Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2011, Article ID 863801, 15 pages
doi:10.1155/2011/863801
Research Article
Oscillation of Second-Order Nonlinear Delay
Dynamic Equations on Time Scales
H. A. Agwa, A. M. M. Khodier, and Heba A. Hassan
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt
Correspondence should be addressed to H. A. Agwa, hassanagwa@yahoo.com
Received 3 May 2011; Accepted 6 June 2011
Academic Editor: Elena Braverman
Copyright q 2011 H. A. Agwa 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.
In this work, we use the generalized Riccati transformation and the inequality technique to
establish some new oscillation criteria for the second-order nonlinear delay dynamic equation
γ Δ
ptxΔ t qtfxτt 0, on a time scale T, where γ is the quotient of odd positive
integers and p(t) and q(t) are positive right-dense continuous rd-continuous functions on T. Our
results improve and extend some results established by Sun et al. 2009. Also our results unify
the oscillation of the second-order nonlinear delay differential equation and the second-order
nonlinear delay difference equation. Finally, we give some examples to illustrate our main results.
1. Introduction
The theory of time scales was introduced by Hilger 1 in order to unify, extend, and
generalize ideas from discrete calculus, quantum calculus, and continuous calculus to
arbitrary time scale calculus. Many authors have expounded on various aspects of this new
theory, see 2–4. A time scale T is a nonempty closed subset of the real numbers, If the
time scale equals the real numbers or integer numbers, it represents the classical theories of
the differential and difference equations. Many other interesting time scales exist and give
rise to many applications. The new theory of the so-called “dynamic equation” not only
unify the theories of differential equations and difference equations, but also extends these
classical cases to the so-called q-difference equations when T qN0 : {qt : t ∈ N0 for
q > 1} or T qZ qZ ∪ {0} which have important applications in quantum theory see
5. Also it can be applied on different types of time scales like T hZ, T N20 , and the
space of the harmonic numbers T Tn . In the last two decades, there has been increasing
interest in obtaining sufficient conditions for oscillation nonoscillation of the solutions of
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different classes of dynamic equations on time scales, see 6–9. In this paper, we deal with
the oscillation behavior of all solutions of the second-order nonlinear delay dynamic equation
γ Δ
qtfxτt 0,
pt xΔ t
t ∈ T, t ≥ t0 ,
1.1
subject to the hypotheses
H1 T is a time scale which is unbounded above, and t0 ∈ T with t0 > 0. We define the
time scale interval t0 , ∞T by t0 , ∞T t0 , ∞ T.
H2 γ is the quotient of odd positive integers.
H3 p and q are positive rd-continuous functions on an arbitrary time scale T, and
∞
t0
Δt
∞
p1/γ t
1.2
H4 τ : T → T is a strictly increasing and differentiable function such that τt ≤ t,
limt → ∞ τt ∞.
H5 f ∈ CR, R is a continuous function such that for some positive constant L, it
0.
satisfies fx/xγ ≥ L for all x /
By a solution of 1.1, we mean that a nontrivial real valued function x satisfies 1.1 for
t ∈ T. A solution x of 1.1 is called oscillatory if it is neither eventually positive nor
eventually negative; otherwise, it is called nonoscillatory. 1.1 is said to be oscillatory if all
of its solutions are oscillatory. We concentrate our study to those solutions of 1.1 which are
not identically vanishing eventually.
It is easy to see that 1.1 can be transformed into a half linear dynamic equation
γ Δ
qtxγ t 0,
pt xΔ t
t ∈ T, t ≥ t0 ,
1.3
where fx xγ , τt t. If γ 1, then 1.1 is transformed into the equation
Δ
ptxΔ t qtfxτt 0,
t ∈ T, t ≥ t0 .
1.4
If pt 1, then 1.4 has the form
xΔΔ t qtfxτt 0,
t ∈ T, t ≥ t0 .
1.5
If fx x, then 1.5 becomes
xΔΔ t qtxτt 0,
t ∈ T, t ≥ t0 .
