Nonlocal fractional functional differential equations with measure of noncompactness in Banach space

Junfei Cao,Qian Tong,Xianyong Huang

doi:10.1007/s40096-015-0150-0

Abstract

In this paper, we are concerned with the following fractional functional differential equations with nonlocal initial conditions in Banach space $$\begin{aligned} \hbox {D}^{\alpha }x(t)=Ax(t)+f(t,x(t),x_{t}),\ \ t\in [0, T], \ \ x(0)=\phi +g(x). \end{aligned}$$ By virtue of the theory of measure of noncompactness associated with Darbo’s fixed point theorem, upon making some suitable assumptions, some existence results of mild solutions are obtained. Moreover the results obtained are utilized to study the existence of solutions to fractional parabolic equations as an illustrative example to show the practical usefulness of the analytical results.

Highlights

We are concerned with the nonlocal initial value problem
The aim of this paper is to study the existence of mild solutions for the fractional functional differential Eq (1.1) in a separable Banach space
For example space-fractional diffusion equations have been used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium [1, 2] or to model activator–inhibitor dynamics with anomalous diffusion [3]

Summary

Introduction

We are concerned with the nonlocal initial value problemDaxðtÞ 1⁄4 AxðtÞ þ f ðt; xðtÞ; xtÞ; t 2 1⁄20; T; xð0Þ 1⁄4 / þ gðxÞ; ð1:1Þ where A is the infinitesimal generator of a strongly are given X-valued functions. Keywords Fractional functional differential equation Á Nonlocal initial condition Á Hausdorff measure of noncompactness Á Mild solution Á Darbo’s fixed point theorem

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Results

Conclusion