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

Hindawi Publishing Corporation Advances in Difference Equations 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. Ouahab 3 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 Copyright q 2010 R. P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any 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: yt ∈  t 0 at − sAysFs, ysds, a.e.t ∈ J,where J 0,b, A is the infinitesimal generator of an integral resolvent family on a separable Banach space E,andF 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 on real compact intervals under different conditions on the kernel see, e.g., 1–4 and references therein. However very few results are available for integral inclusions on compact intervals, see 5–7. Topological structure of the solution set of integral inclusions of Volterra type is studied in 8. 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 integral inclusions of Volterra type of the form y  t  ∈  t 0 a  t − s   Ay  s   F  s, y  s   ds, a.e.t∈ J :  0,b  , 1.1 2 Advances in Difference Equations where a ∈ L 1 0,b, R and A : DA ⊂ E → E is the generator of an integral resolvent family defined on a complex Banach space E,andF : 0,b × E →PE is a multivalued map. In 1980, Da Prato and Iannelli introduced the concept of resolvent families, which can be regarded as an extension of C 0 -semigroups in the study of a class of integrodifferential equations 9. It is well known that the following abstract Volterra equation y  t   f  t    t 0 a  t − s  Ay  s  ds, t ≥ 0, 1.2 where f : R  → E is a continuous function, is well-posed if and only if it admits a resolvent family, that is, there is a strongly continuous family St, t>0, of bounded linear operators defined in E, which commutes with A and satisfies the resolvent equation S  t  x  x   t 0 a  t − s  AS  s  xds, t ≥ 0,x∈ D  A  . 1.3 The study of diverse properties of resolvent families such as the regularity, positivity, periodicity, approximation, uniform continuity, compactness, and others are studied by several authors under different conditions on the kernel and the operator A see 10–24. An important kernel is given by a  t   f 1−α  t  e −kt ,t>0,k≥ 0,α∈  0, 1  , 1.4 where f α  t   ⎧ ⎪ ⎨ ⎪ ⎩ t α−1 Γ  α  if t>0, 0ift ≤ 0 1.5 is the Riemann-Liouville kernel. In this case 1.1 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 for instants 25–27. Our goal in this paper is to complement and extend some recent results to the case of infinite-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, we prove 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 combined the 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 solution set in Section 4.2. Section 5 is concerned with Filippov’s theorem for the problem 1.1. Advances in Difference Equations 3 In Section 6, we discuss the relaxed problem, namely, the density of the solution set of problem 1.1 in that of the convexified problem. 2. Preliminaries In this section, we recall f rom the literature some notations, definitions, and auxiliary results which will be used throughout this paper. Let E, · be a separable Banach space, J 0,b an interval in R and CJ, E the Banach space of all continuous functions from J into E with the norm   y   ∞  sup    y  t    :0≤ t ≤ b  . 2.1 BE refers to the Banach space of linear bounded operators from E into E with norm  N  BE  sup    N  y    :   y    1  . 