Existence And Attractivity Of Mild Solutions For Hilfer Fractional Evolution Equations

The main motivation of studying fractional evolution equations comes from two aspects:(i)One is that many mathematical models in physics and fluid mechanics are characterized by fractional partial differential equations;(ii)many types of fractional partial differential equations can be abstracted as fractional evolution equations.Therefore,the study of fractional evolution equations is of great significance both in terms of theory and practical application.This paper mainly discusses existence and attractivity of mild solutions for Hilfer fractional evolution equations with almost sectorial operators.In Chapter 2,we obtain new sufficient conditions for the existence of mild solutions to a Cauchy problem of Hilfer fractional evolution equation on a finite interval when the semigroup associated with an almost sectorial operator is compact as well as noncompact.Our results improve and extend some recent results in references.In Chapter 3,we investigate the existence and attractivity of mild solutions for a Cauchy problem of Hilfer fractional evolution equations on an infinite interval.The methods are based on the generalized Ascoli–Arzela theorem,Schauder’s fixed point theorem,the Wright function and Kuratowski’s measure of noncompactness.We obtain the global existence and attractivity results of mild solutions when the semigroup associated with an almost sectorial operator is compact as well as noncompact.