Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 969813, 11 pages
http://dx.doi.org/10.1155/2012/969813
Research Article

Local Stability of Period Two Cycles of Second Order Rational Difference Equation

S. Atawna,1 R. Abu-Saris,2 and I. Hashim1

1School of Mathematical Sciences, Universiti Kebangsaan Malaysia, Selangor, 43600 Bangi, Malaysia
2Department of Basic Sciences, King Saud bin Abdulaziz University for Health Sciences, P.O. Box 22490, Riyadh 11426, Saudi Arabia

Received 1 September 2012; Accepted 11 October 2012

Academic Editor: Mustafa Kulenovic

Copyright © 2012 S. Atawna et al. 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 consider the second order rational difference equation   n = 0,1,2,…, where the parameters are positive real numbers, and the initial conditions are nonnegative real numbers. We give a necessary and sufficient condition for the equation to have a prime period-two solution. We show that the period-two solution of the equation is locally asymptotically stable. In particular, we solve Conjecture 5.201.2 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.3 in Kulenović and Ladas monograph (2002).