Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 932362, 12 pages
doi:10.1155/2011/932362
Research Article
On the Behavior of Solutions of the System of
Rational Difference Equations:
xn1 xn−1 /yn xn−1 − 1, yn1 yn−1 /xn yn−1 − 1,
and zn1 zn−1 /yn zn−1 − 1
Abdullah Selçuk Kurbanli
Department of Mathematics, Faculty of Education, Selcuk University, 42090 Konya, Turkey
Correspondence should be addressed to Abdullah Selçuk Kurbanli, agurban@selcuk.edu.tr
Received 23 December 2010; Accepted 26 January 2011
Academic Editor: Ibrahim Yalcinkaya
Copyright q 2011 Abdullah Selçuk Kurbanli. This 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 investigate the solutions of the system of difference equations xn1 xn−1 /yn xn−1 − 1,
yn1 yn−1 /xn yn−1 − 1, zn1 zn−1 /yn zn−1 − 1, where y0 , y−1 , x0 , x−1 , z−1 , z0 ∈ R.
1. Introduction
Recently, there has been great interest in studying difference equation systems. One of
the reasons for this is a necessity for some techniques which can be used in investigating
equations arising in mathematical models describing real life situations in population biology,
economic, probability theory, genetics, psychology, and so forth. There are many papers
related to the difference equations system, for example, the following papers.
In 1, Çinar studied the solutions of the systems of the difference equations
xn1 1
,
yn
yn1 yn
.
xn−1 yn−1
1.1
In 2 Papaschinopoulos and Schinas studied the oscillatory behavior, the boundedness of the solutions, and the global asymptotic stability of the positive equilibrium of the
system of nonlinear difference equations
xn1 A yn
,
xn−p
yn1 A xn
,
yn−q
n 0, 1, . . . , p, q.
1.2
2
Discrete Dynamics in Nature and Society
In 3 Papaschinopoulos and Schinas proved the boundedness, persistence, the
oscillatory behavior, and the asymptotic behavior of the positive solutions of the system of
difference equations
k
Ai
xn1 yn1 pi ,
i0 yn−i
k
Bi
q
i
xn−i
i0
1.3
.
In 4, 5 Özban studied the positive solutions of the system of rational difference
equations
xn xn1 a
yn−3
1
yn−k
yn ,
,
yn1
byn−3
,
xn−q yn−q
1.4
yn
.
xn−m yn−m−k
In 6, 7 Clark and Kulenović investigate the global asymptotic stability
xn1 xn
,
a cyn
yn1 yn
.
b dxn
1.5
In 8 Camouzis and Papaschinopoulos studied the global asymptotic behavior of
positive solutions of the system of rational difference equations
xn1 1 xn
,
yn−m
yn1 1 yn
.
xn−m
1.6
In 9 Yang et al. considered the behavior of the positive solutions of the system of the
difference equations
xn a
yn−p
,
yn byn−p
.
xn−q yn−q
1.7
In 10 Kulenović and Nurkanović studied the global asymptotic behavior of solutions
of the system of difference equations
xn1 a xn
,
b yn
yn1 c yn
,
d zn
zn1 e zn
.
f xn
1.8
In 11 Zhang et al. investigated the behavior of the positive solutions of the system of
difference equations
xn A 1
,
yn−p
yn A yn−1
.
xn−r yn−s
1.9
Discrete Dynamics in Nature and Society
3
In 12 Zhang et al. studied the boundedness, the persistence, and global asymptotic
stability of the positive solutions of the system of difference equations
xn1 A yn−m
,
xn
yn1 A xn−m
.
yn
1.10
In 13 Yalcinkaya studied the global asymptotic behavior of a system of two nonlinear
difference equations.
In 14 Yalcinkaya et al. investigated the solutions of the system of difference equations
2
xn
1
xn1 3
xn
2
2
xn − 1
, xn1 1
xn
k
3
xn − 1
, . . . , xn1 1
xn − 1
1.11
.
