On the Ulam-Hyers stabilities of {\Psi}-Hilfer fractional differential equation by means of abstract Volterra operator

TL;DR

This paper studies the stability of a new class of fractional differential equations involving the {}u{}u-Volterra operator, establishing existence, uniqueness, and Ulam-Hyers stability using fixed point theorems and fractional integrals.

Contribution

It introduces a novel fractional differential equation with the {}u{}u-Volterra operator and analyzes its stability and solutions in Banach spaces.

Findings

Established existence and uniqueness of solutions.

Proved Ulam-Hyers stability on finite and infinite intervals.

Applied results to a specific example.

Abstract

In this paper, we consider the new class of the fractional differential equation involving the abstract Volterra operator in the Banach space and investigate existence, uniqueness and stabilities of Ulam--Hyers on the compact interval $\Delta=[a,b]$ and on the infinite interval $I=[a,\infty )$. Our analysis is based on the application of the Banach fixed point theorem and the Gronwall inequality involving generalized $\Psi$-fractional integral. At last, we performed out an application to elucidate the outcomes got.