Journal of Difference Equations
Volume 2014 (2014), Article ID 936302, 6 pages
http://dx.doi.org/10.1155/2014/936302

Research Article

Asymptotic Behavior of the Solutions of System of Difference Equations of Exponential Form

Vu Van Khuong1,2 and Tran Hong Thai¹

¹Department of Mathematics, Hung Yen University of Technology and Education, Khoai Chau, Hung Yen 393008, Vietnam
²Department of Mathematical Analysis, University of Transport and Communications, Dong Da, Hanoi 10200, Vietnam

Received 31 May 2014; Revised 10 September 2014; Accepted 24 September 2014; Published 13 October 2014

Academic Editor: Honglei Xu

Copyright © 2014 Vu Van Khuong and Tran Hong Thai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The goal of this paper is to study the boundedness, the persistence, and the asymptotic behavior of the positive solutions of the system of two difference equations of exponential form: where , and are positive constants and the initial values are positive real values. Also, we determine the rate of convergence of a solution that converges to the equilibrium of this system.

1. Introduction

In [1], the authors studied the boundedness, the asymptotic behavior, the periodicity, and the stability of the positive solutions of the difference equation:

where are positive constants and the initial values are positive numbers.

Motivated by the above paper we will extend the above difference equation to a system of difference equations; our goal will be to investigate the boundedness, the persistence, and the asymptotic behavior of the positive solutions of the following system of exponential form:

where are positive constants and the initial values are positive real values.

Difference equations and systems of difference equations of exponential form can be found in [2–6]. Moreover, as difference equations have many applications in applied sciences, there are many papers and books that can be found concerning the theory and applications of difference equations; see [7–9] and the references cited therein.

2. Global Behavior of Solutions of System (2)

In the first lemma we study the boundedness and persistence of the positive solutions of (2).

Lemma 1. Every positive solution of (2) is bounded and persists.

Proof. Let be an arbitrary solution of (2). From (2) we can see that

In addition, from (2) and (3) we get

Therefore, from (3) and (4) the proof of the lemma is complete.

In order to prove the main result of this section, we recall the next theorem without its proof. See [10, 11].

Theorem 2. Let and

be a continuous functions such that the following hold:

(a) is decreasing in both variables and is decreasing in both variables for each ;(b)if is a solution of

then and . Then the following system of difference equations,

has a unique equilibrium and every solution of the system (7) with converges to the unique equilibrium . In addition, the equilibrium is globally asymptotically stable.

Now we state the main theorem of this section.

Theorem 3. Consider system (2). Suppose that the following relation holds true:

Then system (2) has a unique positive equilibrium and every positive solution of (2) tends to the unique positive equilibrium as . In addition, the equilibrium is globally asymptotically stable.

Proof. We consider the functions

where

It is easy to see that , are decreasing in both variables for each . In addition, from (9) and (10) we have , as and so , .
Now let be positive real numbers such that

Moreover arguing as in the proof of Theorem 2, it suffices to assume that

From (11) we get

which imply that

Moreover, we get

Then by adding the two relations (14) we obtain

Therefore from (16) we have

Then using (8), (12), and (17) gives us and . Hence from Theorem 2 system (2) has a unique positive equilibrium and every positive solution of (2) tends to the unique positive equilibrium as . In addition, the equilibrium is globally asymptotically stable. This completes the proof of the theorem.

3. Rate of Convergence

In this section we give the rate of convergence of a solution that converges to the equilibrium of the system (2) for all values of parameters. The rate of convergence of solutions that converge to an equilibrium has been obtained for some two-dimensional systems in [12, 13].

The following results give the rate of convergence of solutions of a system of difference equations:

where is a -dimensional vector, is a constant matrix, and is a matrix function satisfying

where denotes any matrix norm which is associated with the vector norm; also denotes the Euclidean norm in given by

Theorem 4 (see [14]). Assume that condition (19) holds. If is a solution of system (18), then either for all large or

exists and is equal to the modulus of one of the eigenvalues of matrix .

Theorem 5 (see [14]). Assume that condition (19) holds. If is a solution of system (18), then either for all large or

exists and is equal to the modulus of one of the eigenvalues of matrix .

The equilibrium point of the system (2) satisfies the following system of equations:

If , we can easily see that the system (23) has an unique equilibrium .

The map associated with the system (2) is

The Jacobian matrix of is

By using the system (23), value of the Jacobian matrix of at the equilibrium point is

Our goal in this section is to determine the rate of convergence of every solution of the system (2) in the regions where the parameters and initial conditions and are arbitrary, nonnegative numbers.

Theorem 6. The error vector of every solution of (2) satisfies both of the following asymptotic relations:

where is equal to the modulus of one of the eigenvalues of the Jacobian matrix evaluated at the equilibrium .

Proof. First, we will find a system satisfied by the error terms. The error terms are given as

By calculating similarly, we get

From (28) and (29) we have

Set

Then system (30) can be represented as

where

Taking the limits of , and as , we obtain

that is

where ,  ,  , and as .
Now, we have system of the form (18):

where

Thus, the limiting system of error terms can be written as

The system is exactly linearized system of (2) evaluated at the equilibrium . Then Theorems 4 and 5 imply the result.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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