Behavior of solutions of a second order rational difference equation

Raafat Abo-Zeid

In this paper, we solve the difference equation $$x_{n+1}=\frac{lpha}{x_nx_{n-1}-1}, \quad n=0,1,\dots,$$ where $\alpha>0$ and the initial values $x_{-1}$, $x_{0}$ are real numbers. We find some invariant sets and discuss the global behavior of the solutions of that equation. We show that when $\alpha>\frac{2}{3\sqrt3}$, under certain conditions there exist solutions, that are either periodic or converging to periodic solutions. We show also the existence of dense solutions in the real line. Finally, we show that when $\alpha<\frac{2}{3\sqrt3}$, one of the negative equilibrium points attracts all orbits with initials outside a set of Lebesgue measure zero.