In 15 Yalcinkaya studied the global asymptotic stability of the system of difference
equations
zn1 tn zn−1 a
,
tn zn−1
zn tn−1 a
.
zn tn−1
tn1 1.12
In 16 Irićanin and Stević studied the positive solutions of the system of difference
equations
2
1
xn1 1
xn1 1
2
xn
1 xn
3
xn−1
3
xn−1
4
xn−2
2
, xn1 2
, xn1 1
3
1 xn
4
xn−1
3
xn
5
xn−2
1
k
, . . . , xn1 4
xn−1
1 xn
2
xn−1
k
, . . . , xn1 1
,
1
xn
2
xn−1
3
xn−2
1.13
.
In 17 Kurbanli et al. studied the behavaior of positive solutions of the system of
rational difference equations
xn1 xn−1
,
yn xn−1 1
yn1 yn−1
.
xn yn−1 1
1.14
Also see references.
In this paper, we investigate the behavior of the solutions of the difference equations
system
xn1 xn−1
,
yn xn−1 − 1
yn1 yn−1
,
xn yn−1 − 1
where the initial conditions are arbitrary real numbers.
zn1 zn−1
,
yn zn−1 − 1
1.15
4
Discrete Dynamics in Nature and Society
Theorem 1.1. Let y0 , y−1 , x0 , x−1 , z−1 , z0 ∈ R be arbitrary real numbers and y0 a, y−1 b, x0 c, x−1 d, z0 e, z−1 f and let {xn , yn , zn } be a solution of the system 1.15. Also, assume that
af /
1 and bc /
1 then all solutions of 1.15 are
⎧
⎪
⎨
d
n-odd,
n,
xn ad − 1
⎪
⎩cbc − 1n , n-even,
⎧
b
⎪
⎨
n-odd,
n,
yn bc − 1
⎪
⎩aad − 1n , n-even,
zn ⎧
f
⎪
⎪
⎪
n
n
n n ,
⎪
⎪
0
1
n
n−1
n−1
n−2
⎪
−1 0 a fd −1 1 a fd · · · −1n−1 n−1 a1 fd0 −1n n
⎪
⎪
⎪
⎪
⎪
⎨
n-odd
⎪
bc − 1n e
⎪
⎪
,
n
n
n
n
⎪
−1n
⎪
⎪
n
1 c0 e · · · −11
n cn−1 e −10
n cn · · · −1n
0 c0
⎪
b
b
b
b
−1
⎪
⎪
n
n
0
1
⎪
⎪
⎩
n-even.
1.16
Proof. For n 0, 1, 2, 3, we have
x2 y2 z2 z3 d
x−1
,
y0 x−1 − 1 ad − 1
y1 y−1
b
,
x0 y−1 − 1 bc − 1
z1 f
z−1
,
y0 z−1 − 1 af − 1
c
x0
cbc − 1
cbc − 1,
y1 x0 − 1 b/bc − 1c − 1 bc − bc 1
y0
a
aad − 1
aad − 1,
x1 y0 − 1 d/ad − 1a − 1 ad − ad 1
e
ebc − 1
ebc − 1
z0
−
,
y1 z0 − 1 b/bc − 1e − 1 be − bc 1
−1 · be bc − 1
x3 y3 x1 d/ad − 1
d
x1
,
y2 x1 − 1 aad − 1 · d/ad − 1 − 1 ad − 12
y1
b/bc − 1
x2 y1 − 1 cbc − 1 · b/bc − 1 − 1
b/bc − 1
b
,
bc − 1
bc − 12
f/ af − 1
f/ af − 1
f
z1
2
2
y2 z1 − 1 aad − 1 · f/ af − 1 − 1
a df − 2af 1
a df − af − af 1 / af − 1
1.17
Discrete Dynamics in Nature and Society
5
for n k assume that
xk yk z2k−1 −1
0
k
0
ak fdk−1
−1
1
⎧
⎪
⎪
⎨
⎪
⎪
⎩
⎧
⎪
⎪
⎨
⎪
⎪
⎩
d
,
ad − 1k
k-odd,
cbc − 1k ,
b
bc − 1k
k-even,
,
k-odd,
aad − 1k ,
f
k
1
k-even,
ak−1 fdk−2
· · · −1k−1
k
k−1
a1 fd0 −1k
k ,
k
k 1, 2, . . . ,
z2k −1k
−1k
k
1
b1 c0 e · · · −11
bc − 1k e
k
bk ck−1 e −10
k
k
k
0
k
bk ck · · · −1k
b1 c1
,
k 1, 2, . . .
