On The Max-Type Equation $x_{n+1}=max\{\frac{A_n}{x_n},x_{n-1}\}$

Abstract

It is shown that every well-defined solution to the second-order difference equation in the title, when $(A_n{)}_{n\in 0}$ is a two-periodic sequence such that $max\{A_0,A_1\}\ge 0$, is eventually periodic with period two. In the case $max\{A_0,A_1\}\le 0$, it is shown the existence of unbounded solutions, by describing all solutions in terms of $A_0$, $A_1$, $x_{-1}$, and $x_0$.