Nonlinear boundary value problems for multivalued differential equations in Banach spaces

0362-546X YQ S.OU+ .W C 1990 Peigamon Press plc

NONLINEAR BOUNDARY VALUE PROBLEMS FOR MULTIVALUED DIFFERENTIAL EQUATIONS IN BANACH SPACES

GIUSEPPE MARINO

Dipartimento di Matematica, Universita della Calabria, 87036 Arcavacata di Rende, Cosenza, Italy

(Received 19 December 1988; received for publication 20 March 1989)

Key words and phrases: Boundary value problems, multivalued differential equations, nonlinear operators, fixed point theorems.

1. INTRODUCTION

THIS PAPER is concerned with the existence of solutions for the multivalued differential system

WL) x’

t

- A(t, x)x E F(f, x)

Lx = H(x)

with F: J x X -0 Xmultivalued map, J = [a, b] compact real interval, X Banach space. A(t, x) is a continuous operator on X for each (t, x) E J x X, L: C(J, X) + X is a linear continuous operator from the space of all continuous functions x: J -+ X, whose range is contained in X, H: C(J, X) + X is a continuous operator, nonlinear in general.

Such problems have been studied repeatedly in the literature with different hypotheses on A, F, L, H. The pioneering work with conditions similar to the ones used in the present paper, but for ordinary differential equations, is the note by Scrucca [l], where the operator A was independent from the state variable x E X. Anichini [2] has studied analogous problems (in the context of ordinary differential systems in IR”) in which the operator A also depends on x E IR”. In the case of multivalued differential systems, Kartsatos [3] has also considered boundary value problems for V-valued differential equations, but over an unbounded time interval. Zecca and Zezza [4] have extended the work of Kartsatos to differential equations in Banach spaces; in both [3] and [4] the operator A is dependent only on t E J.

Recently, Papageorgiu [5] has established the existence of mild solutions on a compact interval with the operator A depending only on t E J but in general is unbounded.

The fundamental tools used in the existence proofs of all above mentioned works are essen- tially fixed-point theorems: Schauder’s theorem in [ 11; Eilenberg-Montgomery’s theorem in [2]; KY-Fan’s theorem in [3, 4, 5). Here we use a fixed-point theorem due to Martelli [6].

The idea of the present paper has originated from the study of an analogous problem examined by Anichini and Conti [7] for ordinary differential systems in m” and Carbone et al. [8] for multivalued differential systems, again in iR”.

2. NOTATIONS AND DEFINITIONS

In the following we will label: J = [a, b], compact interval of the real line IR. X = Banach space with norm Iv!, u E X.

545

546 G. MARINO

C(J, X) = Banach space of continuous functions from J into X with norm

[[xl/, := max((x(t)l: t E JJ.

B(X) = Banach space of bounded linear operators from X into X with norm

IITII := sup((Tul: [VI = 1).

L’(J, X) = Banach space of Bochner integrable functions defined on J with values in X, normed by /IfIll := jfiblf(t)l dt.

AC(J, X) = Banach space of absolutely continuous functions defined on J with values in X. bcf(X) = the set of all bounded, closed, convex and nonempty subsets of X. A multivalued function G: X - X is called upper semicontinuous (USC) on X if for each

u E X the set G(u) is a nonempty, closed subset of X, and if for each open set Vof X containing G(u), there exists an open neighborhood U of u such that G(U) s V.

We want to point out that if the multivalued map G is nonempty compact values, upper semicontinuity is equivalent to the conditions of being a closed graph map (i.e. V, 4 u, y, -+ y, _Y,, E G(u,) imply y E G(u)) and of transforming compact sets into relatively compact sets.

If G: X - X is a multivalued map, by IG(u)l we mean the sup(lyj:y E G(u)). An USC map G: X -0 X is said to be condensing [6] if for any subset Iv C X with c&V) # 0,

we have o(G(N)) < a(N), where CY denotes the Kuratowski’s measure of noncompactness 191. We remark that a compact map, i.e. an USC map G such that G(N) is relatively compact for

every bounded subset N E X, is the easiest example of a condensing map. A multivalued map G: J + bcf(X) is said measurable if for each u E X the distance between

u and G(t) is a measurable function on J. Finally, for each y E C(J, X), the set of L’-selections Sh,Y of a multivalued map G: X - X

is defined by

S’ G,y := (f, E L’(J, X): &(t) E G(t,y(t)) a.e. for t E JJ.

