I have been stuck with this difference equation for a while and have had no success getting a proper discussion on the asymptotic behavior of it or its stability conditions.
Suppose that $(Y_n)_{n\ge 0}$ is a non-negative and decreasing known sequence (a perturbation to the system). Now, for a given $x_0 > Y_0$ and $x_1 > Y_1$ the difference equation is defined uniquely (if defined) as
$$ (\frac{1}{x_{n+1}-x_{n+2}} - \frac{1}{x_{n+1}-Y_{n+1}}) (\frac{1}{x_{n}-x_{n+1}} - \frac{1}{x_{n}-Y_{n}}) = A (\frac{1}{x_{n}-x_{n+1}} - \frac{1}{x_{n+1}-Y_{n}})^2 $$
where $A>1$ is a constant. I've written it this way (not in the familiar form of $x_{n+2}=f_n(x_{n+1},x_{n})$ to show some symmetry in its nature in case it can be useful.
The most basic goal might be to find some conditions on $x_1$ and $x_2$ such that the sequence above is always defined (and preferrably positive, but that one seems to be more straightforward). Somewhat related, how to choose $x_1$ and $x_2$ properly so that $x_{n+1} > \frac{x_n + Y_n}{2}$. This allows us to use standard theorems (from text book sources) on stability and asymptotic behavior because most of those methods require a characterization $x_{n+2}=f_n(x_{n+1},x_{n})$ in which $f_n$ is well-behaved (e.g. is continuous so has no singularities).
In an unperturbed version $Y_n=0$, a bounded solution is given by $x_n=b^n x_0 $ where $b$ is the unique positive solution in the unit circle for the equation $\frac{x^3}{A}-4x^2+4x-1=0$. But nothing I could get for the perturbed version. Any reference is also appreciated.