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In this paper, we determine sufficient conditions for the boundedness, uniformly asymtotically stability of the solutions to a certain fourth-order non-autonomous differential equations with bounded delay by considering second method of Lyapunov. The results obtain essentially improve, include and complement the consequences in the current literature. [ABSTRACT FROM AUTHOR]

The objective of this study was to get some sufficient conditions which guarantee the asymptotic stability and uniform boundedness of the null solution of some differential equations of third order with the variable delay. The most efficient tool for the study of the stability and boundedness of solutions of a given nonlinear differential equation is provided by Lyapunov theory. However the construction of such functions which are positive definite with corresponding negative definite derivatives is in general a difficult task, especially for higher-order differential equations with delay. Such functions and their time derivatives along the system under consideration must satisfy some fundamental inequalities. Here the Lyapunov second method or direct method is used as a basic tool. By defining an appropriate Lyapunov functional, we prove two new theorems on the asymptotic stability and uniform boundedness of the null solution of the considered equation. Our results obtained in this work improve and extend some existing well-known related results in the relevant literature which were obtained for nonlinear differential equations of third order with a constant delay. We also give an example to illustrate the importance of the theoretical analysis in this work and to test the effectiveness of the method employed. [ABSTRACT FROM AUTHOR]

Stochastic delay differential equations with wideband noise perturbations is considered. First it is shown that the perturbed system converges weakly to a stochastic delay differential equation driven by a Brownian motion. Stability and asymptotic properties of stochastic delay differential equations with a small parameter are developed. It is shown that the properties such as stability, recurrence, etc., of the limit system with time lag is preserved for the solution xε(·) of the underlying delay equation for ε > 0 small enough. Perturbed Liapunov function method is used in the analysis. [ABSTRACT FROM AUTHOR]

This paper studies stability problems of impulsive control systems with time delay. Several criteria on asymptotic stability are established using the method of Lyapunov functions and Lyapunov functionals. It is shown that by using impulsive feedback control, a linear delay differential system can be stabilized even if it contains no stable matrix. Some examples are also discussed to illustrate the theorems. [Copyright &y& Elsevier]

In the paper, employing the variational Lyapunov method, stability and instability properties in terms of two measures for impulsive delay differential equations with fixed moments of impulsive effects are discussed. Some stability and instability criteria are obtained. These results much generalize the known ones. Some examples are given to illustrate the advantages of them as well. [Copyright &y& Elsevier]