Abstract and Applied Analysis

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

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

## Dynamics of a Family of Nonlinear Delay Difference Equations

^{1}College of Electrical Engineering, Guangxi University, Nanning, Guangxi 530004, China^{2}College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China^{3}Department 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).

#### References

- A. M. Amleh, D. A. Georgiou, E. A. Grove, and G. Ladas, “On the recursive sequence ${x}_{n+1}=\alpha +{x}_{n-1}$/${x}_{n}$,”
*Journal of Mathematical Analysis and Applications*, vol. 233, no. 2, pp. 790–798, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. S. Berenhaut and R. T. Guy, “Periodicity and boundedness for the integer solutions to a minimum-delay difference equation,”
*Journal of Difference Equations and Applications*, vol. 16, no. 8, pp. 895–916, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. S. Berenhaut, R. T. Guy, and C. L. Barrett, “Global asymptotic stability for minimum-delay difference equations,”
*Journal of Difference Equations and Applications*, vol. 17, no. 11, pp. 1581–1590, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. S. Berenhaut and A. H. Jones, “Asymptotic behaviour of solutions to difference equations involving ratios of elementary symmetric polynomials,”
*Journal of Difference Equations and Applications*, vol. 18, no. 6, pp. 963–972, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Camouzis and G. Ladas, “When does local asymptotic stability imply global attractivity in rational equations?”
*Journal of Difference Equations and Applications*, vol. 12, no. 8, pp. 863–885, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Devault, V. L. Kocic, and D. Stutson, “Global behavior of solutions of the nonlinear difference equation ${x}_{n+1}={p}_{n}+{x}_{n-1}/{x}_{n}$,”
*Journal of Difference Equations and Applications*, vol. 11, no. 8, pp. 707–719, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. El-Metwally, “Qualitative properties of some higher order difference equations,”
*Computers & Mathematics with Applications*, vol. 58, no. 4, pp. 686–692, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Fan, L. Wang, and W. Li, “Global behavior of a higher order nonlinear difference equation,”
*Journal of Mathematical Analysis and Applications*, vol. 299, no. 1, pp. 113–126, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Gelişken, C. Çinar, and A. S. Kurbanli, “On the asymptotic behavior and periodic nature of a difference equation with maximum,”
*Computers & Mathematics with Applications*, vol. 59, no. 2, pp. 898–902, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. D. Iričanin, “Global stability of some classes of higher-order nonlinear difference equations,”
*Applied Mathematics and Computation*, vol. 216, no. 4, pp. 1325–1328, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. R. S. Kulenović, G. Ladas, and C. B. Overdeep, “On the dynamics of ${x}_{n+1}={p}_{n}+{x}_{n-1}/{x}_{n}$ with a period-two coefficient,”
*Journal of Difference Equations and Applications*, vol. 10, no. 10, pp. 905–914, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. S. Kurbanlı, C. Çinar, and G. Yalçinkaya, “On the behavior of positive solutions of the system of rational difference equations ${x}_{n+1}={x}_{n-1}$/$({y}_{n}{x}_{n-1}+1)$, ${y}_{n+1}={y}_{n-1}$/$({x}_{n}{y}_{n-1}+1)$,”
*Mathematical and Computer Modelling*, vol. 53, no. 5-6, pp. 1261–1267, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Ladas, “Open problems and conjecture,”
*Journal of Differential Equations and Applications*, vol. 5, pp. 317–321, 1995. View at Google Scholar - G. Papaschinopoulos, M. A. Radin, and C. J. Schinas, “On the system of two difference equations of exponential form: ${x}_{n+1}=a+b{x}_{n-1}{e}^{-yn}$, ${y}_{n+1}=c+d{y}_{n-1}{e}^{-xn}$,”
*Mathematical and Computer Modelling*, vol. 54, no. 11-12, pp. 2969–2977, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Papaschinopoulos and C. J. Schinas, “On the dynamics of two exponential type systems of difference equations,”
*Computers & Mathematics with Applications*, vol. 64, no. 7, pp. 2326–2334, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - G. Papaschinopoulos, C. J. Schinas, and G. Stefanidou, “On the nonautonomous difference equation ${x}_{n+1}={A}_{n}+{x}_{n-1}^{p}$/${x}_{n}^{q}$,”
*Applied Mathematics and Computation*, vol. 217, no. 12, pp. 5573–5580, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Qian, “Global attractivity of periodic solutions in a higher order difference equation,”
*Applied Mathematics Letters*, vol. 26, pp. 578–583, 2013. View at Publisher · View at Google Scholar - S. Stević, “Boundedness character of a class of difference equations,”
*Nonlinear Analysis. Theory, Methods & Applications A*, vol. 70, no. 2, pp. 839–848, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Periodicity of a class of nonautonomous max-type difference equations,”
*Applied Mathematics and Computation*, vol. 217, no. 23, pp. 9562–9566, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Sun and H. Xi, “Global behavior of the nonlinear difference equation ${x}_{n+1}=f({x}_{n-s},{x}_{n-t})$,”
*Journal of Mathematical Analysis and Applications*, vol. 311, no. 2, pp. 760–765, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Sun, H. Xi, and Q. He, “On boundedness of the difference equation ${x}_{n+1}={p}_{n}+{x}_{n-3s+1}$/${x}_{n-s+1}$
with period-
*k*coefficients,”*Applied Mathematics and Computation*, vol. 217, no. 12, pp. 5994–5997, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - N. Touafek and E. M. Elsayed, “On the solutions of systems of rational difference equations,”
*Mathematical and Computer Modelling*, vol. 55, no. 7-8, pp. 1987–1997, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet