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Advances in Difference Equations

, 2008:143943 | Cite as

On the Solutions of Systems of Difference Equations

  • İbrahim Yalçinkaya
  • Cengiz Çinar
  • Muhammet Atalay
Open Access
Research Article

Abstract

We show that every solution of the following system of difference equations Open image in new window , Open image in new window as well as of the system Open image in new window , Open image in new window is periodic with period 2 Open image in new window if Open image in new window ( Open image in new window 2), and with period Open image in new window if Open image in new window ( Open image in new window 2) where the initial values are nonzero real numbers for Open image in new window .

Keywords

Differential Equation Qualitative Analysis Partial Differential Equation Ordinary Differential Equation Functional Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economy, physics, and so on [1]. So, recently there has been an increasing interest in the study of qualitative analysis of rational difference equations and systems of difference equations. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solutions. (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] and the references cited therein).

Papaschinopoulos and Schinas [9, 10] studied the behavior of the positive solutions of the system of two Lyness difference equations

where Open image in new window are positive constants and the initial values Open image in new window are positive.

In [2] Camouzis and Papaschinopoulos studied the behavior of the positive solutions of the system of two difference equations

where the initial values Open image in new window Open image in new window are positive numbers and Open image in new window is a positive integer.

Moreover, Çinar [3] investigated the periodic nature of the positive solutions of the system of difference equations

where the initial values Open image in new window are positive real numbers.

Also, Özban [7] investigated the periodic nature of the solutions of the system of rational difference equations

where Open image in new window is a nonnegative integer, Open image in new window is a positive integer, and the initial values Open image in new window are positive real numbers.

In [12] Irćianin and Stević studied the positive solution of the following two systems of di¤erence equations

where Open image in new window fixed.

In [11] Papaschinopoulos et al. studied the system of difference equations

where Open image in new window (for Open image in new window are positive constants, Open image in new window is an integer, and the initial values Open image in new window (for Open image in new window are positive real numbers.

It is well known that all well-defined solutions of the difference equation
are periodic with period two. Motivated by (1.8), we investigate the periodic character of the following two systems of difference equations:

which can be considered as a natural generalizations of (1.8).

In order to prove main results of the paper we need an auxiliary result which is contained in the following simple lemma from number theory. Let Open image in new window denote the greatest common divisor of the integers Open image in new window and Open image in new window

Lemma 1.1.

Proof.

Suppose the contrary, then we have Open image in new window for some Open image in new window

Since Open image in new window it follows that Open image in new window is a divisor of Open image in new window On the other hand, since Open image in new window we have Open image in new window which is a contradiction.

Remark 1.2.

From Lemma 1.1 we see that the rests Open image in new window for Open image in new window of the numbers Open image in new window for Open image in new window obtained by dividing the numbers Open image in new window by Open image in new window , are mutually different, they are contained in the set Open image in new window , make a permutation of the ordered set Open image in new window , and finally Open image in new window is the first number of the form Open image in new window such that Open image in new window

2. The Main Results

In this section, we formulate and prove the main results in this paper.

Theorem 2.1.

Consider (1.9) where Open image in new window Then the following statements are true:

(a)if Open image in new window , then every solution of (1.9) is periodic with period 2k,

(b)if Open image in new window , then every solution of (1.9) is periodic with period k.

Proof.

First note that the system is cyclic. Hence it is enough to prove that the sequence Open image in new window satisfies conditions (a) and (b) in the corresponding cases.

Further, note that for every Open image in new window system (1.9) is equivalent to a system of Open image in new window difference equations of the same form, where
On the other hand, we have
(a)Let Open image in new window for Open image in new window be the rests mentioned in Remark 1.2. Then from (2.2) and Lemma 1.1 we obtain that
Using (2.1) for sufficently large Open image in new window we obtain that (2.3) is equivalent to (here we use the condition Open image in new window )

From this and since by Lemma 1.1 the numbers Open image in new window are pairwise different, the result follows in this case.

which yields the result.

Remark 2.2.

In order to make the proof of Theorem 2.1 clear to the reader, we explain what happens in the cases Open image in new window and Open image in new window .

For Open image in new window , system (1.9) is equivalent to the system
where we consider that
From this and (2.2), we have
Using again (2.2), we get Open image in new window which means that the sequence Open image in new window is periodic with period equal to 2. If Open image in new window , system (1.9) is equivalent to system (2.6) where we consider that Open image in new window , and Open image in new window Using this and (2.2) subsequently, it follows that

that is, the sequence Open image in new window is periodic with period 6.

Remark 2.3.

The fact that every solution of (1.8) is periodic with period two can be considered as the case Open image in new window in Theorem 2.1, that is, we can take that

Similarly to Theorem 2.1, using Lemma 1.1 with Open image in new window for Open image in new window the following theorem can be proved.

