Uniform stability theorems for delay differential equations with impulses

In the paper, we obtain sufficient conditions for the uniform stability of the zero solution of the delay differential equation with impulses

     

Asymptotic behavior and stability of second order neutral delay differential equations

We study the asymptotic behavior of a class of second order neutral delay differential equations by both a spectral projection method and an ordinary differential equation method approach. We discuss the relation of these two methods and illustrate some features using examples. Furthermore, a fixed point method is introduced as a third approach to study the asymptotic behavior. We conclude the paper with an application to a mechanical model of turning processes.     

Linear stability of general linear methods for systems of neutral delay differential equations

This paper is concerned with the numerical solution of delay differential equations (DDEs). We focus on the stability of general linear methods for systems of neutral DDEs with multiple delays. A type of interpolation procedure is considered for general linear methods. Linear stability properties of general linear methods with this interpolation procedure are investigated. Many extant results are unified.     

The asymptotic stability of multistep multiderivative methods for systems of delay differential equations

IntroductionFor many years, many papers investigated the linear stabilit}' of delay differential equation(DDE) solvers and a significant number of important results have already been found for bothRunge-Kutta methods and linear multistep methods (see, for example, [l--8]). In this paper,we firstly consider stability of numerical methods with derivative for DDEs. It is shown thatA-stability of multistep multiderivative methods for ordinary differential equations (ODEs) isequit,alent to p-s…     

Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations

There are few results on the numerical stability of nonlinear neutral stochastic delay differential equations (NSDDEs). The aim of this paper is to establish some new results on the numerical stability for nonlinear NSDDEs. It is proved that the semi-implicit Euler method is mean-square stable under suitable condition. The theoretical result is also confirmed by a numerical experiment.     

Oscillations and asymptotic behavior of first-order neutral delay differential equations

Consider the neutral delay differential equation [display math001] In this paper we are concerned with the asymptotic behavior and the oscillatory nature of solutions of Eq. (1).     

Uniform asymptotic stability of impulsive discrete systems with time delay

This paper studies impulsive discrete systems with time delay. Some novel criteria on uniform asymptotic stability are established by using the method of Lyapunov functions and the Razumikhin-type technique. Examples are presented to illustrate the criteria.     

Fixed points and stability of neutral stochastic delay differential equations

In this paper we consider a linear scalar neutral stochastic differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically mean square stable by means of fixed point theory. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved. Some well-known results are improved and generalized.     

A stability property of A-stable collocation-based Runge-Kutta methods for neutral delay differential equations

We consider a linear homogeneous system of neutral delay differential equations with a constant delay whose zero solution is asymptotically stable independent of the value of the delay, and discuss the stability of collocation-based Runge-Kutta methods for the system. We show that anA-stable method preserves the asymptotic stability of the analytical solutions of the system whenever a constant step-size of a special form is used.     

Exponential stability of singularly perturbed impulsive delay differential equations

In this paper, the exponential stability of singularly perturbed impulsive delay differential equations (SPIDDEs) is concerned. We first establish a delay differential inequality, which is useful to deal with the stability of SPIDDEs, and then by the obtained inequality, a sufficient condition is provided to ensure that any solution of SPIDDEs is exponentially stable for sufficiently small ε>0. A numerical example and the simulation result show the effectiveness of our theoretical result.     

A global stability criterion in nonautonomous delay differential equations

Consider the following nonautonomous nonlinear delay differential equation:

     

Nonlinear stability of general linear methods for neutral delay differential equations

This paper is concerned with the numerical solution of neutral delay differential equations (NDDEs). We focus on the stability of general linear methods with piecewise linear interpolation. The new concepts of GS(p)$GS\left(p\right)$-stability, GAS(p)$GAS\left(p\right)$-stability and weak GAS(p)$GAS\left(p\right)$-stability are introduced. These stability properties for (k,p,0)$(k,p,0)$-algebraically stable general linear methods (GLMs) are further investigated. Some extant results are unified.     

On stability in terms of two measures for impulsive systems of functional differential equations

This paper considers the stability problem for a class of impulsive systems of functional differential equations. By using Lyapunov functions and the Razumikhin technique, several stability criteria in terms of two measures are established. Some examples are worked out to illustrate the theorems.     

扰动的中立型时滞差分方程的一致渐近稳定性

本文研究扰动的一维中立型时滞差分方程△(x(n)-cx(n-k))=f(n,xn)+g(n,xn),n∈Z+的稳定性问题.证明了在某些条件下,无扰动方程△(x(n)-cx(n-k))=f(n,xn),n∈Z+零解的一致渐近稳定性蕴涵着上述扰动方程零解的一致渐近稳定性.本文的结果推广并改进了已有的结果.