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

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:

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

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.

Letbe some interval of real numbers and let

(2.1)

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

(2.2)

has a unique solution.

Definition 2.2.

A pointis called an equilibrium point of (2.2) if

(2.3)

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

Definition 2.3.

Letbe two nonnegative integers such that. Splittinginto , wheredenotes a vector with-components of, we say that the function possesses a mixed monotone property in subsetsofifis monotone nondecreasing in each component ofand is monotone nonincreasing in each component offor. In particular, if, then it is said to be monotone nondecreasing in.

Definition 2.4.

Letbe an equilibrium point of (2.2).

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

(ii)is a local attractor if there existssuch thatholds for any initial conditions with.

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

(iv)is a global attractor ifholds 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.

Assume thatand. Then

(2.4)

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

(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 a function defined by

(3.1)

If , then it follows that

(3.2)

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

(3.3)

Moreover, we have that

(3.4)

Thus, the linearized equation of (1.4) aboutis

(3.5)

Theorem 3.1.

Ifand, then the equilibrium pointof (1.4) is locally stable.

Proof.

It is obvious by Lemma 2.5 that (3.5) is locally stable if

(3.6)

that is

(3.7)

from which the result follows.

Theorem 3.2.

Letbe an interval of real numbers and assume thatis a continuous function satisfying the mixed monotone property. If there exists

(3.8)

such that

(3.9)

then there existsatisfying

(3.10)

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

Proof.

Usingandas a couple of initial iteration conditions we construct two sequencesand () from the equation

(3.11)

It is obvious from the mixed monotone property ofthat the sequencesandpossess the following monotone property:

(3.12)

where , and

(3.13)

Set

(3.14)

then

(3.15)

By the continuity of we have

(3.16)

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

Theorem 3.3.

If there exists

(3.17)

such that

(3.18)

then the equilibrium pointof (1.4) is global attractor when.

Proof.

We can easily see that the functiondefined by (3.1) is nondecreasing inand nonincreasing in. Then from (1.4) and Theorem 3.2, there exist satisfying

(3.19)

thus

(3.20)

In view of , we have

(3.21)

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

4. Numerical Simulations

In this section, we give numerical simulations supporting our theoretical analysis. As examples, we consider the following difference equations:

(4.1)

(4.2)

Let. It is obvious that (4.1) and (4.2) satisfy the conditions of Theorem 3.3 when the initial conditions are.

Figure 1 shows the numerical solution of (4.1) with and the relations thatwhen

Figure 1

Chart of (4. 1) with .

Figure 2 shows the numerical solution of (4.2) withand the relations thatwhen

Figure 2

Chart of (4. 2) with .