On the instability of differential systems with varying delay

The unstable properties of the null solution of the nonautonomous delay system x′(t)=A(t)x(t)+B(t)x(t−r₁(t))+f(t,x(t),x(t−r₂(t))) are examined; the nonconstant delays r₁, r₂ are assumed to be continuous bounded functions. The case A=constant is reviewed, where a theorem, recalling the Perron instability theorem for ordinary differential equations, is obtained.     

On Infinite Systems of Linear Autonomous and Nonautonomous Stochastic Equations

The solvability of autonomous and nonautonomous stochastic linear differential equations in is studied. The existence of strong continuous (L^p-continuous) solutions of autonomous linear stochastic differential equations in with continuous (L^p-continuous) right-hand sides is proved. Uniqueness conditions are obtained. We give examples showing that both deterministic and stochastic linear nonautonomous differential equations with the same operator in may fail to have a solution. We also establish existence and uniqueness conditions for nonautonomous equations.     

具无限时滞的非自治捕食者-食饵系统的持久性

本文采用新的方法研究了具无限时滞的非自治、非卷积的捕食者－食饵系统的持久性问题，得到了新的、有趣的结果。本文的结果包含已知的结果作为特例．     

Invariant Sets in Banach Lattices and Linear Differential Equations and Delay

In this paper we characterize the invariance of some sets with respect to a C ₀-semigroup in Banach lattice. An application to a linear differential equation with delay is given.     

Stabilization of Linear Nonautonomous Systems with Norm-Bounded Controls

In this paper, we study the stabilization problem for a class of linear nonautonomous systems with norm-bounded controls. Using the Lyapunov function technique, we establish simple verifiable stabilizability conditions without solving any Riccati differential equation. Numerical examples are given to illustrate the results.Communicated by F. E. UdwadiaThis work was supported by the National Basic Program in Natural Sciences, Vietnam and by a Thailand Research Fund Grant. The authors thank the anonymous referees for valuable comments and remarks which have improved the paper.     

一类具有阶段结构和时滞的非自治捕食系统

本文研究一类具有阶段结构和时滞的非自治捕食系统解的渐近性质 ,在适当条件下 ,得到系统是持续生存的且全局渐近稳定的     

三阶非自治微分方程的不稳定性定理

本文运用构造李雅普诺夫函数的方法，给出了一类三阶非自治微分方程零解不稳定的充分条件．     

具有可变时滞的高阶非线性非自治差分方程的振动性

研究具有可变时滞的高阶非自治中立型差分方程Δm(xn-cnxn-k)+h(n,xn-ln)=0,n=0,1,2,…的振动性.利用方程的最终正解和最终负解导致序列{Δi(xn-cnxn-k)}(i=0,1,2,…,m-1)的最终单调和保号性,得到了这类差分方程振动的一系列充分条件.     

非自治时滞微分方程的正周期解

By utilizing a fixed point theorem on cone,some new results on the existence of positive periodic solutions for nonautonomous differential equations with delay are derived.     

一阶中立型线性微分系统解的振动性

本文给出一类一阶中立型线性微分系统解振动的充分条件 ,改进和推广了文 [1 ]的工作 .     

Asymptotic Behavior of Solutions to a Class of Systems of Delay Differential Equations

The authors investigate the asymptotic behavior of solutions to a class of systems of delay differential equations. It is shown that every bounded solution of such a class of systems tends to a constant vector as t→∞. Our results improve and extend some corresponding ones already known.     

基于一类时滞不等式的时滞微分方程的渐近稳定性

利用Lyapunov的方法讨论了时滞微分方程x.(t)=f(t,x(t),x(t-τ(t)))的全局指数渐近稳定性和全局渐近稳定性.     

Oscillation Theorem of Systems of Quasilinear Impulsive Delay Hyperbolic Equations

In this paper, oscillatory properties for solutions of the systems of certain quasilinear impulsive delay hyperbolic equations with nonlinear diffusion coefficient are investigated. A sufficient criterion for oscillations of such systems is obtained.     

On the Contractivity and Asymptotic Stability of Systems of Delay Differential Equations of Neutral Type

In this paper we investigate both the contractivity and the asymptotic stability of the solutions of linear systems of delay differential equations of neutral type (NDDEs) of the form y(t) = Ly(t) + M(t)y(t – (t)) + N(t)y(t – (t)). Asymptotic stability properties of numerical methods applied to NDDEs have been recently studied by numerous authors. In particular, most of the obtained results refer to the constant coefficient version of the previous system and are based on algebraic analysis of the associated characteristic polynomials. In this work, instead, we play on the contractivity properties of the solutions and determine sufficient conditions for the asymptotic stability of the zero solution by considering a suitable reformulation of the given system. Furthermore, a class of numerical methods preserving the above-mentioned stability properties is also presented.     

拟线性脉冲时滞双曲型方程组的振动性定理

In this paper, oscillatory properties for solutions of the systems of certain quasilinear impulsive delay hyperbolic equations with nonlinear diffusion coefficient are investigated. A sufficient criterion for oscillations of such systems is obtained.     

偶数阶时滞偏微分方程组解的振动性定理

本文获得了一类偶数阶时滞偏泛函微分方程组解振动的若干充分条件     

一类脉冲中立型时滞抛物方程组的振动性

研究一类脉冲中立型时滞抛物方程组解的振动性,得到了该类方程组在两类不同边界条件下所有解振动的若干充分条件.     

On First Range Times of Linear Diffusions

In this paper we consider first range times (with randomised range level) of a linear diffusion on R. Inspired by the observation that the exponentially randomised range time has the same law as a similarly randomised first exit time from an interval, we study a large family of non-negative 2-dimensional random variables (X,X′) with this property. The defining feature of the family is F^c(x,y)=F^c(x+y,0), ∀ x ≥ 0, y ≥ 0, where F^c(x,y):=P (X > x, X′ > y) We also explain the Markovian structure of the Brownian local time process when stopped at an exponentially randomised first range time. It is seen that squared Bessel processes with drift are serving hereby as a Markovian element.     

脉冲时滞抛物型偏微分方程组解的振动性

研究一类脉冲时滞抛物型偏微分方程组解的振动性,利用一阶脉冲时滞微分不等式获得了该类方程组在两类不同边值条件下所有解振动的若干充分条件.所得结果充分反映了脉冲和时滞在振动中的影响作用.     

不确定中立型线性随机时滞系统的鲁棒稳定性

建立了中立型随机微分时滞方程的LaSalle不变原理,然后应用LaSalle不变原理讨论了不确定中立型随机时滞系统的随机渐近稳定和几乎必然指数稳定的代数判据, 同时给出示例说明结果的有效性．