1.6
Recently, Zhang et al. 10 have considered the nonlinear delay 1.1 and established some
sufficient conditions for oscillation of 1.1 when γ ≥ 1. Also Grace et al. 11 introduced
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3
some new sufficient conditions for oscillation of the half linear dynamic equation 1.3. In
2009, Sun et al. 12 extended and improved the results of 6, 13, 14 to 1.1 when γ ≥ 1, but
their results can not be applied for 0 < γ < 1. In 2008, Hassan 15 considered the half linear
dynamic equation 1.3 and established some sufficient conditions for oscillation of 1.3. In
2007, Erbe et al. 13 considered the nonlinear delay dynamic equation 1.4 and obtained
some new oscillation criteria which improve the results of Şahiner 14. In 2005, Agarwal
et al. 6 studied the linear delay dynamic equation 1.6, also Şahiner 14 considered the
nonlinear delay dynamic equation 1.5 and gave some sufficient conditions for oscillation
of 1.6 and 1.5. In this work, we give some new oscillation criteria of 1.1 by using the
generalized Riccati transformation and the inequality technique. Our results are general cases
for some results of 12, 15.
This paper is organized as follows. In Section 2, we present some preliminaries on time
scales. In Section 3, we give several lemmas. In Section 4, we establish some new sufficient
conditions for oscillation of 1.1. Finally, in Section 5, we present some examples to illustrate
our results.
2. Some Preliminaries on Time Scales
A time scale T is an arbitrary nonempty closed subset of the real numbers R. On any time
scale T, we define the forward and backward jump operators by
σt inf{s ∈ T, s > t},
ρt sup{s ∈ T, s < t}.
2.1
A point t ∈ T, t >inf T is said to be left dense if ρt t, right dense if t < sup T and σt t,
left scattered if ρt < t, and right scattered if σt > t. The graininess function μ for a time
scale T is defined by μt σt − t.
A function f : T → R is called rd-continuous provided that it is continuous at rightdense points of T, and its left-sided limits exist finite at left-dense points of T. The set of
1
T, R, we mean the set of functions
rd-continuous functions is denoted by Crd T, R. By Crd
whose delta derivative belongs to Crd T, R.
For a function f : T → R the range R of f may be actually replaced with any Banach
space, the delta derivative f Δ is defined by
f Δ t fσt − ft
,
σt − t
2.2
provided that f is continuous at t, and t is right scattered. If t is not right scattered, then the
derivative is defined by
f Δ t lim
s→t
provided that this limit exists.
fσt − ft
ft − fs
,
lim
s→t
t−s
t−s
2.3
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A function f : a, b → R is said to be differentiable if its derivative exists. The derivative f Δ and the shift f σ of a function f are related by the equation
f σ fσt ft μtf Δ t.
2.4
0 of two
The derivative rules of the product fg and the quotient f/g where gg σ /
differentiable functions f and g are given by
f ·g
Δ
t f Δ tgt f σ tg Δ t ftg Δ t f Δ tg σ t,
Δ
f
f Δ tgt − ftg Δ t
.
t g
gtg σ t
2.5
An integration by parts formula reads
b
Δ
ftg tΔt ftgt
b
a
a
−
b
f Δ tg σ tΔt
2.6
a
or
b
a
b
f σ tg Δ tΔt ftgt a −
b
f Δ tgtΔt
2.7
a
and the infinite integral is defined by
∞
fsΔs lim
t→∞
b
t
2.8
fsΔs.
b
Note that in case T R, we have
σt ρt t,
μt 0,
f Δ t f t,
b
a
ftΔt b
ftdt,
2.9
a
and in case T Z, we have
σt t 1,
ρt t − 1,
μt 1,
b
if a < b,
a
ftΔt f Δ t Δft ft 1 − ft
b−1
ta
2.10
ft.
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Throughout this paper, we use
d t : max{0, dt},
d− t : max{0, −dt},
⎧
⎨αt 0 < γ ≤ 1
βt :
⎩αγ t γ > 1,
2.11
where
Rt
,
αt :
Rt μt
Rt : p
1/γ
t
t
t0
Δs
p1/γ s
,
for t ≥ t0 .
2.12
3. Several Lemmas
In this section, we present some lemmas that we need in the proofs of our results in Section 4.
