In my work, I need to study the convergence of sequence defined by the non-linear recurrence relation
$$
u\_0,u\_1>0, \qquad \forall n\in \mathbb N, \; u\_{n+2}=a\ln(1+u\_n)+b\ln(1+u\_{n+1})
$$
with fixed $a,b> 0$.


For $a=b=1$ Wolfram gives the limit $-1-2 W\_{-1}\left(-\frac{1}{2\sqrt{e}}\right)$. (W is the Lambert function) <https://en.m.wikipedia.org/wiki/Lambert_W_function>. I'm looking for ideas to study convergence of this sequence. I would suspect that non-classical arguments are needed to do so.


Note: I have already asked the question on [Mathematics Stack Exchange](https://math.stackexchange.com/q/3720998/793015), but it was closed quickly for a reason that I do not understand.



Convergence for a non-linear second order difference equation
*I would suspect that non-classical arguments are needed to do so.*


All you need to know is that $t\mapsto \frac 1{1+t}$ is a decreasing function, so for $0<x\le x'$ we have $\frac{\log(1+x')}{\log(1+x)}\le \frac {x'}x$. This immediately implies that the mapping $T:(x,y)\mapsto (y,a\log(1+x)+b\log(1+y))$ is non-expanding in the metric $d((x,y),(x',y'))=max(|\log x-\log x'|,|\log y-\log y'|)$ on $(0,+\infty)^2$ and $T^2$ is a weak contraction ($d(T^2p,T^2q)<d(p,q)$ if $p\ne q$).


The next step is to consider the equation $x=(a+b)\log(1+x)$ and notice that either $a+b\le 1$ (in which case the iterations trivially converge to $(0,0)$, i.e., "escape to infinity" in our metric) or $a+b>1$ in which case there is a positive solution $x\_0$ of that equation and $T$ maps every compact ball $B(p\_0,r)$ into itself where $p\_0=(x\_0,x\_0)$ is a fixed point of $T$. If $(u\_0,u\_1)$ lies in that ball, we can apply the usual result about weak contractions on compact sets to conclude that we have convergence to $p\_0$.


*but it was closed quickly for a reason that I do not understand.*


The closure reason cited on MSE is totally ridiculous IMHO. The very fact that you ask a well-posed mathematical question is a sufficient proof of "relevance to you" and nobody is obliged to verify the "relevance to the community" (whatever it might mean) when asking. The only possible reason for closure I see is that the question is rather trivial, but, given the usual amount of total junk floating on MSE, I doubt that it was what determined its fate. So, please, accept my apologies for the MSE users behavior, ignore this incident and keep asking :-)


