On homoclinic solutions for a second order difference equation with p-Laplacian

TL;DR

This paper establishes conditions for the existence of infinitely many homoclinic solutions in a second order difference equation involving the p-Laplacian, using a variant of the fountain theorem to improve previous results.

Contribution

It introduces a new set of conditions under which homoclinic solutions exist, refining earlier work by addressing inconsistencies in the nonlinear conditions.

Findings

Proves existence of infinitely many homoclinic solutions.

Utilizes a variant of the fountain theorem for the proof.

Improves upon previous results by resolving nonlinear condition inconsistencies.

Abstract

In this paper, we obtain conditions under which the difference equation $-\Delta \left( a(k)\phi _{p}(\Delta u(k-1))\right) +b(k)\phi_{p}(u(k))=\lambda f(k,u(k)),\quad k\in \mathbb{Z}$, has infinitely many homoclinic solutions. A variant of the fountain theorem is utilized in the proof of our theorem. It improves the results in [L.Kong, homoclinic solutions for a second order difference equation with $p-$Laplacian, \textit{Appl. Math. Comput}., \textbf{247} (2014), 1113--1121], where the set of conditions imposed on nonlinearity is inconsistent.