This may be empty. It is nonempty if and only if the function Y: J + II? defined by

Y(t) := inf((ul: u E G(t,y(t))J belongs to L’(J, m) [5].

3. STATEMENT OF THE PROBLEM AND PRELIMINARY RESULTS

Let us consider the following multivalued boundary value problem

WLL) E A(t, x)x + zqt, x)

and we assume that the hypotheses (h,)-(h,) below hold.

(h,). A: J x X --* B(X), (t, u) - A(t, u) is a continuous function on J x X with values in B(X) such that

(i) V r > 0 3 r, = TV > 0 such that Iv/ I r =) (IA(t, u)I( 5 r, v t E J, v u E X.

Remark 1. If X = R”, the hypothesis (hi) (i) is an immediate consequence of the continuity of A. Analogously (hi) (i) is satisfied if A: J x X -, B(X) is a compact map.

Nonlinear boundary value problems 547

Remark 2. From (h,) we are able to claim the existence for any fixed u E C(J, X), of a unique function E,: J x J + B(X), (t, s) y E,(t, s), defined and continuous on J x J such that

(evolution operator of A) A,(t) := A(t, u(t)) (see e.g.

From (3.1) one has

E”(f, t) = 1,

and moreover

s I

E,(t, s) = I + 4,(w, s) dw (3.1) s

where, as usually, I stands for the identity operator on X and

[lOI).

E”V. d&6, r) = Jw, r), (t, s, r) E J x J x J

@W, G/at) = A,(OJ%O, s) a.e. for t E J, v s E J.

From this it follows that the (Caratheodory) solutions of the linear homogeneous equation

x’ = A,(t)x

are defined in J and form a space isomorphic to X via the map

j, : X --t Ker D, , where D, := (d/dt) - AU), defined by is(v) := E,( * , s)v

for every s E J fixed [12]. In particular, for each u E X the function E,( -, a)v satisfies

(d/dt)E,(t, a)u = A.(t)E,(t, a)v.

Remark 3. From (h,) it follows also that u E C(J, X) implies A, E C(J, B(X)) and

IIn, - ~Ollca + 0 =) II4, - Au,&, := max(/AU.(?) - A.,(t)ll: t E J) -+ 0.

(A,). F: J x X + bcf(X), (t, u) - F(t, u) is a multivalued map measurable with respect to t for each u E X, USC with respect to u for each t E J and such that the set of selections Sk,” of F is nonempty for each fixed u E C(J, X).

At this point it is useful to quote the following three lemmas will be crucial in the proof of our main result.

LEMMA 3.1 [6]. Let G: X -* bcf(X) be a condensing map. If the set M := (u E X: JJ.J E G(u) for some A > 1) is bounded, then G has a fixed point, i.e. there exists u E X such that u E G(u).

LEMMA 3.2 [ll]. Let w(t, z) be a continuous real function defined on J x I? such that the initial value problem for the equation

z’(t) = c&9 z(t))

has the unique solution z(t) for t E J. Then, if lx’(t)] I w(l, Ix(t)l) for every 1 E J and if Ix(a)1 5 z(a), we have Ix(t)] I z(t) for every t E J.

548 G. MARINO

LEMMA 3.3 [4]. Let F be a function satisfying (h,) and let I- be a linear continuous mapping fron t’(J, X) to C(J, X), then the operator r * Sk: C(J, X) + bcf(C(J, X)),

Y - (f * G)(Y) := W,,)

is a closed graph operator in C(J, X) x C(J, X).

A fundamental hypothesis for the proof of main result is the following.

W IN, WI + IF(t, u)l 5 o(t, Iv]) v t E J and v u E X, where o is the function ir lemma 3.2.

Now we are able to state the following first result for the initial value nonlinear multivalued problem.

