Solvability of a Second-Order Rational System of Difference Equations

Abstract

In this paper, we represent the admissible solutions of the system of second-order rational difference equations given below in terms of Lucas and Fibonacci sequences: \begin{eqnarray*} \begin{split} x_{n+1}=\dfrac{L_{m+2}+L_{m+1}y_{n-1}}{L_{m+3}+L_{m+2}y_{n-1}},\quad y_{n+1}=\dfrac{L_{m+2}+L_{m+1}z_{n-1}}{L_{m+3}+L_{m+2}z_{n-1}},\\ z_{n+1}=\dfrac{L_{m+2}+L_{m+1}w_{n-1}}{L_{m+3}+L_{m+2}w_{n-1}},\quad w_{n+1}=\dfrac{L_{m+2}+L_{m+1}x_{n-1}}{L_{m+3}+L_{m+2}x_{n-1}}. \end{split} \end{eqnarray*} where $n\in\mathbb{N}_0$, $\{L_m\}_{m=-\infty}^{+\infty}$ is Lucas sequence and the initial conditions $x_{-1}$, $x_{0}$, $y_{-1}$, $y_{0}$, $z_{-1}$, $z_{0}$, $w_{-1}$, $w_{0}$ are arbitrary real numbers such that $\displaystyle v_{-i}\neq-\frac{L_{m+3}}{L_{m+2}}$, where $v_{-i}=x_{-i},y_{-i},z_{-i},w_{-i}$, $i=0,1$ and $m\in\mathbb{Z}$.