Boundedness and asymptotic Behaviour of Solutions of some second-order nonlinear stochastic differential equations with delay

Abstract

This paper considers a certain second-order nonlinear stochastic differential equation with delay. Novel conditions for the existence of solutions that are uniformly bounded and ultimately bounded are obtained. Moreover, we also study the asymptotic behaviour of solutions for the considered equation. We employ Lyapunov’s second method via an appropriate complete Lyapunov functional to achieve these. Obtained results are new, and they improve and complement some existing relatively recent results in the literature. Finally, an example is provided to illustrate the obtained results.