On Stability of Solutions of a Certain Fourth-Order Functional Differential Equations

Using a Razumikhin-type theorem,we obtain sufficient conditions for the global asymptotic stability of the zero solution of a certain fourth order functional differential equations.The result generalizes the well known results.

Global Solutions and Exponential Stability of Stochastic Functional Differential Equations with Infinite Delay

This paper proves that, under the local Lipschitz condition, the stochastic functional differential equations with infinite delay have global solutions without the linear growth condition. Furthermore, the pth moment exponential stability conditions are given. Finally, one example is presented to illustrate our theory.

Asymptotic stability for impulsive functional differential equations

In this paper, we investigate the stability of a class of impulsive functional differential equations by using Lyapunov functional and Jensen＇s inequality. Some new stability theorems are obtained. Examples are given to demonstrate the advantage of the obtained results.

On the Stability and Boundedness of Solutions of Certain Non-Autonomous Delay Differential Equation of Third Order

In this paper, we study certain non-autonomous third order delay differential equations with continuous deviating argument and established sufficient conditions for the stability and boundedness of solutions of the equations. The conditions stated complement previously known results. Example is also given to illustrate the correctness and significance of the result obtained.

Some Stability Theorems for Neutral Functional Differential Equations

In this paper,the author investigates uuiform stability,asymptotic stability,and uniform asymptotic stability forthe zero solution of neutral functional differentialequations.Some well-known results are improved.

On the Stability of Solutions of Nonlinear Functional Differential Equation of the Fifth-Order

The main purpose of this paper is to investigate global asymptotic stability of the zero solution of the fifth-order nonlinear delay differential equation on the following form By constructing a Lyapunov functional, sufficient conditions for the stability of the zero solution of this equation are established.

Practical Stability in the pth Mean of Stochastic Differential Equations with Discontinuous Coefficients

In this paper,we give sufficient conditions to analyze the practical stability in the pth mean of stochastic differential equations with discontinuous coefficients.The Lyapunov-like function plays an important role in analysis.Some numerical computations are carried out to illustrate the theoretical results.

ON STABILITY OF SOLUTIONS OF CERTAIN FOURTH-ORDER DELAY DIFFERENTIAL EQUATIONS

By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established result is less restrictive than those reported in the literature.

Extreme Stability of Functional Differential Equations with Finite Delay

This paper obtained some theorems that can ascertain the zero solution of functional differential equations are extremely uniformly stable, extremely asymptotically stable or extremely uniformly asymptotically stable. In the obtained theorems, the derivative of Liapunov function on t along the solutions of functional differential equations is not required to be always negative, especially, it may be even positive.

Stability of a Certain Retard Functional Differential Equation

The authors obtain some sufficient conditions for the stability of zero solutions to some types of the functional equation. (x)(t)+ p(t)-x(t)+q(t)x(t)+f (t, xt)=0 by transformations and the Liapunov's Second method. The obtained conclusions generalize some results of Stability of Equation (x)(t)+p(t)(x)(t)+q(t)x(t)=0 and Jack Hale in his paper of Theory of Functional Differential Equations.

First integrals and stability of second-order differential equations

The stability of second-order differential equations is studied by using their integrals. A system of second-order differential equations can be considered as a mechanical system with holonomic constraints. A conserved quantity of the mechanical system or a part of the system is obtained by using the Noether theory. It is possible that the conserved quantity becomes a Liapunov function and the stability of the system is proved by the Liapunov theorem.

RAZUMIKHIN-TYPE THEOREM FOR NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several different techniques to investigate stability. To show our idea clearly, we examine neutral stochastic delay differential equations with unbounded delay and linear neutral stochastic Volterra unbounded-delay-integro-differential equations.

The H-stability of the Solution for Infinite Delay Functional Differential Equation

H stability is a new and important concept. In this paper,we discuss the equationx(t)=-a(t)x(t)+∫ t -∞ k(t,s-t,x(s)) d sand we gain a new decision theorem. Using this decision theorem,we obtained a very extensive result of the H uniformly asymptotical stability of this equation. That is,eliminating the restriction that a(t) is bounded.

Exponential Stability of Impulsive Neutral Stochastic Functional Differential Equations with Markovian Switching

The aim of this paper is to the discussion of the exponential stability of a class of impulsive neutral stochastic functional differential equations with Markovian switching.Under the influence of impulsive disturbance,the solution for the system is discontinuous.By using the Razumikhin technique and stochastic analysis approaches,as well as combining the idea of mathematical induction and classification discussion,some sufficient conditions for the pth moment exponential stability and almost exponential stability of the systems are obtained.The stability conclusion is full time-delay.The results show that impulse,the point distance of impulse and Markovain switching affect the stability for the system.Finally,two examples are provided to illustrate the effectiveness of the results proposed.

Stability of Second Order Differential Equation Disturbed with Delays

In this paper, stability problems for the second order nonlinear differential equations disturbed with delays are studied. By means of the new stability theorems and Liapunov functional, the authors obtain some results of the zero solution of the equations, some well-known results are extended.

RAZUMIKHIN-TYPE THEOREMS OF NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

The stability of stochastic functional differential equation with Markovian switching was studied by several authors,but there was almost no work on the stability of the neutral stochastic functional differential equations with Markovian switching.The aim of this article is to close this gap.The authors establish Razumikhin-type theorem of the neutral stochastic functional differential equations with Markovian switching,and those without Markovian switching.

ASYMPTOTIC STABILITIES OF STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

Asymptotic characteristic of solution of the stochastic functional differential equation was discussed and sufficient condition was established by multiple Lyapunov functions for locating the limit set of the solution. Moreover, from them many effective criteria on stochastic asymptotic stability, which enable us to construct the Lyapunov functions much more easily in application, were obtained, The results show that the wellknown classical theorem on stochastic asymptotic stability is a special case of our more general results. In the end, application in stochastic Hopfield neural networks is given to verify our results.

A NEW APPROACH TO STABILITY THEORY OF FUNCTIONAL DIFFERENTIAL EQUATIONS

In this work, a new approach to stability theory of functional differential equations is proposed. Instead of putting all components of the state variable x in one Liapunov-Razumikhin function, several functions of partial components of x, which can be much easier constructed. are used so that the conditions ensuring that stability are simpler and less restrictive. Also, an example is given to illustrate the advantages of the obtained results.

Stability of Neutral Stochastic Differential Equations with Multiple Variable Delays

This paper discusses the pth moment stability of neutral stochastic differential equations with multiple variable delays. The equation has a much more general form than the neutral stochastic differential equations with delay. A new kind of φ-function is introduced to address the stability, which is more general than the exponential stability and polynomial stability. Using a specific Lyapunov function, a stability criteria for the neutral stochastic differential equations with multiple variable delays is established, by which it is relatively easy to verify the stability of such equations. Finally, the proposed theories are illustrated by two examples.

THE STABILITY OF A CLASS OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYS

This paper deals with the problem on the stability for zero solution to a class of functional differential equations with infinite delays. We give up the usual confine to the boundedness of the coefficient matrix of the equations and obtain some new results which guarantee the stability and asymptotic stability for zero solution of the equations. The results are of simple forms, easily checked and applicable, and extend the relative results of [1].