Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 812165, 16 pages
doi:10.1155/2012/812165
Research Article
Nonoscillatory Solutions of
Second-Order Superlinear Dynamic Equations with
Integrable Coefficients
Quanwen Lin1 and Baoguo Jia2
1
2
Department of Mathematics, Maoming University, Maoming 525000, China
School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China
Correspondence should be addressed to Quanwen Lin, linquanwen@126.com
Received 9 September 2011; Revised 15 November 2011; Accepted 23 November 2011
Academic Editor: D. Anderson
Copyright q 2012 Q. Lin and B. Jia. 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.
The asymptotic behavior of nonoscillatory solutions of the superlinear dynamic equation on time
Δ
Δ
t pt|xσt|γ sgnxσt 0, γ > 1, is discussed under the condition that P t scales rtx
τ
limτ → ∞ t psΔs exists and P t ≥ 0 for large t.
1. Introduction
Consider the second-order superlinear dynamic equation
rtxΔ t
Δ
pt|xσt|γ sgn xσt 0,
γ > 1,
1.1
where
P t lim
τ →∞
τ
1.2
psΔs
t
exists and is finite. P t ≥ 0 for large t.
When T R, rt 1, 1.1 is the second-order superlinear differential equation
x t pt|xt|γ sgn xt 0,
γ > 1.
1.3
2
Abstract and Applied Analysis
When T N, rn 1, 1.1 is the second-order superlinear difference equation
Δ2 xn pn|xn 1|γ sgn xn 1 0,
1.4
γ > 1.
The following condition is introduced in 1.
Condition H. We say that T satisfies Condition (H) provided one of the following holds.
1 There exists a strictly increasing sequence {tn }∞
n0 ⊂ T with limn → ∞ tn ∞ and for each
n ≥ 0 either σtn tn1 or the real interval tn , tn1 ⊂ T;
2 T ∩ R T , ∞ for some T ∈ T.
We note that time scales which satisfy Condition H include most of the important
time scales, such as R, Z, and qN0 , where q > 1 and N0 is the nonnegative integers and
harmonic numbers { nk1 1/k : n ∈ N} 2, Example 1.45.
In 3, Naito proved the following result.
∞
Theorem 1.1. If P t t psds ≥ 0 for large t, then a nonoscillatory solution xt of 1.3
satisfies exactly one of the following three asymptotic properties:
lim xt c / 0,
t→∞
lim
t→∞
xt
0,
t
lim
t→∞
lim xt ±∞,
1.5
t→∞
xt
c
/ 0.
t
In this paper, we extend Theorem 1.1 to superlinear dynamic equation 1.1 on time
scale. As an application, we get the asymptotic behavior of each nonoscillation solution of
the difference equation
Δ2 xn a
−1n b
|xn 1|γ sgn xn 1 0,
nc
n1c
γ > 1,
1.6
where b > 0, c > 1, and a/c > b/2.
2. Main Theorems
Consider the second-order nonlinear dynamic equation
Δ
rtxΔ t pt|xσt|γ sgn xσt 0,
where rt, pt ∈ CT, R, rt > 0, t0 ∈ T, and
and is finite.
∞
t0
2.1
γ > 0,
rt−1 dt ∞. limt → ∞
t
t0
psΔs exists
Abstract and Applied Analysis
3
Lemma 2.1. Suppose that T satisfies Condition H. If xt is a positive solution of 1.1 on T, ∞,
then the integral equation
rtxΔ t
α P t xγ t
∞
rsγ
1 0
xs hμsxΔ s
γ−1
2
dh xΔ s
xγ sxγ σs
t
Δs
2.2
is satisfied for t ≥ T , where α is a nonnegative constant.
Proof. Suppose that xt is a positive solution of 1.1 on T, ∞.
In the first place, we will prove
∞
rsγ
1
0
T
2
xh sγ−1 dh xΔ s
Δs < ∞,
xγ sxγ σs
2.3
where xh t xt hμtxΔ t 1 − hxt hxσt > 0.
