Applied Mathematics, 2011, 2, 410-413
doi:10.4236/am.2011.24050 Published Online April 2011 (http://www.SciRP.org/journal/am)
On the Periodicity of Solutions of the System of Rational
Difference Equations
, 
11
11
11
11
nn nn
nn
nn nn
xy y
x
xy
yx xy







Abdullah Selçuk Kurbanli1, Cengiz Çinar1, Dağistan Şımşek2
1Department of Mat hem at ic s, Education Faculty, Selçuk University, Meram, Turkey
2Department of Industrial Engineering, Engineering-Architecture Faculty, Selçuk University, Kampüs, Turkey
Received December 5, 2010; revised Febru a ry 2, 2011; accepted February 5, 2011
Abstract
In this paper, we have investigated the periodicity of the solutions of the system of difference equations
11
11
11
,
11
nn nn
nn
nn nn
x
yyx
xy
yx xy






, where 0101 ,,,xx yy


.
Keywords: Difference Equation, Difference Equation Systems, Solutions
1. Introduction
Recently, there has been great interest in studying dif-
ference equation systems. One of the reasons for this is a
necessity for some techniques which can be used in inve-
stigating equations arising in mathematical models desc-
ribing real life situations in population biology, econom-
ic, probability theory, genetics, psychology etc. There are
many papers with related to the difference equations sys-
tem for example,
In [1] Cinar studied the solutions of the systems of the
difference equations
11
11
1, n
nn
nnn
y
xy
yxy



In [2] Papaschinnopoulos and Schinas studied the os-
cillatory behavior, the boundedness of the solutions, and
the global asymptotic stability of the positive equilibrium
of the system of nonlinear difference equations
11
--
, , 0,1,2,,,
nn
nn
np nq
yx
x
AyAnpq
xy

 .
In [3] Papaschinnopoulos and Schinas proved the
boundedness, persistence, the oscillatory behavior and
the asymptotic behavior of the positive solutions of the
system of difference equations
11
00
,
ii
kk
ii
nn
pq
ii
ni ni
A
B
xy
yx






.
In [4,5] Özban studied the positive solutions of the
system of rational difference equations and
11
, n
nn
nknmnmk
y a
xy
yxy



.
In [6,7] Clark and Kulenović investigate the global
asymptotic stability
11
,
nn
nn
nn
xy
xy
acy bdx



.
In [8] Camouzis and Papaschinnopoulos studied the
global asymptotic behavior of positive solutions of the
system of rational difference equations
11
1, 1
nn
nn
nm nm
x
y
xy
yx


 .
In [9] Kurbanli, Çinar and Yalcinkaya studied On the
behavaior of positive solutions of the system of rational
difference equations
11
11
11
,
11
nn
nn
nn nn
xy
xy
yx xy





.
In [10] Kurbanli studied the behavaior of solutions of
the system of rational difference

A. S. KURBANLI ET AL. 411
11
11
11
,
11
nn
nn
nn nn
xy
xy
yx xy





.
In [11] Yang, Liu and Bai consided the behavior of the
positive solutions of the system of the difference equa-
tions
, np
nn
npnq nq
by
a
xy
y
xy


.
In [12] Kulenović, Nurkanović studied the global asy-
mptotic behavior of solutions of the system of difference
equations
111
, ,
nnn
nnn
nnn
ax cyez
xyz
bydzf x

 


.
In [13] Zhang, Yang, Megson and Evans investigated
the behavior of the positive solutions of the system of
difference equations
1
1, n
nn
npnr ns
y
xA yA
yxy



In [14] Zhang, Yang, Evans and Zhu studied the boun-
dedness, the persistence and global asymptotic stability
of the positive solutions of the system of difference equa-
tions
11
,
nmnm
nn
nn
yx
xA yA
x
y



In [15] Yalcinkaya and Cinar studied the global asm-
ptotic stability of the system of difference equations
11
11
11
,
nn nn
nn
nn nn
tz azt a
zt
tz zt






.
In [16] Yalcinkaya, Cinar and Atalay investigated the
solutions of the system of difference equations
 









23 1
12
11 1
23 1
, ,,
11 1
k
nn n
nn n
nn n
xx x
xx x
xx x
 
 
 
.
In [17] Yalcinkaya studied the global asmptotic sta-
bility of the system of difference equations
11
11
11
,
nn nn
nn
nn nn
tz azt a
zt
tz zt






.
In [18] Irićanin and Stević studied the positive solu-
tions of the system of difference equations
 

 

 

  

  

  

23 1
12
111
34 2
11 1
23 34
12
11
11
45
22
12
1
13
2
11 1
, , ,
11
, ,
1
,
k
nn n
nn n
nn n
nn nn
nn
nn
knn
n
n
x
xx
xx x
xx x
xx xx
xx
xx
xx
xx
 
 






 
 
 





In this paper, we investigated the periodicity of the so-
lutions of the difference equation system
11
11
11
,
11
nn nn
nn
nn nn
x
yyx
xy
yx xy






