A family of nonlinear difference equations: existence, uniqueness, and asymptotic behavior of positive solutions

We study solutions $(x_n)_{n \in \mathbb{N}}$ of nonhomogeneous nonlinear second order difference equations of the type $\ell_n = x_n ( \sigma_{n,1} x_{n+1} + \sigma_{n,0} x_n + \sigma_{n,-1} x_{n-1} ) + \kappa_n x_n$, with given initial data $x_0 \in \mathbb{R}$, $x_1 \in \mathbb{R}^+$ where $(\ell_n)_{n\in\mathbb{N}} \in \mathbb{R}^+$, $(\sigma_{n,0})_{n\in\mathbb{N}} \in \mathbb{R}^+$ and $(\kappa_n)_{n\in\mathbb{N}} \in \mathbb{R}$ and the left and right $\sigma$-coefficients satisfy either $(\sigma_{n,1})_{n\in\mathbb{N}} \in \mathbb{R}^+$ and $(\sigma_{n,-1})_{n\in \mathbb{N}} \in \mathbb{R}^+$ or $(\sigma_{n,1})_{n\in\mathbb{N}} \in \mathbb{R}^+_0$ and $(\sigma_{n,-1})_{n\in\mathbb{N}} \in \mathbb{R}^+_0$. Depending on one's standpoint, such equations originate either from orthogonal polynomials associated with certain Shohat-Freud-type exponential weight functions or from Painlev\'e's discrete equation $\#1$.