Fractional differential inclusions with anti-periodic boundary conditions in Banach spaces.

Ahmed Ibrahim

doi:10.14232/ejqtde.2014.1.65

Abstract

The main purpose of this paper is to provide the theory of differential inclu- sions by new existence results of solutions for boundary value problems of differential inclusions with fractional order and with anti-periodic boundary conditions in Banach spaces. We prove existence theorems of solutions under both convexity and nonconvex- ity conditions on the multivalued side. Meanwhile, the compactness of the set solutions is also established.

Highlights

During the past two decades, fractional differential equations and fractional differential inclusions have gained considerable importance due to their applications in various fields, such as physics, mechanics and engineering
For some recent development on initial value problems for differential equations and inclusions of fractional order we refer the reader to the references [1, 32, 34,35,36,37,38]
Many authors have studied differential inclusions with various boundary conditions. Some of these works have been done in finite dimensional spaces and of positive integer order, for example, Ibrahim et al [25] and Gomma [17] considered a functional multivalued three-point boundary value problem of second-order

Summary

Introduction

During the past two decades, fractional differential equations and fractional differential inclusions have gained considerable importance due to their applications in various fields, such as physics, mechanics and engineering. Some applied problems in physics require fractional differential equations and inclusions with boundary conditions. Many authors have studied differential inclusions with various boundary conditions Some of these works have been done in finite dimensional spaces and of positive integer order, for example, Ibrahim et al [25] and Gomma [17] considered a functional multivalued three-point boundary value problem of second-order. Several results have been obtained for fractional differential equations and inclusions with various boundary value conditions in finite dimensional spaces. For some recent works on boundary value problems for fractional differential inclusions in infinite dimensional spaces, we refer to Benchohra et al [8] who established the existence of solutions of nonlinear fractional differential inclusions with two point boundary conditions. The proofs rely on the methods and results for boundary value fractional differential inclusions, the properties of noncompact measure and fixed point techniques

Preliminaries and notation

Convex case

Nonconvex case

Examples

Conclusion