Differentiability of solutions of the second order abstract Cauchy problem

Semigroup Forum Vol. 64 (2002) 472–488c© 2001 Springer-Verlag New York Inc.

DOI: 10.1007/s002330010092

RESEARCH ARTICLE

Differentiability of Solutions of theSecond Order Abstract Cauchy Problem

Hernan R. Henrıquez and Carlos H. Vasquez∗

Communicated by Jerome A. Goldstein

Abstract

In this paper we study the differentiability of solutions of the second orderabstract Cauchy problem when the underlying space is reflexive or at least hasthe Radon-Nikodym property. We consider the linear and semilinear case.

Key words and phrases: Cosine Functions of Operators; Abstract CauchyProblem; Differentiability of Solutions.2000 Mathematics subject classification: 47D09, 34G10.

1. Preliminaries.

In this work we are concerned with regularity properties of solutions of thesecond order abstract Cauchy problem. We refer the reader to [5], [11] for thetheory of cosine functions of operators and the associated second order Cauchyproblem. Next we establish some basic facts that will be needed in the sequel.

Let X be a Banach space endowed with a norm ‖ · ‖ . Henceforth C(t)is a strongly continuous operator cosine function with infinitesimal generatorA . We denote by S(t) the sine function associated with C which is definedby

S(t)x :=

∫ t

0

C(s)x ds, x ∈ X, t ∈ R. (1.1)

We refer the reader to [9] for the necessary concepts related with the integrationof vector valued functions. In particular, it is well known that A is a closedlinear operator with domain D(A) dense in X . Furthermore, henceforth wewill use frequently the relations

C(t)x− x = A

∫ t

0

S(s)x ds (1.2)

∗This work was supported by DICYT-USACH, Project 04-9633HM and FONDECYT,project 1970716.

Henrıquez and Vasquez 473

for all x ∈ X and

C(t)x− x =

∫ t

0

S(s)Axds (1.3)

for all x ∈ D(A) ([11], Proposition 2.3).

We denote by [D(A)] the space D(A) endowed with the graph norm

‖x‖A = ‖x‖+ ‖Ax‖, x ∈ D(A).

Moreover, in this work the notation E stands for the space formed by the vectorsx ∈ X for which the function C(·)x is of class C1 . It was proved by Kisinsky[8] that E endowed with the norm

‖x‖1 = ‖x‖+ sup0≤t≤1

‖AS(t)x‖, x ∈ E,

is a Banach space. The operator valued function G(t) =[C(t) S(t)AS(t) C(t)

]is a

strongly continuous group of linear operators on the E ×X generated by the

operator A =[

0 IA 0

]defined on D(A)× E .

Additional terminology and notations are those generally used in func-tional analysis. In particular, L(X) stands for the Banach space of boundedlinear operators from X into X and X∗ denotes the dual space of X .

The existence of solutions of the second order abstract Cauchy problem

x′′(t) = Ax(t) + h(t), 0 ≤ t ≤ a, (1.4)

x(0) = x0, x′(0) = x1, (1.5)

where h: [0, a] → X is an integrable function, has been discussed in [12].Similarly, the existence of solutions of the semilinear second order abstractCauchy problem it has been treated in [13]. We only mention here that thefunction x(·) given by

x(t) = C(t)x0 + S(t)x1 +

∫ t

0

S(t− s)h(s) ds, 0 ≤ t ≤ a, (1.6)

is called a mild solution of (1.4)–(1.5). In the case in which x0 ∈ E then x(·)is continuously differentiable and

x′(t) = AS(t)x0 + C(t)x1 +

∫ t

0

C(t− s)h(s) ds. (1.7)

The purpose of this work is to study the existence of the second derivative ofthe mild solution x(·). To get to this goal the paper is organized as follows.In the next section we establish some related properties of differentiability forcosine functions. Section 3 is devoted to discuss a pair of concepts of regularityfor the mild solution of problem (1.4)–(1.5) whilst in section 4 we establish aresult of regularity for mild solutions of the semilinear problem.

474 Henrıquez and Vasquez

2. Differentiability properties of cosine functions

The aim of this section is to characterize the elements x ∈ X for which certaindifferentiability properties of C(·)x or S(·)x is fulfilled. Initially we generalizea well known result for strongly continuous semigroups of linear operators. Butfirst we need some elementary properties of cosine functions.

Proposition 2.1. Suppose C(·) is a strongly continuous cosine function ofoperators with infinitesimal generator A . Then the following properties hold:

(a) For all x ∈ X ,

limt→0

1

t2

∫ t

0

S(s)x ds =1

2x.

(b) An element x ∈ D(A) if and only if there exists y ∈ X such that

C(t)x− x =∫ t

0S(s)y ds .

