Global behavior of a max-type system of difference equations of the second order with four variables and period-two parameters

Abstract

In this paper, we study global behavior of the following max-type system of difference equations of the second order with four variables and period-two parameters

\begin{document}$ \left\{\begin{array}{ll}x_{n} = \max\Big\{A_n , \frac{z_{n-1}}{y_{n-2}}\Big\}, \ y_{n} = \max \Big\{B_n, \frac{w_{n-1}}{x_{n-2}}\Big\}, \ z_{n} = \max\Big\{C_n , \frac{x_{n-1}}{w_{n-2}}\Big\}, \ w_{n} = \max \Big\{D_n, \frac{y_{n-1}}{z_{n-2}}\Big\}, \ \end{array}\right. \ \ n\in \{0, 1, 2, \cdots\}, $\end{document}

where $ A_n, B_n, C_n, D_n\in (0, +\infty) $ are periodic sequences with period 2 and the initial values $ x_{-i}, y_{-i}, z_{-i}, w_{-i}\in (0, +\infty)\ (1\leq i\leq 2) $. We show that if $ \min\{A_0C_1, B_0D_1, A_1C_0, B_1D_0\} < 1 $, then this system has unbounded solutions. Also, if $ \min\{A_0C_1, B_0D_1, A_1C_0, B_1D_0\}\geq 1 $, then every solution of this system is eventually periodic with period $ 4 $.