Global Behavior of the Difference Equation x n + 1 = x n - 1 g ( x n )

Abstract

We consider the following difference equation $x_{n+1}=x_{n-1}g(x_n)$, $n=\mathrm{0,1},\dots ,$ where initial values $x_{-1},x_0\in [0,+\infty )$ and $g:[0,+\infty )\to (\mathrm{0,1}]$ is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges to $a,0,a,0,\dots ,$ or $0,a,0,a,\dots$ for some $a\in [0,+\infty )$. (2) Assume $a\in (0,+\infty )$. Then the set of initial conditions $(x_{-1},x_0)\in (0,+\infty )\times (0,+\infty )$ such that the positive solutions of this equation converge to $a,0,a,0,\dots ,$ or $0,a,0,a,\dots$ is a unique strictly increasing continuous function or an empty set.