Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 798067, 37 pdf

We first present several existence results and compactness of solutions set for the following Volterra type integral inclusions of the form: yt ∈ t 0a t − sAys Fs, ysds, a.e.. However v

Volume 2010, Article ID 798067, 37 pages

doi:10.1155/2010/798067

Research Article

Some Results for Integral Inclusions of

Volterra Type in Banach Spaces

R P Agarwal,1, 2 M Benchohra,3 J J Nieto,4 and A Ouahab3

1 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA

2 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals,

Dhahran 31261, Saudi Arabia

3 Laboratoire de Math´ematiques, Universit´e de Sidi Bel-Abb`es, B.P 89, Sidi Bel-Abb`es 22000, Algeria

4 Departamento de Analisis Matematico, Facultad de Matematicas, Universidad de Santiago de Compostela, Santiago de Compostela 15782, Spain

Correspondence should be addressed to R P Agarwal,agarwal@fit.edu

Received 29 July 2010; Revised 16 October 2010; Accepted 29 November 2010

Academic Editor: M Cecchi

Copyrightq 2010 R P Agarwal et al This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

We first present several existence results and compactness of solutions set for the following Volterra

type integral inclusions of the form: yt ∈ t

0a t − sAys  Fs, ysds, a.e t ∈ J, where

J  0, b, A is the infinitesimal generator of an integral resolvent family on a separable Banach space E, and F is a set-valued map Then the Filippov’s theorem and a Filippov-Wa ˙zewski result

are proved

1 Introduction

In the past few years, several papers have been devoted to the study of integral equations onreal compact intervals under different conditions on the kernel see, e.g., 1 4 and referencestherein However very few results are available for integral inclusions on compact intervals,see5 7 Topological structure of the solution set of integral inclusions of Volterra type isstudied in8

In this paper we present some results on the existence of solutions, the compactness

of set of solutions, Filippov’s theorem, and relaxation for linear and semilinear integralinclusions of Volterra type of the form

y t ∈

t

0

a t − sAy s  Fs, y sds, a.e t ∈ J : 0, b, 1.1

where a ∈ L10, b, R and A : DA ⊂ E → E is the generator of an integral resolvent family defined on a complex Banach space E, and F : 0, b × E → PE is a multivalued map.

In 1980, Da Prato and Iannelli introduced the concept of resolvent families, which can

be regarded as an extension of C0-semigroups in the study of a class of integrodifferentialequations9 It is well known that the following abstract Volterra equation

An important kernel is given by

a t  f1−αte −kt , t > 0, k ≥ 0, α ∈ 0, 1, 1.4where

is the Riemann-Liouville kernel In this case1.1 and 1.2 can be represented in the form

of fractional differential equations and inclusions or abstract fractional differential equations

and inclusions Also in the case where A ≡ 0, and a is a Rieman-Liouville kernel, 1.1 and

1.2 can be represented in the form of fractional differential equations and inclusions, see forinstants25–27

Our goal in this paper is to complement and extend some recent results to the case ofinfinite-dimensional spaces; moreover the right-hand side nonlinearity may be either convex

or nonconvex Some auxiliary results from multivalued analysis, resolvent family theory,and so forth, are gathered together in Sections 2 and 3 In the first part of this work, weprove some existence results based on the nonlinear alternative of Leray-Schauder type

in the convex case, on Bressan-Colombo selection theorem and on the Covitz combinedthe nonlinear alternative of Leray-Schauder type for single-valued operators, and Covitz-Nadler fixed point theorem for contraction multivalued maps in a generalized metric space

in the nonconvex case Some topological ingredients including some notions of measure

of noncompactness are recalled and employed to prove the compactness of the solutionset in Section 4.2 Section 5 is concerned with Filippov’s theorem for the problem 1.1

In Section 6, we discuss the relaxed problem, namely, the density of the solution set ofproblem1.1 in that of the convexified problem.

