In this paper, we consider the Cauchy-type problem for a nonlinear differential equation involving a Ψ-Hilfer fractional derivative and prove the existence and uniqueness of solutions in the weighted space of functions. The Ulam-Hyers and Ulam-Hyers-Rassias stabilities of the Cauchy-type problem is investigated via the successive approximation method. Further, we investigate the dependence of solutions on the initial conditions and their uniqueness using ϵ-approximated solutions. Finally, we present examples to illustrate our main results.

Keywords: Ψ-Hilfer fractional derivative, existence and uniqueness, Ulam-Hyers stability, successive approximations, є,-approximate solution, dependence of solution.