(c) For fixed x ∈ X and t ∈ R ,

limτ→0

∫ t

0

S(s)C(τ)x− x

τ2ds =

C(t)x− x2

.

Proof. Assertion (a) follows easily from (1.1) whereas (b) is an immediateconsequence of (a) and the relation (1.3). To prove part (c), applying (1.2) andthe properties of operator A we can write∫ t

0

S(s)C(τ)x− x

τ2ds =

1

τ2

∫ t

0

S(s)A

∫ τ

0

S(ξ)x dξ ds

=1

τ2A

∫ t

0

S(s)

∫ τ

0

S(ξ)x dξ ds

=1

τ2A

∫ τ

0

S(ξ)

∫ t

0

S(s)x ds dξ

=1

τ2

∫ τ

0

S(ξ)A

∫ t

0

S(s)x ds dξ

=1

τ2

∫ τ

0

S(ξ)(C(t)x− x) dξ

and the result follows now from assertion (a).

If an element x ∈ D(A), then

Ax = 2 limt→0+

C(t)x− xt2

.

Thus, the function C(t)x−xt2 is bounded as t → 0+ . Proceeding as in [1],

Theorem 2.1.2, we can prove the following converse.


Theorem 2.1. Assume that X is reflexive. Let C(·) be a strongly conti-

nuous cosine function with infinitesimal generator A . If lim inft→0+ ‖C(t)x−xt2 ‖

<∞ , then x ∈ D(A) .

Proof. Under the condition of the theorem it follows that there exists asequence (τn)n convergent to 0 such that the set {C(τn)x−x

τ2n

: n ∈ N} is bounded.

Since X is reflexive, by passing to a subsequence we may assume that C(τn)x−xτ2n

converges to some element y ∈ X as n → ∞, in the weak topology. Hence we

also obtain that S(s)C(τn)x−xτ2n

→ S(s)y, as n → ∞, for all s ≥ 0, in the weak

topology. Thus, for a fixed x∗ ∈ X∗, using the statement (c) of the precedingProposition we have∫ t

0

⟨x∗, S(s)

C(τn)x− xτ2n

⟩ds→

⟨x∗,

C(t)x− x2

⟩,

as n → ∞ . Furthermore, applying the Lebesgue’s dominated convergencetheorem on the left hand side we obtain∫ t

0

⟨x∗, S(s)

C(τn)x− xτ2n

⟩ds→

∫ t

0

〈x∗, S(s)y〉 ds =

⟨x∗,

∫ t

0

S(s)y ds

⟩,

as n→∞ . Hence, it follows that

C(t)x− x = 2

∫ t

0

S(s)y ds

which, by Proposition 2.1 (b), implies that x ∈ D(A).

Employing the Eberlein-Smulian theorem and arguing as above we cansubstitute the reflexivity of X by a compactness condition. For the sake ofbrevity we omit the proof.

Proposition 2.2. Let C(·) be a strongly continuous cosine function withinfinitesimal generator A . If { 1

t2 (C(t)x − x): 0 < t ≤ 1} is relatively weaklycompact then x ∈ D(A) .

In this work we have special interest in Lipschitz continuous functions. Inthe sequel, for a strongly continuous function F : [0,∞)→ L(X) and x ∈ X wesay that F (·)x is locally Lipschitz continuous if the function t→ F (t)x satisfiesa Lipschitz condition on each interval [0, a], a > 0. Since cosine functionsare uniformly bounded on bounded intervals, using the d’Alembert functionalrelation we easily can prove the following property.

Remark 2.1. If C(·)x is Lipschitz continuous on an interval [0, b], for someb > 0, then C(·)x is locally Lipschitz continuous.

On the other hand, it is well known that every reflexive space has theRadon-Nikodym property (abbreviated, RNP) and that there are non reflexive


spaces that have the RNP. We refer to [3] for several characterizations of theRNP. By this reason in the sequel we consider spaces that have the RNP.

Theorem 2.2. Assume that X has the RNP. Let C(·) be a strongly con-tinuous cosine function with infinitesimal generator A . Let x ∈ X be anelement such that C(·)x is locally Lipschitz continuous. Then x ∈ E .

Proof. We introduce the set E formed by all elements u ∈ X such that thefunction C(·)u is locally Lipschitz continuous. It is clear that E is a subspaceof X . For u ∈ E we define

‖|u|‖ = ‖u‖+ sup

{‖C(t)u− C(s)u‖|t− s| : 0 ≤ s, t ≤ 1, s 6= t

}.

It is easy to see that E endowed with the norm ‖| · |‖ is a Banach space.Furthermore, if u ∈ E , then u ∈ E and ‖u‖1 = ‖|u|‖ . This shows that E is aclosed subspace of E .