A function y : J → E is called measurable provided for every open subset U ⊂ E, the set

y−1U  {t ∈ J : yt ∈ U} is Lebesgue measurable A measurable function y : J → E

is Bochner integrable ify is Lebesgue integrable For properties of the Bochner integral,

see, for example, Yosida28 In what follows, L1J, E denotes the Banach space of functions

y : J → E, which are Bochner integrable with norm

Denote byPE  {Y ⊂ E : Y / ∅}, PclE  {Y ∈ PE : Y closed}, P b E  {Y ∈ PE : Y

bounded}, PcvE  {Y ∈ PE : Y convex}, PcpE  {Y ∈ PE : Y compact}.

2.1 Multivalued Analysis

LetX, d and Y, ρ be two metric spaces and G : X → PclY be a multivalued map A single-valued map g : X → Y is said to be a selection of G and we write g ⊂ G whenever

g x ∈ Gx for every x ∈ X.

G is called upper semicontinuous (u.s.c for short) on X if for each x0 ∈ X the set Gx0

is a nonempty, closed subset of X, and if for each open set N of Y containing Gx0, there

exists an open neighborhood M of x0such that GM ⊆ Y That is, if the set G−1V   {x ∈

X, G x∩V / ∅} is closed for any closed set V in Y Equivalently, G is u.s.c if the set G1V   {x ∈ X, Gx ⊂ V } is open for any open set V in Y.

The following two results are easily deduced from the limit properties

Lemma 2.1 see, e.g., 29, Theorem 1.4.13 If G : X → Pcp x is u.s.c., then for any x0∈ X,

lim sup

x → x0

Lemma 2.2 see, e.g., 29, Lemma 1.1.9 Let Knn∈N ⊂ K ⊂ X be a sequence of subsets where

K is compact in the separable Banach space X Then

co

lim sup

where co C refers to the closure of the convex hull of C.

G is said to be completely continuous if it is u.s.c and, for every bounded subset A ⊆ X,

G A is relatively compact, that is, there exists a relatively compact set K  KA ⊂ X such that GA  ∪ {Gx, x ∈ A} ⊂ K G is compact if GX is relatively compact It is called locally compact if, for each x ∈ X, there exists U ∈ Vx such that GU is relatively compact.

G is quasicompact if, for each subset A ⊂ X, GA is relatively compact.

Definition 2.3 A multivalued map F : J  0, b → PclY is said measurable provided for every open U ⊂ Y, the set F1U is Lebesgue measurable.

Lemma 2.5 see 31, Theorem 19.7 Let Y be a separable metric space and F : a, b → PY a

measurable multivalued map with nonempty closed values Then F has a measurable selection.

Lemma 2.6 see 32, Lemma 3.2 Let F : 0, b → PY be a measurable multivalued map and

u : a, b → Y a measurable function Then for any measurable v : a, b → 0, ∞, there exists a

measurable selection f v of F such that for a.e t ∈ a, b,

Corollary 2.7 Let F : 0, b → P cp Y be a measurable multivalued map and u : 0, b → E a

measurable function Then there exists a measurable selection f of F such that for a.e t ∈ 0, b,

2.1.1 Closed Graphs

We denote the graph of G to be the set GrG  {x, y ∈ X × Y, y ∈ Gx}.

Definition 2.8 G is closed if GrG is a closed subset of X × Y, that is, for every sequences

x nn∈N ⊂ X and y nn∈N ⊂ Y, if x n → x∗, y n → y∗ as n → ∞ with y n ∈ Fx n, then

y∗∈ Gx∗

We recall the following two results; the first one is classical

Lemma 2.9 see 33, Proposition 1.2 If G : X → PclY is u.s.c., then GrG is a closed subset

of X × Y Conversely, if G is locally compact and has nonempty compact values and a closed graph,

then it is u.s.c.

Lemma 2.10 If G : X → P cp Y is quasicompact and has a closed graph, then G is u.s.c.

Given a separable Banach spaceE,  · , for a multivalued map F : J × E → PE,

denote

Definition 2.11 A multivalued map F is called a Carath´eodory function if

a the function t → Ft, x is measurable for each x ∈ E;

b for a.e t ∈ J, the map x → Ft, x is upper semicontinuous.

