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 $\ell_n = x_n ( σ_{n,1} x_{n+1} + σ_{n,0} x_n + σ_{n,-1} x_{n-1} ) + κ_n x_n$, with given initial data $x_0 \in \mathbb{R}$, $x_1 \in \mathbb{R}^+$ where $(\ell_n)_{n\in\mathbb{N}} \in \mathbb{R}^+$, $(σ_{n,0})_{n\in\mathbb{N}} \in \mathbb{R}^+$ and $(κ_n)_{n\in\mathbb{N}} \in \mathbb{R}$ and the left and right $σ$-coefficients satisfy either $(σ_{n,1})_{n\in\mathbb{N}} \in \mathbb{R}^+$ and $(σ_{n,-1})_{n\in \mathbb{N}} \in \mathbb{R}^+$ or $(σ_{n,1})_{n\in\mathbb{N}} \in \mathbb{R}^+_0$ and $(σ_{n,-1})_{n\in\mathbb{N}} \in \mathbb{R}^+_0$. Depending on one's standpoint, such equations originate either from orthogonal polynomials associated with certain Shohat-Freud-type exponential weight functions or from Painlevé's discrete equation $\#1$.

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