Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 927690, 15 pages
doi:10.1155/2011/927690
Research Article
Oscillation of Second-Order Neutral Functional
Differential Equations with Mixed Nonlinearities
Shurong Sun,1, 2 Tongxing Li,1, 3 Zhenlai Han,1, 3 and Yibing Sun1
1
School of Science, University of Jinan, Jinan, Shandong 250022, China
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO
65409-0020, USA
3
School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China
2
Correspondence should be addressed to Shurong Sun, sshrong@163.com
Received 2 September 2010; Revised 26 November 2010; Accepted 23 December 2010
Academic Editor: Miroslava Růžičková
Copyright q 2011 Shurong 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 study the following second-order neutral functional differential equation with mixed
α−1
nonlinearities rt|ut ptut − σ |α−1 ut ptut − σ q0 t|uτ
∞0 t|1/α uτ0 t β−1
γ−1
q1 t|uτ1 t| uτ1 t q2 t|uτ2 t| uτ2 t 0, where γ > α > β > 0, t0 1/r tdt < ∞.
Oscillation results for the equation are established which improve the results obtained by Sun and
Meng 2006, Xu and Meng 2006, Sun and Meng 2009, and Han et al. 2010.
1. Introduction
This paper is concerned with the oscillatory behavior of the second-order neutral functional
differential equation with mixed nonlinearities
α−1 ut ptut − σ
rt ut ptut − σ q0 t|uτ0 t|α−1 uτ0 t
q1 t|uτ1 t|β−1 uτ1 t q2 t|uτ2 t|γ −1 uτ2 t 0,
1.1
t ≥ t0 ,
where γ > α > β > 0 are constants, r ∈ C1 t0 , ∞, 0, ∞, p ∈ Ct0 , ∞, 0, 1, qi ∈
Ct0 , ∞, Ê, i 0, 1, 2, are nonnegative, σ ≥ 0 is a constant. Here, we assume that there
exists τ ∈ C1 t0 , ∞, Ê such that τt ≤ τi t, τt ≤ t, limt → ∞ τt ∞, and τ t > 0 for
t ≥ t0 .
2
Abstract and Applied Analysis
One of our motivations for studying 1.1 is the application of this type of equations in
real word life problems. For instance, neutral delay equations appear in modeling of networks
containing lossless transmission lines, in the study of vibrating masses attached to an elastic
bar; see the Euler equation in some variational problems, in the theory of automatic control
and in neuromechanical systems in which inertia plays an important role. We refer the reader
to Hale 1 and Driver 2, and references cited therein.
Recently, there has been much research activity concerning the oscillation of secondorder differential equations 3–8 and neutral delay differential equations 9–20. For the
particular case when pt 0, 1.1 reduces to the following equation:
rt|ut|α−1 ut
q0 t|uτ0 t|α−1 uτ0 t
β−1
q1 t|uτ1 t|
uτ1 t q2 t|uτ2 t|
1.2
γ −1
uτ2 t 0,
t ≥ t0 .
Sun and Meng 6 established some oscillation criteria for 1.2, under the condition
∞
t0
1
dt < ∞,
r 1/α t
1.3
they only obtained the sufficient condition 6, Theorem 5, which guarantees that every
solution u of 1.2 oscillates or tends to zero.
Sun and Meng 7 considered the oscillation of second-order nonlinear delay
differential equation
α−1
rtu t u t
q0 t|uτ0 t|α−1 uτ0 t 0,
1.4
t ≥ t0
and obtained some results for oscillation of 1.4, for example, under the case 1.3, they
obtained some results which guarantee that every solution u of 1.4 oscillates or tends to
zero, see 7, Theorem 2.2.
Xu and Meng 10 discussed the oscillation of the second-order neutral delay
differential equation
α−1 rt ut ptut − τ qtfuσt 0,
ut ptut − τ
t ≥ t0
1.5
and established the sufficient condition 10, Theorem 2.3, which guarantees that every
solution u of 1.5 oscillates or tends to zero.
Han et al. 11 examined the oscillation of second-order neutral delay differential
equation
α−1 rtψut ut ptut − τ qtfuσt 0,
ut ptut − τ
t ≥ t0
1.6
Abstract and Applied Analysis
3
and established some sufficient conditions for oscillation of 1.6 under the conditions 1.3
and
σt ≤ t − τ.
