Báo cáo hóa học: " Research Article The Periodic Character of the Difference Equation xn 1 f xn−l 1, xn−2k 1" pdf

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Advances in Difference Equations
Volume 2008, Article ID 143723, 6 pages
doi:10.1155/2008/143723
Research Article
The Periodic Character of the Difference Equation
x
n1
 fx
n−l1
,x
n−2k1

Taixiang Sun
1
and Hongjian Xi
2
1
Department of Mathematics, College of Mathematics and Information Science, Guangxi University,
Nanning 530004, Guangxi, China
2
Department of Mathematics, Guangxi College of Finance and Economics, Nanning 530003,
Guangxi, China
Correspondence should be addressed to Taixiang Sun,
Received 3 February 2007; Revised 18 September 2007; Accepted 27 November 2007
Recommended by H. Bevan Thompson
In this paper, we consider the nonlinear difference equation x
n1
 fx
n−l1
,x

n−2k1
, n  0, 1, ,
where k, l ∈{1, 2, } with 2k
/
 l and gcd 2k, l1 and the initial values x
−α
,x
−α
 1, ,x
0
∈
0, ∞ with α  max{l − 1, 2k − 1}. We give sufficient conditions under which every positive solu-
tion of this equation converges to a  not necessarily prime  2-periodic solution, which extends and
includes corresponding results obtained in the recent literature.
Copyright q 2008 T. Sun and H. Xi. 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.
1. Introduction
In this paper, we consider a nonlinear difference equation and deal with the question of
whether every positive solution of this equation converges to a periodic solution. Recently,
there has been a lot of interest in studying the global attractivity, the boundedness character,
and the periodic nature of nonlinear difference equations e.g., see 1, 2.In3, Grove et al.
considered the following difference equation:
x
n1

p  x
n−2m1
1  x
n−2r

,n 0, 1, , E1
where p ∈ 0, ∞ and the initial values x
−α
,x
−α1
, ,x
0
∈ 0, ∞ with α  max {2r, 2m  1},
and proved that every positive solution of E1 converges to not necessarily prime a2s-
periodic solution with s  gcd m  1, 2r  1.In4, Stevi
´
c investigated the periodic character
of positive solutions of the following difference equation:
x
n1
 1 
x
n−2s1
x
n−2r1s1
,n 0, 1, , E2
2 Advances in Difference Equations
and proved that every positive solution of E2 converges to not necessarily prime a2s-
periodic solution, which generalized the main result of 5. Furthermore, Stevi
´
c 6 studied the
periodic character of positive solutions of the following difference equation:
x
n
 1 


k
i1
α
i
x
n−p
i

m
j1
β
j
x
n−q
j
,n 1, 2, , E3
where α
i
, i ∈{1, ,k},andβ
j
, j ∈{1, ,m}, are positive numbers such that Σ
k
i1
α
i
Σ
m
j1
β

j

1, and p
i
,i∈{1, ,k},andq
j
,j∈{1, ,m}, are natural n umbers such that p
1
<p
2
< ··· <p
k
and q
1
<q
2
< ··· <q
m
. For closely related results, see 7, 8.
In this paper, we consider the more general equation
x
n1
 f

x
n−l1
,x
n−2k1

,n 0, 1, 2, , 1.1

where k,l ∈{1, 2, } with 2k
/
 l and gcd 2k, l1, the initial values x
−α
,x
−α1
, ,x
0
∈
0, ∞ with α  max {l − 1, 2k − 1},andf satisfies the following hypotheses:
H
1
 f ∈ CE×E, 0,∞ with ainf
u,v∈E×E
fu, v∈E,whereE ∈{0,∞, 0,∞};
H
2
 fu, v is decreasing in u and increasing in v;
H
3
 there exists a decreasing function g ∈ Ca, ∞, a, ∞ such that
i for any x>a, ggx  x and x  fgx,x;
ii lim
x→a

gx∞ and lim
x→∞
gxa.
The main result of this paper is the following theorem.
Theorem 1.1. Every positive solution of 1.1  converges to (not necessarily prime) a 2-periodic solu-

tion.
2. Proof of Theorem 1.1
In this section, we will prove Theorem 1.1. Without loss of generality, we may assume l<2k
the proof for the case l>2k is similar ;then
{l, 2l, 3l, ,2kl}  {0, 1, 2, ,2k − 1} mod 2k. 2.1
Lemma 2.1. Let {x
n
}
∞
n−α
be a positive solution of 1.1. Then there exists a real number L ∈ a, ∞
such that L ≤ x
n
≤ gL for all n ≥ 1. Furthermore, let lim sup x
n
 M and lim inf x
n
 m, then
M  gm and m  gM.
Proof. By H
1
 and H
2
,wehave
x
i
 f