Multiplying both sides of 1.1 by 1/xγ σt, we get that
rtxΔ t
xγ t
Δ
−pt −
rtγ
1
0
2
xh tγ−1 dh xΔ t
.
xγ txγ σt
2.4
Integrating from T to t,
rtxΔ t rT xΔ T −
−
xγ t
xγ T t
psΔs −
T
t
rsγ
1
0
T
2
xh sγ−1 dh xΔ s
Δs.
xγ sxγ σs
2.5
If 2.3 fails to hold, that is,
∞
T
rsγ
1
0
2
xh sγ−1 dh xΔ s
Δs ∞,
xγ sxγ σs
2.6
from 2.5, we have
rtxΔ t
−∞.
t → ∞ xγ t
lim
2.7
Without loss of generality, we can assume that for t ≥ T
rT xΔ T −
xγ T t
T
psΔs < −1.
2.8
4
Abstract and Applied Analysis
t
Otherwise, let L maxt≥T | T psΔs|. By 2.7, we can take a large T1 > T such that
rT1 xΔ T1 /xγ T1 < −2L 1. So we have
rT1 xΔ T1 −
xγ T1 t
psΔs < −2L 1 −
t
T1
psΔs −
T
T1
psΔs
T
2.9
≤ −2L 1 − −2L −1.
So we can replace T by T1 > T such that 2.8 still holds.
From 2.5 and 2.8, we get for t ≥ T
rtxΔ t
xγ t
t
rsγ
1
0
T
2
xh sγ−1 dh xΔ s
Δs < −1.
xγ sxγ σs
2.10
In particular, we have
xΔ t < 0,
for t ≥ T.
2.11
Therefore, xt is strictly decreasing.
Assume that t ti−1 < ti σt. Then, xσt < xt, and so
1
γ
0
xh sγ−1 dh γ
1
1 − hxs hxσsγ−1 dh
0
1
1 − hxs hxσsγ 0 xγ σs − xγ s
.
xσs − xs
xσs − xs
2.12
If the real interval ti−1 , ti ⊂ T, then, for s ∈ ti−1 , ti , we have
1
γ
xh sγ−1 dh γxγ−1 s.
2.13
0
Let
yt : 1 t
rsγ
1
0
2
xh sγ−1 dh xΔ s
xγ sxγ−1 σs
T
Δs.
2.14
Hence, from 2.10, we get that
−
rtxΔ t
> yt.
xγ t
2.15
Abstract and Applied Analysis
5
From 2.14 and 2.15, we get that
Δ
y t > yt
1
2
xh tγ−1 dh xΔ t
xγ txγ σt
1
γ 0 xh tγ−1 dh −xΔ t
rtγ
0
xγ σt
2.16
.
Assume that t ti−1 < ti σt. From 2.16 and 2.12, we get that
yσt − yt xγ σt − xγ t xt − xσt
>
.
ytσt − t
xσt − xt xγ σtσt − t
2.17
yσt
xγ t
> γ
,
yt
x σt
2.18
yti xγ ti−1 > γ
.
yti−1 x ti 2.19
So,
that is,
If the real interval ti−1 , ti ⊂ T, then, for t ∈ ti−1 , ti , it follows from 2.16 and 2.13 that
y t γxγ−1 t−x t
>
,
yt
xγ t
2.20
that is,
ln yt > −ln xγ t .
2.21
Integrating from ti−1 to t, we get that
yt
xγ ti−1 >
,
yti−1 xγ t
t ∈ ti−1 , ti .
2.22
Let T tn0 , and let t ∈ T, ∞T . Then, there is an n > n0 such that t ∈ tn−1 , tn T . From 2.22
and 2.19, we get that
ytn0 1 yt
xγ tn0 xγ tn−1 ytn−1 xγ tn−2 >
,
> γ
,...,
> γ
.
