫ T

0

|[ϕp(x′1(t))]′|dt < 2gρT + |e|1 + 2(r0 + ε)T |x1|p−1

0 , (2.12)

where |e|1 =∫ T

0|e(s)|ds.

From ϕp(x′1(0)) = ϕp(x′1(T )), we know that there is η ∈ [0, T ] such that

ϕp(x′1(η)) = 0.

So|ϕp(x′1)|0 ≤ ∫ T

0|ϕp(x′1(t))|′dt

≤ 2gρT + |e|1 + 2(r0 + ε)T |x1|p−10 .

In view of |ϕp(x′1)|0 = |x′1|p−10 , we get

|x′1|p−10 ≤ 2(r0 + ε)T |x1|p−1

0 + 2gρT + |e|1.

By (2.7),

|x′1|p−10 < 2(r0 + ε)T (d +

∫ T

0

|x′1(s)|ds)p−1 + 2gρT + |e|1. (2.13)

Next we prove that there exists a constant R > 0 such that

|x′1|0 ≤ R. (2.14)

Case 1. If 1 < p ≤ 2, i.e., 0 < p− 1 ≤ 1, then from (2.13) we know that

|x′1|p−10 ≤ 2(r0 + ε)Tdp−1 + 2(r0 + ε)T (

∫ T

0|x′1(s)|ds)p−1 + 2gρT + |e|1

≤ 2(r0 + ε)Tdp−1 + 2(r0 + ε)T p|x′1|p−10 + 2gρT + |e|1.

By (2.9), we have

|x′1|p−10 <

2(r0 + ε)Tdp−1 + 2gρT + |e|11− 2(r0 + ε)T p

,

i.e.,

|x′1|0 <

[2(r0 + ε)Tdp−1 + 2gρT + |e|1

1− 2(r0 + ε)T p

] 1p−1

:= R1 (2.15)

Case 2. If p > 2, then p − 1 > 1. We know from [A1] that x1(t) is not constant and then∫ T

0|x′1(s)|ds > 0. In fact, if x1(t) is a constant, then by (2.5), e(t) = c and this contradicts

assumption [A1]. Now it is elementary to check that there is a constant h > 0 (independent of λ)

such that

(1 + u)p−1 < 1 + pu, ∀u ∈ (0, h]. (2.16)

(i) If dT |x′1|0 ≥ h, then

|x′1|0 ≤d

Th. (2.17)

(ii) If dT |x′1|0 < h, then from (2.13) and (2.16),

|x′1|p−10 < 2(r0 + ε)T (d + T |x′1|0)p−1 + 2gρT + |e|1

= 2(r0 + ε)T · T p−1|x′1|p−10 [1 + d

T |x′1|0 ]p−1 + 2gρT + |e|1≤ 2(r0 + ε)T p|x′1|p−1

0 [1 + pdT |x′1|0 ] + 2gρT + |e|1

= 2(r0 + ε)T p|x′1|p−10 + 2pd(r0 + ε)T p−1|x′1|p−2

0 + 2gρT + |e|1,

7


and then

[1− 2(r0 + ε)T p]|x′1|p−10 < 2pd(r0 + ε)T p−1|x′1|p−2

0 + 2gρT + |e|1.

From (2.9) and p− 1 > p− 2, it follows that there exists R2 > 0 such that

|x′1|0 ≤ R2. (2.18)

Let R = maxR1, R2,d

Th. It is easy to see from (2.15), (2.17) and (2.18) that (2.14) holds in any

case. Thus by (2.6),

|x1|0 ≤ d + T |x′1|0 ≤ d + TR := M1. (2.19)

By the first equation of (2.4), we have∫ T

0

|x2(s)|q−2x2(s)ds = 0,

which implies that there is a constant t2 ∈ [0, T ] such that x2(t2) = 0. So

|x2|0 ≤∫ T

0

|x′2(s)|ds. (2.20)

On the other hand, by using the second equation of (2.4), we obtain∫ T

0

|x′2(s)|ds ≤ λ(gM1T + |e|1) < gM1T + |e|1,

where gM1 = max|u|≤M1

|g(u)|. So from (2.20), we have

|x2|0 ≤ gM1T + |e|1 := M2. (2.21)

Let Ω2 = x ∈ Ker L : Nx ∈ Im L. If x ∈ Ω2, then x ∈ Ker L and QNx = 0. From

assumption [A1] we see that |x2|q−2x2 = 0,

g(x1) = 0.(2.22)

So

|x1| ≤ d ≤ M1, x2 = 0 ≤ M2. (2.23)

Let Ω = x = (x1, x2)> ∈ X : |x1|0 < N1, |x2|0 < N2, where N1 and N2 are constants with

N1 > M1, N2 > M2 and (N2)q > dgd, where gd = max|u|≤d

|g(u)|. Then Ω1 ⊂ Ω, Ω2 ⊂ Ω. From (2.19),

(2.21) and (2.23), it is easy to see that conditions (1) and (2) of Mawhin’s Continuation Theorem

are satisfied.

Next, we claim that condition (3) of Mawhin’s Continuation Theorem is also satisfied. For this,

define the isomorphism J : Im Q → Ker L by

J(x1, x2) =

(x2, x1), if ug(u) < 0 for |u| > d,

(−x2, x1), if ug(u) > 0 for |u| > d,

and let H(v, µ) := µv + 1−µT JQNv, (v, µ) ∈ Ω × [0, 1]. By simple calculation, we obtain, for

(x, µ) ∈ ∂(Ω ∩KerL)× [0, 1],

x>H(x, µ) =

µ(x21 + x2

2) + 1−µT (−x1g(x1) + |x2|q) > 0, if ug(u) < 0 for |u| > d,

µ(x21 + x2

2) + 1−µT (x1g(x1) + |x2|q) > 0, if ug(u) > 0 for |u| > d.

8

CHEUNG-REN170

Hence

degJQN, Ω ∩KerL, 0 = degH(x, 0), Ω ∩KerL, 0

= degH(x, 1), Ω ∩KerL, 0 = degI, Ω ∩KerL, 0

6= 0,

and so condition (3) of Mawhin’s Continuation Theorem is also satisfied.

Therefore, by Mawhin’s Continuation Theorem, we conclude that equation

Lx = Nx

has a solution x(t) = (x1(t), x2(t))> on Ω, i.e., Eq.(1.1) has a T−periodic solution x1(t) with

|x1|0 ≤ M2.

THEOREM 2 Suppose assumptions [A1] and [A2] in Theorem 1 hold, and [A3] is replaced by

[A3]′ there is a constant r1 ≥ 0 such that limu→+∞

|g(u)|up−1 = r1,

then Eq. (1.1) has at least one T−periodic solution if 2r1Tp < 1.

In fact, if [A3]′ holds, then (2.10) can be replaced by

|g(u)| ≤ |r1 + ε|up−1, for u > ρ.

So it follows from

(∫

E1

+∫

E2

+∫

E3

)g(x1(t− τ(t)))dt = 0

that

∫

E3

|g(x1(t− τ(t)))|dt = |∫

E1

g(x1(t− τ(t)))dt| ≤∫

E1

|g(x1(t− τ(t)))|dt +∫

E2

|g(x1(t− τ(t)))|dt

and so (2.12) is satisfied. The rest of the proof is analogous to the proof of Theorem 1.

REMARK 1. It is obviously that the one-side linear growth condition [A3] (or [A3]′) is weaker

than the corresponding ones in (1.3), (1.5) and (1.7).

9


Acknowledgements

Research of the first author was partially supported by the Research Grants Council of the

Hong Kong SAR, China (Project No. HKU7040/03P)

Research of the second author was partially supported by the National Natural Science Foun-

dation, China (Project No. 10371006)

10

CHEUNG-REN172

REFERENCES

[1] A. Capietto, J. Mawhin and F. Zanolin, A continuation approach to super-linear periodic

boundary value problems, J. Differential equations 88, 347-395 (1990).

[2] T. Ding and F. Zanoli, Time-maps for the solvability of perturbed nonlinear Duffing equa-

tions, Nonlinear Analysis TMA 17, 635-653 (1991).

[3] T. Din, R. Iannacci, and F. Zanolin, Existence and multiplicity results for periodic solutions

of semi-linear Duffing equations, J. Differential Equations 105, 364-409 (1993).

[4] M.A. Del Pino and R.F. Manasevich, Multiple solutions for the p-Laplacian under global

non-resonance, Proc. Amer. Math. Soc. 112, 131-138 (1991).

[5] M.A. Del Pino, R.F. Manasevich and A. Murua, Existence and multiple of solutions with

prescribed period for a second order quasi-linear ordinary differential equation, Nonlinear Analysis

TMA 18, 79-92 (1992).

[6] C. Fabry and D. Fayyad, Periodic solutions of second order differential equations with a

p-Laplacian and asymmetric nonlinearities, Rend. Ist. Univ. Trieste 24, 207-227 (1992).

[7] R.E. Gaines and J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations,

Springer-Verlag , Berlin, 1977.

[8] X. Huang and Z. Xiang, On the existence of 2π-periodic solutions of Duffing type equation

x′′(t) + g(x(t− τ) = p(t), Chinese Science Bulletin 39, 201-203 (1994) [in Chinese].

[9] B. Liu, Multiplicity results for periodic solutions of a second order quasi-linear ODE with

asymmetric nonlinearities, Nonlinear Analysis TMA 24, 139-160 (1992).

[10] S. Lu and W. Ge, Periodic solutions for a kind of second order differential equations with

multiple deviating arguments, Applied Mathematics and Computation 146, 195-209 (2003).

[11] S. Ma, Z. Wang and J. Yu, An abstract theorem at resonance and its applications, J.

Differential Equations 145, 274-294 (1998).

[12] S. Ma, Z. Wang and J. Yu, Coincidence degree and periodic solutions of Duffing equations,

Nonlinear Analysis TMA 34, 443-460 (1998).

[13] R.F. Manasevich and J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian

like operarors, J. Differential Equations 145, 367-393 (1998).

11


174

BOSBACH AND RIECAN STATES ON RESIDUATED LATTICES

LAVINIA CORINA CIUNGU

Abstract. Residuated lattices were introduced firstly as generalization of ideal lat-tices of rings and they served as algebraic structures for substructural logics. Thenotion of a state is an analogue to probability measure and it has been studied for dif-ferent types of non-commutative fuzzy structures such as pseudo MV-algebras, pseudoBL-algebras and bounded non-commutative R`-monoids. In this paper we investigatethe states on residuated lattices and we show that the extension of Georgescu’s originalproblem from pseudo BL-algebras has negative solution for good residuated lattices.

1. Introduction

The notion of a state is an analogue to probability measure and it has a very im-portant role in the theory of quantum structures ([10]). The state on MV-algebraswas introduced firstly by F.Kopka and F.Chovanec ([21]) and the state on BL-algebraswas introduced by B.Riecan ([23]). In the case of non-commutative fuzzy structures,the states were introduced by A. Dvurecenskij ([7]) for pseudo MV-algebras, by G.Georgescu ([13]) for pseudo BL-algebras and by A.Dvurecenskij and J.Rachunek ([11])for bounded non-commutative R`-monoids.In the case of a pseudo MV-algebra M , A.Dvurecenskij proved in [6] that there is an`-group (G, u) with strong unit u such that M is isomorphic to Γ(G, u) = g ∈ G/0 ≤g ≤ u. This allowed him to define a partial addition +, that is x + y is defined ifx ≤ y− = u − y and the state is a mapping s : M → [0, 1] which preserves the partialaddition + and s(1) = 1. We recall that the elements a and b are orthogonal if a + b isdefined in M .The other non-commutative structures don’t have such a group representation and itwas more difficult to define the notion of states for these structures.We recall that a state on MV-algebras always exists in contrast to pseudo MV-algebras([7]), on the other hand, in [9] it was solved the existence of states for linear pseudoBL-algebras (see also [8]).In the case of pseudo BL-algebras G.Georgescu defined in [13] the Bosbach state andthis definition was generalized by A. Dvurecenskij and J. Rachunek ([11]) for non-commutative R`-monoids.For a good pseudo BL-algebra G.Georgescu proved that any Bosbach state is also aRiecan state, but he formulated as open problem to find an example of Riecan state ona good pseudo BL-algebra which is not a Bosbach state.Inspired by the above mentioned results, in this paper we extend the notion of statesto residuated lattices and the final results consist of proving that any Bosbach state ona good residuated lattice is a Riecan state, but conversely it turns out not to be true.

Date: 2006.03.15.Key words and phrases. Residuated lattice, Bosbach state, Riecan state, orthogonal elements, nor-

mal filter.2000 Mathematics Subject Classification 06B05, 03G25, 28E15.

1


BOSBACH AND RIECAN STATES ON RESIDUATED LATTICES 2

The paper is organized as follows.In Section 2 we give the definition of a residuated lattice and we prove the basic prop-erties of this structure which are also valid for other structures such as pseudo BL-algebras, weak pseudo BL-algebras and bounded non-commutative R`-monoids. Thedistance functions defined in this section are very important for the main results in thenext section.In Section 3 we define the Bosbach and Riecan states on a residuated lattice and weinvestigate their basic properties, proving that any Bosbach state is a Riecan state.As an answer to Georgescu’s open problem, we give an example of a Riecan state on agood residuated lattice which is not a Bosbach state.We refer to [3], [16] and [22] for general notions on lattices theory and for unexplainednotions and results on residuated lattices.

2. Residuated lattices and their basic properties

Definition 2.1. A residuated lattice is an algebra L = (L,∧,∨,,→, , 0, 1) of type(2, 2, 2, 2, 2, 0, 0) satisfying the following conditions:(L1) (L,∧,∨, 0, 1) is a bounded lattice ;(L2) (L,, 1) is a monoid ;(L3) x y ≤ z iff x ≤ y → z iff y ≤ x z for any x, y, z ∈ L.

In the sequel we agree that the operations ∧,∨, have higher priority than theoperations →, .

Examples 2.2. Let’s consider L = 0, a, b, c, 1 with 0 < a < b < c < 1 and theoperations ,→, given by the following tables:

0 a b c 10 0 0 0 0 0a 0 0 0 a ab 0 0 0 b bc 0 a a c c1 0 a b c 1

→ 0 a b c 10 1 1 1 1 1a b 1 1 1 1b b c 1 1 1c 0 a b 1 11 0 a b c 1

0 a b c 10 1 1 1 1 1a b 1 1 1 1b b b 1 1 1c 0 b b 1 11 0 a b c 1

.

Then (L,∧,∨,,→, , 0, 1) is a residuated lattice.

Examples 2.3. [4],[5] Let’s consider a pseudo MV-algebra M = (M,⊕,− ,∼ , 0, 1) withthe additional operation x y = (y− ⊕ x−)∼.The order on M is defined by x ≤ y iff x− ⊕ y = 1 (iff y ⊕ x∼ = 1).Defining x ∧ y = x (x− ⊕ y) and x ∨ y = x ⊕ x∼ y, according to [14], Prop.1.13,(M,∧,∨, 0, 1) is a bounded distributive lattice.Applying [14], Prop.1.7, (M,, 1) is a non-commutative monoid.If we define x → y = y ⊕ x∼ and x y = x− ⊕ y, then according to [14], Prop.1.12 wehave x y ≤ z iff x ≤ y → z iff y ≤ x z.Thus, M = (M,∧,∨,,→, , 0, 1) is a bounded residuated lattice.

Remark 2.4. (1) If additionally, for any x, y ∈ L the structure L satisfies the conditions:(L4) (x → y) x = x (x y) = x ∧ y ,(L5) (x → y) ∨ (y → x) = (x y) ∨ (y x) = 1then L is a pseudo BL-algebra.(2) If L satisfies the conditions (L1), (L2), (L3) and (L5), then it is a weak pseudo BL-algebra (or pseudo MTL-algebra).

CIUNGU176


(3) If L satisfies the conditions (L1), (L2), (L3) and (L4), then it is a bounded R`-monoid(divisible residuated lattice) ([11]).

One can easily prove that the structure L = (L,∧,∨,,→, , 0, 1) from Example2.2 is a pseudo MTL-algebra, but it is not a pseudo BL-algebra because (b → a) b 6=b (b a), so (L4) does not hold.There are residuated lattices that are not pseudo MTL-algebras. Indeed, put L′ =L⊕ (L2×2 ⊕ L3), where L2×2 = L2 × L2 and ⊕ represents the ordinal sum of structures(see [19]). Note that L2×2 ⊕ L3 does not satisfy (L5), therefore L′ doesn’t satisfy iteither. Also, L′ doesn’t satisfy (L4) because L doesn’t. Thus, L′ is a residuated latticethat is neither pseudo MTL-algebra, nor R`-monoid.

Definition 2.5. A residuated lattice is called commutative if x y = y x.

Proposition 2.6. A residuated lattice is commutative iff x → y = x y.

Proof. ” ⇒ ”: For any x, y ∈ L we have the following equivalences:

x ≤ y → z ⇔ x y ≤ z ⇔ x ≤ y z,

hence, y → z = y z.” ⇐ ”: For any z ∈ L, x y ≤ z ⇔ x ≤ y → z ⇔ x ≤ y z ⇔ y x ≤ z.Thus, x y = y x (indeed, for z = y x we have x y ≤ y x and for z = x y weget y x ≤ x y)

In a residuated lattice L = (L,∧,∨,,→, , 0, 1) we define two negations for allx ∈ L : x− = x → 0 and x∼ = x 0.

Proposition 2.7. ([1],[20]) In any residuated lattice the following properties hold:(1) x → (y → z) = (x y) → z;(2) x (y z) = (y x) z;(3) x ≤ y iff x → y = 1 iff x y = 1;(4) x → x = x x = 1;(5) x → 1 = x 1 = 1;(6) 0 → x = 0 x = 1;(7) x 0 = 0 x = 0;(8) (x → y) x ≤ y and x (x y) ≤ y;(9) x ≤ y → (x y) and x ≤ y (y x);(10) x ≤ y implies x z ≤ y z and z x ≤ z y for any z ∈ L;(11) (x → y) x ≤ x ∧ y and x (x y) ≤ x ∧ y;(12) (x → y) x ≤ x ≤ y → (x y) and (x → y) x ≤ y ≤ x → (y x) ;(13) x (x y) ≤ y ≤ x (x y) and x (x y) ≤ x ≤ y (y x);(14) If x ≤ y then z → x ≤ z → y and z x ≤ z y;(15) If x ≤ y then y → z ≤ x → z and y z ≤ x z;(16) 1 → x = x and 1 x = x;(17) x → y ≤ (y → z) (x → z);(18) x y ≤ (y z) → (x z) ;(19) x → y = x → (x ∧ y);(20) x y = x (x ∧ y);(21) y ≤ x → y and y ≤ x y;(22) If x ≤ y then x ≤ z → y and x ≤ z y ;(23) x → y ≤ (x z) → (y z);

177


(24) x y ≤ (z x) (z y) ;(25) 1− = 1∼ = 0 and 0− = 0∼ = 1;(26) x− x = 0 and x x∼ = 0;(27) x ≤ y− iff x y = 0 and x ≤ y∼ = 0 iff y x = 0;(28) x ≤ x−∼ and x ≤ x∼−;(29) x → y− = (x y)− and x y∼ = (y x)∼;(30) x ≤ y− iff y ≤ x∼;(31) If x ≤ y, then y− ≤ x− and y∼ ≤ x∼;(32) x ≤ x∼ → y and x ≤ x− y;(33) x → y ≤ y− x− and x y ≤ y∼ → x∼;(34) x → y∼ = y x− and x y− = y → x∼;(35) x∼−∼ = x∼ and x−∼− = x−;(36) x → x∼ = x x− ;(37) x (∨i∈Iyi) = ∨i∈I(x yi);(38) (∨i∈Iyi) x = ∨i∈I(yi x) ;(39) (∨i∈Ixi) y = ∧i∈I(xi y);(40) (∨i∈Ixi) → y = ∧i∈I(xi → y);(41) y (∧i∈Ixi) = ∧i∈I(y xi);(42) y → (∧i∈Ixi) = ∧i∈I(y → xi) ;(43) x → (y z) = y (x → z);(44) x (y → z) = y → (x z);(45) (x ∨ y) → (x ∧ y) = (x → y) ∧ (y → x);(46) (x ∨ y) (x ∧ y) = (x y) ∧ (y x) ;(47) (x ∨ y)− = x− ∧ y− and (x ∨ y)∼ = x∼ ∧ y∼;(48) (x ∧ y)− ≥ x− ∨ y− and (x ∧ y)− ≥ x∼ ∨ y∼;(49) (x ∨ y)−∼ ≥ x−∼ ∨ y−∼ and (x ∨ y)∼− ≥ x∼− ∨ y∼− ;(50) y− x− = x−∼ → y−∼ = x → y−∼;(51) y∼ → x∼ = x∼− y∼− = x y∼−.

If a residuated lattice is a chain, then it is a weak pseudo BL-algebra.

Definition 2.8. ([4], [5]) A residuated lattice L is good if x−∼ = x∼− for any x ∈ L.

Proposition 2.9. In any good residuated lattice we have (x∼ y∼)− = (x− y−)∼.

Proof. Applying Proposition 2.7(29),(50),(51) we have:

(x∼ y∼)− = x∼ → y∼− = x∼ → y−∼ = y−∼− x∼−

= y− x∼− = y− x−∼ = (x− y−)∼.

(In the last equality we also applied Proposition 2.7(29)).

Proposition 2.10. In any good residuated lattice we have x−∼ y−∼ ≤ (x y)−∼.

Proof. Because the residuated lattice is good and by Proposition 2.7(8), we have:

(x y)−∼ = (x y)∼− ≥ (x y)∼− ∧ x∼− ≥ x∼− (x∼− (x y)∼−)

= x∼− (x∼− (x y)−∼) = x∼− (x∼− (x → y−))∼.

But, applying Proposition 2.7(29) and Proposition 2.7(2) we have:

x∼− (x → y−)∼ = x∼− ((x → y−) 0) = (x∼− → y−) x∼− 0

= ((x∼− → y−) x∼−)∼ ≥ (x∼− ∧ y−)∼ ≥ x∼−∼ ∨ y−∼ = x∼ ∨ y−∼.

CIUNGU178


(By Proposition 2.7(29) we have (x∼− → y−) x∼− ≤ (x∼− ∧ y−), so((x∼− → y−) x∼−)∼ ≥ (x∼− ∧ y−)∼).It follows that

(x y)−∼ ≥ x∼− (x∼ ∨ y−∼) = (x∼− x∼) ∨ (x∼− y−∼)

= 0 ∨ (x∼− y−∼) = x∼− y∼− = x−∼ y−∼.

(We applied Proposition 2.7(37-38) and Proposition 2.7(26)).

Proposition 2.11. Let L be a good residuated lattice. We define a total binary operation⊕ on L by x⊕ y := (y∼ x∼)−, x, y ∈ L. Then for all x, y, z ∈ L we have:(1) x⊕ y := (y− x−)∼;(2) ⊕ is associative;(3) x, y ≤ x⊕ y;(4) x⊕ 0 = x−∼ = 0⊕ x;(5) x⊕ 1 = 1 = 1⊕ x;(6) x⊕ y = x− y−∼ = y∼ → x−∼.

Proof. (1) follows from Proposition 2.9.(2) We have :

(x⊕ y)⊕ z = (y∼ x∼)− z = (z∼ (y∼ x∼)−∼)−

= z∼ → (y∼ x∼)−∼− = z∼ → (y∼ x∼)− = z∼ → (y∼ → x∼−)

(we applied Proposition 2.7(29),(35));

x⊕ (y ⊕ z) = x⊕ (z∼ y∼)− = ((z∼ y∼)−∼ x∼)−

= (z∼ y∼)−∼ → x∼− = (z∼ y∼)−∼ → x−∼

= (z∼ y∼) → x−∼ = z∼ → (y∼ → x−∼) = z∼ → (y∼ → x∼−) = (x⊕ y)⊕ z.

(we applied Proposition 2.7(50),(1));(3) By Proposition 2.7(29),(32) we have:

x⊕ y = (y− x−)∼ = x− y−∼ ≥ x

x⊕ y = (y∼ x∼)− = y∼ → x∼− ≥ y;

(4) Applying Proposition 2.7(26),(16) we get:x⊕ 0 = (0∼ x∼)− = 0∼ → x−∼ = 1 → x−∼ = x∼− = x−∼;0⊕ x = (x− 0−)∼ = x− 0−∼ = x− 0 = x−∼.(5) x⊕ 1 = (1− x−)∼ = x− 1−∼ = x− 1 = 1 (Proposition 2.7(29),(5);1⊕ x = (x− 1−)∼ = 1− x−∼ = 0 x−∼ = 1 (Proposition 2.7(29),(6)).(6) x⊕ y = (y∼ x∼)− = y∼ → x∼− = y∼ → x−∼.x⊕ y = (y− x−)∼ = x− y−∼.It follows that x⊕ y = x− y−∼ = y∼ → x−∼.

In a residuated lattice we can define two distance functions:

d1(x, y) = (x → y) ∧ (y → x) = x ∨ y → x ∧ y

d2(x, y) = (x y) ∧ (y x) = x ∨ y x ∧ y.

(later equalities hold according to Proposition 2.7(45-46)).

179


Proposition 2.12. The two distance functions fulfill the following properties:(1) d1(x, y) = d1(y, x) and d2(x, y) = d2(y, x);(2) d1(x, y) = 1 iff x = y iff d2(x, y) = 1;(3) d1(x, 0) = x− and d2(x, 0) = x∼;(4) d1(x, 1) = x = d2(x, 1);(5) d1(x, y) ≤ d2(x

−, y−);(6) d2(x, y) ≤ d1(x

∼, y∼);(7) d1(x, y) ≤ d1(x

∼−, y∼−);(8) d2(x, y) ≤ d2(x

−∼, y−∼);(9) d2(x

−, y−) = d1(x−∼, y−∼);

(10) d1(x∼, y∼) = d2(x

∼−, y∼−).

Proof. (1) Obvious.(2) d1(x, y) = 1 ⇔ x → y = 1 and y → x = 1 ⇔ x ≤ y and y ≤ x ⇔ x = y;Similarly, d2(x, y) = 1 ⇔ x = y.(3) d1(x, 0) = (x → 0) ∧ (0 → x) = x− ∧ 1 = x−;d2(x, 0) = (x 0) ∧ (0 x) = x∼ ∧ 1 = x∼;(4) d1(x, 1) = (x → 1) ∧ (1 → x) = 1 ∧ x = x;d2(x, 1) = (x 1) ∧ (1 x) = 1 ∧ x = x.(5) By Proposition 2.7(33) we have:d1(x, y) = (x → y) ∧ (y → x) ≤ (y− x−) ∧ (x− y−) = d2(x

−, y−).(6) By Proposition 2.7(33) we have:d2(x, y) = (x y) ∧ (y x) ≤ (y∼ → x∼) ∧ (x∼ → y∼) = d1(x

∼, y∼).(7) By (5) and (6) we get: d2(x, y) ≤ d1(x

∼, y∼) ≤ d2(x∼−, y∼−).

(8) Similarly, d1(x, y) ≤ d2(x−, y−) ≤ d1(x

−∼, y−∼).(9) By above properties we get:d2(x

−, y−) ≤ d1(x−∼, y−∼) ≤ d2(x

−∼−, y−∼−) = d2(x−, y−), hence d2(x

−, y−) = d1(x−∼, y−∼).

(10) Similarly:d1(x

∼, y∼) ≤ d2(x∼−, y∼−) ≤ d1(x

∼−∼, y∼−∼) = d1(x∼, y∼), hence d1(x

∼, y∼) = d2(x∼−, y∼−).

The following result is inspired by [13].

Proposition 2.13. On the residuated lattice L let s : L −→ [0, 1] be a function suchthat s(1) = 1. Then the following are equivalent:(i) 1 + s(x ∧ y) = s(x ∨ y) + s(d1(x, y)) for all x, y ∈ L;(ii) 1 + s(x ∧ y) = s(x) + s(x → y) for all x, y ∈ L;(iii) s(x) + s(x → y) = s(y) + s(y → x) for all x, y ∈ L, where the + is the usualaddition of real numbers.

Proof. (i)⇒(ii) If a ≤ b, then a ∧ b = a, a ∨ b = b, a → b = 1 and

d1(a, b) = (a → b) ∧ (b → a) = 1 ∧ (b → a) = b → a,

hence by hypothesis, 1 + s(a) = s(b) + s(b → a).Letting a = x ∧ y and b = y it follows that

1 + s(x ∧ y) = s(y) + s(y → x ∧ y) = s(y) + s(y → x) (we applied Proposition 2.7(19).)

(ii)⇒(iii) s(x) + s(x → y) = 1 + s(x ∧ y) = 1 + s(y ∧ x) = s(y) + s(y → x).(iii)⇒(i) We have that d1(x, y) = x ∨ y → x ∧ y, hence, applying the hypothesis we get

CIUNGU180


that:

s(x ∨ y) + s(d1(x, y)) = s(x ∨ y) + s(x ∨ y → x ∧ y) =

= s(x ∧ y) + s(x ∧ y → x ∨ y) = s(x ∧ y) + s(1) =

= 1 + s(x ∧ y)

(x ∧ y ≤ x ∨ y ⇒ x ∧ y → x ∨ y = 1).

Proposition 2.14. On the residuated lattice L let s : L −→ [0, 1] be a function suchthat s(1) = 1. Then the following are equivalent:(i) 1 + s(x ∧ y) = s(x ∨ y) + s(d2(x, y)) for all x, y ∈ L;(ii) 1 + s(x ∧ y) = s(x) + s(x y) for all x, y ∈ L;(iii) s(x) + s(x y) = s(y) + s(y x) for all x, y ∈ L.

Proof. Similarly.

3. Bosbach state and Riecan state

Inspired by [13], [11], [12], we introduce the notion of Bosbach state on a residuatedlattice L.

Definition 3.1. A Bosbach state on a residuated lattice L is a function s : L → [0, 1]such that the following conditions hold for any x, y ∈ L :(B1) s(x) + s(x → y) = s(y) + s(y → x);(B2) s(x) + s(x y) = s(y) + s(y x);(B3) s(0) = 0 and s(1) = 1.

Examples 3.2. Let’s consider the set L = 0, a, b, c, 1 with 0 < a < b < c < 1 andthe residuated lattice L = (L,∧,∨,,→, , 0, 1) with 0 < a < b < c < 1 where theoperations ,→, are given in the following tables:

0 a b c 10 0 0 0 0 0a 0 a a a ab 0 a a a bc 0 a b c c1 0 a b c 1

→ 0 a b c 10 1 1 1 1 1a 0 1 1 1 1b 0 b 1 1 1c 0 b b 1 11 0 a b c 1

0 a b c 10 1 1 1 1 1a 0 1 1 1 1b 0 c 1 1 1c 0 a b 1 11 0 a b c 1

.

The function s : L → [0, 1] defined by: s(0) = 0, s(a) = 1, s(b) = 1, s(c) = 1, s(1) = 1is the unique Bosbach state on L.L is actually even a good pseudo-MTL algebra.

Proposition 3.3. Let s be a Bosbach state on L. Then for all x, y ∈ L the followingproperties hold:(1) s(x → y) = s(s y);(2) s(d1(x, y)) = s(d2(x, y));(3) s(x−) = s(x∼) = 1− s(x);(4) s(x−∼) = s(x∼−) = s(x−−) = s(x∼∼) = s(x);(5) x ≤ y implies 1 + s(x) = s(y) + s(y → x) = s(y) + s(y x);(6) x ≤ y implies s(x) ≤ s(y);(7) s(x y) = 1− s(x → y−) and s(y x) = 1− s(x y∼);

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(8) s(x) + s(y) = s(x y) + s(y− → x);(9) s(x) + s(y) = s(y x) + s(y∼ x);(10) s(x− → y−) = s(y−∼ → x−∼);(11) s(x∼ → y∼) = s(y∼− → x∼−)

Proof. (1) By Proposition 2.13 and Proposition 2.14 it follows that:s(x) + s(x → y) = 1 + s(x ∧ y) = s(x) + s(x y), hences(x → y) = s(x y);(2) By Proposition 2.13 and Proposition 2.14, s(d1(x, y)) = 1 + s(x ∧ y) − s(s ∨ y) =s(d2(x, y)).(3) s(x) + s(x−) = s(x) + s(x → 0) = s(0) + s(0 → x) = s(0) + s(1) = 1, hences(x−) = 1− s(x). By (1) we have s(x∼) = s(x−), hence s(x−) = s(x∼) = 1− s(x);(4) Applying (3) twice;(5) Since x ≤ y we have x → y = x y = 1.Applying (B1) and (B3) we get that:s(y) + s(y → x) = s(x) + s(x → y) = s(x) + 1.Similarly, by (B2) and (B3) we obtain s(y) + s(y x) = s(x) + 1;(6) By (5) and (3) we get that s(y)− s(x) = 1− s(y → x) = s((y → x)−) ≥ 0;(7) By (3) s((x y)−) = 1− s(x y).But (x y)− = x → y−, so s(x y) = 1− s(x → y−).Similarly, s(y x) = 1− s((y x)∼) = 1− s(x y∼);(8) Applying (B1) and (7) we have:

s(x y) + s(y− → x) = s(x y) + s(x)− s(x → y−)− s(y−)

= s(x y) + s(x)− (1− s(x y))− 1 + s(y) = s(x) + s(y);

(9) Applying (B2) and (7) we have:

s(y x) + s(y∼ x) = s(y x) + s(x) + s(x y∼)− s(y∼)

= s(y x) + s(x) + 1− s(y x)− 1 + s(y) = s(x) + s(y);

(10) By Proposition 2.7(33) we have x− y− ≤ y−∼ → x−∼, so,by (1) and (6) itfollows that:

s(x− → y−) = s(x− y−) ≤ s(y−∼ → x−∼) ≤ s(x−∼− y−∼−)

= s(x− y−) = s(x− → y−).

Thus, s(x− → y−) = s(y−∼ → x−∼);(11) Similarly.

According to [13], [11], [23] we introduce the following notion.

Definition 3.4. Let L be a good residuated lattice. The elements x, y ∈ L are calledorthogonal, denoted by x ⊥ y iff y−∼ ≤ x−.If the elements x, y ∈ L are orthogonal, we define a partial operation + on L byx + y := x⊕ y.

Proposition 3.5. In a good residuated lattice the following are equivalent:(i) x ⊥ y ;(ii) x−∼ ≤ y∼ ;(iii) y−∼ x−∼ = 0.

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Proof. (i) ⇔ (iii) x ⊥ y ⇔ y−∼ ≤ x− = x−∼− ⇔ y−∼ x−∼ = 0 (by Proposition2.7(27)).(ii) ⇔ (iii) x−∼ ≤ y∼ = y∼−∼ = y−∼− ⇔ y−∼ x−∼ = 0 (by Proposition 2.7(29)).

Proposition 3.6. In a good residuated lattice the following hold:(1) x∼ ⊥ x and x ⊥ x−;(2) If x ≤ y then x ⊥ y− and y∼ ⊥ x;(3) If L is commutative, then x ⊥ y iff y ⊥ x.

Proof. (1) x−∼ = x∼− ⇒ x∼ ⊥ x; x−∼− = x−−∼ = x− ⇒ x ⊥ x−.(2) x ≤ y ⇒ y− ≤ x− ⇒ y−∼− ≤ x− ⇒ y−−∼ ≤ x− ⇒ x ⊥ y−.x ≤ y ⇒ x−∼ ≤ y−∼ = y∼− ⇒ y∼ ⊥ x.(3) x ⊥ y ⇔ y−∼ x−∼ = 0 ⇔ x−∼ y−∼ = 0 ⇔ y ⊥ x.

The following notion of a state was firstly defined for BL-algebras in [23], in [13] forpseudo BL-algebras and in [11] for another more general structure.

Definition 3.7. Let L be a good residuated lattice. A Riecan state on L is a functions : L → [0, 1] such that the following conditions hold for all x, y ∈ L :(R1) If x ⊥ y, then s(x + y) = s(x) + s(y);(R2) s(1) = 1.

Example 3.8. Let’s consider again the residuated lattice in Example 3.2.One can easily check that x−∼ = x∼− for any x ∈ L, so L is a good residuated lattice.We claim that the Bosbach state s : L → [0, 1] defined by: s(0) = 0, s(a) = 1, s(b) =1, s(c) = 1, s(1) = 1 is also Riecan state on L.Indeed, the orthogonal elements of L are the pairs (x, y) of the following table:

x y x− y−∼ x⊕ y0 0 1 0 00 a 1 1 10 b 1 1 10 c 1 1 10 1 1 1 1a 0 0 0 1b 0 0 0 1c 0 0 0 11 0 0 0 1

One cas easy check that s is a Riecan state.

Proposition 3.9. Let s be a Riecan state on the good residuated lattice L. Then thefollowing properties hold for all x, y ∈ L :(1) s(x−) = s(x∼) = 1− s(x);(2) s(0) = 0;(3) s(x−∼) = s(x∼−) = s(x−−) = s(x∼∼) = s(x);(4) If x ≤ y then s(y)− s(x) = 1− s(x⊕ y−) = 1− s(y∼ ⊕ x);(5) If x ≤ y then s(x) ≤ s(y).

Proof. (1) By Proposition 3.6(1) we have x ⊥ x− and x∼ ⊥ x. By (R1) and Proposition2.7(26) it follows that:s(x) + s(x−) = s(x⊕ x−) = s(x−∼ x∼)− = s(x∼− x∼)−) = s(0−) = s(1) = 1, hence

183


s(x−) = 1− s(x).Similarly, s(x∼) = 1− s(x);(2) s(0) = s(1−) = 1− s(1) = 0;(3) s(x−∼) = 1− s(x−) = 1− 1 + s(x) = s(x). Similarly the others;(4) By Proposition 3.6(2),(1) we have :

s(x⊕ y−) = s(x) + s(y−) = s(x) + 1− s(y)

s(y∼ ⊕ x) = s(y∼) + s(x) = 1− s(y) + s(x);

(5) By (4) s(y)− s(x) = 1− s(x⊕ y−) ≥ 0.

Theorem 3.10. Let L be a good residuated lattice. Any Bosbach state on L is a Riecanstate on L.

Proof. Let s be a Bosbach state on L. Assume x ⊥ y, i.e. y−∼ ≤ x−.By Proposition 3.6(3),(5) we have:1 + s(y) = 1 + s(y−∼) = s(x−) + s(x− → y−∼) = 1− s(x) + s(x− → y−∼), hences(x− → y−∼) = s(x) + s(y).On the other hand, x⊕ y = (y∼ x∼)− = (y− x−)∼ = x− y−∼, hences(x⊕ y) = s(x− y−∼) = s(x− → y−∼) = s(x) + s(y).Therefore, s(x⊕ y) = s(x) + s(y) and by hypothesis s(1) = 1.It follows then that s is a Riecan state on L.

By the next example we show that there exists a Riecan state which is not Bosbachstate.

Example 3.11. Let’s consider again the residuated lattice in Example 2.2,L = (L,∧,∨,,→, , 0, 1) where L = 0, a, b, c, 1 with 0 < a < b < c < 1. It is easyto check that L is a good residuated lattice.The function s : L → [0, 1] defined by s(0) = 0, s(a) = 1/2, s(b) = 1/2, s(c) = 1, s(1) = 1is a Riecan state.Indeed, the orthogonal elements of L are the pairs (x, y) of the following table:

x y x− y−∼ x⊕ y0 0 1 0 00 a 1 b b0 b 1 b b0 c 1 1 10 1 1 1 1a 0 b 0 ba a b b 1a b b b 1b 0 b 0 bb a b b 1b b b b 1c 0 0 0 11 0 0 0 1

One can easily check that s is a Riecan state.But the function s above defined is not a Bosbach state.Indeed, trying to check the conditions (B2) of Bosbach state definition we obtain:

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s(a) + s(a b) = s(a) + s(1) = 1/2 + 1 = 3/2s(b) + s(b a) = s(b) + s(b) = 1/2 + 1/2 = 1,so the condition (B2) doesn’t hold.We conclude that s is not a Bosbach state.

Remarks 3.12. (1) In case of good pseudo-BL algebras, in [13] G.Georgescu left as anopen problem to find an example of Riecan state which is not a Bosbach state ;(2) In case of good R`-monoids, A.Dvurecenskij and J.Rachunek ([11]) proved that anyRiecan state is also a Bosbach state, hence it is true for pseudo BL-algebras ;(3) By the above example we proved that in case of good residuated lattices, Riecanstates need not be Bosbach state. Moreover, as the above structure is actually a pseudoMTL-algebra, we can see that this is also valid for the class of pseudo MTL-algebras.

In the sequel we shall write state instead of Bosbach state.

Definition 3.13. Let L be a residuated lattice. A nonempty set F of L is called filterof L if the following conditions hold:(F1) If x, y ∈ F , then x y ∈ F ;(F2) If x ∈ F, y ∈ L, x ≤ y then y ∈ F .

Proposition 3.14. [12] If F is a filter of L then:(F3) 1 ∈ F ;(F4) If x ∈ F, y ∈ L, then y → x ∈ F, y x ∈ F ;(F5) If x, y ∈ F then x ∧ y ∈ F.

Proposition 3.15. [12] For a subset F of L the following are equivalent:(1) F is a filter;(2) 1 ∈ F and if x, x → y ∈ F , then y ∈ F ;(3) 1 ∈ F and if x, x y ∈ F , then y ∈ F .A set F that fulfills (2) and (3) is called deductive system.

Definition 3.16. A filter H of L is called normal if for any x, y ∈ L

x → y ∈ H iff x y ∈ H.

Example 3.17. Let’s consider the good residuated lattice from Example 2.2,L = L,∧,∨,,→, , 0, 1 where L = 0, a, b, c, 1 with 0 < a < b < c < 1.It is easy to check that H = c, 1 is a normal filter of L.

With any normal filter H of L we associate a binary relation ≡H on L by defining

x ≡H y iff d1(x, y) ∈ H iff d2(x, y) ∈ H.

Remark 3.18. x ≡H y iff x → y, y → x ∈ H iff x y, y x ∈ H.

Proposition 3.19. For a given normal filter H of L the relation x ≡H y is an equivalentrelation on L.

Proof. Reflexivity: x ≡H x because x → x = x x = 1 ∈ F.Symmetry: Obviously x ≡H y ⇒ y ≡H x according to the above remark.Transitivity: Let’s consider x ≡H y and y ≡H z.We apply Proposition 2.7(17-18):

x → y ≤ (y → z) (x → z)

x y ≤ (y z) → (x z).

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Because x y ∈ H it follows that (y z) → (x z) ∈ H.Because y z ∈ H and (y z) → (x z) ∈ H, we get z x ∈ H.Also by Proposition 2.7(17-18) changing x and z we have:

z → y ≤ (y → x) (z → x)

z y ≤ (y x) → (z x),

so we get z x ∈ H.Similarly, x → z ∈ H and z → x ∈ H.Thus, x ≡H z and the transitivity is proved.

For any x ∈ L, let x/L be the equivalence class x/ ≡H .L/H becomes a residuated lattice with the natural operation induced from those of L.If x, y ∈ L, then x/H ≤ y/H iff x → y ∈ H iff x y ∈ H.

Definition 3.20. If s : L → [0, 1] is a state on L, we define the kernel Ker(s) by

Ker(s) = x ∈ L | s(x) = 1.

Proposition 3.21. If s is a state on residuated lattice L, then Ker(s) is a normal filteron L.

Proof. Obviously 1 ∈ Ker(s).Assume x, x → y ∈ Ker(s), that is s(x) = s(x → y) = 1.Then, by Proposition 2.13(iii), s(y) + s(y → x) = s(x) + s(x → y) = 2, hences(y) = s(y → x) = 1. It follows that y ∈ Ker(s). Thus, Ker(s) is a filter of L.By Propositions 2.13 and 2.14 we haves(x) + s(x → y) = 1 + s(x ∧ y) = s(x) + s(x y), hence s(x → y) = s(x y).It follows that x → y ∈ Ker(s) iff x y ∈ Ker(s).Thus, Ker(s) is a normal filter of L.

Proposition 3.22. If s is a state on the good residuated lattice L, the following prop-erties hold:(1) x/Ker(s) = y/Ker(s) iff s(x ∧ y) = s(x ∨ y);(2) If s(x ∧ y) = s(x ∨ y), then s(x) = s(y) = s(x ∧ y).

Proof. (1) x/Ker(s) = y/Ker(s) iff d1(x, y) ∈ Ker(s) iff s(d1(x, y)) = 1 iff s(x ∧ y) =s(x ∨ y) (by Proposition 2.13(i));(2) By Proposition 3.9(5) we have s(x∧ y) ≤ s(x), s(y) ≤ s(x∨ y) and by hypothesis itfollows that s(x) = s(y) = s(x ∧ y).

Theorem 3.23. Let L be a good residuated lattice. If s is a Riecan state on L, thenthe function s : L/Ker(s) → [0, 1] defined by s(x/Ker(s)) = s(x) is a Riecan state onL/Ker(s).

Proof. First we prove that s is well-defined.Indeed, if x/Ker(s) = y/Ker(s), then by Proposition 3.22 it follows that s(x ∧ y) =s(x ∨ y). Then by Proposition 2.13(i)we have s(d1(x, y)) = 1.It follows that d1(x, y) ∈ Ker(s) and similarly, d2(x, y) ∈ Ker(s).Thus, x ≡Ker(s) y.Moreover, if x ≡Ker(s) y, then s(x) = s(y).Indeed, x ≡Ker(s) y is equivalent to s(x → y) = s(y → x) = 1 and by Proposition2.13(iii) it follows that s(x) = s(y).Applying the method used in [11] we prove now that s is a Riecan state on L/Ker(s).

CIUNGU186


First we recall that if x ≤ y, then there is an element x1 ∈ x such that x1 ≤ y.Indeed, it is sufficient to take x1 = x ∧ y.Assume that x ⊥ y, that is y−∼ ≤ x−, hence x−∼ ≤ y−∼∼ = y∼−∼ = y∼, so x−∼ ≤ y∼.Let’s take x1 ∈ x−∼ such that x1 ≤ y∼. Hence, x−1

∼ ≤ y∼ and by Proposition 3.5(5) itfollows that x1 ⊥ y.Therefore,

s(x + y) = s(y∼ x∼)− = s((y∼ x∼)−) = s(y∼ x∼)− = s(x⊕ y)

= s(x−∼ ⊕ y−∼) = s(x1 + y−∼) = s(x1 + y−∼) = s(x1) + s(y−∼) = s(x) + s(y)

= s(x) + s(y)

(We took in consideration that x1 ≡Ker(s) x implies s(x1) = s(x)).

Remarks 3.24. Let’s consider the set L = 0, a, b, c, 1 with 0 < a < b < c < 1 andthe residuated lattice L = (L,∧,∨,,→, , 0, 1) with 0 < a < b < c < 1, where theoperations ,→, are given in the following tables:

0 a b c 10 0 0 0 0 0a 0 0 0 0 ab 0 0 0 0 bc 0 0 a a c1 0 a b c 1

→ 0 a b c 10 1 1 1 1 1a c 1 1 1 1b b c 1 1 1c b c c 1 11 0 a b c 1

0 a b c 10 1 1 1 1 1a c 1 1 1 1b c c 1 1 1c a c c 1 11 0 a b c 1

.

One can easily check that L is a not good residuated lattice (a−∼ = a, but a∼− = b).(1) G.Georgescu proved that for pseudo BL-algebras the existence of a state is equivalentwith the existence of a maximal filter which is normal ([13]). In the above example,H = 1 is a maximal and normal filter of L, but there are no states on L.Indeed, assume that A admits a Bosbach state s such that s(0) = 0, s(a) = α, s(b) = β,s(c) = γ, s(1) = 1. From s(x) + s(x → y) = s(y) + s(y → x), taking x = a, y = 0,x = b, y = 0 and respectively x = c, y = 0 we get α = 1/2, β = 1/2, γ = 1/2.On the other hand, taking x = b, y = a we get β + γ = α + 1, so 1 = 3/2 which is acontradiction. Hence, L does not admit a Bosbach state.(2) A.Dvurecenskij proved in [8] that every linear pseudo BL-algebra admits a state.The above example shows that there exist linear residuated lattices having no states.(3) In the case of a BL-algebra A, it is proved that a filter H is normal if and onlyif x H = H x for any x ∈ A ([5],Proposition 1.3). This equality doesn’t hold inthe case of residuated lattices. Indeed, let’s consider the normal filter H = a, b, c, 1of the residuated lattice L in Example 3.2. In this case we have c H = a, c andH c = a, b, c, so cH 6= H c.

Acknowledgment

References

[1] P. Bahls, J. Cole, N. Galatos, P. Jipsen, Cancellative residuated lattices, Algebra Universalis50(2003).

[2] B. Bosbach, Residuation groupoids, Resultate der Mathematik 5(1982), 107-122.

[3] S. Burris, H. P. Sankappanavar, A course in Universal Algebra, Springer-Verlag, New York, (1981).

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[4] A. Di Nola, G. Georgescu, A. Iorgulescu, Pseudo BL-algebras Part.I, Multiple Valued Logic8(2002), 673-714.

[5] A. Di Nola, G. Georgescu, A. Iorgulescu, Pseudo BL-algebras Part.II, Multiple Valued Logic8(2002), 717-750.

[6] A. Dvurecenskij, Pseudo MV-algebras are intervals in `-groups, J.Australian Math.Soc. 70 (2002),427-445.

[7] A. Dvurecenskij, States on pseudo MV-algebras, Studia Logica 68(2001), 301-327.

[8] A. Dvurecenskij, Every linear BL-algebra admits a state, Soft Computing (2006), DOI10.1007/s00500-006-0078-2 (On line April 4,2006).

[9] A. Dvurecenskij, M. Hycko, On the existence of states for linear pseudo BL-algebras, Math.Slovaca56(2006), to appear.

[10] A. Dvurecenskij, S. Pulmannova, New trends in Quantum Structures, Kluwer Acad.Publ.,Dordrecht, Ister Science, Bratislava, (2000).

[11] A. Dvurecenskij, J. Rachunek, On Riecan and Bosbach states for bounded non-commutativeR`-Monoids, Math.Slovaca 56(2006), to appear.

[12] A. Dvurecenskij, J. Rachunek, Probabilistic averaging in bounded non-commutative R`-monoids,Semigroup Forum 72(2006), 190-206.

[13] G. Georgescu, Bosbach states on fuzzy structures, Soft Computing 8(2004), 217-230.

[14] G.Georgescu, A.Iorgulescu, Pseudo MV-algebras, Multiple Valued Logic, 6(2001), 95-135.

[15] G. Georgescu, A. Popescu, Noncommutative fuzzy structures and pairs of weak negations, FuzzySets and Systems, 143(2004), 129-155.

[16] G. Gratzer, Lattice theory, W.H.Freeman and Company, San Francisco, (1979).

[17] A. Iorgulescu, Classes of pseudo BCK-algebras - Part I, Journal of Multiple-Valued Logic andSoft Computing, 12(2006), 71-130.

[18] A. Iorgulescu, Classes of pseudo BCK-algebras - PartII, Journal of Multiple-Valued Logic andSoft Computing, 12(2006).

[19] A. Iorgulescu, Classes of BCK algebras - Part III, Preprint IMAR (2004).

[20] P. Jipsen, C. Tsinakis, A survey of residuated lattices, In:Ordered Algebraic Struc-tures,(J.Martinez,ed) Kluwe Acad.Publ., Dordrecht (2002), 19-56.

[21] F. Kopka, F. Chovanec, D-posets, Mathematica Slovaca, 44(1994), 21-34.

[22] H. Ono, Substructural logics and residuated lattices - an introduction, 50 Years of Studia Logica,Trends in Logic, Kluwer Academic Publishers, 21, 193-228, (2003).

[23] B. Riecan, On the probability on BL-agebras, Acta Math.Nitra, 4(2000), 3-13.

Department of Mathematics, Polytechnical University of Bucharest, Splaiul Independentei313, Bucharest, RomaniaE-mail address: lavinia [email protected]

CIUNGU188

Computing VaR and AVaR

of Skewed-T Distribution

Steftcho DokovFinAnalytica Inc., Seattle, USA


Stoyan V. StoyanovChief Financial Researcher,

FinAnalytica Inc., Seattle, USA and

Department of Econometrics and Statistics,

University of Karlsruhe, D-76128 Karlsruhe, Germany


Svetlozar T. Rachev∗


University of Karlsruhe, D-76128 Karlsruhe, Germany and

Department of Statistics and Applied Probability,

University of California Santa Barbara, CA 93106, USA


December 11, 2007

Abstract

We consider the skewed-T distribution defined as a normal mixture withinverse gamma distribution. Analytical formulas for its value-at-risk, VaRquantile, and average value-at-risk, AVaR conditional mean are derived.High-accuracy approximations are developed and numerically tested.

Keywords: skewed-T distribution, asymmetric, value-at-risk, VaR, AVaR

1 Introduction

The skewed-T distribution is a popular choice for modeling financial time seriesof asset returns. The VaR quantile and the average VaR quantile, i.e. AVaR, of

∗Rachev gratefully acknowledges research support by grants from Division of Mathematical,Life and Physical Sciences, College of Letters and Science, University of California, SantaBarbara, the Deutschen Forschungsgemeinschaft and the Deutscher Akademischer AustauschDienst.

1


2 Computing VaR and AVaR of Skewed-T Distribution

those returns are usually estimated from a large sample of observations. If suchlarge sample is not available, as in a case when only short history of returns ispresent, then we need a reliable way for assessing the magnitude of the VaR andAVaR risk measures. Analytical formulas might help in this case and let thor-ough analysis been performed on the risk measures by varying the distributionparameters and assessing different risk levels.

We note that asymmetric distributions can be defined in different ways fromtheir symmetric counterparts. For one such case of skewed-T distribution we seeanalytical formulas for VaR and AVaR derived in [1] (AVaR is denoted CVaRin [1]). In this case the skewed Student-t density function was proposed byHansen (1994) in [2]. The density is an extension of the conventional symmetricStudent-t distribution. The asymmetry is introduced by weighting differently,multiplying by different weights, the negative and the positive values of thesymmetric Student-t distribution.

We consider skewed-T distribution defined as a normal mixture with inversegamma distribution (e.g. see [3] for details). Such skewed-T random variable,X, is defined by

X = µ + γW + Z√

W,

where W ∼ Ig(ν/2, ν/2), i.e. W is inverse gamma random variable, Z is multi-variate normal random variable Z ∼ Nd(0,Σ), and W , Z are independent. Inthe paper we present the analytical formulas for α-level VaR(X) and AVaR(X)risk measures. We denote the d-dimensional distribution by X ∼ td(ν, µ,Σ, γ)where ν stands for degrees of freedom, ν ≥ 4, µ is a location parameter, and Σ isd-by-d covariance matrix. Finally, the sign of γ controls the distribution asym-metry: positive for skewed to the right, having fat right tail of asset returns,and vice-versa, negative for skewed to the left; except for the case ν = 5.

In the paper we specifically consider the cases for γ 6= 0, that is, the caseswith ”true” asymmetry in X. Nevertheless, we note that for small γ’s (oforder 10−6) our formulas numerically converge to the symmetric, conventionalStudent-t, AVaR formula

AV aRα(X) =Γ(

ν+12

)Γ(

ν2

) √ν

(ν − 1)α√

π

(1 +

(V aRα(X))2

ν

) 1−ν2

for ν > 1.The paper is organized as follows: in next Section 2 we state the classical

definitions of VaR and AVaR. Then we elaborate on the form of the skewed-Tpdf needed for development of the analytical formulas. An integral represen-tation of the Bessel function (involved in the pdf) is utilized. The analyticalformulas for VaR and AVaR are stated in two theorems. Their properties arediscussed and one related proposition is stated. The proofs of the theorems andthe proposition are presented in the appendix with thorough details. Two corol-laries state approximation versions of the theorems statements. The corollariesresults are very useful in numerical implementations. Theire proofs are in theappendix as well. In Section 3 we discuss some issues arising in numerical im-plementations of the developed formulas. The issues are resolved via the Besselfunction asymptotic forms and via locating the spike-like unimodal peak of thequadrature integrand function. The paper is concluded in Section 4.

DOKOV-STOYANOV-RACHEV190

Steftcho Dokov, et al. 3

2 VaR and AVaR for skewed-T distribution

The definition of VaR calls for a confidence level α ∈ (0, 1). Then the VaR ofportfolio return at confidence level α is defined as the smallest number x0 suchthat the probability that the loss X exceeds x0 is not larger than (1−α). Thatis, in general

V aRα(X) = inf x0 : P (X > x0) ≤ 1− α= inf x0 : FX(x0) ≥ α= F−1

X (α)

where FX(·) is the cdf (cumulative distribution function) of X, F−1X is the inverse

function of FX provided one exists, and the last equality holds for continuousdistributions. In probabilistic terms VaR is the α-quantile of the loss distribu-tion. If we consider the random variable X for modeling the asset returns then−X models the asset losses. Here we will not distinguish between the two, butwe will derive analytical formulas for both tails of the skewed-T distribution,that is, formulas for smaller and larger α values for the VaR quantile.

If we let the random variable X denote portfolio loss then the definition ofthe α level AVaR is given by the following conditional expectation

AV aR(X) = E [X|X ≥ V aRα(X)]

=1

1− α

∫X≥x0

xf(x)dx

that is, the α level average value-at-risk AV aR(X) is the average loss largerthan the α level quantile loss V aRα(X). Similarly to the VaR case, we willderive AVaR results for both distribution tails combined with the two casesfor the sign of the asymmetry parameter γ which controls the fatness of thedistribution tails.

For our analytical results we need the probability density function, f(x), ofthe skewed-T random variable X, which is given by

f(x) =21−(ν+d)/2

Γ(ν2 )(πν)d/2 |Σ|1/2

∗ exp((x− µ)′Σ−1γ)(1 + (x−µ)′Σ−1(x−µ)

ν

)(ν+d)/2

∗K(ν+d)/2

(√(ν + (x− µ)′Σ−1(x− µ))γ′Σ−1γ

)(√

(ν + (x− µ)′Σ−1(x− µ))γ′Σ−1γ)−(ν+d)/2

where Kλ(·) is the modified Bessel function of the third kind with index λ (alsoknow as modified Hankel function or Macdonald function). For details see, forexample, [4] and [5]. The skewed-T distribution is in the class of generalizedhyperbolic distributions. When we model asset returns with this distributionthen the resulting portfolio return can be seen as a random variable which is alinear combination of skewed-T returns. Because of a property of the generalizedhyperbolic distributions such linear combination has a generalized hyperbolicdistribution as well (see Corollary 2.2.4 in [8]).

VAR AND AVAR... 191


For assessing the VaR and AVaR of a single asset we consider its return asunivariate, d = 1, skewed-T random variable. The corresponding univariateprobability density (pdf) function is

f(x) =νλγ2λ

Γ(ν2 )√

πν22λ

∫ ∞

0

t−λ−1e−t− νγ2

4t eγx− (γx)2

4t dt

where, for convenience, we set µ = 0, Σ ≡ σ2 = 1, and λ = (ν + 1)/2. That is,we consider X normalized by the standard transformation

X − µ

σ=

γ

σW + N(0, 1)

√W.

where N(0, 1) stands for a random variable from the standard normal distri-bution. For numerical implementations we note that the skewness controllingparameter γ of the normalized skewed-T random variable X−µ

σ actually becomesγσ when the above normalized pdf f(x) is utilized.

In the last form of the pdf, f(x), we applied the following integral represen-tation of the Bessel function (with y =

√(ν + x2)γ2)

Kλ(y) =12

(y

2

)λ∫ ∞

0

t−λ−1e−t− y2

4t dt.

Among other places this representation can be seen in [6]. We note that theskewed-T distribution is also popular under the name asymmetric Laplace dis-tribution, for example, in statistical applications in medical research: see [7]where the inverse gamma random variable W is replaced with a special case ofits reciprocal values.

2.1 VaR formula for skewed-T distribution

In this section we develop a method for computing V aR(X), i.e., the 1 − αquantile of a skewed-T random variable X. We look for a formula and/ornumerical procedure yielding x0 = VaRα which is such that

1− α =∫ ∞

x0

f(x)dx

where f(x) is the univariate density which is also normalized with the notationswe introduced.

For a given skewed-T random variable we know the sign of the skewnesscontrolling parameter γ yielding heavier right or left distribution tail. Similarly,we have to know the sign of x0, the α level V aR(X), in order to distinguishbetween the two distribution tails (that is, we have to know whether we lookfor the VaR quantile for a ”smaller” or for a ”larger” α value). We so, first,compute the above integral on the interval [0,∞], and we set

1− α0 =∫ ∞

0

f(x)dx.

Hence, if the given VaR level α is such that α < α0 then we look for negativex0, otherwise, we look for positive x0. This naturally leads to two cases in



the x0 = VaRα calculation depending on whether the specified VaR level α isgreater than or less than the α0 value we set. These two cases will have tobe combined with the other two cases coming from the sign of the skewnesscontrolling parameter γ. This naturally leads to total of four cases which wedescribe and study in this section and in the next section with respect to theVaR and AVaR formulas.

The formula for computing α0 comes as a corollary of the following theorem.

Theorem 1. The VaR formula for skewed-T random variable, that is, the valueof x0 = VaRα is coming as the unique zero of the following equation (providedγ > 0)

g(x0) = −α +2C√

π

γ

∫ ∞

0

t−(ν+2)/2e−νγ2

4t Φ(

γx0√2t−√

2t

)dt = 0.

For negative skewness the value of x0 = VaRα is coming as the unique zeroof the next equation (i.e., provided γ < 0)

g(x0) = 1− α +2C√

π

γ

∫ ∞

0

t−(ν+2)/2e−νγ2

4t Φ(

γx0√2t−√

2t

)dt = 0,

In both cases the zero, x0 of g, is sought on the interval (−∞, 0] provided α < α0

or, on the interval [0,+∞) provided α > α0.

Proof. For the case γ > 0 see the Appendix.

In the theorem statement the Φ(·) stands for the standard normal cdf, theconstant

C =νλγ2λ

√πνΓ(ν

2 )22λ

depends on the skewed-T distribution parameters: degrees of freedom ν, λ =(ν + 1)/2, and γ is the skewness controlling parameter.

We note that g(·) is an increasing function of x0 from (−α) to 1 − α as x0

ranges from minus to plus infinity. Hence, the unique zero of g can be found byany numerical routine, for example, by one like a bi-sectional search.

The integrals on infinite intervals in the theorem have to be computed nu-merically. Also the search for the zero x0 of g has to be performed on infiniteintervals. We so derive approximations of the theorem statement where the infi-nite intervals are replaced with finite intervals. We prove that the accumulatednumerical error is bounded by 10−9 when the infinite intervals are replaced byfinite intervals.

Corollary 1. The integral on infinite interval in Theorem 1 can be replacedwith the following integral on a finite interval∫ t0(x0)

0

t−(ν+2)/2e−νγ2

4t Φ(

γx0√2t−√

2t

)dt

where

t0(x) =

(3√

2 +√

18 + 2γx

2

)2

VAR AND AVAR... 193


The approximation error R(t0(x0)) brought in

g(x0) = −α +∫ t0(x0)

0

t−(ν+2)/2e−νγ2

4t Φ(

γx0√2t−√

2t

)dt + R(t0(x0))

for γ > 0, and for γ < 0 as well, is less than 10−9.Furthermore, the search for the zero x0 of g is performed on the following

intervals

[−9/γ, 0] if α < α0, γ > 0,

[0,+∞) if α > α0, γ > 0,

(−∞, 0] if α < α0, γ < 0,

[0,−9/γ] if α > α0, γ < 0,

Proof. For the case γ > 0 see the Appendix.

Getting rid of the error term R(t0(x0)) still preserves the increasing natureof g(x0). Hence, the search for the unique zero x0 is fine with the approximationwe propose in Corollary 1. The value of α0 which specifies the sign of x0 comesas corollary from the above approximation after substituting the later with zero.

Corollary 2. The formula for α0 is

α0 =2C√

π

γ

∫ 18

0

t−(ν+2)/2e−νγ2

4t Φ(−√

2t)dt, if γ > 0,

α0 = 1 +2C√

π

γ

∫ 18

0

t−(ν+2)/2e−νγ2

4t Φ(−√

2t)dt, if γ < 0,

Proof. By substitution x0 = 0.

The proofs of the theorem and the first corollary in the appendix are pre-sented for the case γ > 0. The formulas for g(x0) when γ < 0 are derived fromtheir γ > 0 counterparts by substituting X with −X, γ with −γ, and the VaRlevel α of X with 1− α which is the VaR level α for −X.

In both cases, for positive and negative skewness, when the zero x0 of g(·)has to be sought on infinite interval

x0 ∈ (−∞, 0] provided α < α0 and γ < 0,x0 ∈ [0,+∞) provided α > α0 and γ > 0,

we perform some additional analysis which let us do the zero search on a fi-nite interval. Note that these are the cases of the heavy tail in the skewed-Tdistribution. For the case x0 ∈ [0,+∞) let us assume for example that weare looking for Value-at-Risk, x0 = VaRα, at confidence level α less than, say,99.99%. Hence, 1 − α ≥ 0.0001, or in general 1 − α ≥ ε where the ε is a smallpositive number which we can specify in advance. That is, if we choose, forexample, ε = 0.0002 then we will be able to do the search for the zero x0 of g ona finite interval but, the user specified VaR confidence level must not be greaterthan 99.98%. Thus, for a chosen small positive number ε, the −/+ infinity inthe above two intervals can be replaced with −/+ ”big” number M dependingon ε.



Proposition 1. The infinite intervals for the search of the unique α level x0 =VaRα can be replaced by finite intervals such that the −/+ infinity in Corollary1 is replaced with −/ + M given by

M =2d + 3

√8d

∓γ

where

d =νγ2

4

[εΓ(

ν + 22

)]−2/ν

and ε is an arbitrary small positive number specified in advance.

Proof. See the Appendix

Here ν and γ are the skewed-T distribution parameters, and Γ(·) is theGamma function. Technically, in the case x0 ∈ [0,+∞), we have that g(0) =−α + α0 < 0, while g(M) ≥ 1 − α − ε > 0 and g(+∞) = 1 − α > 0. So, theinterval [0,+∞) for the zero x0 is replaced with [0,M ] provided that ε is chosensuch that ε < 1− α. Note, M depends on ε.

Thorough proof for the M formula in the first case, x0 ∈ (−∞, 0], is pre-sented in the Appendix.

2.2 AVaR formula for skewed-T distribution

The AVaR (average VaR, also known as conditional value-at-risk CVaR, orETL, i.e. expected tail loss) is defined as the expectation of a distribution tail.As we already discussed it with respect to the VaR formula, for the skewed-Tdistribution we distinguish four cases depending on whether we are interested incomputing the expectation of the left or the right distribution tail and, on thesign of the skewness controlling parameter γ. As it is also discussed, the interestin computing the conditional expectation of the left or the right distribution tailmight depend on whether the distribution is utilized for modeling asset returnsor asset losses. We so study all possible four cases for the conditional tailexpectations.

Case 1: x0 ≥ 0, γ > 0,

AV aR1 = E[X|X > x0] =1

P (X > x0)

∫ ∞

x0

xf(x)dx

Case 2: x0 ≤ 0, γ > 0,

AV aR2 = E[X|X < x0] =1

P (X < x0)

∫ x0

−∞xf(x)dx

Case 3: x0 ≥ 0, γ < 0,

AV aR3 = E[X|X > x0] =1

P (X > x0)

∫ ∞

x0

xf(x)dx

Case 4: x0 ≤ 0, γ < 0,

AV aR4 = E[X|X < x0] =1

P (X < x0)

∫ x0

−∞xf(x)dx

VAR AND AVAR... 195


The alpha level AVaR depends on x0 which stands for the alpha level VaR,that is, x0 = VaRα is the 1−α quantile of the distribution. The x0 is such that

1− α =∫ ∞

x0

f(x)dx

The VaRα is studied in the previous section.Along with the notations we have so far, like C and t0(x), here we introduce

two additional notations

β =√

νγ2 + (γx0)2 and h0 =γx0√

2t−√

2t

and, an expression which repeatedly appears in the final AVaR formulas

KI =

[Kλ−1(β)

(2β

)λ−1

eγx0 −√

π

∫ ∞

0

t−λ+1/2e−νγ2

4t Φ(h0)dt

]

which is the difference between the modified Bessel function of the third kindwith index (λ− 1) and a finite integral. Moreover,

KI = Kλ−1(β)(

2β

)λ−1

eγx0 , if 9 + γx0 < 0

that is, the integral vanishes (has a value of order 10−9) for γ and x0 such thattheir values satisfy the last inequality.

Theorem 2. The AVaR formula for skewed-T random variable, in each one ofthe four cases we describe, is

AV aR1 =γν

(1− α)(ν − 2)+

4C

(1− α)γ2KI

AV aR2 =−4C

αγ2KI

AV aR3 =4C

(1− α)γ2KI

AV aR4 =γν

α(ν − 2)+−4C

αγ2KI

Proof. See the Appendix for all details about the proof of the AV aR1 for-mula. The AV aR2 formula is derived as a complement for AV aR1 to the mean,E(X) = (γν)/(ν − 2), of the normalized skewed-T random variable X. Theother two formulas, AV aR3 and AV aR4, come from the first two after the sub-stitution of X with −X, the substitution of the skewness parameter γ with −γ,and the substitution of α = αX with (1− α) = α(−X).

We note that the AVaR calculation for the skewed-T distribution requiresnumerical integration on infinite interval. Similarly to the previous section weapproximate it with integration on a finite interval. Next we state this result.



Corollary 3. The integral on infinite interval in the KI expression in Theorem2 can be replaced with the following integral on a finite interval∫ t0(x0)

0

t−λ+1/2e−νγ2

4t Φ(h0)dt

The approximation error is lees than

νγ

(1− α)(ν − 2)10−9.

Proof. See the Appendix for all details about the proof of the AV aR1 approxi-mation.

Theorems 1 and 2 provide the analytical formulas for VaR and AVaR of theskewed-T distribution defined as a normal mixture with inverse Gamma dis-tribution. The approximation versions of the formulas presented in Corollaries1 and 3 allow to carry on numerical tests of those formulas. The results arepresented in the next section.

3 About issues in numerical implementations

In this section we present our findings when the analytical VaR and AVaRformulas are tested in numerical experiments. We, first, generate ten millionvariates from the skewed-T distribution. This let us achieve good sample esti-mates for VaR and AVaR quantities at different confidence levels α. We varythe confidence level from one to ninety nine percent. Then we compare the esti-mated quantities with their analytical counterparts. The percent relative error100 ∗ |(sampleEstimate−analyticalResult)/sampleEstimate| stays below onepercent. But, when we begin vary the distribution parameters significantly thenwe discover that some additional theoretical work must be done.

Two important issues deserve our attention. First, the accuracy of the nu-merical integration, and second, the asymptotic behavior of the modified Besselfunction of the third kind. The later is well studied in the scientific literature.We so only point out the way we utilize this asymptotic behavior for achievinghigh numerical accuracy.

About the first issue, the accuracy in the numerical integration, after exten-sive numerical testing we note that the integrand function, that is, the functiondefining the quadratures has a spike-like shape which can easily destroy theaccuracy in the numerical integration. Finding the exact point at which thespike appears requires additional amount of numerical calculations or theoret-ical analysis. So, we basically determine that the spike must appear near thepoint γ2/2. This fact is stated and proved in the next proposition.

Proposition 2. The integrand function in the quadratures in Theorems 1 and2, respectively, in Corollaries 1 and 3

u(t) = t−θe−νγ2

4t Φ(

γx0√2t−√

2t

)where θ = (ν + 2)/2, or θ = λ − 1/2 = ν/2, has maximum for t > 0 in aneighborhood of t = γ2/2.

VAR AND AVAR... 197


Proof. We use the notation h0 introduced in the previous section. The firstderivative of u(t) becomes

u′(t) = t−θ−2e−νγ2

4t

[(νγ2

4− θt

)Φ(h0)− (γx0 + 2t)

√t

2√

2Φ′(h0)

]The first expression multiplying the cdf Φ(h0) changes sign at t1 = νγ2/(4θ)being positive for t < t1 and negative for t > t1. The second expression multi-plying the pdf Φ′(h0) changes sign at t2 = −γx0/2 only if γ and x0 = V aRα

are such that γx0 < 0 (note, we are interested in partitioning the quadraturesfor t ∈ [0, t0(x0)], that is for positive t’s, on two intervals which union coversthe quadratures interval).

If γx0 > 0 then the second expression in u′(t) does not change sign. Hence,the derivative has a unique zero, respectively the integrand function in thequadratures has a unique maximum, for t ”near” t1, that is, for t < t1 becausethe second expression in u′(t) takes on negative values which shift the t1 zeroto the left. We note that in both cases for θ, i.e. for (ν + 2)/2 and ν/2, the t1value is close to the γ2/2 approximation which we suggest in this proposition,and which is numerically tested.

If γx0 < 0 then the derivative u′(t) can change sign (eventually more thanonce) for t between t1 ≈ γ2/2 and t2 = −γx0/2. Otherwise, for t < min(t1, t2)the derivative u′(t) takes on positive values, and for t > max(t1, t2) the deriva-tive has negative values. Therefore there is at least one maximum for the in-tegrand function u(t) between t1 and t2, that is, for t belonging to the interval[min(t1, t2), max(t1, t2)].

We combine the conclusions from the above two cases, and we approximatethe location of the maximum with t1 ≈ γ2/2 for all cases.

Based on the proposition and the numerical experiments, we conclude thatevery case of numerical integration in the formulas for VaR and AVaR must bepartitioned in two quadratures at γ2/2. This is especially important for nearsymmetric skewed-T distributions, that is when γ goes to zero. This completelyresolves the numerical issues arising in the analytical VaR calculation. However,in the analytical AVaR calculation we have to deal with the Bessel functionevaluation involved in our formula.

We use the following two asymptotic properties of the modified Bessel func-tion of the third kind

Kλ(x) −→√

π

2xe−x

for large x >> |λ2 − 1/4|, and

Kλ(x) → Γ(λ)2

(2x

)λ

for small positive x <<√

λ + 1. The first asymptotic is especially importantfor large γ. For such γ’s we have that β goes to plus infinity and β has the orderof z0. In this case we see that the Beseel part

Kλ−1(β)(

2β

)λ−1

eγx0



in the KI expression tends to zero which significantly helps in the numerical cal-culations because, otherwise, one might have to deal with an undefined numericexpression looking like zero times infinity. On the other side when evaluating theabove expression for small γ then the second asymptotic significantly improvesthe accuracy in the analytical AVaR calculation.

4 Conclusions

We develop analytical formulas for computing the α level VaR and AVaR for arandom variable X having the asymmetric Student-t distribution, also knownas the skewed-T distribution. It is defined as a normal mixture with inverseGamma distribution. The distribution pdf and the AVaR formula require theBessel function calculation.

The analytical formulas are tested and they appear to be accurate for dif-ferent confidence levels α. The parameters of the skewed-T distribution X ∼td(ν, µ, σ, γ) are also varied. For the normalized random variable (X−µ)/σ it isimportant to vary the degrees of freedom parameter ν and the ratio γ/σ whenthe achieved numerical accuracy is tested against very large sample estimates.We tested the analytical formulas for ν in the range [4, 400] where the resultsfor ν > 300 are approximated very well with ν = 300. The test range for theratio γ/σ is [10−4, 102]. In all tested cases the analytical formulas yield resultswhich differ from ten million sample size estimates by less than one percent.

The derived analytical formulas are very useful if only a small number ofsample observations is available. In this case we find that the sample estimatestend to underestimate the heavy tail extreme quantiles. On the other side, forlarge sample size we find that the sample estimates tend to overestimate the light(short) tail extreme quantiles. Further qualitative and quantitative analysiscould be performed on the derived formulas in a future research concerningtheir applications in modeling financial time series.

Appendix

The V aR formula for γ > 0(Proof of Theorem 1)

The result for the α level Value-at-Risk, x0 = V aRα of a skewed-T randomvariable, derived in the main text, says that x0 is the unique zero of the followingequation

g(x0) = −α +2C√

π

γ

∫ ∞

0

t−(ν+2)/2e−νγ2

4t Φ(

γx0√2t−√

2t

)dt = 0, if γ > 0,

where the zero, x0 of g, is sought in the interval [−9/γ, 0] provided α < α0,or in the interval [0,+∞] provided α > α0. Furthermore, g(·) is an increasingfunction of x0 which actually implies the uniqueness of the zero.

Here we prove this fact. We begin with the VaR definition

1− α =∫ ∞

x0

f(x)dx

VAR AND AVAR... 199


which is rewritten in

g(x0) = 1− α−∫ ∞

x0

f(x)dx.

Next, we simplify the integral∫ ∞

x0

f(x)dx =∫ ∞

x0

C

∫ ∞

0


4t eγx− (γx)2

4t dtdx

=C

γ

∫ ∞

0


4t

∫ ∞

x0

eγx− (γx)2

4t d(γx)dt.

The change of variables z = γx, z0 = γx0, in the inner integral yields∫ ∞

x0

f(x)dx =C

γ

∫ ∞

0


4t

∫ ∞

z0

ez− z24t dzdt

=2C√

π

γ

∫ ∞

0

t−λ−1/2e−νγ2

4t

[1− Φ

(γx0√

2t−√

2t

)]dt.

We now use the following identity

2C√

π

γ

∫ ∞

0

t−λ−1/2e−νγ2

4t dt = 1

(note, the above right-hand side, 1, must be replaced with -1 if γ < 0). Wesubstitute back λ = (ν + 1)/2, and we obtain∫ ∞

x0

f(x)dx = 1− 2C√

π

γ

∫ ∞

0

t−(ν+2)/2e−νγ2

4t Φ(

γx0√2t−√

2t

)dt

which completes the proof.

The approximation V aR formula for γ > 0(Proof of Corollary 1)

We note that for t such thatγx0√

2t−√

2t < −6.

the tail of the last integral (in the above proof) becomes infinitely small. Thisis so because the standard normal cdf satisfies Φ

(γx0√

2t−√

2t)

< Φ(−6) < 10−9,and the integral from the remaining integrand function is bounded by one overthe tail of the integral (this statement is rigorously proved below). The tail ison the interval t ∈ [t0(x0),+∞] where t0(x0) satisfies the above inequality. Thelast inequality is equivalent to a quadratic inequality with respect to

√t

2t− 6√

2t− γx0 > 0

which is true for

t > t0(x0) ≡

(3√

2 +√

18 + 2γx0

2

)2



(which coincides with the notation t0(x) in the main text). We partition thelast integral on [0,∞] as integral on [0, t0(x0)] plus integral on [t0(x0),∞]. Forthe latter we argue above that it is infinitely small (basically, it is the error termR(t0(x0))). This is so because we have the following inequality

R(t0(x0)) ≡2C√

π

γ

∫ ∞

t0(x0)

t−(ν+2)/2e−νγ2

4t Φ(

γx0√2t−√

2t

)dt

≤ 2C√

π

γΦ(−6)

∫ ∞

t0(x0)

t−(ν+2)/2e−νγ2

4t dt

The last expression simplifies to, and is bounded by Φ(−6) < 10−9

Φ(−6)γ(

νγ2

4t0(x0); ν

2

)Γ(

ν2

) ≤ Φ(−6)

where γ(

νγ2

4t0(x0); ν

2

)is the lower incomplete Gamma function. Therefore

∫ ∞

x0

f(x)dx ≈ 1− 2C√

π

γ

∫ t0(x0)

0

t−(ν+2)/2e−νγ2

4t Φ(

γx0√2t−√

2t

)dt

which completes the proof of the approximation formula for g(x0). Rigorouslywe have

g(x0) = −α +2C√

π

γ

∫ t0(x0)

0

t−(ν+2)/2e−νγ2

4t Φ(

γx0√2t−√

2t

)dt + R(t0(x0))

and,

g(x0) ≥ −α +2C√

π

γ

∫ t0(x0)

0

t−(ν+2)/2e−νγ2

4t Φ(

γx0√2t−√

2t

)dt

where we chose the lower bound of g(x0) as its approximation.Furthermore, (recall, we are in the case γ > 0), we note that if α > α0,

i.e., x0 > 0, then the quadratic inequality is true for t > t0(x0). And, ifα < α0, i.e., x0 < 0, then the quadratic inequality is true for t > t0(0) provided18 + 2γx0 > 0. Otherwise, if 18 + 2γx0 < 0 then the quadratic inequality istrue for any t. Hence, the search for negative x0 should be performed onlyfor x0 > −9/γ. Finally, the approximation for g(x0) increases in x0 becausefrom the expression we have for g(x0) we see that Φ(·) increases in x0 and, theintegral in the g(x0) expression is an integral from nonnegative function on theinterval [0, t0(x0)] where the right end t0(x0) of the interval also increases withrespect to x0.

