Homoclinic solutions for a second-order singular differential equation

In this paper, the problem of existence of homoclinic solutions is studied for the second-order singular differential equation x′′(t)+f(x(t))x′(t)-g(x(t))-α(t)x(t)1-x(t)=h(t),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} x''(t)+f(x(t))x'(t)-g(x(t))-\frac{\alpha (t)x(t)}{1-x(t)}=h(t), \end{aligned}$$\end{document}where f,g,h,α:R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f,g,h,\alpha : R\rightarrow R$$\end{document} are continuous and α(t+T)≡α(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (t+T)\equiv \alpha (t)$$\end{document} for all t∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in R$$\end{document}. Using the continuation theorem of coincidence degree theory given by Mawhin and Manásevich, a new result on the existence of homoclinic solutions to the equation is obtained.