γ
ytn−1 x t
ytn−2 x tn−1 ytn0 x tn0 1 2.23
6
Abstract and Applied Analysis
Multiplying, we get that
yt
xγ tn > γ 0 .
ytn0 x t
2.24
Using 2.15 again, we get
−
ytn0 xγ tn0 rtxΔ t
>
yt
>
.
xγ t
xγ t
2.25
If we set L : ytn0 xγ tn0 , we get
xΔ t < −
L
.
rt
2.26
Integrating from T to t, we get that
xt − xT < −
t
T
L
Δs −→ −∞,
rs
as t −→ ∞,
2.27
which contradicts xt > 0.
In 2.5, letting t → ∞, replacing T by τ, and denoting α limt → ∞ rtxΔ t/xγ t, we
get that
α
∞
τ
psΔs ∞
rsγ
1
0
τ
2
xh sγ−1 dh xΔ s
rτxΔ τ
Δs
.
xγ sxγ σs
xγ τ
2.28
We need to show that α ≥ 0.
Suppose that α < 0. Then, there exists a large T1 such that, for t > T1 , we have
rtxΔ t α
≤ .
xγ t
2
2.29
αxγ t
.
2rt
2.30
So,
xΔ t ≤
Thus,
MT1 :
∞
rsγ
1
0
T1
≥−
α
2
∞
T1
γ
1
0
2
xh sγ−1 dh xΔ s
Δs
xγ sxγ σs
xh sγ−1 dh −xΔ s
Δs.
xγ σs
2.31
Abstract and Applied Analysis
7
Assume that t ti−1 < ti σt. From 2.12 and xΔ t < 0, we have
σt
γ
1
0
t
xh sγ−1 dh −xΔ s
Δs
xγ σs
1
γ 0 xh tγ−1 dh −xΔ t σt − t
xγ σt
xγ t − xγ σt
xγ σt
xγ t
1
dv
≥
v
xγ σt
ln
ln
2.32
xγ t
xγ σt
xγ ti−1 .
xγ ti If the real interval ti−1 , ti ⊂ T, from 2.13 we have, for t ∈ ti−1 , ti ,
t
γ
ti−1
1
xh sγ−1 dh −xΔ s
Δs
xγ σs
t
γxγ−1 s−x s
ds
xγ s
ti−1
0
ln
2.33
xγ ti−1 .
xγ t
From 2.31, 2.32, 2.33 and the additivity of the integral, it is easy to get
MT1 ≥ −
xγ T1 α
lim ln γ
.
2 u→∞
x u
2.34
So, for large u, we have
ln
xγ T1 2MT1 1.
≤−
xγ u
α
2.35
Thus,
2MT1 x u ≥ x T1 exp
−1 .
α
γ
γ
2.36
8
Abstract and Applied Analysis
By 2.30 and noticing that α < 0, we get that
2MT1 αxγ T1 exp
−1 .
x u ≤
2ru
α
Δ
2.37
Integrating 2.37, we get that xu → −∞, which is a contradiction.
This completes the proof of the lemma.
Consider the second-order superlinear dynamic equation
Δ
rtxΔ t pt|xσt|γ sgn xσt 0,
where rt > 0,
∞
T
2.38
γ > 1,
1/rsΔs ∞,
P t lim
τ →∞
τ
2.39
psΔs
t
exists and is finite, and P t ≥ 0 for t ≥ T .
Theorem 2.2. Suppose that T satisfies Condition H and P t ≥ 0 for t ≥ T . Then each
nonoscillatory solution xt of 2.38 satisfies exactly one of the following three asymptotic properties:
lim xt c / 0,
2.40
t→∞
t→∞ t
T
lim
xt
0,
Δs/rs
xt
t→∞ t
T
lim
Δs/rs
lim xt ±∞,
t→∞
2.41
c
/ 0.
