International Scholarly Research Notices
Volume 2014 (2014), Article ID 857480, 4 pages
http://dx.doi.org/10.1155/2014/857480

Research Article

On Positive Solutions for the Rational Difference Equation Systems , and

Hui-li Ma and Hui Feng

Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, China

Received 3 June 2014; Revised 24 July 2014; Accepted 6 August 2014; Published 29 October 2014

Academic Editor: Dang Dinh Hai

Copyright © 2014 Hui-li Ma and Hui Feng. 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.

Our aim in this paper is to investigate the behavior of positive solutions for the following systems of rational difference equations: , and , where , , , and are positive real numbers and and are positive constants.

1. Introduction

In recent years, with the wide application of computers, difference system has become one of the important theoretical bases for computer, information system, engineering control, ecological balance, and so forth. As typical nonlinear difference equations, rational difference equations have become a research hot spot in mathematical modelling. The behavior of solutions of the system for rational difference equation has received extensive attention.

In [1], Ozban has investigated the periodicity of solutions of the system of difference equations:

In [2], Kurbanlı et al. studied the behavior of the positive solutions of the system of difference equations:

The periodicity of the positive solutions of the rational difference system

has been studied by Çinar in [3].

In [4], Ozban studied the behavior of the positive solutions of the system of difference equations

For similar research on difference systems, we refer the reader to [5, 6] and the references therein.

In this paper, we investigate the behavior of positive solutions for the system of rational difference equations

where , and , , , and .

Before stating our main results, we state some definitions used in this paper.

Definition 1. A pair of sequences of positive real numbers that satisfies (5) is called a positive solution of (5).

Definition 2. A solution of (5) is periodic, if there exists a positive integer such that , , and is called a period.

2. Main Results

First, we study the periodic nature of positive solution of system (5).

Theorem 3. Let , , , and be positive real numbers and let be a solution of system (5). Then for , , , all solutions of system (5) are periodic with period 3.

Proof. For , , it can be seen easily that all solutions of (5) are positive. Thus, by (5) we have the following equality:

Repeating application of (5) yields

Similarly,

The proof is complete.

Theorem 4. Suppose that , , , and are positive real numbers. Let be a solution of system (5) with , , , and . Then for , where and are positive constants, all solutions of system (5) are

Proof. For , , it can be seen easily that all solutions of (5) are positive.
For we have

Now suppose that and that our assumption holds for . One will show that the result holds for . From system (5), we obtain

Then,

In particular, from Theorem 3, we get

Therefore, the proof is complete.

Example 5. Set and and , , , and . Figure 1 describes the periodic nature of system (5).

Figure 1

Next, we consider the case that , where and are positive constants.

Theorem 6. Let be an arbitrary positive solution of (5).

(i)If , then, for each integer , the subsequence , and the subsequence .(ii)If , then, for each integer , the subsequence , and the subsequence .(iii), .

Proof. (i) For every fixed , we will show that

In fact, for , . Assume that (14) holds for ; that is,

For , we have the following:

Similarly, it can be obtained by induction that

If , we can get by (14) and (17) that

(ii) If , by (14) and (17),

(iii) From the proof of Theorem 3, we have

Multiplying both sides, respectively, yields .

Example 7. Let , , , and .

(i)For , , and , Figure 2 and Table 1 describe the behavior of the sequence and (shown by the black spot at the top of every peak).(ii)For , , and , Figure 3 and Table 2 describe the behavior of the sequence and (shown by the black spot at the top of every peak).

Table 1

0
1
2
3
4
5
6
7
⋯

144.00
36.00
9.00
2.25
0.56

⋯

0.17
0.67
2.67
10.67
42.67

⋯

Table 2

0
1
2
3
4
5
6
7
⋯

⋯

⋯

Figure 2

Figure 3

Conflict of Interests

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

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (61363058), the Scientific Research Fund for Colleges and Universities of Gansu Province (2013B-007, 2013A-016), Natural Science Foundation of Gansu Province (145RJZA232, 145RJYA259), and Promotion Funds for Young Teachers in Northwest Normal University (NWNU-LKQN-12-14).

References

A. Y. Ozban, “On the positive solutions of the system of rational difference equations, ,” Journal of Mathematical Analysis and Applications, vol. 323, pp. 26–32, 2006.

A. S. Kurbanlı, C. Çinar, and İ. Yalçinkaya, “On the behavior of positive solutions of the system of rational difference equations ,” Mathematical and Computer Modelling, vol. 53, pp. 1261–1267, 2011.

C. Çinar, “On the positive solutions of the difference equation system ,” Applied Mathematics and Computation, vol. 158, pp. 303–305, 2004.

A. Y. Ozban, “On the system of rational difference equations , ,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 833–837, 2007.

X. Yang, “On the system of rational difference equations ,” Journal of Mathematical Analysis and Applications, vol. 307, no. 1, pp. 305–311, 2005.

X. Yang, Y. Liu, and S. Bai, “On the system of high order rational difference equations ,” Applied Mathematics and Computation, vol. 171, pp. 853–856, 2005.