 (1.1)
where the initial conditions are arbitrary real numbers.
2. Main Results
Theorem 1. Let 0101 , , , yay bxcx d


  be ar-
bitrary real numbers and let

,,
nnn
x
yz be a solutions of
the system (1.1). Also, assume that 1ad  and 1cb

.
All solutions of (1.1) are as following:
,61
1
,62
,63
, 0, 1, 2,
,64
1
,65
,66
n
da nk
adbnk
ank
xk
bc nk
cbdnk
cnk















 (1.2)
,61
1
,62
,63
, 0, 1, 2,
,64
1
,65
,66
n
bc nk
cbdnk
cnk
yk
da nk
adbnk
ank















 (1.3)
Proof: For n = 0, 1, 2, 3, 4, 5, we have
10
1
0111
xyda
xyx ad





,
10
1
01
11
yx bc
y
x
ycb





,

01
2
10
2
22
1
11
1
1
1
,
11
1
bc
c
xy cb
xbc
yx c
cb
ccbb ccbb
cb b
cc
cb














2
01
22
10
1
11
1
1
da
a
yx ad d
ad
yd
da
xy a
a
ad










,

A. S. KURBANLI ET AL. 412

2
12
32
21
1
11
1 11
11
ad
dad
xy ad ad
x
a
da d
yx dad ad



 



,

2
12
32
21
1
1
11
1
1
bc bcb
yx bc
yc
bc
xy b
bbc







,
23
4
32 11
xy bc
xyx cb



,
23
4
32 11
yx da
yxyad



,
2
34
52
43
1
11
1
1
da
a
xy ada da
ad
x
d
da
yx ad aad
a
ad




 




,
2
34
52
43
1
11
1
1
bc
c
yx bbc
cb
yb
bc
xy c
c
cb




 




and

2
45
62
54
1
1
11
1
1
bc bcb
xy cb
x
c
bc
yx b
bcb


 




,
2
45
62
54
1
11
1
1
dad
yx ad a
ad
ya
da
xy d
dad



 




;
for n = 6, 7, 8, 9, 10, 11,
56
71
65 11
xy da
x
x
yx ad



,
56
71
65 11
yx bc
yy
xy cb



,

2
67
82
76
1
1
11
1
1
bc
cbc
xy cb
x
b
bc
yx c
c
cb



 




,
2
67
82
2
76
1
11
1
1
da
a
yx ad d
ad
ydy
da
xy a
a
ad




 




,
2
78
93
2
87
1
1
11
1
1
da
xy ad
ad
x
ax
da
yx d
dad



 




,
2
78
93
2
87
1
11
1
1
bc
yx ccb
cb
ycy
bc
xy b
bcb









,
89
10 4
98 11
xy bc
x
x
yx cb




,
89
10 4
98 11
yx da
yy
xy ad




,
2
910
11 5
2
10 9
1
11
1
1
da
a
xy ad d
ad
x
dx
da
yx a
a
ad










,
2
910
11 5
2
10 9
1
11
1
1
bc
c
yx cb b
cb
yby
bc
xy c
c
cb










and
2
10 11
12 6
2
11 10
1
11
1
1
bc b
xy ccb
cb
x
cx
bc
yx b
bcb










,
2
10 11
12 6
2
11 10
1
11
1
1
dad
yx ad a
ad
yy
da
xy d
dad










.
Also, we have
171361
, 0,1,2,3,
1n
da
xxxxn
ad 

 

2814 62
, 0,1,2,3,
n
xbxxx n


 
391563
, 0,1,2,3,
n
xaxxx n


 
4101664
, 0,1,2,3,
1n
bc
xxxxn
cb 

 

51117 65
, 0,1,2,3,
n
xdx xxn


 
61218 66
, 0,1,2,3,
n
xcx xxn


 
and
171361
, 0,1,2,3,
1n
bc
yyyyn
cb 

 

2814 62
, 0,1,2,3,
n
ydyyy n


 
391563
, 0,1,2,3,
n
ycyyy n


 
4101664
, 0,1,2,3,
1n
da
yyyyn
ad 

 

51117 65
, 0,1,2,3,
n
yby yyn




A. S. KURBANLI ET AL. 413
61218 66
, 0,1,2,3,
n
yayyyn

 .□
Theorem 2. Let 0101 , , , yay bxcxd

 be ar-
bitrary real numbers and let

,,
nnn
x
yz be a solutions of
the system (1.1). Also, assume that 1ad  and cb1

.
The solutions of n
x
and n y are six periodic.
Proof. The proof is clear from Theorem 1.□
Corolla ry 1. If n for 663nk nk xy

.
Proof. The proof is clear from Theorem 1.
3. References
[1] C. Çinar, “On the Positive Solutions of the Difference
Equation System 11
11
1, n
nn
nnn
y
xy
yxy


