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

In this paper, the existence of positive periodic solutions is studied for Liénard equation with a singularity of repulsive type, $x^{″}(t)+f(x(t))x^{\prime }(t)+\phi (t)x^{\mu }(t)-\frac{1}{x^{\gamma }(t)}=e(t),$ \( x''(t)+f(x(t))x'(t)+\varphi (t)x^{\mu}(t)-\frac{1}{x^{\gamma}(t)}=e(t), \) where $f:(0,+\mathrm{\infty })\to R$ $f:(0,+\infty )\rightarrow R$ is continuous, which may have a singularity at the origin, the sign of $\phi (t)$ $\varphi (t)$ , $e(t)$ $e(t)$ 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 $\mu \in [0,+\mathrm{\infty })$ $\mu \in [0,+\infty )$ .