Lemma 1 Bohner and Peterson 3, Theorem 1.90. If x(t) is delta differentiable and eventually
positive or negative, then
xtγ
Δ
γ
1
hxσt 1 − hxtγ−1 xΔ tdh.
3.1
0
Lemma 2 Hardy et al. 16, Theorem 41. If A and B are nonnegative real numbers, then
λABλ−1 − Aλ ≤ λ − 1 B λ ,
3.2
λ > 1,
where the equality holds if and only if A B.
Lemma 3. If (H1 )–(H3 ) and 1.2 hold and 1.1 has a positive solution x on t0 , ∞T , then
γ Δ
pt xΔ t
< 0,
xΔ t > 0,
xt
> αt,
xσ t
for t ∈ t0 , ∞.
3.3
Proof. The proof is similar to the proof of Lemma 2.1 in 15 and, hence, is omitted.
4. Main Results
Theorem 1. Assume that (H1 )–(H5 ), 1.2, Lemma 3 hold and τ ∈ C1rd t0 , ∞T , T, τt0 , ∞T t0 , ∞T . Furthermore, assume that there exists a positive Δ-differentiable function δt such that
⎤
Δ γ1
pτs
δ
s
lim sup ⎣Lαγ τsqsδσ s − γ ⎦Δs ∞.
γ1 t→∞
t0
γ 1
βτsδσ sτ Δ s
t
⎡
Then every solution of 1.1 is oscillatory on t0 , ∞T .
4.1
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Proof. Assume that 1.1 has a nonoscillatory solution on t0 , ∞T . Then, without loss of
generality, we assume that xt > 0, xτt > 0 for all t ∈ t1 , ∞T , t1 ∈ t0 , ∞T , and
there is T ∈ t0 , ∞T such that xt satisfies the conclusion of Lemma 3 on T, ∞T . Consider
the generalized Riccati substitution
wt δtpt
xΔ t
xτt
γ
4.2
.
Using the delta derivative rules of the product and quotient of two functions, we have
Δ
Δ
w t δ t
Δ
γ
pt xΔ t
xτtγ
δ t
wt δt
δ t
σ
γ Δ
pt xΔ t
xτtγ
γ Δ
δσ t pt xΔ t
γ
xτσt
−
Δ
γ δ tpt x t xτt
xτtγ xτσtγ
σ
4.3
γ Δ
,
using the fact fx/xγ ≥ L and xt/xσ t > αt, we have
γ Δ
δσ tpt xΔ t xτtγ
δΔ t
γ
σ
w t ≤
wt − Lα τtδ tqt −
.
δt
xτtγ xτσtγ
Δ
4.4
If 0 < γ ≤ 1, then using the chain rule and the fact that xt is strictly increasing on T, ∞T ,
we obtain
Δ
xτtγ γ
1 γ−1
xτt hμτtxτtΔ
dhxτtΔ
0
γ
1
1 − hxτt hxσ τtγ−1 dhxτtΔ
4.5
0
≥ γxσ τtγ−1 xτtΔ τ Δ t
which implies
γ
γδσ tpt xΔ t xσ τtγ−1 xτtΔ τ Δ t
δΔ t
γ
σ
w t ≤
wt − Lα τtδ tqt −
δt
xτtγ xτσtγ
Δ
≤
γδσ txτtΔ ατtτ Δ t
δΔ t
wt − Lαγ τtδσ tqt −
wt,
δt
δtxτt
4.6
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since ptxΔ tγ Δ < 0, then by integrating from t to τt, we get
Δ
pt
1/γ
Δ
1/γ x t
pτt
xτt > 1/γ Δ
γδσ t pt
x tατtτ Δ t
δΔ t
γ
σ
wt − Lα τtδ tqt −
w t ≤
wt,
1/γ
δt
δtxτt pτt
Δ
4.7
4.8
that is,
γδσ tατtτ Δ t
δΔ t
γ1/γ
wt − Lαγ τtδσ tqt −
t .
1/γ w
δt
δγ1/γ t pτt
wΔ t ≤
4.9
If γ > 1, then using the chain rule and the fact that xt is strictly increasing on T, ∞T ,
we obtain
xτtγ
Δ
≥ xτtγ−1 xτtΔ τ Δ t.
