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

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. Copyright © 2007 L. Gutnik and S. 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. 1