Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 512437, 14 pages
doi:10.1155/2010/512437
Research Article
Oscillation Criteria for Second-Order Quasilinear
Neutral Delay Dynamic Equations on Time Scales
Yibing Sun,1 Zhenlai Han,1, 2 Tongxing Li,1
and Guangrong Zhang1
1
2
School of Science, University of Jinan, Jinan, Shandong 250022, China
School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China
Correspondence should be addressed to Zhenlai Han, hanzhenlai@163.com
Received 25 January 2010; Accepted 24 February 2010
Academic Editor: Gaston Mandata N’Guerekata
Copyright q 2010 Yibing Sun 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.
We establish some new oscillation criteria for the second-order quasilinear neutral delay dynamic
equations rtzΔ tγ Δ q1 txα τ1 t q2 txβ τ2 t 0 on a time scale T, where zt xt ptxτ0 t, 0 < α < γ < β. Our results generalize and improve some known results for oscillation
of second-order nonlinear delay dynamic equations on time scales. Some examples are considered
to illustrate our main results.
1. Introduction
In this paper, we are concerned with oscillation behavior of the second order quasilinear
neutral delay dynamic equations
γ Δ
q1 txα τ1 t q2 txβ τ2 t 0,
rt zΔ t
1.1
on an arbitrary time scale T, where zt xt ptxτ0 t, γ, α, and β are quotient of odd
positive integers such that 0 < α < γ < β, r, p, q1 , and q2 are rd-continuous functions on
T, and r, q1 , and q2 are positive, −1 < −p0 ≤ pt < 1, p0 > 0; the so-called delay functions
τi : T → T satisfy that τi t ≤ t for t ∈ T and τi t → ∞ as t → ∞, for i 0, 1, 2, and there
exists a function τ : T → T which satisfies that τt ≤ τ1 t, τt ≤ τ2 t, and τt → ∞ as
t → ∞.
Since we are interested in the oscillatory and asymptotic behavior of solutions near
infinity, we assume that sup T ∞ and define the time scale interval t0 , ∞T by t0 , ∞T :
t0 , ∞ ∩ T.
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We will also consider the two cases
∞
t0
Δt
∞,
r 1/γ t
t0
r 1/γ t
∞
Δt
1.2
< ∞.
1.3
Recently, there has been a large number of papers devoted to the delay dynamic
equations on time scales, and we refer the reader to the papers in 1–17.
Agarwal et al. 1, Sahiner 10, Saker 11, Saker et al. 12, and Wu et al. 15 studied
the second-order nonlinear neutral delay dynamic equations on time scales
rt
yt ptyτt
Δ γ Δ
f t, yδt 0,
t ∈ T,
1.4
where 0 ≤ pt < 1, and 1.2 holds. By means of Riccati transformation technique, the authors
established some sufficient conditions for oscillation of 1.4.
Sun et al. 14 considered 1.1, where r Δ t ≥ 0, −1 < −p0 ≤ pt ≤ 0, and 1.2 holds.
The authors established some oscillation results of 1.1. To the best of our knowledge, there
are no results regarding the oscillation of the solutions of 1.1 when 1.3 holds.
We note that if T R, 1.1 becomes the second-order Emden-Fowler neutral delay
differential equation
γ q1 txα τ1 t q2 txβ τ2 t 0,
rt z t
t ≥ t0 .
1.5
Chen and Xu 18 as well as Xu and Liu 19 considered 1.5 and obtained some oscillation
criteria for 1.5 when rt 1. Qin et al. 20 found that some results under the case when
−1 < p0 ≤ pt ≤ 0 in 18, 19 are incorrect.
The paper is organized as follows. In the next section, by developing a Riccati
transformation technique some sufficient conditions for oscillation of all solutions of 1.1
on time scales are established. In Section 3, we give some examples to illustrate our main
results.
2. Main Results
In this section, by employing the Riccati transformation technique, we establish some new
oscillation criteria for 1.1. In order to prove our main results, we will use the formula
xγ tΔ γ
1
hxσ t 1 − hxtγ−1 xΔ tdh,
2.1
0
which is a simple consequence of Keller’s chain rule 21, Theorem 1.90. Also, we need the
following lemmas.
