Homoclinic Orbits for second order Hamiltonian Equations in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document}

We are concerned with the existence and multiplicity of homoclinic solutions for the second order Hamiltonian equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\ddot{u}+\omega(t)u=F_u(t,u) \quad t \in \mathbb{R}, \quad\quad\quad(1)$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega \in \mathcal{C}(\mathbb{R})}$$\end{document} is positive and bounded, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F\in \mathcal{C}^1(S^1\times\mathbb{R})}$$\end{document} . Under some growth condition on F, we prove that (1) admits at least two solutions which are homoclinic to zero and do not change sign. We also prove that for every integer k ≥  1, (1) possesses at least two solutions homoclinic to zero changing sign exactly k times, and for k ≥  2 these solutions have at least k and at most k + 2 zeros which are isolated, or ‘isolated from the left’, or ‘isolated from the from right’.