Fast homoclinic orbits for a class of damped vibration systems

We study the existence of fast homoclinic orbits for the following damped vibration system u¨(t)+q(t)u˙(t)+∇V(t,u(t))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{u}(t)+q(t)\dot{u}(t)+\nabla V(t,u(t))=0$$\end{document}; where q∈C(R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\in C(\mathbb {R},\mathbb {R})$$\end{document} and V∈C1(R×RN,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})$$\end{document} is of the type V(t,x)=-K(t,x)+W(t,x). A map K is not a quadratic form in x and W(t, x) is superquadratic in x.