THEOREM 3.1. Assume that F and A satisfy (h,), (h,) and (hJ. Moreover, if x0 E X, (EJt,, , a)x, + jfp E,(t, s)f,(.s) ds: u E B C C(J, X) bounded] is relatively compact for fO E J. Then the system

(NCL) E A(1, x)x + F(t, x)

= x, has at least one solution.

Proof, We define the multivalued map S: C(J, X) - C(J, X) by

S(u) := (

“f S”*& E C(J, X), S,J”(t) := &(f, @$I +

I E,(t> Y)~,(Y) dY: f, E $.u .

L1 1

From remark 2 it follows that each sUJU satisfies the linearized problem

(LC)” x’

I

E A,(t)x + F(t, x)

x(a) = x,

so that any (possible) fixed point for S is solution of (NCL). We will show that S has fixed points.

Step 1. S(U) is convex for every fixed u E C(J, X). Indeed, if s,,~,.~, s~J*.” belong to S(U) and Orc~ l,then

C%J-,.” + (1 - c)k& = %(c/,,“+U-clfi,.)

and this last function belongs S(U) also, being Sk,, convex (because F is convex-values).

Step 2. S sends bounded sets into relatively compact sets. First, we show that S sends bounded sets into bounded sets. For this purpose, it is appropriate to write every sUJU as

t

i

f

%J”U) = x0 + MY)~,~~,(Y) dy + a

/t(u) dy.


In this way we obtain

Is,,&)l 5 I%l + s

‘IIAAy)ll IsUJU(y)I dy + bl.fu(~~I dy 5 (from (M (I I2

5 hl + bLk-e ‘Is,~.(Y)~ dy + llwul!oD I 0

here I(o,IJ, := max(lo(t, u(t)): t E 4. By Gronwall inequality and hypothesis (A,) (i) thus follows

II4.n 5 r =$ IIS”J”II oD s (lx01 + R,exp(r,(b - a)) =: J

here S& := max(Ilo,ll,: I/&_, I r]. We prove now that S sends bounded sets into equicontinuous sets. Let t,, tz E J, t, < t,.

Then

so that

lqf”(4) - %4J”(~2)I 5 r

‘%%,(u)llIs,~.(r)~ dy + ‘21f,(~)l dy 11 s tt

l140a 5 r =D b”J”(M - ~“J”Wl 5 r,Yt* - t*) + Q,(b - t*) hence we have that S sends bounded sets into equicontinuous sets.

Step 3. S has closed graph. Let U, 4 uo, sUmfn E S&J, s,,~” + so. We shall prove that so E S(u,). It is enough to show that there exists f. E Si,U, such that so is solution of

x(t) = xo + ’ A,,(Y)x(Y) dy + I‘

;u,I[,,

s ‘A,(Y) dy

Of course, there exists r > 0 such that I)u,,ll,, IIs .~~,I[_,, llsOllm s r. Moreover, from

I-%“(O~“.J”(~) - h&bo(~)l 5 Ik, - Adollcoll~Ollca + II~“.llaJ~U,,,, - ~oIIoM it follows that as

u,(t) := s

I f A,,(Y)~,,J,(Y) dy and u,(t) :=

a 5 &,(y)so(~) dy, then IIu, - voll -) 0.

0

Hence w,(t) := j:f,(y) dy is such that (Iw,, - (so - x0 - u,)II, -+ 0. From lemma 3.3 applied to integral operator r, we obtain that there existsf, E SL,U, such that

s

t S,(l) = x0 + &,(Y)~o(Y) dy +

s ‘A,(y) dy, c.v.d.

a II

It remains to prove (thank lemma 3.1) that M := {u E C(J, X): AU E S(U), A > I), is bounded to conclude that S has fixed points, i.e. that (NCt) has solutions.

For this, let Au E S(U), A > 1. Then u satisfies the system

t

u’ E A&)u + A-‘F(t, u)

u(a) = Pxo. (3.2)

550 G. MARINO

Let US consider the corresponding initial value problem

t

2’ = o(t, 2)

t(a) = I&l. Now, (3.2) yields

(3.3)

Since lu(a)l _( z(a), from lemma 3.2 we obtain Ilull m 5 I).& where z is the unique solution of (3.3), so that M is bounded. n

We return now to quote the boundary value hypotheses for the main result. From now on we denote zUJU as the function defined by

(h,). L: C(J, X) + X is a linear and continuous operator.