Theorem 2.4.

Consider (1.10) where Open image in new window Then the following statements are true:

(a)if Open image in new window , then every solution of (1.10) is periodic with period 2k,

(b)if Open image in new window , then every solution of (1.10) is periodic with period k.

Proof.

First note that the system is cyclic. Hence, it is enough to prove that the sequence Open image in new window satisfies conditions (a) and (b) in the corresponding cases.

Indeed, similarly to (2.2), we have
(a)Let Open image in new window for Open image in new window be the rests mentioned in Remark 1.2. Then from (2.11) and Lemma 1.1 we obtain that
Using (2.1) for sufficiently large Open image in new window we obtain that (2.12) is equivalent to (here we use the condition Open image in new window )

From this and since by Lemma 1.1 the numbers Open image in new window are pairwise different, the result follows in this case.

which yields the result.

Corollary 2.5.

Let Open image in new window be solutions of (1.9) with the initial values Open image in new window Assume that

then all solutions of (1.9) are positive.

Proof.

We consider solutions of (1.9) with the initial values Open image in new window satisfying (2.15). If Open image in new window , then from (1.9) and (2.15), we have

for Open image in new window and Open image in new window .

If Open image in new window , then from (1.9) and (2.15), we have

for Open image in new window and Open image in new window .

From (2.16) and (2.17), all solutions of (1.9) are positive.

Corollary 2.6.

Let Open image in new window be solutions of (1.9) with the initial values Open image in new window . Assume that

then Open image in new window are positive, Open image in new window are negative for all Open image in new window

Proof.

From (2.16), (2.17), and (2.18), the proof is clear.

Corollary 2.7.

Let Open image in new window be solutions of (1.9) with the initial values Open image in new window . Assume that

then Open image in new window are negative, Open image in new window are positive for all Open image in new window

Proof.

From (2.16), (2.17), and (2.19), the proof is clear.

Corollary 2.8.

Let Open image in new window be solutions of (1.9) with the initial values Open image in new window , then the following statements are true (for all Open image in new window and Open image in new window

(i)if Open image in new window then Open image in new window and Open image in new window ,

(ii)if Open image in new window then Open image in new window and Open image in new window

(iii)if Open image in new window then Open image in new window and Open image in new window

(iv)if Open image in new window then Open image in new window and Open image in new window ,

(v)if Open image in new window then Open image in new window and Open image in new window ,

(vi)if Open image in new window then Open image in new window and Open image in new window .

Proof.

From (2.16) and (2.17), the proof is clear.

Corollary 2.9.

Let Open image in new window be solutions of (1.10) with the initial values Open image in new window . Assume that

then all solutions of (1.10) are positive.

Proof.

We consider solutions of (1.10) with the initial values Open image in new window satisfying (2.20). If Open image in new window , then from (1.10) and (2.20), we have

for Open image in new window and Open image in new window .

If Open image in new window , then from (1.10) and (2.20), we have

for Open image in new window and Open image in new window .

From (2.21) and (2.22), all solutions of (1.10) are positive.

Corollary 2.10.

Let Open image in new window be solutions of (1.10) with the initial values Open image in new window . Assume that

then Open image in new window are positive, Open image in new window are negative for all Open image in new window

Proof.

From (2.21), (2.22) and (2.23), the proof is clear.

Corollary 2.11.

Let Open image in new window be solutions of (1.10) with the initial values Open image in new window . Assume that

then Open image in new window are negative, Open image in new window are positive for all Open image in new window

Proof.

From (2.21), (2.22) and (2.24), the proof is clear.

Corollary 2.12.

Let Open image in new window be solutions of (1.10) with the initial values Open image in new window , then following statements are true (for all Open image in new window and Open image in new window

(i)if Open image in new window then Open image in new window and Open image in new window ,

(ii)if Open image in new window then Open image in new window and Open image in new window

(iii)if Open image in new window then Open image in new window and Open image in new window

(iv)if Open image in new window then Open image in new window and Open image in new window

(v)if Open image in new window then Open image in new window and Open image in new window

(vi)if Open image in new window then Open image in new window and Open image in new window

Proof.

From (2.21), (2.22), and (2.24), the proof is clear.

Example 2.13.

Let Open image in new window . Then the solutions of (1.9), with the initial values Open image in new window and Open image in new window in its invertal of periodicity can be represented by Table 1.

Notes

Acknowledgment

The authors are grateful to the anonymous referees for their valuable suggestions that improved the quality of this study.

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Copyright information

© İbrahim Yalçinkaya et al. 2008

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  • İbrahim Yalçinkaya
    • 1
  • Cengiz Çinar
    • 1
  • Muhammet Atalay
    • 1
  1. 1.Mathematics Department, Faculty of EducationSelcuk UniversityKonyaTurkey

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