The AV aR1 formula for skewed-T(Proof of Theorem 2)

The proof for the following formula

AV aR1 =γν

(1− α)(ν − 2)+

4C

(1− α)γ2KI

VAR AND AVAR... 201


is presented below.

AV aR1 = E[X|X > x0]

=1

P (X > x0)

∫ ∞

x0

xf(x)dx

=1

(1− α)

∫ ∞

x0

xνλγ2λ

Γ(ν2 )√

πν22λ

∫ ∞

0


4t eγx− (γx)2

4t dtdx

=C

(1− α)γ2

∫ ∞

0


4t

∫ ∞

x0

(γx)eγx− (γx)2

4t d(γx)dt

We set z = γx, z0 = γx0, and simplify the inner integral, first, with integra-tion by parts ∫ ∞

z0

zez− z24t dz = −2t

∫ ∞

z0

ezd(e−z24t

)= 2t

(ez0−

z20

4t +∫ ∞

z0

ez− z24t dz

)and, second, with change of variables technique h = z√

2t−√

2t, h0 = z0√2t−√

2t

leading to closed-form result with standard normal cdf

∫ ∞

z0

zez− z24t dz = 2t

(ez0−

z20

4t + et√

2t

∫ ∞

h0

e−h22 dh

)= 2t

(ez0−

z20

4t + et√

2t√

2π(1− Φ(h0)))

Hence, the conditional expectation of the right tail of the skewed-T distributionbecomes

AV aR1 =2C

(1− α)γ2

∫ ∞

0

t−λe−t− νγ2

4t

(ez0−

z20

4t + et√

2t√

2π(1− Φ(h0)))

dt

=2C

(1− α)γ2(E1 + E2 − E3),

where

E1 = ez0

∫ ∞

0

t−λe−t− νγ2

4t −z20

4t dt

E2 = 2√

π

∫ ∞

0

t−λ+1/2e−νγ2

4t dt

E3 = 2√

π

∫ ∞

0

t−λ+1/2e−νγ2

4t Φ(h0)dt

We combine the integral representation of the Bessel function with one earliernotation β =

√νγ2 + (γx0)2 (recall z0 = γx0) which simplifies E1 to a closed-

form result



E1 = ez0

∫ ∞

0

t−λe−t− β2

4t dt

= 2ez0

(2β

)λ−1

Kλ−1(β)

We note that the result for E1 (along with the 2C(1−α)γ2 multiplier) corre-

sponds to the Bessel function term in KI

4C

(1− α)γ2Kλ−1(β)

(2β

)λ−1

eγx0

in the formula for AV aR1.Next, we show that E2 simplifies to the first term in the AV aR1 formula.

We apply the Gamma function definition Γ(u) =∫∞0

vu−1e−vdv for argument

u = λ− 3/2 with change of variables v = νγ2

4t . Hence

E2 = 2√

π

∫ ∞

0

t−λ+1/2e−νγ2

4t dt

= 2√

π

(4

νγ2

)λ−3/2

Γ(λ− 3/2)

=√

π22λ−2Γ(λ− 1/2)

ν(λ−3/2)γ2λ−3(λ− 3/2)

=γ3ν

2(ν − 2)C

where the second to the last equality comes from the following property Γ(u) =(u − 1)Γ(u − 1). For the last equality we use the definition of the notation Cand recall that λ = (ν +1)/2. We note that adjusting the last result for E2 withthe multiplier 2C

(1−α)γ2 yields the first term in the AV aR1 formula.Finally, we deal with the E3 expression

E3

2√

π=∫ ∞

0

t−λ+1/2e−νγ2

4t Φ(h0)dt

which is an integral on infinite interval. We note that the result for E3 (alongwith the 2C

(1−α)γ2 multiplier) corresponds to the third term in the AV aR1 for-mula, that is, to the integral expression in KI. This completes the proof of thetheorem.

The approximation formula for AV aR1

(Proof of Corollary 3)

We keep work on the E3 expression from the end of the previous proof. Wewill approximate the integral with one on a finite interval. We note that theargument, h0, of the standard normal cdf tends to minus infinity as t goes toplus infinity (recall h0(t) = z0√

2t−√

2t). Hence, for sufficiently large t we have

VAR AND AVAR... 203


Φ(h0) going to zero. Technically, sufficiently large t can be determined (as inthe proof of Theorem 1) from Φ(−6) < 10−9. The inequality h0 < −6 is truefor t > t0(x0) where the last notation t0(x0) is already specified in the maintext and in the proof of Theorem 1. We so obtain

E3

2√

π=∫ t0(x0)

0

t−λ+1/2e−νγ2

4t Φ(h0)dt

+∫ ∞

t0(x0)

t−λ+1/2e−νγ2

4t Φ(h0)dt

The first integral corresponds to the third term in the AV aR1 formula, that is,to the integral expression in KI (”here we complete the proof of this formula”).The second integral ”vanishes” because it simplifies similarly to the proof ofTheorem 1. We have that h0(t) ≤ h0(t0(x0)) = −6 for t ≥ t0(x0), hence theintegral is bounded by∫ ∞

t0(x0)

t−λ+1/2e−νγ2

4t Φ(h0)dt ≤ Φ(−6) ∗∫ ∞

t0(x0)

t−λ+1/2e−νγ2

4t dt

As it is already done earlier in this proof, the last expression must be adjustedby the multipliers 2C

(1−α)γ2 and 2√

π. Hence, after simplification (similar to theone in the proof of Theorem 1) we obtain that the second integral is boundedby

Φ(−6) ∗ νγ

(1− α)(ν − 2)∗

γ(

νγ2

4t0(x0); ν

2 − 1)

Γ(

ν2 − 1

)where the last multiplier is bounded by 1, the second multiplier is a constantfor given distribution parameters and AVaR level α, and the first multiplier isbounded by 10−9. Therefore, we assume that the error made in the transitionfrom integral on infinite interval to integral on finite interval is infinitely small.

The ”big” M formula(Proof of Proposition 1)

The proof for the following formula is presented below

M =2d + 3

√8d

(−γ)

where

d =νγ2

4

[εΓ(

ν + 22

)]−2/ν

in the case α < α0 and γ < 0.In this case we look for the unique zero x0 of

g(x0) = 1− α +2C√

π

γ

∫ t0(x0)

0

t−(ν+2)/2e−νγ2

4t Φ(

γx0√2t−√

2t

)dt = 0,

in (−∞, 0], and we prove that for a given small positive number ε such that α > εthe zero search can be performed on [−M, 0] rather than on (−∞, 0]. We note



that we are in the very left end of the heavy left tail of the skewed-T distribution,that is, the small ε represent the probability for being in (−∞,−M ]. In themain text we said that g(·) is an increasing function from g(−∞) = −α < 0 tog(0) = −α + α0 > 0. So, we now prove that

g(−M) ≤ −α + ε,

that is, g(−M) < 0 provided that ε is chosen such that (ε < α) it is less thanthe user specified confidence level α for VaRα.

We utilize the notations

F (t, k) = t−(ν+2)/2e−νγ2

4te−k2/2

√2π

h0(t) =γx0√

2t−√

2t

and the definition of the standard normal cdf Φ(·) to rewrite the integral in theg(x0) expression as follows∫ t0(x0)

0

∫ h0(t)

−∞F (t, k)dkdt.

Next, we change the order of integration, and present the integral as a sum oftwo integrals ∫ −6

−∞

∫ t0(x0)

0

F (t, k)dtdk +∫ ∞

−6

∫ h−10 (k)

0

F (t, k)dtdk

where t = h−10 (k) is the inverse function of k = h0(t). The two functions are

defined on the intervals k ∈ [−6,+∞) and t ∈ [0, t0(x0)] respectively (note, theinverse function exists because h0(·) is a monotone function). The first doubleintegral is bounded by

Γ(

νγ2

4t0(x0); ν

2

)Γ(

ν2

) Φ(−6)

which is a product of (positive) multiplier less than one (i.e. the ratio of upperincomplete Gamma function and the Gamma function) and, technically, thezero value Φ(−6) ≈ 10−9. We so focus only on the second double integral inthe last expression for g(x0). Hence,

g(x0) = 1− α− 1Γ(

ν2

) ∫ ∞

−6

[Γ(ν

2

)− γ

(νγ2

4h−10 (k)

;ν

2

)]e−k2/2

√2π

dk

= −α +1

Γ(

ν2

) ∫ ∞

−6

γ

(νγ2

4h−10 (k)

;ν

2

)e−k2/2

√2π

dk

where γ(·; ν

2

)is the lower incomplete Gamma function, i.e., Γ

(ν2

)= γ

(·; ν

2

)+

Γ(·; ν

2

). Note, for the pair of double integrals in g(x0) we utilized the identity

2C√

π

γ

(νγ2

4

)−ν/2

Γ(ν

2

)= −1

VAR AND AVAR... 205


for γ < 0 (otherwise, the above expression simplifies to plus one for γ > 0).Next we have

g(x0) = −α +1

Γ(

ν2

) ∫ 6

−6

γ(·; ν

2

) e−k2/2

√2π

dk +∫ ∞

6

γ(·; ν

2

)Γ(

ν2

) e−k2/2

√2π

dk

where the second integral is technically equal zero for reasons described earlierwith respect to the first double integral in g(x0). Therefore, we focus on bound-ing the integral on finite interval [−6, 6]. We note that the lower incompletegamma function in this integral, γ

(·; ν

2

), is an increasing function of k in its

argument νγ2

4h−10 (k)

. Hence,

g(x0) ≤ −α +1

Γ(

ν2

) ∫ 6

−6

γ(·; ν

2

)∣∣∣k=6

e−k2/2

√2π

dk

= −α +γ(

νγ2

4h−10 (6)

; ν2

)Γ(

ν2

)Next we bound the lower incomplete gamma function γ(u; a) by

γ(u; a) =∫ u

0

ta−1e−tdt ≤ ua/a.

Hence,

g(x0) ≤ −α +

(νγ2

4h−10 (6)

)ν/2

ν2Γ(

ν2

) ≡ l(x0)

where

h−10 (6) =

(√

k2 + 4γx0 − k)2

8

∣∣∣∣∣k=6

=(√

9 + γx0 − 3)2

2Here we may observe that

h−10 (6)

∣∣x0=−M

= d

where d is specified in the M definition.Finally, some tedious algebraic manipulations show that l(−M) = −α + ε

which completes the proof for the ”big” M formula.Remark: If we decide that we do not have to ignore (we do not want to

ignore) the two infinite small terms (integrals) in the proof (involving a constanttimes Φ(−6) where the absolute value of that constant is less than one) thenwe may choose ε such that ε < α− 2 ∗ 10−9 for some specified in advance VaRlevel α.

References

[1] C. Martins-Filho and F. Yao, Estimation of value-at-risk and expectedshortfall based on nonlinear models of return dynamics and Extreme ValueTheory, ”Studies in Nonlinear Dynamics and Econometrics,” Volume 10,Issue 2, (2006).



[2] B. Hansen, Autoregressive conditional density estimation. ”InternationalEconomic Review,” 35, 705-729, (1994).

[3] S. Demarta and A. J. McNeil, The t Copula and Related Copulas, ”Inter-national Statistical Review,” 73(1), 111-129, (2005).

[4] B. R. Fabijonas, D. W. Lozier and J. M. Rappoport, Algorithms and codesfor the Macdonald function: recent progress and comparisons, ”Journal ofComputational and Applied Mathematics,” Volume 161, Issue 1, 179 - 192,(December 2003).

[5] E. M. Ferreira and J. Sesma, Zeros of the Macdonald function of com-plex order, ”ArXiv Mathematics e-prints,” math/0607471, Provided by theSmithsonian/NASA Astrophysics Data System, (July 2006).

[6] S. Kotz, T. J. Kozubowski and K. Podgorski, An asymmetric multivariateLaplace distribution, ”Technical Report 367,” Working Paper, page 22,(January 2003). http://wolfweb.unr.edu/homepage/tkozubow/0 alm.pdf

[7] E. Purdom and S. P. Holmes, Error Distribution for Gene Expression Data,”Statistical Applications in Genetics and Molecular Biology,” Volume 4,Issue 1, Article 16, (2005).

[8] W. Hu, Calibration of multivariate generalized hyperbolic distributions us-ing the EM algorithm, with applications in risk management, portfoliooptimization and portfolio credit risk, ”PhD Thesis, The Florida StateUniversity,” College of Arts and Science, (2005).

VAR AND AVAR... 207

208

Ball Closure Property In Fuzzy Metric SpacesHakan Efe

Department of Mathematics, Faculty of Science and Arts, GaziUniversity, Teknikokullar, 06500 Ankara, Turkey

[email protected]

The aim of this paper to introduce the ball closure property in fuzzymetric spaces. We also give some theorems on relationship betweenball closure property and density and also compactness.Key Words. Fuzzy metric space, closed ball, compact subset.M.S.C. (2000). 54A40, 54E35.

1. INTRODUCTIONSince the concept of fuzzy set was introduced by Zadeh [10] in 1965,

many authors have introduced the concept of fuzzy metric space indifferent ways [1-5]. George and Veeramani [3] modified the conceptof fuzzy metric space introduced by Kramosil and Michalek [6] anddefined a Hausdorff topology on this fuzzy metric space. Malik andToma [7] introduced the ball closure property in metric spaces.In this paper we study on ball closure property in fuzzy metric spaces

in the sense of George and Veeramani. We also give some relationsbetween ball closure property and density and also compactness.

2. PRELIMINARIES

Definition 1 ([8]). A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] iscontinuous t-norm if ∗ is satisfying the following conditions:

(i) ∗ is commutative and associative;(ii) ∗ is continuous;(iii) a ∗ 1 = a for all a ∈ [0, 1];(iv) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, and a, b, c, d ∈ [0, 1].

Definition 2 ([3]). A triple (X,M, ∗) is said to be a fuzzy metric spaceif X is an arbitrary set, ∗ is a continuous t-norm and M is a fuzzy seton X2× (0,∞) satisfying the following conditions: for all x, y, z ∈ X,s, t > 0,

(i) M(x, y, t) > 0,(ii) M(x, y, t) = 1 if and only if x = y,(iii) M(x, y, t) =M(y, x, t),(iv) M(x, y, t) ∗M(y, z, s) ≤M(x, z, t+ s),(v) M(x, y, .) : (0,∞)→ [0, 1] is continuous.

Remark 1. In fuzzy metric space X, M(x, y, .) is non-decreasing forall x, y ∈ X.


Example 1. Let (X, d) be a metric space. Denote a ∗ b = ab for alla, b ∈ [0, 1] and let Md be a fuzzy set on X2× (0,∞) defined as follows:

Md(x, y, t) =ktn

ktn +md(x, y)

for all k,m, n ∈ R+. Then (X,Md, ∗) is a fuzzy metric space.Remark 2. Note the above example holds even with the t-norm a∗ b =mina, b and henceM is a fuzzy metric with respect to any continuoust-norm. In the above example by taking k = m = n = 1, we get

Md(x, y, t) =t

t+ d(x, y)

We call this fuzzy metric induced by a metric d the standard fuzzymetric.

Definition 3 ([3]). Let (X,M, ∗) be a fuzzy metric space and let r ∈(0, 1), t > 0 and x ∈ X. The set

B(x, r, t) = y ∈ X :M(x, y, t) > 1− ris called the open ball with center x and radius r with respect to t.

Theorem 1 ([3]). Every open ball B(x, r, t) is an open set.

Definition 4 ([3]). Let (X,M, ∗) be a fuzzy metric space and let r ∈(0, 1), t > 0 and x ∈ X. The set

B[x, r, t] = y ∈ X :M(x, y, t) ≥ 1− ris called the closed ball with center x and radius r with respect to t.

Theorem 2 ([3]). Every closed ball B[x, r, t] is a closed set.

Remark 3. Let (X,M, ∗) be a fuzzy metric space. Define τ = A ⊂X : for each x ∈ A, there exist t > 0, r ∈ (0, 1) such that B(x, r, t) ⊂A. Then τ is a topology on X.

Remark 4.

(i) Since B(x, 1n, 1n) : n = 1, 2, ... is a local base at x, the topology

τ is first countable.(ii) Every fuzzy metric space is Hausdorff.(iii) Let (X,M, ∗) be an fuzzy metric space and τ be the topology

on X induced by the fuzzy metric. Then for a sequence (xn)nin X, xn → x if and only if M(xn, x, t)→ 1 as n→∞ and forall t > 0.

(iv) In a fuzzymetric space every compact set is closed and bounded.

Definition 5 ([9]). Let (X,M, ∗) be a fuzzy metric space, x ∈ X andA ⊂ X. The degree of closeness x to A is defined by

M(A, x, t) = supM(y, x, t) : y ∈ Afor all t > 0.

EFE210

3. MAIN RESULTSLet (X,M, ∗) be a fuzzy metric space and A be a subset of X. We

denote

B(A, r, t) = x ∈ X :M(A, x, t) > 1− rB[A, r, t] = x ∈ X :M(A, x, t) ≥ 1− r

for all r ∈ (0, 1) and for all t > 0.Definition 6. Let (X,M, ∗) be a fuzzy metric space. A subset A of Xis said to have ball closure property iff for all r ∈ (0, 1) and t > 0 wehave B(A, r, t) = B[A, r, t].

If each subset A of X has this property we say that the fuzzy metricspace (X,M, ∗) has the ball closure property.Example 2. Let X = [0, 1] ∪ [2, 3]. Define a ∗ b = ab and

M(x, y, t) =t

t+ |x− y|for all t > 0. Let A = 1 and r = 1− t

t+1. Then

B

∙1, 1− t

t+ 1, t

¸=

½x ∈ X :M(1, x, t) ≥ t

t+ 1

¾= [0, 1] ∪ 2.

On the other hand,

B

µ1, 1− t

t+ 1, t

¶=

½x ∈ X :M(1, x, t) > t

t+ 1

¾= (0, 1) = [0, 1].

Hence (X,M, ∗) does not have the ball closure property.Lemma 1. Let (X,M, ∗) be a fuzzy metric space and A be a subset ofX. Then, B(A, r, t) = B(A, r, t) for all r ∈ (0, 1) and t > 0.Proof. Since A ⊂ A, then B(A, r, t) ⊂ B(A, r, t) for all r ∈ (0, 1) andt > 0.Let x be an arbitrary element of B

¡A, r, t

¢. ThenM

¡A, x, t

¢> 1−r

and there exists a point y ∈ A such that M (y, x, t) > 1 − r for r ∈(0, 1) and t > 0. Therefore, there exists a t0, 0 < t0 < t such thatM (y, x, t0) > 1 − r. Let 1 − r0 = M (y, x, t0). Since 1 − r0 > 1 − r,there exists s ∈ (0, 1) such that 1− r0 > 1− s > 1− r. Now for givenr0 and s such that 1− r0 > 1− s we can find r1, 0 < r1 < 1 such that(1− r0) ∗ (1− r1) ≥ 1− s. Now consider the ball B(y, r1, t− t0) ⊂ A.Let z ∈ B(y, r1, t− t0), then M(y, z, t− t0) > 1− r1. Therefore,

M(x, z, t) ≥ M (y, x, t0) ∗M(y, z, t− t0)

≥ (1− r0) ∗ (1− r1) ≥ 1− s

> 1− r

BALL CLOSURE PROPERTY 211

which implies x ∈ B(A, r, t). Then, B(A, r, t) ⊂ B(A, r, t) which com-pletes the proof. ¤Theorem 3. Let (X,M, ∗) be a fuzzy metric space. Then the followingare equivalent:

(i) (X,M, ∗) has the ball closure property.(ii) Every closed subset of X has the ball closure property.(iii) Every countable subset of X has the ball closure property.(iv) For every point x ∈ X, r ∈ (0, 1), t > 0 and every sequence

(xn)n in X, with M(xn, x, t) −→ 1− r, there exists a sequence(yn)n in B((xn)n, r, t) such that yn −→ x.

Proof. (i)⇐⇒(ii) is clear from Lemma 1. (i)=⇒(iii) is obvious. We willprove (iii)=⇒(iv)=⇒(i).(iii)=⇒(iv): Let the assumptions of (iv) be satisfied.Then M((xn)n, x, t) ≥ 1 − r for r ∈ (0, 1) and t > 0, hence x ∈

B[(xn)n, r, t]. Therefore x ∈ B((xn)n, r, t).(iv)=⇒(i): Let A be an arbitrary subset of X and let x ∈ B[A, r, t]

for r ∈ (0, 1) and t > 0. If M(A, x, t) > 1 − r, then x ∈ B(A, r, t) ⊂B(A, r, t) for all r ∈ (0, 1) and t > 0. If M(A, x, t) = 1− r, then thereexists a sequence (xn)n ⊂ A such that M(xn, x, t) −→ 1 − r. From(iv) we have x ∈ B(A, r, t). This implies B[A, r, t] ⊂ B(A, r, t). TheB(A, r, t) ⊂ B[A, r, t] is a consequence of the continuity of the functionx −→M(A, x, t). ¤Theorem 4. Let (X,M, ∗) be a fuzzy metric space satisfying the ballclosure property and let U be an open set in X. Then (U,M, ∗) has theball closure property.

Proof. It is sufficient to show that (U,M, ∗) satisfies the condition (iv)of Theorem 3. Let x ∈ X, r ∈ (0, 1) and (xn)n satisfy the assumptionsof (iv). Then there exists a sequence (yn)n in B((xn)n, r, t) such thatyn −→ x, x ∈ U . Since U is open, we can suppose without loss ofgenerality that (yn)n ∈ B((xn)n, r, t) ∩ U such that yn −→ x whichcompletes the proof. ¤Theorem 5. Let (X,M, ∗) be a fuzzy metric space satisfying the ballclosure property and D be a dense subset of X. Then (D,M, ∗) has theball closure property.

Proof. We again make use of the assertion (iv) of Theorem 3. Letx ∈ D, r ∈ (0, 1), t > 0 and (xn)n ⊂ D satisfy M(xn, x, t) −→ 1 − r.Then there exists a sequence (yn)n in B((xn)n, r, t) such that yn −→ x.Since the D is dense in X there exists a sequence (zn)n in D such thatM(yn, zn, t) > 1− 1

nfor all n ∈ N and t > 0. Obviously zn −→ x. ¤

Theorem 6. Let (X,M, ∗) be a fuzzy metric space. Then the followingare equivalent.

EFE212

(i) Every compact subset of X has the ball closure property.(ii) Every singleton has the ball closure property.(iii) For each x, y ∈ X, r ∈ (0, 1) and t > 0, there exists z ∈ X such

that

M(x, z, t) > 1− r and M(y, z, t) > M(x, y, t).

Proof. (i)=⇒(ii) is clear. Let us prove (ii)=⇒(iii)=⇒(i).(ii)=⇒(iii): From (ii) we have

B(y, 1−M(x, y, t), t) = B[y, 1−M(x, y, t), t]

and x ∈ B[y, 1−M(x, y, t), t]. Since x ∈ B(y, 1−M(x, y, t), t), then

B(x, r, t) ∩B(y, 1−M(x, y, t), t) 6= ∅for all r ∈ (0, 1) and t > 0 which is precisely the assertion of (iii).(iii)=⇒(i): For every compact set A ⊂ X, since M(A, x, t) is a

continuous function, then B(A, r, t) ⊂ B[A, r, t]. We will show theopposite inclusion. Let x be an arbitrary element of B[A, r, t]. IfM(A, x, t) > 1 − r, then x ∈ B(A, r, t) ⊂ B(A, r, t), for all r ∈ (0, 1)and t > 0. If M(A, x, t) = 1− r, then there exists a sequence (xn)n inA such that M(xn, x, t) −→ 1 − r, for all r ∈ (0, 1) and t > 0. SinceA is compact, we can choose a subsequence (xnk)k of (xn)n such thatxnk −→ x0, x0 ∈ A. Making use of continuity the metric, we haveM(x, x0, t) = 1− r, for r ∈ (0, 1) and t > 0. Hence x ∈ B[x0, r, t] i.e.,M(x, x0, t) ≥ 1 − r for r ∈ (0, 1) and t > 0. On the other hand from(iii) there exists z ∈ X such that

M(x, z, t) > 1− r and M(x0, z, t) > M(x, x0, t).

Then, M(x, z, t) > 1− r and M(x0, z, t) > 1− r which implies that

B(x, r, t) ∩B(x0, r, t) 6= ∅

for all r ∈ (0, 1) and t > 0. Hence x ∈ B(x0, r, t) which showsB[x0, r, t] ⊂ B(x0, r, t). SinceB(x0, r, t) ⊂ B(A, r, t), then the inclusionB[A, r, t] ⊂ B(A, r, t) is proved. ¤Theorem 7. Let (X,M, ∗) be a complete fuzzy metric space. Then thefollowing are equivalent.

(i) Every totally bounded subset ofX has the ball closure property.(ii) Every compact subset of X has the ball closure property.(iii) Every singleton in X has the ball closure property.(iv) For each x, y ∈ X, r ∈ (0, 1) and t > 0, there exists z ∈ X such

that

M(x, z, t) > 1− r and M(y, z, t) > M(x, y, t).

Proof. Applying Lemma 1 and Theorem 6 we obtain the equivalenceof the assertions (i) -(iv) as an immediate consequence of the fact that

BALL CLOSURE PROPERTY 213

in a complete fuzzy metric space the closure of each totally boundedset is a compact set. ¤Theorem 8. Let (X1,M1, ∗) and (X2,M2, ∗) are fuzzy metric spacesin which every compact subset has the ball closure property. Then alsothe product space (X1×X2,M, ∗) has this property where M is definedby the

M((x1, x2), (y1, y2), t) = minM1(x1, y1, t),M2(x2, y2, t).Proof. We use the (iii) of Theorem 6. For r ∈ (0, 1) and t > 0, thereexists (z1, z2) ∈ X1 ×X2 such that

M1(x1, z1, t) > 1− r, M2(x2, z2, t) > 1− r

and consequently

M((x1, x2), (z1, z2), t) > 1− r.

Since

M1(y1, z1, t) > M1(x1, y1, t) and M2(y2, z2, t) > M2(x2, y2, t),

therefore

M((y1, y2), (z1, z2), t) > M((x1, x2), (y1, y2), t)

which completes the proof. ¤Acknowledgement. The author would like to thank the referees

for their help in the improvement of this paper.

References

[1] M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl., 69 (1979),205− 230.

[2] A. George and P. Veeramani, On some results in fuzzy metric spaces, FuzzySets and Systems, 64 (1994), 395− 399.

[3] A. George and P. Veeramani, On some results of analysis for fuzzy metricspaces, Fuzzy Sets and Systems, 90 (1997), 365− 368.

[4] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27(1988), 385− 389.

[5] V. Gregori and S. Romaguera, Some Properties of fuzzy metric spaces, FuzzySets and Systems, 115 (2000), 485− 489.

[6] O. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Ky-bernetica, 11 (1975), 326− 334.

[7] J. Malik and V. Toma, Open and Closed Neighbourhoods of Sets in MetricSpaces, Acta Mathematica Universitatis Comenianae, XLV I−XLV II (1985),137− 143.

[8] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math., 10(1960),314− 334.

[9] P. Veeramani, Best approximation in fuzzy metric spaces, J. Fuzzy Math., 9(2001), 75− 80 .

[10] L. A. Zadeh, Fuzzy sets, Inform and Control, 8 (1965), 338− 353.

EFE214

The Canonical Coherent States Associated With Quotients of the

Affine Weyl-Heisenberg Group∗

Stephan Dahlke Dirk A. LorenzPhilipps-Universitat Marburg Technion - Israel Institut of Technology

[email protected] [email protected]

Peter Maass, Chen Sagiv Gerd TeschkeUniversitat Bremen ZIB Berlin

pmaass, [email protected] [email protected]

Abstract

This paper is concerned with uncertainty principles in the context of the affine-Weyl-Heisenberg groupin one and two dimensions. As the representation of this group fails to be square integrable, we explorevarious admissible sections of this group that appear in the context of coorbit and α-modulation spaces,and calculate the resulting uncertainty principles as well as its minimizers with respect to these sections. Inspecial cases, we also consider the relationships of minimzing states with admissibility conditions.

Keywords: Affine Weyl-Heisenberg group, representations via quotients, uncertainty principles, minimizingstates, coorbit spaces.AMS Subject Classification: 22D10, 47B25

1 Introduction

In this paper, we are concerned with the construction of uncertainty principles and the computation of associatedcanonical coherent states. This topic of uncertainty based localization is of interest in abstract as well as inapplied areas and has been therefore extensively considered in an abundant number of papers and books, see,e.g., [1, 2, 4, 5, 7, 8, 9, 12, 15, 16, 19]; even when no uncertainty principle applies, attempts have been madeto derive some sort of optimally localized coherent states, see [14]. The motivation of the present paper isthe interest in very special coherent states of the affine Weyl-Heisenberg group which was introduced by B.Torresani, see [21, 22]. He was the first who constructed irreducible representation and discussed conceptsof square-integrability. Square integrability is always an important aspect in practical applications of grouprepresentations since it ensures the invertibility of the associated integral transform. It is shown in [21] thatthe representations of the affine Weyl-Heisenberg group unfortunately fail to be square integrable, i.e., theredoes not exist any admissible vector. Possible remedies are to factor out a suitable closed subgroup and towork with quotients and/or to weaken the concept of admissibility. These techniques for treating the affineWeyl-Heisenberg group can be nicely used in their full glory for the construction of mixed smoothness spacesand Banach frames for them. These mixed smoothness spaces are specific coorbit spaces that lie in betweenBesov and modulation spaces and coincide with the α-modulation spaces, see [6]. Once this abstract theory isdeveloped, it is quite natural to ask for uncertainty principle and suitable minimizing states for these specialgroup representations modulo quotients. Suitable means here admissible, although we want to emphazise thatuncertainty relations and minimizing functions are clearly of interest by themselves. To these topics we dedicatethis paper.

The most famous uncertainty relation is associated with the Short Time Fourier Transform (STFT). TheSTFT or so-called Gabor transform, see [10], is obtained by applying the action of the Weyl-Heisenberg group toa suitable window function and taking the inner product with the function under consideration. As a well-knownresult, the Gaussian function minimizes the associated Heisenberg uncertainty relation and therefore gives rise

∗This work has been supported through the European Union’s Human Potential Programme, under contract HPRN–CT–2002–00285 (HASSIP), and through DFG, Grants Da 360/4–3, MA 1657/15–1, TE 354/1–2.


to canonical coherent states of the Weyl-Heisenberg group. More recent studies considered the uncertaintyprinciples which are related to the affine group in one dimension and the similitude group as well as the affinegroup in two dimensions [1, 5, 16]. For the one dimensional affine group it was possible to find an analyticalsolution of the form:

ψ(x) = c(x− η)−12−iηµ2+iµ1 , (1)

where c is some constant, η is purely imaginary and µ1, µ2 ∈ R. However, for the multi-dimensional case, itwas not possible to find solutions which simultaneously minimize all appearing uncertainties with respect toall the parameters involved, and therefore solutions that accounted for various subgroups were employed, see,e.g., [6, 16, 19, 20, 21, 22].

In this paper, we follow similar lines, and we are especially interested in the case of the affine Weyl-Heisenberggroup and in the associated mixed smoothness spaces that lie in between Besov and modulation spaces. However,we are not further interested in the spaces themselves but rather in the special representations that pop upin the construction process. For them we consider in detail the computation of canonical coherent states andtheir admissibility properties. The issue of admissibility is discussed in greater detail in Sections 3.2 and 4.2.In Section 3.2, we consider an example where switching to quasi-coherent states allows an easy verificaton ofadmissibility, and it turns out that minimal uncertainty and admissibility fit together quite nicely. Moreover, inSection 4.2 we are able to give explicit and sufficient conditions for admissibility but which can, unfortunately,not be fulfilled by the derived minimizing states. As a suitable alternative, the structure of the coherent states(having exponential decay in Fourier space) suggests the application of a smooth cut off operator that providesus then with admissible and ‘near’ minimizing coherent states.

The paper is organized as follows: In Section 2, we discuss some basic results and summarize related work.We calculate the minimizers for the one dimensional affine Weyl-Heisenberg group and address the issue ofadmissibility in Sections 3. Finally, in Section 4, we continue by analyzing the two-dimensional affine Weyl-Heisenberg group and explore some possible subgroups for obtaining valid minimizers.

2 Background and Related Work

A general theorem which is well-known in quantum mechanics and harmonic analysis [8] relates an uncertaintyprinciple to any two self-adjoint operators and provides a mechanism for deriving a minimizing function for theuncertainty relation. Before repeating this well-known result on uncertainties, let us fix some notation. Let A,B be two self-adjoint operators. Their commutator is defined by

[A,B] := AB −BA,

the expectation of A with respect to some state ψ ∈ dom(A) by

µ(A) := µA := 〈Aψ,ψ〉

and, finally, the variance of A with respect to some state ψ ∈ dom(A) by

∆Aψ := µ((A− µ(A))2).

Theorem 1 Let two self-adjoint infinitesimal generators A and B of a unitary representation of some Lie groupbe given, then for all ψ ∈ dom([A,B]) ∩ dom(A) ∩ dom(B) they obey the uncertainty relation:

∆Aψ∆Bψ ≥ 14|〈[A,B]ψ,ψ〉|2. (2)

A state ψ is said to have minimal uncertainty if the above inequality turns into an equality. This happens iffthere exists an η ∈ iR such that

(A− µA)ψ = η(B − µB)ψ. (3)

For an extensive discussion on the latter theorem concerning the definiteness of ψ, we refer the reader to [4, 15].The Weyl-Heisenberg group as well as the affine group are both related to well-known transforms in signal

processing: the STFT and the wavelet transform, respectively. Both can be derived from square integrablerepresentations of these groups. The windowed Fourier transform is related to the Weyl-Heisenberg group and

DAHLKE ET AL216

the wavelet transform is related to the affine group. The linearized operation of the group at the identityelement can be described by the infinitesimal generators of the related Lie algebra. If the group representationis unitary, then the infinitesimal generators can transformed to be self-adjoint operators. Thus, the generaluncertainty theorem stated above provides a tool for obtaining uncertainty principles using these infinitesimalgenerators. In the case of the Weyl-Heisenberg group, the canonical functions that minimize the correspondinguncertainty relation are Gaussian functions.

The canonical functions that minimize the uncertainty relations for the affine group in one dimension andfor the similitude group in two dimensions, were the subject of previous studies [1, 5]. In these studies, it wasshown that there is no non-trivial canonical function that minimizes the uncertainty relation associated with thesimilitude group of R2, SIM(2). Thus, there is no non-zero solution for the set of differential equations obtainedfor these group generators. Rather than using the original generators of the SIM(2) group, a different set ofoperators was used in [5] that includes elements of the enveloping algebra, i.e., polynomials in the generatorsof the algebra, in order to obtain the 2D isotropic Mexican hat as a minimizer. Further results were achievedin [1] where a symmetry in the set of commutators was obtained for the SIM(2) group and a possible minimizerin the frequency domain for some fixed direction was computed. This solution is a real valued wavelet which isconfined to some convex cone in the positive half-plane of the frequency space with an exponential decay insidethe cone.

The extension of these studies to the affine group in two dimensions resulted in two possible solutions [16].The first accounted for the overall scaling and rotation and utilizes the results of [1, 5]. The second solutionwas obtained by exploiting a symmetry in the group of commutators which led to

ψ(x, y) = (η + x)−12−iµ11+iηµbx eiµbyy. (4)

It belongs to L2 with respect to the variable x if we select Re(η)µbx≥ 0, but it belongs not to L2 in terms of

the variable y, although it is periodic.The affine Weyl-Heisenberg (AWH) group has already been addressed in this context in the early 90′s.

The paper [21] considered wavelets associated with representations of the AWH group. It shows that thecanonical representation of the AWH group is not square integrable, but can be regularized with some densityfunction. This work was later extended to N -dimensional AWH wavelets [22]. In [17] a scaling was introducedin the Heisenberg group with an intertwining operator. More recently, [19] proposed a mechanism to constructgeneralized uncertainty principles and their minimizing wavelets in anisotropic Sobolev spaces. A new set ofuncertainty principles was introduced in this paper by weakening the two operator relations and by introducinga multi-dimensional operator setting. Recently, a study [6] has considered generalizations of the coorbit spacetheory based on group representations modulo quotients. This is based on applying the general theory to theAWH group and obtaining families of smoothness spaces that can be identified with the α-modulation spaces.

3 The 1D Affine Weyl-Heisenberg Group

The affine Weyl-Heisenberg group is generated by time translations b ∈ R, frequency translations ω ∈ R, spatialdilations a ∈ R+, and a toral component φ ∈ R, and is equipped with the group law

(b, ω, a, φ) (b′, ω′, a′, φ′) = (b+ ab′, ω + ω′/a, aa′, φ+ φ′ + ωb′a).

The AWH group can be viewed as the extension of the affine group, incorporating frequency translations or,alternatively, as the extension of the Weyl-Heisenberg group incorporating dilations. The Stone-von-Neumannrepresentation of GAWH on L2(R) is given by:

[U(b, ω, a, φ)ψ](x) = a−12 eiω(x−b)eiφψ(

x− b

a). (5)

This representation, however, fails to be square integrable, see [20]. The AWH group raises a special interest asit “contains” both, the affine group as well as the Weyl-Heisenberg group: If we consider cases where a = 1, weare in the Weyl-Heisenberg framework, and if we consider cases where ω = 0 we are in the affine framework.Two independent studies have regarded these attributes, and suggested a specific section of the AWH [6, 20],where the scale is represented as a function of the frequency. It was proven that this section is admissible.

AFFINE WEYL-HEISENBERG GROUP 217

In addition, this paper introduces a mechanism that starts at the Weyl-Heisenberg case and allows a smoothtransition towards the affine case.

In what follows, we consider a section where the scale is a function of the frequency and calculate theappropriate minimizing functions with respect to the uncertainty principle related to it. Then, we study asection where the frequency is regarded as a function of the scale, consider its admissibility and calculate theappropriate minimizers.

3.1 The Sections Where the Scale is a Function of the Frequency

As suggested in [21, 22] and considered in the context of α-modulation spaces, see [6], we treat the affineWeyl-Heisenberg group and its representation by factoring out a closed subgroup and work with the quotients.

We will consider GAWH/H withH := (0, 0, a, φ) ∈ GAWH

and Borel sections that do not depend on b, namely:

σ(b, ω) = (b, ω, β(ω), 0).

Further, it was shown in [6] that the specific section

β(ω) = ηα(ω)−1 = (1 + |ω|)−α

is admissible.Next, we consider the effect of varying the value of α ∈ [0, 1]. If α = 0 then we obtain:

β(ω) = η0(ω)−1 = (1 + |ω|)0 = 1.

Thus, there are practically no dilations and we obtain Gabor analysis. For α→ 1 we obtain:

β(ω) = ηα(ω)−1 = (1 + |ω|)−α |α|→1−→ 11 + |ω|

.

Thus, the frequency translations and modulations are inversely proportional which is close to wavelet analysis.The intermediate case for which α = 1

2 is known as the Fourier-Bros-Iagolnitzer transform, see [3].The representation for the quotient as a function of α is then given by

[U(b, ω, η−1α (ω))ψ](x) = (1 + |ω|)α

2 eiω(x−b)ψ ((1 + |ω|)α(x− b)) .

As can be seen, this representation is not C1 for ω = 0. Nevertheless, when calculating the infinitesimalgenerators, we may take the one-sided derivatives.

Lemma 2 The infinitesimal operators Tb, Tω associated with the one dimensional GAWH are given by

(Tbψ)(x) = −i ∂∂xψ(x), and (Tωψ)(x) = (i

α

2− x)ψ(x) + iαx

∂

∂xψ(x). (6)

The state ψ which is the minimizer of the associated uncertainty is of the form

ψ(x) = e−ix

α (αλx+ 1)−12−

iµωα +

iµbαλ + i

α2λ , (7)

where λ ∈ iR.

Proof; Taking the (one-sided) derivatives with respect to ω and b and evaluating them at b = 0, ω = 0 leads to

∂

∂bU(b, ω, ηα(ω), 0)ψ|b=0,ω=0(x) = − ∂

∂xψ(x),

∂

∂ωU(b, ω, ηα(ω), 0)ψ|b=0,ω=0(x) = (

α

2+ix)ψ(x)+αx

∂

∂xψ(x).

By its construction, these operators are only skew symmetric and not self-adjoint, but a multiplication with theimaginary unit i assures self-adjointness. We therefore consider operators Tb = i ∂

∂bU and Tω = i ∂∂ωU . This

proves (6).

DAHLKE ET AL218

The commutator between these two operators is non-zero. This implies that we cannot exactly measurethe mean values of the spatial frequency and the position simultaneously. By means of Theorem 1, we maycalculate those states that minimize the corresponding uncertainty principle. Indeed, eq. (3) provides us withthe differential equation

−i ∂∂xψ(x)− µbψ(x) = λ

((iα

2− x)ψ(x) + iαx

∂

∂xψ(x)− µωψ(x)

),

i.e.,∂

∂xψ(x) = iψ(x)

(−λx+ iλα

2 − λµω + µb

αλx+ 1

). (8)

Now (8) can be solved by separation of variables which leads us to (7).

Let λ = iγ, γ ∈ R. If γ < 0 and if γ > 0 respectively, then the solution is contained in L2 if µb > − 1α and if

µb < − 1α respectively. In Figure 1 we have plotted ψ, which is the minimizer for this section of the AWH group

in 1D.

3.2 The Sections where the Frequency is a Function of the Scale

In the previous section we have considered the section

β(ω) = ηα(ω)−1 = (1 + |ω|)−α.

We note that it is not possible to obtain the affine framework using this approach. Hence, let us explore theinverse relationship

ω = ζα(a) = a−1α − 1 ,

which determines the frequency as a function of the scale a.Let us denote κ = 1

α , and restrict the discussion to values of κ ranging between 0 and 1 (corresponding tovalues of α ranging between 1 and ∞). Thus we obtain ω = ζ(a) = a−κ− 1. If κ is selected to be zero, we thenobtain no frequency modulation as then ω = 0, and thus we are in the affine case. If κ is selected to be one, weagain observe reciprocal relations between scale and frequency of the form: |ω| = a−1 − 1, which is the same asa = 1

1+|ω| , and corresponds to the case of Gabor-like wavelets.This concept has again an interpretation in the group theoretical setting. Once more we are working with

the affine Weyl-Heisenberg group GAWH , but this time we consider the subgroup

H := (0, ω, 1, φ) ∈ GAWH

and the associated homogeneous space X = GAWH/H. In order to make this setting well-defined, first of all it isnecessary to establish square-integrability, see, e.g., [1] for a detailed discussion. In general, let a quasi-invariantmeasure µ on X and a section σ be given. Then a unitary representation U of G on a Hilbert space H is calledsquare-integrable modulo (H; σ) if there exists a function ψ ∈ H such that the self-adjoint operator Aσ : H → H(depending on σ and ψ) weakly defined by

Aσf :=∫X

〈f, U(σ(h))ψ〉HU(σ(x))ψdµ(h) (9)

is bounded and has a bounded inverse. The function ψ is then called admissible. If Aσ is a multiple of theidentity then we are in the strictly admissible case. We consider the case of the affine group in the followinglemma.

Lemma 3 Let ψ ∈ L2(R). The operator Aσ in (9) for the affine Weyl-Heisenberg group, i.e.

Aσf(x) =∫ ∫

〈f, ψa,b〉ψa,b(x) dbda

a(10)


α = 0.1 α = 0.5

−0.5

0

0.5

1

−50 −25 0 25 50

−0.5

0

0.5

1

−50 −25 0 25 50

α = 1 α = 5

−0.5

0

0.5

1

−50 −25 0 25 50

−0.5

0

0.5

1

−50 −25 0 25 50

α = 10 α = 50

−0.5

0

0.5

1

−50 −25 0 25 50

−0.5

0

0.5

1

−50 −25 0 25 50

Figure 1: The minimizers of the AWH uncertainty for λ = 0.1i, µb = µω = 0 and different values of α.The real part is plotted solid, the imaginary part is dashed. Note the transition from a Gaussion for small α(Weyl-Heisenberg case) to the Cauchy wavelet for large α (affine case).

DAHLKE ET AL220

with the section σ(a, b) = (b, ζ(a), a, 0), i.e.,

ψa,b(x) =1√ae2πiζ(a)(x−b)ψ

(x− b

a

), (11)

can be written as a Fourier multiplier operator:

Aσf = mζ f (12)

with the symbol

mζ(γ) :=∫

R+

|ψ(a(γ − ζ(a)))|2da. (13)

Proof. We follow the approach of [13]. For the sake of this calculation we use the approximation

ATσ f(x) =∫ ∫

〈f, ψa,b〉ψa,b(x) χ[−T2 ,

T2 ](b) db

da

a. (14)

In order to compute ATσ f(γ), we first derive the Fourier transform of ψa,b:

ψa,b(γ) =1√a

∫ ∞

−∞e2πiζ(a)(x−b)e−2πiγxψ

(x− b

a

)dx. (15)

If we apply the change of variables y = x−ba we obtain

ψa,b(γ) =√ae−2πibγψ(a(γ − ζ(a))). (16)

With the help of Plancherel’s theorem we further obtain

〈f, ψa,b〉 = 〈f , ψa,b〉 =∫ √

af(ω)e2πibω ¯ψ(a(ω − ζ(a)))dω (17)

and thus

ATσ f(γ) =∫ ∫

f(ω) ¯ψ(a(ω − ζ(a)))ψ(a(γ − ζ(a)))

∫e−2πib(γ−ω)χ[−T

2 ,T2 ](b) dbdadω

=∫ψ(a(γ − ζ(a)))

∫f(ω) ¯

ψ(a(ω − ζ(a)))Tsin(π(γ − ω)T )π(γ − ω)T

dωda.

The term

Tsin(π(γ − ω)T )π(γ − ω)T

(18)

can be seen as an approximation of a δ-function when T approaches infinity. Thus, we obtain by standardarguments

Aσ(f)(γ) = f(γ)∫|ψ(a(γ − ζ(a)))|2da = f(γ)mζ(γ). (19)

In order to verify the boundedness of Aσ and its inverse one has to check

C1 ≤ mζ(γ) ≤ C2 (20)

almost everywhere for constants 0 < C1, C2 < ∞. Unfortunately, for the current particular choice of H and βcondition (20) cannot be verified. This can already be seen in the simple situation κ = 1, i. e. ζ(a) = 1

a − 1,then

mζ(γ) =∫ ∞

0

|ψ(a(γ − 1a

+ 1))|2da

=∫ ∞

0

|ψ(a(γ + 1)− 1))|2da

=1

|γ + 1|

∫ ∞

0

|ψ(x)|2dx |γ|→∞−→ 0.


Nevertheless, a possible remedy for the non-admissibility of this section is given in the weaker framework ofquasi-coherent states, see [2]. In this case, we might consider (19) with respect to some positive density ι(a, γ).For the special situation κ = 1 we might choose ι(a, γ) = 1/a, then we obtain

mζ(γ) =∫ ∞

0

|ψ(a(γ − 1a

+ 1))|2ι(a, γ) da =∫ ∞

0

|ψ(x− 1)|2

xdx . (21)

This leads to quasi-coherent states that have the standard properties of the covariant coherent states: overcom-pleteness, resolution of a positive operator Aσ and having a reproducing kernel.

We now turn to calculating the infinitesimal operators and the coherent states minimizing the correspondinguncertainty relation. The choice of the section ω = ζ(a) leads to the representation

[U(b, ζ(a), a, 0) ψ](x) =1√aei(a

−κ−1)(x−b)ψ

(x− b

a

)for which the two infinitesimal generators and the minimizing coherent state are derived in the following lemma.

Lemma 4 The infinitesimal operators with respect to the representation above are

Taψ(x) = (κx− i

2)ψ(x)− ixψx(x) and Tbψ(x) = −iψx(x).

The minimizing states are then given by

ψ(x) = (1− ρx)iµa− 12−i

µbρ −i

κρ e−iκx, (22)

where ρ = ir with r ∈ R. Then, the minimizing states belong to L2 if r < 0 and µb < −κ or if r > 0 andµb > −κ.

Proof: The proof can be performed by following the lines of the proof of Lemma 2. This time, the correspondingdifferential equation is given by

d

dxψ(x) =

(µb − iρ

2 + ρκx− ρµa)

i(ρx− 1)ψ(x), (23)

where ρ is purely imaginary. Eq. (23) can again be solved by separation of variables which leads to (22).

The next lemma verifies the admissibility (19) for the restriction to quasi-coherent states with ι(a, γ) = 1/a andκ = 1. Therefore, at least for κ = 1, minimal uncertainty and admissibility fit together quite nicely.

Lemma 5 Assume 0 < r ∈ R and µb + 1 > r/2 > 0. Then the minimizing elements (22) are admissiblequasi-coherent states, i.e. the integral (21) is positive and bounded.

Proof: For simplifying the notation, we define η := iµa − 1/2− µb/r − 1/r which yields

ψ(x) = (−ρ)η(x+ i/r)ηe−ix.

As long as r > 0 and <(η) < 0 the Fourier transform of ψ is given by

ψ(ω) =c(ω + 1)−η−1e−1(ω+1)/r for ω ≥ −1

0 elsewhere(24)

with c = (−ρ)η√

2π(−i)ηΓ(−η)i. Thus the admissibility condition becomes with η′ = <η

|c|2∫ ∞

0

ω−2(η′+1)e−2ω/r

ωdω = |c|2(r/2)−2(η′+2)Γ(−2(η′ + 1))

provided that µb + 1 > r/2.

DAHLKE ET AL222

Note that the case κ = 0 reproduces the classical pure wavelet conditions. For κ = 1 the current solutionreduces to

ψ(x) = (1− ρx)iµa− 12−i

µbρ −i

1ρ e−ix. (25)

It is interesting to compare this solution to the one obtained for the previously used section a = β(ω) for α = 1,which yields:

ψ(x) = (1 + λx)iµbλ −

12−iµω+ i

λ e−ix . (26)

The constraints for the two solutions to agree for α = 1 are derived as follows

λ = −ρ, µa = −µω .

In Figure 2 we have plotted the minimizing ψ, for this choice of a section for the AWH group in 1D.

4 The 2D Affine Weyl-Heisenberg Group

In this section we are interested in finding uncertainty minimizers for the two-dimensional affine Weyl-Heisenberggroup with a generating element (A,ω, b, φ), ω, b ∈ R2, A ∈ Gl(2,R), φ ∈ R and group law

(b, ω,A, φ) (b′, ω′, A′, φ′) = (b+Ab′, ω +A−1ω′, AA′, φ+ φ′ + ωTAb′).

The Stone-von-Neumann representation of the AWH group in two dimensions is given by:

[U(b, ω,A, φ)ψ](x, y) =1

|det(A)|ei(ωx(x−bx)+ωy(y−by)+φ)ψ (A(x− bx, y − by)) (27)

with the unimodular Haar measure1

|det(A)|2dbdωdm(A)dφ,

where dm(A) denotes a usual measure when parametrizing the matrix A. In our discussion we explore varioussubgroups of the full 2D AWH group, starting from the SIM(2) group, and moving to the full group structure.

The admissibility of the derived minimizing coherent states is in the cases discussed below either difficultto check or not confirmable. For instance in Section 4.2 we are able to give explicit and sufficient conditions(see Theorem 7 stating that a function which has a compactly supported Fourier transform is admissible) foradmissibility but which can, however, not be fulfilled by the derived minimizing coherent states. Nevertheless,the Fourier transform of the minimizing states have its support (up to shifting etc.) in the positive Euclidianhalf space in which they decay exponentially. Hence, chosing the support size large enough and applying asmooth cut off operator provides us with admissible and ‘near’ minimizing coherent states.

4.1 The 2D Similitude Weyl-Heisenberg Subgroup

We start our discussion by considering the similitude group, which only allows rotations and scalings A = A(a, θ).The general representation of the 2D similitude Weyl-Heisenberg subgroup is given by

[U(b, ω, (a, θ), 0)ψ](x, y) =1aei(ωx(x−bx)+ωy(y−by))ψ

(τθ

(x− bxa

,y − bya

)), (28)

where τθ =(

cos(θ) sin(θ)− sin(θ) cos(θ)

). This representation fails to be square integrable [22], therefore, we are faced

with the interesting question of selecting an appropriate section. To this end, let H = (0, 0, (a, 0), φ), considerGAWH \H and take the section

σ(b, ω, θ) = (b, ω, (Φ(ω), θ), 0).


κ = 0.02 κ = 0.1

−0.5

0

0.5

1

−50 −25 0 25 50

−0.5

0

0.5

1

−50 −25 0 25 50

κ = 0.2 κ = 1

−0.5

0

0.5

1

−50 −25 0 25 50

−0.5

0

0.5

1

−50 −25 0 25 50

κ = 2 κ = 10

−0.5

0

0.5

1

−50 −25 0 25 50

−0.5

0

0.5

1

−50 −25 0 25 50

Figure 2: The minimizers of the AWH uncertainty for ρ = 0.1i, µa = µb = 0 and different values of κ. Thereal part is plotted solid, the imaginary part is dashed. Note the transition from a Cauchy wavelet for small κ(affine case) to a Gaussian for large κ (Weyl-Heisenberg case).

DAHLKE ET AL224

We consider a coupling between the frequency ω and the scaling a by a = Φ(ω). More specifically, in thespirit of the α-modulation spaces framework, we assume that the function Φ depends only on the p-norm of thefrequency vector ω

Φ(ω) =1

(1 + ‖ω‖p)α. (29)

As before, we would like to obtain the infinitesimal generators of this group by calculating the appropriatederivatives of the representation of this group at the identity element. Depending on the choice of the sectionΦ(ω), we will obtain different infinitesimal operators from the partial derivatives with respect to ωk:

Tωk= (Tωk

+ Φ′ωk(0)Ta), (30)

for k ∈ x, y. Therefore, we have to estimate the derivative of Φ at ω = 0. Our particular choice of Φ yields

Φ′ωk(ω) = −α(1 + ‖ω‖p)−α−1 1

p‖ω‖1−p

p pωp−1k sign(ωk)

=−α

(1 + ‖ω‖p)α+1

[ωpk∑j |ωj |p

] p−1p

sign(ωk). (31)

Next, we have to evaluate this expression at ωk = 0 for all k. In contrast to the 1D situation, the resultinginfinitesimal operators and the corresponding commutation relations strongly depend on the choice of p. Westart by selecting the L1-norm, as this allows a straight forward calculation of infinitesimal generators.

4.1.1 AWH Minimizers Using the L1-Norm

In this case: a = Φ(ω) = 1(1+|ωx|+|ωy|)α , thus the representation becomes:

[U(b, ω,Φ(ω), θ, 0)ψ](x, y) = (1+|ωx|+|ωy|)αei((x−bx)ωx+(y−by)ωy)ψ ((1 + |ωx|+ |ωy|)ατθ(x− bx, y − by)) . (32)

As before, we would like to obtain the infinitesimal generators of this group by calculating the appropriatederivatives of the representation of this group at the identity element. Depending on the choice of the sectionΦ, we obtain the infinitesimal genrators with respect to ωk from the directional derivative of Φ at 0 in directionk ∈ x, y:

˜Tωk= (Tωk

+ Φωk(0)Ta). (33)

Our particular choice of Φ yieldsΦωk

(0) = −α. (34)

Then the self-adjoint infinitesimal operators are given by:

Tωxψ(x, y) = (iα− x)ψ(x, y) + iα(xψx(x, y) + yψy(x, y)),

Tωyψ(x, y) = (iα− y)ψ(x, y) + iα(xψx(x, y) + yψy(x, y)),

Tbxψ(x, y) = −iψx(x, y),

Tbyψ(x, y) = −iψy(x, y),

Tθψ(x, y) = i(yψx(x, y)− xψy(x, y)). (35)

Out of the ten commutation relations, three vanish, [Tωx, Tby

] = 0, [Tωy, Tbx

] = 0, [Tbx, Tby

] = 0, and we are leftwith seven partial differential equations.

1.

(iα−x)ψ(x, y)+iα(xψx(x, y)+yψy(x, y))−µωxψ(x, y) = λ1

((iα− y)ψ(x, y) + iα(xψx(x, y) + yψy(x, y))− µωy

ψ(x, y))

2.(iα− x)ψ(x, y) + iα(xψx(x, y) + yψy(x, y))− µωx

ψ(x, y) = λ2 (−iψx(x, y)− µbxψ(x, y))


3.

(iα− x)ψ(x, y) + iα(xψx(x, y) + yψy(x, y))− µωxψ(x, y) = λ3 (i(yψx(x, y)− xψy(x, y))− µθψ(x, y))

4.(iα− y)ψ(x, y) + iα(xψx(x, y) + yψy(x, y))− µωy

ψ(x, y) = λ4

(−iψy(x, y)− µby

ψ(x, y))

5.

(iα− y)ψ(x, y) + iα(xψx(x, y) + yψy(x, y))− µωyψ(x, y) = λ5 (i(yψx(x, y)− xψy(x, y))− µθψ(x, y))

6.i(yψx(x, y)− xψy(x, y))− µθψ(x, y) = λ6 (−iψx(x, y)− µbx

ψ(x, y))

7.i(yψx(x, y)− xψy(x, y))− µθψ(x, y) = λ7

(−iψy(x, y)− µby

ψ(x, y))

The only simultaneous solution to these equations is the trivial one, ψ = 0 everywhere. Therefore, we aim atfinding a partial solution to this set of equations, which involves operators from the enveloping algebra. Firstof all, we impose rotational invariance.

Suppose that the minimizer is of the form g(r) where r =√x2 + y2. Then, we consider the following

infinitesimal operators with respect to g(r):

Tθg(r) = 0,

Tbg(r) = (T 2bx

+ T 2by

)g(r) = −d2g

dr2(r)− 1

r

dg

dr(r).

Moreover, the operators Tωx, Tωy

are commuting with respect to g(r),i.e., [Tωx, Tωy

]g(r) = 0. These observationslead to two possible solutions: the first involves defining a new operator: Tω = Tωx

Tωy−Tωy

Tωxand considering

its commutator relations with Tθ and Tb. Then, any function g(r) that is rotation invariant is a valid minimizerof the uncertainties related to these operators.

Another option is to consider Tωxand Tωy

with respect to g(r). The commutators of these operators withTb are not equal to zero and we obtain the differential equation

grr(r) +1rgr(r) + µbg(r) = 0 (36)

whose solution is given by Bessel functions of the first and second kind:

ψ(r) = c1J0(√µbr) + c2Y0(

√µbr). (37)

Nevertheless, this solution does not belong to L2.Another interesting effort is to find a solution for a single differential equation and thus obtain a selective

minimal uncertainty with respect to two operators only. For example, let us consider equation 2 only:

(iα− x)ψ(x, y) + iα(xψx(x, y) + yψy(x, y))− µωxψ(x, y) = λ2 (−iψx(x, y)− µbx

ψ(x, y)) .

Choosing λ2 = 0 yields

(iα− x)ψ(x, y) + iα(xψx(x, y) + yψy(x, y))− µωxψ(x, y) = 0.

A possible solution is given by the expression:

ψ(x, y) = y−iµωx

α −1e−ixα τ

(x

y

), (38)

where τ is an arbitrary function of the variable xy . This function, however, is not even a member of L2.

Therefore we are faced here with an example where minimal uncertainty and admissibility do not really fittogether. Nevertheless, let us again emphasize that the minimizing states are always of interest by themselves.A particular solution is depicted in Figure 3.

DAHLKE ET AL226

alfa =0.1

alfa =0.5

alfa =1

Figure 3: A possible minimizing function for τ(xy

)= 1.

4.1.2 AWH Minimizers Using the L2-Norm

As we have seen in the previous section, the choice a = Φ(ω) =(1 +

√ω2x + ω2

y

)−αproves to be futile. It is

interesting to explore another solution to the problem mentioned in Section 4.1, where the relationship betweenthe scale and frequency is given by

a = Φ(ω) =(1 + ω2

x + ω2y

)−α.

The unitary representation induced by this section of the similitude Weyl-Heisenberg group is then given by

[U(b, ω, (Φ(ω), θ), 0)ψ](x, y) = (1 + ω2x + ω2

y)αei((x−bx)ωx+(y−by)ωy)ψ

((1 + ω2

x + ω2y)ατθ(x− bx, y − by)

), (39)

and τθ is the same as already defined. The infinitesimal generators are then given by:

Tωxψ(x, y) = −xψ(x, y),

Tωyψ(x, y) = −yψ(x, y),

Tbxψ(x, y) = −iψx(x, y),

Tbyψ(x, y) = −iψy(x, y),

Tθψ(x, y) = i(yψx(x, y)− xψy(x, y)). (40)

It is interesting to note that the dependency on the parameter α has disappeared. This means that selectingthis type of section may provide a solution regardless of the smoothness space we are dealing with.

The differential equations resulting from the non-commuting operators are:

−xψ(x, y)− µωxψ(x, y) = λ1(−iψx(x, y)− µbx

ψ(x, y)),−xψ(x, y)− µωx

ψ(x, y) = λ2(i(yψx(x, y)− xψy(x, y))− µθψ(x, y)),−yψ(x, y)− µωy

ψ(x, y) = λ3(−iψy(x, y)− µbyψ(x, y)),

−yψ(x, y)− µωyψ(x, y) = λ4(i(yψx(x, y)− xψy(x, y))− µθψ(x, y)),

−iψx(x, y)− µbxψ(x, y) = λ5(i(yψx(x, y)− xψy(x, y))− µθψ(x, y)),

−iψy(x, y)− µbyψ(x, y) = λ6(i(yψx(x, y)− xψy(x, y))− µθψ(x, y)). (41)

If, again, we search for a solution which is rotational invariant, i.e. ψ(x, y) = g(r) , we may satisfy allequations that involve the operator Tθ. Moreover, applying restrictions to our parameters, e.g. λ1 = λ3 =


λ, µbx= µby

, ωx = ωy, we may obtain a rotationally invariant solution to the first and third equation as well,which is a Gaussian function

ψ(x, y) = e− i

λ

“x2+y2

2

”, with λ ∈ iR. (42)

4.2 AWH Minimizers with Anisotropic Scaling

In the previous treatment of the two-dimensional case, we regard the frequency as a vector, but treat the scaleas a scalar argument. Moreover, we use the SIM(2) group rather than the full affine group. We are interestedto add more degrees of freedom to our setting, and as a first step we observe a relationship where the scale isalso a two dimensional vector and not a scalar

ax = β1(ωx), ay = β2(ωy).

We explore a generalization to two dimensions of the one-dimensional affine Weyl-Heisenberg group, and ignorefor now rotation and shear. We thus consider the group with a generic element g = (bx, by, ωx, ωy, ax, ay, φ, )where bx, by, ωx, ωy, φ ∈ R and ax, ay ∈ R+ equipped with a group law

(bx, by, ωx, ωy, ax, ay, φ) (b′x, b′y, ω

′x, ω

′y, a

′x, a

′y, φ

′) =

(bx + axb′x, by + ayb

′y, ωx + a−1

x ω′x, ωy + a−1y ω′y, axa

′x, aya

′y, φ+ φ′ + ωxaxb

′x + ωyayb

′y).

This is a subgroup of the 2D AWH group. The inverse element of g ∈ G is given by

g−1 = (−a−1x bx,−a−1

y by,−axωx,−ayωy, a−1x , a−1

y ,−φ+ bxωx + byωy). (43)

Let us look at the following representation

[U(bx, by, ωx, ωy, ax, ay, φ)ψ](x, y) =1

√axay

ei((x−bx)ωx+(y−by)ωy)+φψ

(x− bxax

,y − byay

)(44)

which is the 2D extension of the Stone-von-Neumann representation of the 1D AWH group. This representationfails to be square integrable, and therefore we restrict ourselves to the homogeneous space GAWH\H with

H := (0, 0, 0, 0, ax, ay, φ) ∈ GAWH . (45)

Next, we consider the section σ(bx, by, ωx, ωy) = (bx, by, ωx, ωy, βx(ωx), βy(ωy), 0), and would like to prove thatthis section is admissible.

We define a self-adjoint operator Aσf

(Aσf)(x, y) :=∫X

〈f, U(σ(h))ψ〉U(σ(h))ψdµ(h)

=∫ ∫ ∫ ∫

〈f, ψωx,ωy,βx(ωx),βy(ωy),bx,by〉ψωx,ωy,βx(ωx),βy(ωy),bx,by

(x, y)dbxdbydωxdωy. (46)

It can be written as a Fourier multiplier operator

(Aσf) = mβx,βyf (47)

wheremβx,βy

(γx, γy) =∫ ∫

|ψ(βx(ωx)(γx − ωx), βy(ωy)(γy − ωy))|2βx(ωx)βy(ωy)dωxdωy. (48)

Next, we follow the lines of [6] to show that mβx,βyis bounded from above and below, i.e.

C1 ≤ mβx,βy≤ C2 (49)

for constants 0 < C1 < C2 <∞. We start with the following lemma which is a straight forward generalizationof Lemma 5.1 in [6] to the 2D-case. Therefore we omit the details.

DAHLKE ET AL228

Lemma 6 Consider the specific section σ given by the functions

βx(ωx) = βx,αx(ωx) = (1 + |ωx|)−αx ,

βy(ωx) = βy,αy(ωy) = (1 + |ωy|)−αy .

Let us define

rγx(ωx) := βx(ωx)(γx − ωx) = (1 + |ωx|)−αx(γx − ωx),

rγy(ωy) := βy(ωy)(γy − ωy) = (1 + |ωy|)−αy (γy − ωy).

Then, for any fixed A > 0, there exist γx,A, γy,A > 0 such that for all γx ≥ γx,A, γy ≥ γy,A the functions rγx, rγy

are invertible on

Aωx= ωx : rγx

(ωx) ∈ [−A,A] and Aωy= ωy : rγy

(ωy) ∈ [−A,A]

respectively. The inverse functions r−1γx, r−1γy

of rγx, rγy

on [−A,A] have the form

r−1γx

= −xg1(γx, x) + γx, r−1γy

= −yg2(γy, y) + γy,

with some functions g1(γx, x), g2(γy, y) satisfying

xg1(γx, x) + g1(γx, x)1

αx = 1 + γx and yg2(γy, y) + g2(γy, y)1

αy = 1 + γy.

Furthermore, g1, g2 fulfill

limγx→∞

γ−αxx g1(γx, x) = 1, lim

γy→∞γ−αyy g2(γy, y) = 1

uniformly for x, y ∈ [−A,A].

Theorem 7 Let the Borel section σ be given by σ(bx, by, ωx, ωy) = (bx, by, ωx, ωy, βx(ωx), βy(ωy), 0) with βx(ωx) =(1 + |ωx|)−αx , βy(ωy) = (1 + |ωy|)−αy . Let ψ be a non zero L2 function whose Fourier transform is compactlysupported. Then, ψ is admissible, i.e., the condition

C1 ≤ mβx,βy(γx, γy) ≤ C2

is satisfied for 0 < C1 ≤ C2 <∞.

Proof. The proof can be performed by following the lines of the proof of Theorem 5.2 in [6]. For reader’sconvenience, we briefly sketch the arguments. We consider the case where either γx or γy tend to +∞. Let usassume that supp(ψ) ⊂ [−A,A] × [−A,A]. We substitute x = rγx

(ωx), y = rγy(ωy) for γx ≥ γx,A > 0, γy ≥

γy,A > 0 in the expression for mβx,βy(γx, γy) to obtain

mβx,βy(γx, γy) =

∫R

∫R|ψ(rγx

(ωx), rγy(ωy))|2βx(ωx)βy(ωy)dωxdωy

=∫

R

∫R|ψ(x, y)|2βx(r−1

γx(x))βy(r−1

γy(y))(r−1

γx)′(r−1

γy)′dxdy. (50)

Next, we calculate the values of the derivatives of the inverse functions r−1γx, r−1γy

using

r′γx(ωx) = β′x(ωx)(γx − ωx)− βx(ωx) = −βx(ωx)

(αxγx − ωx1 + ωx

+ 1),

r′γy(ωy) = β′y(ωy)(γy − ωy)− βy(ωy) = −βy(ωy)

(αyγy − ωy1 + ωy

+ 1)

to obtain

(r−1γx

)′(x) =1

r′γx(r−1γx (x))

= − 1

βx(r−1γx (x)

(1 + αx

γx−r−1γx (x)

1+r−1γx (x)

) ,(r−1γy

)′(y) =1

r′γy(r−1γy (y))

= − 1

βy(r−1γy (y)

(1 + αy

γy−r−1γy (y)

1+r−1γy (y)

) .


Thus, for values of γx ≥ γx,A > 0, γy ≥ γy,A > 0 we have


∫ A

−A

∫ A

−A|ψ(x, y)|2G(γx, γy, x, y)dxdy (51)

where

G(γx, γy, x, y) =1(

1 + αxγx−r−1

γx (x)

1+r−1γx (x)

) 1(1 + αy

γy−r−1γy (y)

1+r−1γy (y)

) =1

1 + αxxg1(γx, x)1−1

αx

1

1 + αyyg2(γy, y)1− 1

αy

,

where we have used the definitions in the previous lemma. According to this lemma, we may substitute γαxx for

g1(γx, x) when γx goes to infinity, and the same for g2(γy, y) when γy →∞

limγx→∞,γy→∞

G(γx, γy, x, y) = limγx→∞,γy→∞

1

1 + αxxg1(γx, x)1−1

αx

1

1 + αyyg2(γy, y)1− 1

αy

= limγx→∞,γy→∞

1

1 + αxxγαx(1− 1

αx)

x

1

1 + αyyγαy(1− 1

αy)

y

= 1,

and therefore we finally have

limγx→∞,γy→∞


∫ A

−A

∫ A

−A|ψ(x, y)|2dxdy (52)

for any L2-function with compact support in the Fourier domain, and thus we obtain that mβx,βyis bounded

from below and above.

Now, that this section is proven to be admissible, we would like to explore the uncertainty principle min-imizers associated with this representation. We assume that it should be a two-dimensional extension of theone-dimensional solution obtained earlier. The representation for the quotient as a function of αx, αy is thengiven by:

[U(bx, by, ωx, ωy, βx(ωx), βy(ωy), 0)ψ](x, y) =

(1 + |ωx|)αx2 (1 + |ωy|)

αy2 ei(ωx(x−bx)+ωy(y−by))ψ ((1 + |ωx|)αx(x− bx), (1 + |ωy|)αy (y − by)) .

From this representation we may see that the x and y axes are not correlated, and thus we obtain the followinginfinitesimal generators

(Tbxψ)(x, y) = − ∂

∂xψ(x, y),

(Tbyψ)(x, y) = − ∂

∂yψ(x, y),

(Tωxψ)(x, y) = (

αx2

+ i)ψ(x, y) + αxx∂

∂xψ(x, y),

(Tωyψ)(x, y) = (

αy2

+ i)ψ(x, y) + αyy∂

∂yψ(x, y). (53)

In order to make these operators self-adjoint, we multiply them by i. The commutators between the x and yoperators vanish, and we have to solve two independent one-dimensional problems, with the following solutions

ψ(x, y) = (αxλxx+ 1)− 1

2−iµωx

αx+

iµbxαxλx

+ iα2

xλx e−ixαx (αyλyy + 1)

− 12−

iµωyαy

+iµbyαyλy

+ iα2

yλy e−iyαy . (54)

In order for this solution to be a member of L2, the following should be met: we denote λx = i$x, λy = i$y where$x, $y ∈ R. Then, if $x, $y < 0, then µbx

> − 1αx, µby

> − 1αy

. If $x, $y > 0, then µbx< − 1

αx, µby

< − 1αy

.Unfortunately, the Fourier transform of this minimizing state is not compactly supported. However, it has atleast exponential decay. Therefore, by choosing a sufficiently large interval and using a smooth cut off functionwe obtain admissible and ‘almost’ minimizing states.

Acknowledgements: The authors want to thank Y.Y. Zeevi and N.A. Sochen for many fruitful discussions.

DAHLKE ET AL230

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[14] Matthias Holschneider and Gerd Teschke, “Existence and Computation of Optimally Localized CoherentSates”, Journal of Mathematical Physics, to appear, 2006.

[15] Karl Kraus, “A Further Remark on Uncertainty Relations”, Zeitschrift fur Physik, 201, 1967, 134–141.

[16] Chen Sagiv, Nir A. Sochen and Yehoshua Y. Zeevi, “The Uncertainty Principle: Group Theoretic Approach,Possible Minimizers and Scale-Space Properties“, Journal of Mathematical Imaging and Vision, 15(6), 2006,1633–1646.

[17] Joseph Segman and Walter J. Schempp, “Two Ways to Incorporate Scale in the Heisenberg Group Withan Intertwining Operator”, Journal of Mathematical Imaging and Vision, 3(1), 1993, 79–94.

[18] Joseph Segman and Yehoshua Y. Zeevi, “Image Analysis by Wavelet-Type Transform: Group TheoreticApproach”, Journal of Mathematical Imaging and Vision, 3, 1993, 51–75.

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DAHLKE ET AL232

BOUNDARY VALUE PROBLEMS IN PLAN SECTORWITH CORNERS FOR A CLASS OF SOBOLEV SPACES

OF DOUBLE WEIGHT

H. BENSERIDI & M. DILMI

Abstract. In this note, we study some boundary value problems of elasticityin plan domain with corners for a class of Sobolev spaces of double weight. Wegive a generalization of some results of existence and unicity of the solutionobteined by P. Grisvard [6] in classical spaces of nul weight, by D. Teniou[8] in Sobolev spaces of simple weight and by M. Dauge [4] for the system ofStokes.

AMS Classification : Mathematics Subject Classification,35B65.Keywords : Elasticity, Lame, Regularity, Singularity, Sobolev.

1. Notations

Let Ω be a sector in R2 of sides Γ0, Γω, angle ω and the vertexe S. Let L bethe Lamé operator :

L = µ∆+ (λ+ µ)∇div,λ and µ are Lamé coefficients (λ ≥ 0, µ > 0) (ui), (−fi), i = 1, 2 designate re-spectively the components of the displacement vector and the density of externalpowers.

P= (σij), i = 1, 2 , j = 1, 2 designate the stress tensor. The stress

tensor and the displacement vector are related via Hooke’s law :

σij = 2µεij(u) + λ div(u)δij , where εij(u) =1

2(∂jui + ∂iuj), εij(u) is called

tensor of linear deformation associated to u.

x

y

Γω

Γ 0

η

τ

ω

Figure 11


2

Lamé operator being invariant by translation and rotation; we may then suppose,without lass of generality, that S is the origine O and that the first side of Ω is onthe positive x axis. We define B by : B = R×]0 , ω[. Let θ0, θ∞ be two reals suchthat: θ0 ≤ θ∞, we put θ0 = η0 − 1, θ∞ = η∞ − 1.Definition 1.1. For m ∈ N, we define the spaces :Hmθ0,θ∞(Ω) =

nu ∈ L2loc(Ω) : r

θ0−m+|α|(1 + r)θ∞−θ0Dαu(x, y) ∈ L2(Ω),∀α ∈ N2, |α| ≤ mo.

equiped with the inner product :

hu, vi =X|α|≤m

ZZΩ

r2(θ0−m+|α|)(1 + r)2(θ∞−θ0)Dαu(x, y)Dαv(x, y)dxdy.

Hmθ0,θ∞(B) =

©u ∈ L2loc(B) / eθ0 t (1 + et)θ∞−θ0u(t, θ) ∈ Hm(B)

ª.

equiped with the inner product :

hu, vi =X|α|≤m

ZZB

Dα¡eθ0 t (1 + et)θ∞−θ0u(t, θ)

¢Dα

¡eθ0 t (1 + et)θ∞−θ0v(t, θ)

¢dtdθ.

Lemma 1.1.( cf. [4] ). Let θ1, θ2 be two reals, we assume that θ1 ≤ θ2. Let k be apositive integer then: f ∈ Hk

θ1,θ2(Ω) if and only if f ∈ Hk

θ1,θ1(Ω) ∩Hk

θ2,θ2(Ω) and

we havekfkHk

θ1,θ2(Ω) ≤ c

hkfkHk

θ1,θ1(Ω) + kfkHk

θ2,θ2(Ω)

i,

c benig a constant which depends only on θ1, θ2.

Definition 1.2. Let m ∈ N, we define the space V m(B) by :

V m(B) =nu ∈ L2(B) / (1 + ξ2)

k2 u ∈ L2(R ,Hm−k(]0, ω[) ), pour k = 0, 1, ...m

o.

V m(B) is a Hilbert space with the inner product given by :

hu, vi =mXk=0

Z ZB

(1 + ξ2)k¯Dm−kθ u

¯ ¯Dm−kθ v

¯dθdξ.

We define by F the Fourier transform with repect to the first variable in B.The application: F : Hm(B) −→ V m(B) is an isomorphism.

Proposition 1.1 ( cf. [4, 8] ). k ∈ N , η1 ≤ η2. Let Cη1,η2 be the band of thecomplexe C defined by

Cη1,η2 = λ ∈ C / Imλ ∈ [η1, η2] .Let f ∈ Hk

η1,η2(B), then

(1) The Fourier transform exists for all λ ∈ Cη1,η2 .(2) The application

[η1, η2] −→ L2 (B)

η 7−→ bf (·+ iη, ·)is continuous

BENSERIDI-DILMI234

3

(3) The applicationCη1,η2 −→ Hk (]0, ω[)

λ 7−→ bf (λ, ·)is analytical4) There exists a constant c which depends only η1, η2 such that°°° bf (·+ iη, ·)

°°°V k(B)

≤ c kfkHkη1,η2

(B) .

2. Position of the problem P kθ0,θ∞(Ω)

We take into consideration ten types of problems governed by Lamé operator.If f ∈ L2θ0,θ∞(Ω)

2, we look for u if possible in H2θ0,θ∞(Ω)

2 as a solution for thefollowing problem :

P kθ0,θ∞(Ω) :

Lu = f in Ω,BK0 u = 0 on Γ0,

BKω u = 0 on Γω,

, k = 1, 2, ..., 10.

The operators of trace

B10u = u, B1

ωu = u

B20u =

X(u).η , B2

ωu =X(u).η

B30u = u , B3

ω u =X(u).η

B40u =

½u.η

(P(u).η ).τ

, B4ω =

½u.η

(P(u).η).τ

B50u = u , B5

ωu =

½u.η

(P(u).η).τ)

B60u =

½u.η

(P(u).η).τ)

, B6ωu =

X(u).η

B70u = u , B7

ω =

½u.η

(P(u).η).η

B80u =

½u.τ

(P(u).η).η

, B8ωu =

X(u).η

B90u =

½u.τP(u).η).η

, B9ωu =

½u.τ

(P(u).η).η

B100 u =

½u.τ

(P(u).η).η

, B10ω u =

½u.η

(P(u).η).τ

BOUNDARY VALUE PROBLEMS 235

4

3. Usage of polar coordinates

We put x = r cos θ , y = r sin θ , with r = et.After the multiplication by (e2t), the system Lu = f be cames

(E )

£∂2t + ∂2θ + ν0(sin θ.∂t + cos θ.∂)

¤ eu1 +ν0 [cos θ.∂t − sin θ.∂θ] [sin θ.∂t + cos θ.∂θ] eu2 = eg1£

∂2t + ∂2θ + ν0(sin θ.∂t + cos θ.∂θ)¤ eu2 +

ν0 [sin θ.∂t + cos θ.∂θ] [cos θ.∂t − sin θ.∂θ] eu1 = eg2where egi(t, θ) = e2tf i( e

t cos θ, et sin θ ),eui(t, θ) = ui(et cos θ, et sin θ),

ν0 = (1− 2ν )−1.with ν is the coefficient of Poisson defined by ν =

λ

2(λ+ µ)we have

u ∈ H2θ0,θ∞(Ω)

2

f ∈ H2θ0,θ∞(Ω)

2

¾⇒½ eu ∈ H2

η0,η∞(B)2eg ∈ L2η0,η∞(B)2

Let IK1 be the isomorphisme of Hkθ0,θ∞(Ω)

2 on Hkη0,η∞(B )

2 defined by

Ik1 (u1, u2) = (eu1, eu2).Propriety 3.1. The application Ik2 defined by

Ik2 : Hkη0,η∞(B)

2 −→ Hkη0,η∞(B)

2

(eu1, eu2) −→ V = (V1, V2) = (cos θ.eu1 + sin θ.eu2,− sin θ.eu1 + cos θ.eu2)is an isomorphisme.

