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

In this work, we address a nonlinear ψ-Hilfer fractional-order Volterra integro-differential equation that incorporates n-multiple-variable time delays. Employing the ψ-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 ψ-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 ψ-Hilfer fractional-order Volterra integro-differential equations, both including and excluding single delay, by establishing new findings for nonlinear ψ-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.