Furthermore, F is L1-Carath´eodory if it is locally integrably bounded, that is, for each positive

r, there exists h r ∈ L1J, R such that

Ft, xP≤ h r t, for a.e t ∈ J and all x ≤ r. 2.10

For each x ∈ CJ, E, the set

S F,xf ∈ L1J, E : ft ∈ Ft, xt for a.e t ∈ J 2.11

is known as the set of selection functions

Remark 2.12 a For each x ∈ CJ, E, the set S F,x is closed whenever F has closed values It is convex if and only if Ft, xt is convex for a.e t ∈ J.

b From 34 see also 35 when E is finite-dimensional, we know that S F,x is

nonempty if and only if the mapping t → inf{v : v ∈ Ft, xt} belongs to L1J It

is bounded if and only if the mapping t → Ft, xtPbelongs to L1J; this particularly holds true when F is L1-Carath´eodory For the sake of completeness, we refer also to Theorem1.3.5 in36 which states that S F,x contains a measurable selection whenever x is measurable and F is a Carath´eodory function.

Lemma 2.13 see 35 Given a Banach space E, let F : a, b × E → P cp,cv E be an L1 Carath´eodory multivalued map, and let Γ be a linear continuous mapping from L1a, b, E into

-C a, b, E Then the operator

Γ ◦ S F : Ca, b, E −→ P cp,cv Ca, b, E,

y −→ Γ ◦ S Fy

has a closed graph in C a, b, E × Ca, b, E.

For further readings and details on multivalued analysis, we refer to the books byAndres and G ´orniewicz37, Aubin and Cellina 38, Aubin and Frankowska 29, Deimling

33, G´orniewicz 31, Hu and Papageorgiou 34, Kamenskii et al 36, and Tolstonogov

39

2.2 Semicompactness in L10, b, E

Definition 2.14 A sequence {v n}n∈N⊂ L1J, E is said to be semicompact if

a it is integrably bounded, that is, there exists q ∈ L1J, R such that

v n t E ≤ qt, for a.e t ∈ J and every n ∈ N, 2.13

b the image sequence {v n t} n∈Nis relatively compact in E for a.e t ∈ J.

We recall two fundamental results The first one follows from the Dunford-Pettistheoremsee 36, Proposition 4.2.1 This result is of particular importance if E is reflexive inwhich casea implies b inDefinition 2.14

Lemma 2.15 Every semicompact sequence L1J, E is weakly compact in L1J, E.

The second one is due to Mazur, 1933

Lemma 2.16 Mazur’s Lemma, 28 Let E be a normed space and {x k}k∈N ⊂ E be a sequence

weakly converging to a limit x ∈ E Then there exists a sequence of convex combinations y m 

m

k1α mk x k with α mk > 0 for k  1, 2, , m andm

k1α mk  1, which converges strongly to x.

Definition 3.1 Let A be a closed and linear operator with domain D A defined on a Banach space E We call A the generator of an integral resolvent if there exists ω > 0 and a strongly continuous function S :R → BE such that

1

In this case, St is called the integral resolvent family generated by A.

The following result is a direct consequence of16, Proposition 3.1 and Lemma 2.2

Proposition 3.2 Let {St} t≥0 ⊂ BE be an integral resolvent family with generator A Then the

following conditions are satisfied:

a St is strongly continuous for t ≥ 0 and S0  I;

b StDA ⊂ DA and AStx  StAx for all x ∈ DA, t ≥ 0;

c for every x ∈ DA and t ≥ 0,

Remark 3.3 The uniqueness of resolvent is well knownsee Pr ¨uss 24

If an operator A with domain DA is the infinitesimal generator of an integral resolvent family St and at is a continuous, positive and nondecreasing function which

satisfies limt→ 0St B E /a t < ∞, then for all x ∈ DA we have

Ax lim

t→ 0 

S tx − atx

see 22, Theorem 2.1 For example, the case at ≡ 1 corresponds to the generator of

a C0-semigroup and at  t actually corresponds to the generator of a sine family; see

40 A characterization of generators of integral resolvent families, analogous to the

Hille-Yosida Theorem for C0-semigroups, can be directly deduced from22, Theorem 3.4 More

information on the C0-semigroups and sine families can be found in41–43

Definition 3.4 A resolvent family of bounded linear operators, {St} t>0, is called uniformlycontinuous if

lim

Definition 3.5 The solution operator S t is called exponentially bounded if there are constants M > 0 and ω≥ 0 such that

4 Existence Results

4.1 Mild Solutions

In order to define mild solutions for problem1.1, we proof the following auxiliary lemma

Lemma 4.1 Let a ∈ L1J, R Assume that A generates an integral resolvent family {St} t≥0on

E, which is in addition integrable and D A  E Let f : J → E be a continuous function (or

f ∈ L1J, E), then the unique bounded solution of the problem

This lemma leads us to the definition of a mild solution of the problem1.1.

