- Research Article
- Open Access
On the Solutions of Systems of Difference Equations
- İbrahim Yalçinkaya1Email author,
- Cengiz Çinar1 and
- Muhammet Atalay1
https://doi.org/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
.
Keywords
- Differential Equation
- Qualitative Analysis
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
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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