Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 436852, 15 pages
doi:10.1155/2011/436852
Research Article
On a Class of Nonautonomous Max-Type
Difference Equations
Wanping Liu,1 Xiaofan Yang,1 and Stevo Stević2
1
2
College of Computer Science, Chongqing University, Chongqing 400044, China
Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia
Correspondence should be addressed to Wanping Liu, wanping liu@yahoo.cn
Received 15 February 2011; Accepted 14 June 2011
Academic Editor: Gaston Mandata N’Guerekata
Copyright q 2011 Wanping Liu et al. 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.
β
α
, B/xn−m }, n ∈ N0 , where
This paper addresses the max-type difference equation xn max{fn /xn−k
k, m ∈ N, B > 0, and fn n∈N0 is a positive sequence with a finite limit. We prove that every positive
solution to the equation converges to max{limn → ∞ fn 1/α1 , B 1/β1 } under some conditions.
Explicit positive solutions to two particular cases are also presented.
1. Introduction
The study of difference equations, which usually depicts the evolution of certain phenomena
over the course of time, has a long history. Many experts recently pay some attention to socalled max-type difference equations which stem from certain models in control theory, see,
for example, 1–23 and the references therein.
The study of the following family of max-type difference equations
xn max
r1
0
1 xn−p1
Bn , Bn s 1
xn−q1
r2
2 xn−p2
, Bn s 2
xn−q2
rk
k xn−pk
, . . . , Bn sk
xn−qk
,
n ∈ N0 ,
1.1
where pi , qi ∈ N such that 1 ≤ p1 < · · · < pk , 1 ≤ q1 < · · · < qk , ri , si ∈ R , k ∈ N and
i
Bn i 0, 1, . . . , k are real sequences, was proposed by S. Stević at numerous conferences, for
example, 10, 11. For some results in this direction, see 1, 2, 4, 12–23.
2
Abstract and Applied Analysis
In the beginning of the investigation the following equation was studied:
1
2
k
A n An
An
,
,...,
xn max
,
xn−1 xn−2
xn−k
n ∈ N0 ,
1.2
i
where k ∈ N, An n∈N0 , i 1, . . . , k are real sequences and the initial values are nonzero see,
e.g., 3, 5, 6, 9 and the related references therein.
In 22, Sun studied the second-order difference equation
⎧
⎫
⎨ A
B ⎬
, β
,
xn max
α
⎩ xn−1 x ⎭
n ∈ N0 ,
1.3
n−2
with α, β ∈ 0, 1, A, B > 0, and proved that each positive solution to 1.3 converges to
the equilibrium point max{A1/α1 , B1/β1 }, by considering several subcases. However, the
method used there is a bit complicated and difficult for extending. Hence in 14 Stević
extended this, as well as the main result in 13, by presenting a more concise and elegant
proof of the next theorem.
Theorem 1.1 see 14, Theorem 1. Every positive solution to the difference equation
A2
Ak
A1
xn max α1 , α2 , . . . , αk
xn−p1 xn−p2
xn−pk
,
n ∈ N0 ,
1.4
where pi , i 1, . . . , k are natural numbers such that 1 ≤ p1 < · · · < pk , k ∈ N and Ai > 0, αi ∈
1/α 1
−1, 1, i 1, . . . , k, converges to max1≤i≤k {Ai i }.
Definition 1.2. Let F : N0 × Rk → R be a function of k 1 variables, then the difference
equation
xn Fn, xn−1 , . . . , xn−k ,
n ∈ N0 ,
1.5
is called nonautonomous or time variant.
Note that the following nonautonomous difference equation
1
2
l
A n An
An
xn max α1 , α2 , . . . , αl ,
xn−1 xn−2
xn−l
i
n ∈ N0 ,
1.6
where l ∈ N, αi ∈ R, and An n∈N0 , i 1, . . . , l are real sequences not all constant, is a natural
generalization of 1.2, 1.3, and 1.4. It is a special case of 1.1 of particular interest.
