Existence and continuous dependence of mild solutions for fractional neutral abstract evolution equations

Jianxin Cao,Weifeng Yang,Haibo Chen

doi:10.1186/s13662-014-0333-2

Abstract

This paper is mainly concerned with the existence, uniqueness and continuous dependence of mild solutions for fractional neutral functional differential equation with nonlocal initial conditions and infinite delay. The results are obtained by means of the classical fixed point theorems combined with theory of resolvent operators for integral equations.

Highlights

1 Introduction In this paper, we are concerned with the neutral fractional differential equation of the form
There are many papers treating the problem of the existence of a mild solution for abstract semilinear fractional differential equations; see [ – ]
In [ ], the authors utilized an approach based on the well-developed theory of resolvent operators for integral equations to deal with the following abstract fractional equations

Summary

Introduction

We are concerned with the neutral fractional differential equation of the form. A has discrete spectrum with eigenvalues of the form –n , n ∈ N, and corresponding normalized eigenfunctions given by zn(ξ ) From these expressions it follows that (T(t))t≥ is a uniformly bounded compact semigroup, so that R(λ, A) = (λ – A)– is a compact operator for all λ ∈ ρ(A). There are many papers treating the problem of the existence of a mild solution for abstract semilinear fractional differential equations; see [ – ]. In [ ], the authors utilized an approach based on the well-developed theory of resolvent operators for integral equations to deal with the following abstract fractional equations. By considering an integral equation which is given in terms of probability density and semigroup, they establish criteria on the existence and uniqueness of mild solutions.

For any a

Mf q

Lf q