摘要
考虑非线性系统x′=A(t)x+f(t,x)有界解的存在性,其中线性系统x′=A(t)x满足指数型二分性.在f(t,x)关于x不满足Lipschitz条件的情况下,应用Leray-Shauder不动点定理和Arzela-Ascoli定理给出一个有界连续解存在的充分条件.即若f(t,x)∶R×Rn→Rn连续;存在常数m>0及R+=[0,∞)上的连续递增函数g(t)满足limt→∞(g(t))/t=0,使得|f(t,x)|≤m+g(|x|),(t,x)∈R×Rn,则该系统x′=A(t)x+f(t,x)存在有界连续解.
This paper is concerned with the problem of the existence of a bounded solution for a non-linear system x＇=A（t）x＋f（t,z）, where linear system x ＇ = A （t） x satisfies exponential dichotomy. When Lipschitz condition on x does not meet for f（ t ,x）, the following sufficient condition on the existence of bounded continuous solution of the system is obtained on Learay-Shauder fixed point theorem and Arzela-Ascoli theorem ： if（ 1 ）f（ t, x） ：R×R^n→R^n is continuous ; （ 2 ） there exists a constant m〉 0 and a continuous increasing function g（t） which satisfies lim（t→∞）t/t（t）=0 for t∈R^＋=[0,∞] such that │f（f,z）1≤m＋g（│x│）,↓（t,x）∈R×R^n,then x＇=A（t）x＋f（t,x） has a bounded continuous solution.
出处
《闽江学院学报》
2008年第5期9-12,共4页
Journal of Minjiang University
关键词
微分方程
指数型二分性
有界解
存在性
differential equation
exponential dichotomy
bounded solution
existence