2.2 A function y : J → E is called measurable provided for every open subset U ⊂ E,theset y −1 U{t ∈ J : yt ∈ U} is Lebesgue measurable. A measurable function y : J → E is Bochner integrable if y is Lebesgue integrable. For properties of the Bochner integral, see, for example, Yosida 28. In what follows, L 1 J, E denotes the Banach space of functions y : J → E, which are Bochner integrable with norm   y   1   b 0   y  t    dt. 2.3 Denote by PE{Y ⊂ E : Y /  ∅}, P cl E{Y ∈PE : Y closed}, P b E{Y ∈PE : Y bounded}, P cv E{Y ∈PE : Y convex}, P cp E{Y ∈PE : Y compact}. 2.1. Multivalued Analysis Let X, d and Y, ρ be two metric spaces and G : X →P cl Y 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 ∈ Gx for every x ∈ X. G is called upper semicontinuous (u.s.c. for short) on X if for each x 0 ∈ X the set Gx 0  is a nonempty, closed subset of X, and if for each open set N of Y containing Gx 0 , there exists an open neighborhood M of x 0 such that GM ⊆ Y. That is, if the set G −1 V {x ∈ X, Gx ∩V /  ∅} is closed for any closed set V in Y . Equivalently, G is u.s.c.ifthesetG 1 V  {x ∈ X, Gx ⊂ 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 →P cp x is u.s.c., then for any x 0 ∈ X, lim sup x → x 0 G  x   G  x 0  . 2.4 4 Advances in Difference Equations Lemma 2.2 see, e.g., 29, Lemma 1.1.9. Let K n  n∈N ⊂ K ⊂ X be a sequence of subsets where K is compact in t he separable Banach space X.Then co  lim sup n →∞ K n    N>0 co   n≥N K n  , 2.5 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  KA ⊂ X such that GA ∪{Gx,x∈ A}⊂K. G is compact if GX is relatively compact. It is called locally compact if, for each x ∈ X, there exists U ∈Vx such that GU is relatively compact. G is quasicompact if, for each subset A ⊂ X, GA is relatively compact. Definition 2.3. A multivalued map F : J 0,b →P cl Y is said measurable provided for every open U ⊂ Y ,thesetF 1 U is Lebesgue measurable. We have Lemma 2.4 see 30, 31. The mapping F is measurable if and only if for each x ∈ Y, the function ζ : J → 0, ∞ defined b y ζ  t   dist  x, F  t   inf    x − y   : y ∈ F  t   ,t∈ J, 2.6 is Lebesgue measurable. The following two lemmas are needed in this paper. The first one is the celebrated Kuratowski-Ryll-Nardzewski selection theorem. Lemma 2.5 see 31, Theorem 19.7. Let Y be a separable metric space and F : a, b →PY 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 →PY 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,   u  t  − f v  t    ≤ d  u  t  ,F  t   v  t  . 2.7 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,   u  t  − f  t    ≤ d  u  t  ,F  t  . 2.8 2.1.1. Closed Graphs We denote the graph of G to be the set GrG{x, y ∈ X × Y, y ∈ Gx}. Advances in Difference Equations 5 Definition 2.8. G is closed if GrG is a closed subset of X × Y , that is, for every sequences x n  n∈N ⊂ X and y n  n∈N ⊂ Y ,ifx n → x ∗ , y n → y ∗ as n →∞with y n ∈ Fx n , then y ∗ ∈ Gx ∗ . We recall the following two results; the first one is classical. Lemma 2.9 see 33, Proposition 1.2. If G : X →P cl Y is u.s.c., then GrG 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 space E, ·, for a multivalued map F : J × E →PE, denote  Ft, x  P : sup {  v  : v ∈ F  t, x  } . 2.9 Definition 2.11. A multivalued map F is called a Carath ´ eodory function if a the function t → Ft, x is measurable for each x ∈ E; b for a.e. t ∈ J, the map x → Ft, x is upper semicontinuous. Furthermore, F is L 1 -Carath ´ eodory if it is locally integrably bounded, that is, for each positive r, there exists h r ∈ L 1 J, R   such that  Ft, x  P ≤ h r  t  , for a.e.t∈ J and all  x  ≤ r. 2.10 For each x ∈ CJ, E,theset S F,x   f ∈ L 1  J, E  : f  t  ∈ F  t, x  t  for a.e.t∈ J  2.11 is known as the set of selection functions. Remark 2.12. a For each x ∈ CJ, E,thesetS F,x is closed whenever F has closed values. It is convex if and only if Ft, xt 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 ∈ Ft, xt} belongs to L 1 J.It is bounded if and only if the mapping t →Ft, xt