Positive periodic solution to indefinite singular Liénard equation

Abstract

In this paper, we investigate the existence of a positive periodic solution for the following Lienard equation with a indefinite singularity $$\begin{aligned} x''+f(x)x'+\frac{b(t)}{x}=p(t), \end{aligned}$$ where $$b\in C({\mathbb {R}},{\mathbb {R}})$$ is a T-periodic sign-changing function. The novelty of the present article is that for the first time we show that a indefinite singularity enables the achievement of a new existence criterion of positive periodic solutions through a application of a topological degree theorem by Mawhin. Recent results in the literature are generalized and significantly improved, and we give the existence interval of a positive periodic solution of this equation. At last, an example is given to show applications of the theorems.