Notes on Multiple Periodic Solutions for Second-order Discrete Hamiltonian System

TL;DR

This paper introduces a new method using an orthogonal decomposition and a novel functional to find multiple periodic solutions for a second-order discrete Hamiltonian system, extending previous results especially under negativity conditions.

Contribution

It develops an improved approach with a new decomposition and functional to establish multiple solutions, including an example that prior methods couldn't solve.

Findings

Established multiple periodic solutions under negativity hypothesis

Introduced a new orthogonal decomposition method

Provided an example solving an open problem

Abstract

By a new orthogonal direct sum decomposition $E_{M} = Y \oplus Z$, which $Z$ is related to $\Delta u_i(i=1,2,3,....,M)$, and a new functional $I(u)$, the method in [2] is improved to obtain new multiple periodic solutions with negativity hypothesis on $F$ for a second-order discrete Hamiltonian system. Moreover, we exhibit an instructive example to make our result more clear, which hasn't been solved by the known results.