1.7
The condition 1.7 can be restrictive condition, since the results cannot be applied on the
equation
1
1
λ e2t e2t2 ut − 1 0,
e2t ut ut − 2
2
2
t ≥ t0 .
1.8
The aim of this paper is to derive some sufficient conditions for the oscillation
of solutions of 1.1. The paper is organized as follows. In Section 2, we establish some
oscillation criteria for 1.1 under the assumption 1.3. In Section 3, we will give three
examples to illustrate the main results. In Section 4, we give some conclusions for this paper.
2. Main Results
In this section, we give some new oscillation criteria for 1.1.
Below, for the sake of convenience, we denote
zt : ut ptut − σ,
t
ξt : r 1/α τt
t1
t
Rt :
1
rτs
α
Q0 t : 1 − pτ0 t q0 t,
t0
1/α
1
r 1/α s
ds,
τ sds,
β
Q1 t : 1 − pτ1 t q1 t,
γ
Q2 t : 1 − pτ2 t q2 t,
ζ0 t : q0 t
1
1 p ρt
α
,
ζ2 t : q2 t
h0 t : q0 t
1
1 pt
α
1
1 p ρt
ζ1 t : q1 t
γ
1
1 p ρt
,
,
h2 t : q2 t
h1 t : q1 t
1
1 pt
γ
,
1
1 pt
β
,
β
,
4
Abstract and Applied Analysis
δt :
∞
ρt
1
ds,
1/α
r s
ϕt : q0 t
πt :
δt
1 p ρt
∞
t
α
q1 t
1
r 1/α s
k1 :
ds,
γ −β
,
γ −α
β
δt
1 p ρt
q2 t
k2 :
δt
1 p ρt
γ −β
,
α−β
γ
.
2.1
Theorem 2.1. Assume that 1.3 holds, p t ≥ 0, and there exists ρ ∈ C1 t0 , ∞, Ê, such that
ρt ≥ t, ρ t > 0, τi t ≤ ρt − σ,i 0, 1, 2. If for all sufficiently large t1 ,
∞
ατ tRα−1 τtr 1−1/α τt
1/k1
1/k2
dt ∞,
R τt Q0 t k1 Q1 t
−
k2 Q2 t
ξα t
α
2.2
∞
α
ζ0 t k1 ζ1 t1/k1 k2 ζ2 t1/k2 δα t −
α1
α1
ρ t
dt ∞,
δtr 1/α ρt
2.3
then 1.1 is oscillatory.
Proof. Suppose to the contrary that u is a nonoscillatory solution of 1.1. Without loss of
generality, we may assume that ut > 0 for all large t. The case of ut < 0 can be considered
by the same method. From 1.1 and 1.3, we can easily obtain that there exists a t1 ≥ t0 such
that
zt > 0, z t > 0,
α−1
rtz t z t ≤ 0,
2.4
α−1
rtz t z t ≤ 0.
2.5
or
zt > 0, z t < 0,
If 2.4 holds, we have
α
α
rt z t ≤ rτt z τt ,
t ≥ t1 .
2.6
From the definition of z, we obtain
ut zt − ptut − σ ≥ zt − ptzt − σ ≥ 1 − pt zt.
2.7
Define
ωt Rα τt
rtz tα
,
zτtα
t ≥ t1 .
2.8
Abstract and Applied Analysis
5
Then, ωt > 0 for t ≥ t1 . Noting that z t > 0, we get zτi t ≥ zτt for i 0, 1, 2. Thus,
from 1.1, 2.7, and 2.8, it follows that
α
ατ tRα−1 τt rtz tα
α
α − R τt 1 − pτ0 t q0 t
1/α
r τt
zτt
ω t ≤
β
γ
− Rα τt 1 − pτ1 t q1 tzβ−α τt 1 − pτ2 t q2 tzγ −α τt
− αRα τt
rtz tα
zτtα1
2.9
z τtτ t.
By 2.4, 2.9, and τ t > 0, we get
ω t ≤
α
ατ tRα−1 τt rtz tα
α
α − R τt 1 − pτ0 t q0 t
1/α
r τt
zτt
β
γ
− Rα τt 1 − pτ1 t q1 tzβ−α τt 1 − pτ2 t q2 tzγ −α τt .
2.10
In view of 2.4, 2.6, and 2.10, we have
ω t ≤
α
ατ tRα−1 τt rτtz τtα
− Rα τt 1 − pτ0 t q0 t
α
1/α
r τt
zτt
β
γ
− Rα τt 1 − pτ1 t q1 tzβ−α τt 1 − pτ2 t q2 tzγ −α τt .
