Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2013, Article ID 970316, 7 pages
http://dx.doi.org/10.1155/2013/970316
Research Article
On a System of Difference Equations
Ozan Özkan
1
and Abdullah Selçuk Kurbanli
2
1
Department of Mathematics, Faculty of Science, Selcuk University, 42075 Konya, Turkey
2
Mathematics Department, Ahmet Kelesoglu Education Faculty, N. Erbakan University, Meram Yeni Yol, 42090 Konya, Turkey
Correspondence should be addressed to Ozan
¨
Ozkan; oozkan@selcuk.edu.tr
Received 25 December 2012; Accepted 3 February 2013
Academic Editor: Ibrahim Yalcinkaya
Copyright © 2013 O.
¨
Ozkan and A. S. Kurbanli. Tis 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.
We have investigated the periodical solutions of the system of rational diference equations
+1
=
−2
/(−1 ±
−2
−1
),
+1
=
−2
/(−1 ±
−2
−1
), and
+1
= (
−2
+
−2
)/(−1 ±
−2
−1
), where
0
,
−1
,
−2
,
0
,
−1
,
−2
,
0
,
−1
,
−2
∈ R.
1. Introduction
Recently, a great interest has arisen on studying diference
equation systems. One of the reasons for that is the necessity
for some techniques which can be used in investigating equa-
tions which originate in mathematical models to describe
real-life situations such as population biology, economics,
probability theory, genetics, and psychology. Tere are many
papers related to the diference equations system.
In [1], Kurbanli et al. studied the periodicity of solutions
of the system of rational diference equations
+1
=
−1
+
−1
−1
,
+1
=
−1
+
−1
−1
.
(1)
In [2], C ¸inar studied the solutions of the systems of dif-
ference equations
+1
=
1
,
+1
=
−1
−1
. (2)
In [3, 4],
¨
Ozban studied the positive solutions of the sys-
tem of rational diference equations
=
−3
,
=
−3
−
−
,
+1
=
1
−
,
+1
=
−
−−
.
(3)
In [5–16], Elsayed studied a variety of systems of rational
diference equations; for more, see references.
In this paper, we have investigated the periodical solu-
tions of the system of diference equations
+1
=
−2
−1 ±
−2
−1
,
+1
=
−2
−1 ±
−2
−1
,
+1
=
−2
+
−2
−1 ±
−2
−1
, ∈ N
0
,
(4)
where the initial conditions are arbitrary real numbers.
2. Main Results
Teorem 1. Let
0
=,
−1
=,
−2
=,
0
=,
−1
=,
−2
=,
0
=,
−1
=, and
−2
= be arbitrary real
numbers, and let {
,
,
} be a solution of the system
+1
=
−2
−1 +
−2
−1
,
+1
=
−2
−1 +
−2
−1
,
+1
=
−2
+
−2
−1 +
−2
−1
, ∈ N
0
.
(5)
Also, assume that ̸ =0, ̸ =0, ̸ =1, and ̸ =1. Ten, all
six-period solutions of (5) are as follows:
6+1
=
1 −
,
6+1
=
− 1
,
6+1
=−
+
− 1
,