Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 513757, 14 pages
doi:10.1155/2011/513757
Research Article
Oscillation Criteria for
Second-Order Neutral Delay Dynamic Equations
with Mixed Nonlinearities
Ethiraju Thandapani,1 Veeraraghavan Piramanantham,2
and Sandra Pinelas3
1
Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600 005, India
Department of Mathematics, Bharathidasan University, Tiruchirappalli 620 024, India
3
Departamento de Matemática, Universidade dos Açores, 9501-801 Ponta Delgada, Azores, Portugal
2
Correspondence should be addressed to Sandra Pinelas, sandra.pinelas@clix.pt
Received 20 September 2010; Revised 30 November 2010; Accepted 23 January 2011
Academic Editor: Istvan Gyori
Copyright q 2011 Ethiraju Thandapani 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.
This paper is concerned with some oscillation criteria for the second order neutral delay
dynamic equations with mixed nonlinearities of the form rtutΔ qt|xτt|α−1 xτt n
αi −1
xτi t 0, where t ∈ T and ut |xt ptxδtΔ |α−1 xt i1 qi t|xτi t|
Δ
ptxδt with α1 > α2 > · · · > αm > α > αm1 > · · · > αn > 0. Further the results obtained here
generalize and complement to the results obtained by Han et al. 2010 . Examples are provided to
illustrate the results.
1. Introduction
Since the introduction of time scale calculus by Stefan Hilger in 1988, there has been great
interest in studying the qualitative behavior of dynamic equations on time scales, see, for
example, 1–3 and the references cited therein. In the last few years, the research activity
concerning the oscillation and nonoscillation of solutions of ordinary and neutral dynamic
equations on time scales has been received considerable attention, see, for example, 4–8
and the references cited therein. Moreover the oscillatory behavior of solutions of second
order differential and dynamic equations with mixed nonlinearities is discussed in 9–16.
In 2004, Agarwal et al. 5 have obtained some sufficient conditions for the oscillation
of all solutions of the second order nonlinear neutral delay dynamic equation
Δ γ Δ
f t, yt − δ 0
rt yt ptyt − τ
1.1
2
Advances in Difference Equations
on time scale T, where t ∈ T, γ is a quotient of odd positive integers such that γ ≥ 1, rt,
pt are real valued rd-continuous functions defined on T such that rt > 0, 0 ≤ pt < 1, and
ft, u ≥ qt|u|γ .
In 2009, Tripathy 17 has considered the nonlinear neutral dynamic equation of the
form
rt
γ
Δ γ Δ
yt ptyt − τ
qtyt − δ sgn yt − δ 0,
t ∈ T,
1.2
where γ > 0 is a quotient of odd positive integers, rt, qt are positive real valued rdcontinuous functions on T, pt is a nonnegative real valued rd-continuous function on T
and established sufficient conditions for the oscillation of all solutions of 1.2 using Ricatti
transformation.
Saker et al. 18, Şahı́ner 19, and Wu et al. 20 established various oscillation results
for the second order neutral delay dynamic equations of the form
Δ γ Δ
f t, yδt 0,
rt yt ptyτt
t ∈ T,
1.3
where 0 ≤ pt < 1, γ ≥ 1 is a quotient of odd positive integers, rt, pt are real valued
nonnegative rd-continuous functions on T such that rt > 0, and ft, u ≥ qt|u|γ .
In 2010, Sun et al. 21 are concerned with oscillation behavior of the second order
quasilinear neutral delay dynamic equations of the form
γ Δ
q1 txα τ1 t q2 txβ τ2 t 0,
rt zΔ t
t ∈ T,
1.4
where zt xt ptxτ0 t,γ, α, β are quotients of odd positive integers such that 0 < α <
γ < β and γ ≥ 1, rt, pt, q1 t, and q2 t are real valued rd-continuous functions on T.
Very recently, Han et al. 22 have established some oscillation criteria for quasilinear
neutral delay dynamic equation
γ−1 Δ
α−1
β−1
q1 tyδ1 t yδ1 t q2 tyδ2 t yδ2 t 0,
rtxΔ t xΔ
t ∈ T,
1.5
where xt yt ptyτt, α, β, γ are quotients of odd positive integers such that 0 < α <
γ < β, rt, pt, q1 t, and q2 t are real valued rd-continuous functions on T.
