On existence of solution to nonlinear $\psi$-Hilfer Cauchy-type problem

TL;DR

This paper establishes the existence and uniqueness of solutions for a nonlinear Cauchy-type problem involving the $ ext{ extpsi}$-Hilfer fractional derivative, using fixed point and Gronwall inequalities, with an illustrative example.

Contribution

It introduces new existence and uniqueness results for nonlinear $ ext{ extpsi}$-Hilfer fractional differential equations using fixed point and Gronwall methods.

Findings

Existence of unique solutions proven for the nonlinear problem.

Continuous dependence of solutions on data demonstrated.

Illustrative example provided to support theoretical results.

Abstract

The aim of this paper is to obtain the existence of unique solution to nonlinear Cauchy-type problem. We consider the implicit nonlinear Cauchy-type problem with $\psi$-Hilfer fractional derivative. The Banach fixed point theorem is used to obtain the existence of a unique solution whereas the generalized Gronwall inequality is used to discuss continuous data dependence of the solution. The results obtained herein are supported with illustrative example.