关键词: 周期解, 奇性, 拓展定理, 重合度理论

Abstract:

In this paper, we study the existence of positive periodic solutions for a singular Liénard equation $x''(t)+f(x(t))x'(t)-\varphi(t)x^\delta(t)+\frac{\alpha(t)}{x^{\mu}(t)}=0, $ where $f: (0, +\infty)\rightarrow \mathbb{R} $ is continuous which may have a singularity at $x=0$, $\alpha$ and $\varphi$ are $T$ -periodic functions with $\alpha, \varphi\in L([0, T], \mathbb{R})$, $\mu\in(0, +\infty)$ and $\delta\in(0, 1]$ are constants. The signs of weight functions $\alpha(t)$ and $\varphi(t)$ are allowed to change on $[0, T]$. We prove that the given equation has at least one positive $T$ -periodic solution. The method of proof relies on a continuation theorem of coincidence degree principle.

Key words: Periodic solution, Singularity, Continuation theorem, Coincidence degree principle