带反射变量微分方程的极限周期解

Limit-Periodic Solutions of the Differential Equations with Reflection of the Arguments

本文利用给定的周期序列,定义了极限周期函数的一个子集,并讨论了该子集的一些性质。然后,借助Banach压缩映像原理证明了带反射变量的一阶微分方程x′(t)+ax(t)+bx(-t)=F(t,x(t),x(-t)),b≠0,t∈R的极限周期解的存在性及唯一性,其中函数F关于t是一致极限周期的,且F关于后两个变量满足Lipschitz条件。

In this article,a subclass of limit periodic functions was defined by using a given periodic sequence.Some properties of this class were discussed.Then the existence and uniqueness of the limit periodic solutions were proved by Banach contraction mapping principle for the differential equations x′(t)+ax(t)+bx(-t)=F(t,x(t),x(-t)),b≠0,t∈R,where F is uniformly limit-periodic in t and satisfies Lipschitz condition in the last two variables.