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).
In , by analysis of semicycle structure, the authors discussed the global asymptotic stability of rational difference equation where the initial values .
By using the transformation method, Berenhaut et al.  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.  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.
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 Stević  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 point .
Proof. A positive equilibrium point of (9) satisfies the next equation: from which we may get that is, From the above equation, we can get One can see that for any and , (i)If , from (13) and (14), we can get that (9) has a unique positive equilibrium .(ii)If or and , we have Further, we have (iii)If or and , we have It is clear that (9) has a unique positive equilibrium . The proof is complete.
3. Global Asymptotic Stability for the Unique Positive Equilibrium Point
In this section, we give our main result.
Theorem 4. The unique positive equilibrium point of (9) is globally asymptotically stable.
Lemma 5. Consider the difference equation where and is a continuous function with some unique equilibrium . Suppose that there is a such that for all solutions of (18) where equality holds if and only if . Then is globally asymptotically stable.
Proof of Theorem 4. Let be any solution of (9). We have
It follows from (20) that Clearly, from (21), we have From (22), we have If , it is clear that If and , we have and , so that Similarly, if and , we have and , so that Hence, for , we always have Further, from (23) and (27), we have Therefore, where equality holds if and only if . By Lemma 5 and (29), with , it follows that the unique positive equilibrium point of (9) is globally asymptotically stable. The proof is complete.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author is grateful to the referees for their careful reading of the paper and many valuable comments and suggestions that greatly improved the presentation of this work. This paper is supported partly by Hunan Provincial Natural Science Foundation of China (no. 13JJ3075), Soft Science Fund of Science and Technology Department of Hunan Province (no. 2011ZK3066), Start-up Fund of University of South China (no. 2011XQD49), and the construct program in USC.
D. Li, P. Li, and X. Li, “Dynamical properties for a class of fourth-order nonlinear difference equations,” Advances in Difference Equations, vol. 2008, Article ID 678402, 13 pages, 2008.View at: Google Scholar