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arXiv · 2011.12033

A fixed-point approach for decaying solutions of difference equations

Zuzana Došlá·Mauro Marini·Serena Matucci
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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.

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Zuzana Došlá, Mauro Marini, Serena Matucci. 2020-11-24. A fixed-point approach for decaying solutions of difference equations. https://doi.org/10.1098/rsta.2019.0374

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