INFINITE INTERVAL PROBLEMS FOR HILFER FRACTIONAL EVOLUTION EQUATIONS WITH ALMOST SECTORIAL OPERATORS

The purpose of this paper is to investigate the Cauchy problem on an infinite interval for the fractional evolution equation with Hilfer fractional derivative, which is a generalization of both Riemann-Liuoville and Caputo fractional derivatives. Our methods base on the generalized Ascoli-Arzela theorem, Schauder’s fixed point theorem, Wright function and Kuratowski’s measure of noncompactness. We obtain sufficient conditions of the existence for global mild solutions and attractive solutions when the semigroup associated with an almost sectorial operator is compact as well as noncompact. At last, two examples are provided to illustrate the results.