The aforementioned works are mainly devoted to the study of 1.6 with constant or
periodic numerators.
Abstract and Applied Analysis
3
This paper is devoted to the study of the following nonautonomous max-type
difference equation with two delays:
fn
B
xn max α , β
xn−k x
,
1.7
n ∈ N0 ,
n−m
where k, m ∈ N, α, β ∈ R are fixed and fn n∈N0 is a positive sequence with a finite
limit. Inspired by the methods and proofs of the above-mentioned papers, here we try
to find some sufficient conditions such that every positive solution to 1.7 converges to
max{limn → ∞ fn 1/α1 , B1/β1 }.
This paper proceeds as follows. Several useful lemmas are given in Section 2. In
Section 3 we establish three main results about the global attractivity of 1.7 under some
conditions. Finally motivated by a recent theorem in 21, explicit solutions to two particular
cases of 1.7 are presented in Section 4.
2. Auxiliary Results
To establish the main results in Section 3, here we present several lemmas. First we extend
Lemma 2.4 in 21 by proving the following result.
Lemma 2.1. Consider the nonautonomous difference equation
zn min{C1 n − α1 nzn−1 , . . . , Ck n − αk nzn−k },
n ∈ N0 ,
2.1
where k ∈ N and αi n, Ci n, i 1, 2, . . . , k are sequences. If Ci ns are nonnegative sequences and
there always exists i0 ∈ {1, 2, . . . , k} such that Ci0 n 0 for each fixed n ∈ N0 , then
|zn | ≤ max{|α1 n||zn−1 | − C1 n, . . . , |αk n||zn−k | − Ck n},
n ∈ N0 .
2.2
Proof. Suppose that n ∈ N0 is fixed, and denote by S ⊆ {1, . . . , k} the set of all indices for
which the terms in 2.1 are negative.
If S ∅, which means all terms in the right-hand side of 2.1 are nonnegative, then
apparently
0 ≤ zn ≤ −αi0 nzn−i0
2.3
|zn | ≤ |αi0 n||zn−i0 | − Ci0 n.
2.4
which implies
Otherwise, S /
∅, which means that there exist indices such that the corresponding
terms in 2.1 are negative, then we derive
zn min Cj n − αj nzn−j < 0.
j∈S
2.5
4
Abstract and Applied Analysis
Since αj nzn−j must be positive for j ∈ S, it follows from 2.5 that
|zn | max αj nzn−j − Cj n max αj nzn−j − Cj n .
j∈S
j∈S
2.6
Inequality 2.2 follows easily from 2.4 and 2.6.
The following lemma is widely used in the literature.
Lemma 2.2 see 24. Let an n∈N be a sequence of nonnegative numbers which satisfies the inequality
ank ≤ q max{ank−1 , ank−2 , . . . , an },
for n ∈ N,
2.7
where q > 0 and k ∈ N are fixed. Then there exists an M ≥ 0 such that
an ≤ M
n
k
q ,
n ∈ N,
2.8
which implies an → 0 as n → ∞ if 0 < q < 1.
Lemma 2.3. Assume that xn n≥−k is a sequence of nonnegative numbers satisfying the difference
inequality
xn ≤ max γ1 xn−1 − d1 n, . . . , γk xn−k − dk n ,
n ∈ N0 ,
2.9
where k ∈ N, γi ∈ 0, 1, and di n, i 1, . . . , k are nonnegative sequences. If there exists at least one
positive γi , then the sequence xn converges to zero as n → ∞.
Proof. This lemma follows directly from Lemma 2.2 since
xn ≤ max γ1 xn−1 − d1 n, . . . , γk xn−k − dk n
≤ max γ1 xn−1 , . . . , γk xn−k ≤ γ max{xn−1 , . . . , xn−k },
2.10
where 0 < γ max{γ1 , γ2 , . . . , γk } < 1.