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It will be convenient to make the following notations:
a
θa, b; u : ub
d t : max{0, dt},
u
Δs/r 1/γ s
Δs/r 1/γ s
,
αt, u : θτt, σt; u,
β−α β−α
ν : min
,
,
β−γ γ −α
βt, u : θt, σt; u,
α β−γ/β−α β γ−α/β−α
q2 t 1 − pτ2 t
Q1 t : ν q1 t 1 − pτ1 t
αt, T γ ,
2.2
β−γ/β−α γ−α/β−α
Q2 t : ν q1 t
q2 t
αt, T γ ,
Q1∗ t Q1 t − ηΔ t,
Q2∗ t Q2 t − ηΔ t.
Lemma 2.1 see 3, Lemma 2.4. Assume that there exists T ≥ t0 , sufficiently large, such that
xt > 0,
γ Δ
< 0,
rt xΔ t
xΔ t > 0,
t ≥ T.
2.3
for t ≥ T1 ≥ T.
2.4
Then
xτt ≥ αt, T xσ t,
xt ≥ βt, T xσ t,
Lemma 2.2. Assume that 1.2 holds; 0 ≤ pt < 1. Furthermore, x is an eventually positive solution
of 1.1. Then there exists t1 ≥ t0 such that
zt > 0,
zΔ t > 0,
γ Δ
< 0,
rt zΔ t
for t ≥ t1 .
2.5
Proof. Let x be an eventually positive solution of 1.1. Then there exists t1 ≥ t0 such that
xt > 0, and xτi t > 0 for t ≥ t1 , i 0, 1, 2. From 1.1, we have
γ Δ
−q1 txα τ1 t − q2 txβ τ2 t < 0
rt zΔ t
2.6
for all t ≥ t1 , and so rtzΔ tγ is an eventually decreasing function.
We first show that rtzΔ tγ is eventually positive. Otherwise, there exists t2 ≥ t1
such that rt2 zΔ t2 γ c < 0; then from 2.6 we have rtzΔ tγ ≤ rt2 zΔ t2 γ c for
t ≥ t2 , and so
Δ
z t ≤ c
1/γ
1
rt
1/γ
,
2.7
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which implies by 1.2 that
zt ≤ zt2 c
1/γ
t t2
1
rs
1/γ
Δs −→ −∞ as t −→ ∞,
2.8
and this contradicts the fact that zt ≥ xt > 0 for all t ≥ t1 . Hence, we have that 2.5 holds
and completes the proof.
Lemma 2.3. Assume that 1.2 holds, −1 < −p0 ≤ pt ≤ 0, and limt → ∞ pt p > −1. Furthermore,
assume that there exists {ck }k≥0 such that limk → ∞ ck ∞ and τ0 ck1 ck . Then an eventually
positive solution x of 1.1 satisfies eventually 2.5 or limt → ∞ xt 0.
Proof. Suppose that x is an eventually positive solution of 1.1. Then there exists t1 ≥ t0 such
that xt > 0, and xτi t > 0 for t ≥ t1 , i 0, 1, 2. From 1.1, we have that 2.6 holds for all
t ≥ t1 , and so rtzΔ tγ is an eventually decreasing function.
We first show that rtzΔ tγ is eventually positive. Otherwise, there exists t2 ≥ t1
such that rt2 zΔ t2 γ c < 0; then from 2.6 we have rtzΔ tγ ≤ rt2 zΔ t2 γ c for
t ≥ t2 , and so
zΔ t ≤ c1/γ
1
rt
1/γ
2.9
,
which implies by 1.2 that
zt ≤ zt2 c
1/γ
t t2
1
rs
1/γ
Δs −→ −∞ as t −→ ∞.
2.10
Therefore, there exist d > 0 and t3 ≥ t2 such that
xt ≤ −d − ptxτ0 t ≤ −d p0 xτ0 t,
t ≥ t3 .
