几类时滞微分方程解稳定性和有界性的研究

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1、摘要本文研究了两类三阶和一类四阶时滞微分方程零解的渐近稳定性和所有解的有界性，通过Lyapunov第二方法，得到了使它们的零解渐近稳定和所有解有界的充分性准则.主要内容如下：1.介绍了时滞微分方程稳定性研究的背景、意义以及三阶和四阶时滞微分方程的研究现状.2.介绍了时滞微分方程的基本概念、基本理论和Lyapunov泛函方法.3.研究了两类三阶时滞微分方程零解的渐近稳定性和所有解的有界性.(1)+f2(x{t),x(t-r(t)),x'(,),x'(t-r(t)),x"(/),x"{t-r(/)))(x"⑴)2+^i(x'(0)+(,(卜K,)))=^(x(/),x(/-r(?)),x'(/

2、),x'(?-/-(/)),x"(/),x"(/-；?(?)))它将[1]中研究的二阶时滞微分方程推广成三阶时滞微分方程.^"(0+&(x(0'x'(^(0'x'W)x'(0(2)+/(4卜r(0)^'{t-r(0))+h{x(t-r(/)))^p{t,x{t),x'{t),x{t-r(/)),x'{t-r(,)),x"⑴）它是在[3]中研究的方程的基础上加了一项&(x⑷,x'⑷?).4.研究了一类四阶时滞微分方程零解的渐近稳定性.x4(f)+

3、“x(,-关键词渐近稳定，有界性，Lyapunov泛函，时滞微分方程IABSTRACTThispaperstudiestheasymptoticstabilityofthezerosolutionandtheboundednessofallsolutionsofthetwokindsofthird-orderandaclassoffourth-orderdelaydifferentialequations.BythesecondmethodofLyapunov,somesufficientconditionsfortheasymptoticstabilityofthezerosolution

4、andtheboundednessofallsolutionsofthemareestablished.Themaincontentisasfollows：1.Thebackgroundandsignificanceofthestabilityofdelaydifferentialequationsandtheresearchstatusofthird-orderandfourth-orderdelaydifferentialequationsareintroduced.2.Thebasicconcepts,thebasictheoriesandthemethodofLyapunovfu

5、nctionalofdelaydifferentialequationsareintroduced.3.Theasymptoticstabilityofthezerosolutionandtheboundednessofallsolutionsoftwokindsofthird-orderdelaydifferentialequationsarestudied.x"(/)+./；[x{t),x{t-r(t)),x{t),x'[t-r{t)),x"(t),x"{t-r{t))^x"(t)(1)+/2(外)，x(,-/?(/))，(,)，x'(,-/?(/))，x"(/),?('_r(0))

6、(?(0)2+沿(冲))+g2?卜小))）=p[x{t),x{t-r(t)),x'{t),x'{t-r{t)),x"(t),x"[t-r{t)fjThisthird-orderdelaydifferentialequationisextendedbythesecond-orderdelaydifferentialequationstudiedin[1].(2)+f(x{t-r(,))，x'(/-r(/)))+/7(xG-r(,)))Onthebasisoftheequationstudiedin[3],thisequationisstudiedwhichaddsanitemasg2(x(,

7、)’x'(/))x'(/).4.Theasymptoticstabilityofthezerosolutionofacertainfourth-orderdelaydifferentialequationisstudied.xA(t)+(p{x(t)yx(t)+h{x(t))x(t)+￠[x(t-r),x(t-r))+f[x(t-r))=0Onthebasisoftheequationstudiedin[4