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A detailed study on a solvable system related to the linear fractional difference equation
Durhasan Turgut Tollu1, İbrahim Yalçınkaya1, Hijaz Ahmad2
1Department of Mathematics and Computer Sciences, Necmettin Erbakan University, Konya, Turkey.
Abstract:
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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