A DELAY DIFFERENTIAL INEQUALITY

In this paper, we obtain a slight generalization of the celebrated Halanay delay differential inequality[1]. Using this inequality, Liapunov function and vector Liapunov function, we show that our result can be applied to the stability behavior of solutions of some delay differential systems. And also, we attempt to relax some restrictions of dv/dt, which is necessary under Liapunov function's method on the delay differential systems.     

The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag

This paper is concerned with the exponential stability of singularly perturbed delay differential equations with a bounded (state-independent) lag. A generalized Halanay inequality is derived in Section 2, and in Section 3 a sufficient condition will be provided to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is exponentially stable uniformly for sufficiently small ε>0. This type of exponential asymptotic stability can obviously be applied to general delay differential equations with a bounded lag.     

Stability of one-leg θ-methods for nonlinear neutral differential equations with proportional delay

The stability properties of one-leg θ-methods for nonlinear neutral differential equations with proportional delay is investigated. In recent years, the stability of one-leg θ-methods for this class of equations on a quasi-geometric mesh is investigated. Instead, in the present paper, the focus is on stability of one-leg θ-methods for the neutral differential equations with constant delay obtained by applying the approach of transformation to the proportional delay equations. Some sufficient conditions for global stability and asymptotic stability are established. Two numerical examples are also included.     

一类三阶中立型时滞微分方程的渐近稳定性

讨论了一类三阶中立型时滞微分方程的零解的渐近稳定性,借助于构造函数、推广的Halanay一维时滞微分不等式及泰塔格利亚公式,得到了判定其零解是渐近稳定的且与时滞无关的一个充分条件.     

变时滞的退化中立型微分系统的稳定性

研究了具有变时滞的退化中立型微分系统的稳定性．利用退化时滞微分系统的变易公式和Gronwall-Bellman积分不等式给出了该系统的指数估计以及稳定和指数渐近稳定的充分条件．     

Exponential stability of impulsive stochastic functional differential equations

In this paper, we investigate the pth moment and almost sure exponential stability of impulsive stochastic functional differential equations with finite delay by using Lyapunov method. Several stability theorems of impulsive stochastic functional differential equations with finite delay are derived. These new results are employed to impulsive stochastic equations with bounded time-varying delays and stochastically perturbed equations. Meanwhile, an example and simulations are given to show that impulses play an important role in pth moment and almost sure exponential stability of stochastic functional differential equations with finite delay.     

A new criterion to global exponential stability for impulsive neural networks with continuously distributed delays

This paper considers the global exponential stability and exponential convergence rate of impulsive neural networks with continuously distributed delays in which the state variables on the impulses are related to the unbounded distributed delays. By establishing a new impulsive delay differential inequality, a new criterion concerning global exponential stability for these networks is derived, and the estimated exponential convergence rate is also obtained. The result extends and improves on earlier publications. In addition, two numerical examples are given to illustrate the applicability of the result. Copyright © 2010 John Wiley & Sons, Ltd.     

Stochastic functional differential equations with infinite delay

The stability and boundedness of the solution for stochastic functional differential equation with finite delay have been studied by several authors, but there is almost no work on the stability of the solutions for stochastic functional differential equations with infinite delay. The main aim of this paper is to close this gap. We establish criteria of pth moment ψγ(t)-bounded for neutral stochastic functional differential equations with infinite delay and exponentially stable criteria for stochastic functional differential equations with infinite delay, and we also illustrate the result with an example.     

变时滞的退化滞后型微分系统的稳定性

主要研究了具有变时滞的退化时滞微分系统的稳定性.利用退化时滞微分系统的变易公式和Gronwall-Bellman积分不等式给出了该系统的指数估计以及稳定和指数渐近稳定的充分条件.     

Impulsive delay differential inequality and stability of neural networks

In this article, a generalized model of neural networks involving time-varying delays and impulses is considered. By establishing the delay differential inequality with impulsive initial conditions and using the properties of M-cone and eigenspace of the spectral radius of nonnegative matrices, some new sufficient conditions for global exponential stability of impulsive delay model are obtained. The results extend and improve the earlier publications. An example is given to illustrate the theory.     

