A fixed-point approach for decaying solutions of difference equations
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866915247950921728 |
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| author | Došlá, Zuzana Marini, Mauro Matucci, Serena |
| author_facet | Došlá, Zuzana Marini, Mauro Matucci, Serena |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_12033 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A fixed-point approach for decaying solutions of difference equations Došlá, Zuzana Marini, Mauro Matucci, Serena Classical Analysis and ODEs 39A22, 47H10 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. |
| title | A fixed-point approach for decaying solutions of difference equations |
| topic | Classical Analysis and ODEs 39A22, 47H10 |
| url | https://arxiv.org/abs/2011.12033 |