A fixed-point approach for decaying solutions of difference equations

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.