Definition 4.2 A function y ∈ CJ, E is said to be a mild solution of problem 1.1 if there

exists f ∈ L1J, E such that ft ∈ Ft, yt a.e on J such that

y t 

t

0

Consider the following assumptions

B1 The operator solution {St} t ∈J is compact for t > 0.

B2 There exist a function p ∈ L1J, R and a continuous nondecreasing function ψ :

B3 For every t > 0, St is uniformly continuous.

In all the sequel we assume that S· is exponentially bounded Our first main existence result

is the following

Theorem 4.3 Assume F : J × E → P cp,cv E is a Carath´eodory map satisfying B1-B2or

B2-B3 Then problem 1.1 has at least one solution If further E is a reflexive space, then the

solution set is compact in C J, E.

The following so-called nonlinear alternatives of Leray-Schauder type will be needed

in the proofsee 31,44

Lemma 4.4 Let X,  ·  be a normed space and F : X → P cl,cv X a compact, u.s.c multivalued

map Then either one of the following conditions holds.

a F has at least one fixed point,

b the set M : {x ∈ E, x ∈ λFx, λ ∈ 0, 1} is unbounded.

The single-valued version may be stated as follows

Lemma 4.5 Let X be a Banach space and C ⊂ X a nonempty bounded, closed, convex subset Assume

U is an open subset of C with 0 ∈ U and let G : U → C be a a continuous compact map Then

a either there is a point u ∈ ∂U and λ ∈ 0, 1 with u  λGu,

b or G has a fixed point in U.

Proof of Theorem 4.3 We have the following parts.

Part 1: Existence of Solutions

It is clear that all solutions of problem 1.1 are fixed points of the multivalued operator

N : C J, E → PCJ, E defined by

N

y:

f ∈ S F,yf ∈ L1J, E : ft ∈ Ft, y t, for a.e t ∈ J. 4.10

Notice that the set S F,y is nonemptyseeRemark 2.12,b Since, for each y ∈ CJ, E, the

nonlinearity F takes convex values, the selection set S F,y is convex and therefore N has convex

b N maps bounded sets into equicontinuous sets of CJ, E.

Let τ1, τ2∈ J, 0 < τ1< τ2and B q be a bounded set of CJ, E as in a Let y ∈ B q; then for each

c As a consequence of parts a and b together with the Arz´ela-Ascoli theorem, itsuffices to show that N maps Bq into a precompact set in E Let 0 < t ≤ b and let 0 < ε < t For

which tends to 0 as ε → 0 Therefore, there are precompact sets arbitrarily close to the set

H t  {ht : h ∈ Ny} This set is then precompact in E.

Step 2 N has a closed graph Let h n → h∗, h n ∈ Ny n  and y n → y∗ We will prove that

h∗∈ Ny∗ h n ∈ Ny n  means that there exists f n ∈ S F,y n such that for each t ∈ J

Since y n → y∗andΓ ◦ S F is a closed graph operator byLemma 2.13, then there exists f∗ ∈

S F,y∗such that

Step 3 a priori bounds on solutions Let y ∈ CJ, E be such that y ∈ Ny Then there exists

f ∈ S F,ysuch that

and consider the operator N : U → Pcv,cp CJ, E From the choice of U, there is no y ∈ ∂U such that y ∈ γNy for some γ ∈ 0, 1 As a consequence of the Leray-Schauder nonlinear

alternativeLemma 4.4, we deduce that N has a fixed point y in U which is a mild solution

of problem1.1

Part 2: Compactness of the Solution Set

Let

S F  y ∈ CJ, E | y is a solution of problem 1.1. 4.29

From Part 1, S F / ∅ and there exists M such that for every y ∈ S F,y∞ ≤ M Since N is