2.11
Thus, we can choose some positive integer k0 such that ck ≥ t3 for k ≥ k0 , and
xck ≤ −d p0 xτ0 ck −d p0 xck−1 ≤ −d − p0 d p02 xτ0 ck−1 −d − p0 d p02 xck−2 ≤ · · · ≤ −d − p0 d − · · · − p0k−k0 −1 d p0k−k0 xτ0 ck0 1 2.12
−d − p0 d − · · · − p0k−k0 −1 d p0k−k0 xck0 .
The above inequality implies that xck < 0 for sufficiently large k, which contradicts the fact
that xt is eventually positive. Hence zΔ t is eventually positive. Consequently, there are
two possible cases:
i zt is eventually positive, or
ii zt is eventually negative.
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If there exists a t4 ≥ t1 such that case ii holds, then limt → ∞ zt exists, and
limt → ∞ zt l ≤ 0; we claim that limt → ∞ zt 0. Otherwise, limt → ∞ zt < 0. We can choose
some positive integer k0 such that ck ≥ t4 for k ≥ k0 , and we obtain
xck ≤ p0 xτ0 ck p0 xck−1 ≤ p02 xτ0 ck−1 2.13
p02 xck−2 ≤ · · · ≤ p0k−k0 xτ0 ck0 1 p0k−k0 xck0 ,
which implies that limk → ∞ xck 0, and so limk → ∞ zck 0, which contradicts
limt → ∞ zt l < 0. Now, we assert that xt is bounded. If it is not true, then there exists
{tk } with tk → ∞ as k → ∞ such that
xtk max xs,
lim xtk ∞.
t0 ≤s≤tk
2.14
k→∞
From τ0 t ≤ t, we obtain
ztk xtk ptk xτ0 tk ≥ 1 − p0 xtk ,
2.15
which implies that limk → ∞ ztk ∞; it contradicts limt → ∞ zt 0. Therefore, we can assume
that
lim sup xt x1 ,
t→∞
lim inf xt x2 .
2.16
t→∞
By −1 < p ≤ 0, we get
x1 px1 ≤ 0 ≤ x2 px2 ,
2.17
which implies that x1 ≤ x2 , so x1 x2 . Hence, limt → ∞ xt 0. The proof is complete.
Theorem 2.4. Assume that 1.2 holds, 0 ≤ pt < 1, and γ ≥ 1. Furthermore, assume that there
exist positive rd-continuous Δ-differentiable functions δ and η such that, for all sufficiently large T,
for T1 > T
t
⎡
rs
lim sup ⎣δσ sQ1∗ s − δΔ sηs − γ1
t→∞
T1
γ 1
δΔ s
γ1
δσ sγ
βs, T −γ 2
⎤
⎦Δs ∞. 2.18
Then every solution of 1.1 is oscillatory.
Proof. Suppose that 1.1 has a nonoscillatory solution x. We may assume without loss of
generality that xτi t > 0, i 0, 1, 2, for all t ≥ t0 . By Lemma 2.2, there exists T ≥ t0 such that
2.5 holds. Define the function ω by
γ
rt zΔ t
ηt ,
ωt δt
zγ t
t ≥ T.
2.19
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Advances in Difference Equations
Then ωt > 0. By the product rule and the quotient rule, noteing 2.19, we have
⎡
ωΔ t Δ
δ t
⎢
ωt δσ t⎣
δt
γ Δ
rt zΔ t
zσ tγ
⎤
Δ γ γ
Δ
rt z t z t
⎥
−
ηΔ t⎦.
γ
γ
σ
z tz t
2.20
By 1.1 and 2.5, we obtain
γ Δ
α
β
rt zΔ t
≤ −q1 t 1 − pτ1 t zτ1 t − q2 t 1 − pτ2 t zτ2 t < 0.