1.18
are true. Then for n k 1 we will show that 1.16 is true. From 1.15, we have
x2k1 d/ad − 1k
x2k−1
d
.
y2k x2k−1 − 1 aad − 1k · d/ad − 1k − 1 ad − 1k1
1.19
Also, similarly from 1.15, we have
y2k1 y2k−1
b/bc − 1k
b
,
x2k y2k−1 − 1 cbc − 1k · b/bc − 1k − 1 bc − 1k1
n
k
n!
,
n − k! · k!
1.20
n
n
k
n
0
k−1
n
n
n1
k
n1
n1
,
6
Discrete Dynamics in Nature and Society
from properties of Binomial coefficients,
z2k1 z2k−1
y2k z2k−1 − 1
k
a1 fd0 −1k kk
f/ −10 k0 ak fdk−1 · · · −1k−1 k−1
k
aad − 1k f/ −10 k0 ak fdk−1 · · · −1k−1 k−1
a1 fd0 −1k kk − 1
f
−1
0
k1
0
ak1 fdk −1
1
k1
1
ak fdk−1 · · · −1k
k1
k
a1 fd0 −1k!
k1
k1
1.21
written and accurate.
Also, we have
x2k2 cbc − 1k
x2k
cbc − 1k
cbc − 1k1 ,
y2k1 x2k − 1
b/bc − 1k1 cbc − 1k − 1 bc/bc − 1 − 1
y2k2 y2k
aad − 1k
aad − 1k
aad − 1k! ,
x2k1 y2k − 1
d/ad − 1k1 · aad − 1k − 1 ad/ad − 1 − 1
z2k2 z2k
y2k1 z2k − 1
−1k bc − 1k e/ A −10 k0 bk ck · · · −1k kk b0 c0
−1k1
k1
k
k
0 k
k k
k
k
0
0
b/bc − 1
· −1 bc − 1 e/ A −1 0 b c · · · −1 k b c − 1
×
bc − 1k1 e
−1k1
k1
1
b1 c0 e· · ·−11
where A denotes −1k
k
1
k1
k1
bk1 ck e −10
b1 c0 e · · · −11
k
k
k1
0
bk1 ck1 · · ·−1k1
k1
k1
,
b1 c1
1.22
bk ck−1 e.
Corollary 1.2. Let a, b, c, d be only positive real numbers or negative real numbers and let e, f
be arbitrary nonnegative real numbers, and let {xn , yn , zn } be a solution of the system 1.15.
If ad /
1, bc /
1, 0 < ad < 1 and 0 < bc < 1 then one has
lim x2n−1 lim y2n−1 ∞,
n→∞
n→∞
1.23
lim x2n lim y2n 0.
n→∞
n→∞
Discrete Dynamics in Nature and Society
7
Proof. From ad /
1, bc /
1, 0 < ad < 1 and 0 < bc < 1 we have −1 < ad − 1 < 0 and −1 <
bc − 1 < 0. Hence, we have
x2n−1 d
,
ad − 1n
y2n−1 x2n cbc − 1n ,
b
,
bc − 1n
1.24
y2n aad − 1n .
Then,
lim x2n−1 lim
n→∞
n → ∞ ad
lim y2n−1 lim
n→∞
n → ∞ bc
⎧
⎪
∞, d > 0, n-even
⎪
⎪
⎪
⎪
⎪
⎪
⎨∞, d < 0, n-odd
d
− 1n ⎪
⎪
−∞, d > 0, n-odd
⎪
⎪
⎪
⎪
⎪
⎩−∞, d < 0, n-even,
⎧
⎪
⎪
⎪∞, b > 0, n-even
⎪
⎪
⎪
⎪
⎨∞, b < 0, n-odd
1.25
b
− 1n ⎪
⎪
−∞, b > 0, n-odd
⎪
⎪
⎪
⎪
⎪
⎩
−∞, b < 0, n-even.
