Existence and Controllability Results for Nonlocal Fractional Impulsive Differential Inclusions in Banach Spaces

Ahmed G Ibrahim,Jinrong Wang

doi:10.1155/2013/518306

Abstract

We firstly deal with the existence of mild solutions for nonlocal fractional impulsive semilinear differential inclusions involving Caputo derivative in Banach spaces in the case when the linear part is the infinitesimal generator of a semigroup not necessarily compact. Meanwhile, we prove the compactness property of the set of solutions. Secondly, we establish two cases of sufficient conditions for the controllability of the considered control problems.

Highlights

During the past two decades, fractional differential equations and inclusions have gained considerable importance due to their applications in various fields, such as physics, mechanics, and engineering
The theory of impulsive differential equations and inclusions has been an object of interest because of its wide applications in physics, biology, engineering, medical fields, industry, and technology
The reason for this applicability arises from the fact that impulsive differential problems are an appropriate model for describing process which at certain moments change their state rapidly and which cannot be described using the classical differential problems

Summary

Introduction

During the past two decades, fractional differential equations and inclusions have gained considerable importance due to their applications in various fields, such as physics, mechanics, and engineering. In order to do a comparison between our obtained results in this paper and the known recent results in the same domain, we refer to the following: Ouahab [8] proved a version of Filippov’s Theorem for (1) without impulse and A is an almost sectorial operator, Cardinali and Rubbioni [16] proved the existence of mild solutions to (1) when α = 1 and the multivalued function F satisfies the lower ScorzaDragoni property, and {A(t)}t≥0 is a family of linear operator, generating a strongly continuous evolution operators, Fan [17] studied a nonlocal Cauchy problem in the presence of impulses, governed by autonomous semilinear differential equation, Dads et al [20] and Henderson and Ouahab [21] considered (1) when A = 0, and Zhou and Jiao [12, 13] introduced a suitable definition of mild solution for (1) based on Laplace transformation and probability density functions for (1) when F is single-valued function and without impulse. Ibrahim and Al Sarori [22] gave some existence results of mild solutions for nonlocal impulsive differential inclusions with delay and of fractional order in Caputo sense when the semigroup is compact. Our basic tools are the methods and results for semilinear differential inclusions, the properties of noncompact measure, compactness criterion in the piecewise continuous functions of space, and fixed point techniques

Preliminaries and Notation

Application to Controllability

Conclusion