2.11
By 2.4, we obtain
zτt zτt1 zτt1 t
t1
z τsτ sds
t
t1
1
rτs
≥ r 1/α τtz τt
t
t1
1/α
α 1/α rτs z τs
τ sds
1
rτs
1/α
2.12
τ sds,
that is,
zτt ≥ ξtz τt.
2.13
Set
1/k1
,
a : k1 Q1 tzβ−α τt
1/k2
b : k2 Q2 tzγ −α τt
,
p : k1 ,
q : k2 .
2.14
6
Abstract and Applied Analysis
Using Young’s inequality
|ab| ≤
1 p 1 q
|a| |b| ,
p
q
1 1
1,
p q
a, b ∈ Ê, p > 1, q > 1,
2.15
we have
Q1 tzβ−α τt Q2 tzγ −α τt ≥ k1 Q1 t1/k1 k2 Q2 t1/k2 .
2.16
Hence, by 2.11, 2.13, and 2.16, we obtain
ω t ≤
ατ tRα−1 τtr 1−1/α τt
1/k1
1/k2
α
Q
.
Q
Q
−
R
τt
t
k
t
k
t
0
1
1
2
2
ξα t
2.17
Integrating 2.17 from t1 to t, we get
0 < ωt ≤ ωt1 ,
−
t
t1
2.18
ατ sRα−1 τsr 1−1/α τs
1/k1
1/k2
ds.
R τs Q0 s k1 Q1 s
−
k2 Q2 s
ξα s
2.19
α
Letting t → ∞ in 2.19, we get a contradiction to 2.2. If 2.5 holds, we define the function
υ by
υt rt−z tα−1 z t
,
zα ρt
t ≥ t1 .
2.20
Then, υt < 0 for t ≥ t1 . It follows from rt|z t|α−1 z t ≤ 0 that rt|z t|α−1 z t is
nonincreasing. Thus, we have
r 1/α sz s ≤ r 1/α tz t,
s ≥ t.
2.21
Dividing 2.21 by r 1/α s and integrating it from ρt to l, we obtain
zl ≤ z ρt r 1/α tz t
l
ρt
ds
r 1/α s
,
l ≥ ρt.
2.22
Letting l → ∞ in the above inequality, we obtain
0 ≤ z ρt r 1/α tz tδt,
t ≥ t1 ,
2.23
Abstract and Applied Analysis
7
that is,
z t
r 1/α tδt ≥ −1,
z ρt
t ≥ t1 .
2.24
Hence, by 2.20, we have
−1 ≤ υtδα t ≤ 0,
t ≥ t1 .
2.25
Differentiating 2.20, we get
rt−z tα−1 z t zα ρt − αrt−z tα−1 z tzα−1 ρt z ρt ρ t
,
υ t z2α ρt
2.26
by the above equality and 1.1, we obtain
υ t −q0 t
uα τ0 t
uβ τ1 t
uγ τ2 t
− q1 t α − q2 t α α
z ρt
z ρt
z ρt
2.27
αrt−z tα−1 z tzα−1 ρt z ρt ρ t
−
.
z2α ρt
Noticing that p t ≥ 0, from 10, Theorem 2.3, we see that u t ≤ 0 for t ≥ t1 , so by τi t ≤
ρt − σ, i 0, 1, 2, we have
uα τ0 t
zα ρt
uτ0 t
u ρt p ρt u ρt − σ
α
1
u ρt /uτ0 t p ρt uρt − σ/uτ0 t
≥
1
1 p ρt
uβ τ1 t
zα ρt
α
α
,
uτ1 t
u ρt p ρt u ρt − σ
β
zβ−α ρt
1
u ρt /uτ1 t p ρt u ρt − σ /uτ1 t
≥
1
1 p ρt
β
zβ−α ρt ,
β
zβ−α ρt
8
Abstract and Applied Analysis
u τ2 t/z ρt α
γ
γ
uτ2 t
u ρt p ρt u ρt − σ
zγ −α ρt
1
u ρt /uτ2 t p ρt u ρt − σ /uτ2 t
≥
1
1 p ρt
γ
γ
zγ −α ρt
zγ −α ρt .
