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{$\ast$} \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.

accepted for publication on Philosophical Transactions of the Royal Society A. Issue: Topological degree and fixed point theories in differential and difference equations Editors: Maria Patrizia Pera and Marco Spadini