The Stability of High Order Max-Type Difference Equation

Han Cai-hong^*, Li Lue, Tan Xue

School of Mathematics and Statistics, Guangxi Normal University, Guilin, China

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(Han Cai-hong)

^*Corresponding author

To cite this article:

Han Cai-hong, Li Lue, Tan Xue. The Stability of High Order Max-Type Difference Equation. Applied and Computational Mathematics. Vol. 5, No. 2, 2016, pp. 51-55. doi: 10.11648/j.acm.20160502.13

Received: February 5, 2016; Accepted: March 28, 2016; Published: April 7, 2016

Abstract: In this paper, we investigate the stability of following max-type difference equation , where , with ， ,  and , the initial values are positive. By constructing a system of equations and binary function, we show the equation has a unique positive equilibrium solution, and the positive equilibrium solution is globally asymptotically stable. The conclusion of this paper extends and supplements the existing results.

Keywords: Difference Equations, Positive Solution, Convergence, Globally Stable

1. Introduction

In mathematics, recursive relation, which is difference equation, is a kind of recursion formula to define a sequence: the sequence of each item is defined as a function. Difference equation is the discretization of differential equations. The difference system is described the mathematical model of discrete system, it is an important branch of dynamical system, the application of its theory is rapidly broadening to various fields, such as economics, ecology, physics, engineering, control theory, computer science and so on (see [1-4]). The stability and global behavior is one of the hot spots in researches about difference equation model, the conclusion has a certain guiding role to production practices.

In recent years, more and more researches on the dynamic behaviors of higher order nonlinear difference equations have been studied (see [5-19]). One of the classes of such difference equations are max-type difference equations (see [10-19]).

In [16], Amleh studied the nonlinear difference equation , showed that the unique positive equilibrium solution  is globally asymptotically stable:

In [17], Fan studied the higher order difference equation , and gave a sufficient condition for its global asymptotical stability, these results are applied to the difference equation .

In [18], Sun studied global behavior of the max-type difference equation , proved that if  and  is a periodic sequence, then every positive solution of this equation is eventually periodic with period 2m.

In [19], Stević studied behavior of positive solutions of the following max-type system of difference equations,

proved that if  then every positive solution of the system converges to (1,1).

In this paper, we investigate the global stability of following max-type difference equation

(1)

where , ,  with ,  , and  , the initial values are positive. By constructing a system of equations and binary function, we will formulate and prove the equation has a unique positive equilibrium solution, and the positive equilibrium solution is globally asymptotically stable. The conclusion of this paper extends and supplements the existing results, this conclusion has a certain guiding role to production practices as a mathematical model.

For convenience, we denote , , . So .

2. Some Definitions

In this section we will introduce some definitions (see [20]) which will be needed.

Definition A. [20] Let I be some interval of numbers and let  be a continuously difference function.

A difference equation of order (k+1) is an equation of the form

,

A point is called equilibrium solution of the difference equation if , that is for all .

Definition B. [20] The equilibrium  is called locally stable if for every , there exists  such that if  is a solution of difference equation with initial values satisfied , then

 for all.

Definition C. [20] The equilibrium  is called a global attractor if for every solution  of difference equation, we have .

Definition D. [20] The equilibrium  of difference equation is called globally asymptotically stable if  is locally stable, and  is also a global attractor of the difference equation.

3. Main Results

In this section we formulate and prove some lemmas and main theorems in this paper, obtain that every positive solution of (1) has to be the ultimate form of globally asymptotically stable.

Theorem 1. Equation (1) has a unique positive equilibrium solution .

Proof. Since

we have , so equation (1) has a unique positive equilibrium . ＃

Equation , that is , the only fixed point for the solution of this equation is , denote by , that is . Because

so .

Lemma 1. For any real number , if initial values , then  ().

Proo f. Since , so, for any, we have

suppose for every , there is , then

By induction, for every , we have . ＃

Let , , for any , define the system of equations as follows:

(2)

(3)

that is , .

Lemma 2. For every , there is

and .

Proof. Obtained by above definition (2-3), we have

so . Because

so .

By induction, there is  for every . That is

According to the monotone bounded theorem, we know the limits of  are existence. Let , . Take limits on both sides of (2-3), then

that is , , therefore , so . Since , so , that is  . ＃

Theorem 2. The unique equilibrium  of equation (1) is locally stable.

Proof. Set ,  and  as defined in Lemma 2. For every  with , according to Lemma 2 and local boundedness, there exists  such that .

Take , that is  

. Then for every , we have

,

，

that is .

Similarly, by induction, there is   for every . According to theorem B, the equilibrium  is locally stable. ＃

Theorem 3. The unique equilibrium solution of equation (1) is globally asymptotically stable.

Proof. In Theorem 2, we have proved  is locally stable, then we will prove  is global attractor.

Set ,  and  as defined in Lemma 2. Following Lemma 1, for every , there is . So

,

.

By induction, there is  for every.

Similarly, we have  for every . By induction,  for every  where .

Following Lemma 2, we know , so . By Definition C, we know  is global attractor.

According to Definition D, it is obviouslythat the equilibrium  of equation (1) is globally asymptotically stable. ＃

4. Example

Consider one of example of differential equation (1):

(4)

where the initial values . Obviously, it satisfies the conditions of Theorem 3, so the unique equilibrium  of equation (4) is globally asymptotically stable. By giving the initial value assignment, the following figures 1-2 show the global asymptotic stability.

If initial values , equilibrium  is globally asymptotically stable (see Figure 1).

If initialvalues   , equilibrium  is globally asymptotically (see Figure 2).

Figures 1. The solution of equation (4), when initial values .

Figure 2. The solution of equation (4), when initial values , .

5. Conclusion

In this paper, we investigate the characters of positive solution of the max-type difference equation (1).

First, we showed equation (1) has unique positive equilibrium .

Then, we proved two useful lemmas. By citing lemmas we showed the main theorems in this paper, that is the equilibrium solution  of equation (1) is globally asymptotically stable.

At last, we give an example of difference equation (1), draw the trajectory of the solution by giving two different initial values, thus intuitively reflect the global asymptotic stability.

Acknowledgements

Thanks for editors and reviewers' valuable comments and suggestions for improving this paper.

This research was supported by NNSF of China (11461007), Scientific and technological research project of Guangxi colleges and universities funded by Guangxi Department of Education (LX2014048, LX2014055), and Youth Foundation of Guangxi Normal University.

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