LAUSR

In this paper, we investigate global behavior of the following max-type system of difference equations with four variables $$\begin{aligned} \left\{ \begin{array}{ll}x_{n} = \max \Big \{A,\frac{z_{n-1}}{y_{n-2}}\Big \},\\ y_{n} = \max \Big \{B ,\frac{w_{n-1}}{x_{n-2}}\Big \},\\ z_{n} = \max \Big \{C ,\frac{x_{n-1}}{w_{n-2}}\Big \},\\ w_{n} = \max \Big \{D,\frac{y_{n-1}}{z_{n-2}}\Big \},\\ \end{array}\right. \quad n\in \{0,1,2, \ldots \}, \end{aligned}$$ x n = max { A , z n - 1 y n - 2 } , y n = max { B , w n - 1 x n - 2 } , z n = max { C , x n - 1 w n - 2 } , w n = max { D , y n - 1 z n - 2 } , n ∈ { 0 , 1 , 2 , … } , where $$A, B,C,D\in (0,+\infty )$$ A , B , C , D ∈ ( 0 , + ∞ ) with $$A\le B$$ A ≤ B and $$C\le D$$ C ≤ D , and the initial conditions $$x_{-2},y_{-2},z_{-2},w_{-2},x_{-1},y_{-1},z_{-1},w_{-1}\in (0,+\infty )$$ x - 2 , y - 2 , z - 2 , w - 2 , x - 1 , y - 1 , z - 1 , w - 1 ∈ ( 0 , + ∞ ) . We show that: (1) If $$AC< 1$$ A C < 1 , then there exists a solution $$\{(x_n,y_n,z_n,w_n)\}^{+\infty }_{n= -2}$$ { ( x n , y n , z n , w n ) } n = - 2 + ∞ of this system such that $$x_n=A$$ x n = A and $$z_n=C$$ z n = C for any $$n\ge -2$$ n ≥ - 2 and $$\lim _{n\longrightarrow \infty }y_n=\lim _{n\longrightarrow \infty }w_n=\infty $$ lim n ⟶ ∞ y n = lim n ⟶ ∞ w n = ∞ . (2) If $$AC=1$$ A C = 1 , then every solution of this system is eventually periodic with period 4. (3) If $$AC>1$$ A C > 1 , then every solution of this system is eventually periodic with period 1.