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

TL;DR

This paper analyzes the conditions under which positive solutions of a periodic reciprocal max-type difference equation remain bounded or become unbounded, extending previous work by providing sufficient criteria for both outcomes.

Contribution

It establishes new sufficient conditions on the periods of parameters that determine the boundedness or unboundedness of solutions in the difference equation.

Findings

Provides criteria for bounded solutions based on periodic parameters.

Identifies conditions leading to unbounded solutions.

Complements previous work by clarifying when solutions are bounded.

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.