- Research Article
- Open Access

# Asymptotic Behavior of Equilibrium Point for a Family of Rational Difference Equations

- Chang-you Wang
^{1, 2, 3}Email author, - Qi-hong Shi
^{4}and - Shu Wang
^{3}

**2010**:505906

https://doi.org/10.1155/2010/505906

© Chang-you Wang et al. 2010

**Received:**7 August 2010**Accepted:**19 October 2010**Published:**20 October 2010

## Abstract

This paper is concerned with the following nonlinear difference equation
where the initial data
,
,
are nonnegative integers, and
, *B*, *C*, and *D* are arbitrary positive real numbers. We give sufficient conditions under which the unique equilibrium
of this equation is globally asymptotically stable, which extends and includes corresponding results obtained in the work of Çinar (2004), Yang et al. (2005), and Berenhaut et al. (2007). In addition, some numerical simulations are also shown to support our analytic results.

## Keywords

- Asymptotic Behavior
- Equilibrium Point
- Difference Equation
- Asymptotic Stability
- Nonlinear Differential Equation

## 1. Introduction

For more similar work, one can refer to [9–14] and references therein.

with initial data are nonnegative integers, and and are arbitrary positive real numbers. In addition, some numerical simulations of the behavior are shown to illustrate our analytic results.

This paper proceeds as follows. In Section 2, we introduce some definitions and preliminary results. The main results and their proofs are given in Section 3. Finally, some numerical simulations are shown to support theoretical analysis.

## 2. Preliminaries and Notations

In this section, we prepare some materials used throughout this paper, namely, notations, the basic definitions, and preliminary results. We refer to the monographs of Kocić and Ladas [2], and Kulenović and Ladas [3].

Let and be two nonnegative integers such that . We usually write a vector with components into , where denotes a vector with -components of .

Lemma 2.1.

has a unique solution .

Definition 2.2.

Function is called mixed monotone in subset of if is monotone nondecreasing in each component of and is monotone nonincreasing in every component of for .

Definition 2.3.

If there exists a point such that , is called an equilibrium point of (2.2).

Definition 2.4.

- (1)
The equilibrium of (2.2) is (locally) stable if for every , there exists such that for any initial data satisfying holds for all .

- (2)
The equilibrium of (2.2) is a local attractor if there exists such that for any data satisfying .

- (3)
The equilibrium of (2.2) is locally asymptotically stable if it is stable and is a local attractor.

- (4)
The equilibrium of (2.2) is a global attractor if for all , holds.

- (5)
is globally asymptotically stable if it is stable and is a global attractor.

- (6)
is unstable if it is not locally stable.

Lemma 2.5.

## 3. The Main Results and Their Proofs

In this section, we investigate the globally asymptotic stability of the equilibrium point of (1.4).

It is obvious that is a unique equilibrium point of (1.4) provided either or .

By constructing (1.4) and applying Lemma 2.5, we have the following affirmation.

Theorem 3.1.

If , and with , then the unique equilibrium point of (1.4) is locally stable. If is unstable.

Proof.

which implies our claim.

When , note that the solution of (1.4) is , hence , which implies that is a dispersed sequence.

Theorem 3.2.

Moreover, if , then (2.2) has a unique equilibrium point , and every solution of (2.2) converges to .

Proof.

It guarantees one can choose a new sequence satisfying for .

Moreover, if , then , and then the proof is complete.

Theorem 3.3.

If , , and , then the unique equilibrium point of (1.4) is global attractor for any initial conditions .

Proof.

For any nonnegative initial data , it is obvious that the function defined by (3.1) is nondecreasing in and nonincreasing in .

where

By Theorem 3.2, then every solution of (1.4) converges to the unique equilibrium point .

Theorem 3.4.

If , and , then the unique equilibrium point of (1.4) is globally asymptotically stable for any initial conditions .

Proof.

The result holds from Theorems 3.1 and 3.3.

Theorem 3.5.

If , and , then the unique equilibrium point of (1.4) is global attractor for any initial conditions

Proof.

The proof of this Theorem is similar to that of Theorem 3.3. We omit the details.

Theorem 3.6.

If , and , then the unique equilibrium point of (1.4) is globally asymptotically stable for any initial conditions .

Proof.

The result holds from Theorems 3.1 and 3.5.

## 4. Numerical Simulation

Let , it is obvious that (4.1) and (4.2) satisfy the conditions of Theorem 3.6 when the initial datas . Equation (4.3) satisfies the conditions of Theorem 3.1 for the initial data .