(h,). H: C(J, X) --+ X is a continuous operator such that: (i) V r > 0 3 r, = r2(r): jlullrn 5 r =0 Ill(u)) 5 r,.

(ii) 3 11 z=+ 0: 1(&~(u) - K,Lz,&(a)J 5 q vf, E Sk,” and v u E C(J, X), where KU is the operator defined in (h6).

(h6). For every u E C(J, X) fixed, there exists a linear continuous operator Ku: X -+ Ker V, (where VU: C(J, X) -+ C(J, X), y - V,y is the linear continuous operator defined by

(V,u)(t) := u(t) - E”Q, Ma)) such that

(i) K: C(J, X) --t B(X, C(J, X)), u - Ku is a continuous function (ii) V r > 0 3 M = m(r) > 0: IIull_ s: r =0 llK,jj 5 m

(iii) (I - LK,)(H(u) - Lz,~“) = 0 v u E C(J, X). v f, E Sk,, .

(h,) [(&(H(u) - Lz,,~,))(~,,) + ~,,,~(i,,): u E B c C(J, X) bounded] is relatively compact for t, E J.

Remark 4. We note explicitly that if we denote with D, the operator defined by

(Q&t) := (d/dl)M) - &(t)~(t),

then Ker K,, = Ker D, (remark 2).

Definition. A (Caratheodory) solution of the problem (NLL) is any function x E AC(J, X) which satisfies x’ E A(t, x(t))x(t) + F(t, x(t)) a.e. on J and Lx = H(x).

Finally, we quote the following result useful in the proof of main result.


LEMMA 3.4 [Ill. Suppose that g, h E C(J, It?), p E L’(J, IT?), p 5: 0 a.e.,

Then

4. MAIN RESULT

Now we can prove the following.

THEOREM 4.1. Suppose that (h,)-(h,) hold. Then the problem (MU) admits at least one solution.

Proof. Step 1. IIz& 5 r =D 3 r3 = rs(r) > 0: jlEujlm := max(I&(t,s)II: (t,s) E J] 5 r3. Indeed, from (3.1) we obtain if s 5 t (analogously if I < s):

II&U,s)II I 1 + ’ IIA,WllII~,Wd~ dw s 5

which, applying Gronwall’s inequality, yields

IE,(t, s)ll 5 ew( ~~hCw)ll dw) 5 exp( c[~k,twIll dw) . From (h,) (i) we have thus

ll~ll~ 5 r =D ljEUllm 5 exp(r,(b - a)) =: r3.

Step 2. E,,(t, s) is continuous with respect to u, i.e.

IIu, - ~t,!rn + 0 =D II%,, - &,\I, * 0.

Indeed, let IIu, - u,,lloD + 0. Then there exists r > 0 such that IIu,,ll,, IIu~IJ_ 5 r. hloreover, ifs 5 t (analogously if t < s), we have

Ik,(t. ~1 - Et&, 911 5 (f rom (3.1)) 5 s ‘lEun(w, ~)IIIMu,(w) - &,(w)ll dhv 5

+ s ‘bL,(wII b%,(w, ~1 - &c,(w, s>II dw, I

in such a way that, for lemma 3.4 and step 1 we obtain

II&.(t, s) - &,(t, s)ll 5 &IU. - &J,@ - a)(1 + rl exp(r,@ - 4)); the claim follows thus from remark 3.

To prove that (NU) has solutions, we define the map R: C(J, X) -s C(J, X) by

R(u) := lK,(H(u - Lz,J + zuJy: fu E %,l.

We shall prove that R has fixed points and that such fixed-points are solutions of (XX).

552 G. MARINO

Step 3. R(u) is convex for any fixed U. Indeed, this immediately follows from linearity of K, and from the fact that if 0 I c 5 1, then

CZ”.fl.. + (1 - Ck2.” = z~,cfl.“+(1-c)h.u~

Sk,. being a convex set.

Step 4. Every X, E R(u) is solution of the linearized problem

X’

L

E A,(t)x + F(t, u(t))

Lx = H(u).

