Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 461014, 17 pages
doi:10.1155/2010/461014
Research Article
Explicit Conditions for Stability of Nonlinear
Scalar Delay Impulsive Difference Equation
Bo Zheng
College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China
Correspondence should be addressed to Bo Zheng, zhengbo611@yahoo.com.cn
Received 20 March 2010; Accepted 2 June 2010
Academic Editor: Leonid Shaikhet
Copyright q 2010 Bo Zheng. 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.
Sufficient conditions are obtained for the uniform stability and global attractivity of the zero
solution of nonlinear scalar delay impulsive difference equation, which extend and improve the
known results in the literature. An example is also worked out to verify that the global attractivity
condition is a sharp condition.
1. Introduction and Main Results
Let R, N, and Z be the sets of real numbers, natural numbers, and integers, respectively. For
any a, b ∈ Z, define Za {a, a 1, . . .} and Za, b {a, a 1, . . . , b} when a ≤ b.
It is well known that the theory of impulsive differential equations is emerging as an
important area of investigation, since it is not only richer than the corresponding theory of
differential equations without impulse effects but also represents a more natural framework
for mathematical modeling of many world phenomena 1. Moreover, such equations may
exhibit several real-world phenomena, such as rhythmical beating, merging of solutions, and
noncontinuity of solutions. And hence ordinary differential equations and delay differential
equations with impulses have been considered by many authors, and numerous papers have
been published on this class of equations and good results were obtained see, e.g., 1–10
and references therein.
Since the behavior of discrete systems is sometimes sharply different from the behavior
of the corresponding continuous systems and discrete analogs of continuous problems may
yield interesting dynamical systems in their own right see 11–13, many scholars have
investigated difference equations independently. However, there are few concerned with the
impulsive difference equations or delay impulsive difference equations see 14–19. On the
other hand, stability is one of the major problems encountered in applications, but, to the
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best of our knowledge, very little has been done with the stability of impulsive difference
equations see 15, 20. Motivated by this, the aim of this paper is devoted to studying the
uniform stability and global attractivity of the zero solution of the following nonlinear scalar
delay impulsive difference equation:
nj ,
Δxn fn, xn , n ∈ Z0, n /
Δx nj Ij x nj , j ∈ Z1,
1.1
where Δ denotes the forward difference operator defined by Δxn xn 1 − xn, f :
Z0 − {nj } × S → R, S is the set of all functions φ : Z−k, 0 → R for some k ∈ N, and
xn ∈ S is defined by xn m xn m for m ∈ Z−k, 0, Ij : R → R, and 0 ≤ n1 < n2 < · · · <
nj < nj1 < · · · , with nj → ∞ as j → ∞. By a solution of 1.1, we mean a sequence {xn} of
real numbers which is defined for all n ∈ Zn0 − k and satisfies 1.1 for n ∈ Zn0 for some
n0 ∈ Z0. It is easy to see that, for any given n0 ∈ Z0 and a given initial function φ ∈ S,
there is a unique solution of 1.1, denoted by xn, n0 , φ such that
x n0 m, n0 , φ φm,
for m ∈ Z−k, 0.
1.2
We assume that fn, 0 ≡ 0 and Ij 0 ≡ 0, so that xn ≡ 0 is a solution of 1.1, which we call
the zero solution.
For φ ∈ S, define the norm of φ as
φ max φ j : j ∈ Z−k, 0 ,
1.3
SH φ ∈ S : φ < H .
1.4
and for any H > 0, define
Definition 1.1. The zero solution of 1.1 is stable, if, for any ε > 0 and n0 ∈ Z0, there exists a
δ δn0 , > 0 such that φ ∈ Sδ implies that |xn, n0 , φ| < ε for n ∈ Zn0 . If δ is independent
of n0 , we say that the zero solution of 1.1 is uniformly stable.
Definition 1.2. The zero solution of 1.1 is globally attractive, if every solution of 1.1 tends
to zero as n → ∞.
