In this paper, sufficient criteria that guarantee the existence of stochastic asymptotic stability of the zero solution of the nonautonomous second-order stochastic delay differential equation were established with the aid of a suitable Lyapunov functional. Two examples are given in the last section to illustrate our main result. | Turk J Math (2017) 41: 576 – 584 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Asymptotic stability of solutions for a certain non-autonomous second-order stochastic delay differential equation Ahmed Mohamed ABOU-EL-ELA1 , Abdel-Rahiem SADEK1 , Ayman Mohammed MAHMOUD2,∗, Eman Sayed FARGHALY1 1 Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt 2 Department of Mathematics, Faculty of Science, New Valley Branch, Assiut University, New Valley, El-Khargah, Egypt Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, sufficient criteria that guarantee the existence of stochastic asymptotic stability of the zero solution of the nonautonomous second-order stochastic delay differential equation () were established with the aid of a suitable Lyapunov functional. Two examples are given in the last section to illustrate our main result. Key words: Asymptotic stability, nonautonomous second-order stochastic delay differential equation, Lyapunov functional 1. Introduction It is well known that random fluctuations are abundant in natural or engineered systems. Therefore stochastic modeling has come to play an important role in various fields such as biology, mechanics, economics, medicine, and engineering (see [6, 20, 21]). Moreover, these systems are sometimes subject to memory effects, when their time evolution depends on their past history with noise disturbance. Stochastic delay differential equations (SDDEs) give a mathematical formulation for such systems. They can be regarded as a natural generalization of stochastic ordinary differential equations by allowing the coefficients to depend on the past values. Lyapunov’s direct method has been successfully used to investigate stability problems in deterministic/stochastic differential equations and delay differential equations. Many papers dealt with the delay
