On the theory of $\psi$-hilfer nonlocal Cauchy problem

In this paper, we derive the representation formula of the solution for $\psi$-Hilfer fractional differential equation with constant coefficient in the form of Mittag-Leffler function by using Picard's successive approximation. Moreover, by using some properties of Mittag-Leffler function and fixed point theorems such as Banach and Schaefer, we introduce new results of some qualitative properties of solution such as existence and uniqueness. The generalized Gronwall inequality lemma is used in analyze $\mathrm{E}_{\alpha}$-Ulam-Hyers stability. Finally, one example to illustrate the obtained results.