Attractive singularity problems for superlinear Liénard equation

In this paper, we consider the following quasilinear Lienard equation with a singularity $$\begin{aligned} (\phi _p(x'(t)))'+f(x(t))x'(t)+g(t,x(t))=e(t), \end{aligned}$$ where g has a attractive singularity at the origin and satisfies superlinear condition at \(x=+\infty \). By using Manasevich–Mawhin continuous theorem, we prove that this equation has at least one positive T-periodic solution. We solve a difficulty to estimate it a priori bounds of a periodic solution for quasilinear Lienard equation in the case that superlinear condition. At last, example and numerical solution (phase portrait and time series portrait of the positive periodic solution of example) are given to show applications of the theorem.