A detailed study on a solvable system related to the linear fractional difference equation
In this paper, we present a detailed study of the following system of difference equations $ \begin{equation*} x_{n+1} = \frac{a}{1+y_{n}x_{n-1}}, \ y_{n+1} = \frac{b}{1+x_{n}y_{n-1}}, \ n\in\mathbb{N}_{0}, \end{equation*} $ where the parameters $ a $, $ b $, and the initial values $ x_{-1}, \...
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2021
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oai:doaj.org-article:574b60b4e811461a8aea8149e40f1e932021-11-09T02:11:20ZA detailed study on a solvable system related to the linear fractional difference equation10.3934/mbe.20212731551-0018https://doaj.org/article/574b60b4e811461a8aea8149e40f1e932021-06-01T00:00:00Zhttps://www.aimspress.com/article/doi/10.3934/mbe.2021273?viewType=HTMLhttps://doaj.org/toc/1551-0018In this paper, we present a detailed study of the following system of difference equations $ \begin{equation*} x_{n+1} = \frac{a}{1+y_{n}x_{n-1}}, \ y_{n+1} = \frac{b}{1+x_{n}y_{n-1}}, \ n\in\mathbb{N}_{0}, \end{equation*} $ where the parameters $ a $, $ b $, and the initial values $ x_{-1}, \; x_{0}, \ y_{-1}, \; y_{0} $ are arbitrary real numbers such that $ x_{n} $ and $ y_{n} $ are defined. We mainly show by using a practical method that the general solution of the above system can be represented by characteristic zeros of the associated third-order linear equation. Also, we characterized the well-defined solutions of the system. Finally, we study long-term behavior of the well-defined solutions by using the obtained representation forms.Durhasan Turgut Tolluİbrahim YalçınkayaHijaz AhmadShao-Wen YaoAIMS Pressarticlebehavior of solutionscharacteristic equationgeneral solutionsystem of difference equationsperiodic solutionBiotechnologyTP248.13-248.65MathematicsQA1-939ENMathematical Biosciences and Engineering, Vol 18, Iss 5, Pp 5392-5408 (2021) |
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DOAJ |
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DOAJ |
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EN |
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behavior of solutions characteristic equation general solution system of difference equations periodic solution Biotechnology TP248.13-248.65 Mathematics QA1-939 |
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behavior of solutions characteristic equation general solution system of difference equations periodic solution Biotechnology TP248.13-248.65 Mathematics QA1-939 Durhasan Turgut Tollu İbrahim Yalçınkaya Hijaz Ahmad Shao-Wen Yao A detailed study on a solvable system related to the linear fractional difference equation |
| description |
In this paper, we present a detailed study of the following system of difference equations
$ \begin{equation*} x_{n+1} = \frac{a}{1+y_{n}x_{n-1}}, \ y_{n+1} = \frac{b}{1+x_{n}y_{n-1}}, \ n\in\mathbb{N}_{0}, \end{equation*} $
where the parameters $ a $, $ b $, and the initial values $ x_{-1}, \; x_{0}, \ y_{-1}, \; y_{0} $ are arbitrary real numbers such that $ x_{n} $ and $ y_{n} $ are defined. We mainly show by using a practical method that the general solution of the above system can be represented by characteristic zeros of the associated third-order linear equation. Also, we characterized the well-defined solutions of the system. Finally, we study long-term behavior of the well-defined solutions by using the obtained representation forms. |
| format |
article |
| author |
Durhasan Turgut Tollu İbrahim Yalçınkaya Hijaz Ahmad Shao-Wen Yao |
| author_facet |
Durhasan Turgut Tollu İbrahim Yalçınkaya Hijaz Ahmad Shao-Wen Yao |
| author_sort |
Durhasan Turgut Tollu |
| title |
A detailed study on a solvable system related to the linear fractional difference equation |
| title_short |
A detailed study on a solvable system related to the linear fractional difference equation |
| title_full |
A detailed study on a solvable system related to the linear fractional difference equation |
| title_fullStr |
A detailed study on a solvable system related to the linear fractional difference equation |
| title_full_unstemmed |
A detailed study on a solvable system related to the linear fractional difference equation |
| title_sort |
detailed study on a solvable system related to the linear fractional difference equation |
| publisher |
AIMS Press |
| publishDate |
2021 |
| url |
https://doaj.org/article/574b60b4e811461a8aea8149e40f1e93 |
| work_keys_str_mv |
AT durhasanturguttollu adetailedstudyonasolvablesystemrelatedtothelinearfractionaldifferenceequation AT ibrahimyalcınkaya adetailedstudyonasolvablesystemrelatedtothelinearfractionaldifferenceequation AT hijazahmad adetailedstudyonasolvablesystemrelatedtothelinearfractionaldifferenceequation AT shaowenyao adetailedstudyonasolvablesystemrelatedtothelinearfractionaldifferenceequation AT durhasanturguttollu detailedstudyonasolvablesystemrelatedtothelinearfractionaldifferenceequation AT ibrahimyalcınkaya detailedstudyonasolvablesystemrelatedtothelinearfractionaldifferenceequation AT hijazahmad detailedstudyonasolvablesystemrelatedtothelinearfractionaldifferenceequation AT shaowenyao detailedstudyonasolvablesystemrelatedtothelinearfractionaldifferenceequation |
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