Motivated by the above observation, in this paper we consider the following second
order neutral delay dynamic equation with mixed nonlinearities of the form:
rtutΔ qt|xτt|α−1 xτt n
qi t|xτi t|αi −1 xτi t 0,
1.6
i1
where T is a time scale, t ∈ T and ut |xt ptxδtΔ |
this includes all the equations 1.1–1.5 as special cases.
α−1
xt ptxδtΔ , and
Advances in Difference Equations
3
1
t0 , ∞, which
By a proper solution of 1.6 on t0 , ∞T we mean a function xt ∈ Crd
α
1
has a property that rtxt ptxτt ∈ Crd t0 , ∞, and satisfies 1.6 on tx , ∞T .
For the existence and uniqueness of solutions of the equations of the form 1.6, refer to
the monograph 2. As usual, we define a proper solution of 1.6 which is said to be
oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is known
as nonoscillatory.
Throughout the paper, we assume the following conditions:
C1 the functions δ, τ, τi : T → T are nondecreasing right-dense continuous and satisfy
δt ≤ t, τt ≤ t, τi t ≤ t with limt → ∞ δt ∞, limt → ∞ τt ∞, and limt → ∞ τi t ∞ for i 1, 2, . . . , n;
C2 pt is a nonnegative real valued rd-continuous function on T such that 0 ≤ pt < 1;
C3 rt, qt and qi t, i 1, 2, . . . , n are positive real valued rd-continuous functions on
T with r Δ t ≥ 0;
C4 α, αi , i 1, 2, . . . , n are positive constants such that α1 > α2 > · · · > αm > α > αm1 >
· · · > αn > 0 n > m ≥ 1.
We consider the two possibilities
∞
t0
1
Δs ∞,
r 1/α s
t0
r 1/α s
∞
1
1.7
Δs < ∞.
1.8
Since we are interested in the oscillatory behavior of the solutions of 1.6, we may
assume that the time scale T is not bounded above, that is, we take it as t0 , ∞T {t ≥ t0 : t ∈
T}.
The paper is organized as follows. In Section 2, we present some oscillation criteria
for 1.6 using the averaging technique and the generalized Riccati transformation, and in
Section 3, we provide some examples to illustrate the results.
2. Oscillation Results
We use the following notations throughout this paper without further mention:
d t max{0, dt},
α
Qt qt 1 − pτt ,
κt σt
,
t
βt τt
,
σt
d− t max{0, −dt}
α
Qi t qi t 1 − pτi t i ,
βi t τi t
,
σt
i 1, 2, 3, . . . , n,
2.1
zt xt ptxδt.
In this section, we obtain some oscillation criteria for 1.6 using the following lemmas.
Lemma 2.1 is an extension of Lemma 1 of 13.
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Lemma 2.1. Let αi , i 1, 2, . . . , n be positive constants satisfying
α1 > α2 > · · · > αm > α > αm1 > · · · > αn > 0.
2.2
Then there is an n-tuple η1 , η2 , . . . , ηn satisfying
n
αi ηi α
2.3
i1
which also satisfies either
n
ηi < 1,
0 < ηi < 1,
2.4
ηi 1,
0 < ηi < 1.
2.5
i1
or
n
i1
In the following results we use the Keller’s Chain rule 1 given by
yα t
Δ
αyΔ t
1
α−1
σ
dh,
hy t 1 − hyt
2.6
0
where y is a positive and delta differentiable function on T.
Lemma 2.2 see 23. Let fu Bu − Auα1/α , where A > 0 and B are constants, γ is a positive
γ
integer. Then f attains its maximum value on R at u∗ Bγ /Aγ1 , and
γγ
Bγ1
γ1 Aγ .
γ 1
maxf fu∗ u∈R
2.7
Lemma 2.3. Assume that 1.7 holds. If xt is an eventually positive solution of 1.6, then there
α Δ
exists a T ∈ t0 , ∞T such that zt > 0, zΔ t > 0, and rtzΔ t Moreover one obtains
xt ≥ 1 − pt zt,
t ≥ t1 .
< 0 for t ∈ T, ∞T .
2.8
Since the proof of Lemma 2.3 is similar to that of Lemma 2.1 in 6, we omit the details.