Remark 2.4. If in Lemma 2.3, we assume γi 0, i 1, . . . , k, then the statement also holds,
since in this case, if such a sequence exists, then the solution must be trivial, that is, xn 0,
n ∈ N0 for some results on the existence of nontrivial solutions, see, e.g., 25–27 and the
references therein.
Through some simple calculations, we have the following result.
Lemma 2.5. Every positive solution xn n≥−1 to the first-order difference equation
xn A1−1/ω xn−1 1/ω ,
n ∈ N0 ,
2.11
Abstract and Applied Analysis
5
with A > 0, ω > 0, x−1 > 0, has the form
n1
n1
xn A1−1/ω x−1 1/ω ,
n ∈ N0 .
2.12
Note that Lemma 2.5 leads to the following corollary.
Corollary 2.6. Each positive solution Zn n≥−k to the k th-order difference equation
Zn A1−1/ω Zn−k 1/ω ,
n ∈ N0 ,
2.13
where A > 0, ω > 0, k ∈ N and the initial values Z−1 , . . . , Z−k are positive, has the following form:
n/k1
Zn A1−1/ω
Zρn,k−k
1/ωn/k1
,
n ≥ 0,
2.14
where · represents the integer part function and ρn, k n − k · n/k.
Remark 2.7. By Corollary 2.6 we have that for any positive solution Zn n≥−k to 2.13 the
following three statements hold true if ω > 1:
1 limn → ∞ Zn A;
2 if Zi < A for every i ∈ {−k, . . . , −1}, then the subsequences Zjki j≥0 are all strictly
increasing;
3 if Zi > A for every i ∈ {−k, . . . , −1}, then the subsequences Zjki j≥0 are all strictly
decreasing.
3. Main Results
In this section, we prove the main results of this paper, which concern the global attractivity
of positive solutions to 1.7 under some conditions. In the sequel, we assume that there is a
finite limit of the positive sequence fn n∈N0 in 1.7.
Theorem 3.1. Consider 1.7, where fn n∈N0 is a positive monotone sequence with finite limit A > 0.
If |α| < 1, |β| < 1, B > Aβ1/α1 , then every positive solution to 1.7 converges to B1/β1 .
Proof. By the change xn yn B1/β1 , 1.7 is transformed into
Cn
1
yn max
α , β
yn−k
y
,
n ∈ N0 ,
3.1
n−m
with Cn fn /Bα1/β1 , n ∈ N0 . Note that the sequence Cn n∈N0 is also monotone and
limn → ∞ Cn A/Bα1/β1 < 1.
According to the assumption the sequence fn n∈N0 is nondecreasing or nonincreasing.
If fn n∈N0 is nonincreasing, then for some fixed ε ∈ 0, Bα1/β1 − A, there exists a natural
number N such that for every n ≥ N we have fn − A < ε, which implies
0 < Cn < 1,
n ≥ N.
3.2
6
Abstract and Applied Analysis
On the other hand, if fn n∈N0 is nondecreasing then obviously Cn < 1 for each n ∈ N0 ,
hence 3.2 also holds for this case.
Let D ∈ 0, 1 be fixed. Employing the transformation yn Dzn , 3.1 becomes
Dzn max
Cn
1
,
,
Dαzn−k Dβzn−m
n ∈ N0 ,
3.3
which implies
zn min logD Cn − αzn−k , −βzn−m ,
n ∈ N0 .
3.4
Note that logD Cn > 0 for all n ≥ N. From this and by Lemma 2.1 we get
|zn | ≤ max |α||zn−k | − logD Cn , β|zn−m | ,
n ≥ N.
3.5
When both α and β are zero, it is clear that zn is always zero for n ≥ N. Otherwise, it follows
from Lemma 2.3 that limn → ∞ |zn | 0, which implies
lim zn 0.