On the other hand, since C(·)x ∈W 1,∞([0, 1];X) we can write

C(t+ h)x− C(t)x =

∫ t+h

t

f(s) ds,

where f(s) = ddsC(s)x is defined a.e.

From the theory of cosine functions we know that S(r)x ∈ E, for allr ∈ R . Next we prove that 1

rS(r)x → x, r → 0+ , in the topology of E . Infact,

C(t+ h)− C(t)

h

(1

rS(r)x− x

)=

1

h

∫ t+h

t

(1

rS(r)− I

)f(s) ds.

For each ε > 0, from the Lusin’s Theorem ([9]) we infer the existence of acompact set Kε ⊆ [0, 1] with m([0, 1]\Kε) ≤ ε and a continuous functiong: [0, 1]→ X such that f(s) = g(s), for s ∈ Kε , and ‖g‖∞ = ‖f‖∞ . Therefore,from the above relation we obtain

C(t+ h)− C(t)

h

(1

rS(r)x− x

)=

1

h

∫[t,t+h]∩Nε

(1

rS(r)− I

)f(s) ds

+1

h

∫[t,t+h]∩Kε

(1

rS(r)− I

)g(s) ds,

where we have denoted Nε = [0, 1]\Kε . Now, from the theory of cosinefunctions we can assert that 1

rS(r) is uniformly bounded, so that we can take‖ 1rS(r) + I‖ ≤M1, 0 ≤ r ≤ 1, and that 1

rS(r)y → y, r → 0. In addition, thisconvergence is unifom for y in a compact set. Since the range of g is compact


then there is rε > 0 such that ‖ 1rS(r)g(s) − g(s)‖ ≤ ε , for r ≤ rε and all

0 ≤ s ≤ 1. Hence we easily compute∥∥∥∥∣∣∣∣1rS(r)x−x∣∣∣∣∥∥∥∥ =

∥∥∥∥1

rS(r)x−x

∥∥∥∥+ sup0≤t,t+h≤1

∥∥∥∥C(t+ h)−C(t)

h

(1

rS(r)x−x

)∥∥∥∥≤∥∥∥∥1

rS(r)x− x

∥∥∥∥+ εM1‖f‖∞ + ε.

Since ε was arbitrarily chosen this proves our assertion and shows thatx ∈ E .

Related with this result we notice that if a map F : I → L(X), whereI ⊆ R is an interval, is such that F (·)x is Lipschitz continuous for all x ∈ X,then F is Lipschitz continuous for the norm of operators. In particular, if C(·)xis locally Lipschitz continuous for all x ∈ X, then C(·) is uniformly continuousso that X = D(A).

Next we consider similar properties for the sine function. Initially we

present a new approach for the quotients S(t)x−txt3 . In [2] it was proved, for

a general Banach space, that x ∈ D(A) if and only if limt→0S(t)x−tx

t3 exists.Now, for reflexive Banach spaces we can give a weaker condition.

Theorem 2.3. Assume that X is reflexive. Let C(·) be a strongly con-tinuous cosine function with infinitesimal generator A and associated sine

function S(·) . Let x ∈ X be an element such that lim inft→0+ ‖S(t)x−txt3 ‖ <∞,

then x ∈ D(A) .

Proof. Proceeding as in the proof of Theorem 2.1 we can choose a sequence

tn → 0 such that {S(tn)x−tnxt3n

: n ∈ N} is bounded and, by passing to a

subsequence, S(tn)x−tnxt3n

converges in the weak topology to some element y ∈ X .

On the other hand, if we set

xn =2

t3n

∫ tn

0

∫ t/2

0

∫ t/2

0

C(r)C(s)x dr ds dt

it was proved in [2] that xn → 16x, n→∞, and Axn = S(tn)x−tnx

t3n. Since A is

closed we conclude that x ∈ D(A).

Such as it was mentioned for Proposition 2.2, an application of Eberlein-Smulian’s theorem yields the following result.

Proposition 2.3. Let C(·) be a strongly continuous cosine function withinfinitesimal generator A and associated sine function S(·) . Let x ∈ X be an

element such that {S(t)x−txt3 : 0 < t ≤ 1} is relatively weakly compact. Then

x ∈ D(A) .


Proceeding in similar way as above we can obtain the following results, that westate without proof.

Theorem 2.4. Assume that X is reflexive. Let C(·) be a strongly con-tinuous cosine function with infinitesimal generator A and associated sine

function S(·) . Let x ∈ E be an element such that lim inft→0+‖AS(t)x‖

t < ∞,then x ∈ D(A) .