A simple example of 1.1 is given by
Δxn Cnxn − k 0,
Δx nj cj x nj ,
n ∈ Z0, n /
nj ,
j ∈ Z1,
1.5
where k ∈ N, C : Z0 → R, and {cj } is a sequence of real numbers. In 15, the author
studied the stability of the zero solution of 1.5, where Cn ≥ 0 for n ∈ Z0 − {nj } and
cj ∈ −1, 0, and obtained the following result.
Advances in Difference Equations
3
Theorem 1.3. If
n
Ci
1 cj
−1
≤
nj ∈Zi−k,i−1
in−k
3
,
2
for n ∈ Zk,
1.6
then the zero solution of 1.5 is stable.
In this paper, we assume that there exists a positive constant H and a sequence {P n}
of nonnegative real numbers such that
P nM φ ≥ −f n, φ ≥ −P nM −φ , for n ∈ Z0, φ ∈ SH ,
1.7
where Mφ max{0, maxi∈Z−k, 0 φi}. Furthermore, we assume that there is a sequence
{bj } of positive numbers with bj ≤ 1 such that
bj x2 ≤ x x Ij x ≤ x2 ,
for j ∈ Z1, |x| < H.
1.8
The main purpose of this paper is to establish the following theorems.
Theorem 1.4. Assume that 1.7 and 1.8 hold and
n
P i
bj−1 ≤
nj ∈Zi−k,i−1
in−k
1
3
,
2 2k 1
for n ∈ Zk.
1.9
Then the zero solution of 1.1 is uniformly stable.
Remark 1.5. Theorem 1.4 generalizes and improves Theorem 1.3 greatly.
The next theorem provides a sufficient condition for every solution of 1.1 tends to zero as
n → ∞, that is, the zero solution of 1.1 is globally attractive.
Theorem 1.6. Assume that 1.7 and 1.8 hold and
n
P i
bj−1 ≤ α <
nj ∈Zi−k,i−1
in−k
1
3
,
2 2k 1
for n ∈ Zk,
1.10
and assume that, for each bounded solution {xn}, either
lim xn > 0,
n→∞
∞
fn, xn −∞,
for n ∈ Zk,
1.11
n0
or
lim xn < 0,
n→∞
∞
fn, xn ∞,
n0
Then every solution of 1.1 tends to zero as n → ∞.
for n ∈ Zk.
1.12
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Advances in Difference Equations
Remark 1.7. An example is worked out in Section 3 to verify that the upper bound 3/2 1/2k 1 in 1.10 is best possible, that is, the upper bound in 1.10 cannot be improved.
One special form of 1.1 is when
fn, xn m
pn,j x n kn,j ,
1.13
j1
where pn,j ≤ 0, and kn,j ∈ Z−k, 0 for n ∈ Z1, j ∈ Z1, m. Then for any H > 0, 1.7 holds
with P n m
j1 |pn,j |. As a consequence of Theorem 1.4, we have the following.
Corollary 1.8. Assume that 1.8 holds and
m n pi,j in−k j1
bj −1 ≤
nj ∈Zi−k, i−1
1
3
,
2 2k 1
for n ∈ Zk.
1.14
Then the zero solution of the equation
Δxn m
pn,j x n kn,j ,
n ∈ Z0, n /
nj ,
j1
Δx nj Ij x nj ,
1.15
j ∈ Z1,
is uniformly stable.
For the sake of convenience, throughout this paper, we will use the convention
j
P n 0,
whenever j ≤ i − 1,
ni
P i 1,
i∈A
whenever A is an empty set,
1.16
xn x n, n0 , φ .
2. Proofs of Main Results
Define
gj x ⎧
x
⎪
⎪
, x/
0,
⎪
⎨ x I x
j
1
⎪
⎪
⎪
⎩b ,
j
x0
2.1
Advances in Difference Equations
5
for j ∈ Z1. Then 1 ≤ gj x ≤ 1/bj for j ∈ Z1 if |x| < H. Equation 1.1 can be rewritten as
Δxn fn, xn ,
n ∈ Z0, n /
nj ,
x nj 1 gj −1 x nj x nj ,
j ∈ Z1.
