Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2010, Article ID 308357, 14 pages
doi:10.1155/2010/308357
Research Article
Oscillation Criteria for Even Order Neutral
Equations with Distributed Deviating Argument
Siyu Zhang and Fanwei Meng
Department of Mathematics, Qufu Normal University, Qufu 273165, China
Correspondence should be addressed to Siyu Zhang, siyuzhang521@163.com
Received 24 September 2009; Accepted 24 November 2009
Academic Editor: Leonid Berezansky
Copyright q 2010 S. Zhang and F. Meng. 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 present new oscillation criteria for the even order neutral delay differential equations with
b
distributed deviating argument rtψxtZ n−1 t a pt, ξfxgt, ξdσξ 0, t ≥ t0 ,
where Zt xt qtxt − τ. Assumptions in our theorems are less restrictive, whereas the
proofs are significantly simpler compared to those by Wang et al. 2005.
1. Introduction
In this paper, we are concerned with the oscillation behavior of the even order neutral delay
differential equations of the form
b
n−1
rtψxtZ
t pt, ξf x gt, ξ dσξ 0,
t ≥ t0 ,
1.1
a
where Zt xt qtxt − τ, τ 0 and n is an even positive integer. We assume that
∞
A1 r, q ∈ CI, R and 0 ≤ qt ≤ 1, rt > 0 for t ∈ I, 1/rsds ∞, I t0 , ∞;
0;
A2 ψ ∈ C1 R, R, ψx > 0 for x /
0;
A3 f ∈ CR, R, xfx > 0 for x /
A4 p ∈ CI × a, b, 0, ∞ and pt, ξ is not eventually zero on any half linear tu , ∞ ×
a, b, tu ≥ t0 ;
A5 g ∈ CI × a, b, 0, ∞, gt, ξ ≤ t for ξ ∈ a, b, gt, ξ has a continuous and positive
partial derivative on I × a, b with respect to t and nondecreasing with respect to
ξ, respectively, lim inft → ∞ gt, ξ ∞ for ξ ∈ a, b;
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A6 σ ∈ Ca, b, R is nondecreasing, and the integral of 1.1 is in the sense of
Riemann-Stieltijes.
We restrict our attention to those solutions xt of 1.1 which exist on some half linear
0 for any T t0 . As usual, such a solution of 1.1
tx , ∞ and satisfy sup{|xt| : t tx } /
is called oscillatory if the set of its zeros is unbounded from above; otherwise, it is said to be
nonoscillatory. Equation 1.1 is called oscillatory if all its solutions are oscillatory.
The oscillatory behavior of solutions of higher-order neutral differential equations is
of both theoretical and practical interest. There have been some results on the oscillatory and
asymptotic behavior of even order neutral equations. We mention here 1–12. The oscillation
problem for nonlinear delay equation such as
rtx t qtfxσt 0,
1.2
t > t0
as well as for the the linear ordinary differential equation
rtx t ptx t qtxt 0,
1.3
t > t0
and the neutral delay differential equation
xt qtxt − σ
1.4
ptxt − τ 0
has been studied by many authors with different methods. In 13, Rogovchenko established
some general oscillation criteria for second-order nonlinear differential equation:
rtx t ptx t qtfxt 0,
1.5
t ≥ t0 .
In 14, the authors discussed the following neutral equations of the form
xt ctxt − τn b
pt, ξx gt, ξ dσξ 0,
t ≥ t0
1.6
a
and obtained the following results.
Theorem A see 14, Theorem 2. Assume that there exist functions Ht, s ∈ C D; R, ht, s ∈
CD0 ; R, and ρt ∈ C t0 , ∞,0, ∞, such that
I Ht, t 0, Ht, s > 0;
II Hs t, s ≤ 0, and −Hs t, s − Ht, sρ s/ρs ht, s Ht, s, and
Ht, s
≤ ∞;
0 < inf lim inf
s≥t0 t → ∞
Ht, t0 t
ρsh2 t, s
1
lim sup
ds < ∞.
n−2
t→∞
Ht, t0 t0 g s, ag s, a
C1 C2 International Journal of Differential Equations
3
If there exists a function ϕt ∈ Ct0 , ∞, R satisfying
1
lim sup
t→∞
Ht, u
b
t Ht, sρs ps, ξ 1 − c gs, ξ dσξ
u
a
ρsh2 t, s
−
ds ≥ ϕu,
2Mθ g n−2 s, ag s, a
lim sup
t→∞
1
Ht, t0 t
g u, ag n−2 u, a 2
ϕ udu ∞,
ρu
t0
C3 u ≥ t0 ,
ϕ u max ϕu, 0 ,
u≥t0
C4 then every solution of 1.6 is oscillatory.
We will use the function class W to study the oscillation criteria for 1.1. Let D {t, s | t ≥ s ≥ t0 }, and D0 {t, st > s ≥ t0 }. We say that a continuous function Ht, s ⊂
C D, R belongs to the class W if
A7 Ht, t 0 and Ht, s > 0 for −∞ < s < t < ∞;
∂H/∂s satisfying, for some h ∈ Lloc D, R,
A8 H has a continuous partial derivative
the condition ∂H/∂s −ht, s Ht, s.
The purpose of this paper is to further improve Theorem A by Wang et al. 14, using a
generalized Riccati transformation and developing ideas exploited by the Rogovchenko and
Tuncay 13, we establish some new oscillation criteria for 1.1, which remove condition C2 in Theorem A by Wang et al. 14; this complements and extends the results in 14.