2.42
Proof. Let xt be a nonoscillatory solution of 2.38, say, xt > 0 for t ≥ T > 0. From
Lemma 2.1, it is known that xt satisfies the equality
rtxΔ t αxγ t P txγ t
γx t
γ
∞
t
rs
1 γ−1 Δ 2
dh x s
xs hμsxΔ s
0
xγ sxγ σs
2.43
Δs
for t ≥ T . Therefore, we have
rtxΔ t ≥ P txγ t,
2.44
Abstract and Applied Analysis
9
for t ≥ T . Since P t ≥ 0 for t ≥ T , it follows that xΔ t ≥ 0 for t ≥ T . An integration by parts of
2.38 gives
Δ
rux u − P ux u γ
γ
u
Δ
P sx s
t
Δ
1 xs hμsxΔ s
γ−1
dh Δs
0
2.45
rtx t − P tx t,
γ
1
where u ≥ t ≥ T . Let t be fixed. Since P sxΔ s 0 xs hμsxΔ sγ−1 dh is nonnegative, the
integral term in 2.45 has a finite limit or diverges to ∞ as t → ∞. If the latter case occurs,
then ruxΔ u − P uxγ u → −∞ as u → ∞, which is a contradiction to 2.44. Thus, the
former case occurs, that is,
∞
1 γ−1
xs hμsxΔ s
P sx s
dh Δs < ∞.
Δ
t
2.46
0
Define the function k1 t as
k1 t ∞
Δ
P sx s
1 t
xs hμsxΔ s
γ−1
dh Δs < ∞
2.47
0
and the finite constant β limu → ∞ ruxΔ u − P uxγ u. Then, equality 2.45 yields
rtxΔ t β P txγ t γk1 t,
2.48
for t ≥ T . Observe by 2.44 that β ≥ 0. From 2.44 and 2.46, it follows that
∞
t
P 2 sxγ s
rs
1 γ−1
xs hμsxΔ s
dh Δs ≤ k1 t < ∞.
2.49
0
Next, we define the function k2 t by noticing that γ > 1
k2 t ≥
∞
1 t
P 2 sxγ s
rs
t
P 2 sxγ s
rs
t
P 2 sxγ s
rs
t
P 2 sx2γ−1 s
Δs
rs
∞
∞
∞
xs hμsxΔ s
γ−1
dh Δs
0
1
1 − hxs hxσsγ−1 dh Δs
0
1
2.50
1 − hxs hxs
0
γ−1
dh Δs
10
Abstract and Applied Analysis
for t ≥ T . Dividing 2.48 by rt and integrating from T to t, we get
xt xT t
T
β
Δs rs
t
T
P sxγ s
Δs γ
rs
t
T
k1 s
Δs.
rs
2.51
By Schwartz’s inequality and the fact that xΔ t ≥ 0 for t ≥ T , the second integral term in
2.51 can be estimated as follows:
t
T
P sxγ s
Δs ≤
rs
t
T
1/2 t
1/2
P 2 sx2γ−1 s
xs
Δs
Δs
rs
T rs
t
≤ k21/2 T T
Δs
rs
1/2
2.52
x1/2 t,
for t ≥ T . From 2.51 and 2.52 and noticing that k1 t is decreasing, we get that
xt ≤ xT t
T
β
Δs k21/2 T rs
t
T
Δs
rs
1/2
x
1/2
t γk1 T t
T
Δs
.
rs
2.53
The above inequality may be regarded as a quadratic inequality in x1/2 t. Then, we have
x
1/2
t
1/2
Δs
1
1 1/2
D1/2 t
t ≤ k2 T 2
2
T rs
2.54
for t ≥ T , where
Dt k2 T t
T
It is obvious that Dt O
constant m such that
t
T
t Δs
Δs
4 xT β γk1 T .