2.2
Set
yn xn
gj x nj ,
n ∈ Z−k,
nj ∈Z0,n−1
2.3
then 2.2 reduces to
Δyn fn, xn gj x nj ,
n ∈ Z0, n / nj ,
nj ∈Z0,n−1
2.4
Δy nj 0,
j ∈ Z1.
And it is easy to see that
|xn| ≤ yn ≤ |xn|
bj−1 ,
for j ∈ Z1, n ∈ Z−k.
nj ∈Z0,n−1
2.5
To prove Theorems 1.4 and 1.6, we need the following lemma.
Lemma 2.1. Let {xn}, n ∈ Zn0 − k, be a solution of 1.1, {yn} is defined by 2.3, n∗ ∈
Zn0 2k 2, and Bn∗ max{|yn| : n ∈ Zn∗ − 3k − 2, n∗ − 1} < H. If
n
in−k
P i
bj−1 ≤ c nj ∈Zi−k,i−1
k2
,
2k 1
for n ∈ Zk
2.6
holds for some c ∈ k 2/2k 1, 1 and either
yn∗ ≥ 0,
yn∗ > yn∗ − 1
2.7
yn∗ < 0,
yn∗ < yn∗ − 1,
2.8
or
then |yn∗ | ≤ cBn∗ .
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Proof. We assume that yn∗ ≥ 0 and yn∗ > yn∗ − 1. The case where yn∗ < 0 and yn∗ <
yn∗ − 1 is similar and is omitted.
It is easy to see that Lemma 2.1 holds for yn∗ 0.
If k 0, then c 1 and P n ≤ 2 for all n ∈ Z0 by 2.6. Thus we only need to prove
that |yn∗ | ≤ Bn∗ . Since yn∗ > 0, yn∗ > yn∗ − 1, by 1.7 we have
−P n∗ − 1M−xn∗ −1 ≤ −fn∗ − 1, xn∗ −1 < 0,
2.9
−P n∗ − 1 max{0, −xn∗ − 1} ≤ −fn∗ − 1, xn∗ −1 < 0.
2.10
that is,
So xn∗ − 1 < 0 which admits that yn∗ − 1 < 0. Hence
yn∗ − yn∗ − 1 fn∗ − 1, xn∗ −1 gj x nj
nj ∈Z0,n∗ −2
≤ P n∗ − 1 max{0, −xn∗ − 1}
gj x nj
2.11
nj ∈Z0,n∗ −2
−P n∗ − 1yn∗ − 1 ≤ −2yn∗ − 1,
which implies that yn∗ ≤ −yn∗ − 1, so |yn∗ | ≤ |yn∗ − 1| ≤ Bn∗ by the definition of
Bn∗ .
Now, assume that yn∗ > 0, yn∗ > yn∗ − 1, and k ≥ 1. By 1.7, we also have
−P n∗ − 1M−xn∗ −1 ≤ −fn∗ − 1, xn∗ −1 < 0.
2.12
So,
∗
max 0, max − xn − 1 i
i∈Z−k,0
> 0.
