Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 103065, 11 pages
doi:10.1155/2010/103065
Research Article
Oscillation of Second-Order Sublinear Dynamic
Equations with Damping on Isolated Time Scales
Quanwen Lin1, 2 and Baoguo Jia1, 2
1
2
Department of Mathematics, Maoming University, Maoming 525000, China
School of Mathematics and Computer Science, Zhongshan University, Guangzhou 510275, China
Correspondence should be addressed to Quanwen Lin, linquanwen@126.com
Received 8 October 2010; Accepted 27 December 2010
Academic Editor: M. Cecchi
Copyright q 2010 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.
This paper concerns the oscillation of solutions to the second sublinear dynamic equation with
σ
damping xΔΔ t qtxΔ t ptxα σt 0, on an isolated time scale Ì which is unbounded
above. In 0 < α < 1, α is the quotient of odd positive integers. As an application, we get the
difference equation Δ2 xn n−γ Δxn 1 1/nln nβ b−1n /ln nβ xα n 1 0, where
γ > 0, β > 0, and b is any real number, is oscillatory.
1. Introduction
During the past years, there has been an increasing interest in studying the oscillation
of solution of second-order damped dynamic equations on time scale which attempts to
harmonize the oscillation theory for continuousness and discreteness, to include them in
one comprehensive theory, and to eliminate obscurity from both. We refer the readers to the
papers 1–4 and the references cited therein.
In 5 , Bohner et al. consider the second-order nonlinear dynamic equation with
damping
σ
xΔΔ t qtxΔ t pt f ◦ xσ t 0,
1.1
where p and q are real-valued, right-dense continuous functions on a time scale ⊂ , with
sup ∞. f : → is continuously differentiable and satisfies f x > 0 and xfx > 0 for
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x/
0. When fx xα , where 0 < α < 1, α > 0 is the quotient of odd positive integers, 1.1 is
the second-order sublinear dynamic equation with damping
xΔΔ t qtxΔ t ptxα σt 0.
σ
1.2
When qt 0, 1.2 is the second-order sublinear dynamic equation
xΔΔ t ptxα σt 0.
When
1.3
0 , 1.3 is the second-order sublinear difference equation
1.4
Δ2 xn ptxα n 1 0.
In 6 , under the assumption of being
∞ an isolated time scale, we prove that, when pt is
allowed to take on negative values, t1 tα ptΔt ∞ is sufficient for the oscillation of the
dynamic equation 1.3. As an application, we get that, when pn is allowed to take on
α
negative values, ∞
n1 n pn ∞ is sufficient for the oscillation of the dynamic equation
1.4, which improves a result of Hooker and Patula 7, Theorem 4.1 and Mingarelli 8 .
In this paper, we extend the result of 6 to dynamic equation 1.1. As an application,
we get that the difference equation with damping
−γ
Δ xn n Δxn 1 2
1
nln nβ
b
−1n
ln nβ
xα n 1 0,
1.5
where 0 < α < 1, γ > 0, β > 0, and b is any real number, is oscillatory.
For completeness see 9, 10 for elementary results for the time scale calculus, we
recall some basic results for dynamic equations and the calculus on time scales. Let be a
time scale i.e., a closed nonempty subset of with sup ∞. The forward jump operator
is defined by
σt inf{s ∈
: s > t},
1.6
: s < t},
1.7
and the backward jump operator is defined by
ρt sup{s ∈
where sup ∅ inf , where ∅ denotes the empty set. If σt > t, we say t is right scattered,
while, if ρt < t, we say t is left scattered. If σt t, we say t is right dense, while, if ρt t
and t /
inf , we say t is left-dense. Given a time scale interval c, d Ì : {t ∈ : c ≤ t ≤ d} in
the notation c, d κ Ì denotes the interval c, d Ì in case ρd d and denotes the interval
c, dÌ in case ρd < d. The graininess function μ for a time scale is defined by μt σt−t,
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and for any function f : → the notation f σ t denotes fσt. We say that x :
differentiable at t ∈ provided
xΔ t : lim
s→t
xt − xs
t−s
→
is
1.8
exists when σt t here, by s → t, it is understood that s approaches t in the time scale
and when x is continuous at t and σt > t
xΔ t :
xσt − xt
.
μt
1.9
Note that if , then the delta derivative is just the standard derivative, and when the delta derivative is just the forward difference operator. Hence, our results contain the
discrete and continuous cases as special cases and generalize these results to arbitrary time
scales e.g., the time scale qÆ0 : {1, q, q2 , . . .} which is very important in quantum theory
11 .
