� �-Hilfer Fractional Evolution Equation

Abstract

In this article, we obtain the existence and uniqueness of a solution for fractional evolution equation involving 𝜓-Hilfer fractional derivative. 𝜓-Hilfer fractional derivative generalizes several other fractional derivatives such as Riemann-Liouville, Caputo, 𝜓-Riemann-Liouville, � �-Caputo and so on. Using the generalized Laplace transform and probability density function, the fundamental form of mild solution is obtained. then, we establish the existence of a unique mild solution under the assumption that the nonlinear function is Lipschitz continuous by invoking the Banach contraction principle. After that, we slightly reduce the regularity of the nonlinear function and prove the existence of a unique mild solution for the cases of compact and noncompact semigroups. We use Schauder’s fixed point theorem. After that, Mittag Leffler-Ulam-Hyers-Rassias stability is discussed at the end.