completely continuous, then NS F  is relatively compact in CJ, E Let y ∈ S F ; then y ∈ Ny and S F ⊂ NS F  It remains to prove that S F is closed set in CJ, E Let y n ∈ S F such that y n converge to y For every n ∈ N, there exists v n t ∈ Ft, y n t, a.e t ∈ J such that

converges weakly to some limit v ∈ L1J, E Moreover, the mapping Γ : L1J, E → CJ, E

is a continuous linear operator Then it remains continuous if these spaces are endowed with

their weak topologies Therefore for a.e t ∈ J, the sequence y n t converges to yt, it follows

g n·  k n

i1 α n

i v i · converges strongly to v in L1 Since F takes convex values, using

Lemma 2.2, we obtain that

closed, hence compact in CJ, E.

4.2 The Convex Case: An MNC Approach

First, we gather together some material on the measure of noncompactness For more details,

we refer the reader to36,45 and the references therein

Definition 4.6 Let E be a Banach space and A, ≥ a partially ordered set A map β : PE →

A is called a measure of noncompactness on E, MNC for short, if

for everyΩ ∈ PE.

Notice that if D is dense in Ω, then co Ω  co D and hence

Definition 4.7 A measure of noncompactness β is called

a monotone if Ω0, Ω1∈ PE, Ω0⊂ Ω1implies βΩ0 ≤ βΩ1

b nonsingular if β{a} ∪ Ω  βΩ for every a ∈ E, Ω ∈ PE.

c invariant with respect to the union with compact sets if βK ∪ Ω  βΩ for every relatively compact set K ⊂ E, and Ω ∈ PE.

d real if A  R 0, ∞ and βΩ < ∞ for every bounded Ω.

e semiadditive if βΩ0∪ Ω1  maxβΩ0, βΩ1 for every Ω0,Ω1∈ PE.

f regular if the condition βΩ  0 is equivalent to the relative compactness of Ω.

As example of an MNC, one may consider the Hausdorf MNC

Recall that a bounded set A ⊂ E has a finite ε-net if there exits a finite subset S ⊂ E such that

A ⊂ S  εB where B is a closed ball in E.

Other examples are given by the following measures of noncompactness defined on

the space of continuous functions CJ, E with values in a Banach space E:

i the modulus of fiber noncompactness

ϕΩ  sup

where χ Eis the Hausdorff MNC in E and Ωt  {yt : y ∈ Ω};

ii the modulus of equicontinuity

Definition 4.8 Let M be a closed subset of a Banach space E and β : PE → A, ≥ an MNC

on E A multivalued mapF : M → Pcp E is said to be β-condensing if for every Ω ⊂ M, the

relation

implies the relative compactness ofΩ

Some important results on fixed point theory with MNCs are recalled hereaftersee,e.g.,36 for the proofs and further details The first one is a compactness criterion

Lemma 4.9 see 36, Theorem 5.1.1 Let N : L1a, b, E → Ca, b, E be an abstract

operator satisfying the following conditions:

S1 N is ξ-Lipschitz: there exists ξ > 0 such that for every f, g ∈ L1a, b, E

Nf t − Ngt ≤ ξb

a

f s − gs ds, ∀t ∈ a, b. 4.41

S2 N is weakly-strongly sequentially continuous on compact subsets: for any compact K ⊂ E

and any sequence {f n}∞

n1⊂ L1a, b, E such that {f n t}∞

n1⊂ K for a.e t ∈ a, b, the

weak convergence f n  f0implies the strong convergence N f n  → Nf0 as n → ∞.

Then for every semicompact sequence {f n}∞n1⊂ L1J, E, the image sequence N{f n}∞n1 is relatively

compact in C a, b, E.

Lemma 4.10 see 36, Theorem 5.2.2 Let an operator N : L1a, b, E → Ca, b, E satisfy

conditionsS1-S2 together with the following:

S3 there exits η ∈ L1a, b such that for every integrable bounded sequence {f n}∞n1, one has

χ f n t∞n1≤ ηt, for a.e t ∈ a, b, 4.42

where χ is the Hausdorff MNC.

where ξ is the constant inS1.

