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 Li & eacute;nard equation with a singularity of repulsive type, x ''(t)+f(x(t))x '(t)+phi(t)x mu(t)-1x gamma(t)=e(t),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x''(t)+f(x(t))x'(t)+\varphi (t)x<^>{\mu}(t)-\frac{1}{x<^>{\gamma}(t)}=e(t), $$\end{document} where f:(0,+infinity)-> R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f:(0,+\infty )\rightarrow R$\end{document} is continuous, which may have a singularity at the origin, the sign of phi(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varphi (t)$\end{document}, e(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$e(t)$\end{document} is allowed to change, and mu, gamma 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 is an element of[0,+infinity)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu \in [0,+\infty )$\end{document}.