非自治时滞微分方程的扰动全局吸引性<sup>*</sup>

非自治时滞微分方程的扰动全局吸引性^*

基金项目: 湖南省自然科学基金;湖南省教委科研基金

Research Institute of Sciences, Changsha Railway University, Changsha 410075, P. R. China

摘要: 考虑具有扰动项的非自治时滞微分方程x>(t)=-a(t)x(t-τ)+F(t,x_t),t≥0(*)其中F:[0,∞)×C[-δ,0]→R且连续,C[-δ,0]表示将[-δ,0]映射到R的所有连续函数集合.F(t,0)≡0,a(t)∈C((0,∞),(0,∞)),τ≥0.通常文献对a(t)不依赖于t即a(t)为自治情形,研究了方程(*)零解的局部或全局渐近性质[1～5,7].本文对a(t)为非自治即依赖于t之情形,获得了方程(*)零解全局吸引的充分条件,所得结论在某种意义上说是不可改进的.本文改进和推广了已有文献的相应结果,同时本文采用的方法可应用到非自治非线性扰动方程.
- 扰动全局吸引性时滞微分方程渐近稳定性基本解常数变易公式
Abstract: Consider the perturbed nonautonomous linear delay differential equation x>(t)=-a(t)x(t-τ)+F(t,x_t),t≥0(*)Wherex_t(s)=x(t+s) for-δ≤s≤0.Suppose thata(t)∈C((0,∞),(0,∞)),τ≥0,F:[0,∞)×C[-δ,0]→Ris a continous functions and F(t,0)≡0.HereC[-δ,0]is the space of continuous functions Φ[-δ,0]→Rwith||Φ||<Hfor the norm where|·|is any norm in R and 0<H≤+∞.
- perturbational global attractivity

[1]	T.Yoneyama and J.Segie,On the stability region of scalar of delay-differential equations,J.Math.Anal.Appl.,134,3(1988),408-425.
[2]	J.Hale,Theory of Functional Differential Equations,Applied Mathematical Sciences 3,Springer-Verlag,New York(1977).
[3]	R.Bellman and K.L.Cooke,Differential-Difference Equations,Academic Press,New York(1963).
[4]	N.D.Hayes,Roots of the transcendental equation-associated with a certain difference-differential equation.J.London Math.Soc.,25,2(1950),226-232.
[5]	V.B.Kolmovskii and V.R.Nosov,Stabilily of Functional Differential Equations,Academic Press,London,New York(1986).
[6]	T.A.Burton,A.Casal and A.Somolions,Upper and lower bounds for Liapunov functionals,Funkeal.Ekvac.,32,1(1989),23-55.
[7]	I.Gyori,Global attractivity in a perturbed linear delay differential equation,Applicable Analysis,34,2(1989),167-181.
[8]	T.Hara,T.Yonerama and R.Miyazaki,Some relines of Razunlikhins method and their applications,Funkeialaj Ekvacioj,35,2(1992),279-305.
[9]	K.Kobayashi,Stability theorems for functional differential equations,Nonlinear Analysis TMA,20,10(1993),1183-1192.
[10]	刘开宇,非自治线性时滞微分方程的扰动全局渐近稳定性及一致稳定性,湖南大学硕士学位论文(1996).
[11]	V.B.Kolmanovskii,L.Torell and R.Verniglio,Stability of some test equations with delay,SLAM J.Math.Anal.,25,3(1994),948-961.