Global Existence and Ulam--Hyers Stability of $\mathrm{\Psi }$--Hilfer Fractional Differential Equations

In this paper, we consider the Cauchy-type problem for a nonlinear differential equation involving $\mathrm{\Psi }$-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 $ϵ$-approximated solution. An example is provided to illustrate the results we obtained.