,” Applied Ma-
thematics and Computation, Vol. 158, No. 2, 2004, pp.
303-305. doi:10.1016/j.amc.2003.08.073
[2] G. Papaschinopoulos and C. J. Schinas, “On a System of
Two Nonlinear Difference Equations,” Journal of Ma-
thematical Analysis and Applications, Vol. 219, No. 2,
1998, pp. 415-426. doi:10.1006/jmaa.1997.5829
[3] G. Papaschinopoulos and C. J. Schinas, “On the System
of Two Difference Equations,” Journal of Mathematical
Analysis and Applications, Vol. 273, No. 2, 2002, pp.
294-309. doi:10.1016/S0022-247X(02)00223-8
[4] A. Y. Özban, “On the System of Rational Difference
Equations 3
1
3
, n
nn
nnqnq
aby
xy
yxy



 ,” Applied Mathema-
tics and Co m p uta ti o n, Vol. 188, No. 1, 2007, pp. 833-837.
doi:10.1016/j.amc.2006.10.034
[5] A. Y. Özban, “On the Positive Solutions of the System of
Rational Difference Equations
11
, n
nn
nknmnmk
ay
xy
yxy


 ,” Journal of Mathematical
Analysis and Applications, Vol. 323, No. 1, 2006, pp.
26-32. doi:10.1016/j.jmaa.2005.10.031
[6] D. Clark and M. R. S. Kulenović, “A Coupled System of
Rational Difference Equations,” Computers & Mathe-
matics with Applications, Vol. 43, No. 6-7, 2002, pp.
849-867.
[7] D. Clark, M. R. S. Kulenović and J. F. Selgrade, “Global
Asymptotic Behavior of a Two-Dimensional Diserence
Equation Modelling Competition,” Nonlinear Analysis,
Vol. 52, No. 7, 2003, pp. 1765-1776.
doi:10.1016/S0362-546X(02)00294-8
[8] E. Camouzis and G. Papaschinopoulos, “Global Asym-
ptotic Behavior of Positive Solutions on the System of
Rational Difference Equations
11
1, 1
nn
nn
nm nm
x
y
xy
yx


 ,” Applied Mathematics
Letters, Vol. 17, No. 6, 2004, pp. 733-737.
doi:10.1016/S0893-9659(04)90113-9
[9] A. S. Kurbanli, C. Çinar and I. Yalcinkaya, “On the Be-
havaior of Positive Solutions of the System of Rational
Difference Equations11
11
11
,
11
nn
nn
nn nn
xy
xy
yx xy





,”
Mathematical and Computer Modelling, Vol. 53, No. 5-6,
2011, pp. 1261-1267 doi:10.1016/j.mcm.2010.12.009
[10] A. S. Kurbanli, “On the Behavaior of Solutions of the
System of Rational Difference
11
11
11
,
11
nn
nn
nn nn
xy
xy
yx xy





,” World Applied Scien-
ces Journal, 2010. (In Review).
[11] X. Yang, Y. Liu and S. Bai, “On the System of High
Order Rational Difference Equations
, np
nn
npnqnq
by
a
xy
yxy



 ,” Applied Mathematics and
Computation, Vol. 171, No. 2, 2005, pp. 853-856.
doi:10.1016/j.amc.2005.01.092
[12] M. R. S. Kulenović and Z. Nurkanović, “Global Behavior
of a Three-Dimensional Linear Fractional System of Dif-
ference Equations,” Journal of Mathematical Analysis
and Applications, Vol. 310, No. 2, 2005, pp. 673-689.
[13] Y. Zhang, X. Yang, G. M. Megson and D. J. Evans, “On
the System of Rational Difference Equations
1
1, n
nn
npnr ns
y
xA yA
yxy



 ,” Applied Mathematics
and Computation, Vol. 176, No. 2, 2006, pp. 403-408.
doi:10.1016/j.amc.2005.09.039
[14] Y. Zhang, X. Yang, D. J. Evans and C. Zhu, “On the
Nonlinear Difference Equation System
11
,
nm nm
nn
nn
yx
xA yA
x
y


 ,” Computers & Mathe-
matics with Applications, Vol. 53, No. 10, 2007, pp.
1561-1566.
[15] I. Yalcinkaya and C. Cinar, “Global Asymptotic Stability
of Two Nonlinear Difference Equations
11
11
11
,
nn nn
nn
nn nn
tz azta
zt
tz zt








,” Fasciculi Mathematici,
Vol. 43, 2010, pp. 171-180.
[16] I. Yalcinkaya, C. Çinar and M. Atalay, “On the Solutions
of Systems of Difference Equations,” Advances in Diffe-
rence Equations, Article ID: 143943, Vol. 2008, 2008.
doi: 10.1155/2008/143943
[17] I. Yalcinkaya, “On the Global Asymptotic Stability of a
Second-Order System of Difference Equations,” Discrete
Dynamics in Nature and Society, Article ID: 860152, Vol.
2008, 2008. doi: 10.1155/2008/860152
[18] B. Irićanin and S. Stević, “Some Systems of Nonlinear
Difference Equations of Higher Order with Periodic Solu-
tions,” Dynamics of Continuous, Discrete and Impulsive
Systems. Series A Mathematical Analysis, Vol. 13, No.
3-4, 2006, pp. 499-507.