Homoclinic Solutions for a Discrete Periodic Hamiltonian System with Perturbed Terms

The purpose of this work is to obtain, in a more optimal way, sufficient conditions for the existence of at least one homoclinic solution to a periodic discrete Hamiltonian system with perturbed terms, -Delta[p(n)Delta u(n-1)]+L(n)u(n)=del F(n,u(n))+h(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\Delta [p(n)\Delta u(n-1)]+L(n)u(n)=\nabla F(n,u(n))+h(n) \end{aligned}$$\end{document}where n is an element of Z,u is an element of RN,p,L:Z -> RNxN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathbb {Z}, \;u\in \mathbb {R}<^>N, p,L:\mathbb {Z}\rightarrow \mathbb {R}<^>{N\times N}$$\end{document} and F(n,u):ZxRN -> R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(n,u):\mathbb {Z}\times \mathbb {R}<^>{N}\rightarrow \mathbb {R}$$\end{document}. In this paper, p(n) and L(n) are T-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T-$$\end{document}periodic with respect to n is an element of Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathbb {Z}$$\end{document}, and they are not required to be positive definite. Whether at infinity or at the origin, the nonlinearity F need not be superquadratic, but can be asymptotically quadratic or a mixture of them. This character of superquadraticity is essential as observed in previous literature. Moreover, the existence of homoclinic solutions is shown to remain when the discrete Hamiltonian system involves some perturbations h is an element of l1\{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in l<^>1{\setminus }\{0\}$$\end{document}. To the best of our knowledge, this is the first attempt to obtain the existence of homoclinic solutions for a perturbed discrete Hamiltonian system in the case where p and L are non-positive definite. Our work is an improvement and complement to previous work on the existence of homoclinic solutions for discrete or continuous Hamiltonian systems. Furthermore, our superior results may be applicable to other variational problems.