Discrete Dynamics in Nature and Society 
Volume 2007 (2007), Article ID 27562, 14 pages
doi:10.1155/2007/27562
Research Article

On the Behaviour of the Solutions of a Second-Order Difference Equation

Leonid Gutnik1 and Stevo Stević2

 1Moscow State Institute of Electronics and Mathematics, Moscow, Russia
2Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 35/I, Beograd 11000, Serbia
Received 7 December 2006; Accepted 18 February 2007

Abstract

We study the difference equation xn+1=α−xn/xn−1, n∈ℕ0, where α∈ℝ and where x−1 and x0 are so chosen that the corresponding solution (xn) of the equation is defined for every n∈ℕ. We prove that when α=3 the equilibrium x¯=2 of the equation is not stable, which corrects a result due to X. X. Yan, W. T. Li, and Z. Zhao. For the case α=1, we show that there is a strictly monotone solution of the equation, and we also find its asymptotics. An explicit formula for the solutions of the equation are given for the case α=0.