P belongs to L 1 J; this particularly holds true when F is L 1 -Carath ´ eodory. For the sake of completeness, we refer also to Theorem 1.3.5 in 36 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,letF : a, b × E →P cp,cv E be an L 1 - Carath ´ eodory multivalued map, and let Γ be a linear continuous mapping from L 1 a, b,E into 6 Advances in Difference Equations Ca, b,E. Then the operator Γ ◦ S F : C  a, b  ,E  −→ P cp,cv  C  a, b  ,E  , y −→  Γ ◦ S F   y  :Γ  S F,y  2.12 has a closed graph in Ca, b,E × Ca, b,E. For further readings and details on multivalued analysis, we refer to the books by Andres and G ´ orniewicz 37, 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 L 1 0,b,E Definition 2.14. A sequence {v n } n∈N ⊂ L 1 J, E is said to be semicompact if a it is integrably bounded, that is, there exists q ∈ L 1 J, R   such that  v n t  E ≤ q  t  , for a.e.t∈ J and every n ∈ N, 2.13 b the image sequence {v n t} n∈N is relatively compact in E for a.e. t ∈ J. We recall two fundamental results. The first one follows from the Dunford-Pettis theorem see 36, Proposition 4.2.1. This result is of particular importance if E is reflexive in which case a implies b in Definition 2.14. Lemma 2.15. Every semicompact sequence L 1 J, E is weakly compact in L 1 J, 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 k1 α mk x k with α mk > 0 for k  1, 2, ,mand  m k1 α mk  1, which converges strongly to x. 3. Resolvent Family The Laplace transformation of a function f ∈ L 1 loc R  ,E is defined by L  f   λ  :: a  λ    ∞ 0 e −λt f  t  dt, Re  λ  >ω, 3.1 if the integral is absolutely convergent for Re λ >ω. In order to defined the mild solution of the problems 1.1 we recall the following definition. Advances in Difference Equations 7 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  → BE such that  1 aλ I − A  −1 x   ∞ 0 e −λt S  t  xdt, Re λ>ω, x∈ E. 3.2 In this case, St is called the integral resolvent family generated by A. The following result is a direct consequence of 16, Proposition 3.1 and Lemma 2.2. Proposition 3.2. Let {St} t≥0 ⊂ BE be an integral resolvent family with generator A. Then the following conditions are satisfied: a St is strongly continuous for t ≥ 0 and S0I; b StDA ⊂ DA and AStx  StAx for all x ∈ DA,t≥ 0; c for every x ∈ DA and t ≥ 0, S  t  x  a  t  x   t 0 a  t − s  AS  s  xds, 3.3 d let x ∈ DA.Then  t 0 at − sSsxds ∈ DA, and S  t  x  a  t  x  A  t 0 a  t − s  S  s  xds. 3.4 In particular, S0a0. Remark 3.3. The uniqueness of resolvent is well known see Pr ¨ uss 24. If an operator A with domain DA is the infinitesimal generator of an integral resolvent family St and at is a continuous, positive and nondecreasing function which satisfies lim t → 0  St BE /at < ∞, then for all x ∈ DA we have Ax  lim t → 0  S  t  x − a  t  x  a ∗ a  t  , 3.5 see 22, Theorem 2.1. For example, the case at ≡ 1 corresponds to the generator of a C 0 -semigroup and at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 C 0 -semigroups, can be directly deduced from 22, Theorem 3.4.More information on the C 0 -semigroups and sine families can be found in 41–43. Definition 3.4. A resolvent family of bounded linear operators, {St} t>0 , is called uniformly continuous if lim t → s  St − Ss  BE  0. 3.6 8 Advances in Difference Equations Definition 3.5. The solution operator St is called exponentially bounded if there are constants M>0andω ≥ 0 such that  St  BE ≤ Me ωt ,t≥ 0. 