Is there someone who can show me the way helping me to proof this conjecture. If it's not open, at a least show me links or papers which to be helpful.
Conjecture: Assume $c > 0$ and that an equilibrium point $\bar{z}$ of the difference equation:
$$z_{n+1}=\frac{\alpha+z_{n}\beta+z_{n-1}\lambda }{A+Bz_{n}+cz_{n-1}}, \ n=0,1,\dots$$
is locally asymptotically stable. My question is:
Show that $\bar{x}$ is a global attractor of all positive solutions of precedent equation.
Note: this conjecture deal with the following simpler system: $$\begin{cases} x_{n+1}=\dfrac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\dfrac{\alpha_{2}}{z_{n}} \\ z_{n+1}=\dfrac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3}y_{n}+\lambda_{3}z_{n}}{A_{3}+B_{3}x_{n}+c_{3}y_{n}+D_{3}z_{n}}{}\end{cases} \quad n=0,1,\dots,$$ with non-negative parameters and non-negative initial conditions such that the denominators are always positive.
any help is very welcom , thank you for your replies or any comments