Existence of homoclinic solution in first order discrete Hamiltonian system

Abstract

In this paper we consider the first order discrete Hamiltonian system $$\{\begin{array}{llc}{x}_1(n+1)-x_1(n)& =-H_{x_2}(n,x(n)),x_2(n)-x_2(n-1)& =H_{x_1}(n,x(n)).\end{array}$$ Where $n\in \mathbb{Z}$, $x(n)=$ $\left(\genfrac{}{}{0ex}{}{x_1(n)}{x_2(n)}\right)$$\in {\mathbb{R}}^{2N}$, $H(n,z)=\frac{1}{2}S(n)z\cdot z+R(n,z)$ is periodic in $n$ and asymptotically quadratic as $|z|\to \infty$. We will prove the existence of homoclonic solution by critical point theorem for strongly indefinite functional.