2.13
Hence, there is n1 ∈ Zn∗ − k, n∗ such that xn1 − 1 < 0 and xn ≥ 0 for all n ∈ Zn1 , n∗ . So,
yn1 − 1 < 0 and yn ≥ 0 for all n ∈ Zn1 , n∗ . For n ∈ Zn0 , n∗ − 1, by 1.7 and 2.4, one
has
Advances in Difference Equations
gj x nj
Δyn fn, xn 7
nj ∈Z0,n−1
≤ P nM−xn gj x nj
nj ∈Z0,n−1
P n max 0, max −xn i
i∈Z−k,0
nj ∈Z0,n−1
P n max 0, max −xn i
i∈Z−k,0
gj x nj
nj ∈Z0,n−1
P n max 0, max −xi
i∈Zn−k,n
⎧
⎨
2.14
gj x nj
nj ∈Z0,n−1
P n max 0, max −yi
⎩ i∈Zn−k,n
≤ Bn∗ P n max
gj x nj
nj ∈Z0,i−1
gj−1
⎫
⎬ x nj
g x nj
⎭n ∈Z0,n−1 j
j
bj−1
i∈Zn−k,n
nj ∈Zi,n−1
≤ Bn∗ P n
bj−1
nj ∈Zn−k,n−1
provided that Zn0 , n∗ − 1 contains no impulsive points. And since Δyn 0 if n meets one
of impulsive points, we always have
Δyn ≤ Bn∗ P n
Δ
bj−1 P nBn∗ nj ∈Zn−k,n−1
for n ∈ Zn0 , n∗ − 1, where P n P n
2.15
−1
nj ∈Zn−k,n−1 bj .
By the choice of n1 , there is a real number ξ ∈ n1 − 1, n1 such that
yn1 − 1 yn1 − yn1 − 1 ξ − n1 1 0.
2.16
Next, we will show that, for any l ∈ Z0, k,
∗
−yn − l ≤ Bn n −1
1
in−k
P i − n1 − ξP n1 − 1 ,
for n ∈ Zn1 − 1, n∗ − 1.
2.17
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In fact, for any l ∈ Z0, k,
n
1 −2
−yn − l −yn1 − 1 Δyi Δyn1 − 1ξ − n1 1 in−l
∗
≤ Bn ξ − n1 1P n1 − 1 Bn∗ n −1
1
≤ Bn P i
in−l
2.18
P i − n1 − ξP n1 − 1
in−l
∗
Δyi
in−l
n
1 −2
n
1 −2
n −1
1
P i − n1 − ξP n1 − 1 ,
in−k
which shows that 2.17 holds.
Substituting 2.17 into 2.4, it is easy to get
∗
Δyn ≤ Bn P n
n −1
1
for n ∈ Zn1 − 1, n∗ − 1.
P i − n1 − ξP n1 − 1 ,
2.19
in−k
Let
k2
βc
,
2k 1
d
∗
n
−1
P n n1 − ξP n1 − 1.
nn1
There are two cases to consider.
Case 1 d ≤ 1. We have by 2.6, 2.16, and 2.19
yn∗ yn1 ∗
n
−1
Δyn n1 − ξΔyn1 − 1 nn1
∗
n
−1
Δyn
nn1
∗
≤ n1 − ξP n1 − 1Bn n
1 −1
P i − n1 − ξP n1 − 1
in1 −k−1
∗
n
−1
nn1
∗
P nBn n −1
1
P i − n1 − ξP n1 − 1
in−k
≤ Bn n1 − ξP n1 − 1 β − n1 − ξP n1 − 1
∗
∗
n
−1
nn1
P n β −
n
in1
P i − n1 − ξP n1 − 1
2.20
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∗
∗
n
−1
n
n
−1
2 2
∗
Bn βd −
P n P i − n1 − ξP n1 − 1
P n − n1 − ξ P n1 − 1
nn1
nn1
in1
1
1
Bn βd − d2 −
2
2
∗
9
∗
n
−1
2
2
2
P n n1 − ξ P n1 − 1
.
nn1
2.21
Since
2
∗
n
−1
1
P n n1 − ξ P n1 − 1 ≥ ∗
P n n1 − ξP n1 − 1
n − n1 1 nn1
nn1
∗
n
−1
2
2
2
2.22
1
1
d2 ≥
d2 ,
n∗ − n1 1
k1
we have
1
1
k2 2
d2 Bn∗ βd −
d
yn∗ ≤ Bn∗ βd − d2 −
2
2k 1
2k 1
"
!
k2
cBn∗ .
≤ Bn∗ β −
2k 1
2.23
Case 2 d > 1. In this case, there exists n2 ∈ Zn1 , n∗ − 1 such that
∗
n
−1
∗
n
−1
P n ≤ 1,
P n > 1.
