Eventually periodic solutions of a max-type system of difference equations of higher order (Q1727280)

Summary: We study in this paper the following max-type system of difference equations of higher order: \(x_n=\max\{A,y_{n-k}/x_{n-1}\}\) and \(y_n=\max\{B,x_{n-k}/y_{n-1}\}\), \(n\in\{0,1,2,\dots\}\), where \(A\geq B>0\), \(k\geq1\), and the initial conditions \(x_{-k},y_{-k},x_{-k+1},y_{-k+1},\dots,x_{-1},y_{-1}\in(0,+\infty)\). We show that (1) if \(AB>1\), then every solution of the above system is periodic with period 2 eventually. (2) If \(AB=1>B\), then every solution of the above system is periodic with period \(2k\) or 2 eventually. (3) If \(A=B=1\) or \(AB<1\), then the above system has a solution which is not periodic eventually.