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

Abstract

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 Painlevé's discrete equation $\#1$.

Titled has changed (previously: Nonlinear difference equations, I: existence, uniqueness, and asymptotic behavior of positive solutions)

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• Unique positive solution for an alternative discrete Painlevé I equation
• A Generalised Sextic Freud Weight
• On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight
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• Unique special solution for discrete Painlevé II
• Dynamics of Non-polar Solutions to the Discrete Painlevé I Equation