Existence and exponential stability of periodic solution for impulsive delay differential equations and applications

In this paper, we consider a class of nonlinear impulsive delay differential equations. By establishing an exponential estimate for delay differential inequality with impulsive initial condition and employing Banach fixed point theorem, we obtain several sufficient conditions ensuring the existence, uniqueness and global exponential stability of a periodic solution for nonlinear impulsive delay differential equations. Furthermore, the criteria are applied to analyze dynamical behavior of impulsive delay Hopfield neural networks and the results show different behavior of impulsive system originating from one continuous system.     

Dissipativity of delay functional differential equations with bounded lag

Delay functional differential equations are essentially different from ordinary differential equations because their phase space is infinite dimensional. We first establish a sufficient condition for delay functional differential equations with bounded lag to be dissipative. Then we construct a one-leg θ-method to solve such dissipative equations and prove that it is dissipative if θ=1. One numerical example is given to confirm our theoretical result.     

STABILITY ANALYSIS OF FUZZY DIFFERENTIAL EQUATIONS WITH DELAY

In this paper,the exponential stability of fuzzy differential equations with delay is investigated.By employing the formula for the variation of parameters, inequality technique and the norm and measure of matrix,an algebraic criterion for the exponential stability is obtained.     

On the Partial Equiasymptotic Stability in Functional Differential Equations

A system of functional differential equations with delay dz/dt = Z(t, z_t), where Z is the vector-valued functional is considered. It is supposed that this system has a zero solution z = 0. Definitions of its partial stability, partial asymptotical stability, and partial equiasymptotical stability are given. Theorems on the partial equiasymptotical stability are formulated and proved.     

一类无穷时滞微分系统的周期解和全局渐近稳定性

利用重合度理论中的延拓定理和微分不等式讨论一类无穷时滞微分系统的周期解的存在性和全局渐近稳定性,获得了简便的判别条件.     

Stability analysis of impulsive fractional differential systems with delay

In this paper, a class of impulsive fractional differential systems with finite delay is considered. Some sufficient conditions for the finite-time stability of above systems are obtained by using generalized Bellman–Gronwall’s inequality, which extend some known results.     

Multidimensional BSDEs with weak monotonicity and general growth generators

This paper aims at solving a multidimensional backward stochastic differential equation （BSDE） whose generator g satisfies a weak monotonicity condition and a general growth condition in y. We first establish an existence and uniqueness result of solutions for this kind of BSDEs by using systematically the technique of the priori estimation, the convolution approach, the iteration, the truncation and the Bihari inequality. Then, we overview some assumptions related closely to the monotonieity condition in the literature and compare them in an effective way, which yields that our existence and uniqueness result really and truly unifies the Mao condition in y and the monotonieity condition with the general growth condition in y, and it generalizes some known results. Finally, we prove a stability theorem and a comparison theorem for this kind of BSDEs, which also improves some known results.     

STABILITY OF TIME VARYING SINGULAR DIFFERENTIAL SYSTEMS WITH DELAY

In this paper, stability of time varying singular differential systems with delay is considered. Based on variation formula and Gronwall-Bellman integral inequality, we obtain the exponential estimation of the solution and the sufficient conditions under which the considered system is stable and exponentially asymptotically stable. These results will be very useful to further research on Roust stability and control design of uncertain singular control systems with delay.     

滞后型分段连续随机微分方程的稳定性

本文研究了滞后型分段连续随机微分方程的解析稳定性和数值稳定性问题.首先,利用伊藤公式等方法获得了解析解均方稳定的条件,其次,对于包括均方稳定和T-稳定在内的Euler-Maruyama方法的数值稳定性问题,运用不等式技术和随机分析方法获得了一些新的结果,证明了在一定条件下,Euler-Maruyama方法既是均方稳定又是T-稳定的,推广了随机延迟微分方程的数值稳定性结论.