Also,
lim x2n lim cbc − 1n 0,
n→∞
n→∞
1.26
n
lim y2n lim aad − 1 0.
n→∞
n→∞
Example 1.3. If y0 −1/2, y−1 −1/2, x0 −1/2, x−1 −1/2, then the solutions of
1.15 can be represented by Table 1.
Corollary 1.4. Let a, b, c, d be only positive real numbers or negative real numbers and let e, f
be arbitrary nonnegative real numbers, and let {xn , yn , zn } be a solution of the system 1.15.
If ad /
1, bc /
1, 1 < ad < 2 and 1 < bc < 2 then one has
lim x2n−1 lim y2n−1 ∞,
n→∞
n→∞
lim x2n lim y2n 0.
n→∞
n→∞
1.27
8
Discrete Dynamics in Nature and Society
Table 1
i
1
2
3
4
5
6
7
8
x2i−1
0.6667
−0.8889
1.1852
−1.5803
2.1070
−2.8093
3.7458
−4.9944
x2i
0.3750
−0.2813
0.2109
−0.1582
0.1187
−0.0890
0.0667
−0.0501
y2i−1
0.6667
−0.8889
1.1852
−1.5803
2.1070
−2.8093
3.7458
−4.9944
y2i
0.3750
−0.2813
0.2109
−0.1582
0.1187
−0.0890
0.0667
−0.0501
z2i−1
0.6667
−0.8889
1.1852
−1.5803
2.1070
−2.8093
3.7458
−4.9944
z2i
0.3750
−0.2813
0.2109
−0.1582
0.1187
−0.0890
0.0667
−0.0501
Proof. From ad / 1, bc / 1, 1 < ad < 2 and 1 < bc < 2 we have 0 < ad − 1 < 1 and 0 < bc − 1 < 1.
Hence, we have
x2n−1 d
,
ad − 1n
x2n cbc − 1n ,
y2n−1 b
,
bc − 1n
1.28
y2n aad − 1n .
Then,
lim x2n−1
n→∞
⎧
⎨∞, d > 0
d
lim
n → ∞ ad − 1n
⎩
−∞, d < 0,
⎧
⎨∞, b > 0
1.29
b
n → ∞ bc − 1n
⎩
−∞, b < 0.
lim y2n−1 lim
n→∞
Also,
lim x2n lim cbc − 1n 0,
n→∞
n→∞
1.30
n
lim y2n lim aad − 1 0.
n→∞
n→∞
Example 1.5. If y0 3/2, y−1 3/4, x0 3/2, x−1 3/4, z0 3/2, z−1 3/4
then the solutions of 1.15 can be represented by Table 2.
Corollary 1.6. Let a, b, c, d be only positive real numbers or negative real numbers and let e, f
be arbitrary nonnegative real numbers, and let {xn , yn , zn } be a solution of the system 1.15.
Discrete Dynamics in Nature and Society
9
If −∞ < ad < 0 and −∞ < bc < 0 then one has
lim x2n−1 lim y2n−1 0,
n→∞
n→∞
1.31
lim x2n lim y2n ∞.
n→∞
n→∞
Proof. From −∞ < ad < 0 and −∞ < bc < 0 we have −∞ < ad − 1 < −1 and −∞ < bc − 1 < −1.
Hence, we have
x2n−1 d
,
ad − 1n
x2n cbc − 1n ,
y2n−1 b
,
bc − 1n
1.32
y2n aad − 1n .
Then,
lim x2n−1 lim
n→∞
n → ∞ ad
d
0,
− 1n
1.33
lim y2n−1 lim
n→∞
n → ∞ bc
b
0.
− 1n
Also,
lim x2n lim cbc − 1n n→∞
n→∞
lim y2n lim aad − 1n n→∞
n→∞
⎧
∞, c > 0, n-even
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨∞, c < 0, n-odd
⎪
⎪
⎪
−∞, c > 0, n-odd
⎪
⎪
⎪
⎪
⎪
⎩
−∞, c < 0, n-even,
⎧
∞, a > 0, n-even
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨∞, a < 0, n-odd
1.34
⎪
⎪
⎪
−∞, a > 0, n-odd
⎪
⎪
⎪
⎪
⎪
⎩
−∞, a < 0, n-even.