3.6
n→∞
Finally, from the above two transformations we get
lim xn B1/β1 lim yn B1/β1 Dlimn → ∞ zn B1/β1 .
n→∞
n→∞
3.7
The proof is complete.
Theorem 3.2. Consider 1.7. Let fn n≥−k be a positive solution to 2.13 such that fi < A (or
fi > A), i −k, . . . , −1, and denote
Γ sup
n≥m
ln fn−m − ln A
.
ln fn − ln A
3.8
If ω > 1, 0 < αω < 1, |β|Γ < 1, B < Aβ1/α1 , then every positive solution to 1.7 converges to
A1/α1 .
Proof. Employing the transformation xn yn A1/α1 , 1.7 becomes
Cn
λ
yn max
α , β
yn−k
y
n−m
where Cn fn /A, n ∈ N0 and λ B/Aβ1/α1 .
,
n ∈ N0 ,
3.9
Abstract and Applied Analysis
7
Then by the change yn Cnzn , 3.9 is transformed into
Cnzn
Cn
λ
max
αzn−k , βzn−m ,
Cn−k
Cn−m
n ≥ max{k, m}.
3.10
In the sequel, we proceed by considering two cases.
Case 1. Let fi < A, i −k, . . . , −1.
By Remark 2.7, we have 0 < Cn < 1, n ∈ N0 . From 3.10 we get
zn min 1 − α logCn Cn−k zn−k , logCn λ − β logCn Cn−m zn−m
min 1 − αωzn−k , logCn λ − β logCn Cn−m zn−m ,
3.11
for n ≥ max{k, m}. By the change zn gn 1/αω 1, 3.11 becomes
gn min −αωgn−k , Tn − β logCn Cn−m gn−m ,
n ≥ max{k, m},
3.12
where
Tn logCn λ −
βlogCn Cn−m 1
αω 1
3.13
.
Claim 1. There exists an integer M > 0 such that Tn > 0 for every n ≥ M.
Proof. Since fn ACn , we easily have that
βlog Cn−m ≤ βsup log Cj−m βΓ < 1,
Cn
Cj
n ≥ m.
3.14
j≥m
Hence
0<
βlogCn Cn−m 1
αω 1
< 2,
n ≥ m.
3.15
√
On the other hand,
√ for ε A1 − λ, there exists an M > 0 such that for each n ≥ M we have
fn > A − ε A λ, which along with the fact Cn ∈ 0, 1, n ∈ N0 , implies that
logCn λ > 2,
n ≥ M.
The claim follows directly from 3.15 and 3.16, as desired.
3.16
8
Abstract and Applied Analysis
Next, from Lemma 2.1 and 3.12 it follows that
gn ≤ max αωgn−k , β log Cn−m gn−m − Tn
Cn
≤ max αωgn−k , βΓgn−m − Tn ,
3.17
n ≥ max{k, m, M}.
From 3.17 and by Lemma 2.3, we derive limn → ∞ |gn | 0. Hence
lim ln yn lim zn ln Cn lim zn · ln
n→∞
n→∞
n→∞
lim Cn
n→∞
1
· ln 1 0,
αω 1
3.18
and consequently
lim xn A1/α1 lim yn A1/α1 .
n→∞
3.19
n→∞
Case 2. Let fi > A, i −k, . . . , −1.
By Remark 2.7, we have Cn > 1, n ∈ N0 , and 3.10 is transformed into
zn max 1 − α logCn Cn−k zn−k , logCn λ − β logCn Cn−m zn−m
max 1 − αωzn−k , logCn λ − β logCn Cn−m zn−m ,
3.20
for all n ≥ max{k, m}. Then employing the following change
zn −gn 1
,
αω 1
3.21
3.20 is transformed into
gn min −αωgn−k , Tn − β logCn Cn−m gn−m ,
n ≥ max{k, m},
3.22
where Tn −logCn λ βlogCn Cn−m 1/αω 1. In this case, Tn > 0 obviously holds. The rest
of the proof is similar to that of Case 1 so is omitted.