2. Lemmas
We will need the following second mean value theorem see 10, Theorem 5.45 .
Lemma 2.1. Let h be a bounded function that is integrable on a, b Ì. Let mH and MH be the
t
infimum and supremum, respectively, of the function Ht : a hsΔs on a, b Ì. Suppose that
g is nonincreasing with gt ≥ 0 on a, b Ì. Then, there is some number Λ with mH ≤ Λ ≤ MH such
that
b
2.1
htgtΔt gaΛ.
a
Lemmas 2.2 and 2.4 give two lower bounds of definite integrals on time scale,
respectively.
Lemma 2.2. Assume that {ti }∞
i0 , where 1 t0 < t1 < · · · < ti · · · . If there exists a real number
K ≥ 1 such that ti1 − ti /ti − ti−1 K ≥ 1, for all i ≥ 1, then, for ut > 0, t ≥ T, one has
t
T
uΔ s
u1−α σT
.
Δs
≥
−
uα σs
K1 − α
σ
Remark 2.3. It is easy to know that, when
, K 1 and, when
2.2
qÆ0 , K q.
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Proof. For i ≥ 1, using Theorem 1.75 of 9 , we have
ti
uΔ s
ti − ti−1 uΔ ti−1 Δs
uα σs
uα σti−1 σ
ti−1
σ
ti − ti−1 uΔ ti uα ti ti − ti−1 uti1 − uti ti1 − ti uα ti uti1 − uti .
Kuα ti 2.3
We consider the two cases uti ≤ uti1 and uti > uti1 . First, if uti ≤ uti1 , then we
have that
uti1 − uti ≥
uα ti uti1 uti 1
1
u1−α ti1 − u1−α ti .
ds sα
1−α
2.4
1
1
ds u1−α ti − u1−α ti1 ,
sα
1−α
2.5
On the other hand, if uti > uti1 , then
uti − uti1 ≤
uα ti uti uti1 which implies that
uti1 − uti 1
u1−α ti1 − u1−α ti .
≥
uα ti 1−α
2.6
From 2.3–2.6 and the additivity of the integral, we have
t
T
uΔ s
1
u1−α σt − u1−α σT
Δs ≥
α
u σs
K1 − α
σ
2.7
u1−α σT
.
≥−
K1 − α
Lemma 2.4. Assume that
ut > 0, t ≥ T, one has
{ti }∞
i0 , where 1 t0 < t1 < · · · < ti · · · , with tk → ∞. Then, for
t
T
αu1−α T
uα sΔ uσ s
.
Δs
≥
−
uα suα σs
1−α
2.8
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Proof. For i ≥ 1, using Theorem 1.75 of 9 , we have
Li u :
ti
ti−1
1
1
uα sΔ uσ s
Δs − α
uti .
α
α
α
u su σs
u ti−1 u ti 2.9
Setting vti : 1/uα ti , we have that
Li u :
vti−1 − vti .
v1/α ti 2.10
We consider the two possible cases vti−1 ≥ vti and vti−1 < vti . First, if vti−1 ≥ vti ,
we have that
vti−1 − vti ≥
v1/α ti vti−1 vti 1
ds
s1/α
2.11
α
v1−1/α ti − v1−1/α ti−1 .
1−α
On the other hand, if vti−1 < vti , then
vti − vti−1 ≤
v1/α ti vti vti−1 1
s1/α
ds
2.12
α
v1−1/α ti−1 − v1−1/α ti ,
1−α
which implies that
vti−1 − vti α
v1−1/α ti − v1−1/α ti−1 .
≥
1−α
v1/α ti 2.13
Hence, from 2.10–2.13, we have that
Li u ≥
α
v1−1/α ti − v1−1/α ti−1 .
1−α
2.14
From 2.9, 2.10, 2.14, and the additivity of the integral, we have
t
T
α
uα sΔ uσ s
v1−1/α t − v1−1/α T
Δs ≥
uα suα σs
1−α
α
αu1−α T
u1−α t − u1−α T ≥ −
.
1−α
1−α
2.15
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3. Main Theorem
Theorem 3.1. Assume that
{t0 , t1 , . . . , tk . . .}, where 1 t0 < t1 < · · · < ti · · · . Suppose that
i there exists a real number K ≥ 1 such that ti1 − ti /ti − ti−1 K ≥ 1, for all i ≥ 1;
1
ii there exists a Crd
function at such that for t ∈ T, ∞,
aΔ t ≥ 0,
at > 0,
∞
aΔΔ t ≤ 0,
aα σspsΔs T
qt ≥ 0,
∞
T
Δ
qtaσ t ≤ 0,
3.1
1
Δs ∞.
as
Then, 1.1 is oscillatory.