## 5. Conclusions

This paper presents the use of a variational iteration method for systems of nonlinear difference equations. This technique is a powerful tool for solving various difference equations and can also be applied to other nonlinear differential equations in mathematical physics. The numerical simulations show that this method is an effective and convenient one. The variational iteration method provides an efficient method to handle the nonlinear structure. Computations are performed using the software package Matlab 7.1.

We have dealt with the problem of global asymptotic stability analysis for a class of nonlinear difference equation. The general sufficient conditions have been obtained to ensure the existence, uniqueness, and global asymptotic stability of the equilibrium point for the nonlinear difference equation. These criteria generalize and improve some known results. In particular, some examples are given to show the effectiveness of the obtained results. In addition, the sufficient conditions that we obtained are very simple, which provide flexibility for the application and analysis of nonlinear difference equation.

## Declarations

### Acknowledgments

The authors are grateful to the referee for giving us lots of precious comments. This work is supported by Natural Science Foundation Project of CQ CSTC (Grant no. 2008BB 7415) of China, and National Science Foundation (Grant no. 40801214) of China.

## Authors’ Affiliations

## References

- Li W-T, Sun H-R:
**Global attractivity in a rational recursive sequence.***Dynamic Systems and Applications*2002,**11**(3):339-345.MathSciNetMATHGoogle Scholar - Kocić VL, Ladas G:
*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and Its Applications*.*Volume 256*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xii+228.Google Scholar - Kulenović MRS, Ladas G:
*Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures*. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2002:xii+218.MATHGoogle Scholar - El-Owaidy HM, Ahmed AM, Mousa MS:
**On the recursive sequences**.*Applied Mathematics and Computation*2003,**145**(2-3):747-753. 10.1016/S0096-3003(03)00271-6MathSciNetView ArticleMATHGoogle Scholar - Xianyi X-Y, Zhu D-M:
**Global asymptotic stability in a rational equation.***Journal of Difference Equations and Applications*2003,**9**(9):833-839. 10.1080/1023619031000071303MathSciNetView ArticleMATHGoogle Scholar - Çinar C:
**On the positive solutions of the difference equation**.*Applied Mathematics and Computation*2004,**150**(1):21-24. 10.1016/S0096-3003(03)00194-2MathSciNetView ArticleMATHGoogle Scholar - Yang X, Su W, Chen B, Megson GM, Evans DJ:
**On the recursive sequence**.*Applied Mathematics and Computation*2005,**162**(3):1485-1497. 10.1016/j.amc.2004.03.023MathSciNetView ArticleMATHGoogle Scholar - Berenhaut KS, Foley JD, Stević S:
**The global attractivity of the rational difference equation**.*Applied Mathematics Letters*2007,**20**(1):54-58. 10.1016/j.aml.2006.02.022MathSciNetView ArticleMATHGoogle Scholar - Berenhaut KS, Stević S:
**The difference equation****has solutions converging to zero.***Journal of Mathematical Analysis and Applications*2007,**326**(2):1466-1471. 10.1016/j.jmaa.2006.02.088MathSciNetView ArticleMATHGoogle Scholar - Memarbashi R:
**Sufficient conditions for the exponential stability of nonautonomous difference equations.***Applied Mathematics Letters*2008,**21**(3):232-235. 10.1016/j.aml.2007.03.014MathSciNetView ArticleMATHGoogle Scholar - Aloqeili M:
**Global stability of a rational symmetric difference equation.***Applied Mathematics and Computation*2009,**215**(3):950-953. 10.1016/j.amc.2009.06.026MathSciNetView ArticleMATHGoogle Scholar - Aprahamian M, Souroujon D, Tersian S:
**Decreasing and fast solutions for a second-order difference equation related to Fisher-Kolmogorov's equation.***Journal of Mathematical Analysis and Applications*2010,**363**(1):97-110. 10.1016/j.jmaa.2009.08.009MathSciNetView ArticleMATHGoogle Scholar - Wang C-y, Gong F, Wang S, Li L-r, Shi Q-h:
**Asymptotic behavior of equilibrium point for a class of nonlinear difference equation.***Advances in Difference Equations*2009,**2009:**-8.Google Scholar - Wang C-y, Wang S, Wang Z-w, Gong F, Wang R-f:
**Asymptotic stability for a class of nonlinear difference equations.***Discrete Dynamics in Nature and Society*2010,**2010:**-10.Google Scholar

## Copyright

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