Solvability of two-dimensional system of difference equations with constant coefficients

Abstract

In the present paper, the solutions of the following system of difference equations \begin{equation*} u_{n}=\alpha_{1}v_{n-2}+\frac{\delta_{1}v_{n-2}u_{n-4}}{\beta_{1}u_{n-4}+\gamma_{1}v_{n-6}}, \ v_{n}=\alpha_{2}u_{n-2}+\frac{\delta_{2}u_{n-2}v_{n-4}}{\beta_{2}v_{n-4}+\gamma_{2}u_{n-6}}, \ n\in \mathbb{N}_{0}, \end{equation*} where the initial values $u_{-l}$, $v_{-l}$, for $l=\overline{1,6}$ and the parameters $\alpha_{p}$, $\beta_{p}$, $\gamma_{p}$, $\delta_{p}$, for $p\in\{1,2\}$ are non-zero real numbers, are investigated. In addition, the solutions of aforementioned system of difference equations are presented by utilizing Fibonacci sequence when the parameters are equal $1$. Finally, the periodic solutions according to some special cases of the parameters are obtained.