Existence results for higher order fractional differential inclusions with multi-strip fractional integral boundary conditions

Bashir Ahmad,Sotiris Ntouyas

doi:10.14232/ejqtde.2013.1.20

Abstract

This paper investigates the existence of solutions for higher order fractional differential inclusions with fractional integral boundary conditions involving nonintersecting finite many strips of arbitrary length. Our study includes the cases when the right-hand side of the inclusion has convex as well non-convex values. Some standard fixed point theorems for multivalued maps are applied to establish the main results. An illustrative example is also presented.

Highlights

We study a boundary value problem of a fractional differential inclusion with multi-strip fractional integral boundary conditions given by cDqx(t) ∈ F (t, x(t)), t ∈ [0, T ]
We have introduced Riemann-Liouville type fractional integral boundary conditions involving nonintersecting finite many strips of arbitrary length
We assume that F is a compact and convex valued multivalued map

Summary

Introduction

We study a boundary value problem of a fractional differential inclusion with multi-strip fractional integral boundary conditions given by cDqx(t) ∈ F (t, x(t)), t ∈ [0, T ], m x(0) = 0, x′(0) = 0, . We have introduced Riemann-Liouville type fractional integral boundary conditions involving nonintersecting finite many strips of arbitrary length. The first result relies on the nonlinear alternative of Leray-Schauder type. 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 decomposable values, while in the third result, we shall use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler.

Fractional Calculus

Basic Material for Multivalued Maps

The Caratheodory Case

The Lower Semi-Continuous Case

The Lipschitz Case