4.10
From 4.4, 4.7, and 4.10, we have
γδσ tαγ τtτ Δ t γ1/γ
δΔ t
wt − Lαγ τtδσ tqt −
t.
1/γ w
δt
δγ1/γ t pτt
wΔ t ≤
4.11
By 4.9, 4.11, and the definition of βt, we have for γ > 0
δΔ t
w t ≤
δt
Δ
γδσ tβτtτ Δ t λ
w t,
δλ tpλ−1 τt
4.12
pλ−1/λ τt δΔ t 1/λ ,
λ γδσ tβτtτ Δ t
4.13
wt − Lαγ τtδσ tqt −
where λ γ 1/γ. Defining A ≥ 0 and B ≥ 0 by
γδσ tβτtτ Δ t λ
A w t,
δλ tpλ−1 τt
λ
B
λ−1
then using Lemma 2, we get
δΔ t
δt
γ1
pτt δΔ t γδσ tβτtτ Δ t λ
wt −
w t ≤ γ .
γ1 δλ tpλ−1 τt
γ 1
βτtδσ tτ Δ t
4.14
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From this last inequality and 4.12, we get
γ1
pτt δΔ t w t ≤ −Lα τtδ tqt γ .
γ1 γ 1
βτtδσ tτ Δ t
Δ
γ
σ
4.15
Integrating both sides from T to t, we get
⎤
Δ γ1
pτs
δ
s
⎣Lαγ τsδσ sqs − γ ⎦Δs ≤ wT − wt ≤ wT ,
γ1 T
γ 1
βτsδσ sτ Δ s
4.16
t
⎡
which contradicts the assumption 4.1. This contradiction completes the proof.
Theorem 2. Assume that (H1 )–(H5 ), 1.2, Lemma 3 hold and τ ∈ C1rd t0 , ∞T , T, τt0 , ∞T t0 , ∞T . Furthermore, assume that there exist functions H, h ∈ Crd D, R (where D ≡ {t, s : t ≥
s ≥ t0 }) such that
Ht, t 0,
t ≥ t0 ,
Ht, s > 0,
t > s ≥ t0 ,
4.17
and H has a nonpositive continuous Δ-partial derivative with respect to the second variable H Δs t, s
which satisfies
δΔ t
ht, s
−
Hσt, σsγ/γ1 ,
δt
δt
σt
1
lim sup
Kt, sΔs ∞,
t → ∞ Hσt, t0 t0
H Δs σt, s Hσt, σs
4.18
4.19
where δt is positive Δ-differentiable function and
pτsh− t, sγ1
γ .
γ1 γ 1
βτsδσ sτ Δ s
Kt, s Hσt, σsLαγ τsqsδσ s − 4.20
Then every solution of 1.1 is oscillatory on t0 , ∞T .
Proof. Assume that 1.1 has a nonoscillatory solution on t0 , ∞T . Then, without loss of
generality, we assume that xt > 0, xτt > 0 for all t ∈ t1 , ∞T , t1 ∈ t0 , ∞T , and
there is T ∈ t0 , ∞T such that xt satisfies the conclusion of Lemma 3 on T, ∞T . Define
wt as in the proof of Theorem 1. Replacing δΔ t with δΔ t in 4.12, we have
Lαγ τtδσ tqt ≤ −wΔ t γδσ tβτtτ Δ t λ
δΔ t
wt −
w t.
δt
δλ tpλ−1 τt
4.21
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Multiplying 4.21 by Hσt, σs, and integrating with respect to s from T to σt, we get
σt
Hσt, σsLαγ τsδσ sqsΔs
T
≤−
σt
Hσt, σswΔ sΔs
T
σt
T
−
δΔ s
wsΔs
Hσt, σs
δs
σt
Hσt, σs
T
4.22
γδσ sβτsτ Δ s λ
w sΔs.
δλ spλ−1 τs
Integrating by parts and using 4.17 and 4.18, we obtain
σt
Hσt, σsLαγ τsδσ sqsΔs ≤ Hσt, T wT T
σt T
γδσ sβτsτ Δ s λ
h− t, s
1/λ
w s Δs.
