On a generalized fractional differential Cauchy problem

Qualitative results for abstract problems are very important in understanding mathematical analysis on which any application is possible. The focus of this paper is twofold: first, we investigate the existence and uniqueness of mild solutions to a generalized Cauchy problem for the nonlinear differential equation with non-local conditions in a Banach space $X$. This is achievable using some fixed point theorems in infinite dimensional spaces. Secondly, we study the stability results of the system in the sense of Ulam-Hyers-Rassias. Our results improve and generalize most recent related results in the literature.