In addition, we will make use of the following conditions.
S1 There exists a positive real number M such that |f±uv| ≥ Mfufv for uv > 0.
Lemma 1.1. If a > 0, b ≥ 0, then
a
b2
.
−ax2 bx ≤ − x2 2
2a
1.7
Lemma 1.2 Kiguradze 15. Let u(t ) be a positive and n times differentiable function on R. If
un t is of constant sign and identically zero on any ray t, ∞ for t1 > 0, then there exists a
tu ≥ t1 and an integer l 0 ≤ l ≤ n, with n l even for utun t ≥ 0 or n l odd for utun t ≤ 0,
and for t ≥ tu ,
utuk t > 0,
0 ≤ k ≤ l;
−1k−1 utuk t > 0,
l ≤ k ≤ n.
1.8
Lemma 1.3 Philos 16. Suppose that the conditions of Lemma 1.2 are satisfied, and
un−1 tun t ≤ 0,
t ≥ tu ,
1.9
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then there exists a constant θ in (0,1) such that for sufficiently large t, and there exists a constant
Mθ > 0 satisfying
t ≥ Mθ tn−2 un−1 t,
u
2 1.10
where Mθ θ/n − 2!.
2. When fx Is Monotone
In this section, we will deal with the oscillation for 1.1 under the assumptions
A1 –A8 , S1 and the following assumption.
A9 f x exists, f x ≥ K1 and ψx ≤ L−1 for x /
0.
Theorem 2.1. Let S1 , A1 –A9 hold. Equation 1.1 is oscillatory provided that ρt ∈
C1 t0 , ∞, R such that
1
lim sup
t→∞
Ht, t0 t Ht, sQs −
t0
h2 t, sρsrsβ
ds ∞,
K1 LMθ g n−2 s, ag s, a
2.1
where
Qt ρtM
b
pt, ξf 1 − q gt, ξ dσξ −
a
2
ρ t rt
.
K1 LMθ g n−2 t, ag t, aρt
2.2
Proof. Suppose to the contrary that there exists a solution xt of 1.1 such that
xt > 0,
xt − τ > 0,
x gt, ξ > 0,
t ≥ t1 ,
ξ ∈ a, b
for t t1 t0 .
2.3
From 1.1, we also have Zt > 0 and rtψxtZn−1 t ≤ 0 for t ≥ t1 .
It follows that the function rtψxtZn−1 t is decreasing and we claim that
Zn−1 t ≥ 0
for t ≥ t1 .
2.4
Otherwise, if there exist a t1 ≥ t1 such that Zn−1 t1 < 0, then for all t ≥ t1 ,
rtψxtZn−1 t ≤ r t1 ψ x t1 Zn−1 t1 −CC > 0,
2.5
which implies that Zn−1 t ≤ −C/rtψxt, t ≥ t1 ; integrating the above inequality from
t1 to t, we have
t
Zn−2 t ≤ Zn−2 t1 − CL
t1
1
ds.
rt
2.6
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Let t → ∞; from A1 , we get limt → ∞ Zn−2 t −∞, which implies that Zn−1 t and
Zn−2 t are negative for all large t; from Lemma 1.2, no two consecutive derivative can be
eventually negative, for this would imply that limt → ∞ Zt −∞, which is a contradiction.
Hence Zn−1 t ≥ 0 for t ≥ t1 . Using this fact together with xt ≤ Zt, we have that
xt ≥ 1 − qt Zt,
t ≥ t1 .
2.7
Now from A1 , S1 , and 2.7, we get
f x gt, ξ ≥ Mf 1 − q gt, ξ f Z gt, ξ ,
t ≥ t1 ,
2.8
and thus, from1.1, we get
0 rtψxtZ
M
b
n−1
t b
pt, ξf x gt, ξ dσξ ≥ rtψxtZn−1 t
a
2.9
pt, ξf 1 − q gt, ξ f Z gt, ξ dσξ.
a
Further, observing that gt, ξ is nondecreasing with respect to ξ and Zn−1 t > 0 for t ≥ t1 ,
from Lemma 1.2, we have Z t ≥ 0, t ≥ t1 , and so
Z gt, ξ ≥ Z gt, a ,
t ≥ t1 , ξ ∈ a, b.
2.10
So, fZgt, ξ ≥ fZgt, a for t ≥ t1 and ξ ∈ a, b. Thus
b
pt, ξf 1 − q gt, ξ dσξ ≤ 0,
rtψxtZn−1 t Mf Z gt, a
a
t t1 .
2.11
Define
wt ρt
rtψxtZn−1 t
,
f Z gt, a/2
t t1 .
2.12
From 1.1, 2.11, and Lemma 1.3 we get
rtψxtZn−1 t
ρ t
wt ρt
w t ρt
f Z gt, a/2
rtψxtZn−1 t gt, a
gt, a 1 Z
g t, a
− ρt 2 f Z
2
2
2
f Z gt, a/2
≤
ρ t
wt − ρtM
ρt
b
1 K1 LMθ g n−2 t, ag t, a 2
w t.
pt, ξf 1 − q gt, ξ dσξ −
2
ρtrt
a
2.13
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International Journal of Differential Equations
Then, by Lemma 1.1 we get
w t ≤ −ρtM
b
pt, ξf 1 − q gt, ξ dσξ a
−
2
ρ t rt
K1 LMθ g n−2 t, ag t, aρt
1 K1 LMθ g n−2 t, ag t, a 2
w t
4
ρtrt
−Qt −
2.14
1 K1