Existence of Solution , Filippov ’ s Theorem and Compactness of the Set of Solutions for a Third-Order Differential Inclusion with Three-Point Boundary Conditions

where α, β, and η are constants in R, F : [0, 1]×R → P (R) a multi-valued map, and P (R) is the family of all subsets of R. This paper is a continuation of the work in [11], where the authors discussed the existence of solutions of the problem (1)–(2) when the multi-valued map F is nonconvex and lower semi-continuous. The aim of our present paper is to provide some existence results for the problem (1)–(2) under assumptions of convexity and upper semi-continuity of the right-hand side. To this end, we use a nonlinear alternative of Leray-Shauder type, some hypothesis of Carathéodory type, and some facts of the selection theory. More exactly, we discuss the existence of solutions for the problem (1)–(2) when F is convex and upper semi-continuous and satisfies a Carathéodory condition. We also prove that the set of solutions is compact, and we end our results by presenting a Filippov’s-type result concerning the existence of solutions to the considered problem. An illustrative example of a boundary value problem satisfying the mentioned conditions is also given. The paper is divided into three sections. In the second section, we give some necessary background material. In Section 3, we prove our main results.

In the present paper we study third-order differential inclusions of the form − u (t) ∈ F (t, u (t)) , t ∈ (0, 1) , (1) with boundary conditions of the form: where α, β, and η are constants in R, F : [0, 1] × R → P (R) a multi-valued map, and P (R) is the family of all subsets of R.This paper is a continuation of the work in [11], where the authors discussed the existence of solutions of the problem (1)-( 2) when the multi-valued map F is nonconvex and lower semi-continuous.
The aim of our present paper is to provide some existence results for the problem (1)-( 2) under assumptions of convexity and upper semi-continuity of the right-hand side.To this end, we use a nonlinear alternative of Leray-Shauder type, some hypothesis of Carathéodory type, and some facts of the selection theory.More exactly, we discuss the existence of solutions for the problem (1)-(2) when F is convex and upper semi-continuous and satisfies a Carathéodory condition.We also prove that the set of solutions is compact, and we end our results by presenting a Filippov's-type result concerning the existence of solutions to the considered problem.An illustrative example of a boundary value problem satisfying the mentioned conditions is also given.
The paper is divided into three sections.In the second section, we give some necessary background material.In Section 3, we prove our main results.

Preliminaries
In this section we introduce some notations, definitions, and preliminary facts which will be used in the remainder of the paper.Let C ([0, 1] , R) denote the Banach space of all continuous functions from [0, 1] into R, equipped with the norm We also denote the Banach space of measurable functions u : [0, 1] → R which are Lebesgue integrable by L 1 ([0, 1] , R), normed by By AC i ([0, 1], R) we denote the space of i−times differentiable functions u : [0, 1] → R, whose i th derivative, u (i) is absolutely continuous.
Let (X, d) be a metric space induced from a normed space (X, .).
. Then, (P b,cl (X) , H d ) is a metric space and (P cl (X) , H d ) is a generalized metric space (see [12]).
Let E be a separable Banach space, Y a nonempty closed subset of E and G : Y → P cl (E) a multi-valued map.G is said to be upper semi-continuous (u.s.c) at the point y 0 ∈ Y if for every open W ⊆ Y such that G (y 0 ) ⊂ W there exists a neighborhood V(y 0 ) of y 0 such that G (V (y 0 )) ⊂ W. We say that G has a fixed point if there is x ∈ Y such that x ∈ G(x).G is also said to be completely continuous if G (Ω) is relatively compact for every Ω ∈ P b (Y).If the multi-valued map G is completely continuous with nonempty compact values, then G is upper semi-continuous (u.s.c) if and only if G has a closed graph; that is, For more details on the multi-valued maps, see the books of Aubin and Cellina [13], Aubin and Frankowska [14], Deimling [15], Gorniewicz [16], and Hu and Papageorgiou [17].
We recall here some definitions and Lemmas needed below.

Existence of Solutions
Before giving some results on the existence of solutions for the problem (1) and ( 2), let us introduce the following hypotheses which are assumed hereafter: Theorem 1. Assume that (H 1 ) , (H 2 ) hold, then the boundary value problem (1) and (2) has at least one solution on [0, 1] .

Proof. Define the operator T
for f ∈ S F,u , we will show that T satisfies the assumptions of the nonlinear alternative of Leray-Schauder type.The proof consists of several steps.
Step 1: Let us begin by proving that T is convex for each u ∈ C ([0, 1] , R).
Let h 1 , h 2 ∈ Tu.Then, there exist w 1 , w 2 ∈ S F,u such that for each t ∈ [0, 1], we have Step 2: In this step, we prove that T maps bounded sets into bounded sets in C ([0, 1] , R).
Step 3: Here we verify that T maps bounded sets into equicontinuous sets of C ([0, 1] , R).
Obviously, the right-hand side of the above inequality tends to zero independently from u ∈ B r as t 2 − t 1 → 0. As T satisfies the above three assumptions, it follows by Ascoli-Arzela's theorem that Step 4: In this step we prove that T has a closed graph. Let Then, we need to show that h * ∈ Tu * .Associated with h n ∈ T (u n ), there exists w n ∈ S F,u n such that for each t ∈ [0, 1], So, we have to show that there exists w * ∈ S F,u * such that for each t ∈ [0, 1],

Let us consider the continuous linear operator
So, it follows from Lemma 1 that Θ • S F is a closed graph operator.
Further, we have h n (t) ∈ Θ (S F,u n ).Since u n → u * , therefore, for some w * ∈ S F,u * .
Step 5: We end our proof by discussing an a priori bounds on solutions.
Consequently, by the nonlinear alternative of Leray-Schauder type (see [20]), we deduce that T has a fixed point u ∈ U which is a solution of the problem (1) and ( 2).This completes the proof.

Compactness of the Set of Solutions
Theorem 2. Under Assumptions (H 1 ) , (H 2 ), the set of solutions to Problem (1) and (2) is not empty, and it is compact.
Let (u n ) n∈N ∈ S, then there exist v n ∈ S F.u n such that From (H 2 ), we can prove that there exists an M > 0 such that As in Theorem 1, we can show by using (H 2 ) that the set {u n , n ≥ 1} is equicontinuous in C ([0, 1] , R) ; hence, by Arzela-Ascoli's theorem we can conclude that there exists a subsequence u n k such that u n k converges to some u in C ([0, 1] , R).We shall now prove that there exists v (.) ∈ F (., y (.)) such that Additionally, as F (t, .) is upper semi-continuous, then for every > 0, there exists n 0 ( ), such that for every n ≥ n 0 we have Since F (., .)has compact values, there exists a subsequence v n m such that v n m (.) → v (.) as m → ∞ and v (t) ∈ F (t, u (t)) , a.e.t ∈ [0, 1] , and ∀m ∈ N.
Let the function As above, the multi-valued map ) is measurable, so there exists a measurable selection g 3 of U 3 .Consider the function Repeating the process for n = 0, 1, 2, 3, ...., we arrive at the following bound: Suppose that (15) holds for some n, now it is left to check (16) for n + 1.The multi-valued map [8]); then, there exists a function t → g n+1 (t), which is a measurable selection for U n+1 .
Finally, we prove that the solution v (t)