Proposition 2.4. Let C(·) be a strongly continuous cosine function withinfinitesimal generator A and associated sine function S(·) . Let x ∈ E be

an element such that {AS(t)xt : 0 < t ≤ 1} is relatively weakly compact. Then

x ∈ D(A) .

Theorem 2.5. Assume that X has the RNP. Let C(·) be a strongly con-tinuous cosine function with infinitesimal generator A and associated sinefunction S(·) . Let x ∈ E be an element such that AS(·)x is locally Lipschitzcontinuous. Then x ∈ D(A) .

Proof. We begin by observing that AS(r)x ∈ E , for all r ≥ 0. In fact, from

C(t)AS(r)x = AC(t)S(r)x

=1

2[AS(t+ r)x−AS(t− r)]

we infer that C(·)AS(r)x is locally Lipschitz continuous so that this statementfollows from Theorem 2.2.

On the other hand, since X has the RNP, the function AS(·)x is diffe-rentiable a.e. Assume that the derivative there exists at some t > 0. From therelation

AS(h)x = AS(t+ h− t)x= AS(t+ h)C(t)x−AC(t+ h)S(t)x

we obtain that

AS(h)x

h= C(t)A

S(t+ h)x− S(t)x

h− C(t+ h)− C(t)

hAS(t)x.

Taking limit in this expression as h → 0+ , the first term on the right handside converges by our assumption and the second one converges because y =

AS(t)x ∈ E so that C(·)y is continuously differentiable. Consequently, S(h)xh

→ x and AS(h)xh converges as h → 0+ . Since A is closed it follows that

x ∈ D(A).

We complete this section with some applications to the differentiabilityof functions.


Example 2.1. Let X be a Banach space and let T (·) be the group oftranslations on the space Y = Lp(R;X), 1 ≤ p < ∞, defined by T (t)f(s) =f(s + t). Then C(t) = 1

2 (T (t) + T (−t)) yields a strongly continuous cosinefunction with infinitesimal generator A . Since A = B2 , where B is the infi-nitesimal generator of semigroup T (·) ([5]) then A is defined by Af = f ′′ onthe domain D(A) = W 2,p(R;X) and the space E = W 1,p(R;X). Moreover,the associated sine function is given by

[S(t)f ](s) =

[∫ t

0

C(ξ)f dξ

](s)

=

∫ t

0

[C(ξ)f ](s) dξ

=1

2

∫ t

0

(f(s+ ξ) + f(s− ξ)) dξ

=1

2

∫ s+t

s−tf(ξ) dξ. (2.1)

Applying our results to this example we obtain the following properties.

Proposition 2.5. Let f ∈ Lp(R;X), 1 ≤ p <∞ . Then f ∈ W 2,p(R;X) ifand only if

limt→0

f(s+ t)− 2f(s) + f(s− t)t2

(2.2)

exists in Lp(R;X) . In this case the limit is equal to f ′′ .

Proof. The limit in (2.2) is the characterization of elements f ∈ D(A).

Proposition 2.6. Assume that X is reflexive. Let f ∈ Lp(R;X), 1 < p <∞, be a function such that

lim inft→0+

‖f(s+ t)− 2f(s) + f(s− t)‖pt2

<∞,

then f ∈W 2,p(R;X) .

Proof. Since Lp(R;X) is reflexive ([3]) this assertion follows from Theo-rem 2.1.

The case p = 1 requires a different condition.

Proposition 2.7. Let f ∈ L1(R;X) be a function such that the set{f(s+ t)− 2f(s) + f(s− t)

t2: 0 < t ≤ 1

}is relatively weakly compact. Then f ∈W 2,1(R;X) .


Proof. This assertion follows from Proposition 2.2.

We also can generalize a result established in [2].

Proposition 2.8. Assume that X is reflexive. Let f ∈ Lp(R;X), 1 < p <∞, be a function such that

lim infh→0+

1

h3

∥∥∥∥∥∫ s+h

s−hf(ξ) dξ − 2hf(s)

∥∥∥∥∥p

<∞,

then f ∈W 2,p(R;X) .

Proof. This assertion follows from Theorem 2.3 and expression (2.1).

We also can establish a similar property for p = 1.

Proposition 2.9. Let f ∈ L1(R;X) be a function such that the set{1

h3

(∫ s+h

s−hf(ξ) dξ − 2hf(s)

): 0 < h ≤ 1

}

is relatively weakly compact. Then f ∈W 2,1(R;X) .

Proof. This assertion is consequence of (2.1) and Proposition 2.3.

3. Differentiability of solutions of inhomogeneous equations

In this section we study the regularity of solutions of the second order abstractCauchy problem (1.4)–(1.5). Initially we consider the existence of classicalsolutions. At least we explicitly state a different situation. in this part wealways assume that h is a continuous function.

