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

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 Kulenovi？ 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, Poincaré 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, attracted the attention of many researchers recently. For example, see the articles [4–24], monographs Koci？ and Ladas [25], Kulenovi？ and Ladas [26], and Camouzis and Ladas [27], and the references cited therein. We believe that behavior of solutions of rational difference equations provides prototypes towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of order greater than one [26, page 1]. Our aim in this paper is to study the second order bilinear rational difference equation where the parameters are positive real numbers, and the initial conditions are nonnegative real numbers. Our concentration is on the periodic character of the positive solutions of (1.2). Indeed, our interest in (1.2) was stimulated by the following conjecture proposed by Camouzis and Ladas in [27, Conjecture？？5.201.2]. Conjecture 1.1. Show that the period-two solution of (1.2) is locally asymptotically stable. It is worth mentioning that the aforementioned conjecture appeared previously in Conjecture in the Kulenovi？ and Ladas monograph [26]. To this end and using transformations similar to the ones used by [22, 27], let then (1.2) reduces to where are positive real numbers, and the initial conditions are nonnegative real numbers. That being said, the remainder of this paper is organized as follows. In the next section, we present some