# Asymptotic Behavior of Equilibrium Point for a Class of Nonlinear Difference Equation

- Chang-you Wang
^{1, 2, 3}Email author, - Fei Gong
^{2, 4}, - Shu Wang
^{3}, - Lin-rui Li
^{3}and - Qi-hong Shi
^{3}

**2009**:214309

https://doi.org/10.1155/2009/214309

© Chang-you Wang et al. 2009

**Received: **17 February 2009

**Accepted: **17 September 2009

**Published: **9 November 2009

## Abstract

We study the asymptotic behavior of the solutions for the following nonlinear difference equation where the initial conditions are arbitrary nonnegative real numbers, are nonnegative integers, , and are positive constants. Moreover, some numerical simulations to the equation are given to illustrate our results.

## 1. Introduction

Difference equations appear naturally as discrete analogues and in the numerical solutions of differential and delay differential equations having applications in biology, ecology, physics, and so forth [1]. The study of nonlinear difference equations is of paramount importance not only in their own field but in understanding the behavior of their differential counterparts. There has been a lot of work concerning the globally asymptotic behavior of solutions of rational difference equations [2–6]. In particular, Elabbasy et al. [7] investigated the global stability and periodicity of the solution for the following recursive sequence:

In [8] Elabbasy et al. investigated the global stability, boundedness, and the periodicity of solutions of the difference equation:

Yang et al. [9] investigated the global attractivity of equilibrium points and the asymptotic behavior of the solutions of the recursive sequence:

The purpose of this paper is to investigate the global attractivity of the equilibrium point, and the asymptotic behavior of the solutions of the following difference equation

where the initial conditions are arbitrary nonnegative real numbers, , are nonnegative integers, and , are positive constants. Moreover, some numerical simulations to the equation are given to illustrate our results.

This paper is arranged as follows. In Section 2, we give some definitions and preliminary results. The main results and their proofs are given in Section 3. Finally, some numerical simulations are given to illustrate our theoretical analysis.

## 2. Some Preliminary Results

To prove the main results in this paper we first give some definitions and preliminary results [10, 11] which are basically used throughout this paper.

Lemma 2.1.

Definition 2.2.

That is, for is a solution of (2.2), or equivalently, is a fixed point of .

Definition 2.3.

Let be two nonnegative integers such that . Splitting into , where denotes a vector with -components of , we say that the function possesses a mixed monotone property in subsets of if is monotone nondecreasing in each component of and is monotone nonincreasing in each component of for . In particular, if , then it is said to be monotone nondecreasing in .

Definition 2.4.

Let be an equilibrium point of (2.2).

(i) is stable if, for every , there exists such that for any initial conditions with , hold for

(ii) is a local attractor if there exists such that holds for any initial conditions with .

(iii) is locally asymptotically stable if it is stable and is a local attractor.

(iv) is a global attractor if holds for any initial conditions , .

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

(vi) 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).

Let be the equilibrium points of (1.4), then we have

Moreover, we have that

Thus, the linearized equation of (1.4) about is

Theorem 3.1.

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

Proof.

from which the result follows.

Theorem 3.2.

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

Proof.

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

Theorem 3.3.

then the equilibrium point of (1.4) is global attractor when .

Proof.

It follows by Theorem 3.2 that the equilibrium point of (1.4) is global attractor. The proof is therefore complete.

## 4. Numerical Simulations

## Declarations

### Acknowledgment

The authors are grateful to the referees for their comments. This work is supported by the Science and Technology Project of Chongqing Municipal Education Commission (Grant no. KJ 080511) of China, Natural Science Foundation Project of CQ CSTC (Grant no. 2008BB 7415) of China, the NSFC (Grant no.10471009), and the BSFC (Grant no. 1052001) 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.MATHMathSciNetGoogle 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)00194-2MATHMathSciNetView ArticleGoogle Scholar - Cunningham K, Kulenović MRS, Ladas G, Valicenti SV:
**On the recursive sequence .***Nonlinear Analysis: Theory, Methods & Applications*2001,**47**(7):4603–4614. 10.1016/S0362-546X(01)00573-9MATHMathSciNetView ArticleGoogle Scholar - Li L:
**Stability properties of nonlinear difference equations and conditions for boundedness.***Computers & Mathematics with Applications*1999,**38**(2):29–35. 10.1016/S0898-1221(99)00179-0MATHMathSciNetView ArticleGoogle Scholar - Kulenović MRS, Ladas G, Prokup NR:
**A rational difference equation.***Computers & Mathematics with Applications*2001,**41**(5–6):671–678. 10.1016/S0898-1221(00)00311-4MATHMathSciNetView ArticleGoogle Scholar - Yan X-X, Li WT, Sun H-R:
**Global attractivity in a higher order nonlinear difference equation.***Applied Mathematics E-Notes*2002,**2:**51–58.MATHMathSciNetGoogle Scholar - Elabbasy EM, El-Metwally H, Elsayed EM:
**On the difference equation .***Advances in Difference Equations*2006,**2006:**-10.Google Scholar - Elabbasy EM, El-Metwally H, Elsayed EM:
**Global attractivity and periodic character of a fractional difference equation of order three.***Yokohama Mathematical Journal*2007,**53**(2):89–100.MATHMathSciNetGoogle 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.2004.03.023MATHMathSciNetView ArticleGoogle Scholar - Kulenović MRS, Ladas G:
*Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures*. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2001.View ArticleGoogle 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

## Copyright

This article is published under license to BioMed Central Ltd. 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.