rs
T rs
2.55
Δs/rs as t → ∞, and, consequently, there exists a positive
xt ≤ m
t
T
Δs
rs
2.56
for t ≥ T . Let T1 ≥ T be an arbitrary number. It is clear that
0≤
t
T
P sxγ s
Δs rs
T1
T
P sxγ s
Δs rs
t
T1
P sxγ s
Δs
rs
2.57
Abstract and Applied Analysis
11
for t ≥ T1 . Arguing as in 2.52, we find
t
T1
1/2
t
P sxγ s
Δs
Δs ≤ k2 T1 xt
rs
T1 rs
2.58
for t ≥ T1 , which when combined with 2.56 yields
t
T1
1/2
t
P sxγ s
Δs t Δs
Δs ≤ mk2 T1 rs
T rs T1 rs
≤ mk2 T1 1/2
t
T
for t ≥ T1 ≥ T . Using 2.57 and 2.59 and noticing that
0 ≤ lim sup t
t→∞
T
t
1
Δs/rs
T
2.59
Δs
,
rs
∞
T
Δs/rs ∞, we obtain
P sxγ s
Δs ≤ mk2 T1 1/2 .
rs
2.60
Since T1 is arbitary and k2 T1 → 0 as T1 → ∞, letting T1 → ∞ in 2.60, we get
0 ≤ lim t
t→∞
T
1
t
Δs/rs
T
P sxγ s
Δs 0.
rs
2.61
Using L’Hospital’s rule of time scale see Theorem 1.119 of 2, we have
t
lim
t→∞
T k1 s/rsΔs
t
Δs/rs
T
lim k1 t 0.
t→∞
2.62
t
In view of 2.51, 2.61, and 2.62, we find limt → ∞ xt/ T Δs/rs β.
Recall that xt is nondecreasing for t ≥ T . Now, there are three cases to consider:
i β 0 and xt is bounded above,
ii β 0 and xt is unbounded,
iii β > 0 and hence xt is unbounded.
Case i implies 2.40 with c limt → ∞ xt > 0, while case iii implies 2.42 with
c β > 0. It is also clear that case ii implies 2.41. This completes the proof.
The following lemma is from 1.
Lemma 2.3. Suppose that T satisfies Condition (H). xt > 0 is a solution of 1.1. Then, one has
t
T
xΔ s
xγ σs
Δs ≤
x−γ1 T − x−γ1 t x−γ1 T ≤
.
γ −1
γ −1
2.63
12
Abstract and Applied Analysis
Using Lemma 2.1, we can prove the following corollary.
Corollary 2.4. Under the assumptions of Lemma 2.1, if xt is a positive solution of 1.1 on T, ∞,
then the integral equation
Δ
rtx t
α P t xγ σt
is satisfied for t ≥ T , where P t ∞
t
∞ rs 1 γxh sγ−1 dh xΔ s
2
0
xγ sxγ σs
σt
Δs
2.64
psΔs, xh s xs hμsxΔ s.
Proof. In the left side of 2.2, using
1
1
γ
−
γ
x t x σt
1
γ
x σt
1
γ
x t
1
0
Δ
μt
γxh tγ−1 dhxΔ t
xγ txγ σt
2.65
μt
and using 2.2, 2.65, the additivity of the integral, and 2, Theorem 1.75, we have that
rtxΔ t rtxΔ t rt
γ
xγ t
x σt
α P t 1
2
γxh tγ−1 dh xΔ t
μt
xγ txγ σt
2
1
rs 0 γxh sγ−1 dh xΔ s
∞
xγ sxγ σs
σt
0
Δs
2
σt rs 1 γxh sγ−1 dh xΔ s
0
Δs
xγ sxγ σs
2
1
rs 0 γxh sγ−1 dh xΔ s
t
α P t ∞
xγ sxγ σs
2
1
rt 0 γxh tγ−1 dh xΔ t
μt.
xγ txγ σt
2.66
Δs
σt
From 2.66, we get 2.64.