2.28
On the other hand, from rt−z tα−1 z t ≤ 0, ρt ≥ t, we obtain
z ρt ≤
r 1/α t z t.
r 1/α ρt
2.29
Thus, by 2.20 and 2.27, we get
υ t ≤ − ζ0 t ζ1 tzβ−α ρt ζ2 tzγ −α ρt −
αρ t
−υtα1/α .
r 1/α ρt
2.30
Set
1/k1
a : k1 ζ1 tzβ−α ρt
,
1/k2
b : k2 ζ2 tzγ −α ρt
, p : k1 , q : k2 .
2.31
Using Young’s inequality 2.15, we obtain
ζ1 tzβ−α ρt ζ2 tzγ −α ρt ≥ k1 ζ1 t1/k1 k2 ζ2 t1/k2 .
2.32
Hence, from 2.30, we have
υ t ≤ − ζ0 t k1 ζ1 t1/k1 k2 ζ2 t1/k2 −
αρ t
−υtα1/α ,
r 1/α ρt
2.33
that is,
υ t ζ0 t k1 ζ1 t1/k1 k2 ζ2 t1/k2 αρ t
−υtα1/α ≤ 0,
r 1/α ρt
t ≥ t1 .
2.34
Abstract and Applied Analysis
9
Multiplying 2.34 by δα t and integrating it from t1 to t implies that
δα tυt − δα t1 υt1 α
t
t1
α
t
t1
r −1/α ρs ρ sδα−1 sυsds
ζ0 s k1 ζ1 s1/k1 k2 ζ2 s1/k2 δα sds
t
t1
2.35
δα sρ s
−υsα1/α ds ≤ 0.
r 1/α ρs
Set p : α 1/α, q : α 1, and
a : α 1α/α1 δα
2
/α1
tυt,
b :
α
α/α1
α 1
δ−1/α1 t.
2.36
1
.
δt
2.37
Using Young’s inequality 2.15, we get
−αδα−1 tυt ≤ αδα t−υtα1/α α
α1
α1
Thus,
−
α
αρ tδα−1 tυt
δα t−υtα1/α
≤
αρ
ρ
t
t
α1
r 1/α ρt
r 1/α ρt
α1
1
δtr 1/α
.
ρt
2.38
Therefore, 2.35 yields
δα tυt ≤ δα t1 υt1 ,
−
t
t1
α
ζ0 s k1 ζ1 s1/k1 k2 ζ2 s1/k2 δα s −
α1
α1
ρ s
ds.
δsr 1/α ρs
2.39
Letting t → ∞ in the above inequality, by 2.3, we get a contradiction with 2.25. This
completes the proof of Theorem 2.1.
From Theorem 2.1, when ρt t, we have the following result.
Corollary 2.2. Assume that 1.3 holds, p t ≥ 0, and τi t ≤ t − σ, i 0, 1, 2. If for all sufficiently
large t1 such that 2.2 holds and
∞ α
h0 t k1 h1 t1/k1 k2 h2 t1/k2 π α t −
α1
then 1.1 is oscillatory.
α1
1
dt ∞,
πtr 1/α t
2.40
10
Abstract and Applied Analysis
Theorem 2.3. Assume that 1.3 holds, p t ≥ 0, and there exists ρ ∈ C1 t0 , ∞, Ê, such that
ρt ≥ t, ρ t > 0, τi t ≤ ρt − σ, i 0, 1, 2. If for all sufficiently large t1 such that 2.2 holds and
∞
ζ0 t k1 ζ1 t1/k1 k2 ζ2 t1/k2 δα1 tdt ∞,
2.41
then 1.1 is oscillatory.
Proof. Suppose to the contrary that u is a nonoscillatory solution of 1.1. Without loss of
generality, we may assume that ut > 0 for all large t. The case of ut < 0 can be considered
by the same method. From 1.1 and 1.3, we can easily obtain that there exists a t1 ≥ t0 such
that 2.4 or 2.5 holds.
If 2.4 holds, proceeding as in the proof of Theorem 2.1, we obtain a contradiction
with 2.2.
If 2.5 holds, we proceed as in the proof of Theorem 2.1, then we get 2.25 and 2.34.