Lemma 2.4. Assume that 1.7 and
∞
t0
τ α sQsΔs ∞
2.9
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5
hold. If xt is an eventually positive solution of 1.6, then
zΔΔ t < 0,
zt ≥ tzΔ t,
2.10
and zt/t is strictly decreasing.
α Δ
Proof. From Lemma 2.3, we have rtzΔ t < 0 and
α Δ
α
α Δ
rt zΔ t
r Δ t zΔ t rσt zΔ t
.
2.11
α Δ
Since r Δ t ≥ 0, we have zΔ t < 0. Now using the Keller’s Chain rule, we find that
0<
1 α Δ
α−1
ΔΔ
hzΔ t 1 − hzΔ t
αz t
dh
z t
Δ
2.12
0
or zΔΔ t < 0. Let Zt : zt − tzΔ t. Clearly ZΔ t −σtzΔΔ t > 0. We claim that there is
a t1 ∈ t0 , ∞T such that Zt > 0 on t1 , ∞T . Assume the contrary, then Zt < 0 on t1 , ∞T .
Therefore,
zt
t
Δ
Zt
tzΔ t − zt
−
> 0,
tσt
tσt
t ∈ t1 , ∞T ,
2.13
which implies that zt/t is strictly increasing on t1 , ∞T . Pick t2 ∈ t1 , ∞T so that τt ≥ τt2 and τi t ≥ τi t2 for t ≥ t2 . Then zτt/τt ≥ zτt2 /τt2 : d > 0, and zτt/τt ≥
zτt2 /τt2 : di > 0, so that zτt > τt for t ≥ t2 .
Using the inequality 2.8 in 1.6, we have that
n
α Δ
rt zΔ t
Qtzα τt Qi tzαi τi t ≤ 0.
2.14
i1
Now by integrating from t2 to t, we have
n
α
α t
Δ
Δ
α
αi
Qsz τs rt z t − rt2 z t2 Qi sz τi s Δs ≤ 0,
t2
2.15
i1
which implies that
Δ
Δ
rt2 z t2 ≥ rtz t t Qsz τs α
t2
> dα
t
t2
Qsτ α sΔs n
Qi sz τi s Δs
αi
i1
n
i1
diαi
t
t2
Qi sτiαi sΔs
2.16
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Advances in Difference Equations
which contradicts 2.4. Hence there is a t1 ∈ t0 , ∞T such that Zt > 0 on t1 , ∞T .
Consequently,
zt
t
Δ
Zt
tzΔ t − zt
−
< 0,
tσt
tσt
t ∈ t1 , ∞T ,
2.17
and we have that zt/t is strictly decreasing on t1 , ∞T .
Theorem 2.5. Assume that condition 1.7 holds. Let η1 , η2 , . . . , ηn be n-tuple satisfying 2.3 of
Lemma 2.1. Furthermore one assumes that there exist positive delta differentiable function ρt and a
nonnegative delta differentiable function φt such that
⎡
α1 ⎤
α2
rs ρΔ s ρΔ s κ
s
⎦Δs ∞,
φs −
ρσs⎣Q s − φ s −
lim sup
ρσ s
α 1α1 ρσs α1
t→∞
t1
t
∗
Δ
2.18
for all sufficiently large t1 where Q∗ t Qtβα t η
every solution of 1.6 is oscillatory.
n
ηi
αi ηi
t,
i1 Qi tβi
and η n
−ηi
i1 ηi .
Then
Proof. Suppose that there is a nonoscillatory solution xt of 1.6. We assume that xt is an
eventually positive for t ≥ t0 since the proof for the case xt < 0 eventually is similar. From
the definition of zt and Lemma 2.3, there exists t1 ≥ t0 such that, for t ≥ t1 ,
zt > 0,
zδt > 0,
zτt > 0,
zτi t > 0,
zΔ t > 0,
α Δ
≤ 0.
rt zΔ t
2.19
Define
α
rt zΔ t
wt ρt
φt ,
zα t
t ≥ t1 .
2.20
Then from 2.19, we have wt > 0 and
α Δ
rt zΔ t
ρΔ t
wt ρσt
w t ρσtφΔ t
ρt
zα t
Δ
≤
Δ
α Δ
rt zΔ t
ρ t
wt ρσt
ρt
zα σt
α
rt zΔ t zα tΔ
ρσtφΔ t.
