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On the Solutions of Systems of Difference Equations
Advances in Difference Equations volume 2008, Article number: 143943 (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).
Papaschinopoulos and Schinas [9, 10] studied the behavior of the positive solutions of the system of two Lyness difference equations

where are positive constants and the initial values
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 are positive numbers and
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 are positive real numbers.
Also, Özban [7] investigated the periodic nature of the solutions of the system of rational difference equations

where is a nonnegative integer,
is a positive integer, and the initial values
are positive real numbers.
In [12] Irćianin and Stević studied the positive solution of the following two systems of di¤erence equations

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

where (for
are positive constants,
is an integer, and the initial values
(for
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 denote the greatest common divisor of the integers
and
Lemma 1.1.
Let and
then the numbers
(or
) for
satisfy the following property:

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.
Further, note that for every system (1.9) is equivalent to a system of
difference equations of the same form, where

On the other hand, we have

(a)Let for
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 we obtain that (2.3) is equivalent to (here we use the condition
)

From this and since by Lemma 1.1 the numbers are pairwise different, the result follows in this case.
(b)Let for some
By (2.1) we have

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
.
For , 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 which means that the sequence
is periodic with period equal to 2. If
, system (1.9) is equivalent to system (2.6) where we consider that
, and
Using this and (2.2) subsequently, it follows that

that is, the sequence 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 in Theorem 2.1, that is, we can take that

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.
Indeed, similarly to (2.2), we have

(a)Let for
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 we obtain that (2.12) is equivalent to (here we use the condition
)

From this and since by Lemma 1.1 the numbers are pairwise different, the result follows in this case.
(b)Let for some
By (2.1) we have

which yields the result.
Corollary 2.5.
Let be solutions of (1.9) with the initial values
Assume that

then all solutions of (1.9) are positive.
Proof.
We consider solutions of (1.9) with the initial values satisfying (2.15). If
, then from (1.9) and (2.15), we have

for and
.
If , then from (1.9) and (2.15), we have

for and
.
From (2.16) and (2.17), all solutions of (1.9) are positive.
Corollary 2.6.
Let be solutions of (1.9) with the initial values
. Assume that

then are positive,
are negative for all
Proof.
From (2.16), (2.17), and (2.18), the proof is clear.
Corollary 2.7.
Let be solutions of (1.9) with the initial values
. Assume that

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.
Let be solutions of (1.10) with the initial values
. Assume that

then all solutions of (1.10) are positive.
Proof.
We consider solutions of (1.10) with the initial values satisfying (2.20). If
, then from (1.10) and (2.20), we have

for and
.
If , then from (1.10) and (2.20), we have

for and
.
From (2.21) and (2.22), all solutions of (1.10) are positive.
Corollary 2.10.
Let be solutions of (1.10) with the initial values
. Assume that

then are positive,
are negative for all
Proof.
From (2.21), (2.22) and (2.23), the proof is clear.
Corollary 2.11.
Let be solutions of (1.10) with the initial values
. Assume that

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.
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Acknowledgment
The authors are grateful to the anonymous referees for their valuable suggestions that improved the quality of this study.
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Yalçinkaya, İ., Çinar, C. & Atalay, M. On the Solutions of Systems of Difference Equations. Adv Differ Equ 2008, 143943 (2008). https://doi.org/10.1155/2008/143943
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DOI: https://doi.org/10.1155/2008/143943