Journal of Difference Equations
Volume 2015 (2015), Article ID 396757, 7 pages
http://dx.doi.org/10.1155/2015/396757

Research Article

Some General Systems of Rational Difference Equations

Ali Gelisken and Merve Kara

Mathematics Department, Kamil Özdağ Faculty of Science, Karamanoğlu Mehmetbey University, 70100 Karaman, Turkey

Received 17 December 2014; Accepted 18 February 2015

Academic Editor: Mustafa R. S. Kulenović

Copyright © 2015 Ali Gelisken and Merve Kara. 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.

We investigate behavior of solutions of the following systems of rational difference equations: , where is a positive integer and the initial conditions are real numbers. We show that every solution is periodic with period, considerably improving the results in the literature.

1. Introduction

Recently, a great effort has been made in studying qualitative properties of the solutions of systems of rational difference equations.

Çinar [1] studied periodicity of the positive solutions of the system of difference equations

Kurbanli et al. [2] studied the behavior of positive solutions of the following system:

Elsayed [3, 4] obtained the solutions of the following systems of difference equations:

Touafek and Elsayed [5] investigated the periodic nature and form of the solutions of the following systems of rational difference equations:

Mansour et al. [6] investigated form of the solutions and periodic nature of the following systems of rational difference equation:

Özkan and Kurbanli [7] gave the solutions of the systems of the difference equations

Similar systems of rational difference equations were investigated (see, e.g., [8–15]).

Motivated by [7], in this paper we investigate periodic solutions of the following systems of rational difference equations:

where is a positive integer and the initial conditions are arbitrary real numbers. We show that every solution is periodic with period . Furthermore, we give the solutions of some systems explicitly.

2. The Case

In this section, we consider the following systems:

Özkan and Kurbanli studied periodic solutions of the systems of the difference equations

where the initial conditions are arbitrary real numbers. The following two theorems are proved in [7].

Theorem 1. Let , , , , , , , , and be arbitrary real numbers, and let be a solution of the system (16). Also, assume that and . Then all six-period solutions of system (16) are as follows:

Theorem 2. Let , , , , , , , , and be arbitrary real numbers, and let be a solution of system (17). Also, assume that and . Then all six-period solutions of system (17) are as follows:

These theorems give the solutions of systems (12) and (13). So, we do not show the periodic solutions of systems (12) and (13). In this section, we only give the solutions of systems (14) and (15) explicitly.

Theorem 3. Let , , , , , and be arbitrary real numbers, and let be a solution of system (14). Also, assume that and . Then all six-period solutions of system (14) are as follows:

for .

Proof. We prove the theorem by induction. From system (14), we obtain, immediately by iteration, , , , , , , , , , , , and . Now, assume that (20) holds for a positive integer ; that is,

Then, it follows that

So, the proof is finished.

The following theorem can be proved by induction similarly. So, it will be omitted.

Theorem 4. Let , , , , , and be arbitrary real numbers, and let be a solution of system (15). Also, assume that and . Then all six-period solutions of system (15) are as follows:

for .

3. The Case

In this section, we consider the following systems:

We will consider well-defined solutions. So, we will assume that , , , and for system (24), , , , and for system (25), , , , and for system (26), and , , , and for system (27). We give solutions of these systems in the following theorem explicitly.

Theorem 5. Let be a solution of system (24), or (25), or (26), or (27). Then, is periodic with period 12. Morever, the following are true for , , , , , , , , , , , and .

(i)If is a solution of system (24), then all solutions are as follows:

 for .(ii)If is a solution of system (25), then all solutions are as follows:

 for .(iii)If is a solution of system (26), then all solutions are as follows:

 for .(iv)If is a solution of system (27), then all solutions are as follows:

 for .

Proof. (i) Consider system (14). By iteration we obtain immediately , , , , , , , , , , , , , , , , , , , , , , , and . Now, assume that (28) holds for a positive integer ; that is,

Then, it follows that

So, the proof of is finished. Because of the rest of the proof is similar, it is omitted.

4. The General Case

In this section, we investigate systems (8), (9), (10), and (11).

We will consider well-defined solutions. So, we will assume that and , , for system (8), and , , for system (9), and , , for system (10), and and , , for system (11).

We give the following lemma which is useful to prove the following theorem.

Lemma 6. Let be a solution of system (8), or (9), or (10), or (11) where the initial conditions are real numbers and satisfy well-defined solutions. The followings are true for :

(i);(ii).

Proof. (i) Firstly, we prove (i) for system (8). From system (8), we obtain

for . Considering other systems (9), (10), and (11) individually, the proof can be obtained similarly.
(ii) The proof of (ii) can be obtained similarly. So, it is omitted.

Theorem 7. Let be a solution of system (8), or (9), or (10), or (11) which hold Lemma 6. Then, is periodic with period .

Proof. Suppose that holds conditions in Theorem 7. We must show and , for . Firstly considering system (8), we show . From system (8) and Lemma 6(i), we obtain

for . From system (8) and Lemma 6(ii),

So, we have obtained for . Considering the other systems individually, the proof can be obtained similarly. So, the proof is finished.

Conflict of Interests

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

Acknowledgment

The authors are grateful to Karamanoğlu Mehmetbey University Scientific Research Council (BAP no. BAP-01-M-14).

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