On a nonlinear fractional order differential inclusion

Aurelian Cernea

doi:10.14232/ejqtde.2010.1.78

Abstract

The existence of solutions for a nonlinear fractional order differential inclusion is investigated. Several results are obtained by using suitable fixed point theorems when the right hand side has convex or non convex values.

Highlights

Differential equations with fractional order have recently proved to be strong tools in the modelling of many physical phenomena; for a good bibliography on this topic we refer to [17]
As a consequence there was an intensive development of the theory of differential equations of fractional order [2, 15, 20]
The study of fractional differential inclusions was initiated by ElSayed and Ibrahim [11]

Summary

Introduction

Differential equations with fractional order have recently proved to be strong tools in the modelling of many physical phenomena; for a good bibliography on this topic we refer to [17]. Very recently several qualitative results for fractional differential inclusions were obtained in [3, 6, 7, 8, 9, 13, 18]. Where L = Dα−aDβ, Dα is the standard Riemann-Liouville fractional derivative, α ∈ (1, 2), β ∈ (0, α), a ∈ R and F : I × R → P(R) is a set-valued map. The present paper is motivated by a recent paper of Kaufmann and Yao [14], where it is considered problem (1.1)-(1.2) with F single valued and several existence results are provided. The aim of our paper is to extend the study in [14] to the set-valued framework and to present some existence results for 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 and in Section 3 we prove our main results

Preliminaries

The main results