Global Behavior of Solutions to Two Classes of Second Order Rational Difference Equations

TL;DR

This paper analyzes the global behavior of solutions to two classes of second-order rational difference equations, proving convergence to equilibrium or period-two solutions for all positive parameter choices.

Contribution

It provides a comprehensive proof of convergence properties for two classes of second-order rational difference equations with positive parameters.

Findings

Solutions converge to equilibrium or period-two solutions.

The results hold for all positive parameter values.

The paper establishes a unified approach for these classes.

Abstract

For nonnegative real numbers $\alpha$, $\beta$, $\gamma$, $A$, $B$ and $C$ such that $B+C>0$ and $\alpha+\beta+\gamma >0$, the difference equation \begin{equation*} x_{n+1}=\displaystyle\frac{\alpha +\beta x_{n}+\gamma x_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,... %, \quad x_{-1},x_{0}\in [0,\infty) \end{equation*} has a unique positive equilibrium. A proof is given here for the following statements: \medskip \noindent Theorem 1. {\it For every choice of positive parameters $\alpha$, $\beta$, $\gamma$, $A$, $B$ and $C$, all solutions to the difference equation \begin{equation*} x_{n+1}=\displaystyle\frac{\alpha +\beta x_{n}+\gamma x_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in [0,\infty) \end{equation*} converge to the positive equilibrium or to a prime period-two solution.} \medskip \noindent Theorem 2. {\it For every choice of positive parameters $\alpha$, $\beta$, $\gamma$, $A$, $B$ and $C$, all solutions to the difference equation \begin{equation*} x_{n+1}= \displaystyle\frac{\alpha +\beta x_{n}+\gamma x_{n-1}}{B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in (0,\infty) \end{equation*} converge to the positive equilibrium or to a prime period-two solution.}