Fractional Differential Equations with Fractional Impulsive and Nonseparated Boundary Conditions

Xi Fu,Xiaoyou Liu

doi:10.1155/2014/364348

Abstract

This paper studies the existence results for nonseparated boundary value problems of fractional differential equations with fractional impulsive conditions. By means of Schaefer fixed point theorem, Banach fixed point theorem, and nonlinear alternative of Leray-Schauder type, some existence results are obtained. Examples are given to illustrate the results.

Highlights

The subject of fractional differential equations has recently evolved as an interesting and popular field of research
From the counterexamples given in Lemma 3.1 of [22], Section 1 in [23], and Section 3 in [24], we know that if the case (II) was chosen to construct solutions, the solution formula obtained in the fractional integral equation form is not equivalent to the original impulsive fractional differential equation
We investigated the existence and uniqueness of solutions for a class of fractional differential equations with fractional impulsive conditions and nonseparated boundary conditions

Summary

Introduction

The subject of fractional differential equations has recently evolved as an interesting and popular field of research. As pointed out in the papers [22,23,24], the concept of piecewise continuous solutions used in some already published works to handle the impulsive fractional differential equations is not appropriate (see counterexamples given in Lemma 3.1 of [22], Section 1 in [23], and Section 3 in [24]). The papers on this topic cited above except [22] all deal with the Caputo derivative and the impulsive conditions only involve integer order derivatives.

Preliminaries

Main Results

Examples

Conclusions