To illustrate Theorem 3.2, we present the following example.
Example 3.3. Consider the difference equation
fn
B
xn max α , β
xn−1 x
,
n ∈ N0 ,
n−m
n1
where B > 0, m ≥ 2, and fn Aq1/ω , ω > 1, A, q > 0, q /
1, n ∈ N0 .
3.23
Abstract and Applied Analysis
9
By Theorem 3.2 and through some calculations, we obtain
ln fn−m − ln A
Γ sup
ln fn − ln A
n≥m
ωn1
sup
n−m1
n≥m ω
ωm .
3.24
Hence if 0 < α < 1/ω, |β| < 1/ωm , B < Aβ1/α1 , then every positive solution to 3.23
converges to A1/α1 .
Theorem 3.4. Consider 1.7. If fn n∈N0 is an increasing sequence converging to A, |α|, |β| < 1 and
B Aβ1/α1 , then every positive solution to 1.7 converges to A1/α1 .
Proof. By the change xn yn A1/α1 , 1.7 becomes
1
Cn
yn max
α , β
yn−k
y
,
n ∈ N0 ,
3.25
n−m
where Cn fn /A < 1, n ∈ N0 . The rest of the proof is analogous to that of Theorem 3.1 and
thus is omitted.
4. Explicit Solutions
Recently, Stević and Iričanin in 21 proved the following theorem.
Theorem 4.1 see 21, Theorem 2.8. Consider
a1
ak , . . . , xn−k
xn max xn−1
,
n ∈ N0 ,
4.1
where k ∈ N, ai ∈ R, i 1, . . . , k. Then every well-defined solution of the equation has the following
form:
j
in
j1 aj
k
xn dn
1
k
,
4.2
j
where n k/k ≤ in · · · in ≤ n 1, n ∈ N0 , in ≥ 0, j 1, . . . , k, and where dn is equal to
one of the initial values x−k , . . . , x−1 .
The result is interesting since 4.2 holds for all real aj ’s and for all nonzero initial
values if one of these exponents is negative. However, 4.2 does not give explicit solutions
j
to 4.1 since dn ’s and in ’s in 4.2 are uncertain. Thus the problem of finding more explicit
expressions of solutions to 4.1 is of interest.
10
Abstract and Applied Analysis
In this section we find explicit solutions to the next particular cases of 4.1
1
xn max
1
,
p ,
xn−1 xn−2
1
1
,
xn max
, p
xn−1 xn−2
n ∈ N0 ,
4.3
n ∈ N0 ,
4.4
with p > 1 and positive initial values x−2 , x−1 . First we prove a useful lemma.
Lemma 4.2. Let xn n∈N0 be a positive solution to 4.3 or 4.4. If there exists an N ∈ N0 such that
p
p
xN3 xN ,
p
xN4 xN1 ,
xN5 xN2 ,
4.5
then for each k ∈ N the following equalities hold:
pk
pk
xN3k xN ,
pk
xN3k1 xN1 ,
xN3k2 xN2 .
4.6
Proof. We will only consider 4.3, because similar proof can be given to 4.4. The case k 1
obviously holds due to 4.5. Next assume that 4.6 holds for 1 ≤ k ≤ m for some m ∈ N.
Then by 4.3 we derive
xN3m1 max
,
p
xN3m2 xN3m1
max
xN3m11 max
⎧
⎨
1
,
p
xN2
max
pm
1
1
,
p
xN3 xN2
⎫
⎬
N3
N2
p m
p
m
p
xN4 xN1 ,
⎩ xp
p m
1
1
,
p
xN4
1
xN3
4.7
m1
⎧
⎫
⎨
1
1
1 ⎬
max
,
,
m
m1
p
⎩ xp
⎭
xN3m1 ⎭
x
⎧
⎨
N3m11
max
pm1
⎧
⎫
⎨ 1
1 ⎬
,
,
max
⎩ xpm1 xpm ⎭
xN3m2 ⎭
1
N3m1
N1
N2
xN3 xN ,
xN1
1
⎧
⎫
⎨ 1
1 ⎬
max
,
⎩ xpm1 xpm ⎭
p m
1
⎩ xp
xN3m12 max
1
1
⎫
⎬
N4
pm
N3
pm1
xN5 xN2 .
