Abstract:

In this study, we show that the system of difference equations \begin{align} x_{n}=\frac{x_{n-2}y_{n-3}}{y_{n-1}\left(a+bx_{n-2}y_{n-3} \right) }, \nonumber \\ y_{n}=\frac{y_{n-2}z_{n-3}}{z_{n-1}\left(c+dy_{n-2}z_{n-3} \right) },~n\in\mathbb{N}_{0}, ~ \nonumber \\ z_{n}=\frac{z_{n-2}x_{n-3}}{x_{n-1}\left(e+fz_{n-2}x_{n-3} \right) }, \nonumber \\ \end{align} where the initial values $x_{-i}, y_{-i}, z_{-i}$, $i=\overline{1,3}$ and the parameters $a$, $b$, $c$, $d$, $e$, $f$ are non-zero real numbers, can be solved in closed form. Moreover, we obtain the solutions of above system in explicit form according to the parameters $a$, $c$ and $e$ are equal $1$ or not equal $1$. In addition, we get periodic solutions of aforementioned system. Finally, we define the forbidden set of the initial conditions by using the acquired formulas.

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