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Global Existence and Ulam--Hyers Stability of $Ψ$--Hilfer Fractional Differential Equations (1807.10105v2)
Published 26 Jul 2018 in math.DS
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.