Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 750852, 4 pages

http://dx.doi.org/10.1155/2013/750852

Research Article

Global Asymptotic Stability of a Family of Nonlinear Difference Equations

Maoxin Liao

School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, China

Received 17 June 2013; Accepted 9 November 2013

Academic Editor: Cengiz Çinar

Copyright © 2013 Maoxin Liao. 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.

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 Stević [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 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.

In order to prove Theorem 4, we introduce the following lemma by Kruse and Nesemann [3] and make full use of analytical techniques.

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.

Acknowledgments

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.

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