Indeed, from K,(X) E Ker t;, , it follows

Moreover x:,(t) = -&(t)&(t) + f”(t). (4.1)

Lx, = L(K,(H(u) - zufu)) + Lz,~,, = ((h6) (iii)) = H(u) - Lz,,~, + Lz,,, = H(u).

An immediate consequence of step 4 is that the fixed points of R are solutions of (NLL). We shall prove the existence of fixed points for R using lemma 3.1.

Step 5. R has closed graph. Let u, + u,,, y,, E R(u,), y,, -+ yo. We must prove that y,, E R(u,). First of all, let r > 0 such that

II&II,, II%ll% lIY?J~, IlYollm 5 r*

Now, Yn E R(u,) =D 3 f,, E Sk,.. :

Y, = &,,(~(u,) - Lz,,,,-,) + z,,,,/,, .

It is enough to show that there exists& E Sk,U, such that

YO = &,(H(uo) - Lz,o,~o) + z,,,/, . (4.2)

To this purpose, we consider the sequence jjn E C(J, X) defined by

ylz := K,J(Q - &,J%,,,, + zu,,,~n. (4.3)

We claim that Ily, - Y~,II_ + 0. Indeed,

t lu,(t) - _?rD)l 5 I3

E,,,(t, s> - &,V, -r>V,,W d.s + kK&zu,,~~ - &,Lzu,,dt)l (I

5 (from (&N 5 II&,, - &,,llmllq,,Il& - 0) + bL,Uz,,,,~, - L,,~,,)II~

+ IIK,,L - K,,oLkuo.Jlm 5 Wd (iii)

5 IL%, - J%olI,II~,,Ilco(~ - 0) + ~ll~llllZu,f” - Z”O*fJn

+ IlKu, - &II llLll k,~,,lla


= llcd, - c4011&4L”Ilce(~ - 1 a + mllL II /I\ ‘k”“O, 4 - E”,(f, W,(d d.7 LLI I/ m

+ IlKi. - Ku,ll IIL II iii

’ LOU, 4.m d.T I/

5 II&, - E,,n~ll%iim~~ - 4 + mll~lllii. - ~“oll,llW,“ll,~b - 4 + IlKI, - &II lI~IlhL,llcn(~ - 4

and this last expression tend to zero for step 2 and hypothesis (h6) (i), being IJoUJLp --* (Io~~,,)~_, . Thus

IIY, - 9,ll, + 0. (4.4)

We define now the operator r: L’(J, X) + C(J, X) by

I-v := zuo,u - K,&z,o.u (4.5)

is a linear continuous operator, for

ll~4lm 5 (1 + 4wIIZ”0.“Ilm 5 (1 + ~ll~II~lI~~,ollmll~Il~~. From lemma 3.3 it follows that I- * S> is a closed graph operator in C(J, X) x C(J, X). Moreover

kJ%u,) - K,,H(u,)ll m 5 knH(~,) - K,,H(u,)ll, + lk,,N~,) - K,$(u,)L.,

5 II&, - K,,llk-&)l + ~kL,~~~~(~,) - H(u,)l

so that the hypotheses (h,) and (h6) (i) yield

u, -+ uo *KJfW + K,,fOo). (4.6)

From definitions (4.3) and (4.5) we have jj,, - K,,H(u,) -) y, - K,,H(u,) and being I- * Si a closed graph operator, this (together with the hypothesis u, -+ u,) permit to affirm that y, - K,,H(u,) E lT - S)(u,), i.e. the claim (4.2).

Step 6. R maps bounded sets into relatively compact sets. Indeed,

lluII= 5 r =D Ik(H(u) - k,,,) + z,,~& 5 m(lH(u)l + llL1Illz,,&) + ll~,~,,llm

5 (V+)(i)) 5 lflr2 + ~IILIIll&llcDll~ullm + ll&llml141m 5 (Step 1) 5 mr, + r311coUIJ,(m(ILJ( + 1) 5 mr2 + (mllL\I + l)r,Q, := r,.

Moreover, from (4.1) and (h,) (i) it follows that if /lull.. s r, then

kK(H(u) - LG.,,) + z,,,,)k 5 ~1r4 + fir.

