International Scholarly Research Notices
Volume 2014 (2014), Article ID 760502, 5 pages
http://dx.doi.org/10.1155/2014/760502

Research Article

On the System of High Order Rational Difference Equations

Qianhong Zhang,¹ Wenzhuan Zhang,¹ Yuanfu Shao,² and Jingzhong Liu³

¹Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang, Guizhou 550004, China
²School of Science, Guilin University of Technology, Guilin, Guangxi 541000, China
³Department of Mathematics and Physics, Hunan Institute of Technology, Hengyang, Hunan 421002, China

Received 15 March 2014; Revised 2 July 2014; Accepted 22 July 2014; Published 29 October 2014

Academic Editor: Chunrui Zhang

Copyright © 2014 Qianhong Zhang et al. 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.

This paper is concerned with the boundedness, persistence, and global asymptotic behavior of positive solution for a system of two rational difference equations , where and

1. Introduction

In this paper, we study the global behavior of solutions of the following system:

where are positive constants and initial conditions ,  .

A pair of sequences of positive real numbers that satisfies (1) is a positive solution of (1). If a positive solution of (1) is a pair of positive constants , then the solution is the equilibrium solution.

A positive solution of (1) is bounded and persists, if there exist positive constants such that

In 1998, DeVault et al. [1] proved that every positive solution of the difference equation

where , oscillates about the positive equilibrium of (3). Moreover, every positive solution of (3) is bounded away from zero and infinity. Also the positive equilibrium of (3) is globally asymptotically stable.

In 2003, Abu-Saris and DeVault [2] studied the following recursive difference equation:

where ,   are positive real numbers.

Papaschinopoulos and Schinas [3] investigated the global behavior for a system of the following two nonlinear difference equations:

where is a positive real number, are positive integers, and ,   are positive real numbers.

In 2012, Zhang et al. [4] investigated the global behavior for a system of the following third-order nonlinear difference equations:

where , and the initial values ,  . For other related results, the reader can refer to [5–18].

Motivated by the discussion above, we study the global asymptotic behavior of solutions for system (1). More precisely, we prove the following: if then every positive solution of (1) is persistent and bounded. Moreover, we prove that every positive solution of (1) converges the unique positive equilibrium as .

2. Main Results

In the following lemma, we show boundedness and persistence of the positive solutions of (1).

Lemma 1. Consider (1). Suppose that

are satisfied. Then, every positive solution of (1) is satisfied, for

Proof. Let be a positive solution of (1). Since and for all , (1) implies that

Moreover, using (1) and (9), we have

Let be the solution of the system, respectively,

such that

We prove by induction that

Suppose that (13) is true for . Then, from (10), we get

Therefore, (13) is true. From (11) and (12), we have

Then, from (9), (13), and (15), the proof of the relation (8) follows immediately.

Theorem 2. Consider the system of difference equation (1). If relation (7) is satisfied, then the following statements are true.

(i)Equation (1) has a unique positive equilibrium given by

(ii)Every positive solution of system (1) tends to the positive equilibrium of (1) as .

Proof. (i) Let and be positive numbers such that

Then, from (7) and (17), we have that the positive solution is given by (16). This completes the proof of Part (i).
(ii) From (1) and (8), we have

where ,  . Then, from (1) and (18), we get

from which we have

Then, relations (7) and (20) imply that , from which it follows that

We claim that

Suppose on the contrary that . Then, from (21), we have and so which is a contradiction. So . Similarly, we can prove that . Therefore, (22) is true. Hence, from (1) and (22), there exist the and , as and

where is the unique positive equilibrium of (1). This completes the proof of Part (ii). The proof of Theorem 2 is completed.

Theorem 3. Consider the system of difference equation (1). If relation (7) is satisfied and assuming that

then the unique positive equilibrium is locally asymptotically stable.

Proof. From Theorem 2, the system of difference equation (1) has a unique equilibrium . The linearized equation of system (1) about the equilibrium point is

where , and

Let denote the eigenvalues of matrix and let be a diagonal matrix, where ,  , and

Clearly, is invertible. Computing matrix , we obtain that

From and , imply that

Furthermore, noting (7), (24), and (27), we have

It is well known that has the same eigenvalues as ; we have that

This implies that the equilibrium of (1) is locally asymptotically stable.

Combining Theorem 2 with Theorem 3, we obtain the following theorem.

Theorem 4. Consider the system of difference equation (1). If relations (7) and (24) are satisfied, then the unique positive equilibrium is globally asymptotically stable.

3. Some Numerical Examples

In order to illustrate the results of the previous sections and to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to nonlinear difference equations and system of nonlinear difference equations.

Example 1. Consider the following difference equations:

with the initial values . Then, the solution of system (32) is bounded and persists and the system has a unique equilibrium which is globally asymptotically stable (see Figure 1).

Figure 1: The dynamics of system (32).

Example 2. Consider the following difference equations:

with the initial values . Then, the solution of system (33) is bounded and persists and the system has a unique equilibrium which is globally asymptotically stable (see Figure 2).

Figure 2: The dynamics of system (33).

4. Conclusion

In this paper, we study the dynamics of a system of high order difference equation

It concluded that, under condition , the positive solution of this system is bounded and persists; moreover, if , it converges asymptotically the unique equilibrium .

We conclude the paper by presenting the following open problem.

Open Problem. Consider the system of difference equation (1) with and . Find the set of all initial conditions that generate bounded solutions. In addition, investigate global behavior of these solutions.

Conflict of Interests

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

Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (Grant no. 11361012), the Scientific Research Foundation of Guizhou Provincial Science and Technology Department ([2013]J2083, [2009]J2061), and the Natural Science Foundation of Guizhou Provincial Educational Department (no. 2008040).

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