We put g = (g1, g2) with (g1, g2) =1

λ+ µI22 (eg1,eg2).

With these changes, (E) becames

P kη0,η∞(B )

2(1− ν)(∂2V1∂t2 − V1) +

∂2V2∂t∂θ

+ (1− 2ν)∂2V1

∂θ2− (3− 4ν)∂V2

∂θ= g1

(1− 2ν)(∂2V2∂t2

− V2) +∂2V1∂t∂θ

+ 2(1− ν)∂2V2

∂θ2− (3− 4ν )∂V1

∂θ= g2

eBk0V = 0, on R× 0eBkωV = 0, on R× ω

BENSERIDI-DILMI236

5

where eBk0 , eBk

ω are boundary operators of Bk0 , B

kω, after the Fourier transform.

The application Ik2 is an isomorphisme of the space Hkη0,η∞(B) in itself (propriety

3.1). Thus : V ∈ H2η0,η∞(B)

2 and g ∈ L2η0,η∞(B )2.

Propriety 3.2. The problems P kθ0,θ∞(Ω), P

kη0,η∞(B ) are equivalents.

This result, from propriety 3.1.

4. Fourier transforme of Pkθ0,θ∞(Ω )

We have found on what procedes that V ∈ H2η0,η∞(B )

2 . Thus after the propo-sition (1.1), V and its derivatives of order ≤ 2 admits a Fourier transform withrepect to the first variable t in the band Cη0,η∞ defined by

Cη0,η∞ = λ ∈ C / η0 ≤ Imλ ≤ η∞ .We have also: g ∈ L2η0,η∞(B )

2, then g admits a Fourier transform in the sameband.For simplicity, we write: ∂θX = X 0 for all function X(λ, θ) .

Transforming the problem P kη0,η∞(B ) by Fourier, we obtain

P kλ (B )

(1− 2ν)bV 001 − 2(1− ν)(1 + λ2)bV1 − (3− 4ν − iλ)bV 0

2 = bg12(1− ν)bV 00

2 − (1− 2ν)(1 + λ2)bV2 + (3− 4ν + iλ)bV 01 = bg2

Ck0bV = 0, for θ = 0

CkωbV = 0, for θ = ω

where Ck0 , C

kω are boundary operators of eBk

0 , eBkω, after the Fourier transform.

More precisely we have

C10 bV = bV , C1ω bV = bVC20 bV =

((iλ− 1)bV2 + bV 0

1

(iλν + 1− ν)bV1 + (1− ν)bV 02

, C2ω bV =

((iλ− 1)bV2 + bV 0

1

(iλν + 1− ν)bV1 + (1− ν)bV 02

C30 bV = bV , C3ω bV =

((iλ− 1)bV2 + bV 0

1

(iλν + 1− ν)bV1 + (1− ν)bV 02

C40 bV =

( bV2(iλ− 1)bV2 + bV 0

1

, C4ω bV =

( bV2(iλ− 1)bV2 + bV 0

1


( bV2(iλ− 1)bV2 + bV 0

1

C60 bV =

( bV2(iλ− 1)bV2 + bV 0

1

, C6ω bV =

((iλ− 1)bV2 + bV 0

1

(iλν + 1− ν)bV1 + (1− ν)bV 02


( bV1(iλν + 1− ν)bV1 + (1− ν)bV 0

2


6

C80 bV =

( bV1(iλν + 1− ν)bV1 + (1− ν)bV 0

2

, C8ω bV =

((iλ− 1)bV2 + bV 0

1

(iλν + 1− ν)bV1 + (1− ν)bV 02

C90 bV =

( bV1(iλν + 1− ν)bV1 + (1− ν)bV 0

2

, C9ω bV =

( bV1(iλν + 1− ν)bV1 + (1− ν)bV 0

2

C100 bV =

( bV1(iλν + 1− ν)bV1 + (1− ν)bV 0

2

, C100 bV =

( bV2(iλ− 1)bV2 + bV 0

1

.

Finally, we have to study the following problem :

λ is a fixed in the band Cη0,η∞ , bg being in L2(]0, ω[)2, we look for bV if possiblein H2(]0, ω[)2 solution for the problem P k

λ (B) . The study of the homogeneousproblem corresponding to P k

λ (B) gives the following results: ( cf. Benseridi andMerouani [2] ).

Result 4.1. Let Dk be the determinant of Cramer system given by the conditions:(CK

0bV = 0, CK

ωbV = 0), then: Dk = ckΦk, k = 1, 2, ..., 10 .

The (ck) are constants, and (Φk) are the following functions :

Φ1(λ) =1

λ2£(3− 4ν)2sh2(λω)− λ2 sin2(ω)

¤Φ2(λ) = λ2

£sh2(λω)− λ2 sin2(ω)

¤Φ3(λ) =

£4(1− ν)2 + λ2 sin2(ω) + (3− 4ν)sh2(λω)¤

Φ4(λ) = sh(λ− i)ωsh(λ+ i)ω

Φ5(λ) =1

λ[(3− 4ν)sh(2λω) + λ sin(2ω)]

Φ6(λ) = λ(sh(2λω) + λ sin(2ω))

Φ7(λ) =1

λ[(3− 4ν)sh(2λω) + λ sin(2ω)]

Φ8(λ) = λ(sh(2λω)− λ sin(2ω))

Φ9(λ) = sh(λ− i)ωsh(λ+ i)ω

Φ10(λ) = ch(λ− i)ωch(λ+ i)ω

Result 4.2. Let Fk be the set of zeros of Φk(λ), then the homogeneous problemcorresponding to P k

λ (B) admits a unique solution ( the trivial solution ) if and onlyif λ /∈ Fk.

BENSERIDI-DILMI238

7

Proposition 4.1 ( cf. [4] ). For all λ ∈ C/Fk, for all bg ∈ L2(]0, ω[)2, there existsone and only one bVλ ∈ H2(]0, ω[)2 solution for the problem P k

λ (B). In addition, theresolved of P k

λ (B),

Rλ : L2(]0, ω[)2 −→ H2(]0, ω[)2bg 7−→ Rλ(bg) = bVλsuch that the application

C/Fk −→ L¡L2(]0, ω[)2,H2(]0, ω[)2

¢,

λ 7−→ Rλ

is analytical.

5. Existence and uniqueness of η−solutions

In order to the study the existence and uniqueness of the solution for the problemP kθ0,θ∞(Ω), it is important to introduce the following definition :

Defnition 5.1. Let η ∈ [η0, η∞] , we call η−solution for the problem P kθ0,θ∞(Ω),

all elements u of H2η+1,η+1(Ω)

2, verifying

Lu = f in Ω,Bk0u = 0 on Γ0,

Bkωu = 0 on Γω.

The following propriety is a direct result of lemma (1.1).

Propriety 5.1. u is a solution for the problem P kθ0,θ∞(Ω) iff u is a η0−solution

of the P kθ0,θ∞(Ω) and u is a η∞−solution of P k

θ0,θ∞(Ω).

Proposition 5.1. If Φk have no zero of imaginary part η, the problem P kθ0,θ∞(Ω)

has a unique η− solution, in addition there exists a positive constant c such as

kukH2η+1,η+1(Ω)

2 ≤ c kfkL2θ0,θ∞ (Ω)2 .

Remark 5.1. For the demonstration of the proposition 5.1, ( cf. Teniou [8] ).

6. Comparaison of the η−solutions

Let η1, η2 ∈ [η0, η∞] , η1 ≤ η2. We know that if Φk have no zero of imagi-nary part η1(resp η2), the problem P k

θ0,θ∞(Ω) admits a unique η1−solution (respη2−solution) that we not by uη1 (resp uη2).


8

We put Vη1 = I22 I21 (uη1), Vη2 = I22 I21 (uη2), with this notation , we have thefollowing proposition :

Proposition 6.1. We assume that Φk have no zero of imaginary part η1 or η2,then

Vη1 − Vη2 = iX

λ0∈Fk∩η1≤Imλ≤η2Res(ei tλRλ(bg)) |λ=λ0 .

Proof. The proof of this proposition is very classical. We note first that the sumhas a meaning because the set Fk ∩ η1 ≤ Imλ ≤ η2 is finite and the residuals arewell defined (these are the functions on (t, θ) ∈ B ).Let γ be the contour made by the straight line R+ iη1,R+ iη2. We know that

Rλ is analytical on C/Fk, henceRγ

eitλRλ(bg)dλ = 2πi Pλ0∈Fk∩η1≤Imλ≤η2

Res(ei tλRλ(bg)) |λ=λ0and R

γ

eitλRλ(bg)dλ = R[−ε+iη1,ε+iη1]

eitλRλ(bg)dλ+ R[ε+iη1,ε+iη2]

eitλRλ(bg)dλ+

R[ε+iη2,−ε+iη2]

eitλRλ(bg)dλ+ R[−ε+iη2,−ε+iη1]

eitλRλ(bg)dλwith the limit when ε→∞, we obtain

limε→∞

Zγ

eitλRλ(bg)dλ = +∞Z−∞

eit(ξ+iη1)Rξ+iη1(bg)dξ −+∞Z−∞

eit(ξ+iη2)Rξ+iη2(bg)dξthe integrals R

[ε+iη1,ε+iη2]

eitλ Rλ(bg)dλ ,R

[−ε+iη2,−ε+iη1]eitλ Rλ(bg)dλ,

tends to zero, thus

e−η1t2π

+∞R−∞

eitξRξ+iη1(bg)− e−η2t2π

+∞R−∞

eitξRξ+iη2(bg) = iP

λ0∈Fk∩η1≤Imλ≤η2Res(ei tλRλ(bg)) |λ=λ0

but

Vη1 =e−η1t2π

+∞R−∞

eitξRξ+iη1(bg)dξ ,

Vη2 =e−η2t2π

+∞R−∞

eitξRξ+iη2(bg)dξ.Which ends the proof.

BENSERIDI-DILMI240

9

7. Existence and uniqueness of the solution for the problem P kθ0,θ∞(Ω)

Our objective now is to prove a theorem of existence and uniqueness and regu-larity of the solution of the problem P k

θ0,θ∞(Ω).

Theorem 7.1. Let θ0, θ∞ be two reals such that θ0 ≤ θ∞.We assume that Φk have no zero in the band Cη0,η∞ , thus for all f ∈ L2θ0,θ∞(Ω)

2,

there exists one and only one u ∈ H2θ0,θ∞(Ω)

2 solution for the problem P kθ0,θ∞(Ω),and

we havekukH2

θ0,θ∞ (Ω)2 ≤ c kfkL2θ0,θ∞(Ω)2 .

Proof.

(1) Existence. The hypothesis that Φk has no zero on the band Cη0,η∞ ensures theexistence of η0−solution and the η∞−solution of P k

θ0,θ∞(Ω), that we note uη0 , uη∞and in addition: Fk ∩ η0 ≤ Imλ ≤ η∞ = ∅.We put Vη0 = I22 I21 ( uη0), Vη∞ = I22 I21 ( uη∞), proposition (6.1) implies that

Vη0−Vη∞ = iP

λ0∈Fk∩η0≤Imλ≤η∞Res(ei tλRλ(bg)) |λ=λ0 . As a result Vη0 = Vη∞ , this

shows that uη0 = uη∞ . Now, we put now u = uη0 , it is clear that u ∈ H2θ0,θ0

(Ω)2

and u ∈ H2θ∞,θ∞(Ω)

2. The lemma(1.1) shows that u ∈ H2θ0,θ∞(Ω)

2. Thus u is asolution of P k

θ0,θ∞(Ω) by construction.

(2) Uniqueness. We assume that there exist two solutions u1, u2 ∈ H2θ0,θ∞(Ω)

2.thus u1, u2 are η0−solution and η∞−solution ( propriety 5.1 ). Thus from theuniqueness of η−solutions u1 = u2.

(3) Continuity with repect to the data. We deduce from the proposition 5.1 that

kukH2θ0,θ0

(Ω)2 ≤ c kfkL2θ0,θ∞ (Ω)2 and kukH2θ∞,θ∞ (Ω)

2 ≤ c kfkL2θ0,θ∞(Ω)2 ,

and from the lemma(1.1): kukH2θ0,θ∞(Ω)

2 ≤ chkukH2

θ0,θ0(Ω)2 + kukH2

θ∞,θ∞ (Ω)2

ithus

kukH2θ0,θ∞ (Ω)

2 ≤ c kfkL2θ0,θ∞(Ω)2 .

Remark 7.1.

•We can verify that the functions Φ1, Φ2 have no zeros in the band −1 ≤ Imλ < 0if ω ≤ π . This ensures the regularity of the solution u in H2

θ0,θ∞(Ω)2 if θ0 ≥ 0,

θ∞ < 1 and the vertexe is convexe of Dirichlet type or Neumann type.

• When θ0 = θ∞ , we reproduce the results of Teniou [8].


10

References

[1] H. Benseridi, Régularité de quelques problèmes aux limites linéaires et non linéaires dans desdomaines non réguliers et non homogènes, Thèse de Doctorat, Univ- Ferhat Abbas de Sétif,Algérie, Juil.(2005).

[2] H. Benseridi, B. Merouani, Quelques problèmes de transmission liés au système de Lamédans un polyèdre pour une classe d’espaces de Sobolev à doubles poids, Rev. Roum. Sci. Techn.-Méc. Appl., Tom 48, no 1- 6, Bucarest (2003), pp. 21-34.

[3] H. Benseridi, M. Dilmi, Régularité des solutions de quelques problèmes aux limites dans undomaine de R2 non homogène, Anal. Univ. Oradea, fasc. Math. Tom XII (2005), pp. 221-235.

[4] M. DAUGE, Etude de L’opérateur de Stokes dans un Polygone: Régularité, Singularité etThéorèmes d’indices, Thèse de Doctorat de 3ème cycle, Université de Nantes, Mai 1980.

[5] H. Benseridi, B. Merouani, Regularity of the solution of a nonlinear boundary value problemgoverned by Lame operator in an irregular domain, Far East J. Appl. Math. 16(3) (2004), pp.305-314.

[6] P. Grisvard, Boundary value problems in plan polygons, I nstruction for use, E.D.F, Bulletinde la Direction des études et Recherche, série C, Mathématique no.1, (1986), p 21-59.

[7] N. MOSBAH, Etude du Système de Lamé dans un polygone Pour une classe d’espaces deSobolev à double Poids. Thèse de Magister, Université de Ferhat Abbas Sétif, (1998).

[8] D. TENIOU, Divers Problèmes Théoriques et Numériques liés au Système de L’élasticité dansdes Domaines non Réguliers, Thèse de Doctorat d’état, Université de RENNES I, Septembre1987.

Universty Ferhat Abbas, Faculty of Sciences,Department of Mathématis, Sétif, 19000, [email protected]

BENSERIDI-DILMI242

On the Product of Laplace

and Logistic Random Variables

by

Saralees NadarajahSchool of Mathematics

University of ManchesterManchester M60 1QD

United KingdomE-mail: [email protected]

Abstract: The exact distribution of the product | XY | is derived when X and Y are Laplace andlogistic random variables distributed independently of each other. Tabulations of the associatedpercentage points are given.

1 Introduction

For given random variables X and Y , the distribution of the product | XY | is of interest in problemsin biological and physical sciences, econometrics, and classification. The exact distribution of | XY |has been studied by several authors especially when X and Y are independent random variables andcome from the same family. For instance, see Sakamoto (1943) for uniform family, Harter (1951)and Wallgren (1980) for Student’s t family, Springer and Thompson (1970) for normal family,Stuart (1962) and Podolski (1972) for gamma family, Steece (1976), Bhargava and Khatri (1981)and Tang and Gupta (1984) for beta family, Abu-Salih (1983) for power function family, and Malikand Trudel (1986) for exponential family (see also Rathie and Rohrer (1987) for a comprehensivereview of known results).

However, there is relatively little work of the above kind when X and Y belong to differentfamilies. In the applications, it is quite possible that X and Y could arise from different but similardistributions. In this note, we study the exact distribution of | XY | when X and Y are independentrandom variables having the Laplace and logistic distributions specified by the probability densityfunctions (pdfs)

fX(x) =λ

2exp (−λ | x |) (1)

and

fY (y) =µ exp(−µy)

1 + exp(−µy)2 , (2)

respectively, for −∞ < x < ∞, −∞ < y < ∞, λ > 0 and µ > 0. Tabulations of the associatedpercentage points are also provided.

Laplace and logistic distributions have found applications in a variety of areas that range fromimage and speech recognition and ocean engineering to finance. Both are rapidly becoming distri-butions of first choice whenever “something” with heavier than Gaussian tails is observed in the


data. In many of the application areas, one would be interested in products of Laplace and logisticrandom variables. Some examples are:

1. in communication theory, X and Y could represent the random noise corresponding to twodifferent signals.

2. in ocean engineering, X and Y could represent distributions of navigation errors.

3. in finance, X and Y could represent distributions of log-returns of two different commodities.

4. in image and speech recognition, X and Y could represent “input” distributions.

For further discussion of applications, the reader is referred to Balakrishnan (1992) and Kotz et al.(2001).

The results of this note are organized as follows: exact expressions for the pdf and the cumulativedistribution function (cdf) of | XY | are given in Section 2; moment properties of | XY | includingits characteristic function and moments are considered in Section 3; finally, tabulations of thepercentile points of | XY | obtained by inverting the derived cdf are provided in Section 4.

The calculations of this note involve the modified Bessel function of the first kind defined by

Iν(x) =xν

2νΓ (ν + 1)

∞∑

k=0

1

(ν + 1)kk!

(

x2

4

)k

,

and the modified Bessel function of the third kind defined by

Kν(x) =π I−ν(x) − Iν(x)

2 sin(νπ).

The calculations also require the following representation of (2):

f|Y |(y) = 2∞

∑

k=0

(−1)kµ(k + 1) exp −µ(k + 1)y (3)

for y > 0 (the series does not hold for y = 0). The properties of the above special functions can befound in Prudnikov et al. (1986) and Gradshteyn and Ryzhik (2000).

2 PDF and CDF of | XY |

Theorem 1 derives explicit expressions for the pdf and the cdf of | XY | in terms of the modifiedBessel function of the third kind.

Theorem 1 Suppose X and Y are independent random variables distributed according to (1) and(2), respectively. The cdf of Z =| XY | can be expressed as

F (z) = 1 − 4√

λµz∞

∑

k=0

(−1)k√

k + 1K1

(

2√

λµ(k + 1)z)

(4)

NADARAJAH244

for z > 0. The corresponding pdf is:

f(z) = 4λµ∞

∑

k=0

(−1)k(k + 1)

K0

(

2√

λµ(k + 1)z)

+K1

(

2√

λµ(k + 1)z)

2√

λµ(k + 1)z

−2

√

λµ

z

∞∑

k=0

(−1)k√

k + 1K1

(

2√

λµ(k + 1)z)

(5)

for z > 0.

Proof: Using the relationship (3), one can write

Pr (| XY |> z) =

∫ ∞

0Pr

(

| X |> z

y

)

f|Y |(y)dy

= 2

∫ ∞

0exp

(

−λz

y

)

[

∞∑

k=0

(−1)kµ(k + 1) exp −µ(k + 1)y]

dy

= 2∞

∑

k=0

(−1)kµ(k + 1)

∫ ∞

0exp

−λz

y− µ(k + 1)y

dy

= 4∞

∑

k=0

(−1)kµ(k + 1)

√

λz

(k + 1)µK1

(

2√

λµ(k + 1)z)

,

where the last step follows by direct application of equation (3.471.9) in Gradshteyn and Ryzhik(2000). The switching of the integral with the sum can be justified by the dominated convergencetheorem:

2

∫ ∞

0exp

(

−λz

y

)

∣

∣

∣

∣

∣

n∑

k=0

(−1)kµ(k + 1) exp −µ(k + 1)y∣

∣

∣

∣

∣

dy

≤ 2

∫ ∞

0exp

(

−λz

y

) n∑

k=0

µ(k + 1) exp −µ(k + 1)y dy

< 2

∫ ∞

0exp

(

−λz

y

) ∞∑

k=0

µ(k + 1) exp −µ(k + 1)y dy

= 2

∫ ∞

0exp

(

−λz

y

)

µ exp(−µy)

1 − exp(−µy)2 dy

< ∞. (6)

The result of (5) follows by using the property K′

ν(z) = −Kν−1(z) − (ν/z)Kν(z), see http: //functions.wolfram.com / BesselAiryStruveFunctions/BesselK/20/01/02/0001/. The term by term dif-ferentiation of (4) can also be justified by the dominated convergence theorem. ¥

Figure 1 illustrates possible shapes of the pdf (5) for λ = 1 and a range of values of µ. Notethat the shapes are unimodal and that the value of µ largely dictates the behavior of the pdf nearz = 0.

LAPLACE AND LOGISTIC RANDOM VARIABLES 245

0 1 2 3 4 5

0.00.5

1.01.5

2.02.5

z

PDF

0 1 2 3 4 5

0.00.5

1.01.5

2.02.5

0 1 2 3 4 5

0.00.5

1.01.5

2.02.5

0 1 2 3 4 5

0.00.5

1.01.5

2.02.5

µ = 1µ = 2µ = 5µ = 10

Figure 1. Plots of the pdf (5) for µ = 1, 2, 5, 10 and λ = 1.

3 Moment Properties of | XY |

The moment properties of | XY | can be derived by knowing the same for X and Y . It is wellknown (see, for example, Johnson et al. (1994, 1995)) that

E (| X |n) =n!

λn

and

E (| Y |n) = 2µ−nn!

∞∑

k=0

(−1)k

(1 + k)n .

Thus, the nth moment of Z =| XY | is

E (Zn) =2 (n!)2

(λµ)n

∞∑

k=0

(−1)k

(1 + k)n .

In particular,

E (Z) =2

λµ

∞∑

k=0

(−1)k

1 + k,

which can be reduced to

E (Z) =2 log 2

λµ,

NADARAJAH246

and

E(

Z2)

=8

(λµ)2

∞∑

k=0

(−1)k

(1 + k)2,

which can be reduced to

E(

Z2)

=2π2

3(λµ)2.

Using the fact that the characteristic function (chf) of | X | is

E [exp(it | X |)] =λ

λ − it,

where i =√−1 denotes the complex unit, the chf of | XY | can be derived as

E [exp(it | XY |)]

= 2λµ

∫ ∞

0

1

λ − ity

exp(−µy)

1 + exp(−µy)2 dy

= 2λµ

∫ ∞

0

λ + ity

λ2 + t2y2

exp(−µy)

1 + exp(−µy)2 dy

= 2λµ

∫ ∞

0

λ + ity

λ2 + t2y2

[

∞∑

k=0

(−1)kµ(k + 1) exp −µ(k + 1)y]

dy

= 2λ2µ2∞

∑

k=0

(−1)k(k + 1)

∫ ∞

0

exp −µ(k + 1)yλ2 + t2y2 dy

+2itλµ2∞

∑

k=0

(−1)k(k + 1)

∫ ∞

0

y exp −µ(k + 1)yλ2 + t2y2 dy

=2λµ2

t

∞∑

k=0

(−1)k(k + 1)

sinλµ(k + 1)

tci

λµ(k + 1)

t− cos

λµ(k + 1)

tsi

λµ(k + 1)

t

+2iµ2∞

∑

k=0

(−1)k+1(k + 1)∂

∂p

sinpλ

tci

pλ

t− cos

pλ

tsi

pλ

t

∣

∣

∣

∣

p=µ(k+1)

,

where the last step follows by equation (2.3.7.11) in Prudnikov et al. (1986, volume 1). The si(·)and ci(·) are the sine and cosine integrals defined by

si(x) = −∫ ∞

x

sin t

tdt

and

ci(x) = −∫ ∞

x

cos t

tdt,

respectively. The switching of the integral with the sum above can be justified as in the proof ofTheorem 1.


4 Percentiles of | XY |

In this section, we provide tabulations of percentage points zp associated with the cdf (4). Thesevalues are obtained by numerically solving the equation

4√

λµz∞

∑

k=0

(−1)k√

k + 1K1

(

2√

λµ(k + 1)z)

= 1 − p.

Evidently, this involves computation of the modified Bessel function of the third kind and routinesfor this are widely available. We used the function BesselK (·) in the algebraic manipulation package,MAPLE. Table 1 provides the numerical values of zp for µ = 0.1, 0.2, . . . , 10 and λ = 1.

Table 1. Percentage points of Z =| XY | for λ = 1.

µ p = 0.01 p = 0.05 p = 0.1 p = 0.9 p = 0.95 p = 0.99

0.1 0.030 0.209 0.513 35.507 52.939 103.5170.2 0.015 0.104 0.257 17.754 26.470 51.7590.3 0.010 0.070 0.171 11.836 17.646 34.5060.4 0.007 0.052 0.128 8.877 13.235 25.8790.5 0.006 0.042 0.103 7.101 10.588 20.7030.6 0.005 0.035 0.086 5.918 8.823 17.2530.7 0.004 0.030 0.073 5.073 7.563 14.7880.8 0.004 0.026 0.064 4.438 6.617 12.9400.9 0.003 0.023 0.057 3.945 5.882 11.5021 0.003 0.021 0.051 3.551 5.294 10.352

1.1 0.003 0.019 0.047 3.228 4.813 9.4111.2 0.002 0.017 0.043 2.959 4.412 8.6261.3 0.002 0.016 0.039 2.731 4.072 7.9631.4 0.002 0.015 0.037 2.536 3.781 7.3941.5 0.002 0.014 0.034 2.367 3.529 6.9011.6 0.002 0.013 0.032 2.219 3.309 6.4701.7 0.002 0.012 0.030 2.089 3.114 6.0891.8 0.002 0.012 0.028 1.973 2.941 5.7511.9 0.002 0.011 0.027 1.869 2.786 5.4482 0.001 0.010 0.026 1.775 2.647 5.176

2.1 0.001 0.010 0.024 1.691 2.521 4.9292.2 0.001 0.010 0.023 1.614 2.406 4.7052.3 0.001 0.009 0.022 1.544 2.302 4.5012.4 0.001 0.009 0.021 1.479 2.206 4.3132.5 0.001 0.008 0.020 1.420 2.118 4.1412.6 0.001 0.008 0.020 1.366 2.036 3.9812.7 0.001 0.008 0.019 1.315 1.961 3.8342.8 0.001 0.007 0.018 1.268 1.891 3.6972.9 0.001 0.007 0.018 1.224 1.826 3.5703 0.001 0.007 0.017 1.184 1.765 3.451

3.1 0.001 0.007 0.017 1.145 1.708 3.3393.2 0.001 0.007 0.016 1.110 1.654 3.2353.3 0.001 0.006 0.016 1.076 1.604 3.137

NADARAJAH248

3.4 0.001 0.006 0.015 1.044 1.557 3.0453.5 0.001 0.006 0.015 1.014 1.513 2.9583.6 0.001 0.006 0.014 0.986 1.471 2.8753.7 0.001 0.006 0.014 0.960 1.431 2.7983.8 0.001 0.006 0.014 0.934 1.393 2.7243.9 0.001 0.005 0.013 0.910 1.357 2.6544 0.001 0.005 0.013 0.888 1.323 2.588

4.1 0.001 0.005 0.013 0.866 1.291 2.5254.2 0.001 0.005 0.012 0.845 1.260 2.4654.3 0.001 0.005 0.012 0.826 1.231 2.4074.4 0.001 0.005 0.012 0.807 1.203 2.3534.5 0.001 0.005 0.011 0.789 1.176 2.3004.6 0.001 0.005 0.011 0.772 1.151 2.2504.7 0.001 0.004 0.011 0.756 1.126 2.2034.8 0.001 0.004 0.011 0.740 1.103 2.1574.9 0.001 0.004 0.011 0.725 1.080 2.1135 0.001 0.004 0.010 0.710 1.059 2.070

5.1 0.001 0.004 0.010 0.696 1.038 2.0305.2 0.001 0.004 0.010 0.683 1.018 1.9915.3 0.001 0.004 0.010 0.670 0.999 1.9535.4 0.001 0.004 0.010 0.658 0.980 1.9175.5 0.001 0.004 0.009 0.646 0.963 1.8825.6 0.001 0.004 0.009 0.634 0.945 1.8495.7 0.001 0.004 0.009 0.623 0.929 1.8165.8 0.001 0.004 0.009 0.612 0.913 1.7855.9 0.000 0.004 0.009 0.602 0.897 1.7556 0.000 0.004 0.009 0.592 0.882 1.725

6.1 0.000 0.003 0.008 0.582 0.868 1.6976.2 0.000 0.003 0.008 0.573 0.854 1.6706.3 0.000 0.003 0.008 0.564 0.840 1.6436.4 0.000 0.003 0.008 0.555 0.827 1.6176.5 0.000 0.003 0.008 0.546 0.814 1.5936.6 0.000 0.003 0.008 0.538 0.802 1.5686.7 0.000 0.003 0.008 0.530 0.790 1.5456.8 0.000 0.003 0.008 0.522 0.779 1.5226.9 0.000 0.003 0.007 0.515 0.767 1.5007 0.000 0.003 0.007 0.507 0.756 1.479

7.1 0.000 0.003 0.007 0.500 0.746 1.4587.2 0.000 0.003 0.007 0.493 0.735 1.4387.3 0.000 0.003 0.007 0.486 0.725 1.4187.4 0.000 0.003 0.007 0.480 0.715 1.3997.5 0.000 0.003 0.007 0.473 0.706 1.3807.6 0.000 0.003 0.007 0.467 0.697 1.3627.7 0.000 0.003 0.007 0.461 0.688 1.3447.8 0.000 0.003 0.007 0.455 0.679 1.3277.9 0.000 0.003 0.006 0.449 0.670 1.3108 0.000 0.003 0.006 0.444 0.662 1.294

8.1 0.000 0.003 0.006 0.438 0.654 1.278


8.2 0.000 0.003 0.006 0.433 0.646 1.2628.3 0.000 0.003 0.006 0.428 0.638 1.2478.4 0.000 0.002 0.006 0.423 0.630 1.2328.5 0.000 0.002 0.006 0.418 0.623 1.2188.6 0.000 0.002 0.006 0.413 0.616 1.2048.7 0.000 0.002 0.006 0.408 0.608 1.1908.8 0.000 0.002 0.006 0.404 0.602 1.1768.9 0.000 0.002 0.006 0.399 0.595 1.1639 0.000 0.002 0.006 0.395 0.588 1.150

9.1 0.000 0.002 0.006 0.390 0.582 1.1389.2 0.000 0.002 0.006 0.386 0.575 1.1259.3 0.000 0.002 0.006 0.382 0.569 1.1139.4 0.000 0.002 0.005 0.378 0.563 1.1019.5 0.000 0.002 0.005 0.374 0.557 1.0909.6 0.000 0.002 0.005 0.370 0.551 1.0789.7 0.000 0.002 0.005 0.366 0.546 1.0679.8 0.000 0.002 0.005 0.362 0.540 1.0569.9 0.000 0.002 0.005 0.359 0.535 1.04610 0.000 0.002 0.005 0.355 0.529 1.035

We hope these numbers will be of use to the practitioners mentioned in Section 1. Similartabulations could be easily derived for other values of p, µ and λ by using the BesselK (·) functionin MAPLE. A sample program is shown in the Appendix below.

Appendix

The following procedure in MAPLE can be used to generate tables similar to that presented inSection 3.

percent:=proc(lambda,mu,p)

local ff,pp,k,z;

ff:=sum((-1)**k*sqrt(k+1)*BesselK(1,2*sqrt(lambda*mu*(k+1)*z)),k=0..infinity);

ff:=1-4*sqrt(lambda*mu*z)*ff;

pp:=evalf(fsolve(ff=p,z=0..200)):

end proc;

Acknowledgments

The author would like to thank the Editor and the referee for carefully reading the paper and fortheir comments which greatly improved the paper.

References

NADARAJAH250

Abu-Salih, M. S. (1983). Distributions of the product and the quotient of power-function randomvariables. Arab Journal of Mathematics, 4, 77–90.

Balakrishnan, N. (1992). Handbook of the Logistic Distribution. New York: Marcel Dekker.

Bhargava, R. P. and Khatri, C. G. (1981). The distribution of product of independent beta ran-dom variables with application to multivariate analysis. Annals of the Institute of StatisticalMathematics, 33, 287–296.

Gradshteyn, I. S. and Ryzhik, I. M. (2000). Table of Integrals, Series, and Products (sixth edition).San Diego: Academic Press.

Harter, H. L. (1951). On the distribution of Wald’s classification statistic. Annals of MathematicalStatistics, 22, 58–67.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions (volume1, second edition). John Wiley and Sons, New York, NY.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions (volume2, second edition). John Wiley and Sons, New York, NY.

Kotz, S., Kozubowski, T. J. and Podgorski, K. (2001). The Laplace Distribution and Generaliza-tions: A Revisit with Applications to Communications, Economics, Engineering and Finance.Boston: Birkhauser.

Malik, H. J. and Trudel, R. (1986). Probability density function of the product and quotient oftwo correlated exponential random variables. Canadian Mathematical Bulletin, 29, 413–418.

Podolski, H. (1972). The distribution of a product of n independent random variables with gener-alized gamma distribution. Demonstratio Mathematica, 4, 119–123.

Prudnikov, A. P., Brychkov, Y. A. and Marichev, O. I. (1986). Integrals and Series (volumes 1, 2and 3). Amsterdam: Gordon and Breach Science Publishers.

Rathie, P. N. and Rohrer, H. G. (1987). The exact distribution of products of independent randomvariables. Metron, 45, 235–245.

Sakamoto, H. (1943). On the distributions of the product and the quotient of the independent anduniformly distributed random variables. Tohoku Mathematical Journal, 49, 243–260.

Springer, M. D. and Thompson, W. E. (1970). The distribution of products of beta, gamma andGaussian random variables. SIAM Journal on Applied Mathematics, 18, 721–737.

Steece, B. M. (1976). On the exact distribution for the product of two independent beta-distributedrandom variables. Metron, 34, 187–190.

Stuart, A. (1962). Gamma-distributed products of independent random variables. Biometrika, 49,564–565.

Tang, J. and Gupta, A. K. (1984). On the distribution of the product of independent beta randomvariables. Statistics & Probability Letters, 2, 165–168.

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TABLE OF CONTENTS, JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOLUME 3, NO. 2, 2008 MULTI-ANISOTROPIC GEVREY CLASSES AND ULTRADISTRIBUTIONS, D.CALVO, A.MORANDO,………………………………………………………….139 PERIODIC SOLUTIONS FOR P-LAPLACIAN DUFFING EQUATIONS WITH A DEVIATING ARGUMENT,WING-SUM CHEUNG,J.REN,……………………….163 BOSBACH AND RIECAN STATES ON RESIDUATED LATTICES,L.CIUNGU,175 COMPUTING VAR AND AVAR OF SKEWED-T DISTRIBUTION,S.DOKOV, S.STOYANOV,S.RACHEV,………………………………………………………....189 BALL CLOSURE PROPERTY IN FUZZY METRIC SPACES,H.EFE,……………209 THE CANONICAL COHERENT STATES ASSOCIATED WITH QUOTIENTS OF THE AFFINE WEYL-HEISENBERG GROUP,S.DAHLKE,D.LORENZ, P.MAASS,C.SAGIV,G.TESCHKE,…………………………………………………..215 BOUNDARY VALUE PROBLEMS IN PLAN SECTOR WITH CORNERS FOR A CLASS OF SOBOLEV SPACES OF DOUBLE WEIGHT,H.BENSERIDI,M.DILMI,233 ON THE PRODUCT OF LAPLACE AND LOGISTIC RANDOM VARIABLES, S.NADARAJAH,………………………………………………………………………243

Volume 3,Number 3 July 2008 ISSN:1559-1948 (PRINT), 1559-1956 (ONLINE) EUDOXUS PRESS,LLC


SCOPE AND PRICES OF

















Some Applications of Intuitionistic Fuzzy MetricSpaces

Servet KutukcuDepartment of Mathematics, Faculty of Science and Arts,

Ondokuz Mayis University, Kurupelit, 55139 Samsun, Turkey.e-mail: [email protected]

Sushil SharmaDepartment of Mathematics, Madhav Science College,

Ujjain, M.P. 456 001, India.e-mail: [email protected]

January 4, 2007

Abstract. In this paper, we prove that every intuitionistic fuzzy metricspace has a basis that is countably locally finite. We use this result to concludethat every intuitionistic fuzzy metric space is metrizable. We also introducenotions of orbitally complete intuitionistic fuzzy metric spaces and generalizedcontraction mappings, and prove fixed point theorems which are extensions ofthe classical Banach’s fixed point principle and fixed point theorems of Edelstein[4] and Grabiec [7]. Our results also extend, generalize and fuzzify several fixedpoint theorems on metric spaces, Menger probabilistic metric spaces, uniformspaces and fuzzy metric spaces.Keywords. Generalized contraction; Uniformly convergence; Intuitionistic

fuzzy metric space; Orbitally complete intuitionistic fuzzy metric space.AMS [2000]. 46S40, 47H10, 54H25.

1 IntroductionThe concept of fuzzy sets was introduced by Zadeh [16]. Following the concept offuzzy sets, fuzzy metric spaces have been introduced by Kramosil and Michalek[8], and George and Veeramani [5,6] modified the notion of fuzzy metric spaceswith the help of continuous t-norms. Recently, many authors have proved fixedpoint theorems involving fuzzy sets [2-4,7,9,10,12,14].As a generalization of fuzzy sets, Atanassov [1] introduced and studied the

concept of intuitionistic fuzzy sets. Recently, using the idea of intuitionisticfuzzy sets, Park [15] introduced the notion of intuitionistic fuzzy metric spaces

269


with the help of continuous t-norms and continuous t-conorms as a generaliza-tion of fuzzy metric spaces due to George and Veeramani [5], and showed thatevery metric induces an intuitionistic fuzzy metric, every fuzzy metric space is anintuitionistic fuzzy metric space and found a necessary and sufficient conditionfor an intuitionistic fuzzy metric space to be complete.In this paper, we observe that an intuitionistic fuzzy metric space is normal

and prove that in an intuitionistic fuzzy metric space X every open cover admitsa countably locally finite refinement which covers X. Using this result, we provethat every intuitionistic fuzzy metric space has a countably locally finite basis.We introduce notions of orbitally complete intuitionistic fuzzy metric spaces andgeneralized contraction mappings, and prove fixed point theorems. Our resultsalso extend, generalize and fuzzify several fixed point theorems on metric spaces,Menger probabilistic metric spaces, uniform spaces and fuzzy metric spaces.

2 PreliminariesDefinition 1 A binary operation ∗ : [0, 1]×[0, 1]→ [0, 1] is a continuous t-normif ∗ satisfies the following conditions:

(a) ∗ is commutative and associative;(b) ∗ is continuous;(c) a ∗ 1 = a for all a ∈ [0, 1];(d) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, and a, b, c, d ∈ [0, 1].

Definition 2 A binary operation ♦ : [0, 1] × [0, 1] → [0, 1] is a continuous t-conorm if ♦ satisfies the following conditions:

(a) ♦ is commutative and associative;(b) ♦ is continuous;(c) a♦0 = a for all a ∈ [0, 1];(d) a♦b ≤ c♦d whenever a ≤ c and b ≤ d, and a, b, c, d ∈ [0, 1].

Remark 1 The concepts of triangular norms (shortly t-norms) and triangu-lar conorms (shortly t-conorms) are known as the axiomatic skeletons that weuse for characterizing fuzzy intersections and fuzzy unions, respectively. Theseconcepts were originally introduced by Menger [11]. Several examples for theseconcepts were proposed by many authors (see [9-11,15]).

Lemma 1 ([15]) If ∗ is a continuous t-norm, ♦ is a continuous t-conorm andri ∈ (0, 1), 1 ≤ i ≤ 7, then

(a) If r1 > r2, there are r3, r4 ∈ (0, 1) such that r1 ∗ r3 ≥ r2 and r2♦r4 ≤ r1.(b) If r5 ∈ (0, 1), there are r6, r7 ∈ (0, 1) such that r6 ∗ r6 ≥ r5 and r7♦r7 ≤

r5.

KUTUKCU-SHARMA

270

Definition 3 ([15]) A 5-tuple (X,M,N, ∗,♦) is said to be an intuitionisticfuzzy metric space (shortly IFM-space) if X is an arbitrary set, ∗ is a continuoust-norm, ♦ is a continuous t-conorm and M, N are fuzzy sets on X2 × (0,∞)satisfying the following conditions: for all x, y, z ∈ X, s, t > 0,

(IFM-1) M(x, y, t) +N(x, y, t) ≤ 1;(IFM-2) M(x, y, t) > 0;(IFM-3) M(x, y, t) = 1 if and only if x = y;(IFM-4) M(x, y, t) =M(y, x, t);(IFM-5) M(x, y, t) ∗M(y, z, s) ≤M(x, z, t+ s);(IFM-6) M(x, y, .) : (0,∞)→ (0, 1] is continuous;(IFM-7) N(x, y, t) > 0;(IFM-8) N(x, y, t) = 0 if and only if x = y;(IFM-9) N(x, y, t) = N(y, x, t);(IFM-10) N(x, y, t)♦N(y, z, s) ≥ N(x, z, t+ s);(IFM-11) N(x, y, .) : (0,∞)→ (0, 1] is continuous.Then (M,N) is called an intuitionistic fuzzy metric on X. The functions

M(x, y, t) and N(x, y, t) denote the degree of nearness and the degree of non-nearness between x and y with respect to t, respectively.

Remark 2 Every fuzzy metric space (X,M, ∗) is an IFM-space of the form(X,M, 1 − M, ∗,♦) such that t-norm ∗ and t-conorm ♦ are associated, i.e.x♦y = 1− ((1− x) ∗ (1− y)) for all x, y ∈ [0, 1]. But the converse is not true.

Remark 3 ([15]) In an IFM-space (X,M,N, ∗,♦), M(x, y, .) is nondecreasingand N(x, y, .) is nonincreasing for all x, y ∈ X.

Example 1 (Induced intuitionistic fuzzy metric [15]) Let (X, d) be a met-ric space. Denote a ∗ b = ab and a♦b = min1, a+ b for all a, b ∈ [0, 1] and letMd and Nd be fuzzy sets on X2 × (0,∞) defined as follows:

Md(x, y, t) =t

t+ d(x, y), Nd(x, y, t) =

d(x, y)

t+ d(x, y)

Then (Md,Nd) is an intuitionistic fuzzy metric on X. We call this intuitionisticfuzzy metric induced by a metric d the standard intuitionistic fuzzy metric.

Remark 4 Note that the above example holds even with the t-norm a ∗ b =mina, b and the t-conorm a♦b = maxa, b and hence (Md, Nd) is an intu-itionistic fuzzy metric with respect to any continuous t-norm and continuoust-conorm.

Definition 4 ([15]) Let (X,M,N, ∗,♦) be an IFM-space. For t > 0, the openball B(x, r, t) with center x ∈ X and radius r ∈ (0, 1) is defined by B(x, r, t) =y ∈ X :M(x, y, t) > 1− r, N(x, y, t) < r. Let τ (M,N) be the set of all A ⊂ Xwith x ∈ A if and only if there exist t > 0 and r ∈ (0, 1) such that B(x, r, t) ⊂A. Then τ (M,N) is a topology on X (induced by the intuitionistic fuzzy metric

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(M,N)). This topology is Hausdorff and first countable. A sequence xn inX converges to x in X if and only if M(xn, x, t) tends to 1 and N(xn, x, t)tends to 0 as n tends to ∞, for each t > 0. A sequence xn in X is called aCauchy sequence if for each ε > 0 and λ ∈ (0, 1), there exists n0 ∈ N such thatM(xn, xm, ε) > 1− λ and N(xn, xm, ε) < λ for all n,m ≥ n0. An IFM-space iscalled complete if every Cauchy sequence is convergent.

Remark 5 Since ∗ and ♦ are continuous, the limit is uniquely determined from(IFM-5) and (IFM-10).

3 Main ResultsTheorem 2 Every intuitionistic fuzzy metric space is normal.

Proof. Let (X,M,N, ∗,♦) be an intuitionistic fuzzy metric space, and S,Kbe two disjoint closed sets in X. Let x ∈ S then x ∈ Kc. Since Kc is open,there exist rx ∈ (0, 1) and tx > 0 such that B(x, rx, tx) ∩K = ∅ for all x ∈ S.Similarly, there exist ry ∈ (0, 1) and ty > 0 such that B(x, ry, ty) ∩ S = ∅ forall x ∈ K. Let s = min rx, tx, ry, ty. Then we can find a p0 ∈ (0, p) such that(1 − p0) ∗ (1 − p0) > 1 − p and p0♦p0 < p. Define U = ∪x∈SB(x, p0, p/2) andV = ∪y∈KB(y, p0, p/2). Clearly U and V are open sets such that S ⊂ U andK ⊂ V . Now, we claim that U ∩ V = ∅. Let z ∈ U ∩ V . Then there existx ∈ S and y ∈ K such that z ∈ B(x, p0, p/2) and z ∈ B(y, p0, p/2). Therefore,we have

M(x, y, p) ≥ M(x, z, p/2) ∗M(y, z, p/2)

≥ (1− p0) ∗ (1− p0) > 1− p

and

N(x, y, p) ≤ N(x, z, p/2)♦N(y, z, p/2)≤ p0♦p0 < p.

Hence, y ∈ B(x, p, p). But B(x, p, p) ⊂ B(x, rx, tx), since s < tx, rx. Thus,B(x, rx, tx) ∩K is nonempty which is a contradiction. Therefore, U ∩ V = ∅.Hence, X is normal.

Remark 6 From the above theorem, we can easily deduce that every metriz-able space is normal. Since every intuitionistic fuzzy metric space is normal,Urysohn’s lemma and Tietze extension theorem are true in the case of intuition-istic fuzzy metric spaces.

Lemma 3 Let (X,M,N, ∗,♦) be an IFM-space. If A is an open covering of X,then there is an open covering B of X such that B is a countably locally finiterefinement of A.

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Proof. Since A is an open covering of X, by well ordering theorem, we canchoose a well ordering < for the collection A. Choose n ∈ N and for U ∈ A,define Sn(U) = x ∈ X : B(x, 1/n, 1/n) ⊂ U and Rn(U) = Sn(U) − ∪V <UV .If V,W ∈ A with V < W and if x ∈ Rn(V ), y ∈ Rn(W ) then we claim thatM(x, y, 1/n) ≤ 1 − 1/n and N(x, y, 1/n) ≥ 1/n. Since x ∈ Rn(V ), we havex ∈ Sn(V ) and since y ∈ Rn(W ) and V < W , clearly y is not in V and henceM(x, y, 1/n) ≤ 1−1/n and N(x, y, 1/n) ≥ 1/n. We can find a s ∈ (0, 1/n) suchthat (1− s) ∗ (1− s) ∗ (1− s) > 1− 1/n and s♦s♦s < 1/n.

Now, let En(U) = ∪ B(x, s, 1/3n) : x ∈ Rn(U). Clearly En(U)’s are openand we claim that En(U)’s are disjoint. Let V,W ∈ A with V < W and if x ∈En(V ), y ∈ En(W ) then we claim thatM(x, y, 1/3n) ≤ 1−s andN(x, y, 1/3n) ≥s. In fact, assume that M(x, y, 1/3n) ≥ 1 − s and N(x, y, 1/3n) ≤ s. Sincex ∈ En(V ) and y ∈ En(W ), there exist x0 ∈ Rn(V ) and y0 ∈ Rn(W ) suchthat M(x, x0, 1/3n) ≤ 1 − s, M(y0, y, 1/3n) ≤ 1 − s, N(x, x0, 1/3n) ≥ s andN(y0, y, 1/3n) ≥ s. Since V < W , we have M(x0, y01/n) ≤ 1 − 1/n andN(x0, y01/n) ≥ 1/n. But1− 1/n ≥ M(x0, y01/n) ≥M(x, x0, 1/3n) ∗M(x, y, 1/3n) ∗M(y, y0, 1/3n)

≥ (1− s) ∗ (1− s) ∗ (1− s)

> 1− 1/nand

1/n ≤ N(x0, y01/n) ≤ N(x, x0, 1/3n)♦N(x, y, 1/3n)♦N(y, y0, 1/3n)≤ s♦s♦s< 1/n

which are contradictions and henceM(x, y, 1/3n) ≤ 1−s and N(x, y, 1/3n) ≥ s.Define En = En(U) : U ∈ A. If y ∈ En(U), then there exists x in Rn(U)

such that y ∈ B(x, s, 1/3n). But s < 1/n and hence we have y ∈ B(x, s, 1/3n) ⊂y ∈ B(x, 1/n, 1/n) ⊂ U . Since En(U) ⊂ U for all U ∈ A, En refines A. Weclaim that En is locally finite. Since s ∈ (0, 1), we can find a r0 ∈ (0, 1) such that(1−r0)∗(1−r0) > 1−s and r0♦r0 < s. For each x ∈ X, B(x, r0, 1/6n) intersectsalmost one element of En. If B(x, r0, 1/6n) intersect En(U) and En(V ) withU < V , then there exist y ∈ En(U) and z ∈ En(V ) such that M(x, y, 1/6n) >1−r0, M(x, z, 1/6n) > 1−r0, N(x, y, 1/6n) < r0 and N(x, z, 1/6n) < r0. SinceU < V , we have M(y, z, 1/3n) > 1 − s and N(y, z, 1/3n) < s. Therefore, wehave

M(y, z, 1/3n) ≥ M(x, y, 1/6n) ∗M(x, z, 1/6n)

≥ (1− r0) ∗ (1− r0) > 1− s

and

N(y, z, 1/3n) ≤ N(x, y, 1/6n)♦N(x, z, 1/6n)≤ r0♦r0 < s


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which are contradictions and hence En is locally finite.Now consider the family B = ∪n∈NEn and let x ∈ X. Since A is a covering

of X, we can find a U ∈ A such that U is the first element of A that containsx. Since U is open, we can choose n ∈ N such that B(x, 1/n, 1/n) ⊂ U . Hencex ∈ Sn(U), but U is the first element of A that contains x, x ∈ Rn(U) and hencex ∈ En. Thus, we get a family B of sets satisfying the required conditions.

Theorem 4 Every intuitionistic fuzzy metric space has a basis that is countablylocally finite.

Proof. For a given m ∈ N, define Am = B(x, 1/m, 1/m) : x ∈ X, then Am

covers X for eachm. By Lemma 3, we can find an open covering Dm of X whichis a countably locally finite refinement of Am. Let D = ∪m∈NDm, then D iscountably locally finite. We claim that D is a basis for X. Let x ∈ X. For givenr ∈ (0, 1) and t > 0, we can choosem ∈ N such that (1−1/m)∗(1−1/m) > 1−r,1/m♦1/m < r and 1/m < t/2. If B is the element of Dm which contains x andsinceDm refines Am, then we can find a x0 ∈ X such that B ⊂ B(x0, 1/m, 1/m).Therefore, for any y ∈ B, we have

M(x, y, t) > M(x, y, 2/m) ≥M(x, x0, 1/m) ∗M(y, x0, 1/m)≥ (1− 1/m) ∗ (1− 1/m) > 1− r

and

N(x, y, t) < N(x, y, 2/m) ≤ N(x, x0, 1/m)♦N(y, x0, 1/m)≤ 1/m♦1/m < r.

Thus, y ∈ B(x, y, t) and hence B ⊂ B(x, y, t).

Remark 7 Since the topology induced by a metric and the corresponding in-tuitionistic fuzzy metric are same, by using above theorem, we can deduce thatevery metric space has a basis that is countably locally finite.

Corollary 5 Every intuitionistic fuzzy metric space is metrizable.

Since every intuitionistic fuzzy metric space is regular (Theorem 2) and sinceevery intuitionistic fuzzy metric space has a basis that is countably locally finite,the result follows from the Nagata-Smirnov theorem [13].

Definition 5 A sequence of maps Ti : X → X on an IFM-space (X,M,N, ∗,♦)converges uniformly to a map T : X → X if and only if for every ε > 0 andλ > 0, there exists a positive integer k = k(ε, λ) such that

M(Tx, Tix, ε) > 1− λ and N(Tx, Tix, ε) < λ

for every x ∈ X and i ≥ k.

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Corollary 6 Let X be any nonempty set and (Y, d) be a metric space. Let(Y,M,N, ∗,♦) be the induced intuitionistic fuzzy metric space. Then a sequenceof functions fn from X to Y converges uniformly to a function f from X toY with respect to the metric d if and only if fn converges uniformly to f withrespect to the intuitionistic fuzzy metric (M,N).

Proof. Assume that fn converges uniformly to f with respect to theintuitionistic fuzzy metric (M,N). Then, for given r ∈ (0, 1) and t > 0, thereexists k ∈ N such that M(fn(x), f(x), t) > 1 − r and N(fn(x), f(x), t) < r forall n ≥ k. Let ε > 0 be given. Taking r = ε/(t+ε), we haveM(fn(x), f(x), t) >1− ε/(t+ ε) and N(fn(x), f(x), t) < ε/(t+ ε), and hence d(fn(x), f(x)) < ε forall n ≥ k. Therefore, fn converges uniformly to f with respect to the metricd. The converse part can also be proved in the same way.

Definition 6 An orbit of T at a point x0 ∈ X is a sequence xn given by

O(T, x0) = xn : xn ∈ Txn−1, n = 1, 2, 3, ... .

An IFM-space (X,M,N, ∗,♦) is called T -orbitally complete if every Cauchysequence which is a subsequence of an orbit T at each x ∈ X converges to apoint of X.

Definition 7 A mapping T on IFM-space (X,M,N, ∗,♦) will be called a gen-eralized contraction if and only if there exists a constant q ∈ (0, 1) such that forevery x, y ∈ X,

M(Tx, Ty, qt) ≥ minM(x, y, t),M(x, Tx, t),M(y, Ty, t), (2.1)

M(x, Ty, 2t),M(y, Tx, 2t),

N(Tx, Ty, qt) ≤ maxN(x, y, t), N(x, Tx, t),N(y, Ty, t), (2.2)

N(x, Ty, 2t), N(y, Tx, 2t)

for all t > 0.

Throughout this paper, (X,M,N, ∗,♦) will denote the IFM-space with thefollowing conditions: for all x, y ∈ X and t > 0(IFM-7p) N(x, y, t) < 1,(IFM-11p) N(x, y, .) : (0,∞)→ [0, 1) is continuous,(IFM-12) limt→∞M(x, y, t) = 1,(IFM-13) limt→∞N(x, y, t) = 0.

Lemma 7 Let xn be a sequence in an IFM-space (X,M,N, ∗,♦) with theconditions (IFM-12) and (IFM-13). If there exists a constant q ∈ (0, 1) suchthat

M(xn, xn+1, qt) ≥M(xn−1, xn, t) (2.3)

andN(xn, xn+1, qt) ≤ N(xn−1, xn, t) (2.4)


275

for all t > 0 and n = 1, 2, 3, ... then xn is a Cauchy sequence in X.Proof. To prove that xn is a Cauchy sequence, we prove (2.5) and (2.6) aretrue for all n ≥ n0 and for every m ∈ N,

M(xn, xn+m, t) > 1− λ (2.5)

andN(xn, xn+m, t) < λ. (2.6)

Here we use induction method. From (2.3) and (2.4), we have

M(xn, xn+1, t) ≥ M(xn−1, xn, t/q)≥ M(xn−2, xn−1, t/q2)≥ ... ≥M(x0, x1, t/q

n)→ 1

and

N(xn, xn+1, t) ≤ N(xn−1, xn, t/q)≤ N(xn−2, xn−1, t/q2)≤ ... ≤ N(x0, x1, t/q

n)→ 0

as n→∞, i.e. for t > 0 and λ ∈ (0, 1), we can choose n0 ∈ N such thatM(xn, xn+1, t) > 1− λ and N(xn, xn+1, t) < λ.

Thus, (2.5) and (2.6) are true for m = 1. Suppose that (2.5) and (2.6) are truefor m then we shall show that they are also true for m+ 1.Using the definition of intuitionistic fuzzy metric space, (2.3) with (2.5) and

(2.4) with (2.6), we have

M(xn, xn+m+1, t) ≥ minM(xn, xn+m, t/2),M(xn+m, xn+m+1, t/2) > 1− λ

and

N(xn, xn+m+1, t) ≤ maxN(xn, xn+m, t/2), N(xn+m, xn+m+1, t/2) < λ.

Hence (2.5) and (2.6) are true for m+ 1. This completes the proof.

Lemma 8 Let (X,M,N, ∗,♦) be an intuitionistic fuzzy metric space. If thereexists q ∈ (0, 1) such that M(x, y, qt) ≥ M(x, y, t) and N(x, y, qt) ≤ N(x, y, t)for x, y ∈ X. Then x = y.

Proof. Since M(x, y, qt) ≥ M(x, y, t) and N(x, y, qt) ≤ N(x, y, t), thenM(x, y, t) ≥M(x, y, q−1t) and N(x, y, t) ≤ N(x, y, q−1t). By repeated applica-tion of above inequalities, we have

M(x, y, t) ≥ M(x, y, q−1t) ≥M(x, y, q−2t) ≥ ... ≥M(x, y, q−nt) ≥ ...and

N(x, y, t) ≤ N(x, y, q−1t) ≤ N(x, y, q−2t) ≤ ... ≤ N(x, y, q−nt) ≤ ..., n ∈ Nwhich → 1 and → 0 as n → ∞, respectively. Thus M(x, y, t) = 1 andN(x, y, t) = 0 for all t > 0 and we get x = y.

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Theorem 9 Let (X,M,N, ∗,♦) be an IFM-space where t∗t ≥ t and (1−t)♦(1−t) ≤ (1 − t) for all t ∈ [0, 1]. If T : X → X is a generalized contraction on Xand X is T -orbitally complete, then T has a unique fixed point y ∈ X andlimn→∞ Tnx = y for every x ∈ X.

Proof. Let x be an arbitrary point of X. Then we can construct a sequencexn in X as follows:

x0 = x, x1 = Tx0, x2 = Tx1, ..., xn = Txn−1, ... (2.7)

We will show that the sequence is fundamental in X, i.e., for each ε > 0 andλ ∈ (0, 1), there is an integer n0 such that n,m ≥ n0 implyM(xn, xm, ε) > 1−λand N(xn, xm, ε) < λ.Observe that, by (IFM-3) and (IFM-8), x 6= y implies

M(x, y, qt) < M(x, y, t) and N(x, y, qt) > N(x, y, t) (2.8)

for some t > 0. Also (d) in Definition 1 and t ∗ t ≥ t imply

M(x, z, t+ s) ≥ min M(x, y, t),M(y, z, s) (2.9)

and (d) in Definition 2 and (1− t)♦(1− t) ≤ (1− t) imply

N(x, z, t+ s) ≤ max N(x, y, t), N(y, z, s) (2.10)

for all x, y, z ∈ X and s, t > 0.Suppose that in the sequence (2.7) xn−1 6= xn for every integer n, since

xn−1 = xn = Txn−1 for some integer n implies immediately that (2.7) is fun-damental. Then, for xn−1, xn ∈ X, by (2.1)

M(xn, xn+1, qt) = M(Txn−1, Txn, qt) ≥ minM(xn−1, xn, t),M(xn−1, xn, t),M(xn, xn+1, t),M(xn−1, xn+1, 2t),M(xn, xn, 2t)

= minM(xn−1, xn, t),M(xn, xn+1, t),M(xn−1, xn+1, 2t)

and since M(xn−1, xn+1, 2t) ≥ min M(xn−1, xn, t),M(xn, xn+1, t) it followsthat

M(xn, xn+1, qt) ≥ minM(xn−1, xn, t),M(xn, xn+1, t), (2.11)

by (2.2)

N(xn, xn+1, qt) = N(Txn−1, Txn, qt) ≤ maxN(xn−1, xn, t), N(xn−1, xn, t),N(xn, xn+1, t),N(xn−1, xn+1, 2t), N(xn, xn, 2t)

= maxN(xn−1, xn, t), N(xn, xn+1, t), N(xn−1, xn+1, 2t).

and since N(xn−1, xn+1, 2t) ≤ max N(xn−1, xn, t), N(xn, xn+1, t) it followsthat

N(xn, xn+1, qt) ≤ maxN(xn−1, xn, t), N(xn, xn+1, t) (2.12)

for all t > 0.


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Since we assume that xn 6= xn+1 for each integer n, (2.8) implies that

M(xn, xn+1, qt) ≥ M(xn, xn+1, t),

N(xn, xn+1, qt) ≤ N(xn, xn+1, t)

which are impossible. Then,for each integer n, we have

M(xn, xn+1, qt) ≥ M(xn−1, xn, t), (2.13)

N(xn, xn+1, qt) ≤ N(xn−1, xn, t). (2.14)

By (2.13) and (2.14), for an arbitrary integer n, we have

M(xn, xn+1, t) ≥ M(xn−1, xn, t/q) ≥ ... ≥M(x0, x1, t/qn),

N(xn, xn+1, t) ≤ N(xn−1, xn, t/q) ≤ ... ≤ N(x0, x1, t/qn).

By noting that M(x0, x1, t/qn) → 1 and N(x0, x1, t/qn) → 0 as n → ∞, using

inequalities (2.11) and (2.12), we have

M(xn, xn+1, qt) ≥ M(xn−1, xn, t),N(xn, xn+1, qt) ≤ N(xn−1, xn, t)

for each integer n, q ∈ (0, 1) and t > 0. Hence, by Lemma 7, xn is a Cauchysequence in X. Since (2.7) is an orbit of T at x ∈ X and X is T -orbitallycomplete, there is a point y ∈ X such that y = limn→∞ xn = limn→∞ Tnx.Now, we prove that

Ty = limn→∞xn+1 = y. (2.15)

Let B(Ty, λ, ε) be any neighborhood of Ty. Since y = limn→∞ xn, there existsan integer k such that n ≥ k implies

M

µxn, y,

µ1− q

2q

¶ε

¶> 1− λ, N

µxn, y,

µ1− q

2qε

¶¶< λ, (2.16)

M

µxn, xn+1,

µ1− q

2qε

¶¶> 1− λ, N

µxn, xn+1,

µ1− q

2qε

¶¶< λ.(2.17)

Then, by (2.1) and (2.2), we have

M(xn+1, T y, ε) = M(Txn, Ty, qε/ε) ≥ minM(xn, y, ε/q),M(xn, xn+1, ε/q),

M(y, Ty, ε/q),M(xn, Ty, 2ε/q),M(y, xn+1, 2ε/q),

N(xn+1, Ty, ε) = N(Txn, Ty, qε/ε) ≤ maxN(xn, y, ε/q), N(xn, xn+1, ε/q),N(y, Ty, ε/q), N(xn, Ty, 2ε/q), N(y, xn+1, 2ε/q)

for all t > 0. Since by (2.9)

M

µy, Ty,

ε

q

¶= M

µy, Ty,

µ1− q

2q+1 + q

2q

¶ε

¶≥ min

½M

µy, xn+1,

µ1− q

2q

¶ε

¶,M

µxn+1, Ty,

µ1 + q

2q

¶ε

¶¾

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and

M

µxn, Ty,

2ε

q

¶≥ min

½M

µxn, xn+1,

ε

q

¶,M

µxn+1, Ty,

ε

q

¶¾,

we obtain, as M is nondecreasing, that

M(xn+1, T y, ε) ≥ min M

³xn, y,

³1−q2q

´ε´,M

³xn, xn+1,

³1−q2q

´ε´,

M³xn+1, y,

³1−q2q

´ε´,M

³xn+1, Ty,

³1+q2q

´ε´ ,

(2.18)and also since by (2.10)

N

µy, Ty,

ε

q

¶= N

µy, Ty,

µ1− q

2q+1 + q

2q

¶ε

¶≤ max

½N

µy, xn+1,

µ1− q

2q

¶ε

¶, N

µxn+1, T y,

µ1 + q

2q

¶ε

¶¾and

N

µxn, Ty,

2ε

q

¶≤ max

½N

µxn, xn+1,

ε

q

¶, N

µxn+1, Ty,

ε

q

¶¾,

we obtain, as N is nonincreasing, that

N(xn+1, Ty, ε) ≤ max N

³xn, y,

³1−q2q

´ε´, N

³xn, xn+1,

³1−q2q

´ε´,

N³xn+1, y,

³1−q2q

´ε´, N

³xn+1, Ty,

³1+q2q

´ε´ (2.19)

Hence for all n ≥ k, by (2.16) and (2.17), we have

M(xn+1, Ty, ε) > 1− λ and N(xn+1, Ty, ε) < λ or (2.20)

M(xn+1, T y, ε) = M

µxn+1, Ty,

µ1 + q

2q

¶ε

¶and

N(xn+1, T y, ε) = N

µxn+1, Ty,

µ1 + q

2q

¶ε

¶. (2.21)

Thus, we proved (2.15) if (2.20) is valued. If (2.20) were false, then substitutingsin (2.18) and (2.19) ε by ε1 =

1+q2q ε > ε, it would follow

M(xn+1, T y, ε1) = M

µxn+1, Ty,

µ1 + q

2q

¶ε1

¶=M

Ãxn+1, Ty,

µ1 + q

2q

¶2ε

!,

N(xn+1, T y, ε1) = N

µxn+1, Ty,

µ1 + q

2q

¶ε1

¶= N

Ãxn+1, Ty,

µ1 + q

2q

¶2ε

!.


279

Hence

M(xn+1, T y, ε) = M

µxn+1, Ty,

µ1 + q

2q

¶ε

¶=M

Ãxn+1, Ty,

µ1 + q

2q

¶2ε

!,

N(xn+1, T y, ε) = N

µxn+1, Ty,

µ1 + q

2q

¶ε

¶= N

Ãxn+1, Ty,

µ1 + q

2q

¶2ε

!.

Proceeding in this direction, we would obtain that

1− λ > M(xn+1, Ty, ε) = ... =M

Ãxn+1, Ty,

µ1 + q

2q

¶kε

!→ 1,

λ < N(xn+1, Ty, ε) = ... = N

Ãxn+1, Ty,

µ1 + q

2q

¶kε

!→ 0

as k →∞ which are contradictions. Therefore the inequalities (2.20) is correct,which imply (2.15). So, we conclude, there exists a fixed point for T .To prove the uniqueness of the fixed point y in (2.15), suppose that w 6= y

and Tw = w. Then, by (2.1) and (2.2), we have

M(y,w, qt) = M(Ty, Tw, qt) ≥ min½

M(y, w, t),M(y, Ty, t),M(w,Tw, t),M(y, Tw, 2t),M(w, Ty, 2t)

¾= min M(y, w, t), 1, 1,M(y, w, 2t),M(w, y, 2t)= M(y,w, t),

N(y, w, qt) = N(Ty, Tw, qt) ≤ max½

N(y, w, t), N(y, Ty, t), N(w,Tw, t),N(y, Tw, 2t), N(w, Ty, 2t)

¾= max N(y, w, t), 0, 0, N(y, w, 2t), N(w, y, 2t)= N(y, w, t)

hence, by Lemma 8, we have w = y. This completes the proof.Next, we prove Theorem 9 in T -orbitally complete metric space:

Corollary 10 Let (X,d) be a metric space and let T : X → X be a mapping.If

d(Tx, Ty) ≤ qmax

½d(x, y), d(x, Tx), d(y, Ty),[d(x, Ty) + d(y, Tx)] /2

¾for some q < 1, all x, y ∈ X and X is T -orbitally complete, then T has a uniquefixed point z ∈ X and limn→∞ Tnx = z.

Proof. The proof follows from Theorem 9 considering the induced IFM-space(X,M, N, ∗,♦), where a∗b = ab, a♦b = min 1, a+ b , M(x, y, t) = t/t+d(x, y)and N(x, y, t) = d(x, y)/t+ d(x, y).

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Corollary 11 Let (X,M,N, ∗,♦) be a complete IFM-space where t ∗ t ≥ t and(1− t)♦(1− t) ≤ (1− t) for all t ∈ [0, 1]. If T : X → X is a contraction on X,i.e., if for each x, y ∈ X

M(Tx, Ty, qt) ≥M(x, y, t),N(Tx, Ty, qt) ≤ N(x, y, t)

then there is a unique z ∈ X such that z = Tz. Moreover, Tnx → z for everyz ∈ X.

Proof. The proof is easy from Theorem 9.Next, we prove Corollary 11 in complete fuzzy metric space:

Corollary 12 ([7]) Let (X,M, ∗, ) be a complete fuzzy metric space where t∗t ≥t for all t ∈ [0, 1]. If T : X → X is a contraction on X, i.e., if for each x, y ∈ X

M(Tx, Ty, qt) ≥M(x, y, t)

then there is a unique z ∈ X such that z = Tz.

Proof. In the view of Remark 2, the proof follows from Corollary 11.We will denote subset I of X defined by

I = x ∈ X : there is some Ti such that Tix = x .

Theorem 13 Let Tii∈N be a sequence of maps on an IFM-space (X,M,N, ∗,♦)where t ∗ t ≥ t and (1 − t)♦(1 − t) ≤ (1 − t) for all t ∈ [0, 1]. Let T : X → Xbe generalized contraction on X and X be T -orbitally complete. If each Ti hasat least one fixed point yi and if the sequence Tii∈N on the subset I convergesuniformly to T , then the sequence xii∈N converges to a unique fixed point yof T .

Proof. By Theorem 9, the mapping T has a unique fixed point y. To showthat y = limi→∞ yi, let B(y, λ, ε) be an arbitrary neighborhood of y. We willshow that

M(yi, y, ε) > 1− λ and N(yi, y, ε) < λ

for almost all i ∈ N. Since Ti converges uniformly to T , there exists k ∈ Nsuch that

M(Tx, Tix, (1− q)ε/2) > 1− λ and N(Tx, Tix, (1− q)ε/2) < λ (2.22)

for i ≥ k. For arbitrary yi ∈ X for which Tiyi = yi we have by (2.9) and (2.10)

M(yi, y, ε) = M

µyi, y,

µ1− q

2+1 + q

2

¶ε

¶≥ min

½M

µTiyi, Tyi,

µ1− q

2

¶ε

¶,M

µTyi, y,

µ1 + q

2

¶ε

¶¾.(2.23)


281

Since T is a generalized contraction, y = Ty, yi = Tyi,M is nondecreasing andN is nonincreasing, we have

M

µTyi, y,

µ1 + q

2

¶ε

¶=M

µTyi, Ty,

µq1 + q

2q

¶ε

¶

≥ min M

³yi, y,

³1+q2q

´ε´,M

³yi, T yi,

³1+q2q

´ε´,M

³y, Ty,

³1+q2q

´ε´,

M³yi, Ty,

³1+qq

´ε´,M

³y, Tyi,

³1+qq

´ε´

≥ min M

³yi, y,

³1+q2q

´ε´,M

³Tiyi, T yi,

³1+q2q

´ε´,

M³y, Tyi,

³1+qq

´ε´ .

Using that

M (y, Tyi, (1 + q)/qε) ≥ min½M

µyi, y,

µ1 + q

2q

¶ε

¶,M

µTiyi, Tyi,

µ1 + q

2q

¶ε

¶¾we have

M (y, Tyi, (1 + q)/2ε) ≥ min M (yi, y, (1 + q)/2qε) ,M (Tiyi, Tyi, (1 + q)/2qε)≥ min M (yi, y, (1 + q)/2qε) ,M (Tiyi, Tyi, (1− q)/2qε) .

Then (2.23) results in

M(yi, y, ε) ≥ min M (Tiyi, Tyi, (1− q)/2ε) ,M (Tyi, y, (1 + q)/2ε) .

Hence, by (2.22) we have

M(yi, y, ε) > 1− λ and N(yi, y, ε) < λ or (2.24)

M(yi, y, ε) =M (yi, y, (1 + q)/2qε) and N(yi, y, ε) = N (yi, y, (1 + q)/2qε)(2.25)

for all i ≥ k. The assertion of Theorem follows if (2.24) is valid. SinceM(yi, y, ε) ≤ 1 − λ and N(yi, y, ε) ≥ λ imply (as in the proof of Theorem3)

M(yi, y, ε) = M (yi, y, (1 + q)/2qε)→ 1 and

N(yi, y, ε) = N (yi, y, (1 + q)/2qε)→ 0

as n→∞, which is a contradiction, we see that (2.24) is correct. This completesthe proof.

Corollary 14 Let Tii∈N be a sequence of maps on an IFM-space (X,M,N, ∗,♦)such that xi = Txi for some xi ∈ X and let T0 be a contraction mapping on Xwith a fixed point x0 ∈ X. If Ti converges uniformly to T0, then the sequencexi converges to x0.

Proof. The proof follows easily from Theorem 13.

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References[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986),

87-96.

[2] L.B. Ciric, A generalization of Banach’s contraction principle, Proc. Amer.Math. Soc. 45 (1974), 267-273.

[3] L.B. Ciric, On fixed points generalized contractions on probabilistic metricspaces, Publ. De L’institut Mathematique 18 (32) (1975), 71-78.

[4] M. Edelstein, An extension of Banach’s contraction principle, Proc. Amer.Math. Soc. 12 (1961), 7-10.

[5] A. George, P. Veeramani, On some results in fuzzy metric spaces, FuzzySets and Systems 64 (1994), 395-399.

[6] A. George, P. Veeramani, Some theorems in fuzzy metric spaces, J. FuzzyMath. 3 (1995), 933-940.

[7] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems27 (1988), 385-389.

[8] O. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces,Kybernetika 11 (1975), 326-334.

[9] S. Kutukcu, A fixed point theorem in Menger spaces, Int. Math. J., 1(32)(2006), 1543-1554.

[10] S. Kutukcu, A common fixed point theorem for a sequence of self mapsin intuitionistic fuzzy metric spaces, Commun. Korean Math. Soc., 21(4)(2006), 679-687.

[11] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. 28 (1942), 535-537.

[12] S.N. Mishra, N. Sharma, S.L. Singh, Common fixed points of maps on fuzzymetric spaces, Internat. J. Math. & Math. Sci. 17 (1994), 253-258.

[13] J.R. Munkres, Topology-a first course, Prentice-Hall., New Jersey, 1975.

[14] R.P. Pant, Common fixed point theorems for contractive maps, J. Math.Anal. Appl. 226 (1998), 251-258.

[15] J.H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons & Fractals22 (2004), 1039-1046.

[16] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965), 338-353.


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284

Some Theorems in Intuitionistic Fuzzy MetricSpaces

Servet Kutukcu1, Cemil Yildiz21Department of Mathematics, Faculty of Science and Arts,Ondokuz Mayis University, Kurupelit, 55139 Samsun, Turkey.

[email protected] of Mathematics, Faculty of Science and Arts,Gazi University, Teknikokullar, 06500 Ankara, Turkey.

[email protected]

September 17, 2006

Abstract

In this paper, we prove that the topology induced by any complete in-tuitionistic fuzzy metric space is completely metrizable. We also introducethe notions of t-uniformly continuity, t-equicontinuity and t-equinormality,and state new version of the Ascoli-Arzela theorem for intuitionistic fuzzymetric spaces.

Keywords. Topology, intuitionistic fuzzy metric space, t-uniformcontinuity,t-equicontinuity, t-equinormality, intuitionistic fuzzy contrac-tive map, compact.

M.S.C. (2000). 54A40; 03E72; 54H25

1 IntroductionOne of the main problems in the theory of fuzzy topological spaces is to obtaina consistent notion of a fuzzy metric space. Many authors have investigated thisquestion and several different notions of a fuzzy metric space have been defined.In particular, George and Veeramani [5,6] introduced and studied an interestingnotion of fuzzy metric space. Recently, using the idea of intuitionistic fuzzy setsintroduced by Atanassov [1], Park [16] defined the notion of intuitionistic fuzzymetric space as a natural generalization of fuzzy metric space due to Georgeand Veeramani [5], and proved that every intuitionistic fuzzy metric space gen-erates a Hausdorff first countable topology. Saadati and Park [17] obtained astronger result. In fact, they introduced a uniform structure on intuitionisticfuzzy metric space in the sense of Park [16] and proved (Lemma 1) that thetopology generated by any intuitionistic fuzzy metric space is metrizable.

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In this paper, we prove that if intuitionistic fuzzy metric space is complete,then the generated topology is completely metrizable. We introduce the conceptof t-uniform continuity and uniform continuity, and discuss several propertiesof them. We observe that every t-uniformly continuous mapping is uniformlycontinuous but converse does not hold in general. Moreover, we show that everycontinuous mapping on a compact intuitionistic fuzzy metric space is t-uniformlycontinuous and, for each intuitionistic fuzzy metric space (X,M,N, ∗,♦) suchthat every real valued continuous function is uniformly continuous, there is anintuitionistic fuzzy metric on X compatible with the topology generated by(M,N) for which every real valued continuous function is t-uniformly contin-uous. We also introduce the concept of t-equinormality to characterize intu-itionistic fuzzy metric spaces for which every real valued continuous functionis t-uniform continuous. We observe that a compact intuitionistic fuzzy metricspace is separable and introduce the concept of t-equicontinuity. Using these re-sults, we extend the Ascoli-Arzela theorem on intuitionistic fuzzy metric space.We also prove that every intuitionistic fuzzy metric space is normal and as aresult of that Urysohn’s lemma and Tietze extension theorem are true in thecase of intuitionistic fuzzy metric space in the sense of Park [16].

2 PreliminariesDefinition 1 A binary operation ∗ : [0, 1]× [0, 1]→ [0, 1] is continuous t-normif ∗ is satisfying the following conditions: for all a, b, c, d ∈ [0, 1](a) ∗ is commutative and associative;(b) ∗ is continuous;(c) a ∗ 1 = a for all a ∈ [0, 1];(d) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d.

Definition 2 A binary operation ♦ : [0, 1] × [0, 1] → [0, 1] is continuous t-conorm if ♦ is satisfying the following conditions: for all a, b, c, d ∈ [0, 1](a) ♦ is commutative and associative;

(b) ♦ is continuous;

(c) a♦0 = a for all a ∈ [0, 1];(d) a♦b ≤ c♦d whenever a ≤ c and b ≤ d.

Remark 1 The concepts of triangular norms (t-norms) and triangular conorms(t-conorms) are known as the axiomatic skeletons that we use for characteriz-ing fuzzy intersections and unions, respectively. These concepts were originallyintroduced by Menger [15] in his study of statistical metric spaces. Several ex-amples for these concepts were proposed by many authors (see[8,10-14,16,18]).

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Remark 2 ([16])

(a) For any r1, r2 ∈ (0, 1) with r1 > r2, there exist r3, r4 ∈ (0, 1) such thatr1 ∗ r3 ≥ r2 and r1 ≥ r4♦r2.

(b) For any r5 ∈ (0, 1), there exist r6, r7 ∈ (0, 1) such that r6 ∗ r6 ≥ r5 andr5 ≥ r7♦r7.

Definition 3 ([16]) The 5-tuble (X,M,N, ∗,♦) is an intuitionistic fuzzy met-ric space if X is an arbitrary set, ∗ is a continuous t-norm, ♦ is a continuoust-conorm and M,N are fuzzy sets on X2 × (0,∞) satisfying the following con-ditions: for all x, y, z ∈ X, s, t > 0,

(a) M(x, y, t) +N(x, y, t) ≤ 1;(b) M(x, y, t) > 0;

(c) M(x, y, t) = 1 iff x = y;

(d) M(x, y, t) =M(y, x, t);

(e) M(x, z, t+ s) ≥M(x, y, t) ∗M(y, z, s) for all t, s > 0;(f) M(x, y, .) : (0,∞)→ (0, 1] is continuous;

(g) N(x, y, t) > 0;

(h) N(x, y, t) = 0 iff x = y;

(i) N(x, y, t) = N(y, x, t);

(j) N(x, z, t+ s) ≤ N(x, y, t)♦N(y, z, s) for all t, s > 0;

(k) N(x, y, .) : (0,∞)→ (0, 1] is continuous.

