Global Existence and Ulam--Hyers Stability for $ψ$--Hilfer Fractional Differential Equations

Kishor D. Kucche, Jyoti P. Kharade

Published 2018-07-26Version 1

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 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.