Dynamics of a max-type system of difference equations (Q503406)

The authors study the asymptotic behavior of solutions of the system of max-type difference equations \[ x_n=\max\bigg\{\frac{1}{y_{n-m}},\,\frac{\alpha_n}{x_{n-r}}\bigg\}, \quad y_n=\max\bigg\{\frac{1}{x_{n-m}},\,\frac{\beta_n}{y_{n-r}}\bigg\}, \quad n\in{\mathbb N}\cup\{0\}. \] Here \(\alpha_n,\beta_n\in(0,1)\) are sequences such that \(\sup\{\max\{\alpha_n,\beta_n\},\,n\in{\mathbb N}\cup\{0\}\}<1\), \(r,m\in{\mathbb N}\) (with \(r\neq m\)), and the \(2d\) initial values (\(d=\max\{r,m\}\)) of \(x\) and \(y\) are positive numbers. The authors prove two main results (Theorems~2.6 and~2.7): {\parindent=0.6cm\begin{itemize}\item[1.] The sequences \(\{x_{2nm+i}\}_{n=0}^\infty\) and \(\{y_{2nm+i}\}_{n=0}^\infty\) are eventually monotone for every \(i\in\{0,1,\dots,2m-1\}\); \item [2.] if \(\{\alpha_n\}_{n=0}^\infty\) and \(\{\beta_n\}_{n=0}^\infty\) are periodic, then \(\{(x_n,y_n)\}_{n=0}^\infty\) is eventually periodic with period \(2m\). \end{itemize}} The proofs are based on the analysis of subsequences of \(\{x_n\}_{n=0}^\infty\) and \(\{y_n\}_{n=0}^\infty\) with special properties (such as eventually constant subsequences in the second case).