Then (M,N) is called an intuitionistic fuzzy metric on X. The functionsM(x, y, t) and N(x, y, t) denote the degree of nearness and the degree of nonn-earness between x and y with respect to t, respectively.Until now, (X,M,N, ∗,♦) denotes an intuitionistic fuzzy metric space with

the following condition:

(l) N(x, y, t) < 1 for all x, y ∈ X and t > 0.

Remark 3 In a metric space (X, d) if we define a∗b = ab, a♦b = min 1, a+ b ,Md(x, y, t) = t/[t+d(x, y)] and Nd(x, y, t) = d(x, y)/[t+d(x, y)] then (X,M,N, ∗,♦) is an intuitionistic fuzzy metric space [16]. We call this (Md, Nd) as thestandard intuitionistic fuzzy metric induced by d. In the case that d is theEuclidean metric on R, the induced intuitionistic fuzzy metric will be denotedby (M|.|, N|.|) and the corresponding intuitionistic fuzzy metric space, denoted(R,M|.|, N|.|, ∗,♦), will be called the Euclidean intuitionistic fuzzy metric space.In [16], it is proved that every intuitionistic fuzzy metric (M,N) on X generates


287

a topology τ (M,N) on X which has as a base the family of open sets of the formB(x, r, t) : x ∈ X, r ∈ (0, 1), t > 0, where B(x, r, t) = y ∈ X : M(x, y, t) >1− r,N(x, y, t) < r for every r, with r ∈ (0, 1), and t > 0. Clearly, this topol-ogy is Hausdorff and first countable. Moreover, if (X, d) is a metric space, thenthe topology generated by d coincides with the topology τ (Md,Nd) generated by theintuitionistic fuzzy metric (Md, Nd).

3 The ResultsDefinition 4 A topological space (X, τ) is said to be admit a compatible in-tuitionistic fuzzy metric if there is an intuitionistic fuzzy metric (M,N) on Xsuch that τ (M,N) = τ .

Lemma 1 ([17]) Let (X,M,N, ∗,♦) be an intuitionistic fuzzy metric space.Then (X, τ (M,N)) is a metrizable topological space.

Remark 4 By results of Park [16] mentioned above, every metrizable topolog-ical space admits a compatible intuitionistic fuzzy metric. By results of Saa-dati and Park [17], it is easy to see that if (X,M,N, ∗,♦) is an intuitionisticfuzzy metric space then Un : n ∈ N is a base for a uniformity compatible withτ (M,N), where Un = (x, y) ∈ X × X : M(x, y, 1/n) > 1 − (1/n), N(x, y, 1/n)< 1/n for all n ∈ N. Thus, if (X,M,N, ∗,♦) is an intuitionistic fuzzy metricspace then the topological space (X, τ (M,N)) is metrizable.

Corollary 2 A topological space is metrizable if and only if it admits a com-patible intuitionistic fuzzy metric.

Proof. Let (X, τ) be a metrizable topological space and d be a metric on Xcompatible with τ . Then, the intuitionistic fuzzy metric (Md, Nd), induced byd, is compatible with τ [16]. The converse follows immediately from Lemma 1.

Corollary 3 ([16]) Every separable intuitionistic fuzzy metric space is secondcountable.

Proof. Let (X,M,N, ∗,♦) be a separable intuitionistic fuzzy metric space.By Lemma 1, (X, τ (M,N)) is a separable metrizable space. So, it is secondcountable.

Definition 5 ([16]) Let (X,M,N, ∗,♦) be an intuitionistic fuzzy metric space.A sequence xn in X converges to x in X if and only if M(xn, x, t) tendsto 1 and N(xn, x, t) tends to 0 as n tends to ∞, for each t > 0. A sequencexn in X is said to be a Cauchy sequence if for each ε, 0 < ε < 1 and t > 0,there exists n0 ∈ N such that M(xn, xm, t) > 1 − ε and N(xn, xm, t) < ε forall n,m ≥ n0. An intuitionistic fuzzy metric space is said to be complete ifevery Cauchy sequence is convergent. It is called compact, if every sequence inX contains a convergent subsequence.

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Let us recall that a metrizable topological space (X, τ) is said to be com-pletely metrizable if it admits a complete metric [3]. On the other hand, anintuitionistic fuzzy metric space (X,M,N, ∗,♦) is called complete [16] if everyCauchy sequence is convergent. If (X,M,N, ∗,♦) is a complete intuitionisticfuzzy metric space, we will say that (M,N) is a complete intuitionistic fuzzymetric on X.

Theorem 4 Let (X,M,N, ∗,♦) be a complete intuitionistic fuzzy metric space.Then, (X, τ (M,N)) is completely metrizable.

Proof. It follows from the proof of Lemma 1 that Un : n ∈ N is a base fora uniformity U on X compatible with τ (M,N), where Un = (x, y) ∈ X × X :M(x, y, 1/n) > 1 − (1/n),N(x, y, 1/n) < 1/n for every n ∈ N. Then, thereexists a metric d on X whose induced uniformity coincides with U . To showthat d is complete, we will prove that for given a Cauchy sequence (xn) in(X, d), (xn) is Cauchy sequence in (X,M,N, ∗,♦). For given r ∈ (0, 1) andt > 0, choose a k ∈ N such that 1/k ≤ min r, t. Then, there exists a n0 ∈ Nsuch that (xn, xm) ∈ Uk for every n,m ≥ n0. Consequently,

M(xn, xm, t) ≥M(xn, xm, 1/k) > 1− (1/k) ≥ 1− r

andN(xn, xm, t) ≤ N(xn, xm, 1/k) < (1/k) ≤ r

for each n,m ≥ n0.Hence, we have shown that (xn) is Cauchy sequence in complete intuitionistic

fuzzy metric space (X,M,N, ∗,♦), so it is convergent with the respect to τ (M,N).Thus, d is a complete metric on X. We conclude that (X, τ (M,N)) is completelymetrizable.

Corollary 5 A topological space is completely metrizable if and only if it admitsa compatible complete intuitionistic fuzzy metric.

Proof. Let (X, τ) be a completely metrizable space and d be a completemetric on X compatible with τ . Then, the intuitionistic fuzzy metric (Md, Nd)induced by d is complete [16], and it is compatible with τ . The converse followsimmediately from Theorem 4.Since every completely metrizable space is a Baire space [3], we deduce from

Theorem 4 the following.

Corollary 6 ([16]) Every complete intuitionistic fuzzy metric space is a Bairespace.

Definition 6 A mapping f from an intuitionistic fuzzy metric space (X,M,N, ∗,♦) to an intuitionistic fuzzy metric space (Y,O, P, ,M) is said to be uni-formly continuous if for each ε ∈ (0, 1), there exist r ∈ (0, 1) and s > 0 suchthat M(x, y, s) > 1 − r and N(x, y, s) < r imply O(f(x), f(y), t) > 1 − ε andP (f(x), f(y), t) < ε, for all x, y ∈ X and t > 0.


289

Definition 7 A mapping f from an intuitionistic fuzzy metric space (X,M,N, ∗,♦) to an intuitionistic fuzzy metric space (Y,O, P, ,M) is said to be t -uniformly continuous if for each ε ∈ (0, 1), there exists r ∈ (0, 1) such thatM(x, y, t) > 1 − r and N(x, y, t) < r imply O(f(x), f(y), t) > 1 − ε andP (f(x), f(y), t) < ε, for all x, y ∈ X and t > 0.

Remark 5 It is easy to see that every t-uniformly continuous mapping is uni-formly continuous. In example below, we will show that the converse does nothold. It is also clear that every t-uniformly continuous mapping from an in-tuitionistic fuzzy metric space (X,M,N, ∗,♦) to an intuitionistic fuzzy metricspace (Y,O, P, ,M) is continuous from (X, τ (M,N)) to (Y, τ (O,P )). By a compactintuitionistic fuzzy metric space, we mean an intuitionistic fuzzy metric space(X,M,N, ∗,♦) such that (X, τ (M,N)) is a compact topological space. In [7-9],these notions were introduced and studied in fuzzy metric spaces.

Example 1 Let X = 1∪n1− 1

n+1 : n ∈ No, a∗b = ab and a♦b = min1, a+

b for all a, b ∈ [0, 1]. For each x, y ∈ X and t > 0, define

M(x, y, t) =

1, if x = ytxy, if x 6= y and t < 1xy, if x 6= y and t ≥ 1

and

N(x, y, t) =

0, if x = y1− txy, if x 6= y and t < 11− xy, if x 6= y and t ≥ 1

It is easy to check that (X,M,N, ∗,♦) is an intuitionistic fuzzy metric space.Moreover, for each x ∈ X, B(x, 1/2, 1/2) = x, so τ (M,N) is the discretetopology on X.

Now, let f be any continuous real valued function on X. Given ε ∈ (0, 1)and t > 0, choose r = s = 1/2. Then, M(x, y, s) > 1− r and N(x, y, s) < r ifand only if x = y. Therefore, f is uniformly continuous from (X,M,N, ∗,♦)to (R,M|.|, N|.|, ∗,♦).Finally, let f(1) = 1 and f(x) = 0 for all x ∈ X\ 1. Given ε = 1/2

and t = 1, there exists n ∈ N such that 1n+1 < r for each r ∈ (0, 1), so

M(1, 1− 1n+1 , t) > 1−r and N(1, 1− 1

n+1 , t) < r, but M|.|(f(1), f(1− 1n+1), t) =

M|.|(1, 0, 1) = 1/2 = 1 − ε and N|.|(f(1), f(1 − 1n+1), t) = N|.|(1, 0, 1) = 1/2 =

ε. We conclude that f is not t-uniformly continuous from (X,M,N, ∗,♦) to(R,M|.|, N|.|, ∗,♦).Proposition 7 Every continuous mapping from a compact intuitionistic fuzzymetric space (X,M,N, ∗,♦) to an intuitionistic fuzzy metric space (Y,O, P, ,M) is t-uniformly continuous.

Proof. Suppose that there is a continuous mapping f from the compactintuitionistic fuzzy metric space (X,M,N, ∗,♦) to an intuitionistic fuzzy metric

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space (Y,O, P, ,M) which is not t-uniformly continuous. Then there exist ε ∈(0, 1), and two sequences (xn) and (yn) in X such that M(xn, yn, t) > 1− 1/nand N(xn, yn, t) < 1/n but O(f(xn), f(yn), t) ≤ 1−ε and P (f(xn), f(yn), t) ≥ εfor t > 0 and all n ∈ N.By compactness of (X, τ (M,N)), there are subsequences (xnk) and (ynk) of

(xn) and (yn) respectively, and points x, y ∈ X such that xnk → x and ynk → yin (X, τ (M,N)). Since

M(x, y, 3t) ≥M(x, xnk , t) ∗M(xnk , ynk , t) ∗M(ynk , y, t)

andN(x, y, 3t) ≤ N(x, xnk , t)♦N(xnk , ynk , t)♦N(ynk , y, t),

those immediately follow that M(x, y, 3t) = 1 and N(x, y, 3t) = 0, so x = y.Hence, by continuity of f , f(xnk) → f(x) and f(ynk) → f(y) in (Y, τ (O,P )).Choose δ > 0 such that (1 − δ) (1 − δ) > 1 − ε and δ M δ < ε. Then, thereis k0 ∈ N such that O(f(x), f(xnk), t/2) > 1 − δ, O(f(x), f(ynk), t/2) > 1 − δ,P (f(x), f(xnk), t/2) < δ and P (f(x), f(ynk), t/2) < δ for all k ≥ k0. So

O(f(xnk), f(ynk), t) ≥ O(f(xnk), f(x), t/2) O(f(x), f(ynk), t/2)

≥ (1− δ) (1− δ) > 1− ε

and

P (f(xnk), f(ynk), t) ≤ P (f(xnk), f(x), t/2) M P (f(x), f(ynk), t/2)

≤ δ M δ < ε

for all k ≥ k0, which are contradictions. We conclude that f is t-uniformlycontinuous.

Remark 6 Let f be a t-uniformly continuous mapping from the intuitionisticfuzzy metric space X to the intuitionistic fuzzy metric space Y . If (xn) is aCauchy sequence in X, then (f(xn)) is also a Cauchy sequence in Y .

Next, we will characterize intuitionistic fuzzy metric spaces for which realvalued continuous functions are t-uniformly continuous.

Definition 8 An intuitionistic fuzzy metric (M,N) on a set X is said to bet-equinormal if for each pair of disjoint nonempty closed subsets A and B of(X, τ (M,N)) and each t > 0, sup M(a, b, t) : a ∈ A, b ∈ B < 1 and infN(a, b,t) : a ∈ A, b ∈ B > 0.

Theorem 8 For an intuitionistic fuzzy metric space (X,M,N, ∗,♦), the fol-lowings are equivalent.

(a) For each intuitionistic fuzzy metric space (Y,O, P, ,M), any continuousmapping from (X, τ (M,N)) to (Y, τ (O,P )) is t-uniformly continuous as amapping from (X,M,N, ∗,♦) to (Y,O,P, ,M).


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(b) Any real valued continuous function on (X, τ (M,N)) is t-uniformly con-tinuous from (X,M,N, ∗,♦) to the Euclidean intuitionistic fuzzy metricspace (R,M|.|, N|.|, ∗,♦).

(c) The intuitionistic fuzzy metric (M,N) is t-equinormal.

Proof. (a) =⇒ (b). Obvious.(b) =⇒ (c). Let A and B be two disjoint nonempty closed subsets of

(X, τ (M,N)) and fix t > 0. Let f : X → [0, 1] be a continuous function suchthat f(A) ⊆ 0 and f(B) ⊆ 1. Put ε = 1/(t + 1). By assumption, there isr ∈ (0, 1) such that

t

t+ |f(x)− f(y)| > 1− ε and|f(x)− f(y)|

t+ |f(x)− f(y)| < ε

wheneverM(x, y, t) > 1−r andN(x, y, t) < r. Since we have t/(t+|f(a)− f(b)|)= t/(t + 1) = 1 − ε and |f(a)− f(b)| /(t + |f(a)− f(b)|) = 1/(t + 1) = ε, itfollow that M(a, b, t) ≤ 1 − r and N(a, b, t) ≥ r for all a ∈ A and b ∈ B. Weconclude that (M,N) is a t-equinormal intuitionistic fuzzy metric on X.(c) =⇒ (a). Suppose that there is an intuitionistic fuzzy metric space

(Y,O, P, ,M) and a continuous mapping f from (X, τ (M,N)) to (Y, τ (O,P )) whichis not t-uniformly continuous. Then, there exist ε ∈ (0, 1), and two sequences(an) and (bn) such that M(an, bn, t) > 1 − 1/n and N(an, bn, t) < 1/n butO(f(an), f(bn), t) ≤ 1 − ε and P (f(an), f(bn), t) ≥ ε for t > 0 and all n ∈ N.We distinguish two cases:Case I. The sequence (an) has a convergent subsequence (ank) with limit

a ∈ X. If the sequence (bnk) has a cluster point b ∈ X, then as in the proofof Proposition 7, we obtain that M(a, b, 3t) = 1 and N(a, b, 3t) = 0, so a = b,and by continuity of f , O(f(ank), f(bnk), t) > 1−ε and P (f(ank), f(bnk), t) < εwhich provide contradictions. Otherwise, without loss of generality, we maysuppose that a ∪ ank : k ∈ N and bnk : k ∈ N are disjoint closed subsetsof X. But sup M(ank , bnk , t) : k ∈ N = 1 and inf N(ank , bnk , t) : k ∈ N = 0which provide contradictions.Case II. The sequence (an) has no convergent subsequence. Then, without

loss of generality, we may suppose that ank : k ∈ N and cl bnk : k ∈ N aredisjoint closed subsets of X. Thus, we obtain a contradiction again. Thiscompletes the proof.Let us recall that a metric d on a set X is equinormal ([4]) if for each pair

of disjoint nonempty closed subsets A and B of X, d(A,B) > 0.

Proposition 9 Let (X,M,N, ∗,♦) be an intuitionistic fuzzy metric space forwhich every real valued continuous function is uniformly continuous from (X,M,N, ∗,♦) to the Euclidean intuitionistic fuzzy metric space (R,M|.|, N|.|, ∗, ♦).Then, there is an intuitionistic fuzzy metric (O,P ) on X compatible with τ (M,N)

for which every real valued continuous function on (X, τ (O,P )) is t-uniformlycontinuous from (X,O,P, ,M) to the Euclidean intuitionistic fuzzy metric space.

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Proof. The set Un : n ∈ N is a countable base for a uniformity U on Xcompatible with τ (M,N), where Un = (x, y) :M(x, y, 1/n) > 1− (1/n), N(x, y,1/n) < 1/n for all n ∈ N. Let d be any metric on X whose induced uniformitycoincides with U . Obviously, the topology τd generated by d coincides withτ (M,N). Since every real valued continuous function on the metric space (X, d)is uniformly continuous, d is an equinormal metric on X (see [2,4]). Now, let(Md,Nd) be the intuitionistic fuzzy metric induced by d, where a ∗ b = ab anda♦b = min1, a + b for all a, b ∈ [0, 1]. It is clear that (Md,Nd) is compatiblewith τ (M,N). Now, let A and B be two disjoint nonempty closed subsets of Xand t > 0. By equinormality of d, there exists δ > 0 such that d(a, b) ≥ δ for alla ∈ A and b ∈ B. Therefore, for each a ∈ A and b ∈ B, we have Md(a, b, t) =t/(t + d(a, b)) ≤ t/(t + δ) and Nd(a, b, t) = d(a, b)/(t + d(a, b)) ≥ δ/(t + δ).Hence, sup Md(a, b, t) : a ∈ A, b ∈ B < 1 and inf Nd(a, b, t) : a ∈ A, b ∈ B >0. So, by Theorem 8, every real valued continuous function on (X, τ (Md,Nd))is t-uniformly continuous from (X,Md, Nd, ∗,♦) to the Euclidean intuitionisticfuzzy metric space.

Proposition 10 Every compact intuitionistic fuzzy metric space is separable.

Proof. Let (X,M,N, ∗,♦) be any compact intuitionistic fuzzy metric space.Then, for given r ∈ (0, 1) and t > 0, we can find x1, x2, ..., xn in X such thatX = ∪ni=1B(xi, r, t). In particular, for each n ∈ N, we can find a finite set An

such that X = ∪n∈AnB(a, 1/n, 1/n). Let A = ∪∞i=1An, then A is countable. Weclaim that X ⊂ A. Let x ∈ X, then for each n, there exists an ∈ An such thatx ∈ B(an, 1/n, 1/n). Thus, an converges to x but an ∈ A for all n and hencex ∈ A. Therefore, A is dense in X and X is separable.

Definition 9 ([16]) Let X be any nonempty set and (Y,M,N, ∗,♦) be an in-tuitionistic fuzzy metric space. Then a sequence fn of functions from X to Yis called converge uniformly to a function f from X to Y if given r ∈ (0, 1)and t > 0, there exists n0 ∈ N such that M(fn(x), f(x), t) > 1 − r andN(fn(x), f(x), t) < r for all n ≥ n0 and x ∈ X.

Definition 10 A family of mappings F from an intuitionistic fuzzy metricspace (X,M,N, ∗,♦) to an intuitionistic fuzzy metric space (X,O,P, ,M) issaid to be t-equicontinuous if for each r ∈ (0, 1) and t > 0, there exists r0 ∈ (0, 1)such that M(x, y, t) > 1− r0 and N(x, y, t) < r0 imply O(f(x), f(y), t) > 1− rand P (f(x), f(y), t) < r for all f ∈ F .

Lemma 11 Let fn be an t-equicontinuous sequence of mappings from an in-tuitionistic fuzzy metric space (X,M,N, ∗,♦) to a complete intuitionistic fuzzymetric space (X,O,P, ,M). If fn converges for each point of a dense sub-set A of X, then fn converges at each point of X and the limit function iscontinuous.

Proof. Let s ∈ (0, 1) and t > 0. Then, we can find a r ∈ (0, 1) such that(1 − r) (1 − r) (1 − r) > 1 − s and r M r M r < s. Since F = fn is a t-equicontinuous family, for r ∈ (0, 1) and t > 0, we can find r1 ∈ (0, 1) such that


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M(x, y, t/3) > 1 − r1 and N(x, y, t/3) < r1 imply O(fn(x), fn(y), t/3) > 1 − rand P (fn(x), fn(y), t/3) < r for all fn ∈ F . Since A is dense in X, thereexists y ∈ B(x, r1, t) ∩ A and (fn(y)) converges for that y. This (fn(y)) beinga Cauchy sequence, for r ∈ (0, 1) and t > 0, we can find a n0 ∈ N such thatfor all m,n ≥ n0, O(fn(y), fm(y), t/3) > 1 − r and P (fn(y), fm(y), t/3) < r.Therefore, for any x ∈ X, we have

O(fn(x), fm(x), t) ≥ O(fn(x), fn(y), t/3) O(fn(y), fm(y), t/3)

O(fm(x), fm(y), t/3)

≥ (1− r) (1− r) (1− r)

> 1− s

and

P (fn(x), fm(x), t) ≤ P (fn(x), fn(y), t/3) M P (fn(y), fm(y), t/3)

M P (fm(x), fm(y), t/3)

≤ r M r M r

< s.

Thus, (fn(x)) is a Cauchy sequence in Y . Since Y is complete, fn(x) converges.Define f(x) = limn→∞ fn(x). We claim that f is continuous. Let s0 ∈ (0, 1) begiven. Then, we can find a r0 ∈ (0, 1) such that (1− r0) (1− r0) (1− r0) >1 − s0 and r0 M r0 M r0 < s0. Since F is t-equicontinuous, for r0 ∈ (0, 1)we can find r1 ∈ (0, 1) such that M(x, y, t/3) > 1 − r1 and N(x, y, t/3) < r1imply O(fn(x), fn(y), t/3) > 1 − r0 and P (fn(x), fn(y), t/3) < r0 for all fn ∈F . Since fn(x) converges to f(x), for r0 ∈ (0, 1) we can find an i ∈ N suchthat O(fn(x), f(x), t/3) > 1 − r0 and P (fn(x), f(x), t/3) < r0 for all n ≥ i.Since fn(y) converges to f(y), for r0 ∈ (0, 1) we can find a j ∈ N such thatO(fn(x), f(y), t/3) > 1 − r0 and P (fn(x), f(y), t/3) < r0 for all n ≥ j. Hence,for all n ≥ max i, j, we have

O(f(x), f(y), t) ≥ O(f(x), fn(x), t/3) O(fn(x), fn(y), t/3)

O(fn(y), f(y), t/3)

≥ (1− r0) (1− r0) (1− r0)

> 1− s0

and

P (f(x), f(y), t) ≤ P (f(x), fn(x), t/3) M P (fn(x), fn(y), t/3)

M P (fn(y), f(y), t/3)

≤ r0 M r0 M r0

< s0.

Hence, f is continuous.

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Theorem 12 (Intuitionistic fuzzy Ascoli-Arzela theorem) Let (X,M,N,∗,♦) be a compact intuitionistic fuzzy metric space and (Y,O, P, ,M) be a com-plete intuitionistic fuzzy metric space. Let F be a t-equicontinuous family offunctions from X to Y . Let (fn) be a sequence in F such that clfn(x) : n= 1, 2, ... is a compact subset of Y for each x ∈ X. Then, there exists a con-tinuous function f from X to Y and a subsequence (gn) of (fn) such that gnconverges uniformly to f on X.

Proof. Since X is a compact intuitionistic fuzzy metric space, by Proposition10, X is separable. Let A = x1, x2, ... be a countable dense subset of X.Hence, for each i, cl fn(xi) : n = 1, 2, ... is a compact subset of Y . Since everyintuitionistic fuzzy metric space is first countable [16], every compact subsetof Y is sequentially compact. Therefore, we get a subsequence gn of fnsuch that gn(xi) converges for each i, i = 1, 2, ... . By Lemma 11, we get acontinuous function f from X to Y such that gn(x) converges to f(x) for all xin X. We claim that gn converges uniformly for f on X. Let s ∈ (0, 1) and t > 0be given, then we can find a r ∈ (0, 1) such that (1− r) (1− r) (1− r) > 1− sand r M r M r < s. Since F is t-equicontinuous, we can find r1 ∈ (0, 1) such thatM(x, y, t/3) > 1 − r1 and N(x, y, t/3) < r1 imply O(gn(x), gn(y), t/3) > 1 − rand P (gn(x), gn(y), t/3) < r for all n. Since X is compact, by Proposition 7,f is t-uniformly continuous. Hence, for r ∈ (0, 1), we can find r2 ∈ (0, 1) suchthatM(x, y, t/3) > 1−r2 and N(x, y, t/3) < r2 imply O(f(x), f(y), t/3) > 1−rand P (f(x), f(y), t/3) < r. Let r0 = min r1, r2. Since X is compact and Ais dense in X, X = ∪ki=1B(xi, r0, t/3) for some finite k. If x ∈ X, then we canfind i, 1 ≤ i ≤ k, such that M(x, xi, t/3) > 1− r0 and N(x, xi, t/3) < r0. Sincer0 = min r1, r2, by the t-equicontinuity of F , we have O(gn(x), gn(xi), t/3) >1−r and P (gn(x), gn(xi), t/3) < r, and by the uniform continuity of f , we haveO(f(xi), f(x), t/3) > 1− r and P (f(xi), f(x), t/3) < r. Since gn(xj) convergesto f(xj), we can find a n0 ∈ N such that O(gn(xj), f(xj), t/3) > 1 − r andP (gn(xj), f(xj), t/3) < r for all n ≥ n0 and 1 ≤ j ≤ k. Thus, for all x ∈ X, wehave

O(gn(x), f(x), t) ≥ O(gn(x), gn(xi), t/3) O(gn(xi), f(xi), t/3)

O(f(xi), f(x), t/3)

≥ (1− r) (1− r) (1− r)

> 1− s

and

P (gn(x), f(x), t) ≤ P (gn(x), gn(xi), t/3) M P (gn(xi), f(xi), t/3)

M P (f(xi), f(x), t/3)

≤ r M r M r

< s.

Hence, gn converges uniformly to f on X.


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Theorem 13 Every intuitionistic fuzzy metric space is normal.

Proof. Let (X,M,N, ∗,♦) be an intuitionistic fuzzy metric space, and S,Kbe two disjoint closed sets in X. Let x ∈ S then x ∈ Kc. Since Kc is open,there exist rx ∈ (0, 1) and tx > 0 such that B(x, rx, tx) ∩K = ∅ for all x ∈ S.Similarly, there exist ry ∈ (0, 1) and ty > 0 such that B(x, ry, ty) ∩ S = ∅ forall x ∈ K. Let s = min rx, tx, ry, ty. Then we can find a p0 ∈ (0, p) such that(1 − p0) ∗ (1 − p0) > 1 − p and p0♦p0 < p. Define U = ∪x∈SB(x, p0, p/2) andV = ∪y∈KB(y, p0, p/2). Clearly U and V are open sets such that S ⊂ U andK ⊂ V . Now, we claim that U ∩ V = ∅. Let z ∈ U ∩ V . Then there existx ∈ S and y ∈ K such that z ∈ B(x, p0, p/2) and z ∈ B(y, p0, p/2). Therefore,we have

M(x, y, p) ≥ M(x, z, p/2) ∗M(y, z, p/2) ≥ (1− p0) ∗ (1− p0)

> 1− p

and

N(x, y, p) ≤ N(x, z, p/2)♦N(y, z, p/2) ≤ p0♦p0< p.

Hence, y ∈ B(x, p, p). But B(x, p, p) ⊂ B(x, rx, tx), since s < tx, rx. Thus,B(x, rx, tx) ∩K is nonempty which is a contradiction. Therefore, U ∩ V = ∅.Hence, X is normal.

Remark 7 From the above theorem, we can easily deduce that every metriz-able space is normal. Since every intuitionistic fuzzy metric space is normal,Urysohn’s lemma and Tietze extension theorem are true in the case of intuition-istic fuzzy metric spaces.

4 ConclusionsIn this work, we have shown that the topology generated by any complete in-tuitionistic fuzzy metric space is completely metrizable and if the intuitionisticfuzzy metric space is complete then the generated topology is completely metriz-able. We also introduce the concepts of t-uniform continuity, uniform continuity,t-equinormality and t-equicontinuity, and discuss their properties to extend andAscoli-Arzela theorem on intuitionistic fuzzy metric spaces. During this work,we think some arised natural questions: firstly, if every open cover in an in-tuitionistic fuzzy metric space X admits a countably locally finite refinementwhich covers X, then ,in other way, one could show that every intuitionisticfuzzy metric space has a countably locally finite basis which implies every in-tuitionistic fuzzy metric space is also metrizable. Secondly, in [17] it is provedthat if a compact structure on an intuitionistic fuzzy metric space implies itsprecompactedness and completedness, then one could show that the Niemytzki-Tychonoff theorem is also true in the case of intuitionistic fuzzy metric spaces.

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References[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20,87-96(1986).

[2] G. Beer, Topologies on closed and closed convex sets, Kluwer Acad Publ.,Dordrecht, 1993.

[3] R. Engelking, General topology, PWN-Polish Sci Publ., Warsaw, 1977.

[4] P. Fletcher, W.F. Lindgren, Quasi-Uniform spaces, Marcel Dekker, NewYork, 1982.

[5] A. George, P. Veeramani, On some results in fuzzy metric spaces, FuzzySets Syst., 64,395-399(1994).

[6] A. George, P. Veeramani, Some theorems in fuzzy metric spaces, J. FuzzyMath., 3,933-940(1995).

[7] V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces, FuzzySets Syst.,115,485-489(2000).

[8] V. Gregori, A. Sapena, On fixed-point theorems in fuzzy metric spaces,Fuzzy Sets Syst., 125,245-252(2002).

[9] V. Gregori, S. Romaguera, A. Sapena, On t-uniformly continuous mappingsin fuzzy metric spaces, J Fuzzy Math., 12(1),237—43(2004).

[10] S. Kutukcu, D. Turkoglu, C. Yildiz, Common fixed points of compatiblemaps of type (α) on fuzzy metric spaces, J. Concr. Appl. Math, in press.

[11] S. Kutukcu, D. Turkoglu, C. Yildiz, Some fixed point theorems for multi-valued mappings in fuzzy Menger spaces, J. Fuzzy Math., in press.

[12] S. Kutukcu, A fixed point theorem in Menger spaces, Int. Math. J., 1(32),1543-1554(2006).

[13] S. Kutukcu, A common fixed point theorem for a sequence of self maps inintuitionistic fuzzy metric spaces, Commun. Korean Math. Soc., in press.

[14] S. Kutukcu, Topics in intuitionistic fuzzy metric spaces, J. Comput. Anal.Appl., in press.

[15] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci., 28,535-537(1942).

[16] J.H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons & Fractals,22,1039-1046(2004).

[17] R. Saadati, J.H. Park, On the intuitionistic fuzzy topological spaces, Chaos,Solitons & Fractals, 27,331-344(2006).

[18] B. Schweizer, A. Sklar, Statistical metric spaces, Pacific J. Math., 10,314-334(1960).


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298

A general sampling theory in the functional

Hilbert space induced by a Hilbert space valued

kernel

Antonio G. Garcıa∗ Alberto Portal†

* Departamento de Matematicas, Universidad Carlos III de Madrid, Avda. de la Uni-versidad 30, 28911 Leganes-Madrid, Spain. E-mail:[email protected]

† Departamento de Matematicas, Universidad Carlos III de Madrid, Avda. de la Uni-versidad 30, 28911 Leganes-Madrid, Spain. E-mail:[email protected]

Abstract

Let H be a separable Hilbert space and Ω a fixed subset of R. Consider an H-valued function K : Ω −→ H and x ∈ H. Then, the function fx : Ω −→ C given byfx(t) :=

⟨x,K(t)

⟩H is well-defined. Denote by HK the set of functions obtained in this

way. Although a variety of sampling results for HK is known in the literature, thereexist some simple examples where they do not apply because the implicit interpolationcondition that appears does not adjust the former pattern we need. The main aim ofthis paper is to obtain a more general sampling result including most of these specialcases. In this way, the concept of interpolation condition is redefined and we study howto combine two of them in order to obtain a new sampling result. Some examples areobtained in this new framework.

Keywords: Reproducing kernel Hilbert spaces; Sampling formulas; Biorthonormal Rieszbases.AMS: 94A20; 44A05; 46E22.

1 Statement of the problem

For the past few years a significant mathematical literature on the topic of sampling theoremsassociated with differential or difference problems has flourished [1, 4, 5, 6, 7, 8]. See also [15]and the references therein. In turn, we might consider the Weiss-Kramer sampling theoremas the leitmotiv of all these results [11, 13]. Roughly speaking, the common situation forthese sampling problems is the following:

Let f be a function defined on C by f(t) =∫I F (x)K(x, t) dx, F ∈ L2(I), (or f(t) =∑

n F (n)K(n, t), F ∈ `2 ). The kernel K, which belongs to L2(I) (or `2) for each fixed t ∈ C,satisfies the differential (difference) equation appearing in a differential (difference) problem(P ) which has the sequence of eigenvalues tn. Moreover, whenever we substitute in K thespectral parameter t by tn we obtain the sequence of orthogonal eigenfunctions associatedwith (P ) which constitutes an orthogonal basis for L2(I) (`2). Under these circumstances,

299


f is an entire function which can be recovered from its samples f(tn) by means of asampling formula f(t) =

∑n f(tn)Sn(t), where the sampling functions Sn are given by

Sn(t) = ‖K(·, tn)‖−2 〈K(·, t),K(·, tn)〉 (the inner product in L2(I) or `2).

All the results above can be formulated in an abstract way following the approach inSaitoh’s book [12]. Namely, let H be a separable Hilbert space, and Ω a fixed subset ofR. Given an H-valued function K : Ω → H, for x ∈ H, the function f(t) := 〈x,K(t)〉H iswell-defined as a function f : Ω → C. We denote by HK the set of functions obtained inthis way and by T the linear transform

T : H −→ HK

x 7−→ f(1)

Hereafter we refer the function K as the kernel of the transform T . Note that the continuityof the kernel K implies that the functions in HK are continuous in Ω, a natural setting forsampling purposes. If we define in HK the norm ‖f‖HK

= inf‖x‖H : f = T (x) we obtaina reproducing kernel Hilbert space (RKHS hereafter) whose reproducing kernel is given byk(t, s) := 〈K(s),K(t)〉H i.e., for each s ∈ Ω the function ks defined as ks(t) := k(t, s) belongsto HK , and the reproducing property

f(s) = 〈f, ks〉HK= 〈f, k(·, s)〉HK

, s ∈ Ω , f ∈ HK (2)

holds. Recall that the Moore-Aronszajn procedure [2] leads to the same RKHS via thepositive definite (or positive matrix) function k. Under these circumstances it is known thatthe linear operator T is one-to-one if and only if T is an isometry between H and HK ,or, equivalently, if and only if the set of functions K(t)t∈Ω is complete in H [12]. Animportant property of HK is that convergence in its norm implies pointwise convergence.In fact, by the reproducing property, we have that

|f(t)− fn(t)| = |⟨f − fn, k(·, t)

⟩| ≤ ‖f − fn‖HK

‖K(t)‖H .

Notice that the last inequality also implies uniform convergence in subsets of Ω where thefunction k(t, t) = ‖K(t)‖2 is bounded. The RKHS HK has been largely studied in themathematical literature (see the superb monograph [12] and references therein).

A sampling result for HK can be easily established (see [9] or [10]). Namely, let xn∞n=1

and x∗n∞n=1 be a pair of biorthonormal Riesz bases for a Hilbert space H. Assume that, foreach fixed t ∈ Ω,K(t) can be written asK(t) =

∑∞n=1 Sn(t)x

∗n, where the functions Sn ∈ HK

satisfy, for some fixed sequence tn∞n=1 in Ω, the interpolation property: Sn(tm) = anδn,mfor some constants an∞n=1 ⊂ C \ 0. Then, any function f ∈ HK can be expanded as

f(t) =∞∑n=1

f(tn)Sn(t)an

, t ∈ Ω , (3)

where the convergence of the series is absolute and uniform on subsets of Ω where thefunction ‖K(t)‖ is bounded.

Recall that a Riesz basis wn∞n=1 for H is the image of an orthonormal basis by means ofa bounded invertible operator on H. Any Riesz basis wn∞n=1 has a unique biorthonormal(dual) Riesz basis w∗n∞n=1, i.e., such that 〈wn, w∗m〉H = δn,m , and the expansions

x =∞∑n=1

〈x,w∗n〉Hwn =∞∑n=1

〈x,wn〉Hw∗n

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300

hold for every x ∈ H (see [3] or [14] for more details and proofs).Perhaps the most important examples of RKHS HK that verify the mentioned result are

the classical Paley-Wiener spaces of bandlimited functions, i.e., square integrable functionsin R such that their Fourier transforms are zero outside a bounded set in R. For instance,any function of the form

f(t) =1√2π

∫ π

−πF (x)eitxdx , t ∈ R ,

where F ∈ L2[−π, π], can be expanded as the cardinal series

f(t) =∞∑

n=−∞f(n) sinc(t− n) , t ∈ R ,

where sinc stands for cardinal sine function (or sinc function) defined as sinc t = sinπt/πtfor t 6= 0 and sinc 0 = 1.

The sampling series (3) might also contain samples from functions which are related tof in some sense. Thus, this sampling result can be generalized in the following way: Letxn∞n=1 ∪ yn∞n=1 and x∗n∞n=1 ∪ y∗n∞n=1 be a pair of biorthonormal Riesz bases for theHilbert space H. For each fixed t ∈ Ω, K(t) can be written as

K(t) =∞∑n=1

[Sn(t)x∗n + Tn(t)y∗n

],

where Sn(t) and Tn(t) denote the evaluation at t ∈ Ω of the functions Sn = T (xn) ∈ HK

and Tn = T (yn) ∈ HK obtained by means of the linear transform (1) .Assume that there exist two kernels K1,K2 : Ω −→ H each defining a function in the

way K does, i.e., fj(t) := 〈x,Kj(t)〉H, j = 1, 2. Let T1 and T2 be the corresponding lineartransforms. These kernels can be written as

Kj(t) =∞∑n=1

[Sjn(t)x∗n + T jn(t)y∗n

], j = 1, 2 ,

where Sjn(t) = Tj(xn)[t] and T jn(t) = Tj(yn)[t] for t ∈ Ω, j = 1, 2. Suppose there exist twosequences sn∞n=1 and tn∞n=1 in Ω such that the functions Sjn∞n=1 and T jn∞n=1, j = 1, 2,satisfy the interpolation conditions

S1n(sm) = anδn,m; T 1

n(sm) = bnδn,m, an∞n=1, bn∞n=1 ⊂ CS2n(tm) = cnδn,m; T 2

n(tm) = dnδn,m, cn∞n=1, dn∞n=1 ⊂ C ,

where ∆n := andn− bncn 6= 0 for all n ∈ N. Suppose that f and the functions f1 and f2 arerelated in the sense that kerT ⊆ kerT1 ∩ kerT2. This implies that kerT = 0 so that HK

becomes a RKHS under the inner product 〈f, g〉HK:= 〈x, y〉H where Tx = f and Ty = g.

Under these conditions, the technique used in [9] gives the following result:

Theorem 1 The sequence Sn∞n=1∪Tn∞n=1 is a Riesz basis for HK and, expansions withrespect to this basis allow to recover any function f in HK from the samples f1(sn)∞n=1

GENERAL SAMPLING THEORY

301

and f2(tn)∞n=1 of f1 and f2, by means of the sampling formula

f(t) =∞∑n=1

[dnf1(sn)− bnf2(tn)∆n

Sn(t) +anf2(tn)− cnf1(sn)

∆nTn(t)

]=

∞∑n=1

[f1(sn)

dnSn(t)− cnTn(t)∆n

+ f2(tn)anTn(t)− bnSn(t)

∆n

], t ∈ Ω .

(4)

The convergence of the series is absolute and uniform on subsets of Ω where the function‖K(t)‖ is bounded.

Using a matrix notation, formula (4) can be written as

f(t) =∞∑n=1

(f1(sn) f2(tn)

)(an cnbn dn

)−1(Sn(t)Tn(t)

), (5)

from which it is not difficult to derive a more general result involving the samples of Nfunctions related to f .

Multi-channel sampling in Paley-Wiener spaces can be easily derived from this approach[9]. As a different example of Theorem 1 we can obtain a Hermite-type interpolation seriesfor HK , i.e., a sampling series which involves samples of any function f ∈ HK and its firstderivative, and in addition, the sampling functions generalize the classical Hermite inter-polation polynomials. Indeed, let tn∞n=1 be a sequence of distinct nonzero real numberssuch that

∑∞n=1 |tn|−2 < ∞. There exists an entire function P (t) with simple zeros at the

sequence tn∞n=1. Specifically, the function P (t) is given by the canonical product

P (t) =

∞∏n=1

(1− t

tn

)exp(t/tn) if

∞∑n=1

|tn|−1 = ∞∞∏n=1

(1− t

tn

)if

∞∑n=1

|tn|−1 <∞ .

Consider xn∞n=1 ∪yn∞n=1 and x∗n∞n=1 ∪y∗n∞n=1 a pair of biorthonormal Riesz basesfor H and Q(t) := P (t)2, which has double zeros at tn∞n=1. Take the functions

Sn(t) =Q(t)

(t− tn)2and Tn(t) =

Q(t)t− tn

and define the kernels K(t),K1(t) and K2(t), t ∈ R, by

K(t) =∞∑n=1

[ Q(t)(t− tn)2

x∗n +Q(t)t− tn

y∗n

],

K1(t) = K(t) and K2(t) = K ′(t). It is easy to check the interpolation conditions:

Sn(tm) =Q′′(tn)

2δn,m ; Tn(tm) = 0

S′n(tm) =Q′′′(tn)

6δn,m ; T ′n(tm) =

Q′′(tn)2

δn,m .

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302

Taking into account that Q′′(tn) 6= 0 for all n ∈ N, any function f ∈ HK can be expandedas the series

f(t) =∞∑n=1

[f(tn)

(1− Q′′′(tn)

3Q′′(tn)(t− tn)

)+ f ′(tn)(t− tn)

] 2Q(t)Q′′(tn)(t− tn)2

, t ∈ R .

1.1 An easy anomalous example

Although Theorem 1 is a quite general sampling result, the following example exhibits asituation where it does not work. Indeed, consider the usual orthonormal basis for L2[−π, π]given by

1√2π

∪

1√π

cosnx∞n=1

∪

1√π

sinnx∞n=1

and the kernelsK(t) = K1(t) = cos tx+ sin tx andK2(t) = cos tx ,

from which we define, for each φ ∈ L2[−π, π], the transforms

f(t) = Tφ(t) =⟨φ,K(t)

⟩and fi(t) = Tiφ(t) =

⟨φ,Ki(t)

⟩, i ∈ 1, 2 .

Notice that, if φ ∈ L2[−π, π], we obtain that⟨φ, cos(tx)

⟩is an even function of t and that⟨

φ, sin(tx)⟩

is an odd one. Consequently, φ ∈ kerT implies that T2φ(t) =⟨φ, cos(tx)

⟩=

−⟨φ, sin(tx)

⟩is both an odd and an even function of t. Therefore, φ = 0 so that T is

one-to-one.The corresponding sampling functions are given by

S0(t) =⟨ 1√

2π,K(t)

⟩=√

2π sinc t

Sn(t) =⟨ 1√

πcosnx,K(t)

⟩=

2t(−1)n sinπt√π(t2 − n2)

(n ∈ N)

Tn(t) =⟨ 1√

πsinnx,K(t)

⟩=

2n(−1)n sinπt√π(t2 − n2)

(n ∈ N) .

In this case, S10 = S2

0 = S0, S1n = S2

n = Sn, T 1n = Tn and T 2

n = 0 for n ∈ N. For n,m ∈ N,the following interpolation condition holds:(

S1n(m) T 1

n(m)S2n(m) T 2

n(m)

)= δm,n

(√π

√π√

π 0

).

However, what is happening with S10 and S2

0? Even if we decide to define T 10 = T 2

0 := 0, theinterpolation matrix that we used in (5) will be singular and we will not be able to applyTheorem 1.

This example gives us a suitable generalization that, in practice, can be very useful. Aswe only need one coefficient for S0 (T0 is not defined), we can consider the matrix(

S10(m)S2

0(m)

)= δm,0

(√2π√2π

)(6)

where m ∈ N0 := N∪0. The coefficient of S0(t) can be chosen in two ways for the samplingformula, f1(0)/

√2π or f2(0)/

√2π, the choice being at our disposal; we might choose the

simplest, or perhaps the only one available, etc.


303

2 A general sampling result in HK

The aim in this section is to prove a new sampling result which also applies for anomalousexamples. To this end, we need to introduce new interpolation conditions which will becombined in an appropriate way to obtain the desired sampling result. These topics are thesubject of the next three subsections:

2.1 Interpolation condition of type (L, M)

Consider L,M ∈ N such that L ≥M and define M := 1, 2, . . . ,M and L := 1, 2, . . . , L.Consider a linear independent system for H written as

x1,nn∈I ∪ x2,nn∈I ∪ · · · ∪ xM,nn∈I

where I ⊆ N can be a finite set. Suppose that we have L linear transforms T1, T2, . . . , TLwith associate kernels K1(t),K2(t), . . . ,KL(t) and defined as follows

f`(t) = T`(x)[t] =⟨x,K`(t)

⟩, ` ∈ L .

DenoteS`m,n(t) = T`(xm,n)[t] , t ∈ Ω ,

where ` ∈ L, m ∈ M and n ∈ I.

Definition 1 We say that an interpolation condition of type (L,M) is satisfied by theseelements if there exist L sets of points t`nn∈I in Ω, ` ∈ L, such that, for any fixed ` ∈ Land m ∈ M,

S`m,n(t`k) = an`,mδn,k , n, k ∈ I ,

holds, and the coefficients an`,m ∈ C verify that the rank of the matrices

An :=

an1,1 an1,2 · · · an1,Man2,1 an2,2 · · · an2,M...

.... . .

...anL,1 anL,2 · · · anL,M

is just M for all n ∈ I.

Definition 2 Denote by ψ any increasing function from M into L, which we shall call achoice function. For any matrix B with L rows b(1), b(2), . . . , b(L), we define the choice ofM rows of B by means of ψ as

Bψ :=(b[ψ(1)] b[ψ(2)] · · · b[ψ(M)]

)>.

As we can see, Bψ is a matrix obtained from B by choosing the rows of B given byψ(1), ψ(2), . . . , ψ(M).

In these terms, the rank of An is M if and only if there exists a choice function ψn suchthat Anψn

is not singular.

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304

Definition 3 We shall call such a ψn, an An-regular choice function.

Any An-regular choice function ψ is related to An in such a way that Anψ is regular.However, this choice function can be applied to any matrix with L rows and any number ofcolumns.

If we have a vector v =(v1 v2 · · · vL

)>, then we can choose the same rows fromboth An and v. The product of the matrix and the vector obtained in such a way is givenby

Anψvψ =

anψ(1),1 anψ(1),2 · · · anψ(1),M

anψ(2),1 anψ(2),2 · · · anψ(2),M...

.... . .

...anψ(M),1 anψ(M),2 · · · anψ(M),M

vψ(1)

vψ(2)...

vψ(M)

.

2.2 Compatibility

We began this article by showing an example in which two interpolation conditions wereused. For the first one, I1 = 0, the linear independent system was just one vector and wehad only one transform, i.e., it was an interpolation condition of type (1, 1). In the secondone, I2 = N, the partition of the linear independent system had two elements and there weretwo transforms, so it was an interpolation condition of type (2, 2). The goal of this sectionis to establish which properties must be verified by two interpolation conditions in order fora sampling theorem to be possible. It is the topic compatible interpolation conditions.

The fact of working with two interpolation conditions at least makes the notation usedhard. For the sake of clarity, hereafter, an index kj denotes that the indexed elementcorresponds to the j-th interpolation condition of those we are using.

Consider two interpolation conditions of types (L1,M1) and (L2,M2), respectively. Thismeans that we have two linear independent systems in H:

S1 :=x11,n1

n1∈I1 ∪ x12,n1

n1∈I1 ∪ · · · ∪ x1M1,n1

n1∈I1

S2 :=x21,n2

n2∈I2 ∪ x22,n2

n2∈I2 ∪ · · · ∪ x2M2,n2

n2∈I2

where I1, I2 ⊆ N (possibly finite) and M1,M2 ∈ N. Moreover, we suppose the j-th interpo-lation condition, j ∈ 1, 2, has Lj ∈ N transforms T`j , `j ∈ Lj and Mj ≤ Lj , defined fromeach kernel K`j (t), `j ∈ Lj , by

f`j (t) = T`j (x)[t] =⟨x,K`j (t)

⟩, `j ∈ Lj .

DenoteSìmj ,nj

= Tì(xjmj ,nj

)

for ì ∈ Li, mj ∈ Mj , nj ∈ Ij and i, j ∈ 1, 2, i.e., the image of S1 ∪ S2 by each of thetransforms of both the interpolation conditions is calculated.

As they are interpolation conditions, there exist sequences t`jkjkj∈Ij

, where `j ∈ Lj andj ∈ 1, 2, such that

S`jmj ,nj (t

`jkj

) = anj

`j ,mjδnj ,kj


305

holds for mj ∈ Mj , `j ∈ Lj , nj , kj ∈ Ij and j ∈ 1, 2 in such a way the coefficients anj

`j ,mj∈ C

satisfy that the rank of the matrix

Anj :=

anj

1,1 anj

1,2 · · · anj

1,Mj

anj

2,1 anj

2,2 · · · anj

2,Mj

......

. . ....

anj

Lj ,1anj

Lj ,2· · · a

nj

Lj ,Mj

is just Mj for all nj ∈ Ij , j ∈ 1, 2.

Definition 4 Two interpolation conditions are compatible if the following compatibilitycondition holds:

S`2m1,n1(t`2k2) = 0 , S`1m2,n2

(t`1k1) = 0 ,

for `j ∈ Lj, mj ∈ Mj, nj , kj ∈ Ij, j ∈ 1, 2.

Observe that this condition implies that S1∪S2 is a linear independent system. Indeed,suppose there exists x2

m2,n2in S2 such that

x2m2,n2

=M1∑m1=1

αm1x1m1,n1

+N∑k=1

βkxk

where xk ∈ S1 ∪ S2 is not in x2m2,n2

, x11,n1

, x12,n1

, . . . , x1M1,n1

and there exists an indexk1 ∈ M1 = 1, 2, . . . ,M1 such that αk1 6= 0. For all `1 ∈ L1, we obtain that

0 = [T`1x2m2,n2

](t`1n1) =

M1∑m1=1

αm1an1`1,m1

,

i.e., An1 has a zero column or it has at least two linear dependent columns so that its rankcannot be M1.

2.3 The sampling result

Consider the pair of dual Riesz bases for H given by

R⋃r=1

Mr⋃mr=1

xmr,nrnr∈Ir andR⋃r=1

Mr⋃mr=1

x∗mr,nrnr∈Ir (7)

and suppose that R ∈ N interpolation conditions are satisfied, and each is compatible withevery other (see Definition 1 and Definition 4).

Suppose that the following condition is satisfied, as well:

kerT ⊆R⋂r=1

Lr⋂`r=1

kerT`r . (8)

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306

We show the transform T is one-to-one. First, observe that the kernel K can be writtenas

K(t) =R∑r=1

Mr∑mr=1

∑nr∈Ir

Smr,nr(t)x∗mr,nr

where Smr,nr(t) =⟨xmr,nr ,K(t)

⟩. Analogously, for `q ∈ Lq and 1 ≤ q ≤ R, the kernel K`q

can be expanded as

K`q(t) =R∑r=1

Mr∑mr=1

∑nr∈Ir

S`qmr,nr(t)x

∗mr,nr

.

Suppose that x ∈ H verifies Tx = 0. Then, for all q ∈ 1, 2, . . . , R and `q ∈ Lq, we havethat

f`q = T`qx = 0

and therefore, for pq ∈ Iq,

0 = f`q(t`qpq) =

⟨x,K`q(t

`qpq)⟩

=⟨x,

R∑r=1

Mr∑mr=1

∑nr∈Ir

S`qmr,nr(t

`qpq)x

∗mr,nr

⟩.

Thus, compatibility implies

0 =⟨x,

Mq∑mq=1

∑nq∈Iq

S`qmq ,nq(t

`qpq)x

∗mq ,nq

⟩,

and by using the q-th interpolation condition, we have

0 =⟨x,

Mq∑mq=1

apq

`q ,mqx∗mq ,pq

⟩=

Mq∑mq=1

apq

`q ,mq

⟨x, x∗mq ,pq

⟩, 1 ≤ `q ≤ Lq .

Thus, we have a homogeneous linear system with Lq equations and Mq unknowns whoseunique solution is the trivial one. Consequently, we obtain that

⟨x, x∗mq ,pq

⟩= 0 for all

q ∈ 1, 2, . . . R, mq ∈ Mq and pq ∈ Iq. Since x∗mr,nr: 1 ≤ r ≤ R, mr ∈ Mr, nr ∈ Ir is a

Riesz basis, x = 0. Observe that the following sequences

R⋃r=1

Mr⋃mr=1

Smr,nrnr∈Ir andR⋃r=1

Mr⋃mr=1

S∗mr,nrnr∈Ir (9)

are dual Riesz bases of HK .If we do the same for any x ∈ H such that f = Tx, we obtain the consistent linear

system

f`q(t`qpq) =

Mq∑mq=1

apq

`q ,mq

⟨x, x∗mq ,pq

⟩, 1 ≤ `q ≤ Lq ,

which has a unique solution. For each pq ∈ Iq we can find a choice function ψpq such thatApq

ψpqis regular. Thus, we can write the coefficients

⟨x, x∗mq ,pq

⟩with respect to the samples

f`q(t`qpq) by means of(⟨

x, x∗1,pq

⟩ ⟨x, x∗2,pq

⟩· · ·

⟨x, x∗Mq ,pq

⟩)>=(Apq

ψpq

)−1Fpq

ψpq,


307

where F pq

ψpqis obtained from the vector

F pq :=(f1(t1pq

) f2(t2pq) · · · fLq(t

Lqpq ))>

(10)

by using the choice function ψpq .Now, expanding f by using the Riesz basis given by Smr,nr : nr ∈ IrMr

mr=1 and takinginto account that T is an isometry, we have that

f(t) =R∑r=1

Mr∑mr=1

∑nr∈Ir

⟨f, S∗mr,nr

⟩HK

Smr,nr(t)

=R∑r=1

∑nr∈Ir

Mr∑mr=1

⟨x, x∗mr,nr

⟩HSmr,nr(t)

=R∑r=1

∑nr∈Ir

(Fnrψnr

)>[(Anrψnr

)−1]>

Snr(t) ,

where Fnr is given by (10) and

Snr(t) =(S1,nr(t) S2,nr(t) · · · SMr,nr(t)

)>, (11)

which is just the sampling formula we are looking for. Convergence is, as we know, in theHK-norm sense and, also, absolute in Ω and uniform in those subsets of Ω where ‖K(t)‖ isbounded. Consequently, we have proved the next result:

Theorem 2 Consider the dual Riesz bases given by (7). Suppose we have R ∈ N interpo-lation conditions of types (Lr,Mr), where 1 ≤ r ≤ R, each two of them being compatible (inthe sense of Definition 4). Assume condition (8) is satisfied. Then, the sets in (9) are dualRiesz bases for HK and for each set of Anr -regular choice functions

ψnr : Mr −→ Lr | nr ∈

Ir , 1 ≤ r ≤ R, we have that any function f ∈ HK can be recovered from its samples

f`r(t`rnr

) : nr ∈ Ir , `r ∈ LrRr=1

by the following sampling formula

f(t) =R∑r=1

∑nr∈Ir

(Fnrψnr

)>[(Anrψnr

)−1]>

Snr(t) , t ∈ Ω ,

where Fnr and Snr are given by (10) and (11), respectively. Convergence is absolute and,also, uniform in those subsets of Ω where ‖K(t)‖ is bounded.

Notice that the number of transforms we have for each interpolation condition of type(L,M) is M at least. However, it is not important how many of them we have at most. Infact, only the possibility of finding a regular choice of rows of An is needed. Thus, if L isa finite or infinite set of indices, we have an interpolation condition of type (card L,M) ifthere exist some points t`n : n ∈ I`∈L such that, for any fixed ` ∈ L and m ∈ M,

S`m,n(t`k) = an`,mδn,k , n, k ∈ I ,

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308

holds, and for each n ∈ I there exists an An-regular choice function ψn where

An =(an`,1 an`,2 · · · an`,M

)`∈L

represents a function from L into CM for each n ∈ I. If these more general interpolationconditions are used, Theorem 2 still remains valid.

3 Sampling by using linear operators in H

Let H be a separable Hilbert space. Consider two dual Riesz bases of H written as

x1,n∞n=1 ∪ x2,n∞n=1 ∪ · · · ∪ xM,n∞n=1 ,

x∗1,n∞n=1 ∪ x∗2,n∞n=1 ∪ · · · ∪ x∗M,n∞n=1 ,

where M ∈ N. Given a kernel K : Ω ⊂ R −→ H, we define the linear transform T as in(1). Suppose we have a family of bounded linear operators Lλ : H −→ Hλ∈Λ. Related tof = Tx ∈ HK we define the functions fλ(t) :=

⟨Lλx,K(t)

⟩H, where λ ∈ Λ. For any fixed

n ∈ N and 1 ≤ m ≤M , we denote Snλ,m(t) :=⟨Lλxm,n,K(t)

⟩H and, for any n ∈ N, we write

Sm,n(t) :=⟨xm,n,K(t)

⟩H.

In the sequel, we assume that x ∈ kerT implies Lλx ∈ kerT for all λ ∈ Λ (whichoccurs when T commutes with Lλ for all λ ∈ Λ), and that there exists a family of sequencestλkk∈N : λ ∈ Λ

⊂ Ω such that Snλ,m(tλk) = anλ,mδn,k for n, k ∈ N, 1 ≤ m ≤M and λ ∈ Λ.

For each n ∈ N, define the function

An : Λ −→ CM

λ 7−→ (anλ,1, anλ,2, . . . , a

nλ,M )

and suppose that there exists a sequence ψn : M −→ Λn∈N of An-regular choice functions.Then, the following result holds:

Theorem 3 Under the hypotheses as above, any f ∈ HK can be recovered from its samplesfλ(tλn)n∈N : λ ∈ Λ

by means of the sampling formula

f(t) =∑n∈N

(Fnψn

)>[(Anψn

)−1]>

Sn(t) , t ∈ Ω ,

where Fn(λ) := fλ(tλn) and Sn(t) =(S1,n(t) S2,n(t) · · · SM,n(t)

)>. Convergence is ab-solute and uniform in subsets of Ω where ‖K(t)‖ is bounded.

Proof: Defining Kλ(t) := L∗λ[K(t)] for λ ∈ Λ, where L∗λ denotes the adjoint operator of Lλ,we have

fλ(t) := Tλ(x)[t] =⟨Lλx,K(t)

⟩H =

⟨x,Kλ(t)

⟩H , t ∈ Ω .

If x ∈ kerT then, by assumption, Lλx ∈ kerT for all λ ∈ Λ, i.e., 0 =⟨Lλx,K(t)

⟩H =⟨

x,Kλ(t)⟩

H for all λ ∈ Λ. As a consequence,

kerT ⊆⋂λ∈Λ

kerTλ ,

and Theorem 2 implies the desired result.


309

4 Some illustrative examples

In this section we go back to our anomalous example in order to handle it into the newsetting. We also give another example which involves two interpolation conditions of type(2, 2).

4.1 The introductory example revisited

Consider the orthonormal basis of L2[−π, π] given by1√2π

∪

1√π

cosnx∞n=1

∪

1√π

sinnx∞n=1

and the kernels K1(t) = K(t) = cos tx+ sin tx, K2(t) = cos tx− sin tx and K3(t) = cos tx.Notice that, if φ ∈ L2[−π, π], we obtain that

⟨φ, cos(tx)

⟩is an even function of t and that⟨

φ, sin(tx)⟩

is an odd one. Consequently, φ ∈ kerT implies that T3φ(t) =⟨φ, cos(tx)

⟩=

−⟨φ, sin(tx)

⟩is both an odd and an even function of t. Thus, T is injective and trivially

0 = kerT ⊆ kerT1 ∩ kerT2 ∩ kerT3.Sampling functions are given by

S0(t) =⟨ 1√

2π,K(t)

⟩=√

2π sinc t

Sn(t) =⟨ 1√

πcosnx,K(t)

⟩=

2t(−1)n sinπt√π(t2 − n2)

(n ∈ N)

Tn(t) =⟨ 1√

πsinnx,K(t)

⟩=

2n(−1)n sinπt√π(t2 − n2)

(n ∈ N)

Easy calculations show that S1n = S2

n = S3n = Sn, that T 3

n = 0 and that −T 2n = T 1

n = Tnfor n ∈ N and S1

0 = S20 = S3

0 = S0. Thus, we have two interpolation conditions of types(3, 1) and (3, 2), respectively. The first one verifies that:S1

0(m)S2

0(m)S3

0(m)

=

√

2π√2π√2π

δ0,m m ∈ N ∪ 0 ,

and the second one, that:S1n(m) T 1

n(m)S2n(m) T 2

n(m)S3n(m) T 3

n(m)

=

√π √π√

π −√π√

π 0

δn,m m ∈ N ∪ 0 ,

so we have a couple of compatible interpolation conditions.We define f(t) :=

⟨F,K(t)

⟩and fk(t) :=

⟨F,Kk(t)

⟩for k = 1, 2, 3 and F ∈ L2[−π, π].

Finally, Theorem 2 yields the following sampling result:

Corollary 4 Any function f defined as

f(t) =1√π

∫ π

−πF (x)[cos tx+ sin tx]dx , t ∈ R ,

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where F ∈ L2[−π, π], can be recovered from the samples f(0) and fk(n)∞n=1, k = 1, 2, 3,of its related functions f1, f2, f3 by means of the following sampling formula:

f(t) = f(0) sinc t+1√π

∞∑n=1

[G1(n)Sn(t) + G2(n)Tn(t)

], t ∈ R ,

where(G1(n) G2(n)

)is any row of the matrix

f3(n) f3(n)− f2(n)

f3(n) f1(n)− f3(n)

f2(n)+f1(n)2

f1(n)−f2(n)2

. (12)

Convergence is absolute and uniform in subsets of Ω where ‖K(t)‖ is bounded.

This result allows us to recover any classical band-limited function to [−π, π] by meansof the samples of the related functions f1, f2, f3 since band-limited functions can be writtenas

f(t) =1√2π

∫ π

−πF (x)[cos tx+ sin tx]dx , t ∈ R ,

where F ∈ L2[−π, π]. Notice that the above integral representation involves the Hartleytransform of F (see [16]).

4.2 Another example

In this example, we use two compatible interpolation conditions of type (2, 2). To this end,consider xn, yn∞n=1 ∪ xn, yn∞n=1, a Riesz basis for H whose dual Riesz basis is given byx∗n, y∗n∞n=1 ∪ x∗n, y∗n∞n=1. Suppose we have five kernels K,K1,K2, K1, K2 each of themdefining a transform, as usual: f(t) = (Tx)[t] :=

⟨x,K(t)

⟩, fj(t) = (Tjx)[t] :=

⟨x,Kj(t)

⟩,

and fj(t) = (Tjx)[t] :=⟨x, Kj(t)

⟩where t ∈ Ω, x ∈ H and j ∈ 1, 2, denoting a tilde

over an element that it is related to the second interpolation condition. Assume that thefollowing condition holds:

kerT ⊆ kerT1 ∩ kerT2 ∩ ker T1 ∩ ker T2 .

Denote

Sn := Txn , Tn := Tyn ,

Sn := T xn , Tn := T yn ,

and suppose there exist sequences sn∞n=1, tn∞n=1, sn∞n=1 and tn∞n=1 such that((T1(xn))(sm) (T1(yn))(sm)(T2(xn))(tm) (T2(yn))(tm)

)=(an1,1 an1,2an2,1 an2,2

)δm,n =: Anδm,n

for the first interpolation condition, and((T1(xn))(sm) (T1(yn))(sm)(T2(xn))(tm) (T2(yn))(tm)

)=(an1,1 an1,2an2,1 an2,2

)δm,n =: Anδm,n


311

for the second one, being the matrices An and An invertible. For these interpolation condi-tions the compatibility condition reads:(

(T1(xn))(sm) (T1(yn))(sm)(T2(xn))(tm) (T2(yn))(tm)

)=(

(T1(xn))(sm) (T1(yn))(sm)(T2(xn))(tm) (T2(yn))(tm)

)=(

0 00 0

).

Finally, as a consequence of Theorem 2 we deduce the following sampling result for thepreceding compatible interpolation conditions:

Corollary 5 Any function f in the Hilbert space HK can be recovered from the sequences ofsamples f1(sn)∞n=1, f2(tn)∞n=1, f1(sn)∞n=1 and f2(tn)∞n=1 by means of the followingsampling formula

f(t) =∞∑n=1

[f1(sn)

an2,2Sn(t)− an2,1Tn(t)an1,1a

n2,2 − an1,2a

n2,1

+ f2(tn)an1,1Tn(t)− an1,2Sn(t)an1,1a

n2,2 − an1,2a

n2,1

]+

+∞∑n=1

[f1(sn)

an2,2Sn(t)− an2,1Tn(t)an1,1a

n2,2 − an1,2a

n2,1

+ f2(tn)an1,1Tn(t)− an1,2Sn(t)an1,1a

n2,2 − an1,2a

n2,1

].

The convergence of the series above is absolute and uniform in subsets of Ω where ‖K(t)‖is bounded.

4.3 A comment on the choice of the samples

Theorem 2 allows us to combine several interpolation conditions whose types are not neces-sarily equal. Corollary 4 gives a sampling formula for the example of subsection 1.1 whichshows some advantages of our approach. Indeed, for each n ∈ N, we can choose any row ofthe matrix in (12). Thus, if the samples of f2 are lost for n ∈ N ⊆ N, we can still recover fby using the second row of that matrix for n ∈ N.

On the other hand, fixed n ∈ N, the rows of (12) are obtained as solutions of a consistentlinear system with three equations and two unknowns, so we can choose any two of themin order to solve the system. This means that every row of (12) is equal to every other.As a consequence, we can choose any element of the first column and any element (notnecessarily in the same row) of the second one. This remark allows us to avoid cancellationerrors by choosing the appropriate elements of (12). For instance, suppose we have thatf1(n0)f2(n0) > 0 and that f1(n0)f3(n0) < 0 for n0 ∈ N. Then, we can choose the thirdelement of the first column, 1

2 [f1(n0) + f2(n0)], and the second element of the second one,f1(n)− f3(n).

Acknowledgments: This work has been supported by the grant BFM2003–01034 fromthe D.G.I. of the Spanish Ministerio de Ciencia y Tecnologıa.

References

[1] M. H. Annaby. On sampling theory associated with the resolvent of singular Sturm-Liouville problems. Proc. Amer. Math. Soc., 131:1803–1812, 2003.

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[2] N. Aronszajn. Theory of Reproducing Kernels. Trans. Amer. Math. Soc., 68:337–404,1950.

[3] O. Christensen. An Introduction to Frames and Riesz Bases. Birkhauser, Boston, 2003.

[4] W.N. Everitt and G. Nasri Roudsari. Interpolation and sampling theories, and linearordinary boundary value problems. In J.R. Higgins and R.L. Stens, editors, SamplingTheory in Fourier and Signal Analysis, chapter 5, pages 56–77. Oxford University Press,Oxford, 1999.

[5] W.N. Everitt, G. Schottler, and P. L. Butzer. Sturm-Liouville boundary value problemsand Lagrange interpolation series. Rendiconti di Matematica, 14:87–126, 1994.

[6] A. G. Garcıa and M. A. Hernandez-Medina. The discrete Kramer sampling theoremand indeterminate moment problems. J. Comp. Appl. Math., 134:13–22, 2001.

[7] A. G. Garcıa and M. A. Hernandez-Medina. On an integral transform associated withthe regular Dirac operator. Proc. Roy. Soc. Edin., 131:1357–1370, 2001.

[8] A. G. Garcıa and M. A. Hernandez-Medina. Discrete Sturm-Liouville problems, Jacobimatrices and Lagrange interpolation series. J. Math. Anal. Appl., 280(2):221–231, 2003.

[9] A. G. Garcıa and A. Portal. Sampling in the functional Hilbert space induced by aHilbert space valued kernel. Applicable Analysis, 82(12):1145–1158, 2003.

[10] J. R. Higgins. A sampling principle associated with Saitoh’s fundamental theory oflinear transformations. In S. Saitoh et al., editor, Analytic extension formulas andtheir applications. Kluwer Academic, 2001.

[11] H. P. Kramer. A generalized sampling theorem. J. Math. Phys., 63:68–72, 1957.

[12] S. Saitoh. Integral transforms, reproducing kernels and their applications. Longman,Essex, England, 1997.

[13] P. Weiss. Sampling theorems associated with Sturm-Liouville systems. Bull. Amer.Math. Soc., 63:242, 1957.

[14] R. M. Young. An Introduction to Nonharmonic Fourier Series. Academic Press, NewYork, 1980.

[15] A. I. Zayed. Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton FL,1993.

[16] A. I. Zayed. Function and Generalized Function Transformations. CRC Press, BocaRaton FL, 1996.


313

314

PERTURBATIONS OF OPERATORS ON TENSORPRODUCTS AND SPECTRUM LOCALIZATION OF

MATRIX DIFFERENTIAL OPERATORS∗

M. I. Gil’Department of Mathematics

Ben Gurion University of the NegevP.0. Box 653, Beer-Sheva 84105, Israel

E-mail: [email protected]

1

315


Abstract

We investigate spectrum perturbations of a class of operators on the tensor productof a separable Hilbert space and a finite dimensional space. The abstract results areapplied to the higher order matrix nonselfadjoint differential operators on finite andinfinite real segments. Besides, bounds for the spectra and estimates for the norm ofthe resolvents of the differential operators are derived. We also investigate conditions,under which the considered operators generate analytic and stable semigroups. Ourmain tool is a combined use of some properties of operators on tensor products ofHilbert spaces and the recent estimates for norms of resolvents of matrices.

Key words: nonselfadjoint differential operators, higher order operators resolvent,spectrum, tensor product

AMS (MOS) subject classification: 34L15, 47E05

——————-∗ This research was supported by the Kamea Fund of the Israel

2

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316

1 Introduction and notation

The spectrum of ordinary differential operators was considered in a lot of papers and books,cf. [1, 7, 8, 10] and references therein. However, the bounds for the spectrum of the non-selfadjoint higher order differential operators are not enough investigated. The paper [11]should be mentioned. In that paper, the author studies the problem of localization of thespectrum for a class of a scalar differential operator on a finite segment.

In the present paper we establish bounds for the spectrum of a class of the higher ordermatrix nonselfadjoint differential operators on finite and infinite real segments. In addition,we derive estimates for the norm of the resolvents and investigate the conditions, underwhich the considered operators generate analytic and stable semigroups. We consider thedifferential operators as operators on the tensor product of a separable Hilbert space and afinite dimensional space. Our main tool is a combined use of some properties of operators ontensor product of Hilbert spaces and the recent estimates for norms of resolvents of matrices.

A few words about the contents. The paper consists of 11 sections. In Section 2 weformulate the main result of the paper-Theorem 2.1 on spectrum localization of abstractoperators. In Section 3, the proof of Theorem 2.1 is presented. In Sections 4 and 5 weestablish some auxiliary results which are used in the following sections. Section 6 deals withsectorial operators. Note that all the differential operators considered here are sectorial. InSection 7 we derive the conditions, under which the considered operators generate stablesemigroups. Sections 8, 9 and 10 are devoted to differential operators. In Section 8, theperiodic boundary conditions are imposed. The Dirichlet problem on a finite interval isexplored in Section 9. Section 10 is devoted to operators on the whole real line. In Section11 we give an example.

Let Cn be the complex Euclidean space with the scalar product ⊂ ., . n, the norm ‖.‖n =√< ., . >n and the unit operator In. Let J be a (finite or infinite) segment of the real axis.

L2(J) = L2(J,Cn) denotes the complex Hilbert space of functions defined on J with valuesin Cn, the scalar product

(f, h)L2 :=∫

J⊂ f(x), h(x) ndt (f, h ∈ L2(J,Cn))

and the norm |.|L2 . For a linear operator A, Rλ(A) is the resolvent, Dom (A) is the domainand σ(A) is the spectrum. Moreover, β(A) = inf Re σ(A) and θ(A) = inf |σ(A)|.

Let Q be a constant n× n-matrix. Let λk(Q) (k = 1, ..., n) be the eigenvalues of Q withtheir multiplicities. The following quantity plays a key role hereafter:

g(Q) = (N2(Q)−n∑

k=1

|λk(Q)|2)1/2, (1.1)

where N(Q) is the Hilbert-Schmidt (Frobenius) norm of Q, i.e. N2(Q) = Trace(QQ∗). Theasterisk means the adjoint. If Q is a normal matrix: QQ∗ = Q∗Q, then g(Q) = 0. The

3

MATRIX DIFFERENTIAL EQUATIONS

317

following relations are true:

g(Qeiτ + zI) = g(Q) (z ∈ C, τ ∈ R), (1.2)

g2(Q) ≤ N2(Q∗e−iτ −Qeiτ )/2 (τ ∈ R) (1.3)

andg2(Q) ≤ N2(Q)− |TraceQ2|.

If matrices Q and Q1 commute, then g(Q + Q1) ≤ g(Q) + g(Q1). For the proofs see [4,Section 2.1].

2 Statement of the the main result

Let E be a separable Hilbert space with a scalar product < ., . >E, and the norm ‖.‖E =√< ., . >E. Let H = E⊗Cn be the tensor product of E and Cn. IH , IE and In are the unit

operators in H,E and Cn, respectively. The scalar product in H is defined by

< y ⊗ h, y1 ⊗ h1 >H=< y, y1 >E < h, h1 >n (y, y1 ∈ E, h, h1 ∈ Cn)

and the cross norm is ‖.‖H =√< ., . >H . From the theory of tensor products we only need

the basic definition and elementary facts which can be found in [1].Everywhere below S is an invertible normal operator in E and S0 = S⊗In; ck, k = 0, ...,m

are constant n× n-matrix matrices and cm is invertible. Consider the operator

W (S) =m∑

k=0

ck ⊗ Sk (Dom (W (S)) = Dom (Sm0 )).

Let E(s) be the orthogonal resolution of the identity defined on σ(S), such that

S =∫

σ(S)sdE(s).

ThenW (S) =

∫σ(S)

W (s)⊗ dE(s),

where

W (s) =m∑

k=0

cksk (s ∈ σ(S)).

Note that according to (1.1).

g(W (s)) =

√√√√N2(W (s))−n∑

k=1

|λk(W (s))|2, (2.1)

4

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318

where λk(W (s)) are the eigenvalues of W (s) with a fixed s.Furthermore, let T be a linear operator inH having the propertiesDom (T ) = Dom (Sm

0 )and

q := ‖(W (S)− T )S−ν0 ‖H <∞ (2.2)

for some ν ∈ [0,m]. Here S00 = IH .

Now we are in a position to formulate the main result of the paper.

Theorem 2.1 Assume (2.2) and there is a constant v0, such that

g(W (s)) ≤ v0|s|ν (s ∈ σ(S)). (2.3)

Then the spectrum of T lies in the set

∪s∈σ(S);j=1,...,nλ ∈ C : |λ− λj(W (s))| ≤ |s|νy(q, v0), (2.4)

where y(q, v0) is the extreme right-hand root of the algebraic equation

yn = qn−1∑k=0

vk0y

n−k−1

√k!

. (2.5)

The proof of this theorem is presented in the next section. Put

wn =n−1∑j=0

1√k!.

By setting z = v0

z, it follows from (2.5) that

zn =q

v0

n−1∑k=0

zn−k−1

√k!

.

Due to Lemma 1.11.1 from [3], y(q, v0) ≤ δ(q, v0), where

δ(q, v0) =

qwn if qwn ≥ v0,(qvn−1

0 wn)1/n if qwn ≤ v0. (2.6)

This result and Theorem 2.1 imply

Corollary 2.2 Under conditions (2.2) and (2.3), the spectrum of T lies in the set

∪s∈σ(S);j=1,...,nλ ∈ C : |λ− λj(W (s))| ≤ |s|νδ(q, v0). (2.7)

5


319

Note that if all the matrices ck are selfadjoint, then g(W (s)) = 0 and y(q, v0) = q. Sinceν ≤ m, everywhere below, we can take m instead of ν. But in appropriate situations someν < m allows us to derive sharper results.

Take H = L2(J,Cn) = L2(J)⊗Cn. Consider in L2(J,Cn) the operator B defined by

Bu =m∑

k=0

ak(x)Sk0u (x ∈ J, u ∈ Dom (B) = Dom (Sm)), (2.8)

where ak(x), k = 0, ...,m are n× n-matrix-valued functions measurable and bounded on J .In addition am(x) is invertible. It is assumed that for a nonnegative integer ν ≤ m,

ak(x) = ck + bk(x) for k = 0, ..., ν and ak(x) ≡ ck for k = ν + 1, ...,m, (2.9)

where bk(x) are variable n × n-matrices and ck are constant n × n-matrices, again. Put‖bk‖C = supx ‖bk(x)‖n.

Corollary 2.3 Under conditions (2.9) and (2.3), the spectrum of operator B lies in the set(2.4) (and therefore, in the set (2.7)) with q = q0, where

q0 :=ν∑

j=0

‖bj‖C θj−ν(S). (2.10)

In other words, for any λ ∈ σ(B), there is a point s ∈ σ(S) and an eigenvalue λj(W (s)) ofthe matrix W (s), such that

|λ− λj(W (s))| ≤ |s|νy(q0, v0) ≤ |s|νδ(q0, v0).

Indeed, since,

B −W (S) =ν∑

j=0

bj(x)Sj0,

we have

‖(W (S)−B)S−ν0 ‖H ≤

ν∑k=0

‖bk‖C‖Sk−ν‖E ≤ q0.

Now the required result is due to Theorem 2.1 and Corollary 2.2 .

3 Proof of Theorem 2.1

For numbers s ∈ σ(S) and λ ∈ C, put

ρ(W (s), λ) = minj=1,...,n|λj(W (s))− λ|.

6

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320

Lemma 3.1 For a λ ∈ C, let

p(λ, ν) := sups∈σ(S)

|s|νn−1∑k=0

gk(W (s))√k!ρk+1(W (s), λ)

<∞. (3.1)

Then the operator Sν0 (W (S)− IHλ)−1 is bounded and

‖Sν0 (W (S)− IHλ)−1‖ ≤ p(λ, ν).

Proof: Let Q be a linear operator in Cn, and ρ(Q, λ) the distance between σ(Q) of Q anda complex point λ. Then

‖(Q− Inλ)−1‖n ≤n−1∑k=0

gk(Q)√k!ρk+1(Q, λ)

(λ 6∈ σ(Q))

where ρ(Q, λ) = minj=1,...,n|λj(Q)− λ|. For the proof see [4, Section 2.1]. Replacing in thisestimate Q by W (s), we have

‖(W (s)− Inλ)−1‖n ≤n−1∑k=0

gk(W (s))√k!ρk+1(W (s), λ)

. (3.2)

But(W (S)− IHλ)−1 =

∫σ(S)

(W (s)− Inλ)−1 ⊗ dE(s),

andSν

0 (W (S)− IHλ)−1 =∫

σ(S)sν(W (s)− Inλ)−1 ⊗ dE(s).

