Impulsive fractional differential equations in Banach spaces

Djamila Seba,Mouffak Benchohra

doi:10.14232/ejqtde.2009.4.8

Abstract

This paper is devoted to study the existence of solutions for a class of initial value problems for impulsive fractional differential equations involving the Caputo fractional derivative in a Banach space. The arguments are based upon

Highlights

The theory of fractional differential equations is an important branch of differential equation theory, which has an extensive physical, chemical, biological, and engineering background, and has been emerging as an important area of investigation in the last few decades; see the monographs of Kilbas et al [32], Miller and Ross [37], and the papers of Agarwal et al [1, 2], Belarbi et al [11], Benchohra et al [12, 13, 15], Delboso and Rodino [20], Diethelm and Ford [22], El-Sayed et al [23], Furati and Tatar [25, 26, 27], Momani et al [38, 39], and Lakshmikantham and Devi [33]
Applications of the theory of fractional differential equations to different areas were considered by many authors and some basic results on fractional differential equations have been obtained see, for example, Gaul et al [28], Glockle and Nonnenmacher [29], Hilfer [31], Mainardi [35], Metzler et al [36] and Podlubny [42], and the references therein
Benchohra et al [16] applied the measure of noncompactness to a class of Caputo fractional differential equations of order r ∈

Summary

Introduction

The theory of fractional differential equations is an important branch of differential equation theory, which has an extensive physical, chemical, biological, and engineering background, and has been emerging as an important area of investigation in the last few decades; see the monographs of Kilbas et al [32], Miller and Ross [37], and the papers of Agarwal et al [1, 2], Belarbi et al [11], Benchohra et al [12, 13, 15], Delboso and Rodino [20], Diethelm and Ford [22], El-Sayed et al [23], Furati and Tatar [25, 26, 27], Momani et al [38, 39], and Lakshmikantham and Devi [33]. Benchohra et al [16] applied the measure of noncompactness to a class of Caputo fractional differential equations of order r ∈ Lakshmikantham and Devi [33] discussed the uniqueness and continuous dependence of the solutions of a class of fractional differential equations using the Riemann-Liouville derivative of order r ∈ See Akhmerov et al [4], Alvarez [5], Banas et al [6, 7, 8, 9], El-Sayed and Rzepka [24], Guo et al [30], Monch [40], Monch and Von Harten [41] and Szufla [44]

Preliminaries

Existence of Solutions

Nonlocal impulsive differential equations

An Example