For Ascoli-Arzela theorem, this is enough to see that R maps bounded sets into relatively compact sets.

Step 7. R is a compact map (and therefore, a fortiori, a condensing map). This is an immediate consequence of steps 6 and 7. At this point, we introduce the set

M:= (u E C(J,X): lu E: R(u) for some ,J > 1).

554 G. hlARIN0

Step 8. u E M =o /u(a)1 < r;l. Indeed,

uEM=IqfuES~,U: Au = K,(H(u) - Lz,,,,) + zuJu

* 44 = A-‘(K,W(u) - Lz,,~,,) + tufu)(a>

=D ((b) (ii)) lu(a)l < rl.

Step 9. R has fixed points (and therefore, for step 4, (lVLL) has solutions). In order to apply lemma 1 we consider the initial value problem

z’

I

= o(t, t)

z(a) = rl. (4.7)

Now, u E A4 =o ,lu’ = A,(.)h + f, =D

bW)t -( /AU, W))Wt + ~-‘k,W~ < tA(t, W))W~ + t_fu(t)i

5 ((h3)) 5 Ml, IWI).

So, by lemma 2, we have Ilull_ s ]~z~~,, where z is the unique solution of (4.7) in J, thus M is bounded. w

5. APPLICATIONS

Example 1. Let J = [0, 11, X = lR2 normed by Il(xl,x2)]l := Ix,] + IxJ, then B(X) = M2x2, $e ;anr;h ,;lgebra of the real 2 x 2 matrices B = (bjj) normed by (JBIJ = max((b,,( + Ib2,J,

12+ 2.2.

Moreover, let g and p be two functions from J x R into iR. We assume:

(1) g is a bounded continuous function on J x IR,

Ml, := SUP(l&, x)1: ct, x) E J x RI.

(II) Ed is a continuous function for which there exist a function y E C(J, iR) and a constant d > 0 such that

IP(&x)~ 5 y(t) + &I.

We consider the matrix function A: J x iR2 + MzX2 defined by

and let

EuU, 4 = EL’@, s) E,“(t, s)

E,z’(?, s) l!g2(t, s) >

be the evolution operator of A, depending on (u,, u2)’ = u E C(J, IF?‘). Now, we introduce the multivalued map Z? J x R -, R by the law

F(t, x) := [V(f, x) - t, fJ$t, x) + t].

We want to look for the solutions of the second-order nonlinear multivalued differential equation

Y’ - g(t, x)x’ E F(t, x) (5.1)


with boundary conditions of the type

us: expK g(w, x(w)) dw > > do ~(s, x(s)) ds = x(1)

j:(j: exp(j: &I(“‘, x(w)) d+&, x(s)) ds) dt = j.’ x’(t) dt.

(5.2)

The equation (5.1) can be written as

x’ E A(t, x)x + F(?, x)

where x = (x,, x2) belongs to lR* and

Finally, we introduce the operators L and H from C(J, R*) on lR* by

With these positions, the multivalued differential problem (5.1)-(5.2) can be equivalently for- mulated as

WLL) E A(t, x)x + F(t, x)

,

To prove the existence of solutions for the problem (5.1)-(5.2) is thus sufficient to see that are satisfied the hypotheses (hi)-(&).

Step 1. (hi) is satisfied. Trivial by (I).

Step 2. (A,) is satisfied. Easy calculation (we recall that if dimX < co, then Sk,. # 4 for any fixed u E C(J, X), [13]).

Step 3. (h,) is satisfied. Let CY := 1 + llgll_ + d, /? := IIyll_ + 1, cu(t, z) := cxz + p. Then the unique solution of the initial value problem

t

2’ = w(t, z)

z(0) = 20

is given by z(t) = (z. + (P/a) exp(at) - (~/cY). Moreover, if x = (xi, x2)‘, then

II4h x)x11 + IIF(t,x)ll =

5 II-d + Ikllcallxll + 11~11.. + dllxll + 1 = M,x).

556 C. MARINO

Step 4. (h4) is satisfied. Clear.