Thus 4.6 holds for k m 1, finishing the inductive proof of the lemma.
Proposition 4.3. Let xn n∈N0 be a solution to 4.3 with p > 1 and positive initial values x−2 , x−1 ,
then for each k ∈ N0 the following statements hold true.
Abstract and Applied Analysis
11
p
pk1
p
pk1
pk2
pk1
1 If x−2 > x−1 , x−1 ≥ 1, then x3k 1/x−1 , x3k1 x−1 , x3k2 x−1 .
pk
pk1
2 If x−2 > x−1 , x−1 < 1, then x3k 1/x−1 , x3k1 1/x−1 , x3k2 x−1 .
p
pk
p
pk1
3 If x−2 ≤ x−1 , x−2 x−1 ≥ 1 then x3k 1/x−2 , x3k1 x−2 and
pk
x3k2 x−2 if x−2 ≥ 1 or
p
1
pk2
x−2
if x−2 < 1.
pk
p
4.8
pk1
4 If x−2 ≤ x−1 , x−2 x−1 < 1 then x3k1 1/x−1 , x3k2 x−1 and
pk
x3k3 x−1 if x−1 ≥ 1 or
1
pk2
x−1
if x−1 < 1.
4.9
p
Proof. 1 By the assumption x−2 > x−1 and 4.3 it follows that
1
1
x0 max p ,
x
x−1 −2
1
p
x−1
4.10
.
Then by x−1 > 1 and 4.3, we have the following equalities:
1
p2
p2
max x−1 ,
x−1 ,
x−1
⎧
⎫
⎨ 1
⎬
1 1
p
p
x−1 ,
max
x2 max p ,
,
x
⎩ xp3 −1 ⎭
x1 x0
1
1
x1 max p ,
x0 x−1
−1
⎧
⎫
⎨ 1
1
1 1
1 ⎬
p
max
p2 x0 ,
, p2
x3 max p ,
2
p
⎩x x ⎭ x
x2 x1
−1
−1
−1
1 1
1
p3
p3
p
x4 max p ,
max x−1 , p
x−1 x1 ,
x3 x2
x−1
⎧
⎫
⎨
⎬
2
1
1 1
p
p2
p
max
x5 max p ,
x
,
x
x2 .
4
−1
−1
⎩ xp
⎭
x x3
4
4.11
−1
Hence 4.5 is satisfied for N 0. Then by Lemma 4.2 we have that
pk1
x3k 1/x−1 ,
as desired.
pk2
x3k1 x−1 ,
pk1
x3k2 x−1 ,
k ∈ N0 ,
4.12
12
Abstract and Applied Analysis
2 By similar calculations as in 1, the following equalities hold:
x0 1
x1 p ,
x−1
x4 1
,
x−1
1
p
x−1
p
x2 x−1 ,
p
x1 ,
x5 x3 p2
x−1
1
p2
x−1
p
x0 ,
4.13
p
x2 .
