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
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
Copyright © 2007 Leonid Gutnik and Stevo Stević. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.