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 (1.1) were established with the aid of a suitable Lyapunov functional. Two examples are given in the last section to illustrate our main result.

The main purpose of this work is to give sufficient conditions for the uniform stability of the zero solution of a certain fourth-order vector delay differential equation of the following form:X(4)+F(X˙,X¨)X⃛+Φ(X¨)+G(X˙(t-r))+H(X(t-r))=0.By constructing a Lyapunov functional, we obtained the result of stability.

We present two qualitative results concerning the solutions of the following equation: x¨(t)+g(x˙(t))+bx(t-h)+σx(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 case p≡0, while the second one discusses the uniform stochastic boundedness of all solutions in case p≢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.

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In this paper sufficient conditions are given for uniform boundedness and Ž . convergence to zero of all solutions of 1.1 together with their derivatives to the fourth order. ᮊ 1999 Academic Press