Existence and uniqueness of a mild solution for a class of the fractional evolution equation With nonlocal condition involving $\varphi$-Riemann Liouville fractional derivative

Abstract

In this paper, by using the fractional power of operators and theory fixed point theorems, we discuss Existence and uniqueness of mild solution to initial value problems for fractional semilinear evolution equations with compact semigroup in Banach spaces with nonlocal conditions. In particular, we derive the form of fundamental solution in terms of semigroup induced by resolvent and $\varphi$-Riemann-Liouville fractional derivatives. These results generalize previous works where the classical Riemann-Liouville fractional derivative is considered. In the end, we give an example to illustrate the applications of the abstract results.