2.21
In view of γ ≥ 1, from 2.1, we have zγ tΔ ≥ γztγ−1 zΔ t. By 2.20, we obtain
ωΔ t ≤
α zτ1 tα
δΔ t
ωt − δσ tq1 t 1 − pτ1 t
δt
zσ tγ
γ1
rt zΔ t
β zτ2 tβ
σ
− δ tq2 t 1 − pτ2 t
−
γδ
δσ tηΔ t.
t
ztzσ tγ
zσ tγ
2.22
σ
By Young’s inequality
|ab| ≤
1 p 1 q
|a| |b| ,
p
q
a, b ∈ R, p > 1, q > 1,
1 1
1,
p q
2.23
we have
α zτ1 tα γ − α
β zτ2 tβ
β−γ
q1 t 1 − pτ1 t
q
1
−
pτ
t
t
2
2
β−α
β−α
zσ tγ
zσ tγ
γ−α/β−α
α zτ1 tα β−γ/β−α
β zτ2 tβ
q2 t 1 − pτ2 t
≥ q1 t 1 − pτ1 t
zσ tγ
zσ tγ
α β−γ/β−α β γ−α/β−α zτ1 tα β−γ/β−α
q2 t 1 − pτ2 t
q1 t 1 − pτ1 t
zσ tγ
γ−α/β−α
zτ2 tβ
×
zσ tγ
α β−γ/β−α β γ−α/β−α zτt γ
q2 t 1 − pτ2 t
≥ q1 t 1 − pτ1 t
.
zσ t
2.24
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By Lemma 2.1, we have
zτt
≥ αt, T ,
zσ t
zt
≥ βt, T .
zσ t
2.25
Hence, by 2.19 and 2.22, we obtain
ωΔ t ≤
α β−γ/β−α
δΔ t
ωt − νδσ t q1 t 1 − pτ1 t
δt
β γ−α/β−α
× q2 t 1 − pτ2 t
αt, T γ
− γδσ t
2.26
γ1/γ
γ ωt
− ηt
δσ tηΔ t.
βt, T δt
1
rt1/γ
Thus
ωt
− ηt
δt
ω t ≤ −δ t Q1 t − ηΔ t δΔ tηt δΔ t
Δ
σ
− γδσ t
2.27
γ1/γ
γ ωt
− ηt
.
βt, T δt
1
rt1/γ
Set
γ 1
,
λ
γ
B
Aγ
1/λ
δ t
σ
1/λ
1
rt1/γ1
γ 2 /γ1 ωt
βt, T δt − ηt,
γ 3 /γ1
γ γ γ
1
rtγ/γ1
Δ
δ t
.
γ 1 γ γ 2 /γ1 δσ tγ 2 /γ1 βt, T 2.28
Using the inequality
λABλ−1 − Aλ ≤ λ − 1B λ ,
2.29
λ ≥ 1, A ≥ 0, B ≥ 0,
we obtain
rt
ω t ≤ −δ t Q1 t − ηΔ t δΔ tηt γ1
γ 1
Δ
σ
γ1
δΔ t δσ tγ
−γ 2
.
βt, T 2.30
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Integrating the last inequality from T1 > T to t > T1 , we obtain
−ωT1 < ωt − ωT1 ⎡
⎤
Δ γ1
t
2
δ
s
rs
−γ
⎦Δs,
≤ − ⎣δσ s Q1 s−ηΔ s −δΔ sηs− βs, T γ1
δσ sγ
T1
γ 1
2.31
which yields
t
⎡
⎣δσ s Q1 s − ηΔ s − δΔ sηs − rs
γ1
T1
γ 1
δΔ s
γ1
βs, T δσ sγ
−γ 2
⎤
⎦Δs ≤ ωT1 ,
2.32
which leads to a contradiction to 2.18. The proof is complete.
Theorem 2.5. Assume that 1.2 holds, 0 ≤ pt < 1, and γ ≤ 1. Furthermore, assume that there
exist positive rd-continuous Δ-differentiable functions δ and η such that, for all sufficiently large T,
for T1 > T
t
⎡
rs
lim sup ⎣δσ sQ1∗ s − δΔ sηs − γ1
t→∞
T1
γ 1
δΔ s
γ1
δσ sγ
βs, T −γ
⎤
⎦Δs ∞.
2.33
Then every solution of 1.1 is oscillatory.
Proof. Suppose that 1.1 has a nonoscillatory solution x. We may assume without loss of
generality that xτi t > 0, i 0, 1, 2, for all t ≥ t0 .
