On a solvable system of rational difference equations of higher order

Authors: MERVE KARA, YASİN YAZLIK

Abstract: In this paper, we present that the following system of difference equations $$ x_{n}=\frac{x_{n-k}z_{n-l}}{b_{n}x_{n-k}+a_{n}z_{n-k-l}}, \ y_{n}=\frac{y_{n-k}x_{n-l}}{d_{n}y_{n-k}+c_{n}x_{n-k-l}}, \ z_{n}=\frac{z_{n-k}y_{n-l}}{f_{n}z_{n-k}+e_{n}y_{n-k-l}}, $$ where $n\in \mathbb{N}_{0}$, $k,l\in\mathbb{N}$, the initial values $x_{-i},y_{-i},z_{-i}$ are real numbers, for $i \in \overline{1,k+l}$, and sequences $\left( a_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( b_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( c_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( d_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( e_{n}\right) _{n\in \mathbb{N}_{0}}$ and $\left( f_{n}\right) _{n\in \mathbb{N}_{0}}$ are non-zero real numbers, for all $n\in \mathbb{N}_{0}$, which can be solved in closed form. We describe the forbidden set of the initial values using the obtained formulas and also determine the asymptotic behavior of solutions for the case $k=3$, $l=1$, and the sequences $\left( a_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( b_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( c_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( d_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( e_{n}\right) _{n\in \mathbb{N}_{0}}$ and $\left( f_{n}\right) _{n\in \mathbb{N}_{0}}$ are constant. Our results considerably extend and improve some recent results in the literature.

Keywords: System of difference equations, closed form, forbidden set

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