一类线性多步法关于变延迟非线性中立型微分方程的渐近稳定性

1引言中立型微分方程广泛出现于生物学、物理学及工程技术等诸多领域.数值求解中立型微分方程时,数值方法的稳定性研究具有无容置疑的重要性,其中渐近稳定性的研究是其重要组成部分.对于线性中立型延迟微分方程,渐近稳定性研究已有许多重要结果,如文献[1,2,3,4,5,6]等.对于非线性中立型变延迟微分方程,数值方法的稳定性研究近几年才有进展.2000年,Bellen等在文献[7]中讨论了Runge-Kutta法求解一类特殊的中立型延迟微分     

ASYMPTOTIC STABILITY OF SINGULAR NONLINEAR DIFFERENTIAL SYSTEMS WITH UNBOUNDED DELAYS

In this paper,the asymptotic stability of singular nonlinear differential systems with unbounded delays is considered.The stability criteria are derived based on a kind of Lyapunov-functional and some technique of matrix inequalities.The criteria are described as matrix equation and matrix inequalities,which are computationally flexible and efficient.Two examples are given to illustrate the results.     

非线性中立型延迟微分方程单支方法的数值稳定性

本文研究Rα,β类非线性中立型延迟微分方程单支方法的数值稳定性,结果表明:A-稳定的单支方法是数值稳定的,强A-稳定的单支方法是渐近稳定的．最后的数值试验验证了所获理论结果的正确性．     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF NONLINEAR NEUTRAL DIFFERENTIAL EQUATION

The boundedness of the every solution and the asymptotic behavior of all solutions of the nonlinear neutral delay differential equation [x(t) - P(t)x(t - t)]' Q1 (t)f(x{t-σ1))-Q2(t)f(x(t -σ2))=0,t≥t0 are investigated, whereτ,σ1,σ2∈(0,∞), P∈C([t0,∞),R), and Q1,Q2∈C([t0,∞),R), f∈C(R,R). The sufficient conditions obtained improve the existing results in the literatures.     

ASYMPTOTIC STABILITY OF A SINGULAR SYSTEM WITH DISTRIBUTED DELAYS

Based on the stability theory of functional differential equations, this paper studies the asymptotic stability of a singular system with distributed delays by constructing suitable Lyapunov functionals and applying the linear matrix inequalities. A numerical example is given to show the effectiveness of the main results.     

广义中立型系统的渐近稳定性及数值分析

１．引 言 考察如下广义中立型系统：其中,Ｌ,Ｍ,Ｎ ∈ Ｃｄ×ｄ为已知矩阵,　　　为已知向量值函数,　　　　　　　　　　当ｔ＞０时为未知函数,　　　　　　　　　　　　　　　　　　　　　　　　　为常数延时量． 对于　　　　　　　　　　　　　　　　　,１９６７年,Ｂｒａｙｔｏｎ［１］基于Ｌ,Ｍ,Ｎ为实对称矩阵,以及Ｉ± Ｎ和－Ｌ± Ｍ为正定矩阵时,讨论了（１）渐近稳定的充分条件;１９８４年,Ｊａｃｋｉｅｗｉｃｚ［２］基于 Ｌ,Ｍ,Ｎ为复系数时,研究了理论解的渐近稳定性及单步方法的数值稳定性;１９８８年,Ｂ…     

ASYMPTOTIC BEHAVIOR FOR NONOSCILLATORY SOLUTIONS OF REACTION-DIFFUSION DIFFERENTIAL EQUATIONS WITH SEVERAL DELAYS IN THE NEUTRAL TERM

In this paper, we first consider differential equations with several delays inthe neutral term of the formstudy various cases of coefficients in the neutral term and obtain the asymptoticbehavior for nonoscillatory solution of (1~*) under some hypotheses. Moreover,we consider reaction-diffusion differential equations with several delays in theneutral term of the formfor (t, x)∈ R~+ ×Ω. We also study various cases of coefficients in the neutralterm and obtain the asymptotic behavior for nonoscillatory solution of (2~*)under some hypotheses.     

STABILITY OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In the phase space (Ch, |·|h), by using the Liapunov functional approach, sufficient and necessary criteria for the uniform stability and uniformly asymptotic stability of solutions to neutral-functional differential equations with infinite delay are established. We also prove that the uniformly asymptotic stability of the solutions implies the existence of the bounded ones.     

EXPONENTIALLY ASYMPTOTIC STABILITY OF NONLINEAR NEUTRAL DIFFERENTIAL DIFFERENCE SYSTEM

11尬roductlonIn recent ye肘s,there has been extensive literatures(see for lustmce 11－7I)about the oscillations ofneutral dl地rentlal dl征rence equations such asMt)一c川叫一r)]’＋＞Pi川x(t—a;)＝0,(川江1and卜(t)一 ex(t一 r刀’＋ f(t;x(t);x(t一 o－》二 0．(B)But there are few papers about the exponentlal stability for neutral dl洗rentialdl地rence euuatlons because of its difficulty (sc [8－9])．In [9],Li made asurvey on the asymptotlc stability and exponentlally asymptotlc stability inC(‘)I…     

