Existence, Uniqueness, and Asymptotic Behavior of Solutions to a Nonlinear Volterra Integro-Differential Equation with Time-Dependent Delay

Abstract

This paper presents a comprehensive analysis of the asymptotic behavior and stability properties of a novel class of nonlinear Volterra integro-differential equations featuring time-varying delays. The model under investigation incorporates a linear instantaneous dissipation term coupled with a nonlinear integral memory component characterized by a saturation-type response. The nonlinearity is defined by a function  satisfying specific growth and boundedness conditions. We establish the existence and uniqueness of solutions using a constructive framework based on Krasnoselskii's fixed point theorem. Subsequently, we employ the Razumikhin method to derive explicit, computationally verifiable stability criteria. Our main theoretical contribution lies in formulating concrete stability conditions that substantially extend and generalize previous results for similar equation structures. The practical relevance of our theoretical framework is demonstrated through applications to biological systems, such as population dynamics, and engineered systems, including artificial neural networks