Definition 3.1. We say that a function x: [0, a]→ X, a > 0, is a classicalsolution of the Cauchy problem (1.4)–(1.5) if x is a function of class C2 thatsatisfies the equation (1.4) and the initial conditions (1.5).

Applying the reduction to first order system we obtain.

Theorem 3.1. Let x0 ∈ D(A) and x1 ∈ E . Assume that at least one of thefollowing conditions hold:

(a) h ∈ C([0, a];X) ∩ L1([0, a];E) ;

(b) h ∈W 1,1([0, a];X) .

Then the mild solution x(·) of (1.4)–(1.5) is a classical solution and x: [0, a]→E is continuously differentiable.


Proof. We define z(t) =[x(t)x′(t)

]. Then z is the mild solution of the first

order system

z′(t) = Az(t) + h(t), z(0) =

[x0

x1

]∈ D(A),

where h(t) =[

0h(t)

].

If h satisfies condition (a) then h ∈ L1([0, a]; [D(A)]) and the assertionfollows from Corollary 4.2.6 in [10]. In case (b), h ∈ W 1,1([0, a];E × X) andthe assertion is consequence of Theorem 1 in [6]. In both cases the functionx: [0, a]→ E is of class C1 .

To abbreviate our next statements we introduce some additional nota-tions. In the sequel we denote by M a positive constant such that ‖C(t)‖ ≤M,0 ≤ t ≤ a, and we represent by V (h) the variation of h on [0, a] and by V (t, h)the variation of h on the interval [0, t] . In further, for t ≥ 0 we put Tth to des-ignate the translation of h which is defined by Tth(s) = h(s+ t), 0 ≤ s+ t ≤ b,and Tth(s) = h(b), b ≤ s+ t . Finally, for a fixed h , we use the notations

u(t) =

∫ t

0

S(t− s)h(s) ds, (3.1)

v(t) =

∫ t

0

C(t− s)h(s) ds. (3.2)

We begin by establishing some preliminary lemmas.

Lemma 3.1. Assume that X is a reflexive space. Let C(·) be a strong-ly continuous cosine function on X and let h: [0, a] → X be a continuousfunction of bounded variation such that V (Tth − h) → 0, t → 0+ . Then theRiemann-Stieltjes integral

w(t) =

∫ t

0

C(t− s) dsh =

∫ t

0

C(s) dsh(t− s)

exists in the weak topology and defines a continuous function w: [0, a]→ X .

Proof. Let Λ: X∗ → C defined by

Λ(x∗) =

∫ t

0

〈C(t− s)∗x∗, dsh〉.

The Riemann-Stieltjes integral in the above expression exists because C(·)∗x∗is a continuous function ([5]) and h has bounded variation ([7]). Moreover, Λis linear and

|Λ(x∗)| ≤M‖x∗‖V (h).


Consequently, Λ ∈ X∗∗ and in view of that X is reflexive we infer the existenceof w(t) ∈ X such that Λ(x∗) = 〈x∗, w(t)〉 , for all x∗ ∈ X∗ .

On the other hand, from the relations

w(t+ τ)− w(t) =

∫ t+τ

0

C(s) dsh(t+ τ − s)−∫ t

0

C(s) dsh(t− s)

=

∫ t

0

C(t− s) ds[h(τ + s)− h(s)] +

∫ τ

0

C(t+ s) dsh(τ − s)

it follows that

‖w(t+ τ)− w(t)‖ ≤MV (Tτh− h) +MV (τ, h).

Using our hypotheses and Proposition I.2.9 in [7] we obtain from this estimationthat w(·) is right continuous at t . Similarly, one can prove that w is continuousat t .

The following result is a direct consequence of definitions by which weomit the proof.

Lemma 3.2. Let h: [0, a] → X be a step function. Then u(t) ∈ D(A), thefunction Au(·) is continuous, v(·) is piecewise smooth and v′(t) = Au(t)+h(t),a.e.

Theorem 3.2. Assume that X is reflexive. Let x0 ∈ D(A) and x1 ∈ Eand let h be a continuous function of bounded variation on [0, a] such thatV (Tth − h) → 0, t → 0+ . Then the mild solution of (1.4)–(1.5) is a classicalsolution.

Proof. We consider a sequence (hn)n of step functions, where each hn is thefunction h(t1)χ[t0,t1] +

∑ni=2 h(ti)χ(ti−1,ti] . In this expression we have chosen

the points ti = an i, i = 0, 1, . . . , n, and χI represents the characteristic function

associated to an interval I . It is clear that the sequence (hn)n converge uniform-ly to h . Let un, respectively vn , be the functions given by (3.1), respectively(3.2), with hn instead of h . Then, un → u and vn → v, as n→∞, uniformlyon [0, a] . Moreover, by Lemma 3.2 we have that un(t) ∈ D(A).

