A fixed-point approach for decaying solutions of difference equations

TL;DR

This paper introduces a fixed-point method to establish the existence of decaying intermediate solutions for a class of difference equations with advanced arguments, extending techniques from continuous to discrete cases.

Contribution

It develops a novel fixed-point approach for difference equations with advanced arguments, enabling the proof of decaying solutions by reducing to boundary value problems without deviating arguments.

Findings

Existence of intermediate decaying solutions proven.

Reduction of complex difference equations to simpler boundary value problems.

Application of fixed-point theorems from continuous to discrete cases.

Abstract

A boundary value problem associated to the difference equation with advanced argument \begin{equation} \label{*}\Delta\bigl (a_{n}\Phi(\Delta x_{n})\bigr)+b_{n}\Phi(x_{n+p} )=0,\ \ n\geq1 \tag{$*$} \end{equation} is presented, where $\Phi(u)=|u|^{\alpha}$sgn $u,$ $\alpha>0,p$ is a positive integer and the sequences $a,b,$ are positive. We deal with a particular type of decaying solutions of (\ref{*}), that is the so-called intermediate solutions (see below for the definition) . In particular, we prove the existence of these type of solutions for (\ref{*}) by reducing it to a suitable boundary value problem associated to a difference equation without deviating argument. Our approach is based on a fixed point result for difference equations, which originates from existing ones stated in the continuous case. Some examples and suggestions for future researches complete the paper.