Hσt, σs ws − Hσt, σs
δs
δλ spλ−1 τs
4.23
Defining A ≥ 0 and B ≥ 0 by
Aλ Hσt, σs
γδσ sβτsτ Δ s λ
w s,
δλ spλ−1 τs
pλ−1/λ τsh− t, s
Bλ−1 1/λ ,
λ γδσ sβτsτ Δ s
4.24
then using Lemma 2, we get
γδσ sβτsτ Δ s λ
h− t, s
w s
Hσt, σs1/λ ws − Hσt, σs
δs
δλ spλ−1 τs
pτsh− t, sγ1
≤
γ ,
γ1 γ 1
βτsδσ sτ Δ s
4.25
therefore,
σt
Hσt, σsLαγ τsδσ sqsΔs ≤ Hσt, T wT T
σt
T
pτsh− t, sγ1
γ Δs.
γ1 γ 1
βτsδσ sτ Δ s
4.26
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By the definition of Kt, s, we get
σt
Kt, sΔs ≤ Hσt, T wT ,
4.27
T
and this implies that
1
Hσt, T σt
Kt, sΔs ≤ wT ,
4.28
T
which contradicts the assumption 4.19. This contradiction completes the proof.
Theorem 3. Assume that (H1 )–(H5 ), 1.2, Lemma 3 hold and τ ∈ C1rd t0 , ∞T , T, τt0 , ∞T t0 , ∞T . Furthermore, assume that there exists a positive Δ-differentiable function δt such that for
m≥1
1
lim sup m
t→∞ t
⎡
⎤
Δ γ1
pτs
δ
s
t − sm ⎣Lαγ τsqsδσ s −
γ ⎦Δs ∞.
δ 1γ1 βτsδσ sτ Δ s
t0
t
4.29
Then every solution of 1.1 is oscillatory on t0 , ∞T .
Proof. Assume that 1.1 has a nonoscillatory solution on t0 , ∞T . Then, without loss of
generality, we assume that xt > 0, xτt > 0 for all t ∈ t1 , ∞T , t1 ∈ t0 , ∞T , and there
is T ∈ t0 , ∞T such that xt satisfies the conclusion of Lemma 3 on T, ∞T . Proceeding as in
the proof of Theorem 1, we get 4.15 from which we have
γ1
pτt δΔ t Δ
Lα τtδ tqt − γ ≤ −w t,
γ1 σ
Δ
γ 1
βτtδ tτ t
γ
σ
4.30
therefore,
⎛
⎞
Δ γ1
pτs
δ
s
t − sm ⎝Lαγ τsδσ sqs − γ ⎠Δs.
γ1 σ
t1
γ 1
βτsδ sτ Δ s
t
≤−
t
4.31
t − sm wΔ tΔs.
t1
The right hand side of the above inequality gives
t
t1
t − sm wΔ sΔs t − sm wstt1 −
t
t1
t − sm
Δ s
wσsΔs.
4.32
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Since t − sm Δs ≤ −mt − σsm−1 ≤ 0 for t ≥ σs, m ≥ 1, then we have
⎡
⎤
Δ γ1
pτs
δ
s
m
t − sm ⎣Lαγ τsqsδσ s − γ ⎦Δs ≤ t − t1 wt1 ,
γ1 σ
Δ
t1
γ 1
βτsδ sτ s
4.33
t
then,
1
tm
t
t1
⎡
t − s
⎤
γ1
pτs δΔ s t − t1 m
⎦
wt1 Lα τsqsδ s − Δs
≤
γ
γ1 t
γ 1
βτsδσ sτ Δ s
4.34
m⎣
γ
σ
which contradicts 4.29. This contradiction completes the proof.
∞
Theorem 4. Assume that t0 Δt/p1/γ t ∞ and
LRγ τt
lim sup
pτt
t→∞
∞
qsΔs > 1, hold.
4.35
t
Then every solution of 1.1 is oscillatory on t0 , ∞T .
Proof. Assume that 1.1 has a nonoscillatory solution on t0 , ∞T . Then, without loss of
generality, we assume that xt > 0, xτt > 0 for all t ∈ t1 , ∞T , t1 ∈ t0 , ∞T , and
there is T ∈ t0 , ∞T such that xt satisfies the conclusion of Lemma 3 on T, ∞T . From 1.1,
we have
γ Δ
pt xΔ t
−qtfxτt ≤ −Lqtxγ τt.