On the other hand, if we fix 0 < t ≤ a and n ∈ N , then t ∈ (tj−1, tj ], forsome j = 1, . . . , n . From our definitions we can write

Aun(t) = A

j−1∑i=1

∫ ti

ti−1

S(t− s)h(ti) ds+A

∫ t

tj−1

S(t− s)h(tj) ds

=

j−1∑i=1

[C(t− ti−1)− C(t− ti)]h(ti) + [C(t− tj−1)− I]h(tj)


=

j−1∑i=1

C(t− ti−1)[h(ti)− h(ti−1)] + C(t− tj−1)[h(t)− h(tj−1)]

+ C(t− tj−1)[h(tj)− h(t)] + C(t)h(0)− h(tj) (3.3)

so that‖Aun(t)‖ ≤MV (h) + (M + 1)‖h‖∞.

Hence it follows that (Aun(t))n is a bounded sequence. Consequently, there isa subsequence which converges to z(t) ∈ X in the weak topology. This impliesthat u(t) ∈ D(A) and z(t) = Au(t). An standard argument shows that thefull sequence (Aun(t))n converges to Au(t). For each x∗ ∈ X∗ , from (3.3) weobtain

〈x∗, Aun(t)〉 =

j−1∑i=1

〈C(t− ti−1)∗x∗, h(ti)− h(ti−1)〉

+ 〈C(t− tj−1)∗x∗, h(t)− h(tj−1)〉+ 〈C(t− tj−1)∗x∗, h(tj)− h(t)〉+ 〈C(t)∗x∗, h(0)〉 − 〈x∗, h(tj)〉

and taking limit in this expression as n→∞ it follows that

〈x∗, Au(t)〉 =

⟨∫ t

0

C(s)∗x∗, dsh(t− s)⟩

+ 〈C(t)∗x∗, h(0)〉 − 〈x∗, h(t)〉.

An application of Lemma 3.1 yields

Au(t) =

∫ t

0

C(t− s) dsh(s) + C(t)h(0)− h(t)

so that Au(·) is a continuous function. On the other hand, from Lemma 3.2 wehave

v′n(t) = Aun(t) + hn(t), a.e., n ∈ N,so that for each x∗ ∈ X∗ we obtain

d

dt〈x∗, vn(t)〉 = 〈x∗, v′n(t)〉 = 〈x∗, Aun(t)〉+ 〈x∗, hn(t)〉.

Using the fact that Au(·) is a continuous function we infer that

〈x∗, v(t)〉 =

∫ t

0

〈x∗, Au(s) + h(s)〉 ds =

⟨x∗,

∫ t

0

Au(s) + h(s)

⟩ds

which in turn implies that

v(t) =

∫ t

0

(Au(s) + h(s)) ds

so that x(t) = C(t)x0 + S(t)x1 + u(t) is a classical solution of (1.4)–(1.5).


In a particular case we can omit the reflexivity of the space X .

Proposition 3.1. Let x0 ∈ D(A) and x1 ∈ E and let h(t) = ϕ(t)y , whereϕ: [0, a] → C is a continuous function of bounded variation and y ∈ X . Thenthe mild solution of (1.4)–(1.5) is a classical solution.

Proof. We proceed as in the proof of Theorem 3.1 by reducing (1.4)–(1.5)to a first order system and then we apply a result of Eidelman and Tichonov[4].

Related with the cosine function C we introduce a class L of functionsdefined as follows. We say that a function h: [0, a] → X belongs to L if h isintegrable, bounded and there exists a constant L ≥ 0 such that

‖[C(t2)− C(t1)]h(s)‖ ≤ L|t2 − t1|,

for all 0 ≤ s, t1, t2 ≤ a .

Theorem 3.3. Assume that X is reflexive. Let x0 ∈ D(A) and x1 ∈ Eand let h ∈ L be a continuous function. Then the mild solution of (1.4)–(1.5)is a classical solution.

