Existence of Mild Solutions and Controllability of Fractional Impulsive Integrodifferential Systems with Nonlocal Conditions

Haiyong Qin,Zhenyun Gu,Tongxing Li,Youliang Fu

doi:10.1155/2017/6979571

Abstract

This paper is concerned with the existence results of nonlocal problems for a class of fractional impulsive integrodifferential equations in Banach spaces. We define a piecewise continuous control function to obtain the results on controllability of the corresponding fractional impulsive integrodifferential control systems. The results are obtained by means of fixed point methods. An example to illustrate the applications of our main results is given.

Highlights

IntroductionExistence of mild solutions of nonlocal Cauchy problems has been investigated extensively by many researchers (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] and the references cited therein)
In recent decades, existence of mild solutions of nonlocal Cauchy problems has been investigated extensively by many researchers
Using Krasnoselskii’s fixed point theorem, Schauder’s fixed point theorem, and Banach contraction principle, Zhou and Jiao [13] obtained several criteria on the existence and uniqueness of mild solutions of nonlocal Cauchy problems for fractional evolution equations without impulse. Such analysis on nonlocal Cauchy problems is important from an applied viewpoint, since the nonlocal condition has a better effect in applications than a classical initial one

Summary

Introduction

Existence of mild solutions of nonlocal Cauchy problems has been investigated extensively by many researchers (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] and the references cited therein). Using Krasnoselskii’s fixed point theorem, Schauder’s fixed point theorem, and Banach contraction principle, Zhou and Jiao [13] obtained several criteria on the existence and uniqueness of mild solutions of nonlocal Cauchy problems for fractional evolution equations without impulse. Such analysis on nonlocal Cauchy problems is important from an applied viewpoint, since the nonlocal condition has a better effect in applications than a classical initial one.

Preliminaries and Lemmas

Existence and Uniqueness of PC-Mild Solutions

Controllability Results

Example

Conclusions