On the Boundedness of Positive Solutions of the Reciprocal Max-Type Difference Equation $x_{n}=\max\left\{\frac{A^{1}_{n-1}}{x_{n-1}}, \frac{A^{2}_{n-1}}{x_{n-2}}, \ldots, \frac{A^{t}_{n-1}}{x_{n-t}}\right\}$ with Periodic Parameters

Abstract

We investigate the boundedness of positive solutions of the reciprocal max-type difference equation \[ x_{n}=\max\left\{\frac{A_{n-1}^{1}}{x_{n-1}}, \frac{A_{n-1}^{2}}{x_{n-2}}, \ldots, \frac{A_{n-1}^{t}}{x_{n-t}}\right\}, \ \ n=1, 2, \ldots, \] where, for each value of $i$, the sequence $\{A_{n}^{i}\}_{n=0}^{\infty}$ of positive numbers is periodic with period $p_{i}$. We give both sufficient conditions on the $p_{i}$'s for the boundedness of all solutions and sufficient conditions for all solutions to be unbounded. This work essentially complements the work by Biddell and Franke, who showed that as long as every positive solution of our equation is \emph{bounded}, then every positive solution is eventually periodic, thereby leaving open the question as to when solutions are bounded.