Proof. By Theorem 2.2 the values h(s) ∈ E . Thus, if y ∈ R(h) then C(·)yis continuously differentiable and

‖AS(t)y‖ = ‖C ′(t)y‖ ≤ L, (3.4)

for all 0 ≤ t ≤ a . Let fix 0 < t ≤ a . We chose step functions hn, n ∈ N, suchthat R(hn) ⊆ R(h) and the sequence (hn)n converges pointwise to h . By theprevious estimation we have that

‖AS(t− s)hn(s)‖ ≤ L, 0 ≤ s ≤ t, n ∈ N,

which shows that (AS(t−s)hn(s))n is a bounded sequence. Since X is reflexiveand A is closed we conclude that AS(t − s)hn(s) → AS(t − s)h(s), as n →∞ , in the weak topology. Furthermore, the values AS(t − s)h(s) belong tothe clousure of the separable set { 1

τ (C(t − s + τ) − C(t − s))h(s): τ > 0,0 ≤ s ≤ t} . Consequently, from the Pettis theorem we infer that the functions→ AS(t−s)h(s) is measurable. Employing the boundedness condition (3.4) itfollows that s→ AS(t−s)h(s) is integrable. Hence we obtain that u(t) ∈ D(A)and

Au(t) =

∫ t

0

AS(t− s)h(s) ds

is continuous. We complete the proof applying the Proposition 3.6 in [12].

Proceeding in similar way as for first order systems we define a weakerconcept of solution ([10]) .


Definition 3.2. A function x: [0, a]→ X is a strong solution of (1.4)–(1.5) ifx ∈W 2,1([0, a];X), the equation (1.4) is satisfied a.e. and the initial conditions(1.5) are verified.

We begin with some preliminary properties of mild solutions.

Lemma 3.3. Let y(·) be the mild solution of problem (1.4)–(1.5) then∫ t0y(s) ds ∈ D(A) and

A

∫ t

0

y(s) ds = y′(t)− x1 −∫ t

0

h(s) ds,

for all t ≥ 0 .

The proof is an easy consequence of (1.6) and will be omitted.

Proposition 3.2. If x is a strong solution of (1.4)–(1.5) then x is the mildsolution, x ∈ L1([0, a]; [D(A)]) and x0 ∈ E . In further, if x0 ∈ D(A) andx1 ∈ E , then u ∈W 2,1([0, a];X) ∩ L1([0, a]; [D(A)]) .

Proof. Let y(·) be the mild solution of (1.4)–(1.5). Applying the previouslemma we can write

x ′(t)− y ′(t) = A

∫ t

0

(x(s)− y(s)) ds.

Hence, if z(t) =∫ t

0(x(s) − y(s)) ds we have that z′′(t) = Az(t), t ≥ 0 with

z(0) = z′(0) = 0, which implies that z(t) = 0 and x(t) = y(t), a.e. Sincex, y are continuous functions it follows that x(t) = y(t), for all t ≥ 0. Fromthe Definition 3.2 it is clear that x ∈ L1([0, a]; [D(A)]) . Moreover, since x′ iscontinuous we obtain that C(·)x0 is a function of class C1 which implies thatx0 ∈ E . Finally, since u(t) = x(t)−C(t)x0 − S(t)x1 the stated properties of uare immediate consequences of that already established for x .

We also can establish a condition in order the mild solution to be a strongsolution. We need the following extension of Lemma 3.2 in [12].

Lemma 3.4. Let h: [0, a]→ X be an integrable function. Then

limτ→0

1

τ2

[∫ t+τ

t

S(t+ τ − s)h(s) ds+

∫ t−τ

t

S(t+ τ − s)h(s) ds

]= h(t), a.e.

Proposition 3.3. Assume that x0 ∈ E and h ∈ L1([0, a];X) . Let x be themild solution of (1.4)–(1.5). Then x is a strong solution if one of the followingconditions is satisfied:

(i) x ∈W 2,1([0, a];X) ;

(ii) x ∈ L1([0, a]; [D(A)]) .


Proof. Proceeding as in [12], Proposition 3.4, we can write

2C(τ)− I

τ2x(t) =

x(t+ τ)− 2x(t) + x(t− τ)

τ2− 1

τ2

∫ t+τ

t

S(t+ τ − s)h(s) ds

− 1

τ2

∫ t−τ

t

S(t+ τ − s)h(s) ds.

In case (i), applying Proposition 2.5, Lemma 3.4 and the above expression weobtain that x(t) ∈ D(A), a.e. and that Ax(t) = x′′(t)− h(t), a.e.

In case (ii), turning to apply Lemma 3.4 we obtain that

limτ→0

x(t+ τ)− 2x(t) + x(t− τ)

τ2

exists a.e. Moreover, since∥∥∥∥2C(τ)− I

τ2x(t)

∥∥∥∥ ≤M‖Ax(t)‖, 0 ≤ t ≤ a,

applying the Lebesgue’s dominated convergence theorem it follows that the

quotient x(t+τ)−2x(t)+x(t−τ)τ2 converges in L1([0, a];X) to Ax(t)+h(t) as τ → 0.

From Proposition 2.5 we obtain that x ∈ W 2,1([0, a];X) which completes theproof that x is a strong solution.