Multiplying 2.34 by δα1 t and integrating it from t1 to t implies that
δα1 tυt − δα1 t1 υt1 α 1
α
t
t1
t
t1
r −1/α ρs ρ sδα sυsds
ζ0 s k1 ζ1 s1/k1 k2 ζ2 s1/k2 δα1 sds
t
t1
2.42
δα1 sρ s
−υsα1/α ds ≤ 0.
r 1/α ρs
In view of 2.25, we have −υtδα1 t ≤ δt < ∞, t → ∞. From 1.3, we get
t
t1
−r −1/α ρs ρ sδα sυsds ≤
t
t1
t
t1
δα1 sρ s
−υsα1/α ds ≤
r 1/α ρs
r −1/α ρs ρ sds ρt
ρt1 r −1/α udu < ∞,
ρt
ρt1 r −1/α udu < ∞,
t −→ ∞,
t −→ ∞.
2.43
Letting t → ∞ in 2.42 and using the last inequalities, we obtain
∞
ζ0 t k1 ζ1 t1/k1 k2 ζ2 t1/k2 δα1 tdt < ∞,
which contradicts 2.41. This completes the proof of Theorem 2.3.
From Theorem 2.3, when ρt t, we have the following result.
2.44
Abstract and Applied Analysis
11
Corollary 2.4. Assume that 1.3 holds, p t ≥ 0, τi t ≤ t − σ, i 0, 1, 2. If for all sufficiently large
t1 such that 2.2 holds and
∞
h0 t k1 h1 t1/k1 k2 h2 t1/k2 π α1 tdt ∞,
2.45
then 1.1 is oscillatory.
Theorem 2.5. Assume that 1.3 holds, p t ≥ 0, and there exists ρ ∈ C1 t0 , ∞, Ê, such that
ρt ≥ t, ρ t > 0, τi t ≤ ρt − σ, i 0, 1, 2. If for all sufficiently large t1 such that 2.2 holds and
∞
t1
r
−1/α
v
v
t1
1/α
ϕudu
dv ∞,
2.46
then 1.1 is oscillatory.
Proof. Suppose to the contrary that u is a nonoscillatory solution of 1.1. Without loss of
generality, we may assume that ut > 0 for all large t. The case of ut < 0 can be considered
by the same method. From 1.1 and 1.3, we can easily obtain that there exists a t1 ≥ t0 such
that 2.4 or 2.5 holds.
If 2.4 holds, proceeding as in the proof of Theorem 2.1, we obtain a contradiction
with 2.2.
If 2.5 holds, we proceed as in the proof of Theorem 2.1, and we get 2.21. Dividing
2.21 by r 1/α s and integrating it from ρt to l, letting l → ∞, yields
z ρt ≥ −r 1/α tz t
∞
ρt
r −1/α sds −r 1/α tz tδt ≥ −r 1/α t1 z t1 δt : aδt.
2.47
By 1.1, we have
α q0 tuα τ0 t q1 tuβ τ1 t q2 tuγ τ2 t.
rt −z t
2.48
Noticing that p t ≥ 0, from 10, Theorem 2.3, we see that u t ≤ 0 for t ≥ t1 , so by τi t ≤
ρt − σ, i 0, 1, 2, we get
uτi t
uτi t
z ρt
u ρt p ρt u ρt − σ
2.49
1
1
.
≥
u ρt /uτi t p ρt u ρt − σ /uτi t
1 p ρt
Hence, we obtain
α ≥ bϕt,
rt −z t
2.50
12
Abstract and Applied Analysis
where b min{aα , aβ , aγ }. Integrating the above inequality from t1 to t, we have
t
α
α
rt −z t ≥ rt1 −z t1 b
ϕudu ≥ b
t1
t
t1
ϕudu.
2.51
Integrating the above inequality from t1 to t, we obtain
zt1 − zt ≥ b
1/α
t
t1
r
−1/α
v
v
t1
1/α
ϕudu
dv,
2.52
which contradicts 2.46. This completes the proof of Theorem 2.5.
3. Examples
In this section, three examples are worked out to illustrate the main results.
Example 3.1. Consider the second-order neutral delay differential equation 1.8, where λ > 0
is a constant.
Let rt e2t , pt 1/2, σ 2, q0 t λ2e2t e2t2 /2, α 1, τ0 t t − 1, q1 t q2 t 0, and τt τ0 t, then
Rt ξt r
1/α
τt
t
t0
t
t1
−2t
e 0 − e−2t
1
,
ds
2
r 1/α s
1
rτs
q0 t λ 2e2t e2t2
Q0 t ,
2
4
1/α
2t−t 1
−1
e
,
τ sds 2
3.1
2q0 t λ 2e2t e2t2
ζ0 t .
