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.
Highlights
Various aspects of the theory of third-order differential inclusions with boundary conditions attract the attention of many researchers (e.g., [1,2,3,4,5,6,7,8,9,10])
In the present paper we study third-order differential inclusions of the form
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
Summary
Various aspects of the theory of third-order differential inclusions with boundary conditions attract the attention of many researchers (e.g., [1,2,3,4,5,6,7,8,9,10]). 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. F is convex and upper semi-continuous and satisfies a Carathéodory condition. An illustrative example of a boundary value problem satisfying the mentioned conditions is given.
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