Existence of Attractive Solutions for Hilfer Fractional Evolution Equations with Almost Sectorial Operators

The purpose of this paper is to investigate the existence of attractive solutions for a Cauchy problem of fractional evolution equations with Hilfer fractional derivative, which is a generalization of both the Riemann-Liuoville and Caputo fractional derivatives. Our methods are based on the generalized Ascoli-Arzela theorem, Schauder's fixed point theorem, the Wright function and Kuratowski's measure of noncompactness. The symmetric structure of the spaces and the operators defined by us plays a crucial role in showing the existence of fixed points. 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.