Solutions of the Difference Equation x n + 1 = x n x n − 1 − 1

Abstract

Our goal in this paper is to investigate the long-term behavior of solutions of the following difference equation: $x_{n+1}=x_n{x}_{n-1}-1$, $n=0,1,2,\dots ,$ where the initial conditions $x_{-1}$ and $x_0$ are real numbers. We examine the boundedness of solutions, periodicity of solutions, and existence of unbounded solutions and how these behaviors depend on initial conditions.