Existence of positive periodic solutions for Liénard equation with a singularity of repulsive type

Abstract

In this paper, the existence of positive periodic solutions is studied for Liénard equation with a singularity of repulsive type, $$ x''(t)+f(x(t))x'(t)+\varphi (t)x^{\mu}(t)-\frac{1}{x^{\gamma}(t)}=e(t), $$ where $f:(0,+\infty )\rightarrow R$ is continuous, which may have a singularity at the origin, the sign of $\varphi (t)$ , $e(t)$ is allowed to change, and μ, γ are positive constants. By using a continuation theorem, as well as the techniques of a priori estimates, we show that this equation has a positive T-periodic solution when $\mu \in [0,+\infty )$ .