Global Asymptotic Stability of a Family of Nonlinear Difference Equations

In this note, we consider global asymptotic stability of the following nonlinear difference equation , where , , , , and . Our result generalizes the corresponding results in the recent literature and simultaneously conforms to a conjecture in the work by Berenhaut et al. (2007). 1. Introduction The study of dynamical properties of nonlinear difference equations has been an area of intense interest in recent years (e.g., see [1–13]). In [4], by analysis of semicycle structure, the authors discussed the global asymptotic stability of rational difference equation where the initial values . Li [5, 6] investigated the qualitative behavior of the rational difference equations with ？？and？？ via analysis of semicycle structure and verified that every solution of (2) converges to equilibrium 1. By using the transformation method, Berenhaut et al. [1] studied the behavior of positive solutions to the rational difference equation with and ？and proved that every solution of (3) converges to the unique equilibrium 1. Based on the above facts, Berenhaut et al. [1] put forward the following two conjectures. Conjecture 1. Suppose that and that satisfies with . Then, the sequence converges to the unique equilibrium 1. Conjecture 2. Suppose that is odd and , ？and define . If satisfies with , where then the sequence converges to the unique equilibrium 1. Recently, by method used in [4–6], the authors of [12] studied the global asymptotic stability of the following nonlinear difference equation. where the parameter , ？and the initial values . Motivated by the above studies, in this note, we propose and consider the following nonlinear difference equation. where , , , , and . It is noticed that, letting ,？ , ？ ,？ and ？ , (9) reduces to (1); letting ,？？ ,？？ ,？？and？？ and , ？？ , ？？ , ？？ ,？ ？and？？ , (9) reduces to (2); letting ,？？ ,？ ,？？and？？ , (9) reduces to (3); letting , , ？？ ,？？and？？ , (9) reduces to (4); letting ,？？ ,？？ ,？？ ,？？ , , and , (9) reduces to (7); letting be odd, , and , (9) reduces to (5). Clearly, (5) is a special example of (9). In 2007, Berenhaut and Stevi？ [2] had proved Conjecture 1. In this paper, by making full use of analytical techniques, we mainly prove that the unique positive equilibrium point of (9) is globally asymptotically stable. It is clear that our result generalizes the corresponding works in [1, 2, 4–9, 12] and simultaneously conforms to Conjecture 2. 2. Existence of a Unique Positive Equilibrium In this section, we mainly show the existence of a unique positive equilibrium of (9). Theorem 3. In (9) there exists a unique positive equilibrium