2.24
nn2 −1
nn2
Therefore, we may choose a real number η ∈ n2 − 1, n2 such that
∗
n
−1
P n n2 − η P n2 − 1 1.
2.25
nn2
Notice that
∗
yn Δyn1 − 1n1 − ξ n
2 −2
Δyn η − n2 1 Δyn2 − 1
nn1
∗
n
−1
n2 − η Δyn2 − 1 Δyn .
nn2
2.26
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So, by 2.6, 2.15, 2.19, and 2.25, we have
∗
∗
yn ≤ Bn P n1 − 1n1 − ξ ×
n
2 −2
P n η − n2 1 P n2 − 1
nn1
∗
n
−1
P n n2 − η P n2 − 1 Bn∗ n2 − η P n2 − 1
nn2
n
1 −1
×
P i − n1 − ξP n1 − 1 in2 −k−1
∗
Bn ∗
n
−1
P n
n −1
2
≤ Bn Bn P n
n −1
1
∗
n
−1
nn2
P i − n1 − ξP n1 − 1
in−k
P i − n2 − η P n2 − 1
in−k
n2 − η P n2 − 1
∗
∗
nn2
nn2
∗
n
−1
n
2 −1
P i − n2 − η P n2 − 1
in2 −k−1
n
P n β −
P i − n2 − η P n2 − 1
in2
n2 − η P n2 − 1 β − n2 − η P n2 − 1
"
!
1
1
cBn∗ .
≤ Bn β − −
2 2k 1
∗
2.27
The proof is completed by combining Cases 1 and 2.
Proof of Theorem 1.4. By Lemma 2.1, 1.9 implies that 2.6 holds with c 1. For any ε ∈
0, H, assume that Zn0 , n0 2k 1 contains m mn0 , k impulsive points: n0 ≤ nl1 <
nl2 < · · · < nlm ≤ n0 2k 1. Set
β 1
3
,
2 2k 1
ε
δ
1 β
2k .
2.28
We will prove that φ ∈ Sδ implies that
|xn| x n, n0 , φ < ε,
for n ∈ Zn0 .
2.29
for n ∈ Zn0 , n0 2k 1.
2.30
To this end, we first prove that
2k
|xn| < δ 1 β < ε,
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If nl1 n0 , then |xnl1 1| |xn0 I0 xn0 | ≤ |xn0 | < δ < . If nl1 > n0 , then, for
n ∈ Zn0 , nl1 , we claim that
n −n
|xn| < δ 1 β l1 0 < .
2.31
In fact,
|xn0 1| ≤ |xn0 | fn0 , xn0 < δ P n0 δ ≤ δ 1 β < ,
|xn0 2| ≤ |xn0 1| fn0 1, xn0 1 < δ 1 β β xn0 1 ≤ δ 1 β β δ 1 β
2.32
2
δ 1 β < .
In general, we can obtain 2.31 by induction. And so |xn| < δ1 β nl1 −n0 < < H for
either case with n ∈ Zn0 , nl1 . For n ∈ Znl1 1, nl2 , we have
n −n
|xnl1 1| ≤ |xnl1 | |Il1 xnl1 | ≤ |xnl1 | < δ 1 β l1 0 ,
|xnl1 2| ≤ |xnl1 1| fnl1 1, xn 1 l1
nl1 −n0
<δ 1β
β xnl1 1 n −n
n −n
≤ δ 1 β l1 0 β δ 1 β l1 0
2.33
n −n 1
δ 1 β l1 0 < .
And so
n −n −1
|xn| < δ 1 β l2 0 ,
for n ∈ Znl1 1, nl2 ,
2.34
In general, we have
n −n −1
|xn| < δ 1 β li1 0 ,
for n ∈ Znli 1, nli1 , i 1, 2, . . . , m − 1.