x
i−l

,x
i−2k

>f

x
i−l
 1,x
i−2k

≥ a for every 1 ≤ i ≤ α  1. 2.2
Then there exists L ∈ a, ∞ with L<gL such that
L ≤ x
i
≤ gL for every 1 ≤ i ≤ α  1. 2.3
T. Sun and H. Xi 3
It follows from 2.3 and H
3
 that
gLf

L, gL

≥ x
α2
 f

x
α2−l
,x

α2−2k

≥ f

gL,L

 L. 2.4
Inductively, it follows that L ≤ x
n
≤ gL for all n ≥ 1.
Let lim sup x
n
 M and lim inf x
n
 m, then there exist A, B, C, D ∈ m, M and seque-
nces t
n
≥ 1andr
n
≥ 1 such that
lim
n→∞
x
t
n
 M, lim
n→∞
x
t
n

−l
 A, lim
n→∞
x
t
n
−2k
 B,
lim
n→∞
x
r
n
 m, lim
n→∞
x
r
n
−l
 C, lim
n→∞
x
r
n
−2k
 D.
2.5
Thus by 1.1, H
2
,andH

3
,wehave
f

gM,M

 M  fA, B ≤ fm, M,
f

gm,m

 m  fC, D ≥ fM, m,
2.6
from which it follows that gM ≥ m and gm ≤ M. Since g is decreasing, it follows that
m  g

gm

≥ gM,M g

gM

≤ gm. 2.7
Therefore, M  g
m and m  gM. The proof is complete.
Proof of Theorem 1.1. Let {x
n
}
∞
n−α

be a positive solution of 1.1 with the initial conditions
x
0
,x
−1
, ,x
−α
∈ 0, ∞. It follows from Lemma 2.1 that
a<lim inf x
n
 m  gM ≤ lim sup x
n
 M<∞. 2.8
Obviously, every sequence
L, gL,L,gL, 2.9
is a 2-periodic not necessarily prime solution of 1.1,whereL ∈{M, m}.
By taking a subsequence, we may assume that there exists a sequence t
n
≥ 2kl  1 such
that
lim
n→∞
x
t
n
 M,
lim
n→∞
x
t

n
−j
 A
j
∈

gM,M

for j ∈{1, 2, ,2kl}.
2.10
According to 1.1, 2.10,andH
3
,weobtain
f

gM,M

 M  f

A
l
,A
2k

≤ f

gM,M

, 2.11
from which it follows that

A
l
 gM,A
2k
 M. 2.12
4 Advances in Difference Equations
In a similar fashion, we can obtain
f

gM,M

 M  A
2k
 f

A
2kl
,A
4k

≤ f

gM,M

,
f

M, gM

 gMA

l
 f

A
2l
,A
l2k

≥ f

M, gM

,
2.13
from which it follows that
A
4k
 A
2k
 A
2l
 M, A
2kl
 A
l
 gM. 2.14
Inductively, we have
A
j2k
 M for j ∈{1, 2, ,l},

A
jl
 gM for j ∈{1, 3, ,2k − 1},
A
jl
 M for j ∈{0, 2, ,2k},
A
jlr2k
 A
jl
for j ∈{0, 1, ,2k},r∈{0, 1, ,l},jl r2k ≤ 2kl.
2.15
For every r ∈{0, 1, 2, 3, ,2k − 1}, there exist j
r
∈{0, 1, 2, 3, ,2k − 1} and p
r
∈
{0, 1, ,l− 1} such that j
r
l  2kp
r
 r, from which, with 2.15, it follows that
A
2kl−1r
 A
j
r
l



M for r ∈{0, 2, 4, ,2k − 2},
gM for r ∈{1, 3, ,2k − 1},
2.16
lim
n→∞
x
t
n
−2kl−1−j
 M for j ∈{0, 2, ,2k},
lim
n→∞
x
t
n
−2kl−1−j
 gM for j ∈{1, 3, ,2k − 1}.
2.17
In view of 2.17, for any 0 <ε<M− a, there exists some t
β
≥ 4kl such that
M − ε<x
t
β
−2kl−1−j
<M ε if j ∈{0, 2, ,2k},
gM  ε <x
t
β
−2kl−1−j