LINEARIZED OSCILLATIONS FOR ODD-ORDER NEUTRAL DELAY DIFFERENTIAL EQUATIONS WITH ASYMPTOTIC PERIODICCOEFFICIENTS

1IntroductionSince1987,thelinearizedoscillationtheoryofnonlineardelaydifferentialequations(DDEs)hasabsorbedmanymathematicians(see[1,2]),itisparalleltotheso-calledlinearizedstabilitytheory,andhasbecomeamainstudyofoscillationtheoryofDDEs.ByusingthenecessaryandsufficientconditionsfortheoscillationoflinearDDEswithconstantcoefficients,theliteraturesconcentratedtoprovethatcertainnonlinearDDEhasthesameoscillatorycharacterasanassociatedlinearizedDDEwithconstantcoefficients.However)whenthelineari…     

ASYMPTOTIC STABILITY FOR A CLASS OF NONAUTONOMOUS NEUTRAL DIFFERENTIAL EQUATIONS

ＡＳＹＭＰＴＯＴＩＣＳＴＡＢＩＬＩＴＹＦＯＲＡＣＬＡＳＳＯＦＮＯＮＡＵＴＯＮＯＭＯＵＳＮＥＵＴＲＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ＊＊ＹＵＪＩＡＮＳＨＥ＊ＭａｎｕｓｃｒｉｐｔｒｅｃｅｉｖｅｄＪｕｌｙ４，１９９５．ＲｅｖｉｓｅｄＭａｒｃｈ２...     

含多个函数时滞的随机延迟微分方程的矩稳定性

本文考虑具有多个函数时滞的中立型随机延迟微分方程p阶矩稳定性.运用Razumikhin方法,建立了一此新的矩稳定性判别法,并以线性方程为例解释了所得判别法的应用.     

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

1 IntroductionConsider the following functional differential equationX'(t) = A(t)x(t) [' C(t,s)x(s)ds, (1)icwhere x E m; A(t) = (ail(t))... is a n x n function matrix, which continuesin [0, co); C(t,s) = (qj(t,8))... is a n x n function matrix, which conti-nues when 0 5 s 5 t < co, and L oo IIC(u,t)lldu continues in [0, co).The problem on the stability for the zero solution of (1) has been studied bymany papers. But in the known results, the boundedness of j: IIC(t, s)lldsor L " IIC…     

具有扰动的非自治中立型泛函微分方程的渐近稳定性

本文讨论了具有扰动的非自治中立型泛函微分方程的3/2-稳定性,得到了零解一致稳定和渐近稳定的充分条件.     

基于一类积分不等式上的中立型时滞微分方程的稳定性

利用一类积分不等式以及参数变易法,Jensen不等式,给出了更为一般的中立型时滞微分系统的Lyapunov稳定性的判别准则,推广和改进了一些文献中有关中立型微分方程的Lyapunov稳定性的结论.     

STABILITY AND ALMOST PERIODIC SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE

ＳＴＡＢＩＬＩＴＹＡＮＤＡＬＭＯＳＴＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＳＯＦＦＵＮＣＴＩＯＮＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＯＦＮＥＵＴＲＡＬＴＹＰＥＹｕａｎＲｏｎｇ（袁荣）（ＢｅｉｊｉｎｇＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，北京师范大学，...     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF NEUTRAL DIFFERENTIAL EQUATIONS WITH POSITIVE AND NEGATIVE COEFFICIENTS

1 IlltroductionConsider the neutral de1ay differential equations with positive and negativecoefficients of the fOrm[x(t) Ac(t)x(t -- a)]' p(t)x(t -- T) -- Q(t)x(t -- 6) = 0, t 2 to, (1)where A = {--1, 1}, a > 0, T? b 2 0, c,p, Q E C([to, oo), R ), and assume thatthere exists a constant A > 0 such thatQ(t b -- T) 5 Ap*(t), p*(t) = p(t) -- Q(t 6 -- T), for t 2 max{to, to T -- b}.The study of asymptotic behavior of so1ution of (1) has had some work, seefOr example [l-9]. However,…     

ASYMPTOTIC BEHAVIOR OFNONOSCILLATORY SOLUTIONS OF SECOND ORDER NEUTRAL NONLINEAR DIFFERENCE EQUATIONS

1IntroductionConsiderthesecondorderneutralnonlineardifferenceequationwherenEN(no)={no,no 1,'.}(noisanonnegativeinteger),AE{--l,1},AistheforworddifferenceoperatordefinedbyAx.=xu 1--xu.Inrecentyearssthequalitativepropertiesofthesolutionsofsecondorderneutraldifferenceequationswereconsideredinanumberofpapers[1--10].Allthepapersmentioned,however,alwaysassumethatqn50foralllargen.Inthispaperouraimistostudytheasymptoticbehaviorofnonoscillatorysolutionsof(l)whenqn20foralllargen.Byasolutionof(1)weme…     

OSCILLATION OF A NONLINEAR DELAY DIFFERENTIAL EQUATION WITH NONLINEAR NEUTRAL TERM

This paper is concerned with the following nonlinear delay differential equation with nonlinear neutral term[h(t)(y(t) +φ(t,y(τ(t)))']' + q(t)f(y(g(t))) = 0. An oscillation criterion for the above equation is obtained.