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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| Main Authors: | , , , |
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| Format: | article |
| Language: | EN |
| Published: |
AIMS Press
2021
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| Subjects: | |
| Online Access: | https://doaj.org/article/574b60b4e811461a8aea8149e40f1e93 |
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| Summary: | 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. |
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