UPPER AND LOWER SOLUTIONS METHOD FOR DIFFERENTIAL INCLUSIONS WITH INTEGRAL BOUNDARY CONDITIONS

where F : J ×R→ (R) is a compact convex-valued multivalued map, (R) is the family of all nonempty subsets of R, hi : R→ R (i = 1,2) are continuous functions, and ki are nonnegative constants. Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two-, three-, and multipoint and nonlocal boundary value problems as special cases. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers [9, 14, 19] and the references therein. Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors, for example, [4, 6, 16]. The method of upper and lower solutions has been successfully applied to study the existence of multiple solutions for initial and boundary value problems for


Introduction
This paper is concerned with the existence of solutions for the second-order boundary value problem: where F : J × R → ᏼ(R) is a compact convex-valued multivalued map, ᏼ(R) is the family of all nonempty subsets of R, h i : R → R (i = 1,2) are continuous functions, and k i are nonnegative constants.Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems.They include two-, three-, and multipoint and nonlocal boundary value problems as special cases.For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers [9,14,19] and the references therein.Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors, for example, [4,6,16].The method of upper and lower solutions has been successfully applied to study the existence of multiple solutions for initial and boundary value problems for differential equations with nonlinear conditions.This method has been used only in the context of single-valued differential equations.In this regard, we refer the reader to the monographs by Heikkilä and Lakshmikantham [12] and Ladde et al. [17] and to the recent papers by Rahmat [15] and Rahmat and Bashir [1] in which the first-and secondorder differential equations with integral boundary conditions have been considered.Recently this method has been used for initial and nonlinear boundary conditions of differential inclusions in the papers by Benchohra [2], Benchohra and Ntouyas [3], Frigon [8], Halidias and Papageorgiou [11], and Palmucci and Papalini [20].In this paper, we will apply the method of upper and lower solutions combined with the nonlinear alternative of Leray-Schauder type [7] to problem (1.1)- (1.3).These results extend to the multivalued case some results from the literature and complement those related to the application of the method of upper and lower solutions to differential inclusions (see [2,3,8,11,20]).

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper.Let AC 1 (J,R) be the space of differentiable functions y : J → R, whose first derivative, y , is absolutely continuous.
The property We take C(J,R) to be the Banach space of all continuous functions from J into R with the norm and we let L 1 (J,R) denote the Banach space of functions y : J → R that are Lebesgue integrable with norm is measurable.For more details on multivalued maps, see the books of Deimling [5], G órniewicz [10], and Hu and Papageorgiou [13].
For any y ∈ C([0,1],R), we define the set The following lemma is crucial in the proof of our main theorem.
Lemma 2.4 [18].Let X be a Banach space.Let F : [0,b] × X → ᏼ cp,c (X) be an L 1 -Carathéodory multivalued map and let Γ be a linear continuous mapping from

Main result
We are now in a position to state and prove our existence result for the problem (1.1)-(1.3).We first list the following hypotheses: (H1) F : ) where for every fixed t ∈ J.A solution to (3.2)-(3.4) is a fixed point of the operator N : C(J,R) → ᏼ(C(J,R)) defined by where M. Benchohra and A. Ouahab 5 and G(•,•) is the Green function of the problem where Remark 3.2.(i) For each y ∈ C(J,R), the set S 1 F(y) is nonempty (see Lasota and Opial [18]).
(ii) For each y ∈ C(J,R), the set S 1 F1(y) is nonempty.In fact, (i) implies that there exists v ∈ S 1 F(y) , so we set where Then, by the decomposability, w ∈ S 1 F1(y) .
Remark 3.3.Notice that F 1 is an L 1 -Carathéodory multivalued map with compact convex values and there exists φ ∈ L 1 (J,R) such that In order to apply the nonlinear alternative of Leray-Schauder type, we first show that N is completely continuous with convex values.The proof will be given in several steps.
Indeed, if h 1 , h 2 belong to N(y), then there exist g 1 ,g 2 ∈ S 1 F1(y) such that for each t ∈ J, we have Then, for each t ∈ J, we have Since S 1 F1(y) is convex (because F 1 has convex values), we see that 6 Differential inclusions with integral boundary conditions Step 3.5.N maps bounded sets into bounded sets in C(J,R).
It suffices to show that for each q > 0, there exists a positive constant such that for each y ∈ B q = {y ∈ C(J,R) : y ∞ ≤ q}, we have (3.16) Let y ∈ B q and h ∈ N(y); then there exists g ∈ S 1 F1(y) such that for each t ∈ J, we have

G(t,s) g(s) ds
where Step 3.6.N maps bounded sets into equicontinuous sets in C(J,R).
Let u 1 ,u 2 ∈ J, u 1 < u 2 , and let B q be a bounded set in C(J,R) as in Step 3.5.Let y ∈ B q and h ∈ N(y); then there exists g ∈ S 1 F1(y) such that for each t ∈ J, we have where As a consequence of Steps 3.4-3.6,together with the Arzela-Ascoli theorem, we can conclude that N : C(J,R) → ᏼ cp (C(J,R)) is a completely continuous multivalued map.
Let y n → y * , h n ∈ N(y n ), and h n → h * .We will prove that h * ∈ N(y * ).Now h n ∈ N(y n ) implies that there exists g n ∈ S 1 F1(yn) such that for each t ∈ J, where where Since p is continuous, we have Consider the continuous linear operator

G(t,s) g(s) ds
and consider the operator N defined on U. From the choice of U, there is no y ∈ ∂U such that y ∈ λN(y) for some λ ∈ (0,1).As a consequence of the nonlinear alternative of Leray-Schauder type [7], we deduce that N has a fixed point y in U, that is, a solution of the problem (3.2)-(3.4).
If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c.if and only if G has a closed graph (i.e., x n Carathéodory multivalued map;(H2) there exist α and β ∈ AC 1 ([0,1],R), lower and upper solutions, respectively, for the problem (1.1)-(1.3)such that α ≤ β; (H3) h i are continuous and nondecreasing functions, i = 1,2.