Step 5. (h,) is satisfied. First, it is straightforward to verify that -t nu 1

i (I exp g(w, u,(w)) dw du

E,(t, s) = 5 s > t

0 em g(w> u,(W) dw 5 > i

and so H is a continuous operator since

If we note that

(5.3)

for every selection u;‘,f,‘) = f, E Sk,., then (5.3) yields

IIH(u) - ~z,,~,/l 5 ll~IIll~,llm 5 ((4.1)) 5 IILII exp(1 -t lldm).

Step 6. (h6) is satisfied. Let (u,, u2)’ = u E C(J, R2) fixed. We define the operator K,,:R2+KerVUby

where

( b) IR

a, ’ E ‘,

(.i s

pu(s) := exp s(ut u,(u)) du . 0 >

It is a routine calculation to verify that the range of Ku is Ker D, = Ker VU and that LK, = I on R2, so that (h6) (iii) is obviously satisfied. Moreover u - Ku is a continuous function, as it is easy to see by definition of Ku. Finally, for each u E C(J, IR’), we have

llK,ii -( 2 + ikll,(l - exp(- llgll~.) exp(lldl,), so that (h6) (ii) is satisfied also.

Example 2. Let J, g, p, A, p, F be as in the previous example. Let to E J fixed. Then the multivalued differential problem

- g(t, x)x’ E F(t, x)

= sin(ll - x(f,) + ~‘(t~)l)“~

Nonlinear boundary value problems

can be written as x’

i

E A(t, x)x + F(t, x)

Lx = H(x)

where H: C(J, iR2) * C(J, R2) is defined by

557

(5.4)

and L: C(J, R2) 4 C(J, R2) is defined by

Then the hypotheses (h,)-(h6) are satisfied with

and so the problem (5.4) has at least one solution. In general, if f,, f2 are two bounded continuous functions from C(J, IT?) x C(J, 7) into R,

then the multivalued differential system

A-” - g(t, x)x’ E F(r, x)

x(O) = f*kx')

x'(O) =fz(x, x')

(5.5)

can be written in the form (5.4) with H defined by

As above one can verify that (hi)-@,) are satisfied, so that the problem (5.5) admits solutions.

REFERENCES

1. SCRUCCA E., Un problema ai limiti quasi lineare in spazi di Banach, Afri Arcad. nuz. Lincei R. XLII, 361-364 (1967).

2. ANICHINI G., Nonlinear problems for systems of differential equations, Nonlinear Analysis 1, 691-699 (1977). 3. KARTSATOS A., Locally invertible operators and existence problems in differential systems, T&&u .Wrfh. J. 28,

167-176 (1976). 4. ZECCA P. & ZEZZA P. L., Nonlinear boundary value problems in Banach space for multivalued differential equa-

tions in a noncompact interval, Nonlinear Analysis 3, 347-352 (1979). 5. PAPAGEORGW N. S., Boundary value problems for evolution inclusions, Commentat. Math. Univ. Carol. 29.

355-363 (1988). 6. MARTE~LI M., A Rothe’s type theorem for non-compact acyclic-valued maps, Boll. CJn. mar. ital. 4 (Suppl. Fast.

3), 70-76 (1975). 7. ANICHINI G. & CONTI G., Boundary value problems with nontinear boundary conditions, Nonlinearify 1 (1988).

5.58 G. tVIARIN0

8. CARBONE A., CONTI G. & MARINO G., A nonlinear boundary value problem for multivalued differential systems, to appear in Atti Semin. mat. fis. Univ. Modena.

9. KIJRATOWSKI C., Topologie. Monografie Mafhematyczne 20, Warsaw (1958). 10. CON~I R., Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. (in.

mat. ifal. XXII, 135-178 (1967). 11. HALE J. K., Ordinary Differenfial Equarions. Interscience, New York (1969). 12. KREIN S. G., Linear differential equations in Banach spaces, Trans. Mafh. Monagr. Am. mafh. Sot. fransi. 29

(1971). 13. LASO~A A. & OPML Z., An application of the Kakutani-KY-Fan theorem in the theory of ordinary differential

equations, BUN. Acad. Pol. Sci. Se+. Sri. math. ask phys. 13, 781-786 (1965).

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Nonlinear boundary value problems for multivalued differential equations in Banach spaces