Asymptotic Behavior of Equilibrium Point for a Family of Rational Difference Equations
© Chang-you Wang et al. 2010
Received: 7 August 2010
Accepted: 19 October 2010
Published: 20 October 2010
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.
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 , and Kulenović and Ladas .
3. The Main Results and Their Proofs
In this section, we investigate the globally asymptotic stability of the equilibrium point of (1.4).
By constructing (1.4) and applying Lemma 2.5, we have the following affirmation.
which implies our claim.
The result holds from Theorems 3.1 and 3.3.
The proof of this Theorem is similar to that of Theorem 3.3. We omit the details.
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 .
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.
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.
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