4.36
Integrating last equation from τt to ∞, we obtain
∞
γ
γ
Lqsxγ τsΔs < pτt xΔ τt − lim ps xΔ s .
s→∞
τt
4.37
Since psxΔ sγ decreasing and psxΔ sγ > 0, then we have
1
pτt
∞
τt
γ
Lqsxγ τsΔs < xΔ τt .
4.38
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Since xt > RtxΔ t, then xτt > RτtxΔ τt, and consequently
L
pτt
∞
qsxγ τsΔs <
τt
LRγ τt
pτt
∞
xτt
Rτt
γ
,
4.39
qsx τsΔs < x τt,
γ
γ
τt
but
LRγ τt
pτt
∞
qsxγ τsΔs <
t
LRγ τt
pτt
∞
qsxγ τsΔs < xγ τt.
4.40
τt
Since xt and τt are strictly increasing, then we get that
LRγ τt
pτt
∞
qsΔs < 1,
4.41
qsΔs ≤ 1.
4.42
t
therefore,
LRγ τt
pτt
∞
t
This contradiction completes the proof.
5. Examples
In this section, we give some examples to illustrate our main results.
Example 1. Consider the second-order nonlinear delay dynamic equation
γ Δ
tγ xΔ t
λ
xγ τt 0
tαγ τt
for t ∈ t0 , ∞T , t0 ≥ 0,
5.1
where λ is a positive constant,l and γ is the quotient of odd positive integers.
Here,
pt tγ ,
qt λ
,
tαγ τt
fx xγ ,
L 1.
5.2
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If δt 1, then
∞
t0
Δt
1/γ pt
∞
t0
Δt
∞,
t
⎤
Δ γ1
pτs
δ
s
lim sup ⎣Lαγ τsqsδσ s − γ ⎦Δs
γ1 t→∞
t0
γ 1
βτsδσ sτ Δ s
t
⎡
lim sup
t→∞
t
t0
5.3
λ
Δs ∞.
s
Therefore, by Theorem 1, every solution of 5.1 is oscillatory.
Example 2. Consider the second-order nonlinear delay dynamic equation
xΔ t
γ Δ λσ γ−1 t
xγ τt x2γ τt 1 0 for t ∈ t0 , ∞T , t0 ≥ 0,
γ γ
t τ t
5.4
where λ is a positive constant, and 0 < γ ≤ 1 is the quotient of odd positive integers, that is,
αt βt.
Here,
pt 1,
qt λσ γ−1 t
,
tγ τ γ t
fx xγ x2γ 1 ,
L 1,
τt t
.
2
5.5
It is clear that 1.2 holds.
τt
Since Rτt p1/γ τt t0 Δs/p1/γ s τt − t0 , then we can find 0 < b < 1 such
that
ατt τt − t0
Rτt
Rτt μτt τt − t0 στt − τt
τt − t0
bτt
,
>
στt − t0 στt
for t ≥ tb > t0 .
If δt t, then
⎤
Δ γ1
pτs
δ
s
lim sup
⎣Lα τsqsδ s − γ ⎦Δs
γ1 t→∞
t0
γ 1
βτsδσ sτ Δ s
t γ γ
b τ sλσ γ−1 sσs
22γ σ γ τs
> lim sup
−
Δs
γ1 γ γ γ
σ γ τsτ γ ssγ
t→∞
t0
b s σ s
γ 1
t
⎡
γ
σ
5.6
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t γ γ
b λσ τs
22γ σ γ s
> lim sup
Δs
−
γ1 γ γ γ
sγ σ γ τs
t→∞
t0
b s σ s
γ 1
t
1
22γ
γ
lim
sup
b λ− Δs ∞,
γ1 γ
γ
s
t→∞
t0
b
γ 1
5.7
if λ > 22γ /b2γ γ 1γ1 . Then by Theorem 1, every solution of 5.4 is oscillatory if λ >
22γ /b2γ γ 1γ1 .
