On mountain pass theorem and its application to periodic solutions of some nonlinear discrete systems

TL;DR

This paper introduces a new mountain pass theorem that is independent of boundary functional values, leading to improved existence results for nontrivial periodic solutions in nonlinear discrete systems.

Contribution

It develops a novel quantitative deformation lemma and a mountain pass theorem that enhances previous results by removing boundary value dependence.

Findings

Established a new mountain pass theorem independent of boundary functional values.

Proved the existence of nontrivial periodic solutions in certain nonlinear second-order discrete systems.

Significantly improved previous results on periodic solutions in nonlinear discrete systems.

Abstract

We obtain a new quantitative deformation lemma, and then gain a new mountain pass theorem. More precisely, the new mountain pass theorem is independent of the functional value on the boundary of the mountain, which improves the well known results (\cite{AR,PS1,PS2,Qi,Wil}). Moreover, by our new mountain pass theorem, new existence of nontrivial periodic solutions for some nonlinear second-order discrete systems is obtained, which greatly improves the result in \cite{Z04}.