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

Abstract

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.