Proof. For the sake of contradiction, assume that 1.1 is nonoscillatory. Then, without loss of
generality, there is a solution xt of 1.1 and a T ∈ with xt > 0, for all t ∈ T, ∞Ì. Making
the substitution xt atut in 1.1 and noticing that
xΔ t aΔ tuσ t atuΔ t,
σ
σ
σ
xΔ t aΔ tu σ 2 t aσ tuΔ t,
3.2
xΔΔ t aΔΔ tuσ t aΔ tuσ tΔ aΔ tuΔ t aσ tuΔΔ t,
σ
we get that
aΔΔ tuσ t aΔ tuσ tΔ aΔ tuΔ t aσ tuΔΔ t
σ
σ
qta tu σ t qtaσ tuΔ t aα σtptuα σt 0.
Δσ
3.3
2
Multiplying both sides of 3.3 by 1/uα σt, integrating from T to t, and using an integration
by parts formula, we get
t
T
aΔΔ suσ s
Δs uα σs
t
T
t
T
t
T
t
T
aΔ ssuσ sΔ
Δs
uα σs
σ
t t
aΔ suΔ s
as
asuΔ s
−
Δs α s
uα σs
uα s
u
T
Δ
uΔ sΔs
T
t
σ
qsa su σ s
qsaσ suΔ s
Δs
Δs
uα σs
uα σs
T
Δσ
2
aα σspsΔs 0.
3.4
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Next, using the quotient rule and then Pötzsche’s chain rule 9, Theorem 1.90 gives
t
T
as
uα s
Δ
t
uΔ sΔs T
t
T
t
≤
T
aΔ suΔ s
Δs −
uα σs
t
T
aΔ suΔ s
Δs − α
uα σs
asuα s Δ uΔ s
Δs
uα suα σs
2 1
as uΔ s
uh s
uα suα σs 0
t
T
α−1
dhΔs
3.5
aΔ suΔ s
Δs,
uα σs
where we used the fact that uh s : us hμsuΔ s 1 − hus huσ s ≥ 0. Using this
last inequality in 3.4, we get
t
T
aΔΔ suσ s
Δs uα σs
t
T
aΔ ssuσ sΔ
Δs
uα σs
σ
atuΔ t aTuΔ T
−
uα t
uα T
t
T
qsaσ suΔ s
Δs uα σs
σ
t
T
t
σ
qsaΔ su σ 2 s
Δs
uα σs
3.6
aα σspsΔs ≤ 0.
T
Note that
t
T
t
σ
aΔ suσ sΔ
aΔ suσ s
Δs uα σs
uα s
−
t T
sT
aΔ s
uα s
Δ
uσ sΔs
aΔ tuσ t aΔ Tuσ T
−
uα t
uα T
−
t
T
aΔΔ suσ s
Δs uα σs
t
T
3.7
uα sΔ aΔ suσ s
Δs.
uα suα σs
Let us define A : aTuΔ T/uα T, B : aΔ Tuσ T/uα T. Then, we get from 3.6 and
3.7 that
aΔ tuσ t
−B
uα t
t
T
uα sΔ aΔ suσ s
Δs
uα suα σs
atuΔ t
−A
uα t
t
T
t
T
σ
qsaΔ su σ 2 s
Δs
uα σs
qsaσ suΔ s
Δs uα σs
σ
t
T
aα σspsΔs ≤ 0,
3.8
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since aΔ t ≥ 0, for t > T. So the first term of 3.8 is nonnegative. From 3.8, we get that
−B
t
T
atuΔ t
uα sΔ aΔ suσ s
Δs
−A
uα suα σs
uα t
t
T
t
t
σ
σ
qsaΔ su σ 2 s
qsaσ suΔ s
Δs Δs aα σspsΔs ≤ 0.
uα σs
uα σs
T
T
3.9
From aΔΔ t ≤ 0 and qtaσ tΔ ≤ 0, using the second mean value theorem 10, Theorem
5.45 and Lemmas 2.2 and 2.4, we get that
t
T
t
T
αaΔ Tu1−α T
uα sΔ aΔ suσ s
,
Δs
≥
−
uα suα σs
1−α
qsaσ suΔ s
qTaσ Tu1−α σT
.