Consequently‖Sν

0 (W (S)− IHλ)−1‖H = sups∈σ(S)

|s|ν‖(W (s)− Inλ)−1‖n. (3.3)

In particular,‖(W (S)− IHλ)−1‖H = sup

s∈σ(S)‖(W (s)− Inλ)−1‖n.

This and (3.2) prove the lemma. 2

Lemma 3.2 Let the conditions (2.2) and

qp(λ, ν) < 1 (3.4)

hold. Then λ is a regular point for T and

‖(T − IHλ)−1‖H ≤ p(λ, 0)(1− qp(λ, ν))−1. (3.5)

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321

Proof: Since

(W (S)− IHλ)−1 − (T − IHλ)−1 = (T − IHλ)−1(T −W (S))(W (S)− IHλ)−1 =

(T − IHλ)−1(T −W (S))S−ν0 Sν

0 (W (S)− IHλ)−1,

the conditionθ0(λ) := q‖Sν

0 (W (S)− IHλ)−1‖H < 1 (3.6)

implies‖(T − λ)−1‖H ≤ ‖(W (S)− IHλ)−1‖H(1− θ0(λ))−1. (3.7)

Now the previous lemma proves the required result. 2

Proof of Theorem 2.1: Let λ ∈ σ(T ). Due to the previous lemma, for some s ∈ σ(S)and j,

q|s|νn−1∑k=0

gk(W (s))√k!|λj(W (s))− λ|k+1

≥ 1.

Hence due to (2.3),

q|s|νn−1∑k=0

(v0|s|ν)k

√k!|λj(W (s))− λ|k+1

≥ 1.

Consequently, |λj(W (s)) − λ| ≤ z(s), where z(s) is the extreme right-hand root of theequation

q|s|νn−1∑k=0

(v0|s|ν)k

√k!zk+1

= 1. (3.8)

Put in this equation z = |s|νy. Then we have equation (2.5). Hence

z(s) ≤ |s|νy(q, v0).

This finishes the proof. 2

4 Auxiliary inequalities

In the present section we suggest the inequalities, which allow us to check condition (2.3).Let

S = S∗. (4.1)

Then due to (2.1) and (1.3) with τ = 0, we have

√2g(W (s)) ≤

m∑k=0

N(ck − c∗k)|s|k. (4.2)

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322

That is, condition (2.3) holds with ν = m and

v0 =1√2

m∑k=0

N(ck − c∗k)θk−m(S). (4.3)

If, additionally,ck = c∗k, k = ν + 1, ...,m,

then under (4.1) condition (2.3) holds with

v0 =1√2

ν∑k=0

N(ck − c∗k)θk−ν(S).

The caseS = −S∗ (4.4)

can be reduced to (4.1), if we take iS instead of S. Besides,

√2g(W (s)) ≤

m∑k=0

N(ck + c∗k)|s|k.

That is, condition (2.3) holds with ν = m and

v0 =1√2

m∑k=0

N(ck + c∗k)θk−m(S).

Let, additionally, ck = −c∗k, k = ν + 1, ...,m. Then condition (2.3) holds with

v0 =1√2

ν∑k=0

N(ck + c∗k)θk−ν(S).

Consider now the operatorB = Sm

0 + a0(x), x ∈ J (4.5)

with a0(x) = b0(x) + c0 and ν = 0. Take W (S) = Sm0 + c0. According to (1.2) g(W (s)) =

g(c0) = v0. Thanks to (2.7), taking into account that in the considered case ν = 0 we canassert that

σ(B) ⊂ ∪s∈σ(S);j=1,...,nλ ∈ C : |λ− sm − λj(c0)| ≤ δ(q, v0). (4.6)

5 Additional estimates for resolvents

In this section we derive additional estimates for the resolvent of the operator B defined by(2.8), which in appropriate situations, are more useful than Lemma 3.2.

Putζn = (n− 1)−(n−1)/2

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323

and

w(λ, s) :=ζn[N2(W (s))− 2Re (λ Trace (W (s))) + n|λ|2](n−1)/2

|det (λI −W (s))|for s ∈ σ(S), λ ∈ C. It should be noted that

w(λ, s) ≤ w(λ, s), (5.1)

where

w(λ, s) :=ζn[N(W (s)) + |λ|

√n]n−1

|det (λI −W (s))|.

So in the next lemma w(λ, s) can be replaced by the simple function w(λ, s).

Lemma 5.1 Letψ(λ, ν) := sup

σ(S)|s|νw(λ, s) <∞.

Then the operator Sν0 (W (S)− IHλ)−1 is bounded and

‖Sν0 (W (S)− IHλ)−1‖H ≤ ψ(λ, ν).

Proof: We need the following result: let Q be a constant n × n-matrix. Then for allλ 6∈ σ(Q),

‖(Iλ−Q)−1 det (λI −Q)‖n ≤ ζn[N2(Q)− 2Re (λ Trace (Q)) + n|λ|2](n−1)/2. (5.2)

For the proof see [4, p. 28]. Hence,

‖(Iλ−W (s))−1‖ ≤ w(s, λ) ≤ w(s, λ) (λ 6∈ σ(W (s))).

Taking into account (3.3), we get the required result. 2

The latter lemma and relations (3.6), (3.7) yield

Theorem 5.2 Let an operator T satisfy the conditions (2.2) and

qψ(λ, ν) < 1 (5.3)

hold. Then λ is a regular point of operator T and

‖(T − IHλ)−1‖H ≤ ψ(λ, 0)(1− qψ(λ, ν))−1. (5.4)

Recall that B and q0 are defined by (2.8) and (2.10), respectively.

Corollary 5.3 Let the conditions (2.9) and (5.3) hold with q = q0. Then λ is a regularpoint of operator B and

‖(B − IHλ)−1‖H ≤ ψ(λ, 0)(1− q0ψ(λ, ν))−1. (5.5)

Therefore, under conditions (2.9), for any λ ∈ σ(B), there is an s ∈ σ(S), such that

q0ζn|s|ν[N(W (s)) +

√n|λ|]n−1

|det (λI −W (s))|≥ 1.

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6 Sectorial operators

For a natural number µ ≥ 1, letm = 2µ, (6.1)

and the following conditions hold:

am(x) = a∗m(x) (x ∈ J) is a strongly positive definite matrix and Sµ is selfadjoint . (6.2)

We use the notion of a sectorial operator, which can be found, for instance, in [5].

Theorem 6.1 Under conditions (6.1) and (6.2), let S be a normal operator defined by theexpression

Su = du/dx+ w0u (u ∈ D(S))

with a constant w0 ∈ C and

D(S) = u ∈ L2(J,Cn) : du/dx ∈ L2(J,Cn) + selfafjoint boundary conditions.

In addition, letam(x) have bounded derivatives on J up to order µ. (6.3)

Then the operator B defined by (2.8) is sectorial.

Proof: Firstly, let w0 = 0. Take A0 = Sµ0 am(.)Sµ

0 . Clearly,

A0 − am(x)Sm0 =

µ−1∑k=0

(µk)a

(µ−k)k (x)Sk+µ

0

where (nk) = n!/(n − k)!k! are the binomial coefficients. So (A0 − am(x)Sm

0 )S−m+10 is a

bounded operator and

(Sµ0 am(x)Sµ

0 u, u) ≤ β0 (Sm0 u, u) (u ∈ D(B)).

where β0 = supx ‖am(x)‖n. Hence it follows that A0 ≤ β0Sm0 and therefore

S−mτ0 = S−m+1

0 ≤ βτ0A

−τ0 (τ = (m− 1)/m)).

Thus (A0 − am(x)Sm0 )A−τ

0 is a bounded operator. Since the operator

(B − am(x)Sm0 )S−m+1

0

is bounded, we can assert that (A0 − B)A−τ0 is a bounded operator. But A0 is selfadjoint

positive definite. So due to Theorem 1.4.6 from [5] operator B is sectorial. The case w0 6= 0follows by a bounded perturbation argument. 2

Similarly the following theorem can be proved.

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325

Theorem 6.2 Under conditions (6.1) and (6.2), let S be a normal operator defined by theexpression

Su = d2u/dx2 + w0u (u ∈ D(S))

with a constant w0 ∈ C and

D(S) = u ∈ L2(J,Cn) : d2u/dx2 ∈ L2(J,Cn) + selfafjoint boundary conditions.

In addition, let

am(x) have bounded derivatives on J up to order 2µ. (6.4)

Then the operator B defined by (2.8) is sectorial.

Under the hypothesis of Theorem 6.1 or 6.2, thanks to Theorem 1.3.4 [5], the semigroupe−Bt generated by −B is an analytic. Moreover, for any a < β(B), where

β(B) := inf Re σ(B)

there is a constant γa, such that

‖e−Bt‖H ≤ γae−at (t ≥ 0) (6.5)

cf. [5, p. 21].

7 Stable operators

We will say that operator B is stable if β(B) = inf Re σ(B) > 0. Under the hypothesis ofTheorems 6.1 and 6.2, according to (6.5) this means that the semigroup e−Bt is exponentiallystable. The aim of this section is to establish explicit stability conditions for the consideredoperators.

Due to (2.4) and (2.7), for any λ ∈ σ(B), there are s ∈ σ(S) and j = 1, ..., n, such that

Re λj(W (s))−Re λ ≤ |s|νy(q0, v0) ≤ |s|νδ(q0, v0)

where q0 is defined by (2.10). Hence

β(B) = inf Re σ(B) ≥ β(W, ν),

whereβ(W, ν) := inf Re λj(W (s))− |s|νy(q0, v0) : s ∈ σ(S); j = 1, ..., n.

For a matrix Q, put

βR(Q) = minkλk(Q+Q∗)/2 and αR(Q) = max

kλk(Q+Q∗)/2,

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326

and consider the caseS = S∗ > 0. (7.1)

Take into account that for any matrix Q and an eigenvalue λj(Q) we have

< (Q+Q∗)h, h >n= 2Re λj(Q),

where h is the normed eigenvector. Then

βR(W (s)) ≥m∑

k=0

βR(ck)sk.

Hence

β(B) ≥ inf m∑

k=0

βR(ck)sk − sνy(q0, v0) : s ∈ σ(S).

We thus get

Theorem 7.1 Let the conditions (7.1) and

inf m∑

k=0

βR(ck)sk − sνy(q0, v0) : s ∈ σ(S) > 0

hold. Then the operator B defined by (2.8) is stable.

For instance, if

y(q0, v0) <m∑

k=ν

βR(ck)θk−ν(S),

then under (7.1) we get

β(B) ≥m∑

k=0

βR(ck)θk(S)− θν(S)y(q0, v0) > 0

and thus B is stable.

8 Periodic boundary conditions

Consider the operator

A =m∑

k=0

ak(x)dk

dxk(x ∈ (0, 1)) (8.1)

with bounded matrix-valued coefficients ak(x). Put H = L2([0, 1],Cn), E = L2[0, 1] and

Dom (A) = u ∈ H = L2([0, 1],Cn) : u(k) ∈ H : k = 1, ...,m;

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327

u(j)(0) = u(j)(1); j = 0, ...,m− 1. (8.2)

Take ν = m and

Su = −iu′ − π; u ∈ Dom (S) = v ∈ L2[0, 1] : v′ ∈ L2[0, 1]; v(0) = v(1).

Clearly, S is selfadjoint and invertible,

σ(S) = 2πk − π : k = 0,±1,±2, ... and θ(S) = π. (8.3)

Substitute u′ = i(S + π)u into (8.1). Then we have operator B defined by (2.8) with

aj(x) =m∑

k=j

akik(k

j )πk−j (j = 0, ...,m). (8.4)

So am(x) = akim. Under (2.9) due to (4.3), condition (2.3) holds with

v0 =1√2

m∑k=0

N(ck − c∗k)πk−m. (8.5)

In the considered case, according to (2.10)

q0 =m∑

j=0

‖bj‖Cπj−m. (8.6)

Recall that W (s), y(q, v0) and δ(q, v0) are defined in Section 2. For brevity, put

sj = π(2j − 1).

Thanks to Corollary 2.3, we get

Corollary 8.1 Let A be defined by (8.1), (8.2). Then, under (2.9), for any λ ∈ σ(A) thereare an integer k = 0,±1, ... and an eigenvalue λj(W (sk)) of matrix W (sk), such that

|sk|−m|λ− λj(W (sk))| ≤ y(q0, v0) ≤ δ(q0, v0).

Moreover, for a λ ∈ C, let

p1(λ,m) := supj=0,±1,...

n−1∑k=0

vk0 |sj|(k+1)m

√k!ρk+1(W (sj), λ)

<1

q0, (8.7)

where q0 and v0 are is defined by (8.5) and (8.6), respectively. Lemma 3.2 implies

Corollary 8.2 Let A be defined by (8.1), (8.2). In addition, let conditions (2.9) and (8.7)hold. Then λ is a regular point for A and

‖(A− IHλ)−1‖ ≤ p1(λ, 0)(1− q0p1(λ,m))−1.

where

p1(λ, 0) := supj=0,±1,...

n−1∑k=0

vk0 |sj|km

√k!ρk+1(W (sj), λ)

.

Now let conditions (6.1)-(6.3) hold with J = [0, 1] and (8.4) taken into account. Then dueto Theorem 6.1the operator, defined by (8.1), (8.2) generates an analytic semigroup.

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9 Dirichlet conditions

Consider the operator

A =m∑

k=0

ak(x)(−1)k d2k

dx2k(x ∈ (0, 1)), (9.1)

where ak(x) are matrix-valued functions bounded on [0, 1]. Let H and E be the same as inthe previous section. Put

Dom (A) = u ∈ H = L2([0, 1],Cn) : u(k) ∈ H : k = 1, ..., 2m,

u(j)(0) = u(j)(1) = 0, j = 0, ...,m− 1. (9.2)

That is the boundary conditions

u(j)(0) = u(j)(1) = 0; j = 0, ...,m− 1

hold. Take ν = m and Su = u′′ with

Dom (S) = u ∈ E = L2[0, 1] : u′′ ∈ E; u(0) = u(1) = 0.

We have σ(S) = (πk)2 : k = 1, 2, .... So S is invertible and θ(S) = π2. Due to (4.3),condition (2.3) holds with

v0 =1√2

m∑k=0

N(ck − c∗k)π2(k−m). (9.3)

Under consideration, (2.10) implies

q0 =m∑

j=0

‖bj‖Cπ2(j−m). (9.4)

Now Corollary 2.3 implies

Corollary 9.1 Let the operator A be defined by (9.1) and (9.2) and condition (2.9) hold.Then for any λ ∈ σ(A), there are an integer k = 1, 2, ... and an eigenvalue λj(W ((kπ)2) ofmatrix W ((kπ)2), such that

(πk)−2m|λ− λj(W ((kπ)2)| ≤ y(q0, v0) ≤ δ(q0, v0).

Now let conditions (6.1), (6.2) and (6.4) hold with J = [0, 1]. Then the operator, definedby (9.1), (9.2) generates an analytic semigroup. Moreover, Theorem 7.1 yields the followingresult.

Corollary 9.2 Let the conditions (2.9), (6.1),

π2βR(c2k) > αR(c2k−1) (k < m/2) and π2βR(cm) > π2y(q0, v0) + αR(cm−1)

hold. Then the operator A defined by (9.1), (9.2) satisfies the inequality

β(A) >µ∑

k=0

π4kβR(c2k)−µ−1∑k=0

αR(c2k+1)π4k+2 − π2my(q0, v0) > 0.

That is, A is stable.

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10 Operators on the whole line

Put H = L2(R1,Cn), E = L2(R1) and consider the operator

A =m∑

k=0

ak(x)dk

dxk(x ∈ R) (10.1)

with matrix coefficients bounded on R and the domain

Dom (A) = u ∈ H = L2(R1,Cn) : u(k) ∈ H; k = 1, ...,m, (10.2)

Take ν = m and

Su = u′ − u, u ∈ Dom (S) = y ∈ L2(R1) : y′ ∈ L2(R1).

Then S is normal and invertible. Moreover, σ(S) = it− 1 : t ∈ R1 and θ(S) = 1.Substituting u′ = Su+ u in (10.1), we have the operator B defined by (2.8) with

aj =m∑

k=j

(kj )ak

Due to (1.3),√

2g(W (s)) ≤m∑

k=0

N(skck + skc∗k) ≤ 2m∑

k=0

|s|kN(ck)

Thus (2.3) holds with ν = m and

v0 =√

2m∑

k=0

N(ck). (10.3)

According to (2.10),

q0 :=m∑

j=0

‖bj‖C . (10.4)

Thanks to Corollary 2.3, we get the following result.

Corollary 10.1 Let A be defined by (10.1) and (10.2), and condition (2.9) hold. Then forany λ ∈ σ(A) there are a t ∈ R and an eigenvalue λj(W (it− 1)) of matrix W (it− 1), suchthat

(t2 + 1)−m/2|λ− λj(W (it− 1))| ≤ y(q0, v0) ≤ δ(q0, v0),

where v0 and q0 are defined by (10.3) and (10.4).

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11 Example

In space H = L2([0, 1],Cn) consider the operator

(Au)(x) = am(x)dmu(x)

dxm+ a0(x)u(x) (u ∈ D(A); x ∈ (0, 1)) (11.1)

with bounded matrix-valued coefficients am(x), a0(x). and

Dom (A) = u ∈ H = L2([0, 1],Cn) : u(k) ∈ H : k = 1, ...,m;

u(j)(0) = u(j)(1); j = 0, ...,m− 1. (11.2)

Take ν = m and

Su = −iu′ − π; u ∈ Dom (S) = v ∈ L2[0, 1] : v′ ∈ L2[0, 1]; v(0) = v(1).

So σ(S) is defined by (8.3). Substitute u′ = i(S + π)u into (11.1). Then we have operatorB defined by (2.8) with

aj(x) = am(x)im(mj )πm−j (j = 1, ...,m)

anda0(x) = a0(x) + am(x)πmim.

Putcj = aj(0), bj(x) = aj(x)− aj(0) (j = 0, ...,m).

Then v0 and q0 are defined by (8.5) and (8.6), respectively. Recall also that

wn =n−1∑j=0

1√k!.

Assume thatqwn ≤ v0. (11.3)

Then according to (2.6)δ(q0, v0) = (q0v

n−10 wn)1/n.

Thanks to Corollary 8.1, we can assert that for the operator A defined by (11.1), (11.2)under condition (11.3), for any λ ∈ σ(A) there are an integer l = 0,±1, ... and an eigenvalueλj(W (sl)) of the matrix

W (sl) =m∑

k=0

ckskl ,

where sl = π(2l − 1), such that

|sl|−m|λ− λj(W (sl))| ≤ (q0vn−10 wn)1/n.

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References

[1] N. Dunford and Schwartz, J. T. Linear Operators, part III. Spectral Operators.Wiley Interscience publishers, New York, 1966.

[2] D.E. Edmunds, and Evans W.D. Spectral Theory and Differential Operators.Clarendon Press, Oxford, 1990.

[3] M.I. Gil’, Stability of Finite and Infinite Dimensional Systems, Kluwer Ac. Pub-lishers, Boston-Dordrecht-London, 1998.

[4] M.I. Gil’, Operator Functions and Localization of Spectra, Lectures Notes In Math-ematics vol. 1830, Springer-Verlag, Berlin, 2003.

[5] D. Henry. Geometric Theory of Semilinear Parabolic Equations, Springer Verlag,Berlin, 1981.

[6] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag. New York,1966.

[7] P. Kurasov, Naboko, S., On the essential spectrum of a class of singular matrixdifferential operators. I: Quasiregularity conditions and essential self-adjointness.Math. Phys. Anal. Geom. 5, No.3, 243-286 (2002).

[8] J. Locker, Spectral Theory of Nonselfadjoint Two Point Differential Operators.Amer. Math. Soc, Mathematical Surveys and Monographs, Volume 73, R.I, 1999.

[9] R. Nagel (Ed), One-parameter Semigroups of Positive Operators, Springer, Berlin,1986.

[10] M.A. Naimark, Linear Differential Operators, Harrap, London, 1968.

[11] L.A. Shuster, Estimates of eigenfunctions and localization of the spectrum of dif-ferential operators. J. Math. Anal. Appl. 229, No.2, 363-375 (1999).

[12] J. Weidmann, Spectral Theory of Differential Operators, Springer Verlag, Berlin,1987

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Composition Followed by Differentiation

Between Weighted Bergman Spaces and

Bloch Type Spaces

Songxiao LiDepartment of Mathematics, ShanTou University, 515063, Shantou, ChinaDepartment of Mathematics, JiaYing University, 514015, Meizhou, China

E-mail: [email protected]; [email protected]

Stevo StevicMathematical Institute of the Serbian Academy of Science,

Knez Mihailova 35/I, 11000 Beograd, SerbiaE-mail: [email protected]; [email protected]

Abstract: The boundedness of products of differentiation operators and compo-sition operators between the weighted Bergman space and the Bloch type space arediscussed in this paper.

MSC 2000: 47B38, 30H05.

Keywords: differentiation operator, composition operator, Bloch type space,weighted Bergman space.

1 Introduction

Let D be the open unit disk in the complex plane C and H(D) be the class of allfunctions analytic on D. Let dA denote the normalized Lebesgue area measure in Dsuch that A(D) = 1. For 0 < p < ∞ and α > −1, the weighted Bergman space Ap

α isthe set of all functions f analytic on D satisfying

‖f‖pAp

α=

∫

D|f(z)|pdAα(z) < ∞,

where dAα(z) = (α + 1)(1− |z|2)αdA(z).An f ∈ H(D) is said to belong to the Bloch type space, or β-Bloch space Bβ(β > 0)

if‖f‖Bβ = |f(0)|+ sup

z∈D(1− |z|2)β |f ′(z)| < ∞.

Under the norm ‖ ·‖Bβ , Bβ is a Banach space. When β = 1, B1 = B is the well knownBloch space. Let Bβ

0 denote the subspace of Bβ consisting of those f ∈ Bβ for which

(1− |z|2)β |f ′(z)| → 0 as |z| → 1.

333


This space is called the little β-Bloch space(see [9, 10]).Let ϕ be an analytic self map of D. Associated with ϕ is the composition operator

Cϕ defined byCϕf = f ϕ

for f ∈ H(D). It is a well known consequence of the Littlewood’s subordinationprinciple that the composition operator Cϕ is bounded on the classical Hardy andBergman spaces. It is interesting to provide a function theoretic characterization ofwhen ϕ induces a bounded or compact composition operator on various spaces (see,for example, [2] and [10]).

Let D be the differentiation operator. The composition operator is one of thetypical bounded operators, while the differentiation operator is typically unboundedon many analytic function spaces. The products of the composition operator anddifferentiation operator are defined by

DCϕ(f) = (f ϕ)′ = f ′(ϕ)ϕ′, f ∈ H(D)

andCϕD(f) = f ′(ϕ), f ∈ H(D)

respectively. The operator DCϕ was first studied by Hibschweiler and Portnoy in[4], where the boundedness of DCϕ between Hardy space and Bergman space areinvestigated. In [5], the boundedness and the compactness of DCϕ on Bloch typespaces are studied.

In this paper, we study DCϕ between weighted Bergman spaces and Bloch typespaces. Sufficient and necessary conditions for the boundedness of the operator DCϕ

are given. As a byproduct, we obtain the characterization of the boundedness of CϕD.Throughout this paper, constants are denoted by C, they are positive and may

differ from one occurrence to the other. The notation A ³ B means that there is apositive constant C such that B/C ≤ A ≤ CB.

2 Main results and proofs

In this section we give our main results. In order to prove the main results of thispaper, the following auxiliary lemmas are needed.

Lemma 1. Assume that 0 < p < ∞ and α > −1. If f ∈ Apα, then

|f ′(z)| ≤ C‖f‖Ap

α

(1− |z|2)(2+α+p)/p. (1)

Proof. Let β(z, w) denote the Bergman metric between two points z and w in D.It is given by

β(z, w) =12

log1 + |ϕz(w)|1− |ϕz(w)| .

For a ∈ D and r > 0, the set D(a, r) = z ∈ D : β(a, z) < r is a Bergman metric diskwith center a and radius r. It is well known that (see, for example, [10])

(1− |a|2)2|1− az|4 ³ 1

(1− |z|2)2 ³1

(1− |a|2)2 ³1

|D(a, r)| ,

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334

when z ∈ D(a, r), where |D(a, r)| is the hyperbolic area of the disk D(a, r). For0 < r < 1 and z ∈ D, by the subharmonicity of |f ′(z)|p and the well known asymptoticformula (see, for example, [6, 7])

∫

D|f(z)|p(1− |z|2)αdA(z) ³ |f(0)|p +

∫

D|f ′(z)|p(1− |z|2)α+pdA(z),

we obtain

|f ′(z)|p ≤ C

(1− |z|2)2∫

D(z,r)

|f ′(w)|pdA(w)

≤ C

(1− |z|2)2+α+p

∫

D(z,r)

(1− |w|2)α+p|f ′(w)|pdA(w)

≤C‖f‖p

Apα

(1− |z|2)2+α+p,

from which we obtain the desired result.

By a similar argument and using the following asymptotic formula ([6, 7])∫

D|f(z)|p(1− |z|2)αdA(z) ³ |f(0)|p + |f ′(0)|p +

∫

D|f ′′(z)|p(1− |z|2)α+2pdA(z),

we obtain the following lemma.

Lemma 2. Assume that p > 0, α > −1 and f ∈ Apα. Then there is a positive constant

C independent of f such that

|f ′′(z)| ≤ C‖f‖Ap

α

(1− |z|2)(2+α+2p)/p. (2)

Let 0 < p < ∞, µ be a positive Borel measure on D and

Dp(µ) =

f ∈ H(D) | ‖f‖pDp(µ) =

∫

D|f ′(z)|pdµ(z) < ∞

.

Lemma 3. Let µ be a positive measure on D and 0 < p, β < ∞. Then the followingstatements are equivalent:

(a) i : Bβ 7−→ Dp(µ) is bounded;(b) i : Bβ

0 7−→ Dp(µ) is bounded;(c) ∫

D

dµ(z)(1− |z|2)βp

< ∞.

Remark. The above lemma was obtained by Zhao when 0 < β ≤ 1 (see [8]). Infact, his proof implies that the result also holds for β > 1. Partial results can also befound in [1] when β = 1.

Now we are in a position to formulate and prove the main results of this paper.

WEIGHTED BERGMAN AND BLOCH TYPE SPACES

335

Theorem 1. Suppose that 0 < p, β < ∞,−1 < α < ∞ and ϕ is an analytic self-map of the unit disk. Then DCϕ : Ap

α → Bβ is bounded if and only if the followingconditions are satisfied:

(a)

supz∈D

(1− |z|2)β |ϕ′(z)|2(1− |ϕ(z)|2)(2+α+2p)/p

< ∞; (3)

(b)

supz∈D

(1− |z|2)β |ϕ′′(z)|(1− |ϕ(z)|2)(2+α+p)/p

< ∞. (4)

Proof. Suppose that conditions (3) and (4) hold. Then, for arbitrary z ∈ D andf ∈ Ap

α, (1) and (2) imply

(1− |z|2)β |(DCϕf)′(z)|≤ (1− |z|2)β |(f ′(ϕ)ϕ′)′(z)|≤ (1− |z|2)β |ϕ′(z)|2|f ′′(ϕ(z))|+ (1− |z|2)β |ϕ′′(z)||f ′(ϕ(z))|

≤ C|ϕ′(z)|2(1− |z|2)β

(1− |ϕ(z)|2) 2+αp +2

‖f‖Apα

+ C|ϕ′′(z)|(1− |z|2)β

(1− |ϕ(z)|2) 2+αp +1

‖f‖Apα. (5)

In addition, from Lemma 1 we see that

|(DCϕf)(0)| ≤ C|ϕ′(0)|‖f‖Apα

(1− |ϕ(0)|2) 2+αp +1

.

From this, by taking the supremum in the inequality (5) over D, then employingconditions (3) and (4), we obtain that DCϕ : Ap

α → Bβ is bounded.Conversely, suppose that DCϕ : Ap

α → Bβ is bounded, i.e. there exists a constantC such that ‖DCϕf‖Bβ ≤ C‖f‖Ap

αfor all f ∈ Ap

α. Then, taking f(z) = z andf(z) = z2, we obtain that

supz∈D

(1− |z|2)β |ϕ′′(z)| < ∞ (6)

and

supz∈D

(1− |z|2)β |(ϕ′(z))2 + ϕ′′(z)ϕ(z)| < ∞.

Using these facts and the boundedness of the function ϕ(z), we have that

supz∈D

(1− |z|2)β |ϕ′(z)|2 < ∞. (7)

For fixed w ∈ D, we define the test function

fw(z) =(1− |w|2)(α+2)/p

(1− wz)(4+2α)/p.

It is easy to check that fw ∈ Apα and supw∈D ‖fw‖Ap

α≤ C(or see [3]). Moreover

|f ′w(z)| = 4 + 2α

p

(1− |w|2)(2+α)/p|w||1− wz|(4+2α+p)/p

;

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336

|f ′′w(z)| = 4 + 2α

p

4 + 2α + p

p

(1− |w|2)(2+α)/p|w|2|1− wz|(4+2α+2p)/p

.

Hence, we have

C‖DCϕ‖Apα→Bβ ≥ ‖DCϕfϕ(λ)‖Bβ

≥ −4 + 2α

p

4 + 2α + p

p

(1− |λ|2)β |ϕ′(λ)|2|ϕ(λ)|2(1− |ϕ(λ)|2)(2+α+2p)/p

+4 + 2α

p

(1− |λ|2)β |ϕ′′(λ)||ϕ(λ)|(1− |ϕ(λ)|2)(2+α+p)/p

for λ ∈ D. Therefore, we obtain

(1− |λ|2)β |ϕ′′(λ)||ϕ(λ)|(1− |ϕ(λ)|2)(2+α+p)/p

≤ 4 + 2α + p

p

(1− |λ|2)β |ϕ′(λ)|2|ϕ(λ)|2(1− |ϕ(λ)|2)(2+α+2p)/p

+C‖DCϕ‖Apα→Bβ . (8)

Next, set

gw(z) =4 + 2α + p

4 + 2α

(1− |w|2)(α+2)/p

(1− wz)(4+2α)/p− 1− |w|2

1− wz

(1− |w|2)(α+2)/p

(1− wz)(4+2α)/p, w ∈ D.

Then, since

gw(z) =1− |w|21− wz

∈ H∞,

we see that gw ∈ Apα, moreover supw∈D ‖gw‖Ap

α≤ C. Also, we have g′ϕ(λ)(ϕ(λ)) = 0

and

|g′′ϕ(λ)(ϕ(λ))| = 4 + 2α + p

p

|ϕ(λ)|2(1− |ϕ(λ)|2)(2+α+2p)/p

.

Hence, we obtain

∞ > C‖DCϕ‖Apα→Bβ ≥ ‖DCϕgϕ(λ)‖Bβ ≥ 4 + 2α + p

p

(1− |λ|2)β |ϕ′(λ)|2|ϕ(λ)|2(1− |ϕ(λ)|2)(2+α+2p)/p

.

Thus

sup|ϕ(λ)|> 1

2

(1− |λ|2)β |ϕ′(λ)|2(1− |ϕ(λ)|2)(2+α+2p)/p

≤ sup|ϕ(λ)|> 1

2

4(1− |λ|2)β |ϕ′(λ)|2|ϕ(λ)|2(1− |ϕ(λ)|2)(2+α+2p)/p

≤ sup|ϕ(λ)|> 1

2

C‖DCϕ‖Apα→Bβ < ∞. (9)

By (7), we see that

sup|ϕ(λ)|≤ 1

2

(1− |λ|2)β |ϕ′(λ)|2(1− |ϕ(λ)|2)(2+α+2p)/p

≤ 4(2+α+2p)/p

3(2+α+2p)/psup

|ϕ(λ)|≤ 12

(1− |λ|2)β |ϕ′(λ)|2

< ∞.

(10)

Therefore, from (9) and (10) we see that

supλ∈D

(1− |λ|2)β |ϕ′(λ)|2(1− |ϕ(λ)|2)(2+α+2p)/p

< ∞.


337

From this and (8), we obtain

supλ∈D

(1− |λ|2)β |ϕ′′(λ)||ϕ(λ)|(1− |ϕ(λ)|2)(2+α+p)/p

< ∞. (11)

From (11) and (6), we have that

sup|ϕ(λ)|> 1

2

(1− |λ|2)β |ϕ′′(λ)|(1− |ϕ(λ)|2)(2+α+p)/p

≤ 2 sup|ϕ(λ)|> 1

2

(1− |λ|2)β |ϕ′′(λ)||ϕ(λ)|(1− |ϕ(λ)|2)(2+α+p)/p

(12)

and

sup|ϕ(λ)|≤ 1

2

(1− |λ|2)β |ϕ′′(λ)|(1− |ϕ(λ)|2)(2+α+p)/p

≤ 42+α+p

p

32+α+p

p

sup|ϕ(λ)|≤ 1

2

(1− |λ|2)β |ϕ′′(λ)| < ∞. (13)

Combining (12) and (13), we obtain (4). The proof is completed.

Theorem 2. Suppose that 0 < p, β < ∞,−1 < α < ∞ and ϕ is an analytic self-mapof the unit disk. Then, DCϕ : Ap

α → Bβ0 is bounded if and only if DCϕ : Ap

α → Bβ isbounded,

lim|z|→1

(1− |z|2)β |ϕ′(z)|2 = 0; (14)

and

lim|z|→1

(1− |z|2)β |ϕ′′(z)| = 0. (15)

Proof. First assume that DCϕ : Apα → Bβ

0 is bounded. Then, it is clear thatDCϕ : Ap

α → Bβ is bounded. Taking the functions f(z) = z and f(z) = z2 respec-tively, we obtain (14) and (15).

Conversely, assume that DCϕ : Apα → Bβ is bounded, and that (14) and (15) hold.

Then, for each polynomial p(z), we have that

(1− |z|2)β |(DCϕp)′(z)|≤ (1− |z|2)β |ϕ′(z)|2|p′′(ϕ(z))|+ (1− |z|2)β |ϕ′′(z)p′(ϕ(z))|. (16)

Sincesupw∈D

|p′′(w)| < ∞ and supw∈D

|p′(w)| < ∞,

from (14)-(16) it follows that DCϕp ∈ Bβ0 . Since the set of all polynomials is dense

in Apα (see [3]), we have that for every f ∈ Ap

α there is a sequence of polynomials(pn)n∈N such that ‖f − pn‖Ap

α→ 0 as n →∞. Hence

‖DCϕf −DCϕpn‖Bβ ≤ ‖DCϕ‖Apα→Bβ‖f − pn‖Ap

α→ 0

as n → ∞, by using the boundedness of the operator DCϕ : Apα → Bβ . Since Bβ

0

is a closed subset of Bβ , we obtain DCϕ(Apα) ⊂ Bβ

0 . Therefore DCϕ : Apα → Bβ

0 isbounded.

LI-STEVIC

338

Theorem 3. Suppose that ϕ is an analytic self-map of the unit disk. Assume thatp, β > 0 and α > −1. Then the following statements are equivalent:

(a) DCϕ : Bβ → Apα is bounded;

(b) DCϕ : Bβ0 → Ap

α is bounded;(c) ∫

D

|ϕ′(z)|p(1− |ϕ(z)|2)βp

dAα(z) < ∞.

Proof. Let f ∈ Apα. We have

‖DCϕf‖pAp

α=

∫

D|(DCϕf)(z)|pdAα(z)

=∫

D|f ′(ϕ(z))|p|ϕ′(z)|pdAα(z)

=∫

D|f ′(ϕ(z))|pdµ(z) =

∫

D|f ′(z)|pdµ ϕ−1(z),

wheredµ(z) = |ϕ′(z)|pdAα(z).

By Lemma 3, we know that DCϕ : Bβ(orBβ0 ) → Ap

α is bounded if and only if

∞ >

∫

D

dµ ϕ−1

(1− |z|2)βp=

∫

D

|ϕ′(z)|p(1− |ϕ(z)|2)βp

dAα(z),

finishing the proof of the theorem.

Remark 1. Note that by modifying the proofs of Theorems 1-3, we can prove thefollowing results. We omit the details.

Theorem 4. Suppose that 0 < p, β < ∞,−1 < α < ∞ and ϕ is an analytic self-mapof D. Then

(a) CϕD : Apα → Bβ is bounded if and only if

supz∈D

(1− |z|2)β |ϕ′(z)|(1− |ϕ(z)|2)(2+α+2p)/p

< ∞;

(b) CϕD : Apα → Bβ

0 is bounded if and only if ϕ ∈ Bβ0 ;

(c) CϕD : Bβ → Apα is bounded if and only if CϕD : Bβ

0 → Apα is bounded if and

only if ∫

D

(1− |z|2)α

(1− |ϕ(z)|2)βpdA(z) < ∞.

Acknowledgments. The first author of this paper is supported in part by theNNSF China (No.10671115), PHD Foundation (No. 20060560002) and NSF of Guang-dong Province (No. 06105648).


339

References

[1] J. Arazy, S. D. Fisher, and J. Peetre, Mobius invariant function spaces, J. ReineAngew. Math. 363 (1985), 110-145.

[2] C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of AnalyticFunctions, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1995.

[3] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces. GraduateTexts in Mathematics, 199. Springer-Verlag, New York, 2000.

[4] R. A. Hibschweiler and N. Portnoy, Composition followed by differentiation be-tween Bergman and Hardy spaces, Rocky Mountain J. Math. 35 (3) (2005),843-855.

[5] S. Li and S. Stevic, Composition followed by differentiation on Bloch type space,J. Comput. Anal. Appl. 9 (2) (2007), 195-205.

[6] J. H. Shi, Inequalities for the integral means of holomorphic functions and theirderivatives in the unit ball of Cn, Trans. Amer. Math. Soc. 328 (2) (1991), 619-637.

[7] S. Stevic, Weighted integrals of holomorphic functions in Cn, Complex Variables,47 (9) (2002), 821-838.

[8] R. H. Zhao, Composition operators from Bloch type spaces to Hardy and Besovspaces, J. Math. Anal. Appl. 233 (1999), 749-766.

[9] K. Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. Math. 23(3) (1993), 1143-1177.

[10] K. Zhu, Operator Theory on Function Spaces, Marcel Dekker, Inc. Pure andApplied Mathematics 139, New York and Basel, 1990.

LI-STEVIC

340

A NOTE ON INTEGRAL INEQUALITIES INVOLVING THEPRODUCT OF TWO FUNCTIONS ON TIME SCALES

A.TUNA, B. I. YASAR AND S. KUTUKCU

Abstract. In this article we study new integral inequalities involving twofunctions and their derivatives on time scales.

1. Introduction

The theory of dynamic equations on time scales(aka measure chains) was in-troduced by Hilger [3] with the motivation of providing a unied approach tocontinuous and discrete analysis. The generalized derivative or Hilger derivativef4(t) of a function f : T! R, where T is a so-called "time scale" (an arbitraryclosed nonempty subset of R) becomes the usual derivative when T = R, that isf4(t) = f 0(t): On the other hand, if T = Z, then f4(t) reduces to the usual for-ward di¤erence, that is f4(t) = 4f(t): This theory not only brought equationsleading to new applications. Also, this theory allows one to get some insight intoand better understanding of the subtle di¤erences between discrete and continuoussystems [1, 2].In this paper, we establish new integral inequalities involving two functions and

their derivatives on time scales.Now, rst we mention without proof several fundamental denitions and result

from the calculus on time scales in an excellent introductory text by Bohner andPeterson [2].

2. General Definitions

Denition 1. A time scale T is a nonempty closed subset of R.We assume throughout that T has the topology that is inherited from the standard

topology on R. It also assumed throughout that in T the interval [a; b] means the setft 2 T : s < tg for the points a < b in T. Since a time scale may not be connected,we need the following concept of jump operators.

Denition 2. The mappings ; : T! T dened by (t) = inf fs 2 T : s > tg and(t) = sup fs 2 T : s < tg are called the jump operators.

The jump operators and allow the classication of points in T in the followingway:

Denition 3. A nonmaximal element t 2 T is said to be right-dense if (t) = t;right-scattered if (t) > t; left-dense if (t) = t;left-scattered if (t) < t:

In the case T = R, we have (t) = t; and if T =hZ; h > 0; then (t) = t+ h:

2000 Mathematics Subject Classication. Primary 26D15, 39A10,Key words and phrases. Integral inequalities; Product of two functions; Time scales.

1

341


2 A.TUNA, B. I. YASAR AND S. KUTUKCU

Denition 4. The mapping : T! R+ dened by (t) = (t) t is called thegraininess function.

If T = R, then (t) = 0; and when T =Z; we have (t) = 1:

Denition 5. Let f : T! R: f is called di¤erentiable at t 2 Tk; with (delta)derivative f4(t) 2 R if given " > 0 there exists a neighborhood U of t such that,for all s 2 U; f(t) f(s) f4(t)[(t) s] " k(t) sk ;where f = f :If T = R, then f4(t) = df(t)

dt ; and if T = R, then f4(t) = f(t+ 1) f(t):

Some basic properties of delta derivatives are the following [2].

Theorem 1. Assume that f : T! R and let t 2 Tk:(i) If f is di¤erentiable at t; then f is continuous at t:(ii) If f is di¤erentiable at t and t is right-scattered, then f is di¤erentiable at t

with

f4(t) =f(t) f(t)(t) t :

(iii) If f is di¤erentiable at t and t is right-dense, then

f4(t) = limt!s

f(t) f(s)t s :

(iv) If f is di¤erentiable at t; then

f(t) = f(t) + (t)f4(t)

Example 1. (i) If f : T!R is dened by f(t) = for all t 2 T, where 2 R isconstant, then f4(t) 0:(ii) If f : T!R is dened by f(t) = t for all t 2 T, then f4(t) 1

Denition 6. The function f : T! R is said to be rd-continuous (denote f 2Crd(T,R)) if, at all t 2 T,(i) f is continuous at every right-dense point t 2 T,(ii) lims!tf(s) exists and is nite at every left-dense point t 2 T.

Denition 7. Let f 2 Crd(T,R): Then g : T! R is called the antiderivative of fon T if it is di¤erentiable on T and satises g4(t) = f(t) for any t 2 Tk. In thiscase, we dened

tZa

f(s)4 s = g(t) g(a); t 2 T.

Theorem 2. If f is 4-integrable on [a; b], then so is jf j ;andbZa

f(t)4 t

bZa

jf(t)j 4 t:

We assume that T = [a; b] is an arbitrary interval on time scale. Our main resultsare given in the following theorem.

342

INTEGRAL INEQUALITIES ON TIME SCALES 3

3. Main Result

Theorem 3. Let f; g 2 C1rd(T,R). Thenf(t)g(t) 12 [g(t)F + f(t)G](3.1)

1

4

24jg(t)j bZa

f4()4 + jf(t)j bZa

g4()4 35

and

jf(t)g(t) [g(t)F + f(t)G] + FGj(3.2)

1

4

0@ bZa

f4()4 1A0@ bZ

a

g4()4 1A

for all t 2 T, where

(3.3) F =f(a) + f(b)

2; G =

g(a) + g(b)

2:

The constant 14 in (3.1) and (3.2) is sharp.

Proof. From the hypotheses of Theorem 3. we have the following identities

(3.4) f(t) F = 1

2

24 tZa

f4()4 bZt

f4()4

35 ;

(3.5) g(t)G = 1

2

24 tZa

g4()4 bZt

g4()4

35 :Multiplying both sides of (3.4) and (3.5) by g(t) and f(t) respectively, adding

the resulting identities and rewriting we have

f(t)g(t) 12[g(t)F + f(t)G](3.6)

=1

4

24g(t)24 tZa

f4()4 bZt

f4()4

35+f(t)

24 tZa

g4()4 bZt

g4()4

35From (3.6) and using the properties of modulus we havef(t)g(t) 12 [g(t)F + f(t)G]

1

4

24jg(t)j bZa

f4()4 + jf(t)j bZa

g4()4 35 :

This is the required inequality in (3.1).

343


Multiplying the left sides and right sides of (3.4) and (3.5) we get

f(t)g(t) [g(t)F + f(t)G] + FG(3.7)

=1

4

24 tZa

f4()4 bZt

f4()4

3524 tZa

g4()4 bZt

g4()4

35From (3.7) and using the properties of modulus we have

jf(t)g(t) [g(t)F + f(t)G] + FGj 1

4

0@ bZa

f4()4 1A0@ bZ

a

g4(t)4 1A :

This proves the inequality in (3.2).To prove the sharpness of the constant 1

4 in (3.1) and (3.2), assume that theinequalities (3.1) and (3.2) hold with constant c > 0 and k > 0 respectively.That is, f(t)g(t) 12 [g(t)F + f(t)G]

(3.8)

c

24jg(t)j bZa

f4()4 + jf(t)j bZa

g4()4 35 ;

and

jf(t)g(t) [g(t)F + f(t)G] + FGj(3.9)

k

0@ bZa

f4()4 1A0@ bZ

a

g4()4 1A ;

for t 2 T. In (3.8) and (3.9), choose f(t) = g(t) = t and hence f4(t) = g4(t) = 1by Example 1(ii); F = G = a+b

2 : Then by simple computation, we get

(3.10)

t 12(a+ b) 2c(b a);

and

(3.11)

t(t (a+ b)) + (a+ b2 )2 k(b a)2:

By taking t = b; from (3.10) we observe that c 14 and from (3.11) it is easy to

observe that k 14 ; which proves the sharpness of the constants in (3.1) and (3.2).

The proof is complete.

Remark 1. The results of Theorem 3. holds for an arbitrary time scale. Thus forsome peculiar time scales, by Theorem 3., we immediately obtain the following twocorollaries.

344

INTEGRAL INEQUALITIES ON TIME SCALES 5

Corollary 1. Let T = R and f; g 2 C1([a; b],R), [a; b] R; a < b: Thenf(t)g(t) 12 [g(t)F + f(t)G](3.12)

1

4

24jg(t)j bZa

jf 0()j d + jf(t)jbZa

jg0()j d

35 ;and

jf(t)g(t) [g(t)F + f(t)G] + FGj(3.13)

1

4

0@ bZa

jf 0()j d

1A0@ bZa

jg0()j d

1Afor all t 2 [a; b], where

F =f(a) + f(b)

2; G =

g(a) + g(b)

2:

The constant 14 in (3.12) and (3.13) is sharp.

Corollary 2. Let T = Z and fuig ; fvig for i = 0; 1; 2; :::; n; n 2 N be sequences ofreal numbers. Then uivi 12 [viU + uiV ]

(3.14)

1

4

24jvij n1Xj=0

j4uj j+ juijn1Xj=0

j4vj j

35 ;and

juivi [viU + uiV ] + UV j(3.15)

1

4

0@n1Xj=0

j4uj j

1A0@n1Xj=0

j4vj j

1A ;for i = 0; 1; 2; :::; n; where

U =u0 + un2

; V =v0 + vn2

;

and 4 is the forward di¤erence operator. The constant 14 in (3.14) and (3.15)is sharp.

Remark 2. If we take g(t) = 1 and hence g4(t) = 0 in (3.1), then by simplecalculation we get the inequality

(3.16) jf(t) F j 1

2

bZa

f4()4 ;

345


Remark 3. Dividing both sides of (3.6) and (3.7) by (b a), then integrating bothsides with respect to t over T and closely looking at the proof of Theorem 3 we get 1

b a

bZa

f(t)g(t)4 t(3.17)

1

2(b a)

24F bZa

g(t)4 t+GbZa

f(t)4 t

35 1

4(b a)

240@ bZa

jg(t)j 4 t

1A0@ bZa

f4(t)4 t1A

+

0@ bZa

jf(t)j 4 t

1A0@ bZa

g4(t)4 t1A35

and 1

b a

bZa

f(t)g(t)4 t 1

(b a)

24F bZa

g(t)4 t+GbZa

f(t)4 t FG

35(3.18)

1

4

0@ bZa

f4(t)4 t1A0@ bZ

a

g4(t)4 t1A :

Conclusion 1. If we take T = R; we note the inequalities (4.5) and (4.6) aresimilar to those of the well known inequalities due to Grüss and µCebye, see [4, 5]

References

[1] R.P.Agarwal, M. Bohner, D. ORegan, A. Peterson, Dynamic equations on time scale: Asurvey, J. Comput. Appl. Math. 141:1-26 (2002).

[2] M. Bohner, A. Peterson, Dynamic equations on time scale, An Introduction with Applications,Birkhauser, Boston, 2001.

[3] S. Hilger, analysis on measure chains- A unied approach to continuous and discrete calculus,result Math.18:18-56 (1990).

[4] D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, Berlin-New York, 1970.[5] D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Classical and new Inequalities in Analysis,

Kluwer Academic Publishers, Dordrecht, 1993.[6] B. G. Pachpatte, A note on integral inequalities involving the product of two functions, J.

·Ineq. in Pure and Appl. Math., 7 (2006), no.2, Article 78, 4 pp, (electronic).

Department of Mathematics, Faculty of Science and Arts,University of Gazi, BeSevler,06500, Ankara, Turkey

E-mail address : [email protected] address : [email protected]

Department of Mathematics, Faculty of Science and Arts, Ondokuz May¬s University,Kurupelit, 55139, Samsun, Turkey

E-mail address : [email protected]

346

SOME NEW INTEGRAL INEQUALITIES FOR RETARDEDVOLTERRA EQUATIONS

B. I. YASAR AND A.TUNA

Abstract. In this article we study some new integral inequalities for retardedVolterra equations.

1. Introduction

Integral inequalities are very useful in the qualitative analysis of di¤erential andintegral equations. Starting with [6], several recent investigations, see [7, 11, 12, 13,14], were devoted to retarded integral inequalities. Next [5] has considered case ofretarded Volterra integral equations. There, bounds on the solutions and, by meansof examples, established also it is shown the usefulness of results in investigatingthe asymptotic behavior of the solutions.In this article we study some new integral inequalities for retarded Volterra

equations.First we mention several fundamental Theorems without proof by Olivia Lipovan

[5].

2. Preliminaries

2.1. The linear case.

Theorem 1. Let k 2 C(R+;R+); 2 C1(R+;R+); a 2 C(R+ R+;R+) with(t; s) ! @ta(t; s) 2 C(R+ R+;R+): Assume in addition that is nondecreasingand (t) t for t 0: If u 2 C(R+;R+) satises

(2.1) u(t) k(t) +(t)Z0

a(t; s)u(s)ds; t 0;

then

(2.2) u(t) k(t) + e

(t)R0

a(t;s)dstZ0

e(r)R0

a(r;s)ds

@r

(r)R0

a(r; s)k(s)ds

!dr; t 0:

Corollary 1. Assume ; a are as in Theorem 1. and k(t) k > 0: If u 2C(R+;R+) satises (2.1), then

(2.3) u(t) ke

(t)R0

a(t;s)ds

; t 0:

2000 Mathematics Subject Classication. Primary 26D15, 39A10,Key words and phrases. Integral inequalities; Volterra equations; Retarded Volterra

inequalities.

1

347


2 B. I. YASAR AND A.TUNA

Remark 1. We note that for @ta(t; s) 0 in Corollary 1. we get an inequalityobtained in [6]. If, in addition, (t) = t; the inequality given by Corollary 1. reducesto Gronwalls inequality [4].

Theorem 2. Let a; b; k 2 C(R+;R+); 2 C1(R+;R+) and assume that isnondecreasing with (t) t for t 0: If u 2 C(R+;R+) satises

(2.4) u(t) k(t) + a(t)(t)Z0

b(s)u(s)ds; t 0;

then

(2.5) u(t) k(t) + a(t)(t)Z0

e

(t)Rr

a(s)b(s)ds

b(r)k(r)dr; t 0:

Remark 2. Considering (t) = t in Theorem 2., we obtain Morros inequality [4].

2.2. The nonlinear case.

Theorem 3. Let a; be as in Theorem 1. Assume k; ! 2 C(R+;R+) are non-

decreasing functions with k(0) > 0; !(t) > 0 for t > 0 and

1Z1

dt!(t) = 1: If

u 2 C(R+;R+) satises

u(t) k(t) +(t)Z0

a(t; s)!(u(s))ds; t 0;

then

(2.6) u(t) G1

0B@G(k(t)) + (t)Z0

a(t; s)ds

1CA ; t 0:

where G(t) =

tZ1

ds!(s) ; t 0:

Theorem 4. Let a; b; k 2 C(R+;R+); 2 C1(R+;R+) and assume that a; k; are nondecreasing functions with (t) t for t 0: Let also ! 2 C(R+;R+)

be a nondecreasing function such that !(t) > 0 for t > 0 and

1Z1

dt!(t) = 1: If

u 2 C(R+;R+) satises

(2.7) u(t) k(t) + a(t)(t)Z0

b(s)!(u(s))ds; t 0;

then

(2.8) u(t) G1

0B@G(k(t)) + a(t) (t)Z0

b(s)ds

1CA ; t 0:

348

SOME NEW INTEGRAL INEQUALITIES FOR RETARDED VOLTERRA EQUATIONS 3

where G(t) =

tZ1

ds!(s) ; t 0:

3. Main Results

Theorem 5. Let a; c; g; h 2 C(R+;R+) and 2 C1(R+;R+). Assume in additionthat is nondecreasing and (t) t for t 0: If u 2 C(R+;R+) satises

(3.1) (u(t))p a(t) + c(t)

(t)Z0

[g(s)u(s) + h(s)] ds; t 0;

then(3.2)

u(t)

8>><>>:a(t) + c(t)2664e

(t)R0

g(s)c(s)p ds

tZ0

e(r)R0

g(s)c(s)p ds

@r

0B@ (r)Z0

F (s)ds

1CA dr37759>>=>>;

1p

; t 0:

where

(3.3) F (t) = g(t)

p 1p

+a(t)

p

+ h(t):

Proof. Obviously, if (t) = 0; then the inequality (3.2) holds. Thus, in the nextproof, we always assume that (t) t with t > 0;Dene a function z(t) by

(3.4) z(t) =

(t)Z0

[g(s)u(s) + h(s)] ds:

Then (3.1) can be restated as

(3.5) (u(t))p a(t) + c(t)z(t):

Using the elementary inequality (See [8,p,30])

x1p y

1q x

p+y

q;

where x 0; y 0; and 1p +

1q = 1 with p > 1; we observe that

u(t) [a(t) + c(t)z(t)]1p(3.6)

p 1p

+a(t)

p+c(t)

pz(t):

Combining (3.4), (3.5) and (3.6), we have

z0(t) = [g((t))u((t)) + h((t))]0(t) +

(t)Z0

[g(s)u(s) + h(s)] ds; t 0;

hg((t))

hp1p + a((t))

p + c((t))p z((t)))

i+ h((t))

i0(t)

+

(t)Z0

hg(s)

p1p + a(s)

p + c(s)p z(s)

+ h(s)

ids

349


=hg((t))

hp1p + a((t))

p

i+ h((t))

i0(t) +

(t)Z0

hg(s)

p1p + a(s)

p

+ h(s)

ids

+

264 g((t))c((t))p 0(t) +

(t)Z0

g(s)c(s)p ds

375 z(t):or, equivalently,

z0(t) z(t) ddt

0B@ (t)Z0

g(s)c(s)p ds

1CA ddt

0B@ (t)Z0

F (s)ds

1CAwhere

F (t) = g(s)p1p + a(s)

p

+ h(s)

Multiplying the above inequality by e(t)R0

g(s)c(s)p ds

, we get

ddt

0BB@z(t)e(t)R0

g(s)c(s)p ds

1CCA e(t)R0

g(s)c(s)p ds

ddt

0B@ (t)Z0

F (s)ds

1CAConsider now the integral on the interval [0; t] to obtain

(3.7) z(t) e

(t)R0

g(s)c(s)p ds

tZ0

e(r)R0

g(s)c(s)p ds

@r

0B@ (r)Z0

F (s)ds

1CA dr; t 0:Combine the above inequality with (u(t))p a(t) + c(t)z(t) to get (3.2) and,

with this, the proof is complete. Corollary 2. If we take p = 1, h(t) 0; it is clear that we can have the sameTheorem 1.2 and Remark 1.2. in [5].

Theorem 6. Assume that u; a; c; g; h;m 2 C(R+;R+); 2 C1(R+;R+); ! 2C(R+ R+;R+) with (t; s) ! @

@t!(t; s) 2 C(R+ R+;R+): Assume in additionthat is nondecreasing and (t) t for t 0;

(3.8) (u(t))p a(t) + c(t)

(t)Z0

!(t; s) [g(s) (u(s))p+ h(s)u(s) +m(s)] ds; t 0;

implies(3.9)u(t) 8>><>>:a(t) + c(t)

0BB@e(t)R0

!(t;s)A(s)dstZ0

e(r)R0

!(r;s)A(s)ds

@r

0B@ (r)Z0

!(r; s)B(s)ds

1CA dr1CCA9>>=>>;

1p

for t 0: Where

(3.10) A(t) = c(t)

g(t) +

h(t)

p

350

SOME NEW INTEGRAL INEQUALITIES FOR RETARDED VOLTERRA EQUATIONS 5

and

(3.11) B(t) = a(t)g(t) + h(t)

p 1p

+a(t)

p

+m(t):

Proof. Obviously, if (t) = 0; then the inequality (3.9) holds. Thus, in the nextproof, we always assume that (t) t with t > 0;Dene a function z(t) by

(3.12) z(t) =

(t)Z0

!(t; s) [g(s) (u(s))p+ h(s)u(s) +m(s)] ds

As in the proof of Theorem 5., we easily obtain (3.5) and (3.6). Combining(3.12), (3.5) and (3.6), we havez0(t) = !(t; (t)) [g((t)) (u((t)))

p+ h((t))u((t)) +m((t))]0(t)

+

(t)Z0

@@t!(t; s) [g(s) (u(s))

p+ h(s)u(s) +m(s)] ds; t 0;

f!(t; (t)) [g((t))a((t)) + c((t))g((t))z(t)]+h((t))

hp1p + a((t))

p + c((t))p z(t)

i+m((t))

o0(t)

+

(t)Z0

@@t!(t; s) [g(s)a(s) + c(s)g(s)z(s)]

+h(s)p1p + a(s)

p + c(s)p z(s)

+m(s)

ids

h!(t; (t))c((t))

g((t)) + h((t))

p

0(t)

+

(t)Z0

@@t!(t; s)c(s)

g(s) + h(s)

p

ds

375 z(t)+!(t; (t))

ha((t))g((t)) + h((t))

p1p + a((t))

p

+m((t))

i0(t)

+

(t)Z0

@@t!(t; s)

ha(s)g(s) + h(s)

p1p + a(s)

p

+m(s)

ids

or, equivalently,

z0(t) z(t) ddt

0B@ (t)Z0

!(t; s)A(s)ds

1CA ddt

0B@ (t)Z0

!(t; s)B(s)ds

1CAwhereA(t) = c(t)

g(t) + h(t)

p

andB(t) = a(t)g(t) + h(t)

p1p + a(t)

p

+m(t)

Multiplying the above inequality by e(t)R0

!(t;s)A(s)ds

, we get

351


ddt

0BB@z(t)e(t)R0

!(t;s)A(s)ds

1CCA e(t)R0

!(t;s)A(s)dsddt

0B@ (t)Z0

!(t; s)B(s)ds

1CAConsider now the integral on the interval [0; t] to obtain

z(t) e

(t)R0

!(t;s)A(s)dstZ0

e(r)R0

!(r;s)A(s)ds

@r

0B@ (r)Z0

!(r; s)B(s)ds

1CA dr; t 0:

Combine the above inequality with (u(t))p a(t) + c(t)z(t) to get (3.9) and,with this, the proof is complete. Corollary 3. In Theorem 6. if we take p = 1, c(t) 1; g(t) 0; h(t) 1;m(t) 0; we get Theorem 1.1.,Corollary1.1. and Remark 1.1.in [5].

References

[1] R. Bellman, The stability of solutions of linear di¤erential equations, Duke Math. J. 10 (1943)643647.

[2] I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problemsof di¤erential equations, Acta Math. Acad. Sci. Hungar. 7 (1965) 81-94.

[3] C. Corduneanu, Integral Equations and Applications, Cambridge Univ. Press, Cambridge,1991.

[4] T.H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of asystem of di¤erential equations, Ann. of Math. 20 (1919) 292296.

[5] O. Lipovan, Integral inequalities for retarded Volterra equations, J. Math. Anal. Appl. 322(2006) 349-358.

[6] O. Lipovan, A retarded Gronwall-like inequality and its applications, J. Math. Anal. Appl.252 (2000) 389401.

[7] O. Lipovan, A retarded integral inequality and its applications, J. Math. Anal. Appl. 285(2003) 436443.

[8] D.S. Mitrinovic, "Anallytic Inequalities," Springer Verlag, Berlin/New York, 1970.[9] A. Morro, A Gronwall-like inequality and its application to continuum thermodynamics, Boll.

Un. Mat. Ital. B 6 (1982) 553562.[10] B.G. Pachpatte, Inequalities for Di¤erential and Integral Equations, Academic Press, San

Diego, CA, 1998.[11] B.G. Pachpatte, Explicit bounds on certain integral inequalities, J. Math. Anal. Appl. 267

(2002) 4861.[12] B.G. Pachpatte, On some retarded integral inequalities and applications, J. Inequal. Pure

Appl. Math. 3 (2002), Article 18.[13] B.G. Pachpatte, On a certain retarded integral inequality and its applications, J. Inequal.

Pure Appl. Math. 5 (2004), Article 19.[14] B.G. Pachpatte, On some new nonlinear retarded integral inequalities, J. Inequal. Pure Appl.

Math. 5 (2004), Article 80.

Department of Mathematics, Faculty of Science and Arts,University of Gazi, BeSevler,06500, Ankara, Turkey

E-mail address : [email protected] address : [email protected]

352

The Dirichlet-Neumann Problem for the 2-D Laplace Equation in

an Exterior Cracked Domain with Neumann Condition on Cracks.

P.A.Krutitskii

Dept. of Mathematics, Faculty of Physics, Moscow State University,

Moscow 117234, Russia.

Fax: +7-095-9328820


353


2 P.A.Krutitskii

Abstract

The mixed Dirichlet-Neumann problem for the Laplace equation in an unbounded plane do-

main with cuts (cracks) is studied. The Dirichlet condition is given on closed curves making

up the boundary of the domain, while the Neumann condition is specified on the cuts. The

existence of a classical solution is proved by potential theory and a boundary integral equa-

tion method. The integral representation for a solution is obtained in the form of potentials.

The density of the potentials satisfies a uniquely solvable Fredholm integral equation of the

second kind and index zero. Singularities of the gradient of the solution at the tips of the

cuts are investigated.

AMS Subject Classification : 35J05, 35J25.

Key words and Phrases: Laplace equation, Dirichlet–Neumann problem,

boundary integral equation method.

354

The Dirichlet-Neumann Problem 3

1. Introduction.

The Boundary of a 2-D cracked domain consists of both closed curves and open arcs

(cuts). Open arcs model cracks in solids and screens or wings in fluids. Different physical

processes in cracked domains can be described by boundary value problems for the Laplace

equation, for example, distribution of stationary heat and electric fields in cracked solids,

electric flow in cracked semiconductors, flow of an ideal fluid over several obstacles and wings,

etc. Appropriate boundary conditions must be specified on the total boundary, i.e. on both

closed curves and open arcs (cracks). The Neumann boundary condition reflects the nonflow

(of fluid, electric current, etc.) through the boundary. The Dirichlet boundary condition

corresponds to the given temperature in heat theory, fluid pressure in hydrodynamics, electric

potential in electrostatics, etc.

Boundary value problems with mixed boundary conditions were not treated in cracked

domains by rigorous mathematical methods before. Even in the case of Laplace and

Helmholtz equations the problems in domains bounded by closed curves [2], [13–17] and

problems in the exterior of cuts (cracks) [14, 16], [18–20] were treated separately, because

different methods were used in their analysis. Previously the Neumann problem in the exte-

rior of a cut was reduced to a hypersingular integral equation [14, 16, 18, 19] or to an infinite

algebraic system of equations [20], while the Dirichlet problem in domains bounded by closed

curves was reduced to the Fredholm equation of the second kind [13–17]. The combination

of these methods in case of domains bounded by closed curves and cuts leads to an integral

equation, which is algebraic or hypersingular on cuts, and it is an equation of the second

kind with compact integral operators on the closed curves. The integral equation on the

whole boundary is rather complicated to be effectively studied by standard methods. The

355

4 P.A.Krutitskii

approach suggested in the present paper enables us to reduce the mixed Dirichlet–Neumann

problem in a cracked domain to the Fredholm integral equation of the second kind and index

zero on the whole boundary with the help of a nonclassical angular potential. It is shown

that the Fredholm integral equation is uniquely solvable, therefore the integral equation can

be computed by a standard code by discretization and inversion of the matrix. So our ap-

proach is constructive, because it gives the way for finding the numerical solution for mixed

boundary value problem with complicated boundary in an exterior domain. Our approach

is based on [5–6], where the problems in the exterior of cuts were reduced to the Fredholm

integral equations using the angular potential. At first these problems were reduced to the

Cauchy singular integral equation with additional conditions. Next, the singular integral

equation was reduced to the Fredholm integral equation of the 2nd kind and index zero

by regularization. In [7–10] our approach has been applied to the Dirichlet and Neumann

problems for the Laplace and Helmholtz equation in cracked domains. Some nonlinear prob-

lems of fluid dynamics were studied in [4]. Using an integral representation for a solution

of our problem in the form of potentials, obtained in the present paper, we derive explicit

asymptotic formulas for singularities of the gradient of the solution at the tips of the cuts

(cracks).

2. Formulation of the problem.

By a simple open curve we mean a non-closed smooth arc of finite length without self-

intersections [16].

In the plane x = (x1, x2) ∈ R2 we consider the exterior multiply connected domain

bounded by simple open curves Γ11, ...,Γ

1N1∈ C2,λ and simple closed curves

356


Γ21, ...,Γ

2N2∈ C1,λ, λ ∈ (0, 1], so that the curves have no points in common. We put

Γ1 =N1⋃

n=1

Γ1n, Γ2 =

N2⋃

n=1

Γ2n, Γ = Γ1 ∪ Γ2.

The exterior connected domain bounded by Γ2 will be called D. We assume that each curve

Γkn is parametrized by the arc length s : Γk

n = x : x = x(s) = (x1(s), x2(s)), s ∈ [akn, bk

n],

n = 1, ..., Nk, k = 1, 2, so that a11 < b1

1 < ... < a1N1

< b1N1

< a21 < b2

1 < ... < a2N2

< b2N2

and the domain D is on the right when the parameter s increases on Γ2n. Therefore points

x ∈ Γ and values of the parameter s are in one-to-one correspondence except a2n, b2

n, which

correspond to the same point x for n = 1, ..., N2. Below the sets of the intervals on the Os

axisN1⋃

n=1

[a1n, b1

n],N2⋃

n=1

[a2n, b2

n],2⋃

k=1

Nk⋃

n=1

[akn, bk

n]

will be denoted by the same symbols as the corresponding sets of curves, that is, by Γ1, Γ2

and Γ respectively.

We put C0(Γ2n) = F(s) : F(s) ∈ C0[a2

n, b2n], F(a2

n) = F(b2n), and

C0(Γ2) =N2⋂

n=1

C0(Γ2n).

By Dn we denote the interior domain bounded by the curve Γ2n, if n = 1, ..., N2.

The tangent vector to Γ at the point x(s) we denote by τx = (cosα(s), sinα(s)), where

cosα(s) = x′1(s), sinα(s) = x′2(s). Let nx = (sinα(s), − cosα(s)) be the normal vector to

Γ at x(s). The direction of nx is chosen such that it will coincide with the direction of τx if

nx is rotated anticlockwise through an angle of π/2. Therefore nx is the inward normal for

D on Γ2.

We consider the curves Γ1 as a set of cuts. The side of Γ1 which is on the left, when

the parameter s increases, will be denoted by (Γ1)+, and the opposite side will be denoted

357

6 P.A.Krutitskii

by (Γ1)−.

We say, that the function u(x) belongs to the smoothness class K if

1) u ∈ C0(D\Γ1) ∩ C2(D\Γ1),

2) ∇u ∈ C0(D\Γ1\Γ2\X), where X is a point-set, consisting of the end-points of Γ1 :

X =N1⋃

n=1

(x(a1

n) ∪ x(b1n)

),

3) in the neighbourhood of any point x(d) ∈ X for some constants C > 0, ε > −1 the

inequality holds

(1) |∇u| ≤ C |x− x(d)|ε ,

where x → x(d) and d = a1n or d = b1

n, n = 1, ..., N1.

Remark. In the definition of the class K we consider Γ1 as a set of cuts. In particular, by

C0(D\Γ1) we denote a class of functions, which are continuously extended on the cuts Γ1

from the left and right and are continuous at the tips of cuts Γ1. However values of these

functions on Γ1 from the left and right can be different everywhere except at the tips, so

that the functions may have a jump on Γ1.

Let us formulate the mixed Dirichlet-Neumann problem for the Laplace equation in the

domain D\Γ1.

Problem U. Find a function u(x) of class K so that u(x) satisfies the Laplace equation

ux1x1(x) + ux2x2(x) = 0, x ∈ D\Γ1,

the boundary conditions

(2a)∂u(x)∂nx

∣∣∣∣x(s)∈(Γ1)+

= F+(s),∂u(x)∂nx

∣∣∣∣x(s)∈(Γ1)−

= F−(s),

u(x(s))|Γ2 = F (s),

358


and the following conditions as |x| =√

x21 + x2

2 →∞

(2b) |u(x)| ≤ const, |∇u| = o(|x|−1).

All conditions of the problem U must be satisfied in the classical sense.

The edge condition (1) ensures the absence of point sources at the ends of Γ1. It is

assumed that N2 ≥ 1. If N1 = 0 and the cuts Γ1 are absent, then the problem U transforms

to the classical Dirichlet problem in an exterior domain D without cuts.

Using the energy equalities we can prove the following assertion.

Theorem 1. The problem U has at most one solution.

By∫Γk ... dσ we mean

Nk∑

n=1

bkn∫

akn

... dσ.

Proof. Consider the homogeneous problem U and assume that u0(x) is a solution of the

homogeneous problem (with F±(s) ≡ 0, F (s) ≡ 0). Our aims is to show that u0(x) ≡ 0.

According to [3], smoothness of a solution on the part of the boundary with the Dirichlet

condition is the least value among smoothness of the boundary data and smoothness of

the boundary. Therefore, u0(x) ∈ C1,λ(D \ Γ1). Combining this result with the smoothness

ensured for u0(x) by the class K, we have ∇u0(x) ∈ C0(D \ Γ1 \ X), and inequality (1)

holds at the tips of Γ1. We envelope each cut Γ1n (n = 1, ..., N1) by a closed contour so

that all contours lie in D\Γ1. Next we write the energy equalities for a domain, bounded by

our auxiliary contours, Γ2 and the circle of a large enough radius r. We allow the auxiliary

contours shrink to Γ1 and let r tend to infinity. Using the conditions at infinity (2b) and

the smoothness of u0(x) established above, we obtain

‖∇u0‖2L2(D\Γ1) =

∫

Γ1

[u+

0

(∂u0

∂nx

)+

− u−0

(∂u0

∂nx

)−]ds−

∫

Γ2

u0∂u0

∂nxds.

359

8 P.A.Krutitskii

Taking into account the homogeneous boundary conditions (2a), we have

‖∇u0‖2L2(D\Γ1) = 0.

Hence u0(x) ≡ const and const = 0 due to homogeneous Dirichlet boundary condition on Γ2.

Therefore u0(x) ≡ 0, and the theorem is proved thanks to the linearity of the problem U.

3. Integral equations at the boundary.

Below we assume that

(3) F+(s), F−(s) ∈ C0,λ(Γ1), F (s) ∈ C0(Γ2), λ ∈ (0, 1].

Note that the Holder exponent λ in the description of smoothness of these functions and in

the description of smoothness of the boundary Γ is the same. If the exponents are different

in practice, then by λ we denote the least.

If B1(Γ1), B2(Γ2) are Banach spaces of functions given on Γ1 and Γ2, then for functions

given on Γ we introduce the Banach space B1(Γ1) ∩ B2(Γ2) with the norm

‖·‖B1(Γ1)∩B2(Γ2) = ‖·‖B1(Γ1) + ‖·‖B2(Γ2) .

An example of such a Banach space is C0(Γ) = C0(Γ1) ∩ C0(Γ2).

We shall construct the solution of the problem U from the smoothness class K with the

help of potential theory for harmonic functions.

We consider an angular potential [1,5], [12, Appendix] for the Laplace equation:

(4) w1[µ](x) = − 12π

∫

Γ1

µ(σ)V (x, y(σ))dσ.

The kernel V (x, y(σ)) is defined (up to indeterminacy 2πm, m = ±1, ±2, ...) by the formulae

cosV (x, y(σ)) =x1 − y1(σ)|x− y(σ)| , sinV (x, y(σ)) =

x2 − y2(σ)|x− y(σ)| ,

360


where

y = y(σ) = (y1(σ), y2(σ)) ∈ Γ1, |x− y(σ)| =√

(x1 − y1(σ))2 + (x2 − y2(σ))2.

One can see, that V (x, σ) is the angle between the vector−−−→y(σ)x and the direction of the Ox1

axis. More precisely, V (x, y(σ)) is a many-valued harmonic function conjugate to ln |x−y(σ)|.

Below by V (x, y(σ)) we denote an arbitrary fixed branch of this function, which varies

continuously with σ along each curve Γ1n (n = 1, ..., N1) for given fixed x /∈ Γ1.

Under this definition of V (x, y(σ)), the potential w1[µ](x) is a many-valued function. In

order that the potential w1[µ](x) be single-valued, it is necessary to impose the following

additional conditions

(5)

b1n∫

a1n

µ(σ) dσ = 0, n = 1, ..., N1.

Below we suppose that the density µ(σ) belongs to the Banach space Cωq (Γ1), ω ∈ (0, 1],

q ∈ [0, 1) and satisfies conditions (5).

We say, that µ(s) ∈ Cωq (Γ1) if

µ(s)N1∏

n=1

∣∣∣s− a1n

∣∣∣q ∣∣∣s− b1

n

∣∣∣q ∈ C0,ω(Γ1),

where C0,ω(Γ1) is a Holder space with exponent ω and

‖µ(s)‖Cωq (Γ1) =

∥∥∥∥∥∥µ(s)

N1∏

n=1

∣∣∣s− a1n

∣∣∣q ∣∣∣s− b1

n

∣∣∣q

∥∥∥∥∥∥C0,ω(Γ1)

.

As shown in [1], [5], [12, Appendix], for such µ(σ) the angular potential w1[µ](x) belongs

to the class K. In particular, the inequality (1) holds with ε = −q, if q ∈ (0, 1). Moreover,

integrating w1[µ](x) by parts and using (5), we express the angular potential in terms of a

double layer potential

w1[µ](x) =12π

∫

Γ1

ρ(σ)∂

∂nyln |x− y(σ)| dσ,

361

10 P.A.Krutitskii

with the density

ρ(σ) =σ∫

a1n

µ(ξ) dξ, σ ∈ [a1n, b1

n], n = 1, ..., N1.

Consequently, w1[µ](x) satisfies the Laplace equation outside Γ1 and the conditions at in-

finity (2b).

Let us construct a solution of the problem U . We seek a solution of the problem in the

following form

(6) u[ν, µ](x) = v1[ν](x) + w[µ](x) + h[ν, µ](x) ,

where

(7a) w[µ](x) = w1[µ](x) + w2[µ](x) ,

v1[ν](x) = − 12π

∫

Γ1

ν(σ) ln |x− y(σ)| dσ,

w2[µ](x) = − 12π

∫

Γ2

µ(σ)∂

∂nyln |x− y(σ)| dσ,

and w1[µ](x) is given by (4). By h[ν, µ](x) we denote the sum of point sources placed at the

fixed points Yk lying inside Γ2k (k = 1, ..., N2) and a constant:

h[ν, µ](x) = − 12π

N2∑

k=2

∫

Γ2k

µ(σ)dσ ln |x− Yk|+

+12π

∫

Γ2

µ(σ)dσ +∫

Γ1

ν(σ)dσ −∫

Γ21

µ(σ)dσ

ln |x− Y1|+

∫

Γ2

µ(σ)dσ =

=12π

∫

Γ1

ν(σ)dσ ln |x− Y1|+ h2[µ](x).

Here

(7b) h2[µ](x) = − 12π

N2∑

k=2

∫

Γ2k

µ(σ)dσ ln |x− Yk|+

362


+12π

∫

Γ2

µ(σ)dσ −∫

Γ21

µ(σ)dσ

ln |x− Y1|+

∫

Γ2

µ(σ)dσ ; Yk ∈ Dk , k = 1, ..., N2 .

Clearly, h[ν, µ](x) obeys the Laplace equation in R2\N2⋃

k=1

Yk and belongs to

C∞R2\

N2⋃

k=1

Yk

.

Besides, if x(s) ∈ Γ, then h[ν, µ](x(s)) ∈ C1,λ(Γ) in s. We need the function h[ν, µ](x) to

construct a uniquely solvable integral equation. Moreover, h[ν, µ](x) is taken in such a way

that u[ν, µ](x) in (6) satisfies conditions (2b) at infinity.

We will look for the density ν(σ) in the space C0,λ(Γ1).

We will seek µ(s) in the Banach space Cωq (Γ1) ∩ C0(Γ2), ω ∈ (0, 1], q ∈ [0, 1) with the

norm ‖·‖Cωq (Γ1)∩C0(Γ2) = ‖·‖Cω

q (Γ1) + ‖·‖C0(Γ2) . Besides µ(s) must satisfy conditions (5).

It follows from [1,5,17], [12, Appendix] that for such µ(s), ν(s) the function (6) belongs to

the class K and satisfies all conditions of the problem U except the boundary conditions (2a).

To satisfy the boundary conditions, we insert (6) in (2a), use limit formulas for the

normal derivative of the angular potential [1,5], [12, Appendix] and arrive at the system of

integral equations for the densities µ(s), ν(s)

(8a) ±12ν(s) +

12π

∫

Γ1

ν(σ)cosϕ0(x(s), y(σ))|x(s)− y(σ)| dσ−

− 12π

∫

Γ1

µ(σ)sinϕ0(x(s), y(σ))|x(s)− y(σ)| dσ−

− 12π

∫

Γ2

µ(σ)∂

∂nx

∂

∂nyln |x(s)− y(σ)| dσ +

∂

∂nxh[ν, µ](x(s)) = F±(s), s ∈ Γ1,

(8b) − 12π

∫

Γ1

µ(σ)V (x(s), y(σ)) dσ − 12π

∫

Γ1

ν(σ) ln |x(s)− y(σ)| dσ+

363

12 P.A.Krutitskii

+12µ(s)− 1

2π

∫

Γ2

µ(σ)∂

∂nyln |x(s)− y(σ)| dσ + h[ν, µ](x(s)) = F (s), s ∈ Γ2.

By ϕ0(x, y) we denote the angle between the vector −→xy and the direction of the normal nx.

The angle ϕ0(x, y) is taken to be positive if it is measured anticlockwise from nx and negative

if it is measured clockwise from nx. Besides, ϕ0(x, y) is continuous in x, y ∈ Γ if x 6= y. Note,

that for x(s), y(σ) ∈ Γ and x 6= y we have the relationships

∂

∂nxln |x(s)− y(σ)| = ∂

∂τxV (x(s), y(σ)) =

∂

∂sV (x(s), y(σ)) =

= −cosϕ0 (x(s), y(σ))|x(s)− y(σ)| = −sin (V (x(s), y(σ))− α(s))

|x(s)− y(σ)| ,

∂

∂nxV (x(s), y(σ)) = − ∂

∂τxln |x(s)− y(σ)| = − ∂

∂sln |x(s)− y(σ)| =

=sinϕ0 (x(s), y(σ))|x(s)− y(σ)| = −cos (V (x(s), y(σ))− α(s))

|x(s)− y(σ)| ,

where α(s) is the inclination of the tangent τx to the Ox1 axis, and V (x, y(σ)) is the kernel

of the angular potential (4).

