Abstract and Applied Analysis

Volume 2013 (2013), Article ID 456530, 4 pages

http://dx.doi.org/10.1155/2013/456530

Research Article

Dynamics of a Family of Nonlinear Delay Difference Equations

Qiuli He,¹ Taixiang Sun,² and Hongjian Xi³

¹College of Electrical Engineering, Guangxi University, Nanning, Guangxi 530004, China

²College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China

³Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China

Received 16 March 2013; Accepted 18 April 2013

Academic Editor: Zhenkun Huang

Copyright © 2013 Qiuli He 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.

Abstract.

We study the global asymptotic stability of the following difference equation: where and with the initial values are positive, and with . We give sufficient conditions under which the unique positive equilibrium of that equation is globally asymptotically stable.

1. Introduction

In this note, we consider a nonlinear difference equation and deal with the question of whether the unique positive equilibrium of that equation is globally asymptotically stable. Recently, there has been much interest in studying the global attractivity, the boundedness character, and the periodic nature of nonlinear difference equations; for example, see [1–22].

Amleh et al. [1] studied the characteristics of the difference equation:

They confirmed a conjecture in [13] and showed that the unique positive equilibrium of is globally asymptotically stable provided .

Fan et al. [8] investigated the following difference equation:

They showed that the length of finite semicycle of is less than or equal to and gave sufficient conditions under which every positive solution of converges to the unique positive equilibrium.

Kulenović et al. [11] investigated the periodic nature, the boundedness character, and the global asymptotic stability of solutions of the nonautonomous difference equation

where the initial values and is the period-two sequence

Sun and Xi [20] studied the more general equation

where with , the initial values and gave sufficient conditions under which every positive solution of converges to the unique positive equilibrium.

In this paper, we study the global asymptotic stability of the following difference equation:

where and with , the initial values are positive and with and satisfies the following conditions:

 () is decreasing in for any and increasing in for any . () Equation (2) has the unique positive equilibrium, denoted by . () The function has only fixed point in the interval , denoted by . () For any , is nonincreasing in . () If is a solution of the system

then .

2. Main Result

Theorem 1. Assume that hold. Then the unique positive equilibrium of (2) is globally asymptotically stable.

Proof. Let . Since

we have

Claim  1. .

Proof of Claim  1. Assume on the contrary that . Then it follows from (), (), and () that

This is a contradiction. Therefore . Obviously

Claim  1 is proven.

Claim  2. For any is an invariable interval of (2).

Proof of Claim  2. For any , we have from () that

By induction, we may show that for any . Claim  2 is proven.

Let and for any ,

Claim  3. For any , we have

Proof of Claim  3. From Claim  2, we obtain

By induction, we have that for ,

Set

Then

This with () and () implies . Claim  3 is proven.

Claim  4. The equilibrium of (2) is locally stable.

Proof of Claim  4. Let and be the same as Claim  3. For any with , there exists such that

Set . Then for any , we have

In similar fashion,we can show that for any ,

Claim  4 is proven.

Claim  5. is the global attractor of (2).

Proof of Claim  5. Let be a positive solution of (2), and let and be the same as Claim  3. From Claim  2, we have for any . Moreover, we have

In similar fashion, we may show for any . By induction, we obtain

It follows from Claim  3 that . Claim  5 is proven.

From Claims  4 and 5, Theorem 1 follows.

3. Applications

In this section, we will give two applications of Theorem 1.

Example 2. Consider equation

where and with , for any and for any , and the initial conditions with . Write and . If , then the unique positive equilibrium of (20) is globally asymptotically stable.

Proof. Let . It is easy to verify that ), ), and () hold for (20). Note that . Then

has only solution

in the interval , which implies that holds for (20). In addition, let

then

Therefore , which implies that (23) has unique solution

Thus holds for (20). It follows from Theorem 1 that the equilibrium of (20) is globally asymptotically stable.

Example 3. Consider equation

where and with , , for any and for any , and the initial conditions with . Write and . If , then the unique positive equilibrium of (26) is globally asymptotically stable.

Proof. Let . It is easy to verify that )–() hold for (26). In addition, the following equation

has unique solution

which implies that holds for (26). It follows from Theorem 1 that the equilibrium of (26) is globally asymptotically stable.

Acknowledgments

This project is supported by NNSF of China (11261005, 51267001) and NSF of Guangxi (2011GXNSFA018135, 2012GXNSFDA276040).

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