2.35
Now for n ∈ Znlm 1, n0 2k 1,
n −n −1
|xnlm 1| ≤ |xnlm | |Ilm xnlm | ≤ |xnlm | < δ 1 β lm 0 ,
|xnlm 2| ≤ |xnlm 1| fnlm 1, xnlm 1 n −n −1
< δ 1 β lm 0 β xnlm 1 n −n −1
n −n −1
≤ δ 1 β lm 0 β δ 1 β lm 0
n −n
δ 1 β lm 0 < .
2.36
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And so
2k−1
,
|xn| < δ 1 β
for n ∈ Znlm 1, n0 2k 1
2.37
which has proved that 2.30 holds. Now, we will prove that
|xn| < ,
for n ∈ Zn0 2k 2.
2.38
for n ∈ Zn0 2k 2.
2.39
Thus we only need to prove that
yn < ,
For any n∗ ∈ Zn0 2k 2 and Bn∗ < H, we claim that
yn∗ ≤ Bn∗ .
2.40
In fact, we can assume that yn∗ ≥ 0. The case where yn∗ < 0 is similar and is omitted.
If yn∗ ≤ yn∗ − 1, by the definition of Bn∗ , 2.40 holds. If yn∗ > yn∗ − 1, then by
Lemma 2.1 we have that 2.40 holds.
Since
Bn0 2k 2 max yn : n ∈ Z−k, n0 2k 1 < ε < H,
2.41
by 2.40 we have
yn0 2k 2 < ε.
2.42
By repeatedly using 2.40 we get that 2.39 holds.
Combining 2.30 and 2.39, we find that 2.29 holds and the proof of Theorem 1.4 is
complete.
Proof of Theorem 1.6. In view of Theorem 1.4, we see that the zero solution of 1.1 is uniformly
stable. Therefore, for any n0 ∈ Z0, there exists δ > 0 such that φ ∈ Sδ implies that
1
|xn| x n, n0 , φ < H,
2
for n ∈ Zn0 .
2.43
Next, we will prove that
lim xn 0.
n→∞
There are two cases to consider.
2.44
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Case 1. {xn} is eventually positive, that is, there exists n1 ∈ Zn0 such that xn > 0 for all
n ∈ Zn1 . Hence, by 1.7 and 1.8, we have Δxn ≤ 0 for n ∈ Zn1 k. That is, {xn}
is eventually nonincreasing and hence limn → ∞ xn ≥ 0. Thus, by 1.11 we see that 2.44
holds. The case when {xn} is eventually negative is similar and will be omitted.
Case 2. {xn} is oscillatory in the sense that {xn} is neither eventually positive nor
eventually negative, so {yn} is also oscillatory. To prove 2.44, we only need to prove that
lim yn 0.
2.45
n→∞
By the proof of Theorem 1.4, we have
yn < 1 H,
2
2.46
for n ∈ Zn0 .
Let
lim sup yn p,
lim inf yn q,
n→∞
n→∞
2.47
then q ≤ 0 ≤ p. It suffices to show that
p q 0.
2.48
In fact, if 2.48 does not hold, we assume that p ≥ −q and p > 0. The case where p < −q
and q < 0 is similar and is omitted.
Let 0 < ε0 < 1 − c/2p, where c < 1 is given in Lemma 2.1. Then there exists an
integer m0 ∈ Z2k such that
q − ε0 ≤ yn ≤ p ε0 ,
for n ∈ Zm0 .
2.49
Since limn → ∞ sup yn p > 0 and {yn} is oscillatory, there must exist an integer
m1 ∈ Zm0 3k 2 such that
ym1 > p − ε0 ,
ym1 > ym1 − 1.
2.50
By Lemma 2.1 and 2.49 we have
p − ε0 < ym1 ym1 ≤ cBm1 ≤ c p ε0 ,
for n ∈ Zm0 .
2.51
Equations 2.49 and 2.51 imply that p − ε0 < cp ε0 , which contradicts the fact that
ε0 < 1 − c/2p; thus 2.48 holds and the proof is complete.
