On the existence of mild solutions to the Cauchy problem for a class of fractional evolution equation

Jin Liang,Fang Li,Ting-Wen Huang,Sheng-Hua Yan

doi:10.1186/1687-1847-2012-40

Abstract

We are concerned with the existence of mild solutions to the Cauchy problem for fractional evolution equations of neutral type with almost sectorial operators d q d t q ( x ( t ) - h ( t , x ( t ) ) ) = - A ( x ( t ) - h ( t , x ( t ) ) ) + f ( t , x ( t ) ) , t > 0 , x (0) = x 0 , where 0 < q < 1, the fractional derivative is understood in the Caputo sense, A is an almost sectorial operator on a complex Banach space, and f, h are given functions. With the help of the theory of measure of noncompactness and a fixed point theorem of Darbo type, we establish a new existence theorem of mild solutions for the Cauchy problem above. By the way, the global attractive property of the solutions is also obtained. Moreover, we give two examples to illustrate our abstract results.

Highlights

The fractional evolution equations have received increasing attention during recent years and have been studied extensively since they can be used to describe many phenomena arising in engineering, physics, economy, and science.We mention that much of the previous research on the evolution equations was done provided that the operator in the linear part is the infinitesimal generator of a strongly continuous operator semigroup, an analytic semigroup, or a compact semigroup, or is a Hille-Yosida operator
F (t, x(t)), x(0) = x0, t > 0, where 0
With the help of the theory of measure of noncompactness and a fixed point theorem of Darbo type, we establish a new existence theorem of mild solutions for the Cauchy problem above

Summary

Introduction

The fractional evolution equations have received increasing attention during recent years and have been studied extensively (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13] and references therein) since they can be used to describe many phenomena arising in engineering, physics, economy, and science.We mention that much of the previous research on the evolution equations was done provided that the operator in the linear part is the infinitesimal generator of a strongly continuous operator semigroup, an analytic semigroup, or a compact semigroup, or is a Hille-Yosida operator (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,14,15] and references therein). With the help of the theory of measure of noncompactness and a fixed point theorem of Darbo type, we establish a new existence theorem of mild solutions for the Cauchy problem above.

Results

Conclusion