Is there someone who can show me how I can prove this conjecture? Or at least show me how to do the first implication ? 



> 
> **Conjecture**: Assume $\alpha,\beta, \lambda \in [0,\infty)$. Then every positive solution of the difference equation : 
>  $$z\_{n+1}=\frac{\alpha+z\_{n}\beta +z\_{n-1}\lambda}{z\_{n-2}},\quad n=0,1,\ldots$$ 
>  is bounded if and only if $\beta=\lambda$
> 
> 
> 


Any help is very welcome. Thank you for any comments or any replies.


Edit: as mentioned in the comments, this is conjecture 8 in [this paper by Ladas, Lugo and Palladino](http://anubih.ba/Journals/vol.8,no-2,y12/11Ladas-Lugo-Palladino.pdf)



Boundedness of solutions of a difference equation
**Note** :This is comments for the answer related to the question and not lead to the Answer 


of the above question .and may it is helpful to be constrictive for a complet proof 


The discussion of energy functions is at best very difficult to understand and at worst incorrect or incomplete. To illustrate this consider that the Answer above emphasizes that it is enough for $f(x,y,z)$ to go to infinity as x goes to infinity. This of course is patently false for a general system, (consider $x\_{n}=5x\_{n-1}, y\_{n}=2y\_{n-1}, z\_{n}=2z\_{n-1}$, and $f(x,y,z)=x)$, and only true for a delay dynamical system in one dimension. This was perhaps to be assumed by context, but I didn't see it mentioned. Moreover the conjecture wasn't proved.


**"In a short statement"**:The proof is not standard and the theory of Lyapunouv functions cannot solves this type of problems. 


