On the Stochastic Stability and Boundedness of Solutions for Stochastic Delay Differential Equation of the Second Order. (31st March 2015) - Abou-El-Ela, A. M. A.; Sadek, A. I.; Mahmoud, A. M.; Taie, R. O. A.
Abstract : We present two qualitative results concerning the solutions of the following equation:x ¨ ( t ) + g ( x ˙ ( t ) ) + b x ( t - h ) + σ x ( t ) ω ˙ ( t ) = p ( t, x ( t ), x ˙ ( t ), x ( t - h ) ) ; the first result covers the stochastic asymptotic stability of the zero solution for the above equation in casep ≡ 0, while the second one discusses the uniform stochastic boundedness of all solutions in casep ≢ 0 . Sufficient conditions for the stability and boundedness of solutions for the considered equation are obtained by constructing a Lyapunov functional. Two examples are also discussed to illustrate the efficiency of the obtained results.