 Research Article
 Open access
 Published:
Asymptotic Behavior of Equilibrium Point for a Family of Rational Difference Equations
Advances in Difference Equations volumeÂ 2010, ArticleÂ number:Â 505906 (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 Ă‡inar (2004), Yang et al. (2005), and Berenhaut et al. (2007). In addition, some numerical simulations are also shown to support our analytic results.
1. Introduction
Difference equations appear naturally as discrete analogues and in the numerical solutions of differential and delay differential equations, and they have applications in biology, ecology, physics, and so forth [1]. The study of properties of nonlinear difference equations has been an area of intense interest in recent years. There has been a lot of work concerning the globally asymptotic behavior of solutions of rational difference equations (e.g., see [2â€“5]). In particular, Ă‡inar [6] studied the properties of positive solution to
Yang et al. [7] investigated the qualitative behavior of the recursive sequence
More recently, Berenhaut et al. [8] generalized the result reported in [7] to
For more similar work, one can refer to [9â€“14] and references therein.
The main theorem in this paper is motivated by the above studies. The essential problem we consider in this paper is the asymptotic behavior of the solutions for
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 KociÄ‡ and Ladas [2], and KulenoviÄ‡ 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.
Let be some interval of real numbers and
be a continuously differentiable function. Then, for every set of initial conditions ,
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.
Let be an equilibrium point of (2.2).

(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.
Assume that and . Then,
is a sufficient condition for the local stability of the difference equation
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 .
Let be a multivariate continuous function defined by
If we have
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.
Considering the linearized equation of (1.4) with respect to equilibrium point ,
By Lemma 2.5, (1.4) is stable if the following inequality holds
which implies our claim.
When , note that the solution of (1.4) is , hence, which implies that is a dispersed sequence.
Theorem 3.2.
Let defined by (2.2) be a continuous function satisfying the mixed monotone property. If there exists two real numbers satisfying
such that
then there exists satisfying
Moreover, if , then (2.2) has a unique equilibrium point , and every solution of (2.2) converges to .
Proof.
Let and be a couple of initial iteration data, then we construct two sequences and in the form
Note that mixed monotone property of and initial assumption, the sequences and thus possess
It guarantees one can choose a new sequence satisfying for .
Denote
then
By the continuity of , we have
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 .
Let
by view of the assumption we have
From (1.4) and Theorem 3.2, there exists satisfying
Taking the difference
we deduce that
which implies
where
By view of and initial conditions , therefore we have
thus, we have
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
In this section, we give some numerical simulations supporting our theoretical analysis via the software package Matlab 7.1. As examples, we consider
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 .
By employing the software package Matlab 7.1, we can solve the numerical solutions of (4.1), (4.2), and (4.3) which are shown, respectively, in Figures 1, 2, and 3. More precisely, Figure 1 shows the asymptotic behavior of the solution to (4.1) with initial data and , Figure 2 shows the asymptotic behavior of the solution to (4.2) with initial data and , and Figure 3 shows the asymptotic behavior of the solution to (4.3) with initial data and
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.
References
Li WT, Sun HR: Global attractivity in a rational recursive sequence. Dynamic Systems and Applications 2002,11(3):339345.
KociÄ‡ 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.
KulenoviÄ‡ 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.
ElOwaidy HM, Ahmed AM, Mousa MS:On the recursive sequences . Applied Mathematics and Computation 2003,145(23):747753. 10.1016/S00963003(03)002716
Xianyi XY, Zhu DM: Global asymptotic stability in a rational equation. Journal of Difference Equations and Applications 2003,9(9):833839. 10.1080/1023619031000071303
Ă‡inar C:On the positive solutions of the difference equation . Applied Mathematics and Computation 2004,150(1):2124. 10.1016/S00963003(03)001942
Yang X, Su W, Chen B, Megson GM, Evans DJ:On the recursive sequence . Applied Mathematics and Computation 2005,162(3):14851497. 10.1016/j.amc.2004.03.023
Berenhaut KS, Foley JD, SteviÄ‡ S:The global attractivity of the rational difference equation . Applied Mathematics Letters 2007,20(1):5458. 10.1016/j.aml.2006.02.022
Berenhaut KS, SteviÄ‡ S:The difference equation has solutions converging to zero. Journal of Mathematical Analysis and Applications 2007,326(2):14661471. 10.1016/j.jmaa.2006.02.088
Memarbashi R: Sufficient conditions for the exponential stability of nonautonomous difference equations. Applied Mathematics Letters 2008,21(3):232235. 10.1016/j.aml.2007.03.014
Aloqeili M: Global stability of a rational symmetric difference equation. Applied Mathematics and Computation 2009,215(3):950953. 10.1016/j.amc.2009.06.026
Aprahamian M, Souroujon D, Tersian S: Decreasing and fast solutions for a secondorder difference equation related to FisherKolmogorov's equation. Journal of Mathematical Analysis and Applications 2010,363(1):97110. 10.1016/j.jmaa.2009.08.009
Wang Cy, Gong F, Wang S, Li Lr, Shi Qh: Asymptotic behavior of equilibrium point for a class of nonlinear difference equation. Advances in Difference Equations 2009, 2009:8.
Wang Cy, Wang S, Wang Zw, Gong F, Wang Rf: Asymptotic stability for a class of nonlinear difference equations. Discrete Dynamics in Nature and Society 2010, 2010:10.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Wang, Cy., Shi, Qh. & Wang, S. Asymptotic Behavior of Equilibrium Point for a Family of Rational Difference Equations. Adv Differ Equ 2010, 505906 (2010). https://doi.org/10.1155/2010/505906
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/505906