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let $A$ be a rational system of the form :$\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ z_{n+1}=\frac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3}y_{n}+\lambda_{3}z_{n}}{A_{3}+B_{3}x_{n}+c_{3}y_{n}+D_{3}z_{n}}{}\end{cases}$ $n=0,1,....$

with nonnegative paramaters and with nonnegative initial conditions such that the denominators are always positive.

how i could show that :for every positive solution of the difference equation: $z_{n+1}= \frac{\alpha}{1+\prod_{i=0}^{k}z_{n-i}},n=0,1,.... $ has a finit limit ?

I pasted this question in stackexchange but i'm not receive any comments or any replies

REF:https://math.stackexchange.com/q/1047052/156150

Thank you for any replies or any help.

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    $\begingroup$ Three comments: 1) You should wait a bit longer before cross-posting; 2) Why do you expect that this has a positive limit? 3) What is the relationship between the second difference equation and the first? $\endgroup$
    – Anthony Quas
    Commented Dec 2, 2014 at 20:05
  • $\begingroup$ just to define $z_{n+1}$ and the sysetm that i wrote in 3 dimension . $\endgroup$
    – salimmath15
    Commented Dec 2, 2014 at 20:13

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