MMN-5111
On the solvability in closed-form of a rational three-dimensional system of difference equations
Hiba Zabat; Nouressadat Touafek; Imane Dekkar;Abstract
In this paper, we solve in closed-form Consider the following third-order system of nonlinear difference equations
\begin{equation*} x_{n+1}=\frac{a_1x_{n-k}y_{n-k}}{ax_{n-k}+by_{n-k}+cz_{n-k}},\,y_{n+1}=\frac{a_2y_{n-k}z_{n-k}}{ax_{n-k}+by_{n-k}+cz_{n-k}},\, z_{n+1}=\frac{a_3x_{n-k}z_{n-k}}{ax_{n-k}+by_{n-k}+cz_{n-k}},
\end{equation*}
where $n,\,k\in\mathbb{N}_0$, the parameters $a_1$, $a_2$, $a_3$, $a$, $b$, $c$ are real numbers, and the initial values $x_{-k},...$, $x_0$, $y_{-k},...$, $y_0$, $z_{-k},...$, $z_0$, are non-zero real numbers. Firstly, we establish some preliminaries results for the general case, then we solve in a closed form, via some change of variables, our system in the two particular cases $k=0$ and $k=1$.
Vol. 26 (2025), No. 2, pp. 1121-1139