mathematics Article Existence of Solution, Filippov’s Theorem and Compactness of the Set of Solutions for a Third-Order Differential Inclusion with Three- Point Boundary Conditions Ali Rezaiguia 1, * and Smail Kelaiaia 2 ID 1 Department of Mathematics and Computer Science, Faculty of Sciences , University of Souk Ahras, Souk Ahras 41000, Algeria 2 Department of Mathematics, Faculty of Sciences, University of Annaba, P.O. Box 12, Annaba 23000, Algerie; kelaiaiasmail@yahoo.fr * Correspondence: ali_rezaig@yahoo.fr Received: 15 December 2017; Accepted: 16 February 2018; Published: 8 March 2018 Abstract: In this paper, we study a third-order differential inclusion with three-point boundary conditions. We prove the existence of a solution under convexity conditions on the multi-valued right-hand side; the proof is based on a nonlinear alternative of Leray-Schauder type. We also study the compactness of the set of solutions and establish some Filippov’s- type results for this problem. Keywords: differential inclusion; boundary value problem; fixed point theorem; selection theory; Filippov’s Theorem 1. Introduction Various aspects of the theory of third-order differential inclusions with boundary conditions attract the attention of many researchers (e.g., [1–10]). 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: u ′ (0)= u ′ (1)= αu (η ) , u (0)= βu (η ) , (2) 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. Mathematics 2018, 6, 40; doi:10.3390/math6030040 www.mdpi.com/journal/mathematics