The 2nd integral term in (8a) is a Cauchy singular integral. The kernel of the third

integral term in (8b) has a weak singularity as s = σ.

Equation (8a) is obtained as x → x(s) ∈ (Γ1)± and comprises two integral equations.

The upper sign denotes the integral equation on (Γ1)+, the lower sign denotes the integral

equation on (Γ1)−.

In addition to the integral equations written above we have conditions (5).

Subtracting the integral equations (8a), we find

(9) ν(s) =(F+(s)− F−(s)

) ∈ C0,λ(Γ1).

We note that ν(s) is found completely and satisfies all required conditions. Hence, the

364


potential v1[ν](x) is found completely as well. Additionally,

h[ν, µ](x) =12π

∫

Γ1

(F+(σ)− F−(σ))dσ ln |x− Y1|+ h2[µ](x),

where h2[µ](x) is given by (7b).

We introduce the function f(s) on Γ by the formulas

(10a) f(s) =12

(F+(s) + F−(s)

)−

− 12π

∫

Γ1

(F+(σ)− F−(σ)

) cosϕ0 (x(s), y(σ))|x(s)− y(σ)| dσ−

− 12π

∫

Γ1

(F+(σ)− F−(σ))dσ∂

∂nxln |x(s)− Y1|, s ∈ Γ1,

and

(10b) f(s) = F (s)+

+12π

∫

Γ1

(F+(σ)− F−(σ)

)ln |x(s)− y(σ)|dσ−

− 12π

∫

Γ1

(F+(σ)− F−(σ))dσ ln |x(s)− Y1|, s ∈ Γ2,

where F±(s) and F (s) are specified in (2a) and satisfy conditions (3). As shown in [6], if

s ∈ Γ1, then f(s) ∈ C0,λ(Γ1). Consequently,

(10c) f(s) ∈ C0,λ(Γ1) ∩ C0(Γ2).

Adding the integral equations (8a) we obtain the integral equation for µ(s) on Γ1

(11a) − 12π

∫

Γ1

µ(σ)sinϕ0(x(s), y(σ))|x(s)− y(σ)| dσ−

− 12π

∫

Γ2

µ(σ)∂

∂nx

∂

∂nyln |x− y(σ)| dσ +

∂

∂nxh2[µ](x(s)) = f(s), s ∈ Γ1,

where f(s) is given by (10a).

365

14 P.A.Krutitskii

Equation (8b) on Γ2 takes the form

(11b) − 12π

∫

Γ1

µ(σ)V (x(s), y(σ))dσ+

+12µ(s)− 1

2π

∫

Γ2

µ(σ)∂

∂nyln |x(s)− y(σ)| dσ + h2[µ](x(s)) = f(s), s ∈ Γ2,

where f(s) is given in (10b).

Thus, if µ(s) is a solution of equations (11), (5) from the space Cωq (Γ1) ∩ C0(Γ2) with

ω ∈ (0, 1], q ∈ [0, 1), then the potential (6) with ν(s) from (9) satisfies all conditions of the

problem U and belongs to the class K.

The following theorem holds.

Theorem 2. Let Γ1 ∈ C2,λ, Γ2 ∈ C1,λ and conditions (3) hold. If equations (11), (5) have

a solution µ(s) from the Banach space Cωq (Γ1) ∩ C0(Γ2) for some ω ∈ (0, 1] and q ∈ [0, 1),

then the solution of the problem U exists, belongs to the class K and is given by (6), where

ν(s) is defined in (9).

If s ∈ Γ2, then (11b) is an equation of the second kind with a weak singularity in the

kernel. If s ∈ Γ1, then (11a) is a Cauchy singular integral equation of the first kind [16].

Our further treatment will be aimed to the proof of the solvability of the system (11),

(5) in the Banach space Cωq (Γ1) ∩ C0(Γ2). Moreover, we reduce the system (11), (5) to a

Fredholm equation of the second kind and index zero, which can be easily computed by

classical methods.

Equation (11b) on Γ2 can be rewritten in the form

(12) µ(s) +∫

Γ

µ(σ)A2(s, σ)dσ = 2f(s), s ∈ Γ2,

where

A2(s, σ) =− 1

π

(1− δ(Γ2, σ)

)V (x(s), y(σ))−

366


− 1π

δ(Γ2, σ)∂

∂nyln |x(s)− y(σ)| − 1

π

N2∑

k=2

δ(Γ2k, σ) ln |x(s)− Yk|+

+1π

(δ(Γ2, σ)− δ(Γ2

1, σ))

ln |x(s)− Y1|+ 2δ(Γ2, σ)

.

By δ(γ, σ) we denote the characteristic function of the set γ:

δ(γ, σ) =

0, if σ /∈ γ

1, if σ ∈ γ

The kernel A2(s, σ) has a weak singularity if s = σ ∈ Γ2, and A2(s, σ) is continuous if

s 6= σ (s ∈ Γ2, σ ∈ Γ).

Remark. Evidently, f(a2n) = f(b2

n) and A2(a2n, σ) = A2(b2

n, σ) for σ ∈ Γ, σ 6= a2n, b2

n

(n = 1, ..., N2). Hence, if µ(s) is a solution of equation (12) from C0

N2⋃

n=1

[a2n, b2

n]

,

then, according to the equality (12), µ(s) automatically satisfies the matching conditions

µ(a2n) = µ(b2

n) for n = 1, ..., N2 and, therefore, belongs to C0(Γ2). This observation can

be helpful in finding numerical solutions, since we may discard the matching conditions

µ(a2n) = µ(b2

n), (n = 1, ..., N2), which are automatically fulfilled.

It can be easily proved that

− ∂

∂sln|x(s)− y(σ)||s− σ| =

sinϕ0(x(s), y(σ))|x(s)− y(σ)| − 1

σ − s∈ C0,λ(Γ1 × Γ1)

(see [5], [6] for details). Therefore we can rewrite (11a) in the form

(13)1π

∫

Γ1

µ(σ)dσ

σ − s+

∫

Γ

µ(σ)M(s, σ)dσ =

= −2f(s), s ∈ Γ1,

where

M(s, σ) =1π

(1− δ(Γ2, σ)

) (sinϕ0(x(s), y(σ))|x(s)− y(σ)| − 1

σ − s

)+

367

16 P.A.Krutitskii

+ δ(Γ2, σ)∂

∂nx

∂

∂nyln |x(s)− y(σ)|+

N2∑

k=2

δ(Γ2k, σ) ln |x(s)− Yk| −

−(1− δ(Γ21, σ)) ln |x(s)− Y1|

]

and M(s, σ) ∈ C0,λ(Γ1 × Γ).

4. The Fredholm integral equation and the solution of the problem.

Inverting the singular integral operator in (13), we arrive at the following integral equa-

tion of the second kind [5,6]:

(14) µ(s) +1

Q1(s)

∫

Γ

µ(σ)A0(s, σ)dσ +1

Q1(s)

N1−1∑

n=0

Gnsn =

=1

Q1(s)Φ0(s), s ∈ Γ1,

where

A0(s, σ) = − 1π

∫

Γ1

M(ξ, σ)ξ − s

Q1(ξ)dξ,

Q1(s) =N1∏

n=1

∣∣∣∣√

s− a1n

√b1n − s

∣∣∣∣ sign(s− a1n) ,

Φ0(s) =1π

∫

Γ1

2Q1(σ)f(σ)σ − s

dσ,

and G0, ..., GN1−1 are arbitrary constants.

To derive equations for G0, ..., GN1−1, we substitute µ(s) from (14) in the conditions (5),

then we obtain

(15)∫

Γ

µ(σ)ln(σ)dσ +N1−1∑

m=0

BnmGm = Hn, n = 1, ..., N1 ,

where

ln(σ) = −∫

Γ1n

Q−11 (s)A0(s, σ)ds,

368


(16) Bnm = −∫

Γ1n

Q−11 (s)smds,

Hn = −∫

Γ1n

Q−11 (s)Φ0(s)ds.

By B we denote the N1 × N1 matrix with the elements Bnm from (16). As shown in [6,

lemma 7], [11] the matrix B is invertible. The elements of the inverse matrix will be called

(B−1)nm. Inverting the matrix B in (15), we express the constants G0, ..., GN1−1 in terms of

µ(s) as

Gn =N1∑

m=1

(B−1)nm

Hm −

∫

Γ

µ(σ)lm(σ)dσ

.

We substitute Gn in (14) and obtain the following integral equation for µ(s) on Γ1

(17) µ(s) +1

Q1(s)

∫

Γ

µ(σ)A1(s, σ)dσ =1

Q1(s)Φ1(s), s ∈ Γ1,

where

A1(s, σ) = A0(s, σ)−N1−1∑

n=0

snN1∑

m=1

(B−1)nmlm(σ),

Φ1(s) = Φ0(s)−N1−1∑

n=0

snN1∑

m=1

(B−1)nmHm .

It can be verified directly that any solution of (17) in the required space satisfies conditions

(5) automatically. It can be shown using the properties of singular integrals [2], [16], that

Φ0(s), A0(s, σ) are Holder continuous functions if s ∈ Γ1, σ ∈ Γ. Therefore, Φ1(s), A1(s, σ)

are also Holder continuous functions if s ∈ Γ1, σ ∈ Γ. Consequently, any solution of (17)

belongs to Cω1/2(Γ

1), and below we look for µ(s) on Γ1 in this space.

We put

Q(s) =(1− δ(Γ2, s)

)Q1(s) + δ(Γ2, s), s ∈ Γ.

369

18 P.A.Krutitskii

Instead of µ(s) ∈ Cω1/2(Γ

1) ∩ C0(Γ2) we introduce the new unknown function

µ∗(s) = µ(s)Q(s) ∈ C0,ω(Γ1) ∩ C0(Γ2) and rewrite (12), (17) in the form of one equation

(18) µ∗(s) +∫

Γ

µ∗(σ)Q−1(σ)A(s, σ)dσ = Φ(s), s ∈ Γ,

where

A(s, σ) =(1− δ(Γ2, s)

)A1(s, σ) + δ(Γ2, s)A2(s, σ),

Φ(s) =(1− δ(Γ2, s)

)Φ1(s) + 2δ(Γ2, s)f(s).

Thus, the system of equations (5), (11) for µ(s) has been reduced to the equation (18)

for the function µ∗(s). It is clear from our consideration that any solution of (18) gives a

solution of system (5), (11) and conversely.

As noted above, Φ1(s) and A1(s, σ) are Holder continuous functions if s ∈ Γ1, σ ∈ Γ.

More precisely (see [6]), A1(s, σ) belongs to C0,p(Γ1) in s uniformly with respect to σ ∈ Γ,

where p = min1/2, λ. Besides, taking into account (10c) we have Φ1(s) ∈ C0,p(Γ1) .

Consequently, from equation (18) we can conclude the following assertion.

Lemma. Let Γ1 ∈ C2,λ, Γ2 ∈ C1,λ, λ ∈ (0, 1], and Φ(s) ∈ C0,p(Γ1) ∩ C0(Γ2), where

p = minλ, 1/2. If µ∗(s) from C0(Γ) satisfies the equation (18), then µ∗(s) belongs to

C0,p(Γ1) ∩ C0(Γ2).

The condition Φ(s) ∈ C0,p(Γ1) ∩ C0(Γ2) holds if conditions (3) hold.

Hence below we will seek µ∗(s) from C0(Γ).

Consider equation (18). The integral operator

∫

Γ

µ∗(σ)Q−1(σ)A2(s, σ)dσ =∫

Γ1

µ∗(σ)Q−11 (σ)A2(s, σ)dσ +

∫

Γ2

µ∗(σ)A2(s, σ)dσ

is compact from C0(Γ) into C0(Γ2). Indeed, using Arzela theorem one can verify that the

1st term is a compact operator from C0(Γ1) into C0(Γ2), because A2(s, σ) ∈ C0(Γ2 × Γ1).

370


The 2nd term is a compact operator from C0(Γ2) into C0(Γ2), because A2(s, σ) is a polar

kernel [17], i.e. it has a weak singularity as s = σ ∈ Γ2 and it is continuous if s 6= σ

(s, σ ∈ Γ2). Furthermore, using Arzela theorem one can show that the integral operator∫

Γµ∗(σ)Q−1(σ)A1(s, σ)dσ is compact from C0(Γ) into C0(Γ1) since A1(s, σ) ∈ C0(Γ1 × Γ).

Therefore the integral operator from (18):

Aµ∗(s) =∫

Γ

µ∗(σ)Q−1(σ)A(s, σ)dσ

is a compact operator mapping C0(Γ) into itself. Therefore, (18) is a Fredholm equation of

the second kind and index zero in the Banach space C0(Γ).

Let us show that if µ0∗(s) is a solution of the homogeneous equation (18) from C0(Γ), then

it is the trivial solution, i.e. µ0∗(s) ≡ 0. Let µ0∗(s) ∈ C0(Γ) be a solution of the homogeneous

equation (18). According to the Lemma, µ0∗(s) ∈ C0,p(Γ1) ∩ C0(Γ2), p = minλ, 1/2.

Therefore the function µ0(s) = µ0∗(s)Q−1(s) ∈ Cp1/2(Γ

1)∩C0(Γ2) converts the homogeneous

equations (12), (17) into identities. Using the homogeneous identity (17), we check, that

µ0(s) satisfies conditions (5). Besides, acting on the homogeneous identity (17) with a singu-

lar operator with the kernel (s− t)−1, we find that µ0(s) satisfies the homogeneous equation

(13). Consequently, µ0(s) satisfies the homogeneous equations (11). On the basis of Theo-

rem 2, the function u[0, µ0](x) = w[µ0](x) + h2[µ0](x) given by (6), (7) is a solution of the

homogeneous problem U. According to Theorem 1,(

w[µ0](x) + h2[µ0](x))≡ 0, x ∈ D\Γ1.

Using the limit formulas for tangential derivatives of an angular potential [1,5], [12, Ap-

pendix], we obtain

limx→x(s)∈(Γ1)+

∂

∂τx

(w[µ0](x) + h2[µ0](x)

)−

− limx→x(s)∈(Γ1)−

∂

∂τx

(w[µ0](x) + h2[µ0](x)

)= µ0(s) ≡ 0, s ∈ Γ1.

371

20 P.A.Krutitskii

Hence,(

w[µ0](x)+h2[µ0](x))

=(

w2[µ0](x)+h2[µ0](x))≡ 0, x ∈ D, and µ0(s) satisfies

(11b), which takes the form

(19)12µ0(s)− 1

2π

∫

Γ2

µ0(σ)∂

∂nyln |x(s)− y(σ)| dσ+

+h2[µ0](x(s)) = 0, s ∈ Γ2,

where h2[µ](x) is specified in (7b). The Fredholm equation (19) arises when solving the

homogeneous Dirichlet problem for harmonic functions in the exterior domain D by the

double layer potential with the sum of point sources placed inside the curves Γ21, ..., Γ

2N2

.

The equation (19) has only the trivial solution µ0(s) ≡ 0 in C0(Γ2). This is shown in the

appendix.

Consequently, if s ∈ Γ, then µ0(s) ≡ 0, µ0∗(s) = µ0(s)Q−1(s) ≡ 0. Thus, the homoge-

neous Fredholm equation (18) has only the trivial solution in C0(Γ).

We have proved the following assertion.

Theorem 3. If Γ1 ∈ C2,λ, Γ2 ∈ C1,λ, λ ∈ (0, 1], then (18) is a Fredholm equation of

the second kind and index zero in the space C0(Γ). Moreover, equation (18) has a unique

solution µ∗(s) ∈ C0(Γ) for any Φ(s) ∈ C0(Γ).

As a consequence of Theorem 3 and the Lemma we obtain

Corollary. If Γ1 ∈ C2,λ, Γ2 ∈ C1,λ, λ ∈ (0, 1], then equation (18) has a unique solution

µ∗(s) ∈ C0,p(Γ1) ∩ C0(Γ2), for any Φ(s) ∈ C0,p(Γ1) ∩ C0(Γ2), where p = minλ, 1/2.

We recall that Φ(s) belongs to the class of smoothness required in the Corollary if

f(s) ∈ C0,λ(Γ1) ∩ C0(Γ2). As mentioned above, if µ∗(s) ∈ C0,p(Γ1) ∩ C0(Γ2) is a solution

of (18), then µ(s) = µ∗(s)Q−1(s) ∈ Cp1/2(Γ

1) ∩ C0(Γ2) is a solution of system (5), (11). We

obtain the following statement.

372


Theorem 4. If Γ1 ∈ C2,λ, Γ2 ∈ C1,λ, λ ∈ (0, 1], then the system of

equations (5), (11) has a solution µ(s) ∈ Cp1/2(Γ

1) ∩ C0(Γ2), p = min1/2, λ, for

any f(s) ∈ C0,λ(Γ1) ∩ C0(Γ2). Moreover, this solution is expressed by the formula

µ(s) = µ∗(s)Q−1(s), where µ∗(s) ∈ C0,p(Γ1) ∩ C0(Γ2) is the unique solution of the Fredholm

equation (18) in C0(Γ).

Remark. The solution of the system (5), (11) ensured by Theorem 4, is unique in the space

Cpo

1/2(Γ1) ∩ C0(Γ2) for any po ∈ (0, p]. More precisely, the system (5), (11) has at most one

solution in the space Cωq (Γ1) ∩ C0(Γ2) for any ω ∈ (0, 1] and q ∈ [0, 1). The proof of this

fact almost coincides with the proof of Theorem 3.

According to (10), f(s) belongs to C0,λ(Γ1)∩C0(Γ2) if (3) holds. Therefore, the condition

f(s) ∈ C0,λ(Γ1)∩C0(Γ2) in Theorem 4 can be replaced by the condition that (3) holds. On

the basis of Theorem 2 and Theorem 4 we arrive at the final result.

Theorem 5. If Γ1 ∈ C2,λ, Γ2 ∈ C1,λ and condition (3) holds, then the solution of the

problem U exists, belongs to the class K and is given by (6), where ν(s) is defined in (9)

and µ(s) is a solution of system (5), (11) from Cp1/2(Γ

1)∩C0(Γ2), p = min1/2, λ, ensured

by Theorem 4.

It can be checked directly that the solution of the problem U constructed in Theorem 5

satisfies condition (1) with ε = −1/2. Explicit expressions for the singularities of the solution

gradient at the end-points of the open curves will be presented in the next section.

Theorem 5 ensures the existence of a classical solution of the problem U when

Γ1 ∈ C2,λ, Γ2 ∈ C1,λ, and condition (3) holds. The uniqueness of the classical solution fol-

lows from Theorem 1. On the basis of our consideration we suggest the following scheme for

solving the problem U. At first, we find the unique solution µ∗(s) of the Fredholm equation

373

22 P.A.Krutitskii

(18) from C0(Γ). This solution automatically belongs to C0,p(Γ1)∩C0(Γ2), p = minλ, 1/2.

Secondly, we construct the solution of equations (5), (11) from Cp1/2(Γ

1) ∩ C0(Γ2) by the

formula µ(s) = µ∗(s)Q−1(s). Finally, substituting ν(s) from (9) and µ(s) in (6), we obtain

the solution of the problem U.

5. The behaviour of the gradient of the solution at the tips of the cuts Γ1.

In the present section by u(x) = u[ν, µ](x) we denote the solution of problem U con-

structed in the previous section. The integral representation for u(x) obtained in Theorem 5

enables us to derive explicit formulas for the singularities of ∇u at the tips of the cuts Γ1. It

follows from the definition of the class K that the gradient of the solution of the problem U

might be unbounded at the end-points of Γ1, where the estimate (1) holds with ε = −1/2.

Our aim now is to investigate in detail the behaviour of ∇u(x) at the end-points of Γ1. Let

x(d) be one of these points (d = a1n or d = b1

n, where n = 1, ..., N1). In the neighbourhoods

of x(d) we introduce the polar system of coordinates

x1 = x1(d) + |x− x(d)| cosϕ, x2 = x2(d) + |x− x(d)| sinϕ.

We will assume that ϕ ∈ (α(d), α(d)+2π), if d = a1n, and ϕ ∈ (α(d)−π, α(d)+π), if d = b1

n.

Recall that α(s) is the angle between the Ox1 axis and the tangent vector τx drawn at the

point x(s) ∈ Γ. Hence, α(d) = α(a1n + 0), if d = a1

n, and α(d) = α(b1n − 0) if d = b1

n. Thus,

the angle ϕ varies continuously in a neighbourhood of x(d) cut along the contour Γ1.

Let µ1(s) = µ(s)|s− d|1/2 = Q−1(s)µ∗(s)|s− d|1/2 and put µ1(d) = µ1(a1n) = µ1(a1

n +0),

if d = a1n, µ1(d) = µ1(b1

n) = µ1(b1n − 0) if d = b1

n.

Recall that X is the set of end-points of Γ1. The following theorem is easily proved

using the results obtained in [5] and using the properties of Cauchy type integrals near the

374


end-points of the integration line given in [16, section 22], [2, section 8].

Theorem 7. Let x → x(d) ∈ X. Then in the neighbourhood of the point x(d), the

derivatives of the solution of the problem U satisfy the relations

∂

∂x1u(x) = −(−1)m µ1(d)

2|x− x(d)|1/2sin γ−

−(−1)m ν(d)2π

[ln |x− x(d)| cosα(d) + ϕ sinα(d)] + O(1),

∂

∂x2u(x) = (−1)m µ1(d)

2|x− x(d)|1/2cos γ+

+(−1)m ν(d)2π

[− ln |x− x(d)| sinα(d) + ϕ cosα(d)] + O(1),

where m = 0, γ = [ϕ + α(d) − π]/2 if d = a1n, and m = 1, γ = [ϕ + α(d)]/2 if d = b1

n.

Besides, O(1) denotes functions which are continuous at the point x(d). Furthermore, the

functions denoted by O(1) are continuous in the neighbourhood of the point x(d), cut along

the contour Γ1.

This theorem establishes the following curious fact. In the general case, the derivatives

of the solution of the problem U near the end-point x(d) of the contour Γ1 behave as

O(|x− x(d)|−1/2) + O(ln |x− x(d)|−1). However, if µ1(d) = ν(d) = 0, then ∇u(x) will be

bounded and even continuous at the end-point x(d) of Γ1. This effect of disappearence of

singularities happens for certain functions F±(s), F (s) given in the boundary condition

(2a), since the condition µ1(d) = ν(d) = 0 specifies restrictions on these functions.

Appendix.

Here we prove the following assertion.

Proposition A. If Γ2 ∈ C1,λ, λ ∈ (0, 1], then there exists only the trivial solution of the

homogeneous Fredholm equation (19) in C0(Γ2).

375

24 P.A.Krutitskii

Proof. Let µ0(s) ∈ C0(Γ2) be a non-trivial solution of the homogeneous equation (19).

The kernel of the integral term in (19) has a weak singularity. It can be shown with the

help of [16, Sec.51], that the integral term in (19) belongs to C0,λ/4(Γ2) in s, therefore

µ0(s) ∈ C0,λ/4(Γ2). Now we consider the function

(A1) g[µ0](x) = w2[µ0](x) + h2[µ0](x),

where w2[µ0](x) and h2[µ0](x) were introduced in (7a), (7b). The function g[µ0](x) belongs

to C0(D)∩C2(D) and satisfies the following homogeneous Dirichlet problem for the Laplace

equation

∆g = 0 in D, g|Γ2 = 0, |g| ≤ const in D.

Indeed, substituting g[µ0](x) in the boundary condition, we get the identity (19). According

to the uniqueness theorem for the Dirichlet problem, we have

(A2) g[µ0](x) ≡ 0, x ∈ D.

Therefore, letting |x| → ∞ in the expression for g[µ0](x), we obtain

(A3)∫

Γ2

µ0(σ)dσ = 0.

We consider the function

(A4) g∗[µ0](x) = − 12π

−

∫

Γ2

µ0(σ)∂

∂σln |x− y(σ)|dσ+

+N2∑

k=2

∫

Γ2k

µ0(σ)dσ V (x, Yk)−

−

∫

Γ2

µ0(σ)dσ −∫

Γ21

µ0(σ)dσ

V (x, Y1)

,

376


where V (x, y) is the kernel of the angular potential from (4). The function

g∗[µ0](x) is harmonic conjugate to g[µ0](x), i.e. the Cauchy-Riemann relations

∂x1g = ∂x2g∗, ∂x2g = −∂x1g

∗ hold. Consequently, g∗[µ0](x) ≡ Const in D. It is clear from

(A4), that g∗[µ0](x) is a many-valued function, because V (x, Yk) are many-valued functions

(k = 1, ..., N2). Indeed, when passing around the point Yk the value of the function V (x, Yk)

changes for 2π. Evidently, g∗[µ0](x) can be constant in D only if g∗[µ0](x) is single-valued.

In order for g∗[µ0](x) be single-valued, the following N2 conditions must hold

∫

Γ2k

µ0(σ)dσ = 0, k = 2, ..., N2 ,

∫

Γ2

µ0(σ)dσ −∫

Γ21

µ0(σ)dσ = 0.

Along with (A3) we obtain

(A5)∫

Γ2k

µ0(σ)dσ = 0, k = 1, ..., N2.

Under these conditions, g∗[µ0](x) takes the form of the modified single-layer potential

[16]

(A6) g∗[µ0](x) ≡ w∗2[µ0](x) =

12π

∫

Γ2

µ0(σ)∂

∂σln |x− y(σ)|dσ,

and g[µ0](x) transforms to the ordinary double-layer potential

(A7) g[µ0](x) ≡ w2[µ0](x) = − 12π

∫

Γ2

µ0(σ)∂

∂nyln |x− y(σ)|dσ ∈

∈ C0(R2\Γ2) ∩ C2(R2\Γ2).

The potentials (A6) and (A7) are connected by the Cauchy-Riemann relations in R2\Γ2.

Because of µ0(s) ∈ C0,λ/4(Γ2), the potential (A6) is a harmonic function, which belongs

377

26 P.A.Krutitskii

to C0(R2) ∩ C2(R2\Γ2) (see [16] for details). Note, that (A6) is continuous when passing

through Γ2 and is represented on Γ2 by a singular integral (for this we have stressed that

µ0(s) is a Holder continuous function).

As stated above, w∗2[µ0](x) in D is equal to a constant, which is equal to zero due to the

behaviour of this potential at infinity, so that w∗2[µ0](x) ≡ 0 in D.

We consider the internal domain Dk bounded by Γ2k (k = 1, ..., N2). In this domain the

potential (A6) satisfies the Dirichlet problem

∆w∗2 = 0 in Dk, w∗2|Γ2k

= 0,

which has the unique solution

w∗2[µ0](x) ≡ 0, x ∈ Dk, (k = 1, ...N2).

It follows from the Cauchy-Riemann relations and the smoothness of the double-layer

potential that

w2[µ0](x) ≡ ck, x ∈ Dk, k = 1, ...N2,

where c1, ..., cN2 are constants. Using (A2) and the jump relation for the double-layer po-

tential w2[µ0](x) on Γ2, we get

µ0(s)|Γ2k≡ −ck, k = 1, ...N2.

According to (A5), ck = 0, k = 1, ..., N2, and therefore

µ0(s)|Γ2k≡ 0, k = 1, ...N2.

Consequently,

µ0(s) ≡ 0 on Γ2.

378


Hence, the homogeneous equation (19) has only the trivial solution. The proof is completed.

Because of (19) is a Fredholm equation of the second kind, the following Corollary holds.

Corollary A. If Γ2 ∈ C1,λ, λ ∈ (0, 1], then the inhomogeneous Fredholm equation (19) is

uniquely solvable in C0(Γ2) for any right-hand side from C0(Γ2).

The inhomogeneous equation (19) is a particular case of (11b) if the exterior domain D

does not contain cuts.

Acknowledgment.

This research was supported by the RFBR grants No. 06-01-00001, 05-01-00050.

References.

1. Gabov S.A. An angular potential and its applications. Math. U.S.S.R. Sbornik, 1977,

v.32, No.4, p.423-436.

2. Gakhov F.D. Boundary value problems. Pergamon Press, Oxford; Addison-Wesley,

Reading, Mass., 1966.

3. Gilbarg D., Trudinger N.S. Elliptic partial differential equations of second order.

Springer, Berlin, 1983.

4. Krutitskii P.A. On the electric current from electrodes in a magnetized semiconductor

film. IMA J. Appl. Math., 1998, v.60, p.285-297.

5. Krutitskii P.A. Dirichlet problem for the Helmholtz equation outside cuts in a plane.

Comp. Maths. Math. Phys., 1994, v.34, No.8/9, p.1073-1090.

6. Krutitskii P.A. The Neumann problem for the Helmholtz equation outside cuts in the

plane. Comput. Maths. Math. Phys., 1994, No.11, p.1421-1431.

379

28 P.A.Krutitskii

7. Krutitskii P.A. The 2-Dimensional Dirichlet Problem in an External Domain with

Cuts. Zeitscr. Analysis u. Anwend., 1998, v.17, No.2, p.361-378.

8. Krutitskii P.A. The Neumann problem in a 2-D exterior domain with cuts and singu-

larities at the tips. J. Differential Equations, 2001, v.176, No.1, p.269-289.

9. Krutitskii P.A. Wave propagation in a 2 − D external domain bounded by closed

and open curves. Nonlinear Analysis, Theory, Methods and Applications, 1998, v.32, No.1,

p.135-144.

10. Krutitskii P.A. Wave scattering in a 2-D exterior domain with cuts: The Neumann

problem. ZAMM, 2000, v.80, No.8, p.535-546.

11. Krutitskii P.A. An explicit solution of the pseudo-hyperbolic initial boundary value

problem in a multiply connected region. Mathematical Methods in the Applied Sciences,

1995, v.18, No.11, pp.897-925.

12. Krutitskii P.A. The oblique derivative problem for the Helmholtz equation and scat-

tering tidal waves. Proceedings of the Royal Society of London, ser. A, 2001, v.457, p.1735-

1755.

13. Medkova D. Solution of the Dirichlet problem for the Laplace equation. Applications

of Mathematics, 1999, v.44, p.143–168.

14. Lifanov I.K. Singular integral equations and discrete vortices. VSP, Zeist, 1996.

15. Gunter N.M. Potential theory and its application to the basic problems of mathemat-

ical physics. GITTL, Moscow, 1953.

16. Muskhelishvili N.I. Singular integral equations, Noordhoff, Groningen, 1972. (3-d

Russian edition: Nauka, Moscow, 1968).

17. Vladimirov V.S. Equations of Mathematical Physics, Marcel Dekker, N.Y., 1971.

380


18. Durand M. Layer potentials and boundary value problems for the Helmholtz equation

in the complement of a thin obstacle. Math. Meth. Appl. Sci., 1983, v.5, p.389-421.

19. Panasyuk, V.V., Savruk, M.P., Nazarchuk, Z.T. The method of singular integral

equations in two-dimensional diffraction problems, Naukova Dumka, Kiev, 1984. (in Rus-

sian).

20. Tuchkin Y. A., Scattering of waves by an unclosed cylindrical screen of an arbitrary

profile with Neumann boundary condition. Dokl. Akad. Nauk SSSR, 293 (1987), 343-345.

(in Russian).

381

382







383


384


385


386

387

TABLE OF CONTENTS, JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOLUME 3, NO.3, 2008 SOME APPLICATIONS OF INTUITIONISTIC FUZZY METRIC SPACES, S.KUTUKCU, S.SHARMA,……………………………………………………. 269 SOME THEOREMS IN INTUITIONISTIC FUZZY METRIC SPACES, S.KUTUKCU, C.YILDIZ,………………………………………………………..285 A GENERAL SAMPLING THEORY IN THE FUNCTIONAL HILBERT SPACE INDUCED BY A HILBERT SPACE VALUED KERNEL, A.GARCIA, A.PORTAL,299 PERTURBATIONS OF OPERATORS ON TENSOR PRODUCTS AND SPECTRUM LOCALIZATION OF MATRIX DIFFERENTIAL OPERATORS, M.GIL’,…….315 COMPOSITION FOLLOWED BY DIFFERENTIATION BETWEEN WEIGHTED BERGMAN SPACES AND BLOCH TYPE SPACES, S.LI, S.STEVIC,…………333 A NOTE ON INTEGRAL INEQUALITIES INVOLVING THE PRODUCT OF TWO FUNCTIONS ON TIME SCALES, A.TUNA, B.YASAR, S.KUTUKCU,………..341 SOME NEW INTEGRAL INEQUALITIES FOR RETARDED VOLTERRA EQUATIONS, B.YASAR, A.TUNA,………………………………………………347 THE DIRICHLET-NEUMANN PROBLEM FOR THE 2-D LAPLACE EQUATION IN AN EXTERIOR CRACKED DOMAIN WITH NEUMANN CONDITION ON CRACKS, P.KRUTITSKII,…………………………………………………………353

Volume 3,Number 4 October 2008 ISSN:1559-1948 (PRINT), 1559-1956 (ONLINE) EUDOXUS PRESS,LLC


SCOPE AND PRICES OF

















Neumann and Mixed BoundaryValue Problems

AJAY KUMARDepartment of Mathematics,

University of Delhi,Delhi-110007, India.

email [email protected]

RAVI PRAKASHDepartment of Mathematics

Rajdhani College (University of Delhi),Raja Garden, New Delhi-110015, India.

email rprakash [email protected]

Suggested running head: Boundary value problemsContact Author: AJAY KUMAR

Tel No. 00-91-11-27014415/27028581Tel Fax 00-91-11-27020937/27666658

399


Abstract

The Neumann boundary value problem is investigated for the inhomogeneouspolyanalytic equation. The mixed k-Neumann and n − k Schwarz boundaryvalue problem has also been studied.

AMS(2000): 32A30, 30G20, 35J55, 31A10

Keywords and phrases: Neumann problem, Schwarz problem, Gauss theorem,Cauchy-Pompeiu representation, inhomogeneous polyanalytic equation

A.KUMAR AND R.PRAKASH

400

Boundary value problems 3

1. Introduction

The Neumann problem is well studied for harmonic functions and solved un-der certain conditions via the Neumann function. Extending the concept ofNeumann functions for the Laplacian to Neumann functions for powers of theLaplacian, an explicit representation of the solution to the n-Neumann problemfor ∆nu = f has been given in [7]. Integral representations for solutions tohigher order differential equations can be obtained by iterating those represen-tation formulas for first order equations. This procedure has been applied in [3,4, 5, 7, 8] to get explicit representation formula for the solution of higher orderpartial differential equations.Neumann boundary conditions are given via outer normal derviations ∂ν . Forthe unit disc D this is

∂ν = z∂z + z∂z

In [2,6], the following results are proved.

Theorem 1. The Neumann problem wz = f in D, ∂νw = γ on ∂D, w(0) = cfor f ∈ Cα(D,C), 0 < α < 1, γ ∈ C(∂D,C), c ∈ C is solvable if and only if for|z| = 1

12πi

∫

|ζ|=1

γ(ζ)dζ

(1 − zζ)ζ+

12πi

∫

|ζ|=1

f(ζ)dζ1 − zζ

+z

π

∫

|ζ|<1

f(ζ)dξdη(1 − zζ)2

= 0 (1)

The solution then is given as

w(z) = c− 12πi

∫

|ζ|=1

γ(ζ) log(1 − zζ)dζ

ζ− 1

2πi

∫

|ζ|=1

f(ζ) log(1 − zζ)dζ

− z

π

∫

|ζ|<1

f(ζ)dξdηζ(ζ − z)

(2)

Theorem 2. The Schwarz problem for the inhomogeneous polyanalytic equa-tion in the unit disc

∂mz w = f in D, Re ∂s

zw = βs on ∂D, Im ∂szw(0) = bs, 0 ≤ s ≤ m− 1,

is uniquely solvable for f ∈ L1(D,C), βs ∈ C(∂D,R), bs ∈ R, 0 ≤ s ≤ m − 1.The solution is given by

w(z) = i

m−1∑

s=0

bss!

(z + z)s +m−1∑

s=0

(−1)s

2πis!

∫

|ζ|=1

βs(ζ)ζ + z

ζ − z(ζ − z + ζ − z)s dζ

ζ

+(−1)m

2π(m− 1)!

∫

|ζ|<1

(

f(ζ)ζ

ζ + z

ζ − z+f(ζ)ζ

1 + zζ

1 − zζ

)

(ζ − z + ζ − z)m−1dξdη (3)

401

A. Kumar and R. Prakash 4

In [5], the Dirichlet problem and the half-Neumann problem for the inhomo-geneous polyanalytic equation have been studied. In the following section theNeumann problem for the inhomogenous polyanalytic equation is considered.The third section deals with the mixed problem of n-Neumann and m-Schwarzboundary conditions. Our methods are based on repeated applications of Gausstheorem, Cauchy Pompeiu formula and Cauchy Pompeiu operators for the dif-ferential operators ∂m

z ∂nz [1, 3]. For boundary value problems of polyharmonic

functions, see [9, 10].

2. Neumann problem

Let D be a regular domain i.e. a bounded domain in the complex plane witha smooth boundary. Often D will be chosen to be unit disc D in order toreceive explicit formulas. The Neumann problem is improperly formulated forthe first order equation because the differential operator of the equation becomesinvolved into boundary condition. For this reason a half-Neumann problemis considered in [5]. However, in this section we consider the full Neumannproblem. The integrals in the following lemma can be computed using Cauchy-Pompeiu formula [1].

Lemma 1. For r ∈ N ∪ 0 = N0, |z| < 1 and |ζ| < 1, we have

(i) Lr(z, ζ) =1

2πi

∫

|ζ|=1

ζr log(1 − zζ) log(1 − ζ¯ζ)dζ

ζ2

− z

π

∫

|ζ|<1

ζr log(1 − ζ¯ζ)

dξdη

ζ(ζ − z)

= − 1r + 1

(¯ζr+1 − zr+1) log(1 − z

¯ζ)

(ii) Nr(ζ) =1

2πi

∫

|ζ|=1

ζr log(1 − ζ¯ζ)dζ

ζ2= −

¯ζ

r+1

r + 1

Theorem 3. The Neumann problem for the inhomogeneous polyanalyticequation in the unit disc ∂n

z w = f in D, ∂ν(∂rzw) = αr on ∂D, ∂r

zw(0) = ar

for 0 ≤ r ≤ n − 1 is uniquely solvable for f ∈ Cα(D,C), 0 < α < 1,

αr ∈ C(∂D,C), ar ∈ C, 0 ≤ r ≤ n − 1, if and only if for 1 ≤ k ≤ n − 1 and|z| = 1

402


n−1∑

r=k

arzr−k+1

(r − k)!+

12πi

∫

|ζ|=1

αk−1(ζ)ζ − z

d ζ

−n−1∑

r=k

(−1)r−k+1

(r − k + 1)!1

2πi

∫

|ζ|=1

αr(ζ)(ζ − z)r−k[ζ − (r − k + 1)z log(1 − zζ)]dζ

ζ

+(−1)n−k

(n− k)!1

2πi

∫

|ζ|=1

f(ζ)(ζ − z)n−k−1[ζ − (n− k)z log(1 − zζ)]dζ

ζ2

− (−1)n−k

(n− k)!z

π

∫

|ζ|<1

f(ζ)(ζ − z)n−k−1

[

(ζ − z)(1 − zζ)2

+n− k

ζ(1 − zζ)

]

dξdη = 0 (4)

and

12πi

∫

|ζ|=1

αn−1(ζ)dζ

ζ − z− 1

2πi

∫

|ζ|=1

f(ζ)ζdζ

ζ − z− z

π

∫

|ζ|<1

f(ζ)dξdη

(1 − zζ)2

= 0 (5)

The solution then is given by

w(z) =n−1∑

r=0

arzr

r!−

n−1∑

r=0

(−1)r

r!1

2πi

∫

|ζ|=1

αr(ζ)(ζ − z)r log(1 − zζ)dζ

ζ

+(−1)n−1

(n− 1)!1

2πi

∫

|ζ|=1

f(ζ)(ζ − z)n−1 log(1 − zζ)dζ

ζ2

− (−1)n−1

(n− 1)!z

π

∫

|ζ|<1

f(ζ)ζ

(ζ − z)n−1

ζ − zdξdη (6)

Proof: The conditions (5) coincide with (1) and (6) with (2) for n = 1. As-suming Theorem 3 is proved for n− 1 rather than n the problem is decomposedinto the system

∂n−1z w = ω in D, ∂ν(∂r

zw) = αr on ∂D, ∂rzw(0) = ar for 0 ≤ r ≤ n− 2 (7)

∂zω = f in D, ∂νω = αn−1 on ∂D, ω(0) = an−1 (8)

and the solution (6) for n− 1 instead of n and ω instead of f where

ω(z) = an−1 − 12πi

∫

|ζ|=1

(ζαn−1(ζ) − f(ζ)) log(1 − zζ)dζ

ζ2

− z

π

∫

|ζ|<1

f(ζ)ζ(ζ − z)

dξdη (9)

403


The last two terms of (6) with n − 1 instead of n and ω instead of f can bewritten as

an−1J1(z) − 12πi

∫

|ζ|=1

αn−1(ζ)ζ

J2(z, ζ)dζ

+1

2πi

∫

|ζ|=1

f(ζ)ζ2

J2(z, ζ)dζ − 1π

∫

|ζ|<1

f(ζ)ζ

J3(z, ζ)dξdη (10)

where

J1(z) =1

2πi

∫

|ζ|=1

(ζ − z)n−2 log(1 − zζ)dζ

ζ2− z

π

∫

|ζ|<1

(ζ − z)n−2

ζ(ζ − z)dξdη

=(−1)nzn−1

n− 1, (11)

J2(z, ζ) =1

2πi

∫

|ζ|=1

(ζ − z)n−2 log(1 − zζ) log(1 − ζ¯ζ)dζ

ζ2

− z

π

∫

|ζ|<1

(ζ − z)n−2 log(1 − ζ¯ζ)

dξdη

ζ(ζ − z)

Using binomial expansion and expressing J2(z, ζ) in terms of Lr(z, ζ), we obtain

J2(z, ζ) = − 1n− 1

(¯ζ − z)n−1 log(1 − z¯ζ) (12)

Using Cauchy-Pompeiu formula and computing the boundary integrals, we have

J3(z, ζ) = − 12πi

∫

|ζ|=1

ζ(ζ − z)n−2 log(1 − zζ)dζ

ζ2(ζ − ζ)

+z

π

∫

|ζ|<1

ζ(ζ − z)n−2 dξdη

ζ(ζ − z)(ζ − ζ)

= − 1(n− 1)

z

ζ − z(¯ζ − z)n−1 (13)

Substituting (11), (12), (13) in (10) and then subsequently in (6) with n − 1instead of n, we obtain (6).

404


Using (9), the boundary integral in (5) with ω instead of f can be written as

an−1z − 12πi

∫

|ζ|=1

(

αn−1(ζ)ζ

− f(ζ)ζ2

)

Φ1(z, ζ)dζ

− 1π

∫

|ζ|<1

f(ζ)ζ

Φ2(z, ζ)dξdη (14)

where

Φ1(z, ζ) =1

2πi

∫

|ζ|=1

log(1 − ζ¯ζ)

dζ

ζ2(1 − zζ)= −¯

ζ

and Φ2(z, ζ) = − 12πi

∫

|ζ|=1

dζ

(ζ − ζ)ζ(1 − zζ)=

z

1 − zζ

Using Gauss theorem, the area integral in (5) with ω instead of f can be writtenas

− 12πi

∫

|ζ|=1

ω(ζ)ζ − z

dζ

ζ− z

π

∫

|ζ|<1

f(ζ)(ζ − z)

(1 − zζ)2dξdη

Substituting ω from (8) and computing the boundary integrals involved we canwrite the above expression as

12πi

∫

|ζ|=1

αn−1(ζ)ζ

z log(1 − zζ)dζ − z

2πi

∫

|ζ|=1

f(ζ)ζ2

log(1 − zζ)dζ

− z

π

∫

|ζ|<1

f(ζ)(ζ − z)

(1 − zζ)2dξdη (15)

Substituting (14) and (15) in (5) with n− 1 instead of n and ω instead of f weget (4) for k = n− 1.

For 1 ≤ k ≤ n − 2, the last two integrals of (4) with n − 1 instead of n and ωinstead of f can be written as

− 12πi

∫

|ζ|=1

(

αn−1(ζ)ζ

− f(ζ)ζ2

)

ψ1(z, ζ)dζ − 1π

∫

|ζ|<1

f(ζ)ζ

ψ2(z, ζ)dξdη

405


+ zψ3(z) − (n− k − 1)ψ4(z) (16)

where

ψ1(z, ζ) =1

2πi

∫

|ζ|=1

(ζ − z)n−k−1 log(1 − ζ¯ζ)dζ

ζ2

Using binomial expansion, ψ1(z, ζ) can be expressed as linear combination ofzrNr(ζ). Similar using Lemma 1(ii) it follows that

ψ1(z, ζ) = − 1n− k

((¯ζ − z)n−k − (−z)n−k) (17)

ψ2(z, ζ) =1

2πi

∫

|ζ|=1

(ζ − z)n−k−1 ζ dζ

ζ2(ζ − ζ)= 0 (18)

ψ3(z) =1

2πi

∫

|ζ|=1

ω(ζ)(ζ − z)n−k−2 dζ

ζ2

− 1π

∫

|ζ|<1

ω(ζ)(ζ − z)n−k−1

(1 − zζ)2dξdη

Applying Gauss theorem on the area integral and the fact that∂ω

∂z= f in D,

computations similar to that in ψ1(z, ζ) yield

zψ3(z) =1

n− k

12πi

∫

|ζ|=1

(

αn−1(ζ)ζ

− f(ζ)ζ2

)

(z(ζ − z)n−k−1 + (−z)n−k)dζ

+z

n− k

1π

∫

|ζ|<1

f(ζ)(ζ − z)n−k

(1 − zζ)2dξdη

Lastly

ψ4(z) =1

2πi

∫

|ζ|=1

ω(ζ)(ζ − z)n−k−2 log(1 − zζ)dζ

ζ2

+1π

∫

|ζ|<1

ω(ζ)(ζ − z)n−k−2 dξdη

ζ(1 − zζ)(19)

406


Computations similar to those involved in J1(z), J2(z, ζ) and J3(z, ζ) yield,

ψ4(z)=(−1)n−kan−1z

n−k−1

(n− k − 1)+

1n− k − 1

12πi

∫

|ζ|=1

(

αn−1(ζ)ζ

− f(ζ)ζ2

)

(ζ−z)n−k−1

log(1 − zζ)dζ − 1n− k − 1

1π

∫

|ζ|<1

f(ζ)ζ

(ζ − z)n−k−1

1 − zζdξdη (20)

Substituting (15), (16), (17) and (18), (14) can be simplified to

(−1)n−k+1an−1zn−k +

1n− k

12πi

∫

|ζ|=1

(

αn−1(ζ)ζ

− f(ζ)ζ2

)

(ζ − z)n−k−1[ζ − (n− k)z log(1 − zζ)]dζ

+1

n− k

z

π

∫

|ζ|<1

f(ζ)(ζ − z)n−k−1

[

ζ − z

(1 − zζ)2+

n− k

ζ(1 − zζ)

]

dξdη (21)

Inserting (21) in (4) with n− 1 instead of n and ω instead of f and simplifyingterms, we obtain (3) for 1 ≤ k ≤ n − 2. Applying the solvability condition (1)for (8), (5) follows.Using the differentiability of the operators Tm,n [3], it follows that w given by(6) is indeed a solution.

3. Mixed Neumann-Schwarz Problem

In this section, we investigate mixed boundary value problem arising from n-Neumann and m-Schwarz boundary conditions. Since some of the computationsare lengthy, the details have been avoided. All these computations can be madeusing Cauchy-Pompeiu formula, Gauss theorem, binomial and multinomials.

For p, q, r ∈ N0, we denoter!

p!q!(r − p− q)!by N(p, q, r) and the subset (p, q) :

p + q ≤ r of N0 × N0 by A(r).

Lemma 2. For r, s ∈ N0, n ∈ N and |ζ|, |z| < 1, we have

Cr,n(z, ζ) =1

2πi

∫

|ζ|=1

(ζ + ¯ζ − ζ − ζ)r(ζ − z)n−1 log(1 − zζ)

dζ

ζ2

− z

π

∫

|ζ|<1

(ζ + ¯ζ − ζ − ζ)r(ζ − z)n−1 dξdη

ζ(ζ − z)

407


=r∑

j=0

(

r

j

)

(−1)j(ζ + ¯ζ)r−jBj,n(z) (22)

whereBr,n(z) =

12πi

∫

|ζ|=1

(ζ + ζ)r(ζ − z)n−1 log(1 − zζ)dζ

ζ2

− z

π

∫

|ζ|<1

(ζ + ζ)r(ζ − z)n−1 dξdη

ζ(ζ − z)

=∑

(p,q)∈A(r)

N(p, q, r)zr−p−qAp,q+n(z) (23)

and

Ar,n(z) =1

2πi

∫

|ζ|=1

ζr(ζ − z)n−1 log(1 − zζ)dζ

ζ2− z

π

∫

|ζ|<1

ζr(ζ − z)n−1 dξdη

ζ(ζ − z)

=(−1)n+1

n

[

znzr −r−2∑

s=0

(

n

s + 1

)

(−1)s r

r − s− 1zn−(s+1)zr−(s+1)

]

Proof: To express Br,n in terms of Ap,q+n, we write (ζ+ ζ)r = (z+ζ+ ζ− z)r

=∑

(p,q)∈A(r)

N(p, q, r)zr−p−qζp(ζ − z)q

Remaining area integrals and boundary integrals can be computed by usingCauchy Pompeiu formula and Gauss theorem. Using similar arguments as in the above lemma, we obtain the following

Lemma 3. For r ∈ N0, n ∈ N, |ζ|, |z| < 1 we have

(i) If Fr,n(z, ζ) =1

2πi

∫

|ζ|=1

(ζ + ¯ζ − (ζ + ζ))r(ζ − z)n−1 log(1 − zζ)

dζ

ζ2(ζ − ζ)

− z

π

∫

|ζ|<1

(ζ + ¯ζ − (ζ + ζ))r(ζ − z)n−1 dξdη

ζ(ζ − z)(ζ − ζ)

then Fr,n(z, ζ) are given as in (22) with Bj,n(z) replaced by Ej,n(z, ζ) andEr,n(z, ζ) are as in (23) with Ap,q+n(z) replaced by Dp,q+n(z, ζ). TheseDr,n(z, ζ) are given by

Dr,n(z, ζ) =1

2πi

∫

|ζ|<1


ζ2(ζ − z)

408


− z

π

∫

|ζ|<1


ζ(ζ − z)(ζ − ζ)

=ζr

n

[

(−z)n

ζ+z

ζ

(¯ζ − z)n

ζ − z

]

+r−1∑

j=0

ζr−1−jAj,n(z)

(ii) If Hr,n(z, ζ) =1

2πi

∫

|ζ|=1

(ζ + ¯ζ − ζ − ζ)r(ζ − z)n−1 log(1 − zζ)

dζ

ζ2(1 − ζζ)

− z

π

∫

|ζ|<1

(ζ + ¯ζ − ζ − ζ)r(ζ − z)n−1 dξdη

ζ(ζ − z)(1 − ζ¯ζ)

then Hr,n(z, ζ) are given as in (22) with Bj,n(z) replaced by Gj,n(z, ζ) and

Gj,n(z, ζ) are given as in (23) with Ap,q+n(z) replaced by Ip,q+n(z, ζ) where

Ir,n(z, ζ) =1

2πi

∫

|ζ|=1


ζ2(1 − ζ¯ζ)

− z

π

∫

|ζ|<1


ζ(ζ − z)(1 − ζ¯ζ)

= (−1)n−1n−1∑

s=0

(

n− 1s

)

(−1)szn−1−sKr,s(z, ζ)

with

Kr,s(z, ζ) =1

2πi

∫

|ζ|=1

ζr ζs log(1 − zζ)dζ

ζ2(1 − ζ¯ζ)

− z

π

∫

|ζ|<1

ζr ζs dξdη

ζ(ζ − z)(1 − ζ¯ζ)

=

¯ζ

s+1−rlog(1 − z

¯ζ) − 1

s + 11

1 − z¯ζ

[z¯ζ

s+2−r − zs+1zr]

if s ≥ r − 1 or r = 2 and s = 0

−zr−s−1∞∑

k=0

1r − s− 1 + k

(z¯ζ)k − 1

(s + 1)(1 − z¯ζ)

[zr−s−1 − zr zs+1]

if r ≥ 3, 0 ≤ s ≤ r − 2

409


(iii) If Nr,n(z, ζ) =1

2πi

∫

|ζ|=1

(ζ + ¯ζ − ζ − ζ)r(ζ − z)n−1 dζ

ζ2

then Nr,n(z, ζ) are given as in (22) with Bj, n(z) replaced by Mj, n(z) and

Mr, n(z) are given as in (23) with Ap,q+n(z) replaced by Lp,q+n(z). These Lr,n(z)

can be expressed as (−1)n−r(

n−1r−1

)

zn−r.

(iv) If Qr,n(z, ζ) =1

2πi

∫

|ζ|=1

(ζ + ¯ζ − ζ − ζ)r(ζ − z)n−1 dζ

ζ2(ζ − ζ)

then Qr,n (z, ζ) are given as in (22) with Bj,n(z) replaced by Pj,n(z, ζ)

and Pj,n(z, ζ) are given as in (23) with Ap,q+n(z) replaced by Op,q+n(z, ζ).

Op,q+n(z, ζ) can be expressed as Or,n(z, ζ) =1

2πi∫

|ζ|=1

ζr(ζ − z)n−1 dζ

ζ2(ζ − ζ)

= (−1)n−1n−1∑

j=0

(

n−1j

)

(−1)j zn−1−j ζr−j−2 χA(r, s), A = (r, s) : r + s ≥ 2,

χA being the characteristic function of A

(v) If Tr,n(z, ζ) =1

2πi

∫

|ζ|=1

(ζ + ¯ζ − ζ − ζ)r(ζ − z)n−1 dζ

ζ2(1 − ζ¯ζ)

then Tr,n (z, ζ) are given as in (22) with Bj,n(z) replaced by Sj,n(z, ζ) and

Sj,n(z, ζ) are given as in (23) with Ap,q+n(z) replaced by Rp,q+n(z, ζ) where

Rr,n can be expressed as

12πi

∫

|ζ|=1


ζ2(1 − ζ¯ζ)

= (−1)n−1n−1∑

j=0

(

n− 1j

)

(−1)j zn−1−j ¯ζ

j−r+1χA(j, s)

where A is as above in (iv).

(vi) If qr,n(z, ζ) =1

2πi

∫

|ζ|=1

(ζ + ¯ζ − ζ − ζ)r(ζ − z)n−1 dζ

ζ2

− 1π

∫

|ζ|<1

(ζ + ¯ζ − ζ − ζ)r (ζ − z)n

(1 − zζ)2dξdη,

then qr,n(z, ζ) are given as in (22) with Bj,n(z) replaced pj,n(z) where pr,n(z)

can be expressed as

410


12πi

∫

|ζ|=1

(ζ + ζ)r(ζ − z)n−1 dζ

ζ2− 1π

∫

|ζ|<1

(ζ + ζ)2(ζ − z)n

(1 − zζ)2dξdη

=∑

(p,q)∈A(r)

(−1)q+n−pN(p, q, r)(

q+n−1p−1

) q + n

q + n + 1zq+n−p

(vii) If Ur,n(z, ζ) =1

2πi

∫

|ζ|=1

(ζ + ¯ζ − ζ − ζ)r(ζ − z)n−1 dζ

ζ2(ζ − ζ)

− 1π

∫

|ζ|<1

(ζ + ¯ζ − ζ − ζ)r (ζ − z)n−1

(1 − zζ)2dξdη

ζ − ζ,

then Ur,n(z, ζ) are given as in (22) with Bj,n(z) replaced by Tj,n(z, ζ). Tj,n(z, ζ)

are given in (23) with Sp,q+n(z, ζ) instead of Ap,q+n(z). These Sr,n(z, ζ) can be

expressed asn

n + 1Or,n−1(z, ζ) +

1n + 1

ζr (¯ζ − z)n+1

(1 − zζ)2

Or,n(z, ζ) are as in (iv) above.

(viii) If ur,n(z, ζ) =1

2πi

∫

|ζ|=1

(ζ + ¯ζ − ζ − ζ)r (ζ − z)n−1 dζ

ζ2(1 − ζ¯ζ)

− 1π

∫

|ζ|<1

(ζ + ¯ζ − ζ − ζ)r (ζ − z)n−1

(1 − zζ)2dξdη

1 − ζ¯ζ

,

then ur,n(z, ζ) are given as (22) with Bj,n(z) replaced by tj,n(z, ζ) where tj,n(z, ζ)

are given in (23) with sp,q+n(z, ζ) instead of Ap,q+n(z). These sr,n(z, ζ) can

be expressed as1

2πi∫

|ζ|=1


ζ2(1 − ζ¯ζ)

− 1π

∫

|ζ|<1

ζr(ζ − z)n

(1 − zζ)2dξdη

1 − ζ¯ζ

=

n

n + 1Or,n(z, ζ)

Decomposing the term1

ζk(ζ − z)2and using again Gauss theorem we obtain

the following

Lemma 4. For r ∈ N0, n ∈ N, |ζ|, |z| < 1, we have

(i) If nr(z, ζ) = − 12πi

∫

|ζ|=1

(ζ + ¯ζ − ζ − ζ)r ζdζ

ζ − z

411


− z

π

∫

|ζ|<1

(ζ + ¯ζ − ζ − ζ)r dξdη

(1 − zζ)2

then nr(z, ζ) is equal to

r∑

k=0

[(k+1)/2]∑

j=0

( r

k

)

(

k

j + 1

)

(ζ + ¯ζ)r−kzk+1−2j ,

[k] being the greatest integer less than or equal to k.

(ii) If cr(z, ζ) = − 12πi

∫

|ζ|=1

(ζ + ¯ζ − ζ − ζ)r ζdζ

(ζ − z)(ζ − ζ)

− z

π

∫

|ζ|<1

(ζ + ¯ζ − ζ − ζ)r dξdη

(1 − zζ)2(ζ − ζ)

then cr(z, ζ) are expressible as in (22) with Bj,n(z) replaced by bj(z, ζ). The

br(z, ζ) are given byr∑

j=0

(

rj

)

ar−j,j(z, ζ) where

ar,s(z, ζ) = − 12πi

∫

|ζ|=1

ζr ζs+1dζ

(ζ − z)(ζ − ζ)− z

π

∫

|ζ|<1

ζr ζsdξdη

(1 − zζ)2(ζ − ζ)

=(r − 1)zs+2−r − rzs+3−r ζ + zζr ¯

ζs+1

(s + 1)(1 − zζ)2+ χB(r, s)

s+2∑

k=r−1

kζk−r+1

zk−(s+1),

where B = (r, s) : s + 2 < r.

(ii) If fr(z, ζ) = − 12πi

∫

|ζ|=1

(ζ + ¯ζ − ζ − ζ)sζdζ

(ζ − z)(1 − ζ¯ζ)

− z

π

∫

|ζ|<1

(ζ + ¯ζ − ζ − ζ)sdξdη

(1 − zζ)2(1 − ζ¯ζ)

then fr(z, ζ) are expressible as in (22) with Bj,n(z) replaced by ej(z, ζ). These

es(z, ζ) are given ass∑

j=[s/2]

(

sj

)

ds−j,j(z, ζ) where

dr,s(z, ζ) = − 12πi

∫

|ζ|=1

ζr ζs+1dζ

(ζ − z)(1 − ζ¯ζ)

− z

π

∫

|ζ|<1

ζr ζs dξdη

(1 − zζ)2(1 − ζ¯ζ)

=

(

1¯ζ − z

[

¯ζ

s+2−r −(

2s + 3 − r

s + 1

)

zs+2−r

]

412


− z

(s + 1)(¯ζ − z)2

¯ζ

s+2−r − zs+2−r

)

χC(r, s)

where C = (r, s) : s + 2 > r.

Theorem 4. For n,m ≥ 1, the mixed n-Neumann and m-Schwarz problem for

the inhomogeneous polyanalytic equation in the unit disc

∂n+mz w = f in D, ∂ν(∂r

zw) = αr,Re(∂n+sz w) = βs on ∂D, ∂r

zw(0) = ar

and Im ∂n+sz w(0) = bs, where f ∈ Cα(D,C), 0 < α < 1, αr ∈ C(∂D,C),

βs ∈ C(∂D,R), and ar ∈ C, bs ∈ R for 0 ≤ r ≤ n− 1, 0 ≤ s ≤ m− 1 is uniquely

solvable if and only if for 1 ≤ k ≤ m− 1 and |z| = 1,

n−1∑

r=k

arzr−k+1

(r − k)!+

12πi

∫

|ζ|=1

[

αk−1(ζ)ζ(zζ − 1)

+n−1∑

r=k

(−1)r−k

(r − k + 1)!αr(ζ)ζ

(ζ − z)r−k

(ζ − (r − k + 1)z log(1 − zζ))]

dζ

+(−1)n−k

(n− k)!i

m−1∑

s=0

bss!

[Ms,n−k+1(z) +zps,n−k(z) − (n− k)zBs,n−k(z)]

−m−1∑

s=0

(−1)s

s!1

2πi

∫

|ζ|=1

βs(ζ)ζ

[Ns,n−k+1(z, ζ)+zqs,n−k(z, ζ)−(n−k)z cs,n−k(z, ζ)

+2ζQs,n−k+1(z, ζ) +zUs,n−k(z, ζ)−(n− k)zFs,n−k(z, ζ)]dζ

+(−1)m

(m− 1)!1

2π

∫

|ζ|<1

[

f(ζ)ζ

φ(z, ζ)+f(ζ)ζ

ψ(z, ζ)

]

dξdη = 0 (24)

where

φ(z, ζ) = [(−Nm−1,n−k+1 − 2ζQm−1,n−k+1)−z(qm−1,n−k − 2ζUm−1,n−k)

+(n− k) z(Cm−1,n−k + 2ζFm−1,n−k)] (z, ζ)

ψ(z, ζ) = [(2Tm−1,n−k+1 −Qm−1,n−k+1) +z(2um−1,n−k − Um−1,n−k)

−(n− k) z(2Hm−1,n−k − Fm−1,n−k)] (z, ζ)

413


and

12πi

∫

|ζ|=1

[

αn−1(ζ)ζ(zζ − 1)

−m−1∑

s=0

(−1)s

s!βs(ζ)ζ

(ns + 2ζcs)(z, ζ)]

dζ+

im−1∑

s=0

[(s+1)/2]∑

j=0

(

sj+1

) bss!zs+1−2j +

(−1)m

2π(m− 1)!

∫

|ζ|<1

[f(ζ)ζ

(−nm−1−2ζcm−1)(z, ζ)

+f(ζ)ζ

(2fm−1 − nm−1) (z, ζ)] dξdη = 0 (25)

The solution is given as

w(z) =n−1∑

r=0arzr

r!− 1

2πi

∫

|ζ|=1

[ n−1∑

r=0

(−1)r

r!αr(ζ)ζ

(ζ − z)r log(1 − zζ)+

(−1)n−1

(n− 1)!

m−1∑

s=0

(−1)s

s!βs(ζ)ζ

(Cs,n + 2ζFs,n) (z, ζ)] dζ

+i(−1)n−1

(n− 1)!

m−1∑

s=0

bss!Bs,n(z) +

(−1)m

(m− 1)!1

2π

∫

|ζ|<1

[

f(ζ)ζ

(−Cm−1,n−2ζFm−1,n)(z, ζ)

+f(ζ)ζ

(−Cm−1,n +2Hm−1,n) (z, ζ)]

dξdη (26)

Proof: The problem is decomposed into the system

∂nz w = ω in D, ∂ν(∂r

zw) = αr on ∂D, ∂rzw(0) = ar for 0 ≤ r ≤ n− 1 (27)

and ∂mz ω = f in D, Re (∂s

zω) = βs on ∂D, Im ∂szω(0) = bs for 0 ≤ s ≤ m−1.(28)

Using Theorem 3, the problem (27) is uniquely solvable if and only if (4) and

(5) hold with ω instead of f and w is given by (6) with ω instead of f .

The problem (28) is uniquely solvable and ω is given as in (3). Substituting this

value of ω in (6) with ω instead of f , the last two integrals of the expression

can be written as

im−1∑

s=0(−1)s bs

s!

12πi

∫

|ζ|=1

W (ζ, 0, s) log(1 − zζ)dζ

ζ2− z

π

∫

|ζ|<1

W (ζ, 0, s)dξdη

ζ(ζ − z)

414


+m−1∑

s=0

(−1)s

s!1

2πi

∫

|ζ|=1

Bs(ζ)ζ

[

12πi

∫

|ζ|=1

ζ + ζ

ζ − ζW (ζ, ζ, s) log(1 − zζ)

dζ

ζ2

− z

π

∫

|ζ|<1

ζ + ζ

ζ − ζ

W (ζ, ζ, s)ζ(ζ − z)

dξdη

]

dζ

+(−1)m

(m− 1)!1

2π

∫[

f(ζ)ζ

[

12πi

∫

|ζ|= 1

ζ + ζ

ζ − ζW (ζ, ζ,m− 1) log(1 − zζ)

dζ

ζ2

− z

π

∫

|ζ|< 1

ζ + ζ

ζ − ζ

W (ζ, ζ,m− 1)ζ(ζ − z)

dξ dη

]

+f(ζ)

ζ[

12πi

∫

|ζ|= 1

1 + ζ¯ζ

1 − ζ¯ζW (ζ, ζ,m− 1)

log(1 − zζ)dζ

ζ2− z

π

∫

|ζ|< 1

1 + ζ¯ζ

1 − ζ¯ζ

W (ζ, ζ,m− 1)ζ(ζ − z)

dξ dη]]

dξ dη (29)

where W (ζ, ζ, s) = (ζ + ¯ζ − ζ − ζ)s (ζ − z)n−1

The integral in the first sum is equal to B(s,n)(z) (Lemma 2). The inte-

gral in the second summand can be expressed as −Cs,n (z, ζ) −2ζFs,n(z, ζ)

(Lemma 2, 3). The last integrals in (29) withf(ζ)

¯ζ

is equal to −Cm−1,n (z, ζ)

+2Hm−1,n (z, ζ) (Lemma 2, 3). Substituting these values in (6) with ω instead

of f , we obtain (26).

The last two integrals of (4) with ω instead of f are expressible as L1 + zL2

−(n− k)zL3.

Using Lemma 3 (iii), (v) L1 can be written as

L1 = im−1∑

s=0

bss!

Ms,n−k+1(z)

−m−1∑

s=0

(−1)s

s!1

2πi

∫

|ζ|=1

βs(ζ)ζ

[Ns,n−k+1 + 2ζQs,n+k+1] (z, ζ)dζ

+(−1)m−1

2π(m− 1)!

∫

|ζ|<1

f(ζ)ζ

[(Nm−1,n−k+1 +2ζQm−1,n−k+1)(z, ζ)

+f(ζ)ζ

(2Tm−1,n−k+1 −Nm−1,n−k+1) (z, ζ) ] dξdη

415


Using Lemma 3 (vi), (vii), L2 can be expressed as

L2 = im−1∑

s=0

bss!ps,n−k(z) −

m−1∑

s=0

(−1)s

s!1

2πi

∫

|ζ|=1

βs(ζ)ζ

(qs,n−k+2ζ Us,n−k) (z, ζ)dζ

+(−1)m

2π(m− 1)!

∫

|ζ|<1

[

f(ζ)ζ

(−qm−1,n−k −2ζUm−1,n−k) (z, ζ)

+f(ζ)ζ

(2um−1,n−k −qm−1,n−k) (z, ζ)]

dξdη

Lemma 2, Lemma 3 (i), (ii) enable us to write

L3 = im−1∑

s=0

bss!

Bs,n−k(z)

−m−1∑

s=0

(−1)s

s!1

2πi

∫

|ζ|=1

βs(ζ)ζ

[Cs,n−k + 2ζFs,n−k] (z, ζ)dζ

+(−1)m+1

(m− 1)!1

2π

∫

|ζ|<1

[

f(ζ)ζ

(Cm−1,n−k +2ζFm−1,n−k) (z, ζ)

+f(ζ)ζ

(Cm−1,n−k −2Hm−1,n−k) (z, ζ)]

dξdη

Substituting L1 + zL2 −(n− k)zL3 in (6) with ω instead of f , we obtain (24).

Applying Lemma 2, the last two integrals of (5) with ω instead of f can be

written as

im−1∑

s=0

bss!

[(s+1)/2]∑

j=0

(

sj+1

)

zs+1−2j −m−1∑

s=0

(−1)s

s!

∫

|ζ|=1

βs(ζ)ζ

(ns + 2ζcs) (z, ζ) dζ

+(−1)m+1

2π(m− 1)!

∫

|ζ|<1

[

f(ζ)ζ

(nm−1 + 2ζCm−1) (z, ζ)

+f(ζ)ζ

(nm−1 − 2fm−1)(z, ζ)]

dξdη

Substituting this value in (5) with ω instead of f , we obtain (25).

Remark 3.5: The mixed boundary value problem arising with first n-Schwarzand last m-Neumann boundary conditions can be solved in a similar way. Thishas been avoided here due to lack of space.

416


References

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[3] H. Begehr, G.N. Hile, A hierarchy of integral operators. The Rocky Moun-tain Journal of Mathematics, 27, 669–706, 1997.

[4] H. Begehr, A. Kumar, Boundary value problem for bi-polyanalytic func-tions. Applicable Analysis, to appear.

[5] H. Begehr, A. Kumar, Boundary value problems for the inhomogeneouspolyanalytic equation, I, II, Analysis, to appear

[6] H. Begehr, D. Schmersau, The Schwarz problem for polyanalytic function.ZAA, to appear.

[7] H. Begehr, C.J. Vanegas, Iterated Neumann problem for the higher orderPoisson equation. Math. Nachr, to appear.

[8] A.Kumar, R. Prakash, Boundary value problems for the Poisson equationand bi-analytic functions. Complex Variables 50, 597-609, 2005.

[9] M. Nicolescu, Les functions polyharmonique, Hermann, Paris, 1936.

[10] I.N. Vekua, New methods for solving elliptic equations. North Holland,Amsterdam; John Wiley, New York, 1967.

417

418

The Construction of a Kind of Quadrature Formulas1

Ming-Cai Liu, Pi-Xi Zhao, and Wei-Ming YangSchool of Computer Science and Engineering, Dalian Nationalities University, Dalian 116600, P.R.China

AbstractIn this paper, a kind of quadrature formulas are constructed by using the Euler-Maclaurin summation formula. It has the same nodes as composite trapezoidalrule. But its convergence order is very high. Numerical results are presentedwhich also show that they are simple and efficient numerical integration rules.Keywords: Numerical integration; Convergence order; Absolute error.

1. Introduction

Numerical integrations are often encountered in practice, for example, in wavelet-Galerkin methods for integral equations, we need to calculate a lot of numerical integra-tions(see [1,3,4]). The composite trapezoidal rule and Simpson’s rule are simple, and haverecursive relations. But their convergence order is very low. Gaussian rule has high alge-braic accuracy ,but has no recurrence relations. In paper[5], a method of construction ofquadrature formulas for the calculation of inner products of smooth function and scalingfunctions is presented. In this paper, we use the Euler-Maclaurin summation formulato construct a kind of quadrature formulae, we call it modified composite trapezoidalrule. It has the same nodes as composite trapezoidal rule, and has recurrence relations.Furthermore, its convergence order is very high.

2. The Construction of the Quadrature formulas

Suppose f(x) ∈ C2k+3[a, b], where k = 1, 2, · · ·, and let h = b−an , n ≥ 2k, and B2j be

Bernoulli numbers, i.e. B2 = 16 , B4 = − 1

30 , B6 = 142 , B8 = − 1

30 , · · · . Set

p2k+3(x) = (−1)k∞∑

n=1

2 sin 2πnx

(2πn)2k+3,

then, we have the Euler-Maclaurin summation formula[2]:

h

2[f(a) + f(b) + 2

n−1∑

i=1

f(a + ih)] =∫ b

af(x)dx +

k+1∑

i=1

B2i

(2i)!h2i[f (2i−1)(b)− f (2i−1)(a)]

+h2k+3∫ b

ap2k+3(n

x− a

b− a)f (2k+3)(x)dx, (1)

1This paper is supported by the National Natural Science Foundation of China (60372071).

1419


and have the estimation:∣∣∣∫ b

ap2k+3(n

x− a

b− a)f (2k+3)(x)dx

∣∣∣ ≤ M2k+32−2k−2(π)−2k−3(b− a)∞∑

j=1

1j2k+3

, (2)

where M2k+3 = maxa≤x≤b

|f (2k+3)(x)|,∞∑

j=11/(j2k+3) is Riemann series. Set

Tn =h

2[f(a) + f(b) + 2

n−1∑

i=1

f(a + ih)],

Pk =k∑

i=1

B2i

(2i)!h2i[f (2i−1)(a)− f (2i−1)(b)], (3)

and

Ek =B2k+2

(2k + 2)!h2k+2[f (2k+1)(a)− f (2k+1)(b)]− h2k+3

∫ b

ap2k+3(n

x− a

b− a)f (2k+3)(x)dx,

then, (1) can be rewritten as∫ b

af(x)dx = Tn + Pk + Ek. (4)

From Ek, we know that the convergence order of Tn + Pn is O(h2k+2), Consequently,Eq.(4)can be rewritten as

∫ b

af(x)dx = Tn + Pk + O(h2k+2). (5)

The idea is to generate an approximation to derivative in Pk by the vales of the functionat the given nodes, and to ensure that the convergence order is also O(h2k+2).

For f(x) ∈ C2k+1[a, b], using the Taylor formula, we have

f(x + ih)− f(x) =2k∑

j=1

f (j)(x)j!

(ih)j +f (2k+1)(ξi(x))

(2k + 1)!(ih)2k+1, (6)

where i = 1, 2, · · · , 2k, x ≤ ξi(x) ≤ x + ih.

Let

εi(x) = −f (2k+1)(ξi(x))i2k+1

(2k + 1)!, E = (ε1(x), ε2(x), · · · , ε2k(x))>,

yi = f (i)(x)hi, Y = (y1, y2, · · · , y2k)>, fi = f(x + ih)− f(x),

F = (f1, f2, · · · , f2k)>, A = (aij), aij =ij

j!, i, j = 1, 2, · · · , 2k,

then (6) can be rewritten asAY = F + h2k+1E. (7)

Proposition 1. The matrix A in (7) is nonsingular.

Proof. Let the ith row of A be divided by i, (i = 1, 2, · · · , 2k), the jth columnof A be divided by j!, (j = 1, 2, · · · , 2k), then we get a matrix whose determinant is aVandermode determinant. Therefor, the matrix A is nonsingular.

2

LIU ET AL

420

By(7), we haveY = A−1F + h2k+1A−1E. (8)

Let A−1 = (bij), then (8)can be rewritten as

f (i)(x)hi =2k∑

j=1

bij [f(x + jh)− f(x)] + h2k+12k∑

j=1

bijεj(x), i = 1, 2, · · · , 2k. (9)

Let x = a, we have

f (i)(a)hi =2k∑

j=1

bij [f(a + jh)− f(a)] + h2k+12k∑

j=1

bijεj(a). (10)

Let x = b, and replace h by −h,(now b− ih ≤ ξi(b) ≤ b), we have

f (i)(b)(−h)i =2k∑

j=1

bij [f(b− jh)− f(b)]− h2k+12k∑

j=1

bijεj(b). (11)

Substituting (10) and (11)into Pk in (3) gives

Pk = hk∑

i=1

B2i

(2i)!

2k∑

j=1

bij

[f(a+ jh)−f(a)+f(b− jh)−f(b)+(εj(a)−εj(b))h2k+1

]. (12)

Let

Pk = hk∑

i=1

B2i

(2i)!

2k∑

j=1

bij [f(a + jh)− f(a) + f(b− jh)− f(b)], (13)

Ek = h2k+2k∑

i=1

B2i

(2i)!

2k∑

j=1

bij [εj(a)− εj(b)]. (14)

andT (k)

n = Tn + Pk. (15)

then (4) can be rewritten as∫ b

af(x)dx = T (k)

n + Ek + Ek. (16)

where T(k)n is a modified composite trapezoidal rule which we want to construct, and

Ek = Pk − Pk. Now, we estimate the Ek + Ek in (16):

|Ek| ≤ 2h2k+2M2k+1

k∑

i=1

|B2i|(2i)!

( 2k∑

j=1

|bij |) i2k+1

(2k + 1)!

= h2k+2 2M2k+1

(2k + 1)!

k∑

i=1

|B2i|i2k+1

(2i)!

( 2k∑

j=1

|bij |), (17)

where M2k+1 = maxa≤x≤b

|f (2k+1)(x)|.Consequently,

|Ek + Ek| ≤ h2k+2[2M2k+1|B2k+2|

(2k + 2)!+

2M2k+1

(2k + 1)!

k∑

i=1

|B2i|i2k+1

(2i)!

