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

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.

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 [2–5]). In particular, Çinar [6] studied the properties of positive solution to

(1.1)

Yang et al. [7] investigated the qualitative behavior of the recursive sequence

(1.2)

More recently, Berenhaut et al. [8] generalized the result reported in [7] to

(1.3)

For more similar work, one can refer to [9–14] 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

(1.4)

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.

Let be some interval of real numbers and

(2.1)

be a continuously differentiable function. Then, for every set of initial conditions ,

(2.2)

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,

(2.3)

is a sufficient condition for the local stability of the difference equation

(2.4)

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

(3.1)

If we have

(3.2)

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 ,

(3.3)

By Lemma 2.5, (1.4) is stable if the following inequality holds

(3.4)

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

(3.5)

such that

(3.6)

then there exists satisfying

(3.7)

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

(3.8)

Note that mixed monotone property of and initial assumption, the sequences and thus possess

(3.9)

It guarantees one can choose a new sequence satisfying for .

Denote

(3.10)

then

(3.11)

By the continuity of , we have

(3.12)

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

(3.13)

by view of the assumption we have

(3.14)

From (1.4) and Theorem 3.2, there exists satisfying

(3.15)

Taking the difference

(3.16)

we deduce that

(3.17)

which implies

(3.18)

where

By view of and initial conditions , therefore we have

(3.19)

thus, we have

(3.20)

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

(4.1)

(4.2)

(4.3)

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.