具周期参数的倒易极大型差分方程$x_{n}=\max\left\{\frac{A^{1}_{n-1}}{x_{n-1}}, \frac{A^{2}_{n-1}}{x_{n-2}}, \ldots, \frac{A^{t}_{n-1}}{x_{n-t}}\right\}$正解的有界性
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
摘要：
研究了一类倒数极大型差分方程\[ x_{n}=\max\left\{\frac{A_{n-1}^{1}}{x_{n-1}}, \frac{A_{n-1}^{2}}{x_{n-2}}, \ldots, \frac{A_{n-1}^{t}}{x_{n-t}}\right\}, \ \ n=1, 2, \ldots, \]正解的有界性，其中对于$i$的每一个值，正数序列$\{A_{n}^{i}\}_{n=0}^{\infty}$是周期的，周期为$p_{i}$ .给出了关于$p_{i}$的所有解有界的充分条件和所有解无界的充分条件。这项工作基本上补充了比德尔和弗兰克的工作，他们证明了只要我们方程的每个正解都是\emph{bounded}，那么每个正解最终都是周期的，从而留下了解何时有界的问题。
英文摘要：
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.
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