Example 1.7. If y0 −3/2, y−1 −3/4, x0 3/2, x−1 3/4, z0 3/2, z−1 3/4 then the solutions of 1.15 can be represented by Table 3.
10
Discrete Dynamics in Nature and Society
Table 2
i
x2i−1
x2i
y2i−1
y2i
z2i−1
z2i
1
6
0.188
6
0.188
6
0.188
2
48
0.023
48
0.023
48
0.023
3
384
0.003
384
0.003
384
0.003
4
3072
0.0004
3072
0.0004
3072
0.0004
5
24576
0.00004
24576
0.00004
24576
0.00004
6
1.967·105
0.00001
1.967·105
0.00001
1.967·105
0.00001
7
1.573 · 106
7.153 · 10−7
1.573 · 106
7.153 · 10−7
1.573 · 106
7.153 · 10−7
8
1.258 · 107
8.941 · 10−8
1.258 · 107
8.941 · 10−8
1.258 · 107
8.941 · 10−8
Table 3
i
x2i−1
x2i
y2i−1
y2i
z2i−1
z2i
1
−0.3529
−3.1875
0.3529
3.1875
−0.3529
−3.1875
2
0.1661
6.7734
−0.1661
−6.7734
0.1661
6.7734
3
−0.0782
−14.3936
0.0782
14.3936
−0.0782
−14.3936
4
0.0368
30.5863
−0.0368
−30.5863
0.0368
30.5863
5
−0.0173
−64.9959
0.0173
64.9959
−0.0173
−64.9959
6
0.0081
138.1163
−0.0081
−138.1163
0.0081
138.1163
7
−0.0038
−293.4971
0.0038
293.4971
−0.0038
−293.4971
8
0.0018
623.6813
−0.0018
−623.6813
0.0018
623.6813
Corollary 1.8. Let a, b, c, d be only positive real numbers or negative real numbers and let e, f be
arbitrary nonnegative real numbers, and let {xn , yn , zn } be a solution of the system 1.15. If 2 <
ad < ∞ and 2 < bc < ∞ then one has
lim x2n−1 lim y2n−1 0,
n→∞
n→∞
1.35
lim x2n lim y2n ∞.
n→∞
n→∞
Proof. From 2 < ad < ∞ and 2 < bc < ∞ we have 1 < ad − 1 < ∞ and 1 < bc − 1 < ∞.
Hence, we have
x2n−1 d
,
ad − 1n
x2n cbc − 1n ,
y2n−1 b
,
bc − 1n
1.36
y2n aad − 1n .
Then,
lim x2n−1 lim
n→∞
n → ∞ ad
d
0,
− 1n
1.37
lim y2n−1 lim
n→∞
n → ∞ bc
b
0.
− 1n
Discrete Dynamics in Nature and Society
11
Table 4
i
x2i−1
x2i
y2i−1
y2i
z2i−1
z2i
1
0.546
4.125
0.909
6.875
0.546
4.125
2
0.198
11.344
0.331
18.906
0.198
11.344
3
0.072
31.195
0.120
51.992
0.072
31.195
4
0.026
85.787
0.044
142.979
0.026
85.787
5
0.010
235.915
0.016
393.191
0.010
235.915
6
0.004
648.765
0.006
1081.275
0.004
648.765
7
0.001
1784.104
0.002
2973.506
0.001
1784.104
8
0.0005
4906.285
0.001
8177.142
0.0005
4906.285
Also,
lim x2n lim cbc − 1n n→∞
n→∞
lim y2n lim aad − 1n n→∞
n→∞
⎧
⎨∞, c > 0
⎩−∞, c < 0,
⎧
⎨∞, a > 0
1.38
⎩−∞, a < 0.
Example 1.9. If y0 5/2, y−1 5/2, x0 3/2, x−1 3/2, z0 3/2, z−1 3/2
then the solutions of 1.15 can be represented by Table 4.
Acknowledgment
The authors are grateful to the anonymous referees for their valuable suggestions that
improved the quality of this study.
References
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