Example 3. Consider the second-order nonlinear delay dynamic equation
γ Δ
tγ−1 xΔ t
λ γ
x τt 0
tσt
for t ∈ t0 , ∞T , t0 ≥ 0,
5.8
where λ is a positive constant and γ ≥ 1 is the quotient of odd positive integers.
Here,
pt tγ−1 ,
λ
,
tσt
qt fx xγ ,
L 1.
5.9
∞
∞
It is clear that t0 Δt/p1/γ t t0 Δt/tγ−1/γ ∞, for γ ≥ 1, i.e., 1.2 holds and Rτt ≥
τt − t0 ≥ kτt for 0 < k < 1, and t ≥ t0 ≥ 1.
Then,
lim sup
t→∞
LRγ τt
pτt
∞
qsΔs ≥ lim sup
λ lim supkγ τt
t→∞
t→∞
t
∞ t
−1
s
Δ
kγ τ γ t
τ γ−1 t
∞
t
λ
Δs
sσs
λkγ τt
> 1,
Δs t
5.10
if λ > t/kγ τt. Then by Theorem 4, every solution of 5.8 is oscillatory if λ > t/kγ τt.
Remarks 1. 1 The recent results due to Hassan 15, Grace et al. 11 and Agarwal et al. 7
cannot be applied to 5.1, 5.4, and 5.8 as they deal with ordinary equations without delay.
2 If 0 < γ ≤ 1, the results of Sun et al. 12 cannot be applied to 5.1 and 5.4.
References
1 S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,”
Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.
2 R. Agarwal, M. Bohner, D. O’Regan, and A. Peterson, “Dynamic equations on time scales: a survey,”
Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 1–26, 2002.
3 M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser Boston Inc., Boston, Mass,
USA, 2001, An introduction with application.
4 M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston Inc.,
Boston, Mass, USA, 2003.
International Journal of Differential Equations
15
5 V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, New York, NY, USA, 2002.
6 R. P. Agarwal, M. Bohner, and S. H. Saker, “Oscillation of second order delay dynamic equations,”
The Canadian Applied Mathematics Quarterly, vol. 13, no. 1, pp. 1–18, 2005.
7 R. P. Agarwal, D. O’Regan, and S. H. Saker, “Philos-type oscillation criteria for second order halflinear dynamic equations on time scales,” The Rocky Mountain Journal of Mathematics, vol. 37, no. 4,
pp. 1085–1104, 2007.
8 M. Bohner and S. H. Saker, “Oscillation of second order nonlinear dynamic equations on time scales,”
The Rocky Mountain Journal of Mathematics, vol. 34, no. 4, pp. 1239–1254, 2004.
9 L. Erbe, A. Peterson, and S. H. Saker, “Oscillation criteria for second-order nonlinear dynamic
equations on time scales,” Journal of the London Mathematical Society. Second Series, vol. 67, no. 3, pp.
701–714, 2003.
10 Q. Zhang, L. Gao, and L. Wang, “Oscillation of second-order nonlinear delay dynamic equations on
time scales,” Computers and Mathematics with Applications, vol. 61, no. 8, pp. 2342–2348, 2011.
11 S. R. Grace, R. P. Agarwal, M. Bohner, and D. O’Regan, “Philos type criteria for second-order halflinear dynamic equations,” Mathematical Inequalities and Applications, vol. 14, no. 1, pp. 211–222, 2011.
12 S. Sun, Z. Han, and C. Zhang, “Oscillation of second-order delay dynamic equations on time scales,”
Journal of Applied Mathematics and Computing, vol. 30, no. 1-2, pp. 459–468, 2009.
13 L. Erbe, A. Peterson, and S. H. Saker, “Oscillation criteria for second-order nonlinear delay dynamic
equations,” Journal of Mathematical Analysis and Applications, vol. 333, no. 1, pp. 505–522, 2007.
14 Y. Şahiner, “Oscillation of second-order delay differential equations on time scales,” Nonlinear
Analysis, Theory, Methods and Applications, vol. 63, no. 5–7, pp. e1073–e1080, 2005.
15 T. S. Hassan, “Oscillation criteria for half-linear dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 176–185, 2008.
16 G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge
University Press, Cambridge, UK, 1988.
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