By Lemma 2.2, there exists T ≥ t0 such that 2.5 holds. Defining the function ω as
2.19, we proceed as in the proof of Theorem 2.4, and we get 2.20. In view of γ ≤ 1, using
2.1, we have zγ tΔ ≥ γzσ tγ−1 zΔ t. From 2.20 we obtain
ωΔ t ≤
α zτ1 tα
δΔ t
ωt − δσ tq1 t 1 − pτ1 t
δt
zσ tγ
γ1
rt zΔ t
β zτ2 tβ
σ
σ
δσ tηΔ t.
− δ tq2 t 1 − pτ2 t
− γδ t
zγ tzσ t
zσ tγ
The remainder of the proof is similar to that of Theorem 2.4, and hence it is omitted.
2.34
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Theorem 2.6. Assume that 1.3 holds, 0 ≤ pt < 1, limt → ∞ pt p1 < 1, and γ ≥ 1. Furthermore,
assume that there exist positive rd-continuous Δ-differentiable functions δ, η, and φ such that φΔ t ≥
0, then for all sufficiently large T, for T1 > T, one has that 2.18 holds, and
∞
t0
1
φsrs
s
φ τ q1 τ q2 τ Δτ
1/γ
σ
Δs ∞.
2.35
t0
Then every solution of 1.1 is either oscillatory or converges to zero.
Proof. We proceed as in Theorem 2.4, and we assume that xτi t > 0, i 0, 1, 2, for all t ≥ t0 .
From the proof of Lemma 2.2, we see that there exist two possible cases for the sign of zΔ t.
If zΔ t is eventually positive, we are then back to the proof of Theorem 2.4 and we
obtain a contradiction with 2.18.
If zΔ t < 0, t ≥ t1 ≥ t0 , then there exist constants c > 0, a > 0 such that zt ≤ c,
xt ≤ zt ≤ c, t ≥ t1 , and limt → ∞ zt a ≥ 0. Since x is bounded, we let lim supt → ∞ xt x1 ,
lim inft → ∞ xt x2 . From definition of zt, noting 0 ≤ p1 < 1, we have x1 p1 x2 ≤ a ≤
x2 p1 x1 ; hence, we have x1 ≤ x2 .
On the other hand, x1 ≥ x2 ; hence, limt → ∞ xt a/1 p1 . Assume that a > 0. Then
there exist a constant b > 0 and t2 ≥ t1 such that xα τ1 t ≥ b, xβ τ2 t ≥ b for t ≥ t2 . Define
the function
γ
ut φtrt zΔ t .
2.36
Then ut < 0 for t ≥ t2 . From 1.1 we have
γ
γ Δ
γ Δ
≤ φσ t rt zΔ t
uΔ t φΔ trt zΔ t φσ t rt zΔ t
−φ t q1 txα τ1 t q2 txβ τ2 t ≤ −bφσ t q1 t q2 t .
2.37
σ
Integrating the above inequality from t2 to t, we obtain
ut ≤ ut2 − b
t
φ s q1 s q2 s Δs ≤ −b
σ
t2
t
φσ s q1 s q2 s Δs,
2.38
t2
that is,
Δ
z t ≤ −b
1/γ
1
φtrt
t
t2
1/γ
φ s q1 s q2 s Δs
σ
.
2.39
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Integrating the last inequality from t2 to t, we get
zt ≤ zt2 − b
1/γ
t t2
1
φsrs
s
φ τ q1 τ q2 τ Δτ
1/γ
Δs.
σ
2.40
t2
We can easily obtain a contradiction with 2.35. Hence, limt → ∞ xt 0. This completes the
proof.
From Theorem 2.6, we have the following result.
Theorem 2.7. Assume that 1.3 holds, 0 ≤ pt < 1, limt → ∞ pt p1 < 1, and γ ≤ 1. Furthermore,
assume that there exist positive rd-continuous Δ-differentiable functions δ, η, and φ such that, for all
sufficiently large T, for T1 > T, one has that 2.33 and 2.35 hold. Then every solution of 1.1 is
either oscillatory or converges to zero.