The next result is concerned with the nonlinear alternative for β-condensing u.s.c.

multivalued maps

Lemma 4.11 see 36 Let V ⊂ E be a bounded open neighborhood of zero and N : V → P cp,cv E

a β-condensing u.s.c multivalued map, where β is a nonsingular measure of noncompactness defined

on subsets of E, satisfying the boundary condition

for all x ∈ ∂V and 0 < λ < 1 Then Fix N / ∅.

Lemma 4.12 see 36 Let W be a closed subset of a Banach space E and F : W → P cp E is a

closed β-condensing multivalued map where β is a monotone MNC on E If the fixed point set FixF

is bounded, then it is compact.

4.2.1 Main Results

In all this part, we assume that there exists M > 0 such that

Let F : J × E → P cp,cv E be a Carath´eodory multivalued map which satisfies Lipschitz

conditions with respect to the Hausdorf MNC

B4 There exists p ∈ L1J, R such that for every bounded D in E,

Lemma 4.13 Under conditions B2 and B4, the operator N is closed and Ny ∈ P cp,cv CJ, E,

for every y ∈ CJ, E where N is as defined in the proof of Theorem 4.3

Proof We have the following steps.

Step 1 N is closed Let h n → h∗, h n ∈ Ny n , and y n → y∗ We will prove that h∗∈ Ny∗

h n ∈ Ny n  means that there exists f n ∈ S F,y n such that for a.e t ∈ J

h∗t 

t

0

As a consequence, h∗∈ Ny∗, as claimed

Step 2 N has compact, convex values The convexity of Ny follows immediately by the convexity of the values of F To prove the compactness of the values of F, let Ny ∈ PE for some y ∈ CJ, E and h n ∈ Ny Then there exists f n ∈ S F,ysatisfying4.47 Arguingagain as inStep 1, we prove that{f n} is semicompact and converges weakly to some limit

f∗ ∈ Ft, yt, a.e t ∈ J hence passing to the limit in 4.47, h n tends to some limit h∗ in

the closed set Ny with h∗satisfying4.49 Therefore the set Ny is sequentially compact,

hence compact

Lemma 4.14 Under the conditions B2 and B4, the operator N is u.s.c.

Proof Using Lemmas 2.10 and 4.13, we only prove that N is quasicompact Let K be a compact set in CJ, E and h n ∈ Ny n  such that y n ∈ K Then there exists f n ∈ S F,y n such that

h n t 

t

0

Since K is compact, we may pass to a subsequence, if necessary, to get that {y n} converges

to some limit y∗ in CJ, E Arguing as in the proof of Theorem 4.3 Step 1, we can provethe existence of a subsequence{f n } which converges weakly to some limit f∗and hence h n converges to h∗, where

We are now in position to prove our second existence result in the convex case

Theorem 4.15 Assume that F satisfies Assumptions B2 and B4 If

q :  2M

b

0

then the set of solutions for problem1.1 is nonempty and compact.

Proof It is clear that all solutions of problem1.1 are fixed points of the multivalued operator

N defined inTheorem 4.3 By Lemmas4.13and 4.14, N· ∈ P cv,cp CJ, E and it is u.s.c Next, we prove that N is a β-condensing operator for a suitable MNC β Given a bounded subset D ⊂ CJ, E, let mod C D the modulus of quasiequicontinuity of the set of functions

It is well knownsee, e.g., 36, Example 2.1.2 that modC D defines an MNC in CJ, E

which satisfies all of the properties inDefinition 4.7except regularity Given the Hausdorff

MNC χ, let γ be the real MNC defined on bounded subsets on CJ, E by

whereΔCJ, E is the collection of all countable subsets of B Then the MNC β is monotone,

regular and nonsingularsee 36, Example 2.1.4

To show that N is β-condensing, let B ⊂ be a bounded set in CJ, E such that

a β-condensing u.s.c multivalued map, where β is a nonsingular measure of noncompactness defined

on subsets of E, satisfying the boundary condition

for... β-condensing multivalued map where β is a monotone MNC on E If the fixed point set FixF

is bounded, then it is compact.

4.2.1 Main Results

In all... n ∈ Ny Then there exists f n ∈ S F,ysatisfying4.47 Arguingagain as inStep 1, we prove that{f n} is semicompact and converges weakly