Stability of the fractional Volterra integro-differential equation by means of $\psi-$Hilfer operator

TL;DR

This paper investigates the stability of fractional Volterra integro-differential equations using the $ ext{ extpsi}$-Hilfer operator, introducing new stability results and applying fixed-point theorems in Banach and Sobolev spaces.

Contribution

It proposes a novel fractional Volterra integro-differential equation framework utilizing the $ ext{ extpsi}$-Hilfer operator and establishes Ulam-Hyers stability results with applications in Sobolev spaces.

Findings

Established Ulam-Hyers stability for the new fractional equations.

Applied Banach fixed-point theorem to prove stability in Banach spaces.

Demonstrated stability using the $ ext{ extalpha}$-resolvent operator in Sobolev space.

Abstract

In this paper, using the Riemann-Liouville fractional integral with respect to another function and the $\psi-$Hilfer fractional derivative, we propose a fractional Volterra integral equation and the fractional Volterra integro-differential equation. In this sense, for this new fractional Volterra integro-differential equation, we study the Ulam-Hyers stability and, also, the fractional Volterra integral equation in the Banach space, by means of the Banach fixed-point theorem. As an application, we present the Ulam-Hyers stability using the $\alpha$-resolvent operator in the Sobolev space $W^{1,1}(\mathbb{R}_{+},\mathbb{C})$.