A Note on Impulsive Fractional Evolution Equations with Nondense Domain

Zufeng Zhang,Bin Liu

doi:10.1155/2012/359452

Abstract

This paper is concerned with the existence of integral solutions for nondensely defined fractional functional differential equations with impulse effects. Some errors in the existing paper concerned with nondensely defined fractional differential equations are pointed out, and correct formula of integral solutions is established by using integrated semigroup and some probability densities. Sufficient conditions for the existence are obtained by applying the Banach contraction mapping principle. An example is also given to illustrate our results.

Highlights

As pointed in 14, 1.4 is not the mild solution. Motivated by these papers and the fact that impulse effects exist widely in the realistic situations, we give the definition of integral solution and prove the existence results for impulsive semilinear fractional differential equations with nondensely defined operators
An essence error of the formula of solutions which appeared in the recent work on the nondensely defined fractional evolution differential equations is reported in this work
A correct formula of integral solutions for nondensely defined fractional evolution equations could be obtained from the results in this paper

Summary

Introduction

Dqy t Ay t f t, yt , t ∈ J : 0, b , t / tk, k 1, . . . , m, Δy t tk Ik y t−k , k 1, . . . , m, 1.1 y t φ t , t ∈ −τ, 0 , where 0 < q < 1, Dq is the Caputo fractional derivative. f : J × D → E is a given function,. Many authors were devoted to mild solutions to fractional evolution equations, and there have been a lot of interesting works. In 12 , Zhou and Jiao concerned the existence and uniqueness of mild solutions for fractional evolution equations by some fixed point theorems. Existence results for integral solutions of nondensely defined fractional evolution equations were established in some papers 9, 26. As pointed in 14 , 1.4 is not the mild solution Motivated by these papers and the fact that impulse effects exist widely in the realistic situations, we give the definition of integral solution and prove the existence results for impulsive semilinear fractional differential equations with nondensely defined operators.

Preliminaries

Main Results

An Example

Conclusions