脉冲条件下Liénard方程周期解的保持性

Persistence of Periodic Solutions of Liénard Equations under Impulses

本文研究带有脉冲的Lienard方程的周期解的存在性问题.我们通过分析Poincaré映射在脉冲点处的变化特征,利用Poincare-Bohl不动点定理证明:在一串脉冲点于时间轴上具有周期分布特征的情况以及适当的脉冲条件之下,如果位势函数满足Lipschitz条件,而强迫项又是周期函数,则Lienard方程x″+f (x)x′+g(x)=p(t)仍然保持周期解的存在性.另外,我们给出了一个具体的Liénard方程例子,来佐证本文中主要结果的有效性.

In this paper,we are concerned with the existence problems of periodic solutions for the Lienard equation with impulses.By analyzing the variation characteristics of Poincare mapping at impulse points and using Poincare-Bohl fixed point theorem,we prove that if a series of impulse points have periodic distribution characteristics on the time axis and under appropriate impulse conditions,the potential function satisfies the Lipschitz condition and the forcing term is a periodic function,then the existence of periodic solutions for Lienard equation x″+f(x)x′+g(x)=p(t)is still maintained.A concrete example of Lienard equation is provided to show the effectiveness of the main results of present paper.