The proof is similar to that of the proof of Theorem 2.6; hence, we omit the details.
In the following, we give some new oscillation results of 1.1 when pt < 0.
Theorem 2.8. Assume that 1.2 holds, −1 < −p0 ≤ pt ≤ 0, limt → ∞ pt p2 > −1, and γ ≥ 1.
Furthermore, there exists {ck }k≥0 such that limk → ∞ ck ∞ and τ0 ck1 ck . If there exist positive
rd-continuous Δ-differentiable functions δ and η such that, for all sufficiently large T, for T1 > T ,
t
⎡
rs
lim sup ⎣δ sQ2∗ s − δ sηs − γ1
t→∞
T1
γ 1
Δ
σ
δΔ s
γ1
δσ sγ
βs, T −γ 2
⎤
⎦Δs ∞,
2.41
then every solution of 1.1 is oscillatory or tends to zero.
Proof. Suppose that 1.1 has a nonoscillatory solution x. We may assume without loss of
generality that xτi t > 0, i 0, 1, 2, for all t ≥ t0 . By Lemma 2.3, there exists T ≥ t0 such that
2.5 holds, or limt → ∞ xt 0. Assume that 2.5 holds. Define the function ω as 2.19, and
then we get 2.20. By 1.1, we obtain
γ Δ
rt zΔ t
≤ −q1 tzτ1 tα − q2 tzτ2 tβ < 0.
2.42
In view of γ ≥ 1, from 2.1, we have zγ tΔ ≥ γztγ−1 zΔ t. By 2.20, we obtain
ωΔ t ≤
δΔ t
zτ1 tα
zτ2 tβ
ωt − δσ tq1 t
− δσ tq2 t
γ
σ
δt
z t
zσ tγ
γ1
rt zΔ t
σ
− γδ t
δσ tηΔ t.
ztzσ tγ
2.43
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By Young’s inequality 2.23, we have
β−γ
zτ1 tα γ − α
zτ2 tβ
q1 t
q
t
2
γ
β−α
β−α
zσ t
zσ tγ
zτ1 tα
≥ q1 t
zσ tγ
q1 t
γ−α/β−α
β−γ/β−α zτ2 tβ
q2 t
zσ tγ
γ−α/β−α
γ−α/β−α zτ1 tα β−γ/β−α zτ2 tβ
q2 t
zσ tγ
zσ tγ
β−γ/β−α β−γ/β−α γ−α/β−α zτt γ
q2 t
≥ q1 t
.
zσ t
2.44
By Lemma 2.1, we have
zτt
≥ αt, T ,
zσ t
zt
≥ βt, T .
zσ t
2.45
Hence, by 2.19 and 2.43, we obtain
ωΔ t ≤
β−γ/β−α γ−α/β−α
δΔ t
ωt − νδσ t q1 t
q2 t
αt, T γ
δt
− γδσ t
γ1/γ
γ ωt
− ηt
δσ tηΔ t.
βt, T δt
1
rt1/γ
2.46
Thus
ωt
− ηt
δt
ω t ≤ −δ t Q2 t − ηΔ t δΔ tηt δΔ t
Δ
σ
− γδ t
γ1/γ
γ ωt
− ηt
.
βt, T δt
1
σ
rt1/γ
2.47
Set
γ 1
,
λ
γ
B
Aγ
1/λ
δ t
σ
1/λ
1
rt1/γ1
γ 2 /γ1 ωt
βt, T δt − ηt,
γ 3 /γ1
γ γ γ
1
rtγ/γ1
Δ
δ t
.
γ 1 γ γ 2 /γ1 δσ tγ 2 /γ1 βt, T 2.48
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Using the inequality 2.29, we obtain
rt
ω t ≤ −δ t Q2 t − ηΔ t δΔ tηt γ1
γ 1
Δ
σ
γ1
δΔ t δσ tγ
−γ 2
.