( 2k∑

j=1

|bij |)]

3

QUADRATURE FORMULAS

421

+h2k+3M2k+32−2k−2(π)−2k−3(b− a)∞∑

j=1

1j2k+3

. (18)

Then we haveProposition 2. If f(x) ∈ C2k+3[a, b], then the convergence order of the modified

composite trapezoidal rule T(k)n is O(h2k+2). That is∫ b

af(x)dx = T (k)

n + O(h2k+2). (19)

3. Some examples of the modified composite trapezoidal rule

Suppose f(x) ∈ C2k+3[a, b], k = 1, 2, · · · , h = b−an , n ≥ 2k. Let fi = f(a+ih), f−i =

f(b − ih), i = 0, 12 , 1, 3

2 , · · · , n. Notice that f0 = f(a) = f−n, f−0 = f(b) = fn. Let

Tn = h2 (f0 + f−0 + 2

n−1∑i=1

fi). Now, we use the method introduced above to constructed

some formulas of T(k)n for different k:

k=1:T (1)

n = Tn +h

24[−3(f0 + f−0) + 4(f1 + f−1)− (f2 + f−2)]. (20)

The formula of T(1)n has the following properties:

(i) When n = 2, T(1)2 is the Simpson’s rule; when n = 3, T

(1)3 is the Newton-Cotes

formula with n = 3(see [2]).(ii) When n ≥ 6

T (1)n = h

n−3∑

i=3

+h[38(f0 + f−0) +

76(f1 + f−1) +

2324

(f2 + f−2)]. (21)

(iii) By(21), we have a recurrence relations of T(1)2n and T

(1)n :

T(1)2n =

12T (1)

n +h

2

n−2∑

i=1

fi+ 12

+h

48

[(f2 + f−2)− 5(f1 + f−1) + 28(f 1

2+ f− 1

2)]. (22)

k=2:

T (2)n = Tn − h

1440[245(f0 + f−0)− 462(f1 + f−1)

+336(f2 + f−2)− 146(f3 + f−3) + 27(f4 + f−4)]. (23)


(i) T(2)4 and T

(2)5 are Newton-Cotes formulas with n = 4 and n = 5(see [2]), respec-

tively.(ii) When n ≥ 10, we have

T (2)n = h

n−5∑

i=5

fi + h[ 95288

(f0 + f−0) +317240

(f1 + f−1) +2330

(f2 + f−2)

+793720

(f3 + f−3) +157160

(f4 + f−4)]. (24)

4

LIU ET AL

422

(iii) T(2)2n and T

(2)n have a recurrence relations

T(2)2n =

12T (2)

n +h

2

n−3∑

i=2

fi+ 12

+h

2

[317240

(f 12

+ f− 12)− 133

240(f1 + f−1) +

793720

(f 32

+f− 32) +

103480

(f2 + f−2)− 73720

(f3 + f−3) +3

160(f4 + f−4)

]. (25)

k=3:

T (3)n = Tn − h

362880[71043(f0 + f−0)− 167064(f1 + f−1)

+198327(f2 + f−2)− 171072(f3 + f−3) + 94569(f4 + f−4)

−29928(f5 + f−5) + 4125(f6 + f−6)]. (26)


(i) T(3)6 and T

(3)7 are Newton-Cotes formulas with n = 6 and n = 7(see [2]),respec-

tively.(ii) When n ≥ 14, we have

T (3)n = h

n−7∑

i=7

fi + h[ 525717280

(f0 + f−0) +2208115120

(f1 + f−1)

+54851120960

(f2 + f−2) +10370

(f3 + f−3) +89437120960

(f4 + f−4)

+1636715120

(f5 + f−5) +2391724192

(f6 + f−6)]. (27)

(iii) T(3)2n and T


T(3)2n =

12T (3)

n +h

2

n−4∑

i=3

fi+ 12

+h

2

[2208115120

(f 12

+ f− 12)− 121797

120960(f1 + f−1) +

10370

(f 32

+ f− 32)

+34586120960

(f2 + f−2) +1636715120

(f 52

+ f− 52)− 408793

846720(f3 + f−3)

+31523120960

(f4 + f−4)− 124715120

(f5 + f−5) +275

24192(f6 + f−6)

]. (28)

k=4:

T (4)n = Tn − h[

1908789600

(f0 + f−0)− 427487725760

(f1 + f−1) +34982173628800

(f2 + f−2)

−500327403200

(f3 + f−3) +64675670

(f4 + f−4)− 26161613628800

(f5 + f−5)

+2401980640

(f6 + f−6)− 2630773628800

(f7 + f−7) +8183

1036800(f8 + f−8)]. (29)


5

QUADRATURE FORMULAS

423

(i) When n ≥ 18, we have

T (4)n = h

n−9∑

i=9

fi + h[2571389600

(f0 + f−0) +1153247725760

(f1 + f−1) +1305833628800

(f2 + f−2)

+903527403200

(f3 + f−3)− 7975670

(f4 + f−4) +62449613628800

(f5 + f−5)

+5662180640

(f6 + f−6) +38918773628800

(f7 + f−7) +10286171036800

(f8 + f−8)]. (30)

(ii) Let h ≤ 1, by(18), we can obtain

∣∣∣∫ b

af(x)dx− T (4)

n

∣∣∣ ≤ h9

100max

a≤x≤b|f (9)(x)|

(iii) T(4)2n and T


T(4)2n =

12T (4)

n +h

2

n−5∑

i=4

fi+ 12

+h

2

[1153247725760

(f 12

+ f− 12)− 1408913

907200(f1 + f−1) +

903527403200

(f 32

+ f− 32)

+18215877257600

(f2 + f−2) +62449613628800

(f 52

+ f− 52)− 310211

201600(f3 + f−3)

+38918773628800

(f 72

+ f− 72) +

82204797257600

(f4 + f−4)− 26161613628800

(f5 + f−5)

+2401980640

(f6 + f−6)− 2630773628800

(f7 + f−7) +8183

1036800(f8 + f−8)

]. (31)

4. Numerical examples

Now we use the modified composite trapezoidal rule T(k)n , k = 1, 2, 3, 4, composite

trapezoidal rule Tn, composite Simpson’s rule Sn, and composite two-point Gaussian ruleGn, i.e. using two-point Gaussian rule on each subinterval, to approximate integrals. T

(k)n

and Tn as before, Sn and Gn as following:

∫ b

af(x)dx ≈ Sn :=

h

3

[f(x0) + f(x2n) + 4

n∑

i=1

f(x2i−1) + 2n−1∑

i=1

f(x2i)], (32)

where h = b−a2n , xi = a + ih, i = 0, 1, · · · , n.

∫ b

af(x)dx ≈ Gn :=

n−1∑

i=0

h

2[(f(

h

2t1 +

xi + xi+1

2) + f(

h

2t2 +

xi + xi+1

2)], (33)

where h = b−an , xi = a + ih, i = 0, 1, · · · , n, t1 = 0.577350269189628, t2 = −t1.

Example 1.∫ π0 sinxdx = 2.

Example 2.∫ 10

x1+ex = 0.1705573495024382 · · · .

Example 3.∫ 10

11+10x2 = 0.3998760050557660 · · · .

6

LIU ET AL

424

Example 4.∫ 3

2π

0 cos 15xdx = 115 .

Example 5.∫ 2π0 x cos x sin 30xdx = −0.2096724796611 · · · .

Example 6.∫ 10

11−0.98x2 dx = 2.67096531488 · · · .

The numerical results are shown in Table 1. The first column are the ordinal numbersof these examples, the second are numbers of nodes, and the others are absolute errors,which correspond to quadrature Formulas.

Table 1.

example n T(1)n T

(2)n T

(3)n T

(4)n Tn Sn Gn

1 10 4.9E-04 2.3E-05 1.1E-06 5.2E-08 1.6E-02 1.1E-04 4.5E-061 20 3.2E-05 4.1E-07 6.3E-09 1.0E-10 4.1E-03 6.8E-06 2.8E-072 10 6.9E-07 5.6E-09 5.4E-11 1.6E-12 3.5E-04 1.7E-07 7.1E-092 20 4.7E-08 1.1E-10 3.6E-13 2.2E-15 8.9E-05 1.1E-08 4.5E-103 80 1.6E-09 7.8E-10 6.1E-11 6.1E-12 2.1E-06 2.0E-10 8.3E-123 160 5.4E-10 6.4E-12 1.4E-13 4.5E-15 5.4E-07 1.3E-11 5.2E-134 160 3.9E-05 3.2E-07 4.6E-07 1.2E-07 1.0E-03 1.4E-05 5.9E-074 320 3.4E-06 6.2E-08 9.8E-10 2.7E-12 2.4E-04 8.9E-07 3.7E-085 640 4.0E-05 1.6E-06 7.2E-08 2.7E-09 1.5E-03 8.9E-06 3.7E-075 1280 2.6E-06 2.9E-08 3.9E-10 5.6E-12 3.8E-04 5.5E-07 2.3E-086 1280 2.2E-06 8.8E-08 7.1E-09 8.9E-10 2.5E-04 5.8E-07 2.5E-086 2560 1.6E-07 1.9E-09 6.0E-11 9.2E-12 6.2E-05 3.7E-08 1.5E-09

From Table 1, we can see that the modified composite trapezoidal rule T(4)n is signifi-

cantly better than the others.

REFERENCES

1. Z. Chen, C. A. Micchelli, and Y. Xu, The Petro-Galerkin methods for second kindintegral equations II: Multiwavelet scheme,Adv. Comp. Math. 7(1997),199-233.

2. P. J. Davis, and P. Rabinowitz, Methods of numerical integration, (Second edition),Academic Press, New York, 1984.

3. X. Liang, M. Liu, X. Che,Solving second-kind integral equation by Galerkin methodwith continuous orthogonal wavelets, Joural of Computational and Applied Mathe-

matics 136(2001),149-161.

4. C. A. Micchelli, and Y.Xu, Weakly singular Fredholm integral equations I:Singularitypreserving wavelet-Galerkin methods, in Wavelets and Multilevel Approximation(C.K.Chuiand L.Schumaker, Eds), Wlod Sci. Pub. Co., 1995.

5. W. Sweldens, and R. Piessens, Quadrature formulae and asymptotic error expansionsfor wavelet approximations of smooth functions, SIAM J. Numer. Anal.31(1994),1240-1264.

7

QUADRATURE FORMULAS

425

426

Stability of a numerical algorithm for non-stationary transport

equation

Olga Martin Department of Mathematics

University “Politehnica” of Bucharest Splaiul Independentei 313,Bucharest 16

ROMANIA e-mail: [email protected]

Abstract. The initial-boundary value problem for one-dimensional linear transport equation with a source term is considered. This is rewritten as a Cauchy problem: dw /dt + Aw = F, wt = 0 = w0, where w represents a suitable subset of a Hilbert space, whose elements are pairs of real-valued functions depending on three variables: a space variable z∈[0, H], an angle variable ν, with µ = cos ν ∈ [-1, 1] and a time variable t∈[0, T]. A is a linear strictly positive operator. A difference scheme is given in order to approximate the space derivatives appearing in A. Then, the operator A is decomposed as A = A1 + A2 (where both A1 and A2 are positive operators) and another difference scheme is given to approximate the time derivatives. Finally, the numerical integration with respect to µ is carried out. One obtains an algorithm, which is stable and approximates the exact solution with an accuracy of second order in time step τ and in space step h. Key words: transport equation, difference scheme, Krank-Nicholson scheme, bicycle splitting-up method.

MSC : 35J99, 65N99.

1 Introduction The main problem in the nuclear physics is to find the neutrons distribution in the reactor, hence its density, ϕ. This is a scalar function, which is studied in a plan-parallel geometry and depends on the next variables: the position of the neutron on the z – axis, the neutron speed, vc,, which makes an angle ν with Oz and the time, t. The density is the solution of an integral – differential equation, named the neutron transport equation. Many authors paid attention to this problem, [1],[2],[4]–[7], but their papers are theoretical studies. In this paper we present a numerical algorithm in order to find the solution of a boundary value and initial value problem for the non – stationary transport equation. We prove that the iterative process is stable and the approximation of the solution with respect to time step, τ, is of the τ2 order. 2 Problem Formulation Let us consider a transport equation in a plan – parallel geometry:

),,(2

1 1

1

tzfdztv

s

c

µµϕσ

ϕσϕ

µϕ

+=⋅+∂∂

+∂∂

⋅ ∫−

(1)

with the following boundary conditions:

0, if 0

0,0 if 0

<==>==

µϕµϕ

Hz

z (2)

and the initial condition: .0 if 0 == tϕϕ (3)

427


2

In the right-hand side of (1), f is the radioactive source function, the functions σ , σs are continuous in the interval [0, H] and satisfy the conditions:

sccss σσσσσσσσσ −=≤<∞<′≤≤∞<≤≤<0

0;0;0 10

(4)

Further on, we consider for simplicity, vc = 1. Using the notations: ϕ + = ϕ (z, µ, t) ; ϕ - = ϕ (z, -µ, t), where µ > 0, (5) the equation (1) can be written in the form:

∫

∫

−−+−−−

+−++++

++=⋅+∂

∂−

∂∂

++=⋅+∂

∂+

∂∂

1

0

1

0

)(2

)(2

fdzt

fdzt

s

s

µϕϕσ

ϕσϕµϕ

µϕϕσ

ϕσϕµϕ

(6)

Substituting: µ’ = - µ > 0, we get:

∫∫∫∫−

−=′′−=′′−−=

1

0

1

0

0

1

0

1

),,(),,(),,( µϕµµϕµµϕµµϕ ddtzdtzdtz .

The boundary value problem becomes:

],0[],1,0[,0),,(;0),,0( TttHt ∈∀∈∀== −+ µµϕµϕ (7)

Adding and subtracting the equations (6) and introducing the notations:

)(2

1)(

2

1

)(2

1)(

2

1

−+−+

−+−+

+=−=

+=+=

ffrv

ffgu

ϕϕ

ϕϕ (8)

we obtain the following system:

.

1

0

rvz

u

t

v

gduuz

v

t

us

=⋅+∂∂

⋅+∂∂

+′=⋅+∂∂

⋅+∂∂

∫

σµ

µσσµ (9)

The boundary- initial conditions are: u + v = 0 for z = 0 u - v = 0 for z = H (10) and respectively: u = u0, v = v0 for t = 0. (11)

Now we rewrite the problem (9)-(11) in a operator form. For this purpose, we introduce the vector functions having two scalar components:

MARTIN

428

3

=

=

=

r

gF

v

uw

v

uw ,,

0

00 . (12)

and the operator

1

0

∂∂

∂∂′−

=∫

σµ

µµσσ

z

zd

As

(13)

Let us define in the measurable set D = [0, H]×[0, 1], a Hilbert space L2(D), (the functions quadratically integrable), with the scalar product:

∑ ∫∫=

=2

1 0

1

0

),,(),,())(),((i

Hii dztztzdtt µβµαµβα (14)

where α i, β i are the components of the vector functions α, β. Here, the scalar product is a function of time. For solving the problem (9)-(11), we consider that the vector function w is defined on the set [0,T] and has the values in the Hilbert space L2(D). The notation w(t) defines an element of L2(D), which corresponds to a function (z, µ) → w(z,µ,t) with t fixed.

Let us consider Φ, the set of functions w(t), which have the components u, v and

z

v

z

u

∂∂

∂∂

, continuous on D. Then, the domain of definition D(A) = Φ0 is the subset of Φ with the

elements w that verify conditions (10) and have t

w

∂∂

continuous on D.

Let us now define the operator

At

L +∂∂

= (15)

with the domain D(L) = Φ0. Consequently, the problem (9)-(11) becomes:

]1,0[],0[],,0[),,(, ×=×∈=+∂∂

HDTDtzFAwt

w µ (16)

Dzww t ∈∀== ),(,00 µ (17)

where

.)(,]),,0[( 00

2 Φ∈Φ∈×∈ twwTDLF

We prove that A is a positive operator, namely, (Aw, w) > 0 for each w ≠ 0, w∈ Φ0. We have

dzvz

uv

z

vuuduudwAw

H

s∫ ∫∫

+

∂∂

+∂∂

+′−=0

21

0

21

0

),( σµµµσσµ (18)

Using the Hölder inequality we obtain

∫∫∫∫ =

≤

⋅

1

0

21

0

21

0

21

0

11 µµµµ dududdu

Finally, for σs ≤ σ we get

STABILITY OF A NUMERICAL ALGORITHM

429

4

( ) ( ) 0)()(4

1)()(

4

1

)(),(

1

0

220

1

0

22

0

21

00

21

0

21

0

>+=−≥

≥

∂∂

++

−

≥

∫∫

∫∫∫ ∫∫

−+−+ µϕϕµµϕϕµ

µσµµµσ

dd

dzuvz

vddzduduwAw

H

HH

(19)

according with (7). If the operator L is positive, the equation: Lw = F has only one solution. Indeed, let w1 be an element such that Lw1 = F Hence, L(w – w1) = 0 ⇒ 0),(0 =⇒= wwLwL . But the operator L is positive definite, such that

0=w ⇒ w = w1.

In order to get a solution of the problem (16)-(17), we go through three stages. First, a difference scheme is given in order to approximate the space derivatives which appearing in A. We consider on z – axis two points systems: - a principal system, zkk, k ∈0,1,...,N with z0 = 0 and zN = H; - a secondary system,zk+1/2k, k ∈0,1,...,N-1, which verifies the inequality: zk -1/2 < zk < zk+1/2. Integrating the first equation (9) on the intervals: ( z0, z1/2 ), ( zk - 1/2, zk+1/2 ), k∈1, 2,…, N-1, ( zN - 1/2, zN ) and the second equation on ( zk - 1, zk ), k∈1, 2,…, N-1, the system can be written in the form:

∫∫∫∫

∫∫ ∫∫∫∫

=+∂∂

+∂∂

+=+∂∂

+∂∂

1

0

1

0

1

0

1

0

2/1

0

2/1

0

2/1

0

2/1

0

2/1

0

'1

0

z

z

z

z

z

z

z

z

z

z

z

z

s

z

z

z

z

z

z

dzrdzvdzz

udzv

t

dzgdudzdzudzz

vdzu

t

σµ

µσσµ

∫∫∫∫

∫∫ ∫∫∫∫

++++

+

−

+

−

+

−

+

−

+

−

=+∂∂

+∂∂

+=+∂∂

+∂∂

1111

2/1

2/1

2/1

2/1

2/1

2/1

2/1

2/1

2/1

2/1

'

.....................................................................................1

0

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

z

z

z

z

z

z

z

z

z

z

z

z

s

z

z

z

z

z

z

dzrdzvdzz

udzv

t

dzgdudzdzudzz

vdzu

t

σµ

µσσµ (20)

∫∫∫∫

∫∫ ∫∫∫∫

−−−−

−−−−−

=+∂∂

+∂∂

+=+∂∂

+∂∂

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

z

z

z

z

z

z

z

z

z

z

z

z

s

z

z

z

z

z

z

dzrdzvdzz

udzv

t

dzgdudzdzudzz

vdzu

t

1111

2/12/12/12/12/1

'

.......................................................................................1

0

σµ

µσσµ

With the following notations:

Nkzzzzzz

Nkzzzzzz

kkkNNN

kkk

,...,2,1,,

1,..,2,1,,

12/12/1

2/12/102/10

=−=∆−=∆−=−=∆−=∆

−−−

−+ (21)

we define the mean values:

MARTIN

430

5

∫∫∫+

−

++

−∆

=∆

=∆

=+

+

2/1

2/1

12/1

2/1

1,

1,

1

2/12/1

k

k

k

k

k

k

k

z

z

s

k

s

z

zk

k

z

zk

k dzz

dzz

dzz

σσσσσσ

∫∫++

− ++ ∆

=∆

=12/1

2/1 2/12/1

1,

1 k

k

k

k

z

zk

k

z

zk

k dzz

rdzgz

g σ (22)

Let ),,,(max 2/1010

+−≤≤

∆∆∆∆= kkNNk

zzzzh . Dividing the first equation (20) by ∆ z0 and the

second by ∆ z1/2 we obtain:

[ ]

[ ] ∫∫∫

∫∫ ∫

∫∫

∆=

∆+

∆+

∆∂∂

∆+′

∆=

=∆

+∆

+

∆∂∂

1

0

1

0

1

0

1

0

2/1

0

2/1

0

2/1

0

2/1

0

2/1

0

2/12/12/12/1

0

1

0 0

000

1111

11

),,()(111

z

z

z

z

z

z

z

z

z

z

z

zs

z

z

z

z

z

z

dzrz

dzvz

uz

dzvzt

dzgz

ddzuz

dztzuzz

vz

dzuzt

σµ

µσ

µσµ

(23)

In accordance with the boundary conditions:

( ) −=

−+= −=−== ϕϕϕ

2

1

2

1000 zzvv (24)

( ) −=

−+= =+== ϕϕϕ

2

1

2

1000 zzuu (25)

we have: v0 = - u0 and the relations (23) can be rewritten in the form:

2/12/12/12/1

012/1

000

1

0000

0

02/102/1

0

0

1,

rvz

uu

t

v

gdzz

ggduuz

uv

t

u z

z

s

=+∆

−+

∂∂

∆++′=+

∆+

+∂

∂∫∫

σµ

µσσµ (26)

where the functions u, v are replaced by their values in the points: z = 0, z = 1/2 , z = 1. Similarly, we get

1,...,2,1,2/12/12/12/1

12/1

1

0

2/12/1

−==+∆

−+

∂∂

+′=+∆

−+

∂∂

++++

++

−+∫

Nkrvz

uu

t

v

gduuz

vv

t

u

kkk

k

kkk

kkskk

k

kkk

k

σµ

µσσµ (27)

2/12/12/12/1

12/1−−−

−

−− =+∆

−+

∂∂

NNN

N

NNN rvz

uu

t

vσµ (28)

∫∫−

∆=+′=+

∆−

+∂

∂ −N

N

N

z

zN

NNNsNN

N

NNN dzgz

ggduuz

vu

t

u

2/1

1,

1

0

2/1 µσσµ

where uN = vN.


431

6

Let us consider M(0, 2N), the Hilbert space of the vector functions α = ( α0, α1/2, α1,… αN ) with the scalar product:

µβαβα dz i

N

iii 2/

2

0

1

02/2/),( ∑∫

=∆= (29)

and the norm: )2,0(,),( NM h∈= αααα . (30)

We define the vector functions:

),,.,....,,,(

),,,....,,,(

),,,....,,,(

002/1

01

01

02/1

00

0

2/1112/10

2/1112/10

NNN

NNN

NNN

uvuuvu

grggrgf

uvuuvu

−−

−−

−−

=

==

ϕ

ϕ (31)

and the operator: A = L – S, where

+∆∆

−

∆

∆∆−

+∆

=

−−

N

NN

N

N

zz

z

zz

z

L

σµµ

µσ

µσµσµσµ

L

L

LLLLLLL

L

L

0000

0000

000

0000

2/12/1

2/12/1

2/1

00

0

(32)

==

= ∫ number. rational is 2/ if0

numberentier is 2/ if1;2,...,1,0, 2/

1

02/2/ i

iNiddiagS iisi

γµγσ (33)

Then, the system (26) – (28) has the form:

0)0,,(

],0[,

ϕµϕ

ϕϕ

=

∈=+

z

TtfAtd

d

(34)

where ϕ, f, ϕ 0 were defined by (31). In spite of the fact that we use, for simplicity of the writing, the same notation as (1) – (3), here ϕ is the numerical solution for our problem.

It is seen that the operator S is positive and we shall prove that L and A are positive operators. Indeed, let w∈ M and then

( )

0

)(

,

20

22/

2

1

1

02/0

22/1

2

1

1

02/2/

1

0

220

1

0

1

1

12/1

1

02/12/1

2/1

12/1

1

1

1

0

2/12/1

2/1

1

000

2/1

012/10

1

000

0

02/10

>=∆>

>∆++=

+

∆+

∆+

+

+

∆−

∆+

+

∆+

∆+

+

+

∆−

∆+

+

∆+

∆=

∑ ∫

∑ ∫∫∫

∑ ∫∑ ∫

∫∫

=

=

−

−

=+++

+

++

−

=

−+

wdwz

dwzdwwdwwz

wwz

dwwz

wwzdww

z

wwz

dwwz

wwzdww

z

wwzwLw

i

N

ii

N

iiiNNNN

N

NNN

N

iiii

i

iii

N

iiii

i

iii

σµσ

µσµµµσµ

µσµµσµ

µσµµσµ

MARTIN

432

7

because continuous functions on [0,H],σ i, are bounded on [0,H] and by hypothesis σ ≥ σ 0 > 0. Using above results, we obtain

22

2/2/2/

2

0

1

02/

1

0

220

1

02/2/2/

2

0

1

02/

22/2/2/

1

0

220

0)()(

)()(),(

wdwzdww

ddwwwzdwwwAw

ciis

N

iiN

iii

N

isiiiN

i

i

σµσσµµ

µµγσσµµ

≥∆−++≥

≥′−∆++=

∑ ∫∫

∫∑ ∫∫

=

=

In the second stage a difference scheme is given to approximate the time derivatives. This is used together with the bicycle splitting-up method, which writes the operator A as a sum:

A = A1 + A2 (35)

where

∆∆−

∆∆−

∆∆−

∆∆

=

−−

NN

NN

zz

zz

zz

zz

A

µµ

µµ

µµ

µµ

0000

0000

0000

0000

2/12/1

2/12/1

00

1

L

L

MMMLMMM

L

L

and

−= ∫

1

02/2/2/2 µγσσ ddiagA isi i

.

In the following, we shall prove that A1 is a positive operator. If α = ( α0, α1/2, α1,… αN )∈ Mh, we have

( )

0)(

)(,

21

0

20

1

02/1

2/1

1

0 2/11

2/12/1

2/1

1

01

2/10

2/12/1

0

1

02/1

00

00

2

0

1

02/2/12/1

>+=

∆+

∆−

∆+

+

∆+

∆−

∆+

+

∆+

∆−

∆+

+

∆+

∆∆=∆=

∫∫

∫

∫

∫∑ ∫

−

−−

−−

−

=

µααµµααµαµ

µααµαµ

µααµαµ

µααµαµµαααα

ddzz

z

dzz

z

dzz

z

dzz

zdAzA

NNNN

NN

N

NNN

NN

N

N

iiii

L

L

We shall also prove that A2 is a positive definite operator. Indeed,

0)()(1

0

22/

22/

1

02/

1

02/2/2/

1

0

22/2/ 02/2/

>≥−≥′− ∫∫∫∫ µασµασσµµγαασασ dddd icisiiiisii ii


433

8

In these conditions we can apply the bicycle splitting-up method. Let us divide the close interval [0, T] into n subintervals by choosing points: t0 = 0, t1, …., tn =T. Next, we take an arbitrary subinterval: [tj-1, tj+1] = [ tj-1, tj-1/2 ]∪ [ tj-1/2, tj ]∪ [ tj, tj+1/2 ]∪[ tj+1/2, tj+1 ], which has the length equal to 4τ, where τ is the time step. Approximating the operators A1, A2 on this the subinterval by: 2,1),( ==Λ ktA jk

jk , we

shall obtain from (34) a difference system using the Krank-Nicholson scheme, [5]:

02

12/1

1

12/1

=+

Λ+− −−−− jj

jjj ϕϕ

τϕϕ

(36)

jjjjj

jjj

ff 2

2/1

2

2/1

22Λ+=

+Λ+

− −− τϕϕτϕϕ

(37)

jjjjj

jjj

ff 2

2/1

2

2/1

22Λ−=

+Λ+

− ++ τϕϕτ

ϕϕ (38)

02

2/11

1

2/11

=+

Λ+− ++++ jj

jjj ϕϕ

τϕϕ

(39)

with f j= f(tj). Adding (37) and (38) we get

j

jjj

jjj

f=

++

Λ+−

+−

−+

22

2

2/12/1

2

2/12/1ϕϕϕ

τϕϕ

(40)

and the system (36)-(39) can be rewritten:

2/11

11

22/1

2

2/122

11

2/11

22

)(22

2)(

2

22

++

+

−

−−

Λ−=

Λ+

+

Λ−=

Λ+

Λ−=−

Λ+

Λ−=

Λ+

jjjj

jjjjj

jjjjj

jjjj

EE

fEE

EfE

EE

ϕτϕτ

τϕτϕτ

ϕττϕτ

ϕτϕτ

(41)

where E is unit matrix. Let us define:

)2

()2

( 1 jk

jk

jk EET Λ−Λ+= − ττ

(42)

and from (41) we have:

jjjjj

jjjj

jjj

fTT

Tf

T

τϕϕ

ϕτϕ

ϕϕ

222/1

2/12

11

2/1

+=

+=

=

+

−

−−

jjjjjjjjjjjjjjjj

jjjjjjjjj

fTTTfTTTTTTfTT

fTTTTT

211

211

122121

21212/1

11

2τϕτϕτ

τϕϕϕ

+=++=

=+==−−

++

(43)

where jjjjj TTTTT 1221= .

Now we prove that the algorithm is stable and leads to a numerical solution, which approximates the exact solution with an accuracy of second order in time step τ.

MARTIN

434

9

Approximation

For the estimation of approximation order, we shall expand with respect to the power of small τ, the expression

...)(222

221

j

k

j

k

j

k

j

k

j

k EEET Λ+Λ−=

Λ−

Λ+=− ττττ

when .2,1,12

=<Λ kjk

τ Then

( ) )()(2

31221

2

21

2

21 τττ oETTjjjjjjjjj +

ΛΛ−ΛΛ+Λ+Λ+Λ−=

where 2/)( 21jjj Λ+Λ=Λ . If jjjj

1221 ΛΛ=ΛΛ , we get

)()(2

322

21 τττ oETT jjjj +Λ+Λ−=

When the operators are non-commutative, the approximation with the splitting-up algorithm is

of the first order with respect to τ. Let us now consider

)()(2

)2(2 32

2

1221

1

2

2

1τττ oETTTTTTT jjjjjj

k

jk

k

jk

j +Λ+Λ−=== ∏∏==

Hence, the following estimation is valid in the interval [tj-1, tj+1]:

)()(2)(2

)2(2 312

21 τττϕττϕ ofEE jjjjjj +Λ−+

Λ+Λ−= −+

and

)()()(2

21

111

ττϕττϕϕ

OfEE jjjjjjj

+Λ−=Λ−Λ+− −

−+

(44)

Using the Taylor series expansion of the solution ϕ in the neighborhood of the point tj-1 and

substituting tj for t, we can write:

)(),,( 2

1

1 ττϕϕµϕϕ ot

tz

j

jj

j +

∂∂

+==−

− (45)

Then, we eliminate 1−

∂∂

j

t

ϕ, writing the transport equation (35) in the point tj-1 in the form:

)(21

1

τϕϕof

t

jjj

j

++Λ−=

∂∂ −

−

and (45) becomes:

)()( 21 τττϕϕ ofE jjjj ++Λ−= − .

Finally, we get )(2

21

τϕτ

ϕϕof jjj

jj

+=Λ+−+

(46)


435

10

This relation is an approximation with the accuracy of second order in time step τ of the initial equation (35) on the interval [tj-1, tj+1].

Stability

The algorithm: ,on )0,,(],,0[on 01 DzTDfT jjjj ϕµϕτϕϕ =×+=+ is stable, if for any step h,

which characterizes the first difference scheme, and for any j ≤ T /τ , τ - the time step, we have the inequality:

τ

ϕϕ,

20

1hhh

fCCj

ΦΦΦ+≤ (47)

where C1, C2 are the independent constants of h,τ, ϕ 0, f and Φh,,Φh,τ are the systems of these reticular functions.

Turning to the formula (43) we get:

],[for 121211

++ ∈+= jj

jjjjjjj tttfTTTT τϕϕ (48)

First, we prove that:

11 )2

)(2

()2

()2

( −− Λ+Λ−=Λ−Λ+= jk

jk

jk

jk

jk EEEET

ττττ (49)

using the identity: )2

()2

( 1 jk

jk EE Λ+Λ+ − ττ

= E. Indeed, multiplying to the left with

)2

()2

( 1 jk

jk EE Λ−Λ+ − ττ

and using the next commutative property:

)2

)(2

()2

)(2

( jk

jk

jk

jk EEEE Λ−Λ+=Λ+Λ−

ττττ (50)

we obtain (49). Then, we have

)2

)(2

()2

)(2

(

)2

()2

)(2

()2

(

122

111

21

211

121

−−

−−

Λ+Λ−Λ+Λ−=

=Λ−Λ+Λ−Λ+=

jjjj

jjjjjj

EEEE

EEEETT

ττττ

ττττ

(51)

Now, we make use of the Kellog theorem,[5]: if jk

Λ is a positive definite operator and τ /2 ≥ 0, then we

have:

1)2

)(2

( 1 ≤Λ+Λ− −jk

jk EE

ττ (52)

In accordance with (51) and (52) we get:

111)2

)(2

()2

)(2

(

)2

()2

)(2

()2

(

122

111

21

211

121

=⋅≤Λ+Λ−⋅Λ+Λ−≤

≤Λ−Λ+Λ−Λ+=

−−

−−

jjjj

jjjj

EEEE

EEEETT

ττττ

ττττ

(53)

Then, the equality (48) becomes:

MARTIN

436

11

( ) jjjjjjjjjjjjj ffTTfTTTT τϕτϕτϕϕ +≤+⋅≤+≤+212121

1 (54)

Using the recurrence formula (54) we obtain:

fTgff jjjjj ⋅+≤≤+≤+≤ −+ .......211 τϕτϕϕ (55)

where j

jff max= and T is the total length of the time interval. Hence, a sufficient condition that

the algorithm to be stable is the following:

njTj

k ,...,2,1,1 =∀≤ (56)

which is always true when we can applied the Kellog theorem. We shall now summarize the above results in the following theorem. Theorem

Suppose the following conditions hold: (1) the solution ϕ of the problem (34) has bounded derivatives with respect to t up to the second

order; (2) the operators A1 and A2 are positive definite;

(3) ,12

<Λ jk

τ where τ is time step.

Then the numerical algorithm (41) is stable and approximates the exact solution with an accuracy of second order in τ. In the practical application, we use instead of the system (36) – (39) on the interval [tj-1, tj+1],the following splitting-up method:

02

3/21

1

13/2

=+

Λ+− −−−− jj

jjj ϕϕ

τϕϕ

02

3/13/2

2

3/23/1

=+

Λ+− −−−− jj

jjj ϕϕ

τϕϕ

jjj

f=− −−

τϕϕ

2

13/2

(57)

02

02

3/21

1

3/21

3/13/2

2

3/13/2

=+

Λ+−

=+

Λ+−

++++

++++

jjj

jj

jjj

jj

ϕϕτϕϕ

ϕϕτ

ϕϕ

or in other form:

3/21

1

3/12

3/2

3/13/1

3/22

3/1

111

1

13/2

2

22

++

++

−+

−−

−−

−

=

=

+=

=

=

Λ−

Λ+=

jjj

jjj

jjj

jjj

jjjjj

T

T

f

T

TEE

ϕϕ

ϕϕ

τϕϕ

ϕϕ

ϕττϕ

(58)


437

12

Similarly, we obtain for 0)(,0)( 2211 ≥=Λ≥=Λ jj

jj tAtA and the recurrence formula

jjjjjjjjj fTTTTTT 211

12211 2τϕϕ += −+

that the schema (57) is stable. At the beginning, for determining the solution of the system (57), we consider the first equation, for a fixed µ, and the operators (32), (33). We get:

=

⋅

+

−

−−

−

−

−

∆∆−

∆∆−

∆−

∆∆−

∆∆+

−−3/2

3/22/1

3/21

3/22/1

3/20

22

22

12

21

2

221

10...000

1...000

000...0

000...

000...0

2/12/1

1

2/12/1

0 0

jN

jN

j

j

j

u

v

u

v

u

NN

NN

zz

zz

z

zz

zz

MMMMMMMM

µτµτ

µτµτ

µτ

µτµτ

µτµτ

⋅

−

=

−

−−

−

−

−

∆∆

∆−

∆

∆

∆−

∆

∆−

∆−

−−1

12/1

11

12/1

10

22

22

12

21

2

221

10...000

1...000

000...0

000...

000...0

2/12/1

1

2/12/1

0 0

jN

jN

j

j

j

u

v

u

v

u

NN

NN

zz

zz

z

zz

zz

MMMMMMMM

µτµτ

µτµτ

µτ

µτµτ

µτµτ

.

We obtain an analogous relation from the last equation. For the second and the fourth equation we have:

=

⋅

−

−

−

−

∫−+

∫−+

+

∫−+

3/1

3/11

3/12/1

3/10

1

0

1

011

2/1

1

000

2

2

21

2

1000

0100

000

0001

jN

j

j

j

u

u

v

u

d

d

d

sNN

s

s

M

L

MMMMM

L

L

L

µσστ

µσστ

στ

µσστ

MARTIN

438

13

⋅

=

−

−

−

−

∫−−

∫−−

−

∫−−

3/2

3/21

3/22/1

3/20

1

0

1

011

2/1

1

000

2

2

21

2

1000

0100

000

0001

jN

j

j

j

u

u

v

u

d

d

d

sNN

s

s

M

L

MMMMM

L

L

L

µσστ

µσστ

στ

µσστ

We obtain the following relations for the numerical solution:

3/22

1

23/1

22−

−−

−

+= jjjj AEAE ϕττϕ

Elements of the product are:

Nidji

siji

i

ic

ji ,...,1,0,

221

21

1 1

0

3/23/23/1 =

+

−+

= ∫−−− µϕ

τσϕ

τστσ

ϕ (59)

Niji

i

iji

,...,2,1,2/1

2/1 3/2

2/1

2/13/12/1 =⋅

+−

= −

−

−−− ϕ

τστσ

ϕ (60)

Analogously, we have

Nidji

siji

i

ic

ji ,...,1,0,

221

21

1 1

0

3/13/13/2 =

+

−+

= ∫+++ µϕ

τσϕ

τστσ

ϕ (61)

Nij

i

i

ij

i ,...,2,1,2/1

2/1 3/12/1

2/1

2/13/22/1 =⋅

+−

= +−

−

−+− ϕ

τστσ

ϕ (62)

At the third stage, we consider the points: µ0 = 0, µ1, ..., µm = 1, in the interval [0, 1] and compute the integrals with respect to µ, using a numerical integration (trapezoidal approximation):

)(,)(1

1

0ll

m

lllSd µψψψµµψ =≈∑∫

= (63)

Then, the system (57) can be written in the form:


439

14

3/2,1

1,1

3/1,2

3/2,2

3/13/1

3/2,2

3/1,2

1,1

3/2,1

)2

()2

(

)2

()2

(

2

)2

()2

(

)2

()2

(

++

++

−+

−−

−−

−=+

−=+

+=

−=+

−=+

jl

jl

jl

jl

j

l

j

l

j

l

j

l

jl

jl

jl

jl

jl

jl

jl

j

l

j

l

j

l

j

l

AEAE

AEAE

f

AEAE

AEAE

ϕτϕτ

ϕτϕττϕϕ

ϕτϕτ

ϕτϕτ

(64)

In this choice of the steps, which correspond to the variables z, t , we use the condition: )(min 2/i

iz∆≤τ (65)

3 Numerical Example We wish to find the solution of the problem:

]2,0[]1,0[]4,0[),,(),,,(),,(),,(

××∈=+ tztzftzAtd

tzd µµµϕµϕ

(66)

),()0,,( 0 µϕµϕ zz = (67)

Considering the partition of [0,4] into four subintervals of equal length by points: z0 = 0 < z1/2 < z1 < z3/2 < z2 = 4 with: ∆ z0 = z1/2 - z0 = 1; ∆ z1/2 = z1 - z0 = 2; ∆ z1 = z3/2 - z1/2 = 2;

∆ z3/2 = z2 - z1 = 2; ∆ z2 = z2 - z3/2 = 1. The partition of the interval [0, 1] is: µ0 = 0 < µ1 = 1/2 < µ2 = 1.

For the variable t, we consider the regular partition of the interval [0,2] by the points: t0 = 0 < t1/3 < t2/3 < t1 < t4/3 < t5/3 < t2 = 2. The initial value problem is defined by: ( ) ( )1,1,1,1,1,,,, 0

20

2/301

02/1

00

0 == uvuvuϕ (68)

The functions σ (z), σs(z) and f, which here depends only of µ are defined with the help of the table 1. The values of 2,2/3,1,2/1,0, ∈iiϕ with respect to µl and tj are presented in table 2.

Table 1

z 0 1 1/2 3/2 2 σ(z) 1 1.4 1.65 1.76 1.9 σs(z) 0.9 0.95 1 0.8 0.7

From the relations (8) and using the mean values for u1/2, v1, u3/2 we obtain the density, ϕ +, for µ > 0 and the density, ϕ -, for µ < 0 for each value of zi and tj :

MARTIN

440

15

2/321

2/32/11

212/3

2/1102/32/1

1

010

2/1

3

2),,2/3(0),,0(,2),,2(

2),,1(

3

2),,2/3(

3

2),,2/1(

2),,1(

0),,2(,2),,0( 3

2),,2/1(

vuu

ttut

vvut

uuvt

vuu

tvv

ut

tutuu

vt

N −+⋅

===

+−=

+⋅+=

−⋅+

=+

+=

=−=−⋅+

+=

−++

−+

−+

−−+

µϕµϕµϕ

µϕµϕ

µϕµϕ

µϕµϕµϕ

Table 2 ϕ t = 1/3 t = 2/3 t = 4/3 t = 5/3 t = 2

µ = 0 µ =

1/2 µ = 1 µ = 0 µ =

1/2 µ = 1 µ = 0 µ =

1/2 µ = 1 µ =

1/2 µ = 0 µ = 1 µ = 0 µ =

1/2 µ = 1

u0 1 0.69 0.44 0.92 0.67 0.46 1.25 0.97 0.69 1.17 0.94 0.71 1.17 0.73 0.38

v1/2 1 0.98 0.95 0.62 0.62 0.59 0.95 0.9 0.82 0.59 0.56 0.5 0.59 0.54 0.43

u1 1 0.99 0.99 0.87 0.85 0.87 1.2 1.17 1.09 1.05 1.03 0.98 1.05 1.04 0.98

v3/2 1 1. 1. 0.56 0.56 0.56 0.89 0.86 0.79 0.5 0.48 0.44 0.5 0.5 0.5

u2 1 1. 1. 0.69 0.69 0.69 0.99 0.99 0.92 0.68 0.68 0.64 0.68 0.65 0.59

Table 3 ϕ + t = 1/3 t = 2/3 t = 4/3 t = 5/3 t = 2

µ = 0 µ =

1/2 µ = 1 µ = 0 µ =

1/2 µ = 1 µ = 0 µ =

1/2 µ = 1 µ = 0 µ =

1/2 µ = 1 µ = 0 µ =

1/2 µ = 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1/2 2 1.88 1.76 1.52 1.4 1.32 2.17 2 1.77 1.7 1.56 1.35 1.7 1.47 1.2

1 2 1.99 1.97 1.5 1.44 1.44 2.12 2.05 1.9 1.6 1.55 1.45 1.6 1.6 1.45

3/2 2 2 2 1.37 1.36 1.37 2 1.97 1.83 1.4 1.39 1.31 1.4 1.4 1.35

2 2 2 2 1.38 1.38 1.38 1.98 1.98 1.84 1.36 1.36 1.3 1.36 1.3 1.2

Table 4 ϕ - t = 1/3 t = 2/3 t = 4/3 t = 5/3 t = 2

µ = 0 µ =

1/2 µ = 1 µ = 0 µ =

1/2 µ = 1 µ = 0 µ =

1/2 µ = 1 µ = 0 µ =

1/2 µ = 1 µ = 0 µ =

1/2 µ = 1

0 2 1.4 0.87 1.85 1.34 0.92 2.5 1.94 1.38 2.34 1.88 1.42 2.34 1.46 1.96

1/2 0 0 0 0.27 0.18 0.14 0.27 0.2 0.14 0.5 0.44 0.39 0.5 0.29 0.35

1 0 0.4 0.02 0.28 0.26 0.29 0.3 0.28 0.29 0.51 0.51 0.51 0.51 0.52 0.52

3/2 0 0.06 0 0.25 0.24 0.24 0.24 0.25 0.25 0.41 0.43 0.43 0.42 0.4 0.35

2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

We mention that, the numerical values of ϕ + and ϕ - were obtained for the neutron speed, vc =1. Also, in accordance with [3], we multiply with a factor 2 the number of the particles, which pass through a surface with unitary area. Finally, to obtain the true values of ϕ + and ϕ -, we will multiply these with 2vc.

The results of this numerical example prove its practical importance: how depends the density in a point z at the time t for different values of angle ν.

References

[1] K. M. Case and P. F. Zweifel: Linear Transport Theory, Addison-Wesley, Massachusetts, 1967. [2] W. R. Davis: Classical Fields, Particles and the Theory of Relativity , Gordon and Breach,


441

16

New York, 1970. [3] R. Feynman, Lectures on physics, Addison-Wesley, Massachusetts, 1969. [4] S. Glasstone and C. Kilton : The Elements of Nuclear Reactors Theory, Van Nostrand, Toronto – New York – London, 1982. [5] G. Marchouk : Méthodes de calcul numérique, Édition MIR de Moscou, 1980. [6] G. Marchouk and V. Shaydourov : Raffinementdes solutions des schémas aux différences, Édition MIR de Moscou, 1983. [7] G. Marciuk and V. Lebedev : Cislennie metodî v teorii perenosa neitronov, Atomizdat, Moscova, 1971. [8] O. Martin : A numerical solution of a two-dimensional transport equation, Central European

Journal of Mathematics, Vol. 2, No. 2, (2004), pp. 191 – 198. [9] N. Mihailescu : Oscillations in the power distribution in a reactor, Rev. Nuclear Energy,Vol.9, No.1-4, (1998), pp.37-41. [10] E. Lewis, W. Miller, Computational Methods of Neutron Transport, Am. Nucl. Soc., New York, 1993. [11] H. Pilkuhn : Relativistic Particle Physics, Springer Verlag, New York - Heidelberg-Berlin, 1980. [12] A. Yamamoto, Y. Kitamura, T. Ushio, N. Sugimura, Convergence improvement of Coarse Mesh

Rebalance Method for Neutron Transport Calculations, Journal of Nuclear Science and Technology, vol.41, nr.8, 781-789, 2004.

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442

Asymptotic distribution of the sample averagevalue-at-risk in the case of heavy-tailed returns

Stoyan V. StoyanovChief Financial Researcher,

FinAnalytica Inc., Seattle, USA


Svetlozar T. Rachev∗


University of Karlsruhe, D-76128 Karlsruhe, Germany and

Department of Statistics and Applied Probability,

University of California Santa Barbara, CA 93106, USA


November 6, 2007

∗Rachev gratefully acknowledges research support by grants from Division of Mathe-matical, Life and Physical Sciences, College of Letters and Science, University of California,Santa Barbara, the Deutschen Forschungsgemeinschaft and the Deutscher AkademischerAustausch Dienst.

1

443


Abstract

In this paper, we provide a stable limit theorem for the asymptoticdistribution of the sample average value-at-risk when the distributionof the underlying random variable X describing portfolio returns isheavy-tailed. We illustrate the convergence rate in the limit theoremassuming that X has a stable Paretian distribution and Student’s tdistribution.

Keywords average value-at-risk, risk measures, heavy-tails, asymptoticdistribution, Monte Carlo

1 Introduction

The average value-at-risk (AVaR) risk measure has been proposed in theliterature as a coherent alternative to the industry standard Value-at-Risk(VaR), see Artzner et al. (1998) and Pflug (2000). It has been demonstratedthat it has better properties than VaR for the purposes of risk managementand, being a downside risk-measure, it is superior to the classical standarddeviation and can be adopted in a portfolio optimization framework, seeRachev et al. (2006), Stoyanov et al. (2007), Biglova et al. (2004), and Rachevet al. (2008).

The AVaR of a random variable X at tail probability ε is defined as

AV aRε(X) = −1

ε

∫ ε

0

F−1(p)dp.

where F−1(x) is the inverse of the cumulative distribution function (c.d.f.)of the random variable X. The random variable may describe the return ofstock, for example. A practical problem of computing portfolio AVaR is thatusually we do not know explicitly the c.d.f. of portfolio returns. In orderto solve this practical problem, the Monte Carlo method is employed. Thereturns of the portfolio constituents are simulated and then the returns ofthe portfolio are calculated. In effect, we have a sample from the portfolioreturn distribution which we can use to estimate AVaR. The sample AVaRequals,

AV aRε(X) = −1

ε

∫ ε

0

F−1n (p)dp.

where F−1n (p) denotes the inverse of the sample c.d.f. Fn(x) = 1

n

∑ni=1 IXi ≤

x in which IA denotes the indicator function of the event A, and X1, . . . , Xn

STOYANOV-RACHEV

444

is a sample of independent, identically distributed (i.i.d.) copies of a randomvariable X.

Under a very general regularity condition, the larger the sample, the closerthe estimate to the true value. Suppose that E max(−X, 0) < ∞. Then, itis easy to demonstrate that the following relation holds,

E max(−X, 0) < ∞ ⇐⇒ AV aRε(X) < ∞.

Thus, by the strong law of large numbers, the condition E max(−X, 0) < ∞is necessary and sufficient for the almost sure convergence of the sampleAVaR to the true one,

AV aRε(X)a.s.−→ AV aRε(X) as n →∞. (1)

However, with any finite sample, the sample AVaR will fluctuate aboutthe true value and, having only a sample estimate, we have to know theprobability distribution of the sample AVaR in order to build a confidenceinterval for the true value. The problem of computing the distribution ofthe sample AVaR is a complicated one even if we know the distribution ofX. From a practical viewpoint, X describes portfolio return which can be acomplicated function of the joint distribution of the risk drivers. Therefore,we can only rely on large sample theory to gain insight into the fluctuationsof sample AVaR. That is, for a large n, we can use a limiting distributionto calculate a confidence interval. In this respect, a limit theorem for thedistribution of the sample AVaR can be regarded as a way to describe thespeed of convergence in (1).

Concerning the finite sample properties, the estimator AV aRε(X) has anegative bias,

AV aRε(X) ≤ AV aRε(X).

The asymptotic bias is of order O(n−1) and we consider it negligible for thepurposes of our study. For further details, see Trindade et al. (2007).

In this paper, we discuss the asymptotic distribution of the sample AVaRassuming that the random variable X can be heavy-tailed and may have aninfinite second moment. In such a case, we cannot take advantage of theclassical Central Limit Theorem (CLT) to establish a limit theorem. Forthis reason, we resort to the Generalized CLT and the characterization ofthe domains of attraction of stable distributions which appear as limitingdistribution in it.

Stable distributions are introduced by their characteristic functions. Therandom variable Z is said to have a stable distribution if its characteristicfunction ϕ(t) = EeitZ has the form

2

ASYMPTOTIC DISTRIBUTION OF SAMPLE AVERAGE

445

ϕ(t) =

exp−σα|t|α(1− iβ t

|t| tan(πα2

)) + iµt, α 6= 1

exp−σ|t|(1 + iβ 2π

t|t| ln(|t|)) + iµt, α = 1

(2)

and is denoted by Z ∈ Sα(σ, β, µ). The parameter α ∈ (0, 2] is called thetail index and governs the tail behavior and the kurtosis of the distribution.Smaller α indicates heavier tails and higher kurtosis. If α < 2, then Z hasinfinite variance. If 1 < α ≤ 2, then Z has finite mean and the AVaR of Zcan be calculated. The Gaussian distribution appears as a stable distributionwith α = 2. The stable distributions with α < 2 are referred to as stableParetian distributions. The parameter β ∈ [−1, 1] is a skewness parameter.If β = 0, the distribution is symmetric with respect to µ. Positive β indicatesthat the distribution is skewed to the right and negative β indicates that thedistribution is skewed to the left. The parameter σ > 0 is a scale parameterand µ ∈ R is a location parameter.

The notion of slowly varying functions is extensively used in the paper. Apositive function L(x) is said to be slowly varying at infinity if the followinglimit relation is satisfied,

limx→∞

L(tx)

L(x)= 1, ∀t > 0. (3)

The main result concerning the domains of attraction of stable distribu-tions is given in the following theorem.

Theorem 1. Let X1, . . . , Xn be i.i.d. with c.d.f. F (x). There existan > 0, bn ∈ R, n = 1, 2, . . ., such that the distribution of

a−1n [(X1 + . . . + Xn)− bn]

converges as n →∞ to Sα(1, β, 0) if and only if both

(i) xα[1− F (x) + F (−x)] = L(x) is slowly varying at infinity.

(ii)1− F (x)− F (−x)

1− F (x) + F (−x)→ β as x →∞

The an must satisfy

limn→∞

nL(an)

aαn

=

(Γ(1− α) cos(πα/2))−1 if 0 < α < 1,2/π if α = 1,(

Γ(2−α)α−1

| cos πα2|)−1

if 1 < α < 2.

(4)

The bn may be chosen as follows:

3

STOYANOV-RACHEV

446

bn =

0 if 0 < α < 1,

nan

∫ ∞

−∞sin(x/an)dF (x) if α = 1,

n

∫ ∞

−∞xdF (x) if 1 < α < 2.

(5)

In all cases, an = n1/αL0(n) where L0(n) is slowly varying at infinity.

For further information about stable distributions and their properties, seeSamorodnitsky and Taqqu (1994).

The result in Theorem 1 characterizes the domains of attraction of stableParetian laws. If the index α characterizing the tails of the c.d.f. F (x) incondition (i) satisfies α ≥ 2, then the tail index of the limiting distributionequals α∗ = 2. Thus, the relationship between the tail index of the limitingdistribution, which we denote by α∗, and the tail index in condition (i) canbe generalized as α∗ = min(α, 2). If α > 2, then EX2

1 < ∞ and we arein the setting of the classical CLT. The centering and normalization canbe done bn = nEX1 and an = n1/2σX1 , where σX1 denotes the standarddeviation of X1. The case α = 2 is more special because the variance ofX1 is infinite and an cannot be chosen in this fashion. Moreover, the propernormalization cannot be obtained by computing the limit α → 2 in equation(4). Under the more simple assumptions that the function L(x) in condition(i) equals a constant A, Zolotarev and Uchaikin (1999) provide the formulaan = (n log n)1/2A1/2.

The paper is organized in the following way. Section 2 provides a stablelimit theorem for the asymptotic distribution of the sample AVaR. In Section3, we apply the theorem assuming that the random variable X has a stableParetian distribution and also Student’s t distribution. Under these assump-tions, we study the effect of skewness and heavy tails on the convergence ratein the limit theorem.

2 A stable limit theorem

In order to develop the limit theorem, we need a few additional facts relatedto building a linear approximation to AVaR and estimating the rate of im-provement of the linear approximation. They are collected in the followingproposition.

Proposition 1. Suppose X is a r.v. with c.d.f. F which satisfies thecondition E max(−X, 0) < ∞ and F is differentiable at the ε-quantile of X.Denote by Fn the sample c.d.f. of X1, . . . , Xn which is a sample of i.i.d.

4


447

copies of X. There exists a linear functional ∆ defined on the differenceG− F where the functions G and F are c.d.f.s, such that

|φ(Fn)− φ(F )−∆(Fn − F )| = o(ρ(Fn, F )) (6)

where ρ(Fn, F ) = supx |Fn(x)− F (x)| stands for the Kolmogorov metric and

φ(G) = −1

ε

∫ ε

0

G−1(p)dp

in which G−1 is the inverse of the c.d.f. G. The linear functional ∆ has theform

∆(Fn − F ) =1

ε

∫ qε

−∞(qε − x)d(Fn(x)− F (x)). (7)

where qε is the ε-quantile of X.

Proof. The condition E max(−X, 0) < ∞ guarantees φ(F ) < ∞. Note thatφ(Fn) is convergent with any finite sample.

Consider the difference φ(Fn)− φ(F ).

φ(Fn)− φ(F ) = −1

ε

∫ ε

0

F−1n (p)dp +

1

ε

∫ ε

0

F−1(p)dp

= −1

ε

∫ F−1n (ε)

−∞pdFn(p) +

1

ε

∫ qε

−∞pdF (p)

= −1

ε

∫ qε

−∞pdFn(p)− 1

ε

∫ F−1n (ε)

qε

pdFn(p) +1

ε

∫ qε

−∞pdF (p)

= −1

ε

∫ qε

−∞pd(Fn(p)− F (p))− Cn

ε(F (qε)− Fn(qε))

= −1

ε

∫ qε

−∞pd(Fn(p)− F (p)) +

Cn

ε(Fn(qε)− F (qε))

where, by the mean-value theorem, the constant Cn is between qε and F−1n (ε).

For example if we assume, for the sake of being particular, that qε ≤ F−1n (ε),

then qε ≤ Cn ≤ F−1n (ε). Due to the assumption that F is differentiable at

qε, F−1n (ε) → qε in almost sure sense as n increases indefinitely. As a result,

Cn → qε in almost sure sense.Choose the linear functional ∆(Fn−F ) as in equation (7). The fact that

it is linear with respect to the difference of the c.d.f.s is a property of theintegral. Consider the left-had side of (7), which we denote by LHS, havingin mind the expression derived above. We obtain

5

STOYANOV-RACHEV

448

LHS =

∣∣∣∣−1

ε

∫ qε

−∞xd(Fn(x)− F (x)) +

Cn

ε(Fn(qε)− F (qε))− L(Fn − F )

∣∣∣∣

=

∣∣∣∣−1

ε

∫ qε

−∞qεd(Fn(x)− F (x)) +

Cn


∣∣∣∣

=

∣∣∣∣−qε

ε(Fn(qε)− F (qε)) +

Cn


∣∣∣∣

=|Cn − qε|

ε|Fn(qε)− F (qε)|

≤ |Cn − qε|ε

supx|Fn(x)− F (x)|

=|Cn − qε|

ερ(Fn, F )

As a result,

|φ(Fn)− φ(F )− L(Fn − F )|ρ(Fn, F )

→ 0, as n →∞

in almost sure sense. As a result we obtain the asymptotic relation in equa-tion (6).

Corollary 1. Under the assumptions in the proposition,

|φ(Fn)− φ(F )−∆(Fn − F )| = o(n−1/2). (8)

Proof. By the Kolmogorov theorem, the metric ρ(Fn, F ) approaches zero ata rate equal to n−1/2 which indicates the rate of improvement of the linearapproximation ∆(Fn − F ).

The main result is given in the theorem below. The idea is to use thelinear approximation ∆(Fn − F ) of the AVaR functional in order to obtainan asymptotic distribution as n →∞.

Theorem 2. Suppose that X is random variable with c.d.f. F (x) whichsatisfies the following conditions

a) xαF (−x) = L(x) is slowly varying at infinity

b)

∫ 0

−∞xdF (x) < ∞

c) F (x) is differentiable at x = qε, where qε is the ε-quantile of X.

6


449

Then, there exist cn > 0, n = 1, 2 . . ., such that for any 0 < ε < 1,

c−1n

(AV aRε(X)− AV aRε(X)

)w→ Sα∗(1, 1, 0), (9)

in whichw→ denotes weak limit, 1 < α∗ = min(α, 2), and cn = n1/α∗−1L0(n)/ε

where L0 is slowly varying at infinity. Furthermore, the cn are representableas cn = an/nε where an stands for the normalizing sequence in Theorem 1and must satisfy the condition in equation (4).

Proof. By the result in Proposition 1,

φ(Fn)− φ(F ) = ∆(Fn − F ) + o(n−1/2) (10)

where φ is the AVaR functional and ∆(Fn − F ) is given in (6). Simplifyingthe expression for ∆(Fn − F ), we obtain

φ(Fn)− φ(F ) =1

nε

n∑i=1

[(qε −Xi)+ − E(qε −Xi)+] + o(n−1/2) (11)

It remains to apply the domains of attraction characterization in Theorem1 to the right-hand side of equation (11). For this purpose, consider theexpression

n∑i=1

Yi − nEY1 (12)

where Yi = (qε−Xi)+ are i.i.d. random variables. Denote by FY (x) the c.d.f.of Y . The left-tail behavior of X assumed in a) implies xα(1−FY (x)) = L(x)as x →∞ where L(x) is the slowly varying function assumed in a). This isdemonstrated by

xα(1− FY (x)) = xαP (max(qε −X, 0) > x)

= xαP (X < qε − x)

∼ xαP (X < −x)

(13)

Furthermore, the asymptotic behavior of the left tail of Y is FY (−x) = 0which holds for any x ≥ −qε. As a result, condition (i) from Theorem 1holds.

Condition b) implies that the tail exponent α in a) must satisfy the in-equality α > 1. Therefore, subtracting nEY1 in (12) is a proper centeringof the sum as suggested in (5) in Theorem 1. Note that if α ≥ 2, then Y is

7

STOYANOV-RACHEV

450

in the domain of attraction of the normal distribution and the same choiceof centering is appropriate. Thus, the tail index of the limiting distributionsatisfies 1 < α∗ = min(α, 2).

Finally, computing condition (ii) in Theorem 1 from the tail behavior ofY yields β = 1. Essentially, this follows because FY (−x) = 0 if x ≥ −qε.

Therefore, all conditions in Theorem 1 are satisfied and as, a result, thereexists a sequence of normalizing constants an satisfying (4), such that

a−1n

(n∑

i=1

Yi − nEY1

)w→ Sα∗(1, 1, 0). (14)

as n → ∞. In order to apply this result to sample AVaR, we need (14)reformulated for the average rather than the sum of Yi. Thus, a more suitableform is

nεa−1n

(1

nε

n∑i=1

(Yi − EYi)

)w→ Sα∗(1, 1, 0). (15)

as n →∞.As a final step, we apply the limit result in (15) to equation (11). Multi-

plying both sides of (11) by nεa−1n yields the limit

nεa−1n (φ(Fn)− φ(F ))

w→ Sα∗(1, 1, 0) (16)

as n → ∞. It remains only to verify if the normalization does not lead toexplosion of the residual. Indeed,

nεa−1n o(n−1/2) =

n1/2

an

o(1) = o(1),

because the factor n1/2/an approaches zero by the asymptotic behavior of an

given in the domains of attraction characterization in Theorem 1.

A number of comments are collected in the following remarks.

Remark 1. By definition, the AVaR is the negative of the average of thequantiles of X beyond a reference quantile qε. For this reason, it is only thebehavior of the left tail of X which matters and the assumptions a) and b)in Theorem 2 concern the left tail only. Condition c) is technical and allowsthe calculation of the influence function of AVaR.

Remark 2. If α > 2 in condition a), then∫ 0

−∞ x2dF (x) < ∞ and the limitingdistribution is the standard normal distribution. In this case, the normalizingsequence cn should be calculated using σ2

ε = D(qε −X)+,

8


451

−5 −3 −1 0 1 3 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

α = 1.2α = 1.5α = 1.8α = 2

Figure 1: Densities of the limiting stable distribution corresponding to dif-ferent tail behavior.

cn = n−1/2σε/ε.

The case α > 2 is considered in detail in Stoyanov and Rachev (2007).

Remark 3. The limiting stable distribution is totally skewed to the right,β = 1. However, the observed skewness in the shape of the distributiondecreases as α → 2, see Figure 1. At the limit, when α = 2, the limitingdistribution is Gaussian and is symmetric irrespective of the value of β.Therefore, the degree of the observed skewness in the limiting distribution isessentially determined by the tail behavior of X, or by the value of α, and isnot influenced by any other characteristic.

Remark 4. When ε → 1, then AVaR approaches the mean of X (or thesample average if we consider the sample AVaR),

limε→1

AV aRε(X) = EX.

Unfortunately, there is no such continuity in equation (9) unless X has finitevariance. That is, generally it is not true that the weak limit in equation (9)holds for the sample average letting ε → 1. The reason is that if ε = 1, thenboth tails of the distribution of X matter and the limiting stable distribution

9

STOYANOV-RACHEV

452

can have any β ∈ [−1, 1]. The condition DX < ∞ is sufficient to guaranteethat the limiting distribution is normal for any ε ∈ (0, 1] and in this casethere is continuity in equation (9) as ε → 1.

As an illustration of the singularity at ε = 1, consider the following ex-ample. Suppose that the right tail of X is heavier than the left tail and as aconsequence,

∫ qε

−∞x2dF (x) < ∞, for any ε < 1,

but EX2 = ∞. Under this assumption, the limiting distribution of thesample AVaR is normal for any ε < 1. If ε = 1, then the limiting distributionbecomes stable non-Gaussian due to the heavier right tail. Thus, there is achange in the limiting distribution of the sample AVaR with ε < 1 and thesample average.

3 Examples

The result in Theorem 2 provides the limiting distribution but does notprovide any insight on the rate of convergence. That is, it does not give ananswer to the question how many observations are needed in order for thedistribution of the left-had side in equation (9) to be sufficiently close to thedistribution of the right-hand side in terms of a selected probability metric.In this section, we provide illustrations of the stable limit theorem and therate of convergence assuming particular distributions of X.

3.1 Stable Paretian Distributions

We remarked that stable Paretian distributions are stable distributions withtail index α < 2. This distinction is made since their properties are verydifferent from the properties of the normal distribution which appears asa stable distribution with α = 2. For example, in contrast to the normaldistribution, stable Paretian distributions have heavy tails exhibiting powerdecay. In the field of finance, stable Paretian distribution were proposed as amodel for stock returns and other financial variables, see Rachev and Mittnik(2000).

Denote by X the random variable describing the return of a given stock.In this section, we assume that X ∈ Sα(σ, β, µ) with 1 < α < 2, β 6= 1,and our goal is to apply the result in Theorem 2 which provides a tool ofcomputing the confidence interval of the sample AVaR of X on condition thatthe Monte Carlo method is used with a large number of scenarios. Since by

10


453

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

n = 250n = 1,000n = 10,000n = 100,000S

1.5(1,1,0)

Figure 2: The density of the sample AVaR as n increases with β = 0.7 andε = 0.01.

assumption α > 1, which guarantees convergence of the sample AVaR to thetheoretical AVaR in almost sure sense. In the case of stable distributions,the quantity AV aRε(X) can be calculated using a semi-analytic expressiongiven in Stoyanov et al. (2006).

In order to apply the result in Theorem 2, first we have to check if theconditions are satisfied and then choose the scaling constants cn. For thispurpose, we use the following property of stable Paretian distributions, seeSamorodnitsky and Taqqu (1994).

Property 1. Let X ∈ Sα(σ, β, µ) 0 < α < 2. Then

limλ→∞

λαP (X > λ) = Cα1 + β

2σα

limλ→∞

λαP (X < −λ) = Cα1− β

2σα

where

Cα =

(∫ ∞

0

x−α sin(x)dx

)−1

=

1−αΓ(2−α) cos(πα/2)

, α 6= 1

2/π, α = 1

11

STOYANOV-RACHEV

454

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

n = 250n = 1,000n = 10,000n = 100,000S

1.5(1,1,0)

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

n = 250n = 1,000n = 10,000n = 100,000S

1.5(1,1,0)

Figure 3: The density of the sample AVaR as n increases with β = 0.7 (top)and β = −0.7 (bottom) and ε = 0.05.

12


455

This property provides the asymptotic behavior of the left tail of the dis-tribution. We further assume that β 6= 1 since in this case the asymptoticbehavior of the left tail is different, see Samorodnitsky and Taqqu (1994).Condition b) is satisfied because of the assumption 1 < α < 2 and, finally,condition c) is satisfied for any choice of 0 < ε < 1 since all stable distribu-tions have densities. Therefore, all assumptions are satisfied and the result inTheorem 2 holds with α∗ = α and the scaling constants cn should be chosenin the following way,

cn = n1/α−1

(1− β

2

)1/ασ

ε.

Note that in this case, the skewness in the distribution of X translates intoa different scaling of the normalizing constants. If X is negatively skewed(β < 0), the scaling factor is larger than if X is skewed positively (β > 0).

We carry out a Monte Carlo study assuming X ∈ S1.5(β, 1, 0) whereβ = ±0.7 and two choices of the tail probability ε = 0.01 and ε = 0.05. Wegenerate 2,000 samples from the corresponding distribution the size of whichequals n = 250, 1, 000, 10, 000, and 100, 000.

Figure 2 illustrates the convergence rate for the case ε = 0.01 as the num-ber of observations increases. While from the plot it seems that n = 100, 000results in a density which is very close to that of the limiting distribution, butthe Kolmogorov test fails. The convergence rate is much slower in the heavy-tailed case than in the setting of the classical CLT. Stoyanov and Rachev(2007) suggest that about 5,000 simulations are sufficient for the purposesof confidence bounds estimation when the distribution has bounded support.Apparently, much more observations are needed in this heavy-tailed case.

The plots in Figure 3 indicate that as the tail probability ε increases, thebehavior of the sample AVaR distribution improves. Furthermore, the thebehavior improves when X turns from being negatively to positively skewed.

3.2 Student’s t distribution

Student’s t distribution is a widely used model for a stock return distribution.X has Student’s t distribution, X ∈ t(ν), with ν > 0 degrees of freedom ifthe density of X equals,

fν(x) =Γ

(ν+12

)

Γ(

ν2

) 1√νπ

(1 +

x2

ν

)− ν+12

, x ∈ R.

A few simple properties of Student’s t distribution are collected in thenext proposition.

13

STOYANOV-RACHEV

456

Proposition 2. Suppose that X ∈ t(ν) and denote the c.d.f. of X byF (x). Then, xνF (−x) = L(x) where L(x) is a slowly varying function atinfinity and also

limx→∞

xνF (−x) = νν/2−1 Γ(

ν+12

)

Γ(ν/2)√

π. (17)

Proof. The fact that L(x) is a slowly varying function is checked directly ap-plying the definition and the limit in (17) is obtained by applying l’Hospital’srule.

The result in this proposition and Theorem 2 imply that for ν > 2, thelimiting distribution of the sample AVaR is the Gaussian distribution. If1 < ν ≤ 2, then the limiting distribution is stable with α∗ = ν. If ν ≤ 1,then the AVaR of X diverges. The scaling constants cn should be chosen ina different way depending on the value of ν,

cn =

n−1/2σε/ε, if ν > 2n1/ν−1Aν/ε, if 1 < ν < 2

(18)

where σ2ε = D(qε −X)+ and

Aνν = νν/2−1 Γ

(ν+12

)

Γ(ν/2)√

π

Γ(2− ν)

ν − 1| cos(πν/2)|.

The value of the constant Aν is obtained by taking into account the limit in(17) and the condition in equation (4). Stoyanov and Rachev (2007) considerin detail the case ν > 2 and provide the formula for σε. This case is in theclassical setting of the CLT as the variance of X is finite.

We carry out a Monte Carlo experiment in order to study the convergencerate of the sample AVaR distribution to the limiting distribution. We fix thedegrees of freedom, the number of simulations to 100,000, and ε = 0.05. Nextwe generate 2,000 samples from which the sample AVaR is estimated. Thuswe obtain 2,000 estimates of AV aRε(X), X ∈ t(ν). Finally, we calculate theKolmogorov distance

ρ(Gν , G) = supx|Gν(x)−G(x)|

where Gν is the c.d.f. of the sample AVaR approximated by the samplec.d.f. obtained with the 2,000 estimates, and G is the c.d.f. of the limitingdistribution Sα∗(1, 1, 0) where α∗ = min(ν, 2).

Figure 4 shows the values of ρ(Gν , G) as ν varies from 1.05 to 3. Thehorizontal line shows the critical value of the Kolmogorov statistic: if the

14


457

1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Degrees of freedom

Kolmogorov distanceCritical value

Figure 4: The Kolmogorov distance between the sample AVaR distributionof X ∈ t(ν) obtained with 100,000 simulations and the limiting distribution.

calculated ρ(Gν , G) is below the critical value, we accept the hypothesis thatthe sample AVaR distribution is the same as the limiting distribution, oth-erwise we reject it. Since we use a sample c.d.f. to approximate Gν(x), thesolid line fluctuates a little but we notice that for ν ≤ 1.5 and ν ≥ 2.5 itseems that 100,000 observations are enough in order to accept the limitingdistribution as a model. For the middle values, larger samples are needed.This observation indicates that the rate of convergence of the sample AVaRdistribution to the limiting distribution deteriorates as ν approaches 2 andis slowest for ν = 2. This finding can be summarized in the following way byconsidering all possible cases for ν:

• ν > 2. As ν decreases from larger values to 2, the tail thickness in-creases which results in higher absolute moments becoming divergent,E|X|δ = ∞, δ ≥ ν. The limiting distribution is the Gaussian distri-bution but the tails becoming thicker results in deterioration of theconvergence rate to the Gaussian distribution.

• ν = 2. The limiting distribution is the Gaussian distribution eventhough the variance of X is infinite. This case is not covered by thelimit theory behind the classical CLT.

15

STOYANOV-RACHEV

458

• 1 < ν < 2. We continue decreasing ν and the tails become so thick thatthey start influencing the limiting distribution which is stable Paretian,Sν(1, 1, 0), and depends on ν. However, the convergence rate startsimproving.

• 0 < ν ≤ 1. The tails of X become so heavy that AV aRε(X) = ∞.

4 Conclusion

In the paper, we study the asymptotic distribution of the sample AVaR.We provide a stable limit theorem describing all possible asymptotic lawsdepending on the behavior of the left tail of the random variable X. If weassume that X describes the return distribution of a stock, then the left taildescribes losses. Intuitively, the asymptotic distribution of the sample AVaRis determined by the behavior of extreme losses.

Furthermore, in order to adopt the asymptotic law and draw conclusionsbased on it, we need insight on the rate of convergence in the stable limittheorem. We illustrate the rate of convergence by Monte Carlo experimentsassuming a stable distribution and Student’s t distribution for X. In sum-mary, the convergence rate deteriorates as the tail exponent α → 2 and itimproves as the distribution of X becomes more positively skewed. Generally,the skewness of X does not influence the asymptotic law.

16


459

References

Artzner, P., F. Delbaen, J.-M. Eber and D. Heath (1998), ‘Coherent measuresof risk’, Math. Fin. 6, 203–228.

Biglova, A., S. Ortobelli, S. Rachev and S. Stoyanov (2004), ‘Different ap-proaches to risk estimation in portfolio theory’, The Journal of PortfolioManagement Fall 2004, 103–112.

Pflug, G. (2000), ‘Some remarks on the value-at-risk and the conditionalvalue-at-risk’, In: Uryasev, S. (Ed.), Probabilistic Constrained Optimiza-tion: Methodology and Applications. Kluwer Academic Publishers, Dor-drecht .

Rachev, S., D. Martin, B. Racheva-Iotova and S. Stoyanov (2006), ‘Stableetl optimal portfolios and extreme risk management’, forthcoming in De-cisions in Banking and Finance, Springer/Physika, 2007 .

Rachev, S. T., Stoyan V. Stoyanov and F. J. Fabozzi (2008), Advanced sto-chastic models, risk assessment, and portfolio optimization: The ideal risk,uncertainty, and performance measures, Wiley, Finance.

Rachev, S.T. and S. Mittnik (2000), Stable Paretian Models in Finance, JohnWiley & Sons, Series in Financial Economics.

Samorodnitsky, G. and M.S. Taqqu (1994), Stable Non-Gaussian RandomProcesses, Chapman & Hall, New York, London.

Stoyanov, S., G. Samorodnitsky, S. Rachev and S. Ortobelli (2006), ‘Comput-ing the portfolio conditional value-at-risk in the α-stable case’, Probabilityand Mathematical Statistics 26, 1–22.

Stoyanov, S. and S. Rachev (2007), ‘Asymptotic distribution of the sampleaverage value-at-risk’, forthcoming in Journal of Computational Analysisand Applications .

Stoyanov, S., S. Rachev and F. Fabozzi (2007), ‘Optimal financial portfolios’,forthcoming in Applied Mathematical Finance .

Trindade, A. A., S. Uryasev, A. Shapiro and G. Zrazhevsky (2007), ‘Financialprediction with constrained tail risk’, forthcoming in Journal of Bankingand Finance .

Zolotarev, V. M. and V. V. Uchaikin (1999), Chance and stability, stabledistributions and their applications, Brill Academic Publishers.

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460

TRANSPORT PROCESSES IN NETWORKS WITH SCATTERINGRAMIFICATION NODES

AGNES RADLMATHEMATISCHES INSTITUT, AUF DER MORGENSTELLE 10, D-72076 TUBINGEN,

GERMANY

E-MAIL: [email protected]

Dedicated to Rainer Nagel on the occasion of his 65th birthday.

Abstract. We investigate the streaming of particles with different velocities

in a network. In the vertices of the network the particles are scattered, i.e. they

change their velocity but obey a Kirchhoff law. This situation will be formu-lated as an abstract Cauchy problem for an operator (A, D(A)) on a suitable

Banach space X. Then the problem is studied using semigroup methods. The

main emphasis is on the asymptotic behaviour.

keywords: transport processes, networks, semigroups, positivity, spectral theory

1. Statement of the problem

We consider a transport process with absorption and scattering as described bythe classical linear Boltzmann equation, see [7], [11], [14]. As many authors before,e.g. [26], [27], [13], [28], [29], [18], we use the theory of strongly continuous oper-ator semigroups, see [8], [10], [22], in particular the theory of positive semigroupson Banach lattices, see [20] to show wellposedness and to discuss the asymptoticbehaviour of the solutions. However, while the problem is usually considered ona domain in Rn, we study the transport process in a network. This seems to bephysically relevant, and it is mathematically interesting to discuss how the networkstructure influences the process. Moreover, we assume that absorption and scatter-ing takes place only in the ramification nodes of the network and that a Kirchhofflaw holds in each node. As predecessors we mention papers studying transportequations in slab geometry as e.g. [4], [5], [6] and [16]. Closer to our setting is [2]who concentrates on the wellposedness of a similar problem and discusses some ap-plications to physics. Our paper is mainly inspired by [15] and [17]. These authorsassume that all particles move with the same speed in the network. However, indoing so they developed the semigroup techniques we will use.

Our network is represented by a simple, directed and weighted graph G = (V,E),where V = v1, . . . , vn is the set of vertices (or nodes) and E = e1, . . . , em is theset of edges (or arcs). If two vertices are connected by an edge, then the particlescan move between the vertices in the direction given by the edge. The velocityof each particle is constant during its motion along an edge, however, for differentparticles this velocity can vary between a minimal speed vmin > 0 and a maximalspeed vmax > vmin. By the assumption on the minimal speed, each particle willreach a vertex after a finite time. In these vertices the particles are scattered, i.e.they change their velocity, or will be absorbed. Thereafter, they are distributed to

1

461


2 Transport Processes in Networks with Scattering Ramification Nodes

the outgoing edges of the vertex according to the (positive) weight of the outgoingedge. We consider only the case that each vertex has at least one incoming and oneoutgoing edge.

This physical situation will now be modelled in mathematical terms. The edgesej , j = 1, . . . ,m, are parameterised over the intervals [0, lj ] where ej(0) is the tailof the edge ej and ej(lj) is the head of the edge ej : ej(0) -p ej(lj).ej

If edge ej is an outgoing edge of vertex vi, then ωij gives the weight of the edgeej . In each vertex vi the weights of the outgoing edges shall sum up to 1, i.e.

m∑j=1

ωij = 1, (1)

for each i ∈ 1, . . . , n.Our transport process is then described by the equations

(F )

∂

∂tuj(x, v, t) = −v ∂

∂xuj(x, v, t), x ∈ (0, lj), v ∈ [vmin, vmax], t ≥ 0,

uj(x, v, 0) = fj(x, v), x ∈ (0, lj), v ∈ [vmin, vmax], (IC)

φ−ijuj(0, ·, t) = ωijJm∑

k=1

φ+ikuk(lk, ·, t), t ≥ 0, (BC)

where j = 1, . . . ,m, i = 1, . . . , n.Here, uj gives the density of the particles on edge ej depending on the position x,

the velocity v and the time t. The first equation is the well-known one-dimensionaltransport equation without scattering and absorption effects, while (IC) is the usualinitial condition for t = 0. The equation (BC) is a condition in the vertices of thegraph and models the scattering, absorption, and redistribution of particles in thevertices. The operator J appearing in (BC) is called scattering operator. It converts,in each vertex vi, the incoming velocity profile

∑mk=1 φ

+ikuk(lk, ·, t) into an outgoing

velocity profile. Then the ωthij part of this velocity profile is leaving vertex vi into

edge ej . For the scattering operator J we assume the following.

General Assumption 1.1. The operator J is a positive contraction from Y :=L1[vmin, vmax] to Y .

Since ‖f‖1 =∫ vmax

vminf(v) dv gives the total number of the particles for f ∈ Y+,

where

Y+ = f ∈ Y : f(v) ≥ 0 for almost all v ∈ [vmin, vmax],

this assumption means that no particles can enter the system.The properties of J will play an important role for the asymptotics of the process

and we will later make additional assumptions on J , see Sections 3, 5 and 6, withinteresting consequences on the spectral properties and the asymptotic behaviourof the corresponding semigroup.

The coefficients φ−ij and φ+ik in (BC) arise from matrices coding the structure

of the graph and are defined below. In this way, the equations in (BC) relate theone-dimensional particle transport to the underlying network.

To describe the graph we use the following matrices, see also [15].

462

Agnes Radl 3

Definition 1.2. (1) The outgoing incidence matrix Φ− = (φ−ij)n×m is definedby

φ−ij :=

1, vi = ej(0), i.e. vi

-q ej ,

0, otherwise.

(2) The weighted outgoing incidence matrix Φ−w = (φ−ij,w)n×m is defined by

φ−ij,w :=

ωij , vi = ej(0), i.e. vi

-q ejωij

,

0, otherwise.

(3) The incoming incidence matrix Φ+ = (φ+ij)n×m is defined by

φ+ij :=

1, vi = ej(lj), i.e. -ej q vi ,

0, otherwise.

(4) The weighted transposed adjacency matrix A = (αij)n×n is defined by A :=Φ+(Φ−w)T , i.e.

αij =

ωjk, if vj = ek(0) and vi = ek(lk), i.e. vj

-q ek q vi ,

0, otherwise.

(5) The weighted transposed adjacency matrix B = (βij)m×m of the line graphis defined by B := (Φ−w)T Φ+, i.e.

βij =

ωki, if ei(0) = ej(lj) = vk, i.e. -ej qvk -ei

ωki,

0, otherwise.

These matrices determine the structure of the graph completely, see [3] and [9].However, we need the following operator version of the above defined (scalar)

matrices.

Definition 1.3. Let IdY denote the identity operator on Y . We introduce thefollowing operator matrices.

(1) Φ− := (φ−ijIdY )n×m,(2) Φ−w := (φ−ij,wIdY )n×m,(3) Φ+ := (φ+

ijIdY )n×m,(4) Φ+

J := (φ+ijJ)n×m,

(5) A := (αijIdY )n×n,(6) B := (βijIdY )m×m,(7) BJ := (βijJ)m×m.

These operator matrices define operators in the canonical way on products ofthe space Y .

We close this section with a useful observation, see [15, Sect. 2].

Remark 1.4. In each column of Φ−, Φ−w , Φ−, Φ−w , Φ+, Φ+ and Φ+J there is exactly

one non-zero entry. Furthermore, an easy computation using the condition (1)yields

Φ−(Φ−w)T = IdCn

andΦ−(Φ−w)T = IdY n .

Moreover, it follows that A and B are column stochastic matrices.

