两类二阶有理差分方程解的整体性态
Global Behavior of Solutions to Two Classes of Second Order Rational Difference Equations
摘要：
对于非负实数$α$，$β$，$γ$，$A$，$B$和$C$使得$B+C>0$和$α+β+γ>0$，差分方程\begin{equation*} x_{n+1}=\displaystyle\frac{α+βx_{n}+γx_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,... %, \quad x_{-1},x_{0}\in [0,\infty) \end{equation*}有唯一的正平衡点.本文给出了下列命题的证明：\medskip \noindent定理1。{ \it对于每个正参数$α$、$β$、$γ$、$A$、$B$和$C$的选择，差分方程\begin{equation*} x_{n+1}=\displaystyle\frac{α+βx_{n}+γx_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in [0,\infty) \end{equation*}的所有解都收敛到正平衡或素数周期2解。} \medskip定理2. { \it对于每个正参数$α$、$β$、$γ$、$A$、$B$和$C$的选择，差分方程\begin{equation*} x_{n+1}= \displaystyle\frac{α+βx_{n}+γx_{n-1}}{B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in (0,\infty) \end{equation*}的所有解都收敛到正平衡或素数周期2解。}
英文摘要：
For nonnegative real numbers $α$, $β$, $γ$, $A$, $B$ and $C$ such that $B+C>0$ and $α+β+γ>0$, the difference equation \begin{equation*} x_{n+1}=\displaystyle\frac{α+βx_{n}+γx_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,... %, \quad x_{-1},x_{0}\in [0,\infty) \end{equation*} has a unique positive equilibrium. A proof is given here for the following statements: \medskip \noindent Theorem 1. {\it For every choice of positive parameters $α$, $β$, $γ$, $A$, $B$ and $C$, all solutions to the difference equation \begin{equation*} x_{n+1}=\displaystyle\frac{α+βx_{n}+γx_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in [0,\infty) \end{equation*} converge to the positive equilibrium or to a prime period-two solution.} \medskip \noindent Theorem 2. {\it For every choice of positive parameters $α$, $β$, $γ$, $A$, $B$ and $C$, all solutions to the difference equation \begin{equation*} x_{n+1}= \displaystyle\frac{α+βx_{n}+γx_{n-1}}{B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in (0,\infty) \end{equation*} converge to the positive equilibrium or to a prime period-two solution.}
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