βt, T 2.49
Integrating the last inequality from T1 > T to t > T1 , we obtain
−ωT1 < ωt − ωT1 ⎡
⎤
Δ γ1
t
2
δ
s
rs
−γ
⎦Δs,
≤ − ⎣δσ s Q2 s−ηΔ s −δΔ sηs− βs, T γ1
δσ sγ
T1
γ 1
2.50
which yields
t
⎡
⎣δ s Q2 s − ηΔ s − δΔ sηs − rs
γ1
T1
γ 1
δΔ s
σ
γ1
βs, T δσ sγ
−γ 2
⎤
⎦Δs ≤ ωT1 ,
2.51
which leads to a contradiction with 2.41. The proof is complete.
Theorem 2.9. Assume that 1.2 holds, −1 < −p0 ≤ pt ≤ 0, limt → ∞ pt p2 > −1, and γ ≤ 1.
Furthermore, there exists {ck }k≥0 such that limk → ∞ ck ∞ and τ0 ck1 ck . If there exist positive
rd-continuous Δ-differentiable functions δ and η such that, for all sufficiently large T, for T1 > T ,
t
⎡
rs
lim sup ⎣δσ sQ2∗ s − δΔ sηs − γ1
t→∞
T1
γ 1
δΔ s
γ1
δσ sγ
βs, T −γ
⎤
⎦Δs ∞,
2.52
then every solution of 1.1 is oscillatory or tends to zero.
Proof. Suppose that 1.1 has a nonoscillatory solution x. We may assume without loss of
generality that xτi t > 0, i 0, 1, 2, for all t ≥ t0 . By Lemma 2.3, there exists T ≥ t0 such
that 2.5 holds, or limt → ∞ xt 0. Assume that 2.5 holds.
Define the function ω as 2.19, and then we get 2.20. In view of γ ≤ 1, using 2.1,
we have zγ tΔ ≥ γzσ tγ−1 zΔ t. From 2.20 we obtain
δΔ t
zτ1 tα
zτ2 tβ
σ
ωt − δσ tq1 t
−
δ
tq
t
2
δt
zσ tγ
zσ tγ
γ1
rt zΔ t
− γδσ t
δσ tηΔ t.
zγ tzσ t
ωΔ t ≤
The remainder of the proof is similar to that of Theorem 2.8, and hence it is omitted.
2.53
Advances in Difference Equations
13
Remark 2.10. One can easily see that the results obtained in 1, 10–12, 15 cannot be applied
in 1.1, so our results are new.
3. Examples
In this section, we will give some examples to illustrate our main results.
Example 3.1. Consider the second-order quasilinear neutral delay dynamic equations on time
scales
Δ Δ
1
σt 1/3
σt 5/3
tσt xt xτ0 t
x τt x τt 0,
2
τt
τt
3.1
∞
where t ∈ t0 , ∞T , and we assume that t0 Δt/tσt ∞.
Let rt tσt, pt 1/2, q1 t q2 t σt/τt, γ 1, α 1/3, β 5/3, and
τ1 t τ2 t τt. Take δt ηt φt 1. It is easy to show that 2.18 and 2.35 hold.
Hence, by Theorem 2.6, every solution of 3.1 oscillates or tends to zero.
Example 3.2. Consider the second-order quasilinear neutral delay dynamic equations on time
scales
1
xt − xτ0 t
2
Δ Δ
σt 1/3
σt 5/3
x τt x τt 0,
tτt
tτt
3.2
where t ∈ t0 , ∞T , and we assume there exists {ck }k≥0 such that limk → ∞ ck ∞ and τ0 ck1 ck .
Let rt 1, pt −1/2, q1 t q2 t σt/tτt, γ 1, α 1/3, β 5/3, τ1 t τ2 t τt. Take δt ηt 1. It is easy to show that 2.41 holds. Hence, by Theorem 2.8,
every solution of 3.2 oscillates or tends to zero.
Acknowledgment
This research is supported by the Natural Science Foundation of China 60774004, 60904024,
China Postdoctoral Science Foundation funded project 20080441126, 200902564, Shandong
Postdoctoral funded project 200802018, the Natural Science Foundation of Shandong
Y2008A28, ZR2009AL003, and also the University of Jinan Research Funds for Doctors
B0621, XBS0843.
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