Square integrable solutions and stability of a second-order stochastic integro-differential equation

This article investigates the stochastic asymptotic stability, boundedness, and square integrability of solutions to a class of second-order nonlinear stochastic integro-differential equations with multiple variable delays. The analysis is conducted through the construction of an appropriate Lyapunov-Krasovskii functional (L-KF), tailored to handle the combined effects of stochastic perturbations, time-varying delays, and integro-differential memory terms. Unlike many existing studies, our framework accommodates all these complexities simultaneously, thereby generalizing and extending recent contributions in the field while relaxing several restrictive assumptions. To validate the theoretical results and illustrate their practical applicability, numerical simulations are provided.