Corollary 3.1. Let h ∈ W 1,1([0, a];X) and h = h , a.e. Then the mildsolution x of (1.4)–(1.5) is a strong solution.

Proof. From Theorem 3.1 it follows that x(·) is the classical solution ofproblem

x′′(t) = Ax(t) + h(t), x(0) = x0, x′(0) = x1.

This shows that x satisfies the conditions of the preceding proposition.

4. Differentiability of solutions of semilinear equations

We complete this work with an application of the preceding results to theexistence of solutions of the semilinear problem

x′′(t) = Ax(t) + f(t, x(t), x′(t)) t ≥ 0, (4.1)

with initial condition (1.5). In the sequel we assume that A generates a cosinefunction C and that f : [0, a]×X ×X → X is a continuous function.

Definition 4.1. A function x: [0, a]→ X is a mild solution of (4.1)–(1.5) ifx is continuously differentiable and satisfies the integral equation

x(t) = C(t)x0 + S(t)x1 +

∫ t

0

S(t− s)f(s, x(s), x′(s)) ds, t ≥ 0.


The question of existence of solutions of problem (4.1)–(1.5) it has been studiedin [13] under several conditions on A and f . In particular, if x0 ∈ E and f islocally Lipschitz continuous, which means that for all r > 0, there is a constantK ≥ 0 such that

‖f(s, x, y)− f(t, x, y)‖ ≤ K (|t− s|+ ‖x− x‖+ ‖y − y‖) ,

for every 0 ≤ s, t ≤ a and every x, y, x, y ∈ X such that ‖x − x‖ ≤ r and‖y − y‖ ≤ r , there is a unique mild solution.

Next, proceeding as in [6] for first order systems, we establish a result ofexistence of classical solution.

Theorem 4.1. Assume that X has the RNP, x0 ∈ D(A), x1 ∈ E and thatf is locally Lipschitz continuous. Then the mild solution x(·) of (4.1)–(1.5) isa classical solution.

Proof. Since

x′(t) = AS(t)x0 + C(t)x1 +

∫ t

0

C(t− s)f(s, x(s), x′(s)) ds

using the Lipschitz condition of f and the Gronwall lemma it follows that x′(·)is locally Lipschitz continuous which in turn implies that h(t) = f(t, x(t), x′(t))is also Lipschitz continuous. Since X has the RNP then h ∈ W 1,1([0, a];X)and the assertion is now consequence of Theorem 3.1.

Acknowledgment

The authors wish to thank to the referees for their comments and suggestions.

References

[1] Butzer P. L. and H. Berens, “Semi-Groups of Operators and Approxima-tion”, Springer-Verlag, Berlin, 1967.

[2] Cioranescu, I., On twice differentiable functions, Result. Math. 16 (1/2)(1989), 49–53.

[3] Diestel J. and J. J. Uhl, “Vector Measures”, Amer. Math. Society, 1972.

[4] Eidelman, Y. S. and I. V. Tichonov, A note on the abstract Cauchy problem,Semigroup Forum 54 (1997), 112–116.

[5] Fattorini, H. O., “Second Order Linear Differential Equations in BanachSpaces”, North-Holland, Amsterdam, 1985.

[6] Goldstein G. R. and J. A. Goldstein, Regularity for semilinear abstractCauchy problems, Lect. Notes in Pure and Applied Maths. 178, MarcelDekker, New York, 1996, 99–105.


[7] Honig, C. S., The abstract Riemann-Stieltjes integral and its applicationsto linear differential equations with generalized boundary conditions, Notasdo Instituto de Matematica E Estatıstica da Universidade de Sao Paulo,Serie Matematica No1, 1973.

[8] Kisynski, J., On second order Cauchy’s problem in a Banach space, Bull.Acad. Polon. Sci. Ser. Sci. Math. Astronm. Phys. 18 (1970), 371–374.

[9] Marle, C. M., “Mesures et Probabilites”, Hermann, Paris, 1974.

[10] Pazy, A., “Semigroups of Linear Operators and Applications to PartialDifferential Equations”, Springer-Verlag, New York, 1983.

[11] Travis, C. C. and G. F. Webb, Second order differential equations in Banachspace, Proc. Internat. Sympos. on Nonlinear Equations in Abstract Spaces,Academic Press, New York, 1987, 331–361.

[12] Travis, C. C. and G. F. Webb, Compactness, regularity, and uniform con-tinuity properties of strongly continuous cosine families, Houston J. Math.3 (4) (1977), 555–567.

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Universidad de SantiagoDepartamento de MatematicaCasilla 307, Correo 2Santiago, [email protected]@impa.br

Received October 23, 2000and in final form June 1, 2001Online publication October 1, 2001

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Differentiability of solutions of the second order abstract Cauchy problem