Let $\{x_n\} \subset \mathbb R$ be defined recursively as
$$ x_{n+1} = x_{n} - \frac{ \rho^{4n}}{x_n (A \sum_{k=0}^{n} x_k - \sum_{k=0}^{n-1} \frac{\rho^{2k}}{x_k} + Bx_0)^2} $$
and $x_0 > 0 $, for some $0<\rho<1$. My conjecture is that, for any choice of constants $A,B$, there exists a unique $x_0 >0$ for which $x_n > 0$ and $\lim _{n\rightarrow\infty }x_{n+1} = 0$. Any hint on where to start?
An observation is that for any $n$, $x_n$ is increasing in $x_0$. So probably there should be an $\bar x_0 > 0$ above which all the trajectories diverge and below which all the trajectories enter the negative zone. Does that help?