The following theorem can be regarded as a time scale version of 4, Theorem 1.
∞
Theorem 2.5. Suppose that T satisfies Condition (H), rt > 0 with T rt−1 Δt ∞, and suppose
∞
t
that limt → ∞ T psΔs exists and is finite. Let P t t psΔs. Then, the superlinear dynamic
equation 1.1 is oscillatory if
t
lim sup
t→∞
T
P s
Δs ∞.
rs
2.67
Abstract and Applied Analysis
13
Proof. Suppose that xt is a nonoscillatory solution of 1.1 on T, ∞. Without loss of
generality, assume that xt is positive for t ∈ T, ∞. From Corollary 2.4, xt satisfies the
integral equation 2.64. Dropping the last integral term in 2.64, we have the inequality
rtxΔ t
≥ P t.
xγ σt
2.68
Dividing 2.68 by rt, integrating from T to t, and using Lemma 2.3, we find
x−γ1 T ≥
γ −1
t
xΔ s
Δs ≥
γ
x σs
T
t
T
P s
Δs.
rs
2.69
This contradicts 2.67, and so 1.1 is oscillatory.
Consider the second-order superlinear dynamic equation with forced term
where rt > 0,
∞
T
rtxΔ t
Δ
pt|xσt|γ sgn xσt ht,
γ > 1,
2.70
1/rsΔs ∞, and
τ
P t lim
psΔs
τ →∞
2.71
t
exists and is finite.
Lemma 2.6. Suppose that
∞
|hs|Δs < ∞.
2.72
T
If xt is a positive solution of 2.70 and lim inft → ∞ xt > 0, then
∞
rsγ
1
0
T
rtxΔ t
α
xγ σt
2
xh sγ−1 dh xΔ s
Δs < ∞,
xγ sxγ σs
∞
ps −
t
∞
σt
rsγ
2.73
hs
Δs
xγ σs
1
2
γ−1
dh xΔ s
0 xh s
Δs
xγ sxγ σs
2.74
are satisfied for sufficiently large t, where xh s xs hμsxΔ s, α is a nonnegative constant.
14
Abstract and Applied Analysis
Proof. The fact that lim inft → ∞ xt > 0 implies the existence of t1 ≥ T and m > 0 such that
xt ≥ m for t ≥ t1 . Then, using 2.72, we find
t
t hs
1 t
hs Δs ≤ γ
|hs|Δs ≤ M,
Δs ≤ mγ
t1 xγ σs t1 x σs
t1
t ≥ t1 ,
2.75
where M is some finite
τ positive constant.
So, limτ → ∞ t ps − hs/xγ σsΔs exists and is finite.
Similar to the proof of Lemma 2.1 and Corollary 2.4, it is easy to know that 2.73 and
2.74 hold.
For subsequent results, we define
Φ0 t ∞
ps − k|hs| Δs,
t ≥ T,
2.76
t
where k is a positive constant. It is noted that, if 2.71 and 2.72 hold, then Φ0 t is finite
for any k. Assume that Φ0 t > 0 for sufficiently large t. Define, for a positive integer n and a
positive constant ρ, the following functions:
Φ1 t ∞
σt
Φn1 t Φ0 s2
Δs,
rs
∞ σt
2
Φ0 s ρΦn s
Δs.
rs
2.77
We introduce the following condition.
Condition A. For every ρ > 0, there exists a positive integer N such that Φn t is finite for
n 1, 2, . . . , N − 1 and ΦN t is infinite.