Thus 4.5 holds for N 0, and the result follows again by Lemma 4.2.
p
3 By the assumption x−2 ≤ x−1 and 4.3 it follows that
1
x0 max p ,
x−1 x−2
1
1
.
x−2
4.14
p
Then from x−2 x−1 ≥ 1 and 4.3, we get
1
1
x1 max p ,
x0 x−1
1
p
p
max x−2 ,
x−2 .
x−1
4.15
Now we will consider two cases x−2 ≥ 1 and x−2 < 1. When x−2 ≥ 1, the following equalities
hold:
1
1
x2 max p ,
x1 x0
⎧
⎫
⎨ 1
⎬
max
,
x
x−2 ,
−2
⎩ x p2
⎭
1
4
−2
1
1
1
1
p
x3 max p ,
max p , p
p x0 ,
x2 x1
x−2 x−2
x−2
1
1 1
p2
p2
p
max x−2 ,
x−2 x1 ,
x4 max p ,
x
x
−2
x3 2
⎧
⎫
⎨ 1
⎬
1 1
p
p
p
max
x−2 x2 .
x5 max p ,
,
x
3
−2
p
⎩
⎭
x
3
x
x
4.16
−2
On the other hand, the case x−2 < 1 leads to
x2 1
p2
x−2
,
x3 1
p
x−2
p
x0 ,
p2
p
x4 x−2 x1 ,
x5 1
p3
x−2
p
x2 .
4.17
Hence 4.5 holds for N 0 no matter the value of x−2 is bigger or less than one. From this
the result follows by Lemma 4.2.
Abstract and Applied Analysis
13
4 Through analogous calculations to 3, if x−1 ≥ 1 then
x0 1
,
x−2
x4 1
p
x−1
x1 1
,
x−1
p
x2 x−1 ,
p2
p
x1 ,
p
x3 x−1 ,
p
x5 x−1 x2 ,
4.18
p
x6 x−1 x3 .
If x−1 < 1 then
x0 1
,
x−2
x4 1
p
x−1
x1 p
x1 ,
1
,
x−1
x5 p
x2 x−1 ,
p2
x−1
p
x2 ,
x3 x6 1
p3
x−1
1
p2
,
x−1
4.19
p
x3 .
Hence 4.5 holds for N 1 and any x−1 > 0. Hence, the results also follow from Lemma 4.2,
finishing the proof of the proposition.
The next proposition can be similarly proved as the proof of Proposition 4.3, hence the
proof is omitted here.
Proposition 4.4. Let xn n∈N0 be a solution to 4.4 with p > 1 and positive initial values x−2 , x−1 ,
then for each k ∈ N0 the following statements hold true.
p
pk
p
pk
pk
pk1
1 If x−2 ≥ x−1 , x−1 ≥ 1, then x3k 1/x−1 , x3k1 x−1 , x3k2 −1 .
pk1
pk1
2 If x−2 ≥ x−1 , x−1 < 1, then x3k 1/x−1 , x3k1 1/x−1 , x3k2 x−1 .
pk1
p
pk1
3 If x−2 < x−1 , x−2 x−1 ≥ 1 then x3k 1/x−2 , x3k1 x−2 , and
x3k2
⎧ pk2
⎪
⎪
⎨x−2
1
⎪
⎪
⎩ pk1
x−2
if x−2 ≥ 1,
pk1
p
4.20
if x−2 < 1.
pk1
4 If x−2 < x−1 , x−2 x−1 < 1 then x3k1 1/x−1 , x3k2 x−1 , and
x3k3
⎧ pk2
⎪
⎪
⎨x−1
1
⎪
⎪
⎩ pk1
x−1
if x−1 ≥ 1,
if x−1 < 1.
4.21
Remark 4.5. From the above propositions, we know that any positive solution xn n∈N0 to 4.3
or 4.4 can be divided into three subsequences which have explicit expressions. If we regard
the sequence . . . , 0, ∞, ∞, 0, ∞, ∞, . . . as a general periodic solution to 4.3, then the solution
xn n∈N0 eventually converges to the general period-three solution 0, ∞, ∞.
14
Abstract and Applied Analysis
Acknowledgments
This paper is partially supported by the Fundamental Research Funds for the Central
Universities Project no. CDJXS10181130, the New Century Excellent Talent Project of China
no. NCET-05-0759, the National Natural Science Foundation of China no. 10771227, and
the Serbian Ministry of Science Projects III41025 and III44006.
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