A fixed-point approach for decaying solutions of difference equations

TL;DR

The paper addresses the existence of decaying intermediate solutions for nonlinear discrete equations with advanced argument by reducing the problem to a boundary-value problem without deviating arguments and applying a fixed-point framework in a Fréchet space. It establishes a sharp comparison between the original equation and an associated half-linear model, proving that intermediate-solution existence for one implies existence for the other. The approach hinges on a discrete fixed-point theorem and carefully constructed function spaces, providing concrete examples and outlining avenues for extending fixed-point methods to broader discrete problems and asymptotic analyses. This work contributes a rigorous, flexible toolkit for analyzing decaying solutions in nonlinear difference equations with advanced shifts, with potential impact on discretizations of elliptic-type problems and related nonlinear dynamics.

Abstract

A boundary value problem associated to the difference equation with advanced argument \begin{equation} \label{*}Δ\bigl (a_{n}Φ(Δx_{n})\bigr)+b_{n}Φ(x_{n+p} )=0,\ \ n\geq1 \tag{$*$} \end{equation} is presented, where $Φ(u)=|u|^α$sgn $u,$ $α>0,p$ 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.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Corollary 1
• Theorem 3
• Theorem 4
• Lemma 1
• Corollary 2
• Corollary 3