− ρσt
zα tzα σt
2.21
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From Keller’s chain rule, we have, from Lemma 2.1,
zα tΔ ≥
⎧
⎨αzα−1 tzΔ t,
α ≥ 1,
⎩αzα−1 σtzΔ t,
0 < α < 1.
2.22
Using 2.22 and the definition of κt in 2.21, we obtain
n
ρΔ t
ρσt
α
αi
wt
Qtz τt Qi tz τi t w t ≤ − α
z σt
ρt
i1
Δ
α1/α
αρσt 1 wt
− 1/α
−
φt
ρσtφΔ t.
α
r t κ t ρt
2.23
From Lemma 2.4, we see that zt/t is strictly decreasing on t1 , ∞T , and therefore
zτi t zσt
≥
τi t
σt
2.24
zτi t τi t
≥
,
zσt σt
2.25
or
since τi t ≤ σt for all i 1, 2, . . . , n. Using 2.25 in 2.23, we have
Δ
w t ≤ −ρσt Qtβ t α
n
i1
Qi tβiαi tzαi −α σt
ρΔ t
wt
ρt
α1/α
αρσt 1 wt
− 1/α
− φt
ρσtφΔ t.
α
r t κ t ρt
2.26
Now let ui t 1/ηi Qi tβiαi zαi −α σt, i 1, 2, . . . , n. Then 2.26 becomes
Δ
w t ≤ −ρσt Qtβ t α
n
i1
ηi ui t ρΔ t
wt
ρt
α1/α
αρσt 1 wt
− 1/α
− φt
ρσtφΔ t.
α
r t κ t ρt
2.27
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Advances in Difference Equations
By Lemma 2.1 and using the arithmetic-geometric inequality
obtain
n
i1
ηi ui ≥
n
ui
i1 ηi
in 2.27, we
ρΔ t
wt
wΔ t ≤ −ρσt Q∗ t − φΔ t ρt
α1/α
αρσt 1 wt
− 1/α
−
φt
ρσtφΔ t
α
r t κ t ρt
2.28
or
wΔ t ≤ −ρσt Q∗ t − φΔ t ρΔ t φt
α1/α
wt
αρσt 1 wt
Δ
− φt − 1/α
− φt
ρ t ,
α
ρt
r t κ t ρt
t ≥ t1 .
2.29
Set γ α, A αρσt/r1/αt1/κα t, B ρΔ t , and ut |wt/ρt − φt| and applying Lemma 2.2 to 2.29, we have
wΔ t ≤ −ρσt Q∗ t − φΔ t ρΔ t φt α1
rt ρΔ t 2
α κα t.
α1
ρσt
α 1
1
2.30
Now integrating 2.30 from t1 to t, we obtain
⎤
Δ α1
Δ
rs
ρ
ρ
s
s
1
2
φs −
ρσ s⎣Q∗ s − φΔ s −
κα s⎦Δs ≤ wt1 ,
ρσ s
α 1α1 ρσs α1
t1
t
⎡
2.31
which leads to a contradiction to condition 2.18. The proof is now complete.
By different choices of ρt and φt, we obtain some sufficient conditions for the
solutions of 1.6 to be oscillatory. For instance, ρt 1, φt 1 and ρt t, φt 1/t
in Theorem 2.5, we obtain the following corollaries:
Corollary 2.6. Assume that 1.7 holds. Furthermore assume that, for all sufficiently large T , for
T ≥ t0 ,
∞
lim sup
t→∞
Q∗ sΔs ∞,
T
where Q∗ t is as in Theorem 2.5. Then every solution of 1.6 is oscillatory.
2.32
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Corollary 2.7. Assume that 1.7 holds. Furthermore assume that, for all sufficiently large T , for
T ≥ t0 ,
∞
2
rtσtα −α
∗
σsQ s −
lim sup
Δs ∞,
tα2
t→∞
T
2.33
where Q∗ t is as in Theorem 2.5. Then every solution of 1.6 is oscillatory.
Next we establish some Philos-type oscillation criteria for 1.6.