3.7 4. Existence Results 4.1. Mild Solutions In order to define mild solutions for problem 1.1, we proof the following auxiliary lemma. Lemma 4.1. Let a ∈ L 1 J, R. Assume that A generates an integral resolvent family {St} t≥0 on E, which is in addition integrable and DAE.Letf : J → E be a continuous function (or f ∈ L 1 J, E), then the unique bounded solution of the problem y  t    t 0 a  t − s  Ay  s  ds   t 0 a  t − s  f  s  ds, t ∈ J, 4.1 is given by y  t    t 0 S  t − s  f  s  ds, t ∈ J. 4.2 Proof. Let y be a solution of the integral equation 4.2, then y  t    t 0 S  t − s  f  s  ds. 4.3 Using the fact that S is solution operator and Fubini’s theorem we obtain  t 0 a  t − s  Ay  s  ds   t 0 a  t − s  A  s 0 S  s − r  f  r  dr ds   t 0  t r a  t − s  AS  s − r  dsf  r  dr   t 0  t−r 0 a  t − s − r  AS  s  dsf  r  dr   t 0  S  t − r  − a  t − r  f  r  dr   t 0 S  t − s  f  s  ds −  t 0 a  t − s  f  s  ds. 4.4 Advances in Difference Equations 9 Thus y  t    t 0 a  t − s  Ay  s  ds   t 0 a  t − s  f  s  ds, t ∈ J. 4.5 This lemma leads us to the definition of a mild solution of the problem 1.1. Definition 4.2. A function y ∈ CJ, E is said to be a mild solution of problem 1.1 if there exists f ∈ L 1 J, E such that ft ∈ Ft, yt a.e. on J such that y  t    t 0 S  t − s  f  s  ds, t ∈ J. 4.6 Consider the following assumptions. B 1  The operator solution {St} t∈J is compact for t>0. B 2  There exist a function p ∈ L 1 J, R   and a continuous nondecreasing function ψ : 0, ∞ → 0, ∞ such that  Ft, x  P ≤ p  t  ψ   x   for a.e.t∈ J and each x ∈ E 4.7 with Me ωb  b 0 p  s  ds <  ∞ 0 du ψ  u  . 4.8 B 3  For every t>0, St 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 B 1 -B 2 or B 2 -B 3 . 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 proof see 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 ∈ λFx,λ∈ 0, 1} is unbounded. The single-valued version may be stated as follows. 10 Advances in Difference Equations 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  λGu, 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 →PCJ, E defined by N  y  :  h ∈ C  J, E  | h  t    t 0 S  t − s  f  s  ds, for t ∈ J  4.9 where f ∈ S F,y   f ∈ L 1  J, E  : f  t  ∈ F  t, y  t   , for a.e.t∈ J  . 4.10 Notice that the set S F,y is nonempty see Remark 2.12,b. Since, for each y ∈ CJ, E,the nonlinearity F takes convex values, the selection set S F,y is convex and therefore N has convex values. Step 1 N is completely continuous. a N sends bounded sets into bounded sets in CJ, E. Let q>0, B q : {y ∈ CJ, E : y ∞ ≤ q} be a bounded set in CJ, E,andy ∈ B q . Then for each h ∈ Ny, there exists f ∈ S F,y such that h  t    t 0 S  t − s  f  s  ds, for t ∈ J. 4.11 Thus for each t ∈ J,  h  ∞ ≤ e ωb ψ  q   b 0 p  t  dt. 4.12 b N maps bounded sets into equicontinuous sets of CJ, E. Let τ 1 ,τ 2 ∈ J,0<τ 1 <τ 2 and B q be a bounded set of CJ, E as in a.Lety ∈B q ; then for each t ∈ J  h  τ 2  − h  τ 1   ≤ ψ  q   τ 2 τ 1  S  τ 2 − s   BE p  s  ds  ψ  q   τ 1 0  Sτ 1 − s − Sτ 2 − s  BE p  s  ds. 4.13 The right-hand side tends to zero as τ 2 − τ 1 → 0sinceSt is uniformly continuous. [...]... Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, UK, 1991 4 G Gripenberg, S.-O Londen, and O Staffans, Volterra Integral and Functional Equations, vol 34 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1990 5 J Appell, E de Pascale, H T Nguyˆ n, and P P Zabre˘ko, “Nonlinear integral inclusions of e˜ ı Hammerstein type,”... Advances in Difference Equations Performing an integration by parts, we obtain, since P is a nondecreasing function, the following estimates: y3 t − y2 t ≤ ≤ ≤ ≤ ≤ M3 2 t s 2p s ds e2 P γ u 0 s −P u du 0 t e2P s s ds e−2P γ u 0 u du 5.18 0 e2P t t e−2P γ s s ds − 0 t γ s ds 0 t M3 2 du ds 0 t M3 2 s −P u eP γ u 0 M3 2 M3 2 s 2p s e2 P γ s t −P s ds 0 F t, y3 t ∩ B f3 t , p t y3 t −y2 t Then, arguing... function f · in E In addition, since f0 a.e t ∈ J fn t ≤ n fk t − fk−1 t f0 t k 1 ≤ n p t yk−1 t − yk−2 