Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2012, Article ID 969813, 11 pages doi: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. Hashim 1 1 School of Mathematical Sciences, Universiti Kebangsaan Malaysia, Selangor, 43600 Bangi, Malaysia 2 Department of Basic Sciences, King Saud bin Abdulaziz University for Health Sciences, P.O. Box 22490, Riyadh 11426, Saudi Arabia Correspondence should be addressed to S. Atawna, s atawna@yahoo.com Received 1 September 2012; Accepted 11 October 2012 Academic Editor: Mustafa Kulenovic Copyright q 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. We consider the second order rational difference equation x n1  α  βx n  γx n-1 /A  Bx n  Cx n-1 ,n  0, 1, 2,..., where the parameters α,β,γ,A,B,C are positive real numbers, and the initial conditions x -1 ,x 0 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 Kulenovi´ c and Ladas monograph 2002. 1. Introduction Difference equations proved to be effective in modelling and analysing discrete dynamical systems that arise in signal processing, populations dynamics, health sciences, economics, and so forth. They also arise naturally in studying iterative numerical schemes. Furthermore, they appear when solving differential equations using series solution methods or studying them qualitatively using, for example, Poincar´ e maps. For an introduction to the general theory of difference equations, we refer the readers to Agarwal 1, Elaydi 2, and Kelley and Peterson 3. Rational difference equations; particularly bilinear ones, that is, x n1  α  ∑ k i0 α i x n-i β  ∑ ℓ j 0 β j x n-j , n  0, 1, 2, 3 ... 1.1