Theorem 2.8. Assume that 1.7 holds. Suppose that there exists a function H ∈ Crd D, R, where
D ≡ {t, s/t, s ∈ t0 , ∞T and t > s} such that
Ht, t 0,
t ≥ t0 ,
Ht, s ≥ 0,
t > s ≥ 0,
2.34
and H has a nonpositive continuous Δ-partial derivative H Δs with respect to the second variable such
that
ρΔ s ht, s
Hσt, σsα/α1 ,
ρs
ρs
2.35
ht, sα1 rs
ρ sQ s −
α Δs ∞,
α 1α1 ρσ s
2.36
H Δs σt, s Hσt, σs
and for all sufficiently large T ,
1
lim sup
t → ∞ Hσt, T t T
σ
∗
where Q∗ t is same as in Theorem 2.5. Then every solution of 1.6 is oscillatory.
Proof. We proceed as in the proof of Theorem 2.5 and define wt by 2.20. Then wt > 0
and satisfies 2.28 for all t ∈ t1 , ∞T . Multiplying 2.28 by Hσt, σs and integrating,
we obtain
t
Hσt, σsρσ s Q∗ s − φΔ t Δs
t1
≤−
t
Hσt, σswΔ sΔs t1
−
Hσt, σs
t1
t
Hσt, σs
t1
t
ρΔ t
wtΔs
ρs
α1/α
αρσt
1 wt
−
φt
Δs.
α
1/α
α1/α
r tρ
t κ t ρt
2.37
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Advances in Difference Equations
Using the integration by parts formula, we have
t
t1
Hσt, σswΔ sΔs Ht, sws |tt1 −
−Ht, t1 wt1 −
t
H Δs σt, swsΔs
t1
t
2.38
Δs
H σt, swsΔs.
t1
Substituting 2.38 into 2.37, we obtain
t
Hσt, σsρσ s Q∗ s − φΔ t Δs
t1
≤ Ht, t1 wt1 t t1
−
ρΔ t
H σt, s Hσt, σs
wsΔs
ρs
Δs
t
Hσt, σs
t1
2.39
α1/α
αρσt
1 wt
−
φt
Δs.
α
1/α
α1/α
r tρ
t κ t ρt
From 2.35 and 2.39, we have
t
Hσt, σsρσ sQt, sΔs
t1
≤ Ht, t1 wt1 t
ht, s α/α1
H
σt, σswsΔs
t1 ρs
t
α1/α
αρσt
1 wt
−
φt
Hσt, σs 1/α
Δs
−
α
r tρα1/α t κ t ρt
t1
2.40
or
t
Hσt, σsQt, sΔs
t1
≤ Ht, t1 wt1 t
ws
ht, s α1/α
H
− φsΔs
σt, σs
ρs
ρs
t1
t
α1/α
αρσt
1 wt
−
− φt
Hσt, σs 1/α
Δs.
α1/α t κα t ρt
r
tρ
t1
where Qt, s ρσ sQt, s − ht, s/ρsH 1/α σt, σsφs.
2.41
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By setting B ht, s/ρsH α1/α σt, σs and A αρσt/r 1/α tρα1/α t1/
κ t in Lemma 2.2, we obtain
α
t
2
Hσt, σs ρ sQt, s −
σ
t1
hα1 t, srsκα t
α 1α1 ρα σsHσt, σs
Δs
2.42
≤ Ht, t1 wt1 ,
which contradicts condition 2.35. This completes the proof.
Finally in this section we establish some oscillation criteria for 1.6 when the condition
1.8 holds.
Theorem 2.9. Assume that 1.8 holds and limt → ∞ pt p < 1. Let η1 , η2 , . . . , ηn be n-tuple
satisfying 2.3 of Lemma 2.1. Moreover assume that there exist positive delta differentiable functions
ρt and θt such that θΔ t ≥ 0 and a nonnegative function φt with condition 2.30 for all t ≥ t1 .
If
∞ t0
where Qt Qt zero as t → ∞.
n
i1
1
θsrs
1/α
s
Δs ∞,
θσvQvΔv
2.43
t0
Qi t holds, then every solution of 1.6 either oscillates or converges to
Proof. Assume to the contrary that there is a nonoscillatory solution xt such that xt > 0,
xδt > 0, xτt > 0, and xτi t > 0 for t ∈ t1 , ∞T for some t1 ≥ t0 . From Lemma 2.3 we
can easily see that either zΔ t > 0 eventually or zΔ t < 0 eventually.