Δs
≥
−
uα σs
K1 − α
σ
3.10
From aΔ t ≥ 0 and qt ≥ 0, the fifth term of 3.9 is nonnegative. From 3.9, and 3.10, we
get that
−B−
αaΔ Tu1−α T atuΔ t
−A
1−α
uα t
qTaσ Tu1−α σT
−
K1 − α
Since
∞
T
t
3.11
aα σspsΔs ≤ 0.
T
aα σspsΔs ∞, from 3.11, there exists T1 ≥ T such that, for t ≥ T1 , we have
atuΔ t
≤ −1.
uα t
3.12
Dividing both sides of this last inequality by at and integrating from T1 to t, we get, using
inequality 2.11 in 12 , that
1
u1−α t − u1−α T1 ≤
1−α
t
T1
uΔ s
Δs ≤ −
uα s
t
T1
1
Δs.
as
3.13
∞
Since T1 1/asΔs ∞, we get u1−α t < 0, for large t, which is a contradiction. Thus, 1.1
is oscillatory.
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When , qt 0, and at ρt, it is easy to get that aΔ t 1, aΔΔ t 0. So we
have the following corollary see Corollary 2.4 of 6 . Corollary 3.2 shows that, with no sign
assumption on pn, the condition
∞
nα pn ∞
3.14
is sufficient for the oscillation of the difference equation 1.4.
. If
Corollary 3.2. Assume that
∞
nα pn ∞,
3.15
n1
then 1.4 is oscillatory.
By using the idea in Theorem 3.1, we can also consider the differential equation
x t qtx t ptxα t 0,
3.16
0 < α < 1,
where pt, qt ∈ C1, ∞, . It is easy to get the following.
Theorem 3.3. Suppose that there exists a function at ∈ C2 1, ∞, R such that
q t ≤ 0,
at > 0,
∞
1
a t qtat ≤ 0,
a t qtat ≥ 0,
1
dt at
∞
3.17
a tptdt ∞.
α
1
Then, the differential equation 3.16 is oscillatory.
Example 3.4. Consider the sublinear difference equation
1
−γ
Δ xn n Δxn 1 2
nln nβ
b
−1n
ln nβ
xα n 1 0,
3.18
where 0 < α < 1, β > 0, γ > 0, and b is any real number.
Take an lnn − 1β/α , n > 2. We have
an > 0,
Δan ln nβ/α − lnn − 1β/α > 0,
for n > 2,
3.19
β/α
Δ an lnn 1
2
β/α
− 2ln n
lnn − 1
β/α
.
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Let ht ln tβ/α . Then, we have h t < 0, for large t. So ht is concave for large t. Therefore,
we have
hn 1 hn − 1
≤ hn,
2
3.20
∞
α
for large n. That means Δ2 an ≤ 0. It is easy to get that ∞
n1 a n 1pn n1 1/n b−1n ∞ and qnan1 n−γ ln nβ/α is nonincreasing for large n. So from Theorem 3.1,
3.18 is oscillatory.
Example 3.5. Let
qÆ0 , q > 1, and consider the q-difference equation
xΔΔ t t−γ xΔ t σ
1 b−1n α x qt 0,
t1β
3.21
where 0 < α < 1, 0 < β ≤ α, γ ≥ β/α, b > 1 is any real number. Take at t/qβ/α . We have
β/α
− 1 tβ/α−1
q
a t > 0,
q − 1 qβ/α
∞
1
∞
qβ/α − 1 qβ/α−1 − 1 β/α−2
t
≤ 0,
t 2
q − 1 qβ/α
Δ
ΔΔ
a
∞
1
Δt q2β/α 1 − q−1
q1−β/αn ∞,
at
n1
3.22
∞
1 b−1n q − 1 ∞,
aα qt ptΔt 1
n1
and qtaσ t tβ/α−γ is nonincreasing. So from Theorem 3.1, 3.21 is oscillatory.
Example 3.6. Let
1, ∞, and consider the differential equation
−γ
x t t x t 1
tln tβ
b sin t
ln tβ
xα t 0,
3.23
where 0 < α < 1, γ ≥ 0, β ≥ 0, and b is any real number.
Take at ln tβ/α . It is easy to know that
q t ≤ 0,
at > 0,
∞
1
a t qtat > 0,
1
dt ∞,
at
∞
a t qtat < 0,
aα tptdt 1
So from Theorem 3.3, 3.23 is oscillatory.
∞
1
for large t,
1
b sin t dt ∞.
t
3.24
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Acknowledgment
This work was supported by the National Natural Science Foundation of China no.
10971232.
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