<gM − ε if j ∈{1, 3, ,2k − 1}.
2.18
By 1.1 and 2.18,wehave
x
t
β
−2kl−11
 f

x
t
β
−2kl−1−l1
,x
t
β
−2kl1

<f

M − ε, gM − ε

 gM − ε. 2.19
Also 1.1, 2.18,and2.19 imply that
x
t
β
−2kl−12
 f


x
t
β
−2kl−1−l2
,x
t
β
−2kl2

>f

gM − ε,M− ε

 M − ε. 2.20
Inductively, it follows that
x
t
β
−2kl−12n
>M− ε ∀n ≥ 0,
x
t
β
−2kl−12n1
<gM − ε ∀n ≥ 0.
2.21
T. Sun and H. Xi 5
Therefore,
lim
n→∞

x
2n
 M, lim
n→∞
x
2n1
 gM
2.22
or
lim
n→∞
x
2n
 gM, lim
n→∞
x
2n1
 M.
2.23
The proof is complete.
Remark 2.2. 1 The proofs of Lemma 2.1 and Theorem 1.1 draw on ideas from the proofs of
Theorems 2.1 and 2.2 in 6.
2 Consider the nonlinear difference equation
x
n1
 f

x
n−ls1
,x

n−2ks1

,n 0, 1, , 2.24
where s, k, l ∈{1, 2, } with 2k
/
 l and gcd 2k, l1, the initial values x
−α
,x
−α1
, ,x
0
∈
0, ∞ with α  max {ls − 1, 2ks − 1},andf satisfies H
1
–H
3
.Lety
i
n1
 x
nsi1
for every
0 ≤ i ≤ s − 1andn  0, 1, 2, ,then2.24 reduces to the equation
y
i
n1
 f

y
i

n−l1
,y
i
n−2k1

, 0 ≤ i ≤ s − 1,n 0, 1, 2, 2.25
It follows from Theorem 1.1 that for any 0 ≤ i ≤ s − 1, every positive solution of the equation
y
i
n1
 fy
i
n−l1
,y
i
n−2k1
 converges to not necessarily prime a 2-periodic solution. Thus every
positive solution of 2.24 converges to not necessarily prime a2s-periodic solution.
3. Examples
To illustrate the applicability of Theorem 1.1, we present the following examples.
Example 3.1. Consider the equation
x
n1

p 

m1
i1
x
i

n−2k1

m
i0
x
i
n−2k1
 x
n−l1
,n 0, 1, , 3.1
where m, k, l ∈{1, 2, } with 2k
/
 l and gcd 2k, l1 and the initial values x
−α
,x
−α1
, ,
x
0
∈ 0, ∞ with α  max {l − 1, 2k − 1},0<p≤ 1. Let E 0, ∞ and
fx, y
p 

m1
i1
y
i

m
i0

y
i
 x
x ≥ 0,y≥ 0,gx
p
x
x>0. 3.2
6 Advances in Difference Equations
It is easy to verify that H
1
–H
3
 hold for 3.1. It follows from Theorem 1.1 that every solution
of 3.1 converges to not necessarily prime a 2-periodic solution.
Example 3.2. Consider the equation
x
n1
 1 
x
m1
n−2k1

m
i1
x
i
n−2k1
 x
n−l1
,n 0, 1, , 3.3

where m, k, l ∈{1, 2, } with 2k
/
 l and gcd 2k, l1 and the initial values x
−α
,x
−α1
, ,
x
0
∈ 0, ∞ with α  max {l − 1, 2k − 1}.LetE 0, ∞ and
fx, y1 
y
m1

m
i1
y
i
 x
x>0,y>0,gx
x
x − 1
x>1. 3.4
It is easy to verify that H
1
–H
3
 hold for 3.3. It follows from Theorem 1.1 that every solution
of 3.3 converges to not necessarily prime a 2-periodic solution.
Acknowledgments

The authors would like to thank the referees for some valuable and constructive comments
and suggestions. The project is supported by NNSF of China 10461001 and NSF of Guangxi
0640205, 0728002.
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 gx
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,x
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/Ax
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,” Applied Mathematics Letters, vol. 15, no. 3,
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´
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n
 1 

k
i1
α
i
x
n−p
i
/

m
j1
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j
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n−q
j

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
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n
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´
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n
 1  y
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/y
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