If zΔ t > 0 eventually, then the proof is the same as in Theorem 2.5, and therefore we
consider the case zΔ t < 0.
If zΔ t < 0 for sufficiently large t, it follows that the limit of zt exists, say a. Clearly
a ≥ 0. We claim that a 0. Otherwise, there exists M > 0 such that zα τt ≥ M and
zαi τi t ≥ M, i 1, 2, . . . , n, t ∈ t1 , ∞T . From 1.6 we have
Δ
rt z t
α Δ
≤ −M Qt n
Qi t −MQt.
2.44
i1
Define the supportive function
α
ut θtrt zΔ t ,
t ∈ t1 , ∞T ,
2.45
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and we have
α
α Δ
uΔ t θΔ trt zΔ t θσt rt zΔ t
α Δ
≤ θσt rt zΔ t
2.46
−MθσtQt.
Now if we integrate the last inequality from t1 to t, we obtain
ut ≤ ut1 − M
t
θσsQsΔs
2.47
t1
or
zΔ t
α
≤ −M
1
θtrt
t
θσsQsΔs.
2.48
t1
Once again integrate from t1 to t to obtain
M
1/α
t t1
1
θsrs
1/α
s
θσξQξΔξ
Δs ≤ zt1 ,
2.49
t1
which contradicts condition 2.43. Therefore limt → ∞ zt 0, and there exists a positive
constant c such that zt ≤ c and xt ≤ zt ≤ c. Since xt is bounded, lim supt → ∞ xt x1
and lim inft → ∞ xt x2 . Clearly x2 ≤ x1 . From the definition of zt, we find that x1 px2 ≤
0 ≤ x2 px1 ; hence x1 ≤ x2 and x1 x2 0. This completes proof of the theorem.
Remark 2.10. If qi t ≡ 0, i 1, 2, . . . , n, or δt t − δ, τt t − τ, and qi t ≡ 0, i 1, 2, . . . , n,
then Theorem 2.5 reduces to a result obtained in 20 or 24, respecively. If pt ≡ 0, or pt ≡
0, and α 1, or pt ≡ 0, and τt τi t t, i 1, 2, . . . , n, then the results established here
complement to the results of 5, 9, 15 respectively.
3. Examples
In this section, we illustrate the obtained results with the following examples.
Example 3.1. Consider the second order delay dynamic equation
ΔΔ
√ λ
√ 1
λ1 √ λ2
3
xt 2 xδt
3/2 x t x5/3 t 2 x1/3 t 0,
t
t
t
t
3.1
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for all t ∈ 1, ∞T . Here α 1, α1 1/3, α2 5/3, pt 1/t2 , qt λ1 /t3/2 , q1 t λ2 /t, and
q2 t λ3 /t2 . Then η1 η2 1/2. By taking ρt t, and φt 0, we obtain
⎡
α1 ⎤
rs ρΔ s
lim sup
ρ s⎣Q s −
σ α1 ⎦Δs
α 1α1
t→∞
t1
ρ s
t
λ2 λ3
1
λ1
1
1
−
lim sup
1−
1−
Δs
s
s
s
s
4σs
t→∞
t0
t 1 1 λ1 λ2 λ3
≥ lim sup
Δs
−
λ1 λ2 λ3 −
4 s
s2
t→∞
t0
→ ∞ if λ1 λ2 λ3 > 1/4.
t
σ
∗
1
By Theorem 2.5, all solutions of 3.1 are oscillatory if λ1 3.2
λ2 λ3 > 1/4.
Example 3.2. Consider the second order neutral delay dynamic equation
⎛
⎝
⎞
Δ 3 Δ
3
1
⎠ σ t x3 t σt x5 t σt x1/3 t 0,
xt xδt
2
2
3
3
t2
t2
t4
3.3
for all t ∈ 1, ∞T . Here rt 1, pt 1/2, qt σ 3 t/t4 , τt t/2, τ1 t τ2 t t/3,
α 3, α1 5, α2 1/3. From Corollary 2.6, every solution of 3.3 is oscillatory.
Acknowledgment
The authors thank the referees for their constructive suggestions and corrections which
improved the content of the paper.
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