t γ t g t 5.26 k 1 ≤p t ∞ yk t − yk−1 t γ t g t k 1 Hence fn t ≤ MH t p t γ t g t , 5.27 30 Advances in Difference Equations where t H t : M t −P s exp MeP γ s 5.28 ds 0 From the Lebesgue dominated convergence theorem, we deduce that {fn } converges to f in L1 J, E Passing to the limit in 5.22 , we... obtained by interchanging the roles of y and y, we finally arrive at Hd N y , N y ≤ 1 y−y τ ∗ 4.92 , where τ > 1 and y ∗ sup e−τL t y t :t∈J , Lt Meωb t l s ds 0 4.93 Advances in Difference Equations 25 is the Bielecki-type norm on C J, E So, N is a contraction and thus, by Lemma 4.23, N has a fixed point y, which is a mild solution to 1.1 Arguing as in Theorem 4.3, we can also prove the following result... solution set of integral inclusions,” Journal of Integral Equations and Applications, vol 12, no 1, pp 85–94, 2000 9 D Prato and G M Iannelli, “Linear integrodifferential equations in Banach space,” Rendiconti del Seminario Matematico della Universit` di Padova, vol 62, pp 207–219, 1980 a 10 D Araya and C Lizama, “Almost automorphic mild solutions to fractional differential equations, ” Nonlinear Analysis:... has a continuous selection, that is, there exists a continuous single-valued function f : X → L1 J, E such that f x ∈ N x for every x ∈ X Let us introduce the following hypothesis H1 F : J × E → P E is a nonempty compact valued multivalued map such that a the mapping t, y → F t, y is L ⊗ B measurable; b the mapping y → F t, y is lower semicontinuous for a.e t ∈ J The following lemma is crucial in the... follows that t y t S t − s v s ds 4.32 0 It remains to prove that v ∈ F t, y t , for a.e t ∈ J Lemma 2.16 yields the existence of k n αn ≥ 0, i n, , k n such that i 1 αn 1 and the sequence of convex combinaisons i i 14 Advances in Difference Equations k n n 1 gn · i 1 αi vi · converges strongly to v in L Since F takes convex values, using Lemma 2.2, we obtain that {gn t }, v t ∈ a.e t ∈ J n≥1 ⊂ co{vk... D supχ D t t∈J 4.54 Finally, define the following MNC on bounded subsets of C J, E by β D max D∈Δ C J,E γ D , mod C D , 4.55 where Δ C J, E is the collection of all countable subsets of B Then the MNC β is monotone, regular and nonsingular see 36, Example 2.1.4 To show that N is β-condensing, let B ⊂ be a bounded set in C J, E such that β B ≤β N B 4.56 Advances in Difference Equations 19 We will show... }∞ 1 n supχ {hn t }∞ 1 ≤ qγ n t∈J yn ∞ n 1 4.64 20 Advances in Difference Equations Since 0 < q < 1, we infer that γ γ yn 0 implies that χ {yn t } χ yn ∞ n 1 0 4.65 0, for a.e t ∈ J In turn, 4.62 implies that fn t 0, for a.e t ∈ J 4.66 0 To show that modC B 0, i.e, the set {hn } is Hence 4.60 implies that χ {hn }∞ 1 n equicontinuous, we proceed as in the proof of Theorem 4.3 Step 1 Part b It follows... MeP s −P u ds du 0 5.30 Passing to the limit as n → ∞, we get x t −y t ≤η t , a.e t ∈ J 5.31 with t η t : M γ u 0 exp 2MeP t −P s ds 5.32 Advances in Difference Equations 31 6 The Relaxed Problem More precisely, we compare, in this section, trajectories of the following problem, t y t ∈ a t − s Ay s F s, y s ds, a.e t ∈ J, 6.1 0 and those of the convexified Volttera integral inclusion problem y t ∈ t a . Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 798067, 37 pages doi:10.1155/2010/798067 Research Article Some Results for Integral Inclusions. an L 1 - Carath ´ eodory multivalued map, and let Γ be a linear continuous mapping from L 1 a, b,E into 6 Advances in Difference Equations Ca, b,E. Then the operator Γ ◦ S F : C  a,. s  f n  s  ds. 4.50 18 Advances in Difference Equations Since K is compact, we may pass to a subsequence, if necessary, to get that {y n } converges to some limit y ∗ in CJ, E. Arguing as in the proof