463


2. Equation (F) as an abstract Cauchy problem

Since we want to treat the problem (F) with semigroup methods, we rewrite itas an abstract Cauchy problem on a suitable state space X. As the state space forour problem we choose

X := L1([0, l1], Y )× · · · × L1([0, lm], Y )

which is isomorphic to

L1([0, l1]× [vmin, vmax])× · · · × L1([0, lm]× [vmin, vmax]).

If all arc lengths are equal to l, then

X ∼= L1([0, l], Y m) ∼= (L1([0, l], Y ))m ∼= (L1([0, l]× [vmin, vmax]))m.

The space X is endowed with the norm

‖ · ‖1 : X → R, ‖u‖1 :=m∑

j=1

∫ lj

0

∫ vmax

vmin

|uj(x, v)| dv dx,

where u = (uj)1≤j≤m ∈ X. In the spirit of [15] we choose an abstract “boundaryspace” as

∂X := Y n,

endowed with the norm

‖ · ‖1 : ∂X → R, ‖f‖1 :=n∑

i=1

∫ vmax

vmin

|fi(v)| dv,

where f = (fi)1≤i≤n ∈ ∂X.Furthermore, we define

W := W 1,1([0, l1], Y )× · · · ×W 1,1([0, lm], Y )

which is a Banach space for the norm

‖ · ‖W : W → R, u 7→ ‖u‖W := ‖u‖1 + ‖ ∂∂xu‖1.

The trace operatorsΓ0,Γl : W → Y m

are defined byΓ0u := (uj(0))1≤j≤m,

andΓlu := (uj(lj))1≤j≤m,

respectively, where u = (uj)1≤j≤m ∈ W , and give the velocity profiles at theendpoints of the edges. Both operators are continuous on (W, ‖ · ‖W ).

To formulate (F) as an abstract Cauchy problem we proceed as in [15], and startfrom the following “maximal” operator on X.

Definition 2.1. The operator (Aw, D(Aw)) is defined by

D(Aw) := u ∈W : Γ0u ∈ rg(Φ−w)T ,(Awu)j(x, v) := −v ∂

∂xuj(x, v), x ∈ [0, lj ], v ∈ [vmin, vmax], j = 1, . . . ,m.

The condition Γ0u ∈ rg(Φ−w)T means that the proportion of the mass leavingvertex vi over edge ej is determined by the weight ωij . However, this does notcontain the complete boundary condition (BC) from (F). To formulate a conditionequivalent to (BC) we introduce the following continuous operators on (W, ‖ · ‖W ).

464

Agnes Radl 5

Definition 2.2. The outgoing boundary operator L is defined by

L : W → ∂X, u 7→ Φ−Γ0u,

while for the incoming boundary operator MJ we take

MJ : W → ∂X, u 7→ Φ+J Γlu.

Note that (Φ+Γlu)i gives the velocity profile coming into vertex vi. Then, (MJu)i

gives the velocity profile in vertex vi after the scattering and (Lu)i gives the velocityprofile leaving vertex vi. Thus, the condition

Lu = MJu (2)

expresses the Kirchhoff law.The operator corresponding to our original problem (F) is now given as follows.

Definition 2.3. The operator (A,D(A)) is defined by

D(A) := u ∈ D(Aw) : Lu = MJu,Au := Awu.

To show the equivalence fix t in (BC). Then (uj(0, ·, t))1≤j≤m ∈ rg(Φ−w)T . Takingthe sum over j in (BC) yields the Kirchhoff law.

On the other hand, let us require that Lv = MJv and Γ0v ∈ rg(Φ−w)T forv ∈ D(Aw). Then there exists d = (di)1≤i≤n ∈ Y n such that Γ0v = (Φ−w)T d.Since in each row of (Φ−w)T there is exactly one non-zero entry, it follows from thecondition Γ0v ∈ rg(Φ−w)T that for every j ∈ 1, . . . ,m there exists exactly onei ∈ 1, . . . , n such that

vj(0, ·) = ωijdi. (3)With that we compute for i = 1, . . . , n

Jm∑

j=1

φ+ijvj(lj , ·)

MJv=Lv=m∑

j=1

φ−ijvj(0, ·)(3)=

m∑j=1

φ−ijωijdi =m∑

j=1

ωijdi(1)= di. (4)

Combining (4) and (3) yields

vj(0, ·) = ωijJ

m∑i=1

φ+ijvj(lj , ·).

If we multiply both sides by φ−ij and remember that ωij 6= 0 if and only if φ−ij 6= 0,we see that (BC) is fulfilled.

Thus, (F) can equivalently be formulated as the abstract Cauchy problem

(ACP )

u(t) = Au(t), t ≥ 0,

u(0) = u0,

for the operator (A,D(A)) in the Banach space X and the initial value u0 =(fj)1≤j≤m.

Proposition 2.4. The operator (A,D(A)) is closed and densely defined.

Proof. Consider the norm ‖ · ‖G on W given by

‖ · ‖G : W → R, u 7→ ‖u‖G := ‖u‖1 +m∑

j=1

∫ lj

0

∫ vmax

vmin

v| ∂∂xuj(x, v)| dv dx.

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Since ‖ · ‖W and ‖ · ‖G are equivalent, also (W, ‖ · ‖G) is a Banach space.To prove the closedness of A we have to show that (D(A), ‖ · ‖G) is a Banach

space. Therefore, suppose that (u(n))n∈N ⊆ D(A) is a Cauchy sequence convergingto u ∈W . Then for all n ∈ N there exists an fn ∈ ∂X such that

Γ0u(n) = (Φ−w)T fn.

By the continuity of Γ0 on (W, ‖ · ‖G) it follows that

fn = Φ−(Φ−w)T fn = Φ−Γ0u(n) −→ Φ−Γ0u =: f.

Hence,Γ0u = lim

n→∞Γ0u

(n) = limn→∞

(Φ−w)T fn = (Φ−w)T f,

i.e.Γ0u ∈ rg(Φ−w)T .

Since L and MJ are continuous operators on (W, ‖ · ‖G), also the condition Lu =MJu is fulfilled and therefore u ∈ D(A).

An easy computation shows that the set

K := u ∈W : Γ0u = Γlu = 0

is dense in W with respect to the norm ‖ · ‖1. Since K ⊆ D(A) ⊆ W ⊆ X and Wis dense in X, also D(A) is dense in X.

This is the basis to prove the generator property of A in Section 4. Before doingso we investigate its spectral properties.

3. Spectral properties

In this section we apply the method from [15] to determine the spectrum σ(A)of A.

To do so we try to characterise σ(A) by a characteristic equation in the boundaryspace ∂X. We use operator matrix techniques developed by R. Nagel and A.Rhandi, see [19] and [23] and refer to [15] where this has been done for a finitedimensional boundary space ∂X.

We start with the decomposition of D(Aw) as in [12] for which it is essential thatL|D(Aw) is surjective.

Proposition 3.1. The operator L is surjective from D(Aw) to ∂X.

Proof. Let f ∈ ∂X. Then g = (gj)1≤j≤m := (Φ−w)T f ∈ Y m. Now, consider theelement u = (uj)1≤j≤m ∈ X where

uj : [0, lj ] → Y, x 7→ gj

is a constant function for 1 ≤ j ≤ m. Clearly, we have u ∈ D(Aw). Applying L tou yields

Lu = Φ−Γ0u = Φ−(uj(0))1≤j≤m = Φ−g = Φ−(Φ−w)T f = f.

Next, we consider the operator Aw with homogeneous boundary conditions.

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Agnes Radl 7

Definition 3.2. The operator (A0, D(A0)) is defined by

D(A0) := u ∈ D(Aw) : Lu = 0,A0u := Awu.

Lemma 3.3. The domain D(A0) of A0 coincides with K := u ∈W : Γ0u = 0.

Proof. The inclusion K ⊆ D(A0) is clear.To show the other inclusion, suppose that u ∈ D(A0). Then, by the condition

Γ0u ∈ rg(Φ−w)T , there exists f ∈ ∂X such that

Γ0u = (Φ−w)T f.

Therefore, and since Lu = 0, we obtain

0 = Lu = Φ−Γ0u = Φ−(Φ−w)T f = f,

henceΓ0u = (Φ−w)T f = (Φ−w)T 0 = 0.

Hence, it is clear that A0 can be written as an m ×m operator matrix whoseentries in the off-diagonal are 0 and with the same operator in each entry in thediagonal. Its domain is given by the product of the domain of the operator in thediagonal. Each of the diagonal entries is the generator of a strongly continuoussemigroup, see [24, Sect. 3.1], and the semigroup (T0(t))t≥0 generated by A0 is justthe direct sum of these semigroups. More precisely, it is given by

(T0(t)u)j(x, v) := χj(x, v, t)uj(x− vt, v),

where

χj(x, v, t) := 1, if 0 ≤ x− vt ≤ lj ,

0, otherwise,

j = 1, . . . ,m. Similarly, the resolvent of A0 is obtained as

(R(λ,A0)u)j(x, v) =∫ x

0

1v e−λ x−r

v uj(r, v) dr,

j = 1, . . . ,m. From this representation one can easily see that T0(t) and R(λ,A0)are positive for t ≥ 0 and λ ∈ R, respectively. It is also clear that the semigroup(T0(t))t≥0 is nilpotent. This implies that the spectrum of A0 is empty. Hence, by[12, Lemma 1.2], we can decompose the domain of Aw for any λ ∈ C as

D(Aw) = D(A0)⊕ ker(λ−Aw). (5)

By Prop. 3.1 the operator L is surjective. Therefore, the restriction of L to ker(λ−Aw) is bijective. By the open mapping theorem, its inverse Dλ is bounded forevery λ ∈ C. Before we give the explicit form of Dλ we first introduce the followingnotation.

Definition 3.4. The operator ελ ∈ L(Y m, X), λ ∈ C, is defined by

ελ : Y m → X, (ελf)j(x, v) := e−λv xfj(v),

where f = (fj)1≤j≤m ∈ Y m, x ∈ [0, lj ], v ∈ [vmin, vmax].

We now define an operator which turns out to be the inverse of L|ker(λ−Aw).

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Definition 3.5. For λ ∈ C the operator

Dλ : ∂X → ker(λ−Aw)

is defined byf 7→ Dλf := ελ(Φ−w)T f.

It is clear that Dλ maps to ker(λ − Aw). So it suffices to check that Dλ is theinverse of L|ker(λ−Aw).

Proposition 3.6. For λ ∈ C we have

LDλ = Id∂X (6)

andDλL = Idker(λ−Aw). (7)

Proof. Let f ∈ ∂X and recall that Φ−(Φ−w)T = Id∂X , see Remark 1.4. Thus,

LDλf = Φ−Γ0ελΦ−wf = Φ−(Φ−w)T f = f,

and (6) is satisfied. To show (7) take an element u = (uj)1≤j≤m ∈ ker(λ − Aw).The functions w = (wj)1≤j≤m ∈W of the form

wj(x, v) = fj(v)e−λv x,

where x ∈ [0, lj ], v ∈ [vmin, vmax], fj ∈ Y , and Γ0w ∈ rg(Φ−w)T compose the kernelof λ − Aw. Therefore, there exists d ∈ ∂X such that Γ0u = (Φ−w)T d. Thus, u canbe written as u = ελ(Φ−w)T d. Hence,

DλLu = ελ(Φ−w)T Φ−Γ0u = ελ(Φ−w)T Φ−(Φ−w)T d = ελ(Φ−w)T d = u.

To prove a characteristic equation for the spectrum of A we work on the productspace X × ∂X and extend the given operators, see also [15, Sect. 3].

Definition 3.7. (1) X := X × ∂X.

(2) A0 :=(Aw 0−L 0

), D(A0) := D(Aw)× 0n.

(3) X0 := X × 0n = D(Aw)× 0n = D(A0).

(4) B :=(

0 0MJ 0

), D(B) := W × ∂X.

(5) A := A0 + B =(

Aw 0MJ − L 0

), D(A) := D(Aw)× 0n.

Remark 3.8. (1) An easy computation shows that the resolvent of A0 is givenby

R(λ,A0) =(R(λ,A0) Dλ

0 0

),

for each λ ∈ C.(2) The part A|X0 of A in X0 is given by

D(A|X0) = D(A)× 0n, A|X0 =(A 00 0

).

Hence, A|X0 can be identified with the operator (A,D(A)).

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Agnes Radl 9

The next proposition shows that the spectrum of A is characterised by the spec-trum of an operator on the boundary space ∂X. Moreover, an explicit form of theresolvent of A is given.

Proposition 3.9. For λ ∈ C we have

λ ∈ σ(A) ⇐⇒ λ ∈ σ(A) ⇐⇒ 1 ∈ σ(MJDλ). (CE)

For λ ∈ ρ(A) = ρ(A) the resolvent operators are

R(λ,A) = (IdX +Dλ(Id∂X −MJDλ)−1MJ)R(λ,A0),

and

R(λ,A) =(R(λ,A) Dλ(Id−MJDλ)−1

0 0

),

respectively.

Proof. To show the equivalence

λ ∈ σ(A) ⇔ 1 ∈ σ(MJDλ), (8)

we proceed as [15, Prop. 3.3]. First, we decompose

λ−A = λ−A0 − B = (I − BR(λ,A0))(λ−A0).

Note that ρ(A0) = C. From this we see that λ − A is invertible if and only ifI − BR(λ,A0) is invertible. Since

I − BR(λ,A0) =(

IdX 0−MJR(λ,A0) Id∂X −MJDλ

),

one can easily see that the invertibility of I − BR(λ,A0) is equivalent to 1 /∈σ(MJDλ) and (8) is shown. The inverse is then given by

(I − BR(λ,A0))−1 =(

IdX 0(Id∂X −MJDλ)−1MJR(λ,A0) (Id∂X −MJDλ)−1

)and the resolvent of A is

R(λ,A) =(R(λ) Dλ(Id∂X −MJDλ)−1

0 0

),

where R(λ) = (IdY +Dλ(Id∂X −MJDλ)−1MJ)R(λ,A0).If λ > 0, then by Proposition 3.10 (1) ‖MJDλ‖ < 1 and therefore 1 6∈ σ(MJDλ).

Hence, the resolvent set of A is non-empty. Furthermore, A can be identifiedwith the part A|X0 of A in X0. So we can apply [8, Prop. IV.2.17] to prove thatσ(A) = σ(A).

Since (R(λ) 0

0 0

)= R(λ,A)|X0 = R(λ,A|X0),

it follows that

R(λ,A) = R(λ)

for λ ∈ ρ(A).

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The condition 1 ∈ σ(MJDλ) will be called “characteristic equation”. Indeed, itis a condition in ∂X, hence in a space much smaller than the state space X. Touse it we compute MJDλ as

MJDλ = Φ+J

Qe−λ· l1

0. . .

0 Qe−

λ· lm

(Φ−w)T .

Here, and in the followingQg denotes the multiplication by a function g ∈ L∞[vmin, vmax],i.e.

Qg : Y → Y, f 7→ Qgf := gf.

This form of MJDλ (and Proposition 3.9) immediately allows the following con-clusions.

Proposition 3.10. (1) Let λ ∈ C. If <λ > 0 then ‖MJDλ‖ < 1. Thus, thespectral bound of A satisfies s(A) ≤ 0.

(2) If ‖Jf‖1 = ‖f‖1 holds for all f ≥ 0, then s(A) = 0.(3) The resolvent fulfills R(λ,A) ≥ 0 for all λ > 0.

Proof. (1), (2) First, using that J is a contraction, we estimate the norm of MJDλ

as

‖MJDλ‖ = ‖Φ+J

Qe−λ· l1

0. . .

0 Qe−

λ· lm

(Φ−w)T ‖

≤ ‖Φ+J ‖

∥∥∥∥∥∥∥Qe−

λ· l1

0. . .

0 Qe−

λ· lm

∥∥∥∥∥∥∥ ‖(Φ−w)T ‖

= ‖J‖ max1≤j≤m

‖Qe−

λ· lj‖ ≤ max

1≤j≤m‖Q

e−λ· lj‖.

Suppose now that <λ > 0. Then

‖MJDλ‖ ≤ e−<λ

vmaxmin1≤j≤m lj < 1,

and therefore 1 /∈ σ(MJDλ) which is equivalent to λ /∈ σ(A) by (CE). Moreover, ifλ = 0 then

σ(MJD0) = σ(Φ+J

Qe−0· l1

0. . .

0 Qe−

0· lm

(Φ−w)T )

= σ(Φ+J (Φ−w)T ) = σ((αijJ)n×n),

where A = (αij)n×n. By [21, Sect. 4], this can be further decomposed into

σ(MJD0) = σ(A)σ(J).

By the assumption in (2) on J , we have that r(J) = 1 and from the positivity ofJ we know that r(J) ∈ σ(J), see [25, Prop. V.4.1]. Since A is a column stochasticmatrix, 1 ∈ σ(A) and again by (CE) it follows that 0 ∈ σ(A). So we conclude thats(A) = 0.

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Agnes Radl 11

(3) If λ > 0 then R(λ,A0), Dλ, MJ , and MJDλ are positive operators. Since‖MJDλ‖ < 1, the inverse of Id−MJDλ is given by the Neumann series, i.e.

(Id−MJDλ)−1 =∞∑

n=0

(MJDλ)n.

From this representation we see that it is also a positive operator. So R(λ,A) =R(λ,A0) +Dλ(1−MJDλ)−1MJR(λ,A0) consists only of positive operators and istherefore also positive.

Note that assertion (3) in the above proposition also follows from Theorem 4.6.In order to use (CE) we investigate σ(MJDλ) in more detail.

Lemma 3.11. For λ ∈ C the following holds.(1)

σ(MJDλ) \ 0 = σ(

Qe−λ· l1J 0

. . .0 Q

e−λ· lm

J

B) \ 0

= σ(B

Qe−λ· l1J 0

. . .0 Q

e−λ· lm

J

) \ 0.

(2) If all arc lengths are equal to l, then

σ(MJDλ) = σ(A)σ(JQe−

λ· l).

Proof. (1) The first assertion follows from the fact that

σ(EF ) \ 0 = σ(FE) \ 0 for E ∈ L(X1, X2) and F ∈ L(X2, X1), (9)

where X1 and X2 are arbitrary Banach spaces.(2) If all arc lengths are equal to l, then we have

MJDλ = Φ+J

Qe−λ· l 0

. . .0 Q

e−λ· l

(Φ−w)T

=

JQe−λ· l 0

. . .0 JQ

e−λ· l

Φ+(Φ−w)T

=

JQe−λ· l 0

. . .0 JQ

e−λ· l

A

= A

JQe−λ· l 0

. . .0 JQ

e−λ· l

= (αijJQ

e−λ· l)n×n,

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where A = (αij)n×n. The spectrum of operator matrices of this special form isgiven by

σ(MJDλ) = σ(A)σ(JQe−

λ· l),

see [21, Sect. 4].

We now make additional assumptions on J and discuss the spectrum of A withthe help of the characteristic equation (CE) and the above lemma. First, we con-sider the case that the operator J is compact.

Proposition 3.12. If J is a compact operator, then

σp(A) = σp(A) = σ(A).

Proof. Clearly, the point spectra of A and A coincide, i.e.

σp(A) = σp(A). (10)

Let now λ ∈ σ(A). By (CE), this means 1 ∈ σ(MJDλ). Since J is a compactoperator, also MJDλ is compact and therefore σ(MJDλ) = σp(MJDλ) ∪ 0. SoId∂X −MJDλ is not injective, i.e. there exists fλ ∈ ∂X, fλ 6= 0, such that

(Id∂X −MJDλ)fλ = 0.

Using that Dλfλ ∈ ker(λ−Aw) and that LDλ = Id∂X we obtain

(λ−A)(Dλfλ

0

)=(λ−Aw 0L−MJ λ

)(Dλfλ

0

)=(

(λ−Aw)Dλfλ

LDλfλ −MJDλfλ

)=(

0fλ −MJDλfλ

)=(

00

).

So λ − A is not injective, and therefore λ ∈ σp(A). Combining this with (10) weobtain

σ(A) ⊆ σp(A) = σp(A) ⊆ σ(A)

as claimed.

A physically realistic assumption is that the scattering operator J is a compactintegral operator with a strictly positive kernel. More precisely, we assume thatJ ∈ L(Y ) is given by

Jf :=∫ vmax

vmin

k(·, w)f(w) dw, f ∈ Y.

The measurable kernel

k : [vmin, vmax]× [vmin, vmax] → R

fulfills k(v, w) > 0 for almost all v, w ∈ [vmin, vmax]. Moreover, we assume that∫ vmax

vmin

k(v, w) dv = 1 for all w ∈ [vmin, vmax] (11)

so that our General Assumption 1.1 is satisfied. Moreover, these assumptions implythe irreducibility of J , see [25, Example V.6.4] and Definition 5.1 below.

Under these assumptions we can show that 0 is the only spectral value of A onthe imaginary axis.

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Agnes Radl 13

Theorem 3.13. Suppose that all the arc lengths are equal to l and suppose thatthe scattering operator J is as above. Then

σ(A) ∩ iR = 0.

Proof. By the proof of Proposition 3.10 we already know that 0 ∈ σ(A).From assumption (11) follows that

‖Jf‖1 = ‖f‖1 for all f ≥ 0. (12)

Hence, the adjoint operator J ′ ∈ L(Y ′) where Y ′ ∼= L∞[vmin, vmax] satisfies

J ′1 = 1,

where 1 denotes the constant one function. By the irreducibility of J and [25, Thm.V.5.2] we then obtain that there exists g ∈ Y+ such that Jg = g and g(v) > 0 for al-most all v ∈ [vmin, vmax]. Consider the Banach space Y := L1([vmin, vmax], g(v)dv).The positive operator J := Qg−1JQg ∈ L(Y ) is similar to J and satisfies

J1 = 1. (13)

Since J is irreducible, the same holds for J , and also

‖Jf‖Y = ‖f‖Y

remains true for f ∈ Y+. This again implies for the adjoint operator J ′ ∈ L(Y ′) ofJ that

J ′1 = 1. (14)

Suppose now that there is a spectral value λ ∈ iR\0 of A. Define the operatorJλ := Qg−1JQ

e−λ· lQg ∈ L(Y ). Note that Jλ is similar to JQ

e−λ· l ∈ L(Y ). There-

fore, their spectra coincide. We know from the characteristic equation (CE) withthe help of Lemma 3.11 (2) that there must exist an α ∈ σ(Jλ) such that |α| = 1.Since J is compact, α ∈ σp(Jλ). So there exists f ∈ Y , f 6= 0, such that

Jλf = αf.

Since|f | = |αf | = |Jλf | ≤ |Jλ||f | = J |f |,

we have|J |f | − |f || = J |f | − |f |.

From

〈1, |J |f | − |f ||〉 = 〈1, J |f |〉 − 〈1, |f |〉 = 〈J ′1, |f |〉 − 〈1, |f |〉 (14)= 0,

it follows that J |f | = |f |. By [25, Thm. V.5.2] the fixed space of J is one-dimensional and by (13) we conclude that it is spanned by 1. Therefore, we canassume that |f | = 1. Thus, f is a unimodular eigenfunction of Jλ.

If we take h ∈ L∞[vmin, vmax] ⊆ Y , then

0 ≤ |Jλh| ≤ |Jλ||h| = J |h| ≤ J(‖h‖∞1) = ‖h‖∞J1 = ‖h‖∞1.

Therefore, Jλ(L∞[vmin, vmax]) ⊆ L∞[vmin, vmax].By Gelfand’s theorem

L∞[vmin, vmax] ∼= C(K)

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holds for a suitable compact space K. So far, we have shown that all the assump-tions of [25, Prop. V.7.4] are fulfilled. Hence,

Jλ|L∞[vmin, vmax] = αQf JQf−1 |L∞[vmin, vmax].

This implies

k(v, w)e−λw l = αf(v)k(v, w)f(w)

for almost all v, w ∈ [vmin, vmax]. Since k is strictly positive, this means that

f(v) = αeλw lf(w)

has to be fulfilled for almost all v, w ∈ [vmin, vmax]. Evidently, this is not possible,hence there is no spectral value λ 6= 0 on the imaginary axis.

4. Wellposedness

In this section we show the generator property of A and hence the wellposednessof (F). We first renorm the space X and then check that A fulfills all the conditionsin the Phillips generation theorem, see [20, Thm. C-II 1.2]. Therefore, A is thegenerator of a contraction semigroup on X for this norm.

Since J is contractive on Y+ also BJ is contractive on Y m+ as is shown in the

following lemma.

Lemma 4.1. If f ∈ Y m+ , then

‖BJf‖1 − ‖f‖1 ≤ 0.

Proof. Let f ∈ Y m+ . Then the following computation shows the assertion.

‖BJf‖1 − ‖f‖1 =m∑

j=1

∫ vmax

vmin

(BJf − f)j(v) dv

=m∑

j=1

∫ vmax

vmin

[J(Bf)j − fj ](v) dv

=m∑

j=1

∫ vmax

vmin

[J

(m∑

k=1

bjkfk

)− fj

](v) dv

=∫ vmax

vmin

J m∑

k=1

fk

m∑j=1

bjk

−m∑

j=1

fj

(v) dv

B column stochastic=∫ vmax

vmin

J ( m∑k=1

fk

)−

m∑j=1

fj

(v) dv

=m∑

j=1

(‖Jfj‖1 − ‖fj‖1)

Gen. Ass. 1.1≤ 0.

There is an alternative way of writing the domain of A which uses the operatormatrix BJ .

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Agnes Radl 15

Proposition 4.2. The domain of A is given by

D(A) = u ∈W : Γ0u = BJΓlu.

Proof. If u ∈ D(A) then Φ−Γ0u = Φ+J Γlu and there exists f ∈ ∂X such that

Γ0u = (Φ−w)T f . With that we compute

Φ+J Γlu = Φ−Γ0u = Φ−(Φ−w)T f = f.

This implies

Γ0u = (Φ−w)T f = (Φ−w)T Φ+J Γlu = BJΓlu.

On the other hand, if for u ∈ W the condition Γ0u = BJΓlu is fulfilled, thenΓ0u ∈ rg(Φ−w)T holds since BJ = (Φ−w)T Φ+

J Moreover,

Lu = Φ−Γ0u = Φ−BJΓlu = Φ−(Φ−w)T Φ+J Γlu = Φ+

J Γlu = MJu.

This representation of D(A) is needed in Lemma 4.5 to show the dispersivity ofA where dispersive means the following.

Definition 4.3. An operator (B,D(B)) on a Banach lattice Z is called dispersiveif for every z ∈ D(B) one has < < Bz, ψ >≤ 0 for some ψ ∈ Z ′+ such that ‖ψ‖ ≤ 1and < z, ψ >= ‖z+‖.

If X is endowed with the following norm, then A is dispersive.

Definition 4.4. The norm ‖ · ‖1,v on X is

‖ · ‖1,v : X → R, u = (uj)1≤j≤m 7→ ‖u‖1,v :=m∑

j=1

∫ lj

0

∫ vmax

vmin

1v|uj(x, v)| dv dx.

Since1

vmax‖ · ‖1 ≤ ‖ · ‖1,v ≤ 1

vmin‖ · ‖1,

the norm ‖ · ‖1,v is equivalent to the original norm ‖ · ‖1 on X.Now we check the dispersivity of A.

Lemma 4.5. The operator (A,D(A)) is dispersive on the Banach lattice (X, ‖ · ‖1,v).

Proof. The dual space of X is

X ′ ∼= L∞([0, l1], Y ′)× · · · × L∞([0, lm], Y ′)∼= L∞([0, l1]× [vmin, vmax])× · · · × L∞([0, lm]× [vmin, vmax])

where Y ′ = L∞[vmin, vmax]. Let u ∈ D(A) and let Ψ = (Ψk)1≤k≤m ∈ X ′ bedefined by

Ψk(x, ·) =

v 7→ 1

v , uk(x, ·) = u+k (x, ·),

v 7→ 0, else,

where x ∈ [0, lk]. Clearly, ‖Ψ‖ ≤ 1 for Ψ ∈ (X, ‖ · ‖1,v)′.

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Next, we compute

〈u,Ψ〉 =m∑

k=1

∫ lk

0

〈uk(x, ·),Ψk(x, ·)〉 dx

=m∑

k=1

∫ lk

0

∫ vmax

vmin

uk(x, v)Ψk(x, v) dv dx

=m∑

k=1

∫ lk

0

∫ vmax

vmin

uk(x, v) 1vχk(x) dv dx

= ‖u+‖1,v,

where

χk(x) =

1, if uk(x, ·) = u+

k (x, ·),0, else.

We then obtain

〈Au,Ψ〉

=m∑

k=1

∫ lk

0

∫ vmax

vmin

1v (−v) ∂

∂xu+k (x, v) dv dx

= −m∑

k=1

∫ vmax

vmin

∫ lk

0

∂∂xu

+k (x, v) dx dv

=m∑

k=1

∫ vmax

vmin

(u+

k (0, v)− u+k (lk, v)

)dv

Prop. 4.2=

m∑k=1

∫ vmax

vmin

((BJ(Γlu))+k − u+

k (lk, ·))(v) dv

≤m∑

k=1

∫ vmax

vmin

((BJ(Γlu)+)k(v)− u+

k (lk, v))dv

=∫ vmax

vmin

m∑k=1

J

m∑j=1

βkju+j (lj , ·)

(v) dv −m∑

k=1

∫ vmax

vmin

u+k (lk, v) dv

=∫ vmax

vmin

m∑j=1

J

[(m∑

k=1

βkj

)u+

j (lj , ·)

] (v) dv −m∑

k=1

∫ vmax

vmin

u+k (lk, v) dv

B column stochastic=∫ vmax

vmin

m∑j=1

Ju+j (lj , ·)

(v) dv −m∑

k=1

∫ vmax

vmin

u+k (lk, v) dv

=m∑

j=1

‖Ju+j (lj , ·)‖1 −

m∑k=1

‖u+k (lk, ·)‖1

Gen. Ass. 1.1≤ 0.

This shows that all the conditions of Definition 4.3 are fulfilled, hence A is disper-sive.

476

Agnes Radl 17

We now obtain the generator property of A.

Theorem 4.6. The operator (A,D(A)) on X is the generator of a positive andbounded strongly continuous semigroup (T (t))t≥0 with bound vmax

vmin.

Proof. By Proposition 2.4 and Lemma 4.5 it follows that A is a densely defined,dispersive operator and by Proposition 3.10 the operator λ − A is surjective forλ > 0. Therefore, the Phillips theorem, see [20, Thm. C-II 1.2], implies that A isthe generator of a positive contraction semigroup on (X, ‖ · ‖1,v). Returning to ouroriginal norm ‖ · ‖1 on X we obtain that the semigroup is bounded by vmax

vmin.

5. Irreducibility of the semigroup

Irreducibility of the semigroup is an important property. It turns out that weneed both a condition on the structure of the graph and on the scattering operatorJ in the vertices to obtain irreducibility. In Section 6, this will lead to a precisedescription of the asymptotic behaviour of the semigroup. We briefly recall thebasic definitions, see [20] and [25].

Definition 5.1. (1) A positive linear operator S on a Banach lattice E iscalled irreducible if there is no closed ideal in E which is invariant under Sapart from 0 and E.

(2) A positive semigroup (S(t))t≥0 on a Banach lattice E is called irreducible ifthere is no closed ideal in E which is invariant under (S(t))t≥0 apart from0 and E.

The irreducibility of our semigroup (T (t))t≥0 on the Banach lattice X can becharacterised in the following way, cf. [20, Def. C-III 3.1].

Proposition 5.2. The following assertions are equivalent.

(1) The semigroup (T (t))t≥0 on X is irreducible.(2) If u ∈ X and u 0, then R(λ,A)u 0 for all λ > 0.

Here, u 0 means that u 6= 0 and u is positive (u ≥ 0), i.e. uj(x, v) ≥ 0 foralmost all x ∈ [0, lj ] and v ∈ [vmin, vmax] where j = 1, . . . ,m. R(λ,A)u 0means that R(λ,A)u is strictly positive, i.e. (R(λ,A)u)j(x, v) > 0 for almost allx ∈ [0, lj ] and v ∈ [vmin, vmax] where j = 1, . . . ,m. The same notation will be usedto indicate positivity and strict positivity for functions in Y .

To show irreducibility for our semigroup we need the following concept fromgraph theory.

Definition 5.3. A directed graph is called strongly connected if for any two verticesv, w of the graph there exists a path from v to w and from w to v.

We obtain irreducibility of our semigroup combining two assumptions on thegraph G and the scattering operator J .

Proposition 5.4. Let G be strongly connected and suppose that

Jf 0 if f 0. (15)

Then the semigroup (T (t))t≥0 generated by A is irreducible.

477


Proof. Suppose that λ > 0 and let u 0. Then alsoR(λ,A0)u 0 andMJR(λ,A0)u 0. The inverse of Id∂X −MJDλ is given by the Neumann series

(Id∂X −MJDλ)−1 =∞∑

n=0

(MJDλ)n.

The operator MJDλ has the same zero pattern as the adjacency matrix A. Observethat Ak has a non-zero entry at position ij if there is a path from vertex vj to vertexvi of length k. Since G is assumed to be strongly connected, for every pair i, j thereexists k ∈ N such that the entry ij of Ak and thus of (MJDλ)k is nonzero. Thisentry can be written as the composition of J with an operator composed of J andmultiplictions by strictly positive functions. By assumption (15) we conclude that

(Id∂X −MJDλ)−1MJR(λ,A0)u 0

and therefore by the special form of Dλ also

Dλ(Id∂X −MJDλ)−1MJR(λ,A0)u 0.

This impliesR(λ,A)u 0,

which is by Proposition 5.2 equivalent to the irreducibility of the semigroup.

In the following examples we show that only the combination of the two assump-tions in Proposition 5.4 leads to irreducibility.

Example 5.5. If we drop the assumption of the strong connectivity of the graph,then the semigroup need not be irreducible.

To prove this we decompose the graph into its strongly connected components.Assuming the graph to be not strongly connected, there exists a strongly connectedcomponent C = (V ′, E′), V ′ ⊆ V,E′ ⊆ E such that there is no edge e ∈ E \E′ thatis an incoming edge for a vertex v ∈ V ′. Without loss of generality we can assumethat V ′ = vr, . . . , vn for some 2 < r < n, and E′ = es, . . . , em for some 1 <s < m− 1. The incidence matrices have the form

Φ−w =(

Φ−11 0Φ−21 Φ−22

)and Φ+ =

(Φ+

11 Φ+12

0 Φ+22

)respectively,

where Φ−11 and Φ+11 are (r− 1)× (s− 1)−, Φ−12 and Φ+

12 are (r− 1)× (m− s+ 1)−,Φ−21 and Φ+

21 are (n−r+1)×(s−1)− and Φ−22 and Φ+22 are (n−r+1)×(m−s+1)−

operator matrices. Moreover, since there is no path from a vertex vi, 1 ≤ i ≤ r− 1,leading into the subgraph C, we have

(MJDλ)n =(M11 M12

0 M22

), n ∈ N,

where M11 is an (r − 1) × (r − 1)−, M12 is an (r − 1) × (n − r + 1)−, and M22

is an (n − r + 1) × (n − r + 1)− operator matrix. Take an element u ∈ X suchthat u = (uj)1≤j≤m 0 and uj = 0 for j ∈ r, . . . , n. Then, it follows from thespecial form of the operators appearing in the resolvent of A that (R(λ,A)u)j = 0for λ > 0 and j ∈ r, . . . , n. By Proposition 5.2, the semigroup is not irreducible.

Note that in (15) we do not only require the irreducibility of J but a strongercondition. If G is strongly connected and J is only assumed to be irreducible, thenthe semigroup generated by A is not necessarily irreducible as the following exampleshows.

478

Agnes Radl 19

Example 5.6. Define

k : [vmin, vmax]× [vmin, vmax] → R,

k(v, w) :=

c if v ∈ [vmin, v

′] and w ∈ [v′, vmax]

or if v ∈ (v′, vmax] and w ∈ [vmin, v′],

0 else,

where 0 6= c ∈ R and vmin < v′ < vmax. Then the integral operator

J : Y → Y,

(Jf)(v) :=∫ vmax

vmin

k(v, w)f(w) dw =

c

∫ vmax

v′f(w) dw if vmin ≤ v ≤ v′,

c

∫ v′

vmin

f(w) dw if v′ < v ≤ vmax,

is irreducible which can be shown by an easy computation.Consider a graph with the incidence matrices Φ−w = ( 1 0

0 1 ) and Φ+ = ( 0 11 0 )

and suppose that both arcs have length l. Let u = (u1, u2) ∈ X such thatu1 0, u1(x, v) = 0 if 0 ≤ x ≤ l and vmin ≤ v ≤ v′ and u2 ≡ 0. Then also(R(λ,A0)u)2 = 0 and

(f1, f2) := MJR(λ,A0)u = (0, J(∫ l

0

1· e−λ

l−r· u1(r, ·) dr)).

Observe that if f ∈ Y with f |[vmin,v′] = 0, then

(J2k+1f)|[v′,vmax] = 0 (16)

and(J2kf)|[vmin,v′] = 0 for k ∈ N. (17)

Therefore, f2|[v′,vmax] = 0 holds. Suppose that λ > 0. Then the inverse of Id∂X −MJDλ is given by the Neumann series

(Id∂X −MJDλ)−1 =∞∑

k=0

(MJDλ)k

=

( ∑∞k=0(JQe−

λ· l)2k

∑∞k=0(JQe−

λ· l)2k+1∑∞

k=0(JQe−λ· l)2k+1

∑∞k=0(JQe−

λ· l)2k

).

For f ∈ Y the function Qe−

λ· lf vanishes on the same set as f . since these operators

are multiplications by positive functions. Therefore we conclude, using (16) and(17), that

((Id∂X −MJDλ)−1MJR(λ,A0)u)1|[v′,vmax] = 0

and, by the definition of Dλ, also

(Dλ(Id∂X −MJDλ)−1MJR(λ,A0)u)1|[v′,vmax] = 0

holds. With these considerations it follows that(R(λ,A)u)1|[v′,vmax]

=(R(λ,A0)u)1 +Dλ(Id∂X −MJDλ)−1MJR(λ,A0)u)1|[v′,vmax] = 0.

By Proposition 5.2 the semigroup (T (t))t≥0 is not irreducible.

479


6. Asymptotic behaviour

As our final result we describe the asymptotic behaviour of the solutions usingthe theory of positive and irreducible semigroups from [20].

We use the following notation. We denote by [−u, u] for u = (uj)1≤j≤m ∈ Xorder intervals, i.e.

[−u, u] = w = (wj)1≤j≤m ∈ X : −uj(x) ≤ wj(x) ≤ uj(x) for almost allx ∈ [0, lj ], j = 1, . . . ,m,

and the absolute value of u is |u| = (|uj |)1≤j≤m where |uj |(x, v) = |uj(x, v)| forx ∈ [0, lj ] and v ∈ [vmin, vmax]. A lattice norm ‖ · ‖X on X is called strictlymonotone if |u| < |w| implies ‖u‖X < ‖w‖X for all u,w ∈ X. The notation is takenfrom [20] and also the definitions can be found in this book. Moreover, the fixedspace of the semigroup (T (t))t≥0 is

fix(T (t))t≥0 =⋂t≥0

fix(T (t)) = u ∈ X : T (t)u = u for all t ≥ 0.

By [8, Cor. IV.3.8 (i)] the equality

fix(T (t))t≥0 = kerA

holds.To treat the asymptotic behaviour of the semigroup the following compactness

property of the semigroup is important.

Lemma 6.1. Let 0 ∈ σp(A) and suppose that the semigroup (T (t))t≥0 is irreducible.Then T (t) : t ≥ 0 ⊆ L(X) is relatively compact for the weak operator topology,hence it is mean ergodic, i.e.

limr→∞

1r

∫ r

0

T (s)u ds,

exists for all u ∈ X, see [8, Def. V.4.3].

Proof. Since 0 ∈ σp(A) we know from [8, Cor. IV.3.8] that there exists 0 6= u ∈fix(T (t))t≥0. Then, from the positivity of the semigroup, the inequality

|u| = |T (t)u| ≤ T (t)|u| (18)

follows for t ≥ 0. Suppose that |u| < T (t)|u|. Since (T (t))t≥0 is a contractionsemigroup with respect to the strictly monotone norm ‖ · ‖1,v from Definition 4.4on X, we obtain

‖u‖1,v < ‖T (t)|u|‖1,v ≤ ‖u‖1,v

which is a contradiction. Thus, in (18) we have equality and we can assume in thefollowing without loss of generality that u 0. Since the semigroup is irreduciblewe obtain from [20, Prop. C-III 3.5 (a)] that u is a quasi-interior point of X whichimplies that

Xu :=⋃n≥1

[−nu, nu]

is dense in X.Let n ∈ N and take w ∈ [−nu, nu], i.e. −nu ≤ w ≤ nu. Then

−nu = −nT (t)u ≤ T (t)w ≤ nT (t)u = nu,

480

Agnes Radl 21

for all t ≥ 0. Since the order interval [−nu, nu] is weakly compact in X, see [25, p.92], the orbit T (t)w : t ≥ 0 is relatively weakly compact in X. So far, we haveshown that the orbits of elements w from the dense subset Xu of X are relativelyweakly compact. Since the semigroup (T (t))t≥0 is bounded, this suffices to provethat T (t) : t ≥ 0 ⊆ L(X) is relatively weakly compact, see [8, Lem. V.2.7].

The mean ergodicity of (T (t))t≥0 follows from [8, Lem. V.4.6].

The mean ergodicity of the semigroup allows a decomposition of X into thedirect sum of kerA and rgA. If the semigroup is irreducible, then kerA is one-dimensional. If in addition the scattering operator is as in Theorem 3.13, then thesemigroup converges strongly to the one-dimensional projection onto kerA. This isshown in the next theorem.

Theorem 6.2. Let G be strongly connected. Then, under the assumptions of The-orem 3.13, the space X can be decomposed into the direct sum

X = X1 ⊕X2

where X1 = fix(T (t))t≥0 = kerA is one-dimensional and spanned by a strictlypositive eigenvector u ∈ kerA of A, u 0, and (T (t)|X2)t≥0 is strongly stable.

Proof. Observe first that all the assumptions of Proposition 5.4 are fulfilled andhence (T (t))t≥0 is irreducible. Since (T (t))t≥0 is mean ergodic by Lemma 6.1, thespace X can be decomposed into

X = kerA⊕ rg(A) =: X1 ⊕X2,

where kerA = fix(T (t))t≥0, see [8, Lem. V.4.4]. From Proposition 3.12 it is clearthat 0 ∈ σp(A). As in the proof of Lemma 6.1 we can show that there existsu ∈ kerA such that u 0. Moreover, we find by the same construction as in theproof of [8, Lem. V.2.20 (i)] φ ∈ X ′ such that φ 0 and A′φ = 0. Thus, by [20,Prop. C-III 3.5] we obtain that

dim kerA = 1,

and that u is strictly positive, i.e. u 0.Both spaces X1 and X2 are invariant under (T (t))t≥0. Consider now the re-

stricted semigroup (T2(t))t≥0 where T2(t) := T (t)|X2 . Its generator (A2, D(A2)) isgiven by

D(A2) = D(A) ∩X2,

A2v = Av.

In the next step we show that σp(A′2) ∩ iR = ∅. From [8, Prop. IV.1.12] we havethat

σp(A′2) = σr(A2),where σr(A2) = λ ∈ C : rg(λ − A2) is not dense in X2 denotes the residualspectrum. Since σr(A2) ⊆ σ(A) and σ(A) ∩ iR = 0 by Theorem 3.13, we onlyhave to prove that 0 /∈ σp(A′2) = σr(A2). Clearly, (T2(t))t≥0 is a mean ergodicbounded semigroup on X2. So, by [8, Thm. V.4.5], kerA2 separates kerA′2. ButkerA2 = 0 and thus kerA′2 = 0. Hence, it follows that σp(A′2) ∩ iR = ∅. Nowwe can apply the Arendt-Batty-Lyubich-Vu Theorem, see [1, Thm. 5.5.5], to showthe strong stability of (T2(t))t≥0.

We reformulate the above theorem as our final result.

481


Corollary 6.3. Under the conditions of the above theorem the semigroup (T (t))t≥0

converges to the one-dimensional projection P ∈ L(X) onto fix(T (t))t≥0, i.e.

limt→∞

‖T (t)w − Pw‖ = 0 for all w ∈ X.

Here, P = u⊗φ where u and 0 φ ∈ X ′ are as in (the proof of) Theorem 6.2 and〈u, φ〉 = 1.

7. Acknowledgements

The author thanks Rainer Nagel sincerely for many helpful discussions and sug-gestions.

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483

484

(α, β)− Lp 2 -Norm Orthogonality andCharacterizations of 2 - Inner Product Spaces

Vinai K.Singh, S. Kumar* and A. K. Singh**

Department of Mathematics,

R.D.Engineering College,

N.H.58 Delhi Meerut Road,

Duhai, Ghaziabad, INDIA.

*Department of Applied Mathematics

Inderprastha Engineering College,

Ghaziabad 201010,INDIA.

[email protected];[email protected]

**Department of Science and Technology,

Technology Bhawan, New Mahrauli Road,

New Delhi-100016, INDIA.

ABSTRACT

In the present paper we have characterised (α, β)−Lp orthogonality in a 2-

normed linear space. In some way the results proved in this paper generalize

some of the similar characterization of generalized Lp- orthogonality derived

earlier by Zheng Liu[8].

Mathematics Subject Classification 2000: Primary 46C15.

Key words and Phrases: (α, β)−Lp orthogonality, Birkhoff orthogonality,

2-normed linear spaces, 2-inner product spaces, Homogeniety.

INTRODUCTION

Recently there has been special interest to deal with certain analytic

functional aspects in 2-normed spaces of finite or infinite dimensional type.

485


Usually orthogonality is dealt in inner product spaces but there is a concept

like orthogonality in normal linear spaces ([2], [3], [5], [6],[7] and [8]). As

has been noted earlier (for example see reference [7]) Birkhoff orthogonality

plays a typical role in a normed linear space. In some analytic consideration

also Birkhoff orthogonality is important.

In the present paper we have introduced (α, β) − Lp m-orthogonality for

a pair (x,z) and (y,z) in 2-normed spaces. We have also developed certain

properties in the line of those given earlier by Liu [8] as was given in the

normed spaces to be carried over in the setting of 2-normed space and 2-

inner product spaces.

PRELIMINARIES AND NOTATIONS

DEFINITION 1. Let p > 1 be a fixed real number. If (x, z) ∈ X ×X, then we say that (x,z) is Lp-orthogonal and we denote (x, z) ⊥Lp (y, z)

provided

‖ x + y, z ‖p=‖ x, z ‖p + ‖ y, z ‖p is called left Lp orthogonality. In a similar

way(x, y) ⊥Lp (x, z) provided ‖ x, y + z ‖p=‖ x, y ‖p + ‖ x, z ‖p is called

right Lp -orthogonality in 2-normed spaces.

DEFINITION 2. Let p ≥ 1 and α, β 6= 1 be fixed real numbers. If

(x, z) ∈ X ×X then (x,z) is 2-norm (α, β)−Lp− orthogonal to (y,z) denote

by(x, z) ⊥Lp (y, z)(α, β) provided that

‖ x + y, z ‖p + ‖ αx + βy, z ‖p=‖ αx + y, z ‖p + ‖ x + βy, z ‖p

and z /∈ V (x, y) (where V(x,y) is the linear span of x, y ∈ X. Similarly we say

(x,z) is (α, β)− Lp− orthogonal to (y,z) denoted by ((x, z) ⊥Lp (y, z))(α, β)

provided that

‖ x, y + z ‖p + ‖ x, αy + βz ‖p=‖ x, αy + z ‖p + ‖ x, y + βz ‖p .

LEMMA 1. For all (x, z), (y, z) ∈ X × X,α, β 6= 1, ((x, z) ⊥Lp

(y, z))(α, β) if and only ((y, z) ⊥Lp (x, z))(α, β).

The following theorem and corollary demonstrate that the concept of (α, β)−Lp− orthogonality is non-vacuous.

SINGH ET AL

486

THEOREM 1. Let p > 1 and α, β 6= 1 be a fixed real numbers.

If (x, z) 6= 0, (y, z) ∈ X × X then there exists a real number such that

((x, z) ⊥Lp (ax + y, z))(α, β).

PROOF. Set

f(t) =‖ x+tx+y, z ‖p + ‖ ax+β(tx+y), z ‖p − ‖ ax+tx+y, z ‖p − ‖ x+β(tx+y), z ‖p .

Clearly, f is a continuous function on −∞ < t < ∞, and we have, for t 6= 0

f(t) = |t|p[(‖ x+1

t(x+y), z ‖p − ‖ x, z ‖)p−(‖ βx+

1

t(αx+βy), z ‖p − ‖ βx, z ‖)p

−(‖ x+1

t(αx+y), z ‖p − ‖ x, z ‖)p−(‖ βx+

1

t(x+βy), z ‖p − ‖ βx, z ‖)p].

Then for t 6= 0

f(t)

|t|p−1sgnt=‖ x + 1

t(x + y), z ‖p − ‖ x, z ‖p

1t

+‖ βx + 1

t(αx + βy), z ‖ − ‖ βx, z ‖p

1t

−‖ x + 1

t(αx + y), z ‖2 − ‖ x, z ‖p

1t

−‖ βx + 1

t(x + βy), z ‖ − ‖ βx, z ‖p

1t

,

and hence

limt→

+−∞

f(t)

|t|p−1sgnt= p ‖ x, z ‖p−1 J+

−(x, z)(x+y)+p ‖ βx, z ‖p−1 J+

−(βx, z)(αx+βy)

−p ‖ x ‖p−1 J+−(x, z)(αx+y)−p ‖ βx, z ‖p−1 J+

−(βx, z)(x+βy),

where J+((x, z)(y, z)) and J−((x, z)(y, z)) are respectively the right and left

Gateaux derivative of the norm at (x,z), keeping second co-ordinate as fixed

in the direction of (y,z). By James [5] we see that

J+−(x, z)(rx + sy) = r ‖ x, z ‖ +sJ+

−(x, z)(y, z)

for some s ≥ 0 and r, therefore,

limf(t)

p− 1= p ‖ x, z ‖p (1 + αβp−1 − α− βp−1)

p ‖ x, z ‖p (1− α)(1− βp−1).

NORM ORTHOGONALITY AND INNERPRODUCT SPACES

487

Thus for any fixed real number α, β 6= 1 we have either f(t) →∞ as t → +∞or f(t) → −∞. Hence there is a real number a such that f(a) = 0, which

was to be proved.

COROLLARY 1. Let p > 1 and α, β 6= 1 be the fixed real numbers.

If x 6= 0, z 6= 0, (y, z) ∈ X, then there exist a real number a such that

((ax + yz) ⊥Lp (x, z)(α, β)).

PROOF. The result follows from Theorem 1 and Lemma 1.

Lemma 2. Let (x,z) or (y, z) ∈ X ×X and α, β 6= 1

(i) α, β 6= 0, ((x, z) ⊥Lp (y, z))(α, β) if and only if ((αx, z) ⊥Lp (βy, z))( 1α, 1

β),

(ii) if β 6= 0, ((x, z) ⊥Lp (y, z))(α, β)) if and only if ((x, z) ⊥Lp β((y, z))(α, 1β),

(iii)if α 6= 0, ((x, z) ⊥Lp (y, z))(α, β) if and only if ((αx, z) ⊥Lp (y, z))( 1α, β).

Homogenity, symmetry and left and right additivity of (α, β)−Lp− orthog-

onality are defined in usual way, i.e. 2-norm (α, β) − Lp− orthogonality is

homogeneous provided for all x, y, z,∈ X and real numbers a, b, ((x, z) ⊥Lp

(y, z))(α, β) implies ((ax, z) ⊥Lp (by, z))(α, β); 2-norm (α, β)−Lp−2 norm or-

thogonality is symmetric provided for all x, y, z ∈ X, ((x, z) ⊥Lp (y, z))(α, β)

implies (y, z) ⊥Lp (x, z))(α, β); 2 -norm (α, β) − Lp− orthogonality is

left additive if and only if for all x, y, w ∈ X, ((x, z) ⊥Lp (w, z))(α, β)

and ((y, z) ⊥Lp (w, z))(α, β) implies ((x + y, z) ⊥Lp (w, z))(α, β) and 2-

norm (α, β) − Lp− orthogonality is right additive if and only if for all

(x, z), (y, z), (w, z) ∈ X × X, ((x, z) ⊥Lp (y, z))(α, β) and if ((y, z) ⊥Lp

(w, z))(α, β) imply ((x, z) ⊥Lp (y + w, z))(α, β).

The following two corollaries are immediate consequences of the definition

of homogeneity and Lemma 1 and Lemma 2.

COROLLARY 2. For all α, β 6= 1,2-norm (α, β) − Lp− orthogonality

is homogeneous if and only if 2-norm (α, β)− Lp− orthogonality is homoge-

neous.

COROLLARY 3. Suppose 2-norm (α, β)−Lp− orthogonality is homo-

geneous .

(i) If α, β 6= 0, ((x, z) ⊥Lp (y, z))(α, β) if and only if ((x, z) ⊥Lp (y, z))( 1α, 1

β),

(ii) if β 6= 0, ((x, z) ⊥Lp (y, z))(α, β) if and only if ((x, z) ⊥Lp ((y, z))(α, 1β),

SINGH ET AL

488

(iii)if α 6= 0, ((x, z) ⊥Lp (y, z))(α, β) if and only if ((x, z) ⊥Lp (y, z))( 1α, β).

Now let us further study some consequences of homogeneity.

LEMMA 3. If α 6= −1 and (α, β) − Lp − 2−norm orthogonality is

homogeneous, then ((x, z) ⊥Lp (y, z))(α, β) implies

‖ x + y, z ‖p= (1− |β|p) ‖ y, z ‖p + ‖ x + βy, z ‖p .

PROOF. From Corollary 3, it suffices to assume |α| < 1. Suppose

((x, z) ⊥Lp (y, z))(α, β). Then ‖ x + y, z ‖p + ‖ αx + βy, z ‖p=‖ αx + y, z ‖p

+ ‖ x + βy, z ‖p keeping second co-ordinate as fixed. Since the result is

immediate for α = 0. We may assume that α 6= 0. We are denoting the

statement by P (n) i.e.

P (n) :‖ x + y, z ‖p + ‖ αnx + βy, z ‖p=‖ αnx + y, z ‖p + ‖ x + βy, z ‖p .

Clearly P (1) is true and if P (n) is true for some positive integer n. Since 2-

norm (α, β)−Lp− orthogonality is homogeneous, ((αnx, z) ⊥Lp (y, z))(α, β),

we have

‖ αnx + y, z ‖p + ‖ αn+1x + βy, z ‖p=‖ αn+1x + y, z ‖p + ‖ αnx + βy, z ‖p .

Adding this to P (n) we obtain

‖ x + y, z ‖p + ‖ αn+1x + βy, z ‖p=‖ αn+1x + y, z ‖p + ‖ x + βy, z ‖p,

which is P (n + 1). Thus P (n) is true for all positive integer n, but

limt→∞+

αn = 0,

so in the limit, by continuity of the norm, we have

‖ x + y, z ‖p +|β|p ‖ y, z ‖p=‖ y, z ‖p + ‖ x + βy, z ‖p,

and the conclusion of the lemma follows.

THEOREM 2. If α, β 6= −1 and 2-norm (α, β)− Lp− orthogonality is

homogeneous, then ((x, y) ⊥Lp (y, z))(α, β) implies

‖ x + y, z ‖p=‖ x, z ‖p + ‖ y, z ‖p


489

i.e. 2 - norm (α, β)− Lp− orthogonality implies Lp orthogonality.

PROOF. By Corollary 6, we may assume |β| < 1. Suppose ((x, z) ⊥Lp

(y, z))(α, β) and let Q(n) denote the statement

Q(n) :‖ x + y, z ‖p= (1− |βn|p) ‖ y, z ‖p + ‖ x + βny, z ‖p .

The statement Q(1) is Lemma 3. If we assume Q(n) is true for some positive

integer n, since ((x, z) ⊥Lp (y, z))(α, β) by homogeneity, we have by Lemma3,

‖ x + βny, z ‖p= (1− |β|p) ‖ βny, z ‖p + ‖ x + βn+1y, z ‖p

Substituting this in Q(n) we obtain

‖ x + y, z ‖p= (1− |βn+1|p) ‖ y, z ‖p + ‖ x + βn+1y, z ‖p

or Q(n + 1).

Hence Q(n) holds for all positive integer n. Since βn → 0 as n → +∞ by

taking limit in Q(n) we obtain

‖ x + y, z ‖p=‖ x, z ‖p + ‖ y, z ‖p .

THEOREM 3. If 2 - norm (α, β)−Lp− orthogonality is homogeneous,

then ((x, z) ⊥Lp (y, z))(α, β) implies ‖ x − y, z ‖=‖ x + y, z ‖ i.e. 2 - norm

(α, β)− Lp− orthogonality implies 2 - norm isosceles orthogonality.

PROOF. If (α, β) = −1 the result is follows. Otherwise by Lemma 1 we

may assume without loss of generality that α 6= −1. If β = −1 the result is

immediate from Lemma 7. If β 6= −1, by Theorem 4, we have

‖ x + y, z ‖p=‖ x, z ‖p + ‖ y, z ‖p . But by homogeneity ((x, z) ⊥Lp

(y, z))(α, β) holds, so ‖ x − y, z ‖p=‖ x, z ‖p + ‖ −y, z ‖p and the result

follows.

LEMMA 4. For all α, β 6= 1, each one of the following:

(i) (α, β)− Lp 2 - norm orthogonality is symmetric and left additive,

(ii) (α, β)− Lp 2 - norm orthogonality is symmetric and right additive,

(iii) (α, β)− Lp 2 - norm orthogonality is left and right additive,

implies that (α, β)− Lp 2 - norm orthogonality is homogeneous.

SINGH ET AL

490

PROOF. Suppose (i) holds and ((x, z) ⊥Lp (y, z))(α, β), where x , y

and z are arbitrary elements. Since the result is obvious for x = 0 or y =

0. We will assume x 6= 0 and y 6= 0. By Corollary 1, there exist a real

number a such that ((ay − x, z) ⊥Lp (y, z))(α, β). Left additive then gives

((ay, z) ⊥Lp (y, z))(α, β) and hence a = 0. Thus ((−x, z) ⊥Lp (y, z))(α, β),

using left additive and symmetry, we find it now follows that ((nx, z) ⊥Lp

(my, z))(α, β) for all integers m and n, i.e.

‖ nx + my, z ‖p + ‖ αnx + βmy, z ‖p=‖ αnx + my, z ‖p + ‖ nx + βmy, z ‖p

or

‖ x +m

ny, z ‖p + ‖ αx + β

m

ny, z ‖p=‖ αx +

m

ny, z ‖p + ‖ x + β

m

ny, z ‖p .

From the continuity of the norm it follows that

‖ x + ky, z ‖p + ‖ αx + βky, z ‖p=‖ αx + ky, z ‖p + ‖ x + βky, z ‖p

for all real numbers k or ((x, z) ⊥Lp (ky, z))(α, β) for all k. So (α, β)−Lp- 2

- norm orthogonality is homogeneous. By similar reasoning we can also get

that (ii) and (iii) imply (α, β)−Lp− 2 - norm orthogonality is homogeneous.

By similar reasoning, we can also get that (ii) and (iii) imply (α, β)−Lp− 2

- norm orthogonality is homogeneous.

The result may be summarized as follows:

THEOREM 4. Let p > 1 and α, β 6= 1. The following are equivalent.

(i) (α, β)− Lp 2 - norm orthogonality is homogeneous,

(ii) (α, β)− Lp 2 - norm orthogonality is symmetric and left additive,

(iii)(α, β)− Lp 2 - norm orthogonality is symmetric and right additive,

(iv) (α, β)− Lp 2 - norm orthogonality is left and right additive.

Finally we give two characterizations of inner product spaces based on

the relation between (α, β) − Lp, 2 - norm orthogonality and Birkhoff 2 -

norm orthogonality.

DEFINITION 3. If (x, z), (y, z) ∈ X × X, we say (x, z) is Birkhoff

orthogonal to (y, z), denoted (x, z) ⊥β (y, z) provided ‖ x + ky, z ‖≥‖ x, z ‖for all real numbers k.


491

THEOREM 5. Let 1 < p ≤ 2 and 0 < α, β < 1 be fixed real num-

bers. Then Birkhoff 2 - norm orthogonality implies (α, β) − Lp 2 - norm

orthogonality in X if and only if X is an 2 - inner product space and p = 2.

PROOF. Let (x, z) ⊥β (y, z). By assumption and homogeneity of

Birkhoff 2 - norm orthogonality we get,

‖ x + y, z ‖p=‖ αx + y, z ‖p + ‖ x + βy, z ‖p − ‖ αx + βy, z ‖p

= (‖ α2x + y, z ‖p + ‖ αx + βy, z ‖p − ‖ α2x + βy, z ‖p)

+(‖ αx + βy, z ‖p + ‖ x + β2y, z ‖p − ‖ αx + βy, z ‖p)− ‖ αx + βy, z ‖p

= (‖ α2x+y, z ‖p + ‖ x+β2y, z ‖p − ‖ α2x+βy, z ‖p)− ‖ αx+β2y, z ‖p + ‖ αx+βy, z ‖p

=‖ α2x + y, z ‖p + ‖ x + β2y, z ‖p − ‖ α2x + βy, z ‖p − ‖ αx + β2y, z ‖p

+(‖ α2x + βy, z ‖p + ‖ αx + β2y, z ‖p − ‖ α2x + β2y, z ‖p)

=‖ α2x + y, z ‖p + ‖ x + β2y, z ‖p − ‖ α2x + β2y, z ‖p .

Thus by induction we see that (x, z) ⊥β (y, z). implies

‖ x + y, z ‖p=‖ αnx + y, z ‖p + ‖ x + βny, z ‖p − ‖ αnx + βny, z ‖p

for n ≥ 1. In the limit this yields (x, z) ⊥p (y, z). implies

‖ x + y, z ‖p=‖ x, z ‖p + ‖ y, z ‖p (A)

If for p = 2 then (A)yields

‖ x + y, z ‖2=‖ x, z ‖2 + ‖ y, z ‖2 (B)

From the definition of 2 - inner product space, we have

‖ x + y, z ‖2= (x + y, x + y/z)

and

‖ x, z ‖2= (x, x/z)

‖ y, z ‖2= (y, y/z)

SINGH ET AL

492

From (B) (x + y, x/z) + (x + y, y/z) = (x, x/z) + (y, y/z)

i.e. (x, x/z) + y, x/z) + (x, y/z)(y, y/z)

= (x, x/z) + (y, y/z) + 2(x, y/z) = 0 ⇒ (x, y/z) = 0

Although in 1 - norm space the proof that identity (A) implies

µp(X) = sup(x,z)⊥B(y,z)

(‖ x, z ‖p +(‖ y, z ‖p)1p

(‖ x + y, z ‖p= 1

which in turn implies that X is an inner product space by the technique

of Amir[1]. But one has to explore whether the same proof will work for

1 < p ≤ 2 in the context of 2 - norm spaces.

The other part is obvious.

THEOREM 6. Let 1 < p ≤ 2 and 0 < α, β < 1, be fixed real num-

bers. Then (α, β) − Lp 2 - norm orthogonality implies Birkhoff 2 - norm

orthogonality in X if and only if X is an inner product space and p = 2.

PROOF. We first prove that if (α, β)−Lp 2 - norm orthogonality implies

Birkhoff orthogonality then X is strict convex. If not then we can choose

x 6= y as extreme points of the units ball of X such that ‖ x, z ‖=‖ y, z ‖=‖ x+y

z, z ‖= 1. Then

‖ x + y

z+y, z ‖p + ‖ α

x + y

z+βy, z ‖p 6=‖ α

x + y

z+y, z ‖p + ‖ x + y

z+βy, z ‖p .

For otherwise 2p + (α + β)p = (α + 1)p + (β + 1)p which requires α = 1 or

β = 1, i.e. (x+yz

, z) is not (α, β)− Lp 2 - norm orthogonal to (y,z). Without

loss of generality we assume α ≥ β. By Theorem 2 we can choose a 6= 0

such that ((x+yz

, z) ⊥Lp (x+yz

+ y, z))(α + β). Hence (x+yz

, z) ⊥Lp ax+yz

+ y, z)

i.e.‖ x+yz

+ k(ax+yz

+ y), z ‖≥‖ x+yz

, z ‖= 1 for all real numbers k. Putting

k = −1/2 yields |α| ≤ 1, and then k = − 1a+2

yields |a+2| ≤ 1. Thus a = −1.

But then (x+yz

, z ⊥Lp ay−xz

, z)(α, β) and then it gives

1+ ‖ α− β

zx+

α + β

zy, z ‖p=‖ α− 1

zx+

α + 1

zy, z ‖p + ‖ 1− β

zx+

1 + β

zy, z ‖p .

So we have

αp = αp ‖ α− β

2αx +

α + β

2αy, z ‖p=‖ α− 1

zx +

α + 1

zy, z ‖p


493

and it follows ‖ α−12α

x + α+12α

y, z ‖= 1.

Writing (y, z) = 1−α1+α

(x) + (1− α−1α+1

)(α−12α

x + α+12α

y).

We see that y is a convex combination of two points of the unit sphere

which is false since y was taken to be an extrem point of the unit ball X.

Thus X must be strictly convex.

Now we prove that if (α, β)−Lp 2 - norm orthogonality implies (α, β)−Lp

2 - norm orthogonality. If not, then there exists (x, z)(y, z) ∈ X × X such

that (x, z) ⊥β (y, z) and (x,z) is not (α, β) − Lp 2 - norm orthogonolity

to (y,z). By Corollary 1 we can choose b 6= 0 such that ((by + x, z) ⊥Lp

(y, z))(α, β). But then (by + x, z) ⊥β (y, z). Thus we have (x, z) ⊥β (y, z)

and (by + x, z) ⊥β (y, z) which cotradicts the left uniqueness of 2 - norm

Birkhoff orthogonality in strict convex space [5], hence 2 - norm Birkhoff

orthogonality implies (α, β) − Lp 2 - norm orthogonality, which is sufficient

for X to be an inner product space and p = 2 by Theorem 5.

The other part is also obvious.

REFERENCES

1. D. Amir, Characterizations of inner product space, Birkhauser, 1986.

2. E. Z. Andalafte, C. R. Deminnie and R. W. Freese, (α, β)- orthogonality

and a characterization of inner product spaces, Math. Japonica, 30

(1985), 341 -344.

3. K. Iseki, Mathematics on 2 - normed spaces, Bull. Korean Math. Soc.,

13 (1976), 127 - 136.

4. R. C. James, Orthogonality in normed linear spaces, Duke Math. J.,

12 (1945), 291 - 302.

5. R. C. James, Orthogonality and linear functions in normed linear

spaces, Trans. Amer. Math. Soc., 61 (1947), 265 - 292.

6. O. P. Kapoor and J. Prasad, Orthogonality and characterization of

inner product spaces, Bull. Austral. Math. Soc., 19(1978), 403 - 416.

SINGH ET AL

494

7. A. Khan and A. Siddiqi, β− orthogonality in 2-normed spaces, Bull

Calcutta Math. Soc., 74(1982), 216 - 222.

8. Zheng Liu, (α, β)− orthogonality and characterization of inner product

spaces, Math. Japonica, 39(1)(1994), 81 - 87.


495

496

The ”pseudo-remaining Cauchy equations”

D.Vivona - M.Divari

Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate

Facolta di Ingegneria

Sapienza - Universita di Roma

16, Via A.Scarpa - 00161 Roma (ITALY)

email:[email protected]

ABSTRACT. In this paper, given a pseudo-addition ⊕ and its corresponding ⊕-

fitting pseudo-multiplication , we consider and solve the pseudo-remaining Cauchy

equations. They are the remaining Cauchy equations, considered by Aczel, where

the ordinary sum and multiplication are replaced by the pseudo-operations. This

is a further application of the pseudo-operations, already used in the solution of

Cauchy equation and in the axiomatic theory of generalized integrals.

A.M.S. SUBJECT CLASSIFICATION (2000): 39B22, 39B62.

KEY WORDS: Cauchy equation, pseudo-operations.

1 Introduction

After the equation f(x + y) = f(x) + f(y), which is considered as the Cauchy

equation, Aczel in [1] studied the three following equations, which are known as the

remaining Cauchy equations:

(I) f(x + y) = f(x) · f(y) ,

(II) f(x · y) = f(x + y) ,

(III) f(x · y) = f(x) · f(y) .

We call them classical remaining equations because they are expressed using

classical operations + and ·.

In the previous paper [4], Benvenuti, Vivona and Divari have studied the Cauchy

497


equation:

f(x⊕ y) = f(x)⊕ f(y) ,

in which the classical operation + is replaced by the pseudo-operation ⊕. We call

it the pseudo-Cauchy equation.

In this paper, using the pseudo-arithmetical operations ⊕ and , we shall study

the following equations:

(I’) f(x⊕ y) = f(x) f(y) ,

(II’) f(x y) = f(x)⊕ f(y) ,

(III’) f(x y) = f(x) f(y),

which we shall call the pseudo-remaining Cauchy equations.

We obtain the solutions of the pseudo-remaining equations by reduction to the

classical remaining Cauchy equations. However, differently from the classical solu-

tions of the remaining Cauchy equations, we consider only positive solutions.

2 Preliminaries.

Let F be the family of all continuous functions f : [0, F ] → [0, F ], with 0 < F ≤+∞.

A binary operation ⊕ : [0, F ]2 → [0, F ] is called pseudo-addition on [0, F ] ([2]) if

the following properties are satisfied:

(A1) x⊕ y = y ⊕ x (commutativity),

(A2) x ≤ x′, y ≤ y′ ⇒ x⊕ y ≤ x′ ⊕ y′ (monotonicity),

(A3) (x⊕ y)⊕ z = x⊕ (y ⊕ z) (associativity),

(A4) xn → x, yn → y ⇒ xn ⊕ yn → x⊕ y (continuity),

(A5) x⊕ 0 = x (neutral element).

These axioms ensure that the pair ([0, F ],⊕) is an I-semigroup. The structure

of I-semigroup is known by virtue of the representation theorem of Mostert and

Shield in [6] (see also [5, 3]). This theorem asserts that there always exist a finite

VIVONA-DIVARI

498

or countable system of open disjoint intervals (αh, βh), h ∈ K, K and corresponding

continuous, bijective and increasing maps gh : [αh, βh] → [0, +∞], with gh(αh) = 0,

such that

x⊕ y =

g−1h

[gh(x) + gh(y) ∧ gh(βh)

], (x, y) ∈ (αh, βh)

2

x ∨ y , otherwise

(1)

and the function gh is determined uniquely up to a multiplicative positive con-

stant.

The interval Ih = (αh, βh) is of two different types with respect to the operation

⊕: in fact, we can have gh(βh) = +∞ or gh(βh) < +∞ [4, 2].

We recall, now, the definition of a ⊕-fitting pseudo-multiplication [2].

Let ⊕ be a given pseudo-addition on [0, F ]. A binary operation : [0, F ] ×[0, F ] → [0, F ] is called a ⊕-fitting pseudo-multiplication if the following properties

are satisfied:

(M1) x 0 = 0 x = 0 (zero element),

(M2) x ≤ x′, y ≤ y′ ⇒ x y ≤ x′ y′ (monotonicity),

(M3) (x⊕ y) z = (x z)⊕ (y z) (left distributivity),

(M4) supn,m (xn ym) = supn(xn) supm(ym) (left continuity).

More, we shall assume

(M5) there exists u ∈ (0, F ] such that u y = y (left unit element).

As we have seen in [2], if ⊕ 6= ∨ and the operation is a ⊕-fitting pseudo-

multiplication with left unit u, there exists k such that u ∈ Ik, gk(βk) = +∞ and

the pseudo-multiplication is uniquely determined on the strip Ik × [0, F ] by :

x y = g−1h

[gk(x) · gh(y)/gk(u)

], (x, y) ∈ Ik × Ih, (2)

h ∈ K.

We look for solutions of the equations (I’), (II’), (III’) satisfying the following

condition:

(∗) if x ∈ Ik then f(x) ∈ Ik .

CAUCHY EQUATIONS

499

Moreover, we shall find only the restrictions of f |Ik.

For the particular index k, the formula (1) becomes

x⊕ y = g−1k

[gk(x) + gk(y)

], (x, y) ∈ (αk, βk)

2. (3)

3 The equation f (x⊕ y) = f (x) f (y)

From now on we shall assume x, y ∈ Ik.

The equation (I’), using (2) and (3), becomes

f

[g−1

k

gk(x) + gk(y)]

= g−1k

[gk(f(x)) · gk(f(y))/gk(u)

].

Putting

gk(x) = ξ and gk(y) = η , (4)

we obtain

fg−1

k (ξ + η) = g−1

k

[gkf(g−1

k (ξ)) · gkf(g−1k (η))/gk(u)

],

and, immediately,

gkfg−1k (ξ + η) = gkfg−1

k (ξ) · gkfg−1k (η)/gk(u). (5)

Set, now,

gkfg−1k = Ψk and gk(u) = λk > 0 . (6)

So, the equation (5) becomes

Ψk(ξ + η) =1

λk

Ψk(ξ) ·Ψk(η) , (7)

which reduces to the remaining equation (I), when λk = 1.

In order to solve the equation (7), it is easy to prove the following

VIVONA-DIVARI

500

Lemma 3.1 The function h(ξ) is a solution of the equation (I)

h(ξ + η) = h(ξ) · h(η) (8)

if and only if the function

Ψk(ξ) = λk · h(ξ) , (9)

is a solution of the equation (7), with λk = gk(u).

We know [1] that the solutions of (8) are given by

∀ ξ > 0 : h(ξ) = ecξ , c ∈ IR . (10)

Now, we are ready to get the first main result:

Theorem 3.2 The class of the solutions of the equation

(I′) f(x⊕ y) = f(x) f(y)

is given by the functions

∀ x ∈ Ik : f(x) = g−1k

λk ec gk(x), c ∈ IR, or f(x) ≡ ak . (11)

Proof . From (6) and (9), we get

gkfg−1k (ξ) = Ψk(ξ) = λk · h(ξ) ,

and from (4)

gkfg−1k (gk(ξ)) = gkf(x) = λk · h(gk(x)) .

So, the class of the solutions of the equation (I’) is given by

f(x) = g−1k

λk · h(gk(x)) .

Replacing the function h(x) with the functions (11), we obtain the assertion. 2

It is easy to see that the functions (11) satisfy the condition (*).

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501

4 The equation f (x y) = f (x)⊕ f (y)

The equation (II’) , using the pseudo-operations (2) and (3), becomes

f[g−1

k

gk(x) · gk(y)/gk(u)]

= g−1k

[gk(f(x)) + gk(f(y))

],

and so

gkf

[g−1

k


= gk(f(x)) + gk(f(y)).

With the same notations as in (4) and (6), we obtain

Ψk

ξ · ηλk

= Ψk(ξ) + Ψk(η) , (12)

which is the remaining equation (II), when λk = 1.


Lemma 4.1 The function h(ξ) is a solution of the equation (II)

h(ξ · η) = h(ξ) + h(η) (13)


Ψk(ξ) = h ξ

λk

, (14)


We know [1] that the solutions of (13) are given by

∀ ξ > 0 : h(ξ) = c log ξ , c ∈ IR . (15)

Now, we are ready to get the second main result:


(II′) f(x y) = f(x)⊕ f(y)


∀x ∈ Ik : f(x) = g−1k

c loggk(x)

λk

, c ∈ IR . (16)

VIVONA-DIVARI

502

Proof. From (6) and (14), we get

gkfg−1k (ξ) = Ψk(ξ) = h

ξ

λk

and from (4)

gkfg−1k (gk(x)) = gkf(x) = h

g(x)

λk

.

So, the class of the solutions of the equation (II’) is given by

f(x) = g−1k h

g(x)

λk

.



5 The equation f (x y) = f (x) f (y)

The equation (III’) , using (2) and (3), becomes

f

[g−1

k


= g−1k

gk(f(x)) · gk(f(y))/gk(u),

and so

gkf

[g−1

k


= gk(f(x)) · gk(f(y))/gk(u).

With the same notations as in (4), we obtain

Ψk

ξ · ηλk

=Ψk(ξ) ·Ψk(η)

λk

, (17)

which is the remaining equation (III), when λk = 1.


Lemma 5.1 The function h(ξ) is a solution of the equation (III)

h(ξ · η) = h(ξ) · h(η) (18)


Ψk(ξ) = λk h ξ

λk

, (19)


CAUCHY EQUATIONS

503

We know [1] that class of the solutions of (18) are given by the functions

∀ ξ > 0 : h(ξ) = ξc , c ∈ IR . (20)

Now, we are ready to get the third main result:


(III′) f(x y) = f(x) f(y)


∀x ∈ Ik : f(x) = g−1k (λ1−c

k gck(x)), c ∈ IR+

0 . (21)

with λk = gk(u).

Proof. From (6) and (19), we get

gkfg−1k (ξ) = Ψk(ξ) = λk h

ξ

λk

and from (4)

gkfg−1k (gk(x)) = gkf(x) = λk h

gk(x)

λk

.

So, the class of the solution of the equation (III’) is given by

f(x) = g−1k

[λk h

gk(x)

λk

].



6 Conclusion

In this paper we have solved the so called ”Pseudo-Remaining Cauchy Equation”,

using the pseudo-operations. We have given three main theorems, based on three

lemmas, which express the solutions of these equations using the solutions of the

classical remaining Cauchy equations.

VIVONA-DIVARI

504

References

[1] J. ACZEL, Lectures on Functional Equations and their Applications, Academic

Press, New Jork, (1969).

[2] P.BENVENUTI - R.MESIAR - D.VIVONA, Monotone set-functions-based in-

tegrals, Handbook in Measure Theory, E.Pap Ed. Elsevier, Cap 33 , (2002),

1329-1379.

[3] P.BENVENUTI - R.MESIAR, Pseudo-arithmetical operations as a basic for

general measure and integration theory, Inf. Sciences, 160 (2004), 1-11.

[4] P.BENVENUTI - D.VIVONA - M.DIVARI, The Cauchy equation on I-

semigroup, Aequationes Math. XX (2002), 1-11.

[5] E.P.KLEMENT - R.MESIAR - E.PAP, Triangular Norms, Kluver Academic

Publishers, Dordrecht (2000).

[6] P.S.MOSTERT - A.L.SHIELDS, On the structure of semigroup on a compact

manifold with boundary, Ann.of Math. 65 (1957), 117-143.

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TABLE OF CONTENTS, JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOLUME 3, NO.4, 2008 NEUMANN AND MIXED BOUNDARY VALUE PROBLEMS, A.KUMAR, R.PRAKASH,………………………………………………………..399 THE CONSTRUCTION OF A KIND OF QUADRATURE FORMULAS, MING-CAI LIU, P.ZHAO, W.YANG,……………………………………………419 STABILITY OF A NUMERICAL ALGORITHM FOR NON-STATIONARY TRANSPORT EQUATION, O.MARTIN,………………………………………...427 ASYMPTOTIC DISTRIBUTION OF THE SAMPLE AVERAGE VALUE-AT-RISK IN THE CASE OF HEAVY-TAILED RETURNS, S.STOYANOV, S.RACHEV,……………………………………………………...443 TRANSPORT PROCESSES IN NETWORKS WITH SCATTERING RAMIFICATION NODES, A.RADL,…………………………………………………………………461 (a,b)-Lp 2-NORM ORTHOGONALITY AND CHARACTERIZATIONS OF 2-INNER PRODUCT SPACES, V.SINGH, S.KUMAR, A.SINGH,…………………………485 THE “PSEUDO-REMAINING CAUCHY EQUATIONS”, D.VIVONA, M.DIVARI,…………………………………………………………...497

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JOURNAL OF APPLIED FUNCTIONAL ANALYSIS · Approximation Theory,Nonlinear Operators. 28) Vassilis Papanicolaou Department of Mathematics National Technical University of Athens Zografou