Periodicities of a system of difference equations (Q680367)

Summary: We study the periodicities of a system of difference equations \(x_{n + 1} = \max \{1 / x_n, A_n / y_{n - k}\}\), \(y_{n + 1} = \max \{1 / y_n, B_n / x_{n - k}\}\), where initial values \((x_{- k}, y_{- k}), \dots,(x_0, y_0) \in(0, +\infty) \times(0, +\infty)\). We show that if \(A_n, B_n \in(0,1)\) are two periodic sequences, then every solution of the above system is eventually periodic with period 2. If \(k\) is even, there must be one in \(\{x_n\}\) and \(\{y_n\}\) converges to period two solution.