A note on nonlocal discrete problems involving sign-changing Kirchhoff functions

TL;DR

This paper proves a multiplicity theorem for a nonlocal discrete boundary value problem involving sign-changing Kirchhoff functions, extending previous results to cases where the parameter b is negative.

Contribution

It introduces a novel multiplicity theorem for nonlocal discrete problems with sign-changing Kirchhoff functions, specifically when the parameter b is negative.

Findings

Established a multiplicity theorem for the problem

Extended analysis to cases with negative parameter b

First to consider sign-changing Kirchhoff functions in this context

Abstract

In this note, we establish a multiplicity theorem for a nonlocal discrete problem of the type $$\cases{-\left(a\sum_{m=1}^{n+1}|x_m-x_{m-1}|^2+b\right)(x_{k+1}-2x_k+x_{k-1})=h_k(x_k)\hskip 10pt k=1,...,n, \cr & \cr x_0=x_{n+1}=0\cr}$$ assuming $a>0$ and (for the first time) $b<0$.