Existence of homoclinic solution in first order discrete Hamiltonian system

TL;DR

This paper proves the existence of homoclinic solutions in a first-order discrete Hamiltonian system using critical point theory for strongly indefinite functionals, under periodic and asymptotic quadratic conditions.

Contribution

It establishes the existence of homoclinic solutions in a discrete Hamiltonian system with new conditions and applies critical point theory to strongly indefinite functionals.

Findings

Existence of homoclinic solutions proven.

Application of critical point theorem to discrete systems.

Results extend understanding of discrete Hamiltonian dynamics.

Abstract

In this paper we consider the first order discrete Hamiltonian system $$\begin{cases} x_1(n+1)-x_1(n)& =- H_{x_2}(n,x(n)), x_2(n)-x_2(n-1)& =\ \ H_{x_1}(n,x(n)). \end{cases} $$ Where $n\in \mathbb{Z}$, $x(n)=$ $x_1 (n) \choose x_2 (n)$$ \in \mathbb{R}^{2N}$, $ H(n,z)= \frac12S(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.