On the Existence of Solutions and Ulam-Type Stability for a Nonlinear Ψ-Hilfer Fractional-Order Delay Integro-Differential Equation

Abstract

In this work, we address a nonlinear psi-Hilfer fractional-order Volterra integro-differential equation that incorporates n-multiple-variable time delays. Employing the psi-Hilfer fractional derivative operator, we investigate the existence of a unique solution, as well as the Ulam-Hyers-Rassias stability, semi-Ulam-Hyers-Rassias stability, and Ulam-Hyers stability of the proposed psi-Hilfer fractional-order Volterra integro-differential equation through the fixed-point approach. In this study, we enhance and generalize existing results in the literature on psi-Hilfer fractional-order Volterra integro-differential equations, both including and excluding single delay, by establishing new findings for nonlinear psi-Hilfer fractional-order Volterra integro-differential equations involving n-multiple-variable time delays. This study provides novel theoretical insights that deepen the qualitative understanding of fractional calculus.