Theorem 2.7. Suppose that 2.71, 2.72, and Condition (A) hold. Then, every solution xt of
2.70 is either oscillatory or satisfies
lim inf xt 0.
t→∞
2.78
Proof. Suppose on the contrary that xt is a nonoscillatory solution of 2.70 and
lim inft → ∞ |xt| > 0. Without loss of generality, let xt be eventually positive. By Lemma 2.6,
xt satisfies 2.73 and 2.74. Further, there exist t1 ≥ T and m > 0 such that xt ≥ m for
t ≥ t1 . Let
Φ0 t ∞
t
|hs|
ps −
Δs.
mγ
2.79
Abstract and Applied Analysis
15
Then, from 2.74 we find
rtxΔ t
≥ Φ0 t xγ σt
∞
rsγ
1
σt
0
2
xh sγ−1 dh xΔ s
Δs
xγ sxγ σs
2.80
≥ Φ0 t > 0,
for t ≥ t1 . From 2.80, we get
xΔ t ≥
Φ0 txγ σt
> 0,
rt
t ≥ t1 .
2.81
Applying 2.81 and noticing that γ > 1, we find for t ≥ t1
∞
rsγ
1
0
σt
≥
∞
σt
≥ γm
2
xh sγ−1 dh xΔ s
Δs
xγ sxγ σs
γxsγ−1 Φ0 sxγ σs2
Δs
rsxγ sxγ σs
γ−1
∞
σt
2.82
Φ0 s2
Δs γmγ−1 Φ1 t.
rs
If N 1 in Condition A, then the right side of 2.82 is infinite. This is a contradiction to
2.73.
Next, it follows from 2.80 and 2.82 that
rtxΔ t
≥ Φ0 t γmγ−1 Φ1 t.
xγ σt
2.83
Using a similar technique and relations 2.83, we get
∞
rsγ
1
0
σt
2
xh sγ−1 dh xΔ s
Δs
xγ sxγ σs
≥ γmγ−1
∞ σt
2
Φ0 s γmγ−1 Φ1 s
Δs γmγ−1 Φ2 t,
rs
2.84
t ≥ t1 .
If N 2 in Condition A, then the right side of 2.84 is infinite. This again contradicts 2.73.
A similar argument yields a contradiction for any integer N > 2. This completes the
proof of the theorem.
Example 2.8. We have
Δ xn 2
a
b−1n
|xn 1|γ sgn xn 1 0,
nc
n1c
γ > 1,
2.85
16
Abstract and Applied Analysis
where b > 0, c > 1, and a/c > b/2. It is easy to see that
∞
∞
1
1
1
≥
dt c .
1c
1c
cn
k
t
n
kn
By 5, we have
∞
kn −1
n
2.86
/nc ∼ −1n /2nc . So,
∞
−1k
kn
kc
1 o1
−1n
.
2nc
2.87
Using 2.86 and 2.87, we get that, for large n,
P n ∞
kn
a
k1c
b−1k
kc
≥
a/c b/21 o1−1n
> 0.
nc
2.88
By Theorem 2.2, each nonoscillatory solution xt of 2.85 satisfies exactly one of the
following three asymptotic properties:
lim xn c /
0,
n→∞
lim
n→∞
xn
0,
n
lim
n→∞
lim xn ±∞,
n→∞
2.89
xn
c/
0.
n
Acknowledgment
The paper is supported by the National Natural Science Foundation of China no. 10971232.
References
1 B. Baoguo, L. Erbe, and A. Peterson, “Kiguradze-type oscillation theorems for second order superlinear
dynamic equations on time scales,” Canadian Mathematical Bulletin, vol. 54, no. 4, pp. 580–592, 2011.
2 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications,
Birkhäuser, Boston, Mass, USA, 2001.
3 M. Naito, “Nonoscillatory solutions of second order differential equations with integrable coefficients,”
Proceedings of the American Mathematical Society, vol. 109, no. 3, pp. 769–774, 1990.
4 J. S. W. Wong, “Oscillation criteria for second order nonlinear differential equations with integrable
coefficients,” Proceedings of the American Mathematical Society, vol. 115, no. 2, pp. 389–395, 1992.
5 L. Erbe, J. Baoguo, and A. Peterson, “Nonoscillation for second order sublinear dynamic equations on
time scales,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 594–599, 2009.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014