The Behavior of Positive Solutions of a Nonlinear Second-Order Difference Equation

Abstract

This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equation $x_n=f\left(x_{n-2}\right)/g\left(x_{n-1}\right)$, $n\in {\mathbb{N}}_0$, where $f,g\in C\left[\left(0,\infty \right),\left(0,\infty \right)\right]$. It is shown that if $f$ and $g$ are nondecreasing, then for every solution of the equation the subsequences $\left\{x_{2n}\right\}$ and $\left\{x_{2n-1}\right\}$ are eventually monotone. For the case when $f\left(x\right)=\alpha +\beta x$ and $g$ satisfies the conditions $g\left(0\right)=1$, $g$ is nondecreasing, and $x/g\left(x\right)$ is increasing, we prove that every prime periodic solution of the equation has period equal to one or two. We also investigate the global periodicity of the equation, showing that if all solutions of the equation are periodic with period three, then $f\left(x\right)=c_1/x$ and $g\left(x\right)=c_{2}x$, for some positive $c_1$ and $c_2$.