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3. Example
In this section we will give a result which guarantees that the upper bound 3/2 1/2k 1
is best possible in Theorem 1.6.
Example 3.1. Consider the delay impulsive difference equation
Δxn P nxn − k 0,
n ∈ Z0, n / 6k 1i,
i ∈ Z0, ni 6k 1i,
Δxni bi xni ,
3.1
where bi −1/i 22 , i ∈ Z0, k ∈ Z1, and {P n} is a sequence of nonnegative real
numbers defined by
P n ⎧
⎪
0,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
n ∈ Z6k 1i, k 6k 1i
#
Z3k 2 6k 1i, 6k 6k 1i,
1
, n ∈ Zˇk 1 6k 1i, 2k 6k 1i
⎪
⎪
k
1
⎪
#
⎪
⎪
⎪
Z2k 2 6k 1i, 3k 1 6k 1i,
⎪
⎪
⎪
⎪
⎪
⎩
c,
n 2k 1 6k 1i,
3.2
where i ∈ Z0 and c > 0 is an undetermined constant. In view of Theorem 1.6, if
n
in−k
P i
1 bj
−1
≤c
nj ∈Zi−k,i−1
3
1
k
< ,
k 1 2 2k 1
for n ∈ Zk,
3.3
or equivalently
c<
k4
,
2k 1
3.4
then every solution of 3.1 tends to zero as n → ∞.
The following theorem shows that if 3.4 does not hold, then there is a solution of 3.1
which does not tend to zero as n → ∞. This shows that the upper bound 3/2 1/2k 1
cannot be improved.
Theorem 3.2. Assume that
c≥
k4
.
2k 1
Then there exists a solution of 3.1 which does not tend to zero as n → ∞.
3.5
Advances in Difference Equations
15
Proof. Let {xn} be a solution of 3.1 with initial condition of the form
xn ,
for n ∈ Z−k, 0,
3.6
where > 0 is a given constant. Then by 3.1 and the definition of {P n}, we have
xn 1 b0 ε,
Δxn −
for n ∈ Z1, k 1,
1
1 b0 ,
k1
for n ∈ Zk 1, 2k.
3.7
And hence
!
xn 1 b0 ε 2 −
"
n
,
k1
for n ∈ Zk 1, 2k 1,
!
x2k 2 x2k 1 − cxk 1 1 b0 ε
"
1
−c .
k1
3.8
3.9
By virtue of 3.1 and 3.8 we get
!
"
n−k
1
Δxn −1 b0 ε
2−
,
k1
k1
for n ∈ Z2k 2, 3k 1.
3.10
Summing up from 2k 2 to 3k 1 and using 3.9, we have
"
!
n−k
1 b0 3k1
2−
k 1 n2k2
k1
"
!
1
k
− c − 1 b0 1 b0 k1
2k 1
"
!
k−2
def
−1 b0 c ≡ 1 b0 1 .
2k 1
x3k 2 x2k 2 −
3.11
Furthermore, we have, by the definition of {P n},
xn 1 b0 1 ,
for n ∈ Z3k 2, 6k 1.
3.12
Define the sequence {i } as follows:
!
0 ,
i1 −i
"
k−2
,
c
2k 1
for i ∈ Z0.
3.13
16
Advances in Difference Equations
Then we have by 3.5
k − 2 ≥ |i |,
|i1 | |i |c 2k 1 for i ∈ Z0,
3.14
which implies that {|i |} is a non-decreasing sequence and does not tend to zero as n → ∞.
Repeating the above argument, we find that, for n ∈ Z3k 2 i6k 1, i 16k 1, i ∈ Z0,
xn 1 b0 1 b1 · · · 1 bi i1 .
By the definition of {bi }, we know that
The proof is complete.
n
i0 1
3.15
bi 0 as n → ∞, so xn 0 as n → ∞.
Acknowledgments
The author would like to express her thanks to the referees for helpful suggestions. This
research is supported by Guangdong College Yumiao Project 2009.
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