非自治时滞微分方程的扰动全局吸引性  

On the Perturbational Global Attractivity of Nonautomous Delay Differential Equations

作　　者：罗交晚[1] 刘再明[1] 

机构地区：[1]长沙铁道学院科研所

出　　处：《应用数学和力学》1998年第12期1107-1111,共5页Applied Mathematics and Mechanics

基　　金：湖南省自然科学基金;湖南省教委科研基金

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，ｘｔ），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·Consider the perturbed nonautonomous linear delay differential equation (t)=-a(t)x(t-τ)+F(t,x t), t≥0(*) Where x t(s)=x(t+s) for -δ≤s≤0. Suppose that a(t)∈C([0,∞),(0,∞)),τ≥0,F:[0,∞)×C→R is a continous functions and F(t,0)≡0. Here C is the space of con￣tinuous functions Φ:[-δ,0]→R with ‖Φ‖<H for the norm ‖Φ‖= sup -δ≤s≤0|Φ(s)|, where |·| is any norm in R and 0<H≤+∞. Most of the known papers have been concerned with the local or global asymptotic behavior of the zero solution of Eq.(*) when a(t) is independent of t i.e., a(t) is autonomous. The aim in this paper is to derive the sufficient conditions for the global attractivity of the zero solution of of Eq.(*) When a(t) is nonautomous. Our results, which extend and improve the known results, are even “sharp”. At the same time, the method used in this paper can be applicable to the perturbed nonlinear equation.

关 键 词：扰动全局吸引性 时滞微分方程 渐近稳定性 

分 类 号：O175.13[理学—数学]