非自治时滞微分方程的扰动全局吸引性*

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.

On the Perturbational Global Attractivity of Nonautomous Delay Differential Equations

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≤+∞.