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

Abstract

In this paper, we consider the Cauchy-type problem for a nonlinear differential equation involving 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 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 ǫ-approximated solution. An example is provided to illustrate the results we obtained.