  In this paper, we consider the Cauchy-type problem for a nonlinear
differential equation involving $\Psi$-Hilfer fractional derivative and prove
the existence and uniqueness of solutions in the weighted space of functions.
The Ulam--Hyers and Ulam--Hyers--Rassias stability of Cauchy--type problem is
investigated via successive approximation method. Further, we investigate the
dependence of solutions on the initial conditions and uniqueness via
$\epsilon$-approximated solution. An example is provided to illustrate the
results we obtained.
