On the Solutions of Systems of Difference Equations
 İbrahim Yalçinkaya^{1}Email author,
 Cengiz Çinar^{1} and
 Muhammet Atalay^{1}
DOI: 10.1155/2008/143943
© İbrahim Yalçinkaya et al. 2008
Received: 19 March 2008
Accepted: 19 May 2008
Published: 21 May 2008
Abstract
We show that every solution of the following system of difference equations , as well as of the system , is periodic with period 2 if ( 2), and with period if ( 2) where the initial values are nonzero real numbers for .
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 [1–11] and the references cited therein).
where are positive constants and the initial values are positive.
where the initial values are positive numbers and is a positive integer.
where the initial values are positive real numbers.
where is a nonnegative integer, is a positive integer, and the initial values are positive real numbers.
where fixed.
where (for are positive constants, is an integer, and the initial values (for are positive real numbers.
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 denote the greatest common divisor of the integers and
Lemma 1.1.
Proof.
Suppose the contrary, then we have for some
Since it follows that is a divisor of On the other hand, since we have which is a contradiction.
Remark 1.2.
From Lemma 1.1 we see that the rests for of the numbers for obtained by dividing the numbers by , are mutually different, they are contained in the set , make a permutation of the ordered set , and finally is the first number of the form such that
2. The Main Results
In this section, we formulate and prove the main results in this paper.
Theorem 2.1.
Consider (1.9) where Then the following statements are true:
(a)if , then every solution of (1.9) is periodic with period 2k,
(b)if , 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 satisfies conditions (a) and (b) in the corresponding cases.
From this and since by Lemma 1.1 the numbers 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 and .
that is, the sequence is periodic with period 6.
Remark 2.3.
Similarly to Theorem 2.1, using Lemma 1.1 with for the following theorem can be proved.
Theorem 2.4.
Consider (1.10) where Then the following statements are true:
(a)if , then every solution of (1.10) is periodic with period 2k,
(b)if , 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 satisfies conditions (a) and (b) in the corresponding cases.
From this and since by Lemma 1.1 the numbers are pairwise different, the result follows in this case.
which yields the result.
Corollary 2.5.
then all solutions of (1.9) are positive.
Proof.
for and .
for and .
From (2.16) and (2.17), all solutions of (1.9) are positive.
Corollary 2.6.
then are positive, are negative for all
Proof.
From (2.16), (2.17), and (2.18), the proof is clear.
Corollary 2.7.
then are negative, are positive for all
Proof.
From (2.16), (2.17), and (2.19), the proof is clear.
Corollary 2.8.
Let be solutions of (1.9) with the initial values , then the following statements are true (for all and
(i)if then and ,
(ii)if then and
(iii)if then and
(iv)if then and ,
(v)if then and ,
(vi)if then and .
Proof.
From (2.16) and (2.17), the proof is clear.
Corollary 2.9.
then all solutions of (1.10) are positive.
Proof.
for and .
for and .
From (2.21) and (2.22), all solutions of (1.10) are positive.
Corollary 2.10.
then are positive, are negative for all
Proof.
From (2.21), (2.22) and (2.23), the proof is clear.
Corollary 2.11.
then are negative, are positive for all
Proof.
From (2.21), (2.22) and (2.24), the proof is clear.
Corollary 2.12.
Let be solutions of (1.10) with the initial values , then following statements are true (for all and
(i)if then and ,
(ii)if then and
(iii)if then and
(iv)if then and
(v)if then and
(vi)if then and
Proof.
From (2.21), (2.22), and (2.24), the proof is clear.
Example 2.13.
Let . Then the solutions of (1.9), with the initial values and in its invertal of periodicity can be represented by Table 1.
Table 1
i  1  2  3  4  5  6  7  8  9  10  11  12 


 r 
 q 
 p 
 r 
 q 
 p 

 p 
 r 
 q 
 p 
 r 
 q 

 q 
 p 
 r 
 q 
 p 
 r 
Declarations
Acknowledgment
The authors are grateful to the anonymous referees for their valuable suggestions that improved the quality of this study.
Authors’ Affiliations
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