Some remarks on a fractional differential inclusion with non-separated boundary conditions

Aurelian Cernea

doi:10.14232/ejqtde.2011.1.45

Abstract

We study a boundary value problem for a fractional differential inclusion of order � 2 (1,2] with non-separated boundary conditions involving a nonconvex set-valued map. We establish a Filippov type existence theorem and we prove the arcwise connectedness of the solution set of the problem considered.

Highlights

In this paper we study the following problemDcαx(t) ∈ F (t, x(t)) a.e. ([0, T ]), (1.1)x(0) − k1x(T ) = c1, x′(0) − k2x′(T ) = c2, (1.2)where α ∈ (1, 2], Dcα is the Caputo fractional derivative of order α, F : [0, T ] × R → P(R) is a set-valued map and c1, c2, k1, k2 ∈ R, k1, k2 = 1.The present paper is motivated by a recent paper of Ahmad and Ntouyas ([1]) where it is studied problem (1.1)-(1.2) and several existence results for this problem are obtained using nonlinear alternative of Leray Schauder type and some suitable theorems of fixed point theory
We show that Filippov’s ideas ([5]) can be suitably adapted in order to obtain the existence of solutions for problem (1.1)-(1.2)
On the other hand, following the approach in [10] we prove the arcwise connectedness of the solution set of problem (1.1)-(1.2)

Summary

Introduction

We show that Filippov’s ideas ([5]) can be suitably adapted in order to obtain the existence of solutions for problem (1.1)-(1.2). We recall that for a differential inclusion defined by a lipschitzian set-valued map with nonconvex values, Filippov’s theorem ([5]) consists in proving the existence of a solution starting from a given almost solution. On the other hand, following the approach in [10] we prove the arcwise connectedness of the solution set of problem (1.1)-(1.2). The paper is organized as follows: in Section 2 we recall some preliminary facts that we need in the sequel, Section 3 is devoted to the Filippov type existence theorem and in Section 4 we obtain the arcwise connectedness of the solution set

Preliminaries

A Filippov type existence result

Arcwise connectedness of the solution set