Functional Differential Inclusions with Integral Boundary Conditions

In this paper, we investigate the existence of solutions for a class of second order functional differential inclusions with integral boundary conditions. By using suitable fixed point theorems, we study the case when the right hand side has convex as well as nonconvex values.


Introduction
This paper is concerned with the existence of solutions of second order functional differential inclusions with integral boundary conditions.More precisely, in Section 3, we consider the second order functional differential inclusion, with initial function values, and integral boundary condtions, where F : [0, 1]×C([−r, 0], IR) → P(IR) is a compact valued multivalued map, P(IR) is the family of all subsets of IR, λ < 0, φ ∈ C([−r, 0], IR) and g : IR → IR is continuous.Here y t (•) represents the history of the state from t − r, up to the present time t.Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems.They include two, three, 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 by Gallardo [13], Karakostas and Tsamatos [16], Lomtatidze and Malaguti [21] and the references therein.Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors, for instance, Brykalov [6], Denche and Marhoune [10], Jankowskii [15] and Krall [19].Recently Ahmad, Khan and Sivasundaram [1,17] have applied the generalized method of quasilinearization to a class of second order boundary value problem with integral boundary conditions.Some results on the existence of solutions for a class of boundary value problems for second order differential inclusions with integral conditions have been obtained by Belarbi and Benchohra [4].In this paper we shall present three existence results for the problem (1)-( 3), when the right hand side is convex as well as nonconvex valued.The first result relies on the nonlinear alternative of Leray-Schauder type.In the second result, we shall use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler, while in the third result, we shall combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposables values.These results extend to the multivalued case some previous results in the literature.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts from multivalued analysis which are used throughout this paper.
L 1 ([0, 1], IR) denotes the Banach space of measurable functions x : [0, 1] −→ IR which are Lebesgue integrable and normed by AC 1 ((0, 1), IR) is the space of differentiable functions x : (0, For a normed space (X, | • |), let P cl (X) = {Y ∈ P(X) : Y closed}, P b (X) = {Y ∈ P(X) : Y bounded}, P cp (X) = {Y ∈ P(X) : Y compact} and P cp,c (X) = {Y ∈ P(X) : Y compact and convex}.A multivalued map G : G is called upper semi-continuous (u.s.c.) 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 X containing G(x 0 ), there exists an open neighborhood N 0 of x 0 such that G(N 0 ) ⊆ N. G is said to be completely continuous if G(B) is relatively compact for every B ∈ P b (X).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. The fixed point set of the multivalued operator G will be denoted by F ixG.A multivalued map G : [0, 1] → P cl (IR) is said to be measurable if for every y ∈ IR, the function is measurable.For more details on multivalued maps see the books of Aubin and Cellina [2], Aubin and Frankowska [3], Deimling [9] and Hu and Papageorgiou [14].
(iii) for each q > 0, there exists Let E be a Banach space, X a nonempty closed subset of E and G : by letting The operator F is called the Nymetzki operator associated with F. Definition 2.3 Let F : [0, 1] × IR → P(IR) be a multivalued function with nonempty compact values.We say F is of lower semi-continuous type (l.s.c.type) if its associated Nymetzki operator F is lower semi-continuous and has nonempty closed and decomposable values.
Let (X, d) be a metric space induced from the normed space (X, | • |).Consider and (P cl (X), H d ) is a generalized metric space (see [18]).Definition 2.4 A multivalued operator N : X → P cl (X) is called a) γ-Lipschitz if and only if there exists γ > 0 such that The following lemmas will be used in the sequel.

Main Results
In this section, we are concerned with the existence of solutions for the problem (1)-( 3) when the right hand side has convex as well as nonconvex values.Initially, we assume that F is a compact and convex valued multivalued map.
We need the following auxiliary result.Its proof uses a standard argument.
where We shall show that N satisfies the assumptions of the nonlinear alternative of Leray-Schauder type.The proof will be given in several steps.
Indeed, if h 1 , h 2 belong to N(x), then there exist v 1 , v 2 ∈ S F,x such that for each t ∈ [0, 1] we have Step 2: N maps bounded sets into bounded sets in C([−r, 1], IR).
We must show that there exists h * ∈ S F,x * such that, for each t ∈ [0, 1], where Clearly we have Consider the continuous linear operator From Lemma 2.1, it follows that Γ • S F is a closed graph operator.Moreover, we have for some v * ∈ S F,x * .
Step 5: A priori bounds on solutions.
The operator N : U → P(C([0, 1], IR)) is upper semicontinuous and completely continuous.From the choice of U, there is no x ∈ ∂U such that x ∈ λN(x) for some λ ∈ (0, 1).As a consequence of the nonlinear alternative of Leray-Schauder type [11], we deduce that N has a fixed point x in U which is a solution of the problem (1)-( 3).This completes the proof.
We present now a result for the problem (1)-( 3) with a nonconvex valued right hand side.Our considerations are based on the fixed point theorem for multivalued map given by Covitz and Nadler [8].We need the following hypotheses: , IR), the set S F,x is nonempty since by (H5), F has a measurable selection (see [7], Theorem III.6).
Proof.We shall show that N satisfies the assumptions of Lemma 2.3.The proof will be given in two steps.
, IR) and there exists v n ∈ S F,x such that, for each t ∈ [0, 1], Using the fact that F has compact values and from (H6), we may pass to a subsequence if necessary to get that v n converges to v in L 1 ([0, 1], IR) and hence v ∈ S F,x .Then, for each t ∈ [0, 1],
We have By an analogous relation, obtained by interchanging the roles of x and x, it follows that So, N is a contraction and thus, by Lemma 2.3, N has a fixed point x which is solution to (1)-( 3).The proof is complete.
In this part, by using the nonlinear alternative of Leray Schauder type combined with the selection theorem of Bresssan and Colombo for semi-continuous maps with decomposable values, we shall establish an existence result for the problem (1)-(3).We need the following hypothesis: The following lemma is of great importance in the proof of our next result.
It is clear that if x ∈ C([−r, 1], IR) ∩ AC 1 ((0, 1), IR) is a solution of ( 4)-( 6), then x is a solution to the problem ( 1 We can easily show that N is continuous and completely continuous.The remainder of the proof is similar to that of Theorem 3.1.

3 Definition 2 . 2
where χ J stands for the characteristic function of J .EJQTDE, 2007 No. 15, p.Let Y be a separable metric space and let N : Y → P(L 1 ([0, 1], IR)) be a multivalued operator.We say N has property (BC) if 1) N is lower semi-continuous (l.s.c.); 2) N has nonempty closed and decomposable values.Let F : [0, 1] × C([−r, 0], IR) → P(IR) be a multivalued map with nonempty compact values.Assign to F the multivalued operator H d : P(X) × P(X) −→ IR + ∪ {∞} given by H d (A, B) = max sup a∈A d(a, B), sup b∈B d(A, b) , where d(A, b) = inf a∈A d(a, b), d(a, B) = inf b∈B d(a, b).Then (P b,cl (X), H d ) is a metric space