3
3
Setting ρt t 1, we have τ0 t t − 1 ≤ ρt − σ, δt e−2t−2 /2. Therefore, for all
sufficiently large t1 ,
∞
∞
ατ tRα−1 τtr 1−1/α τt
1/k1
1/k2
R τt Q0 t k1 Q1 t
−
dt ∞,
k2 Q2 t
ξα t
α
α
ζ0 t k1 ζ1 t1/k1 k2 ζ2 t1/k2 δα t −
α1
∞
α1
ρ t
dt
δtr 1/α ρt
λ 2e−2 1 − 3
dt ∞
6
if λ > 3/2e−2 1. Hence, by Theorem 2.1, 1.8 is oscillatory when λ > 3/2e−2 1.
3.2
Abstract and Applied Analysis
13
Note that 11, Theorem 2.1 and 11, Theorem 2.2 cannot be applied in 1.8, since
τ0 t > t − 2. On the other hand, applying 11, Theorem 3.2 to that 1.8, we obtain that 1.8
is oscillatory if λ > 3/e−2 2e−4 . So our results improve the results in 11.
Example 3.2. Consider the second-order neutral delay differential equation
√
√
π
65
1 1
et ut u t −
12 65et u t − arcsin
2
4
8
65
0,
t ≥ t0 .
3.3
√
Let rt √et , pt 1/2, σ π/4, q0 t 12 65et , q1 t q2 t 0, α 1, τ0 t t − arcsin 65/65/8, ρt t π/4, and τt t − π/4, then
t
Rt t0
1
ds e−t0 − e−t ,
r 1/α s
Q0 t √
q0 t
6 65et ,
2
t
1
rτs
ξt r
1/α
ζ0 t √
2q0 t
8 65et ,
3
τt
t1
1/α
τ sds et−t1 − 1,
δt e−t−π/4 .
3.4
Therefore, for all sufficiently large t1 ,
∞
∞
ατ tRα−1 τtr 1−1/α τt
1/k1
1/k2
R τt Q0 t k1 Q1 t
−
dt ∞,
k2 Q2 t
ξα t
α
α
ζ0 t k1 ζ1 t1/k1 k2 ζ2 t1/k2 δα t −
α1
∞
α1
ρ t
dt
δtr 1/α ρt
√ −π/4 1 dt ∞.
8 65e
−
4
3.5
Hence, by Theorem 2.1, 3.3 oscillates. For example, ut sin 8t is a solution of 3.3.
Example 3.3. Consider the second-order neutral differential equation
t e z t e2λ∗ t uλ0 t q1 tu1/3 λ1 t q2 tu5/3 λ2 t 0,
t ≥ t0 ,
3.6
where zt ut ut − 1/2, λi > 0 for i 0, 1, 2, are constants, q1 t > 0, q2 t > 0 for t ≥ t0 .
Let rt et , σ 1, q0 t e2λ∗ t , λ∗ max{λ0 , λ1 , λ2 }, τi t λi t, τt λt, 0 < λ <
min{λ0 , λ1 , λ2 , 1}, ρt λ∗ t 1, α 1, β 1/3, and γ 5/3, then k1 k2 2,
Rt ξt r
1/α
τt
t
t1
t
t0
1
rτs
1
r 1/α s
1/α
ds e−t0 − e−t ,
3.7
λt−t1 τ sds e
− 1,
δt e
−λ∗ t−1
.
14
Abstract and Applied Analysis
It is easy to see that 2.2 and 2.41 hold for all sufficiently large t1 . Hence, by Theorem 2.3,
3.6 is oscillatory.
4. Conclusions
In this paper, we consider the oscillatory behavior of second-order neutral functional
differential equation 1.1. Our results can be applied to the case when τi t > t, i 0, 1, 2;
these results improve the results given in 6, 7, 10, 11.
Acknowledgments
The authors sincerely thank the reviewers for their valuable suggestions and useful
comments that have led to the present improved version of the original paper. This research is
supported by the Natural Science Foundation of China nos. 11071143, 60904024, 11026112,
China Postdoctoral Science Foundation funded Project no. 200902564, the Natural Science
Foundation of Shandong nos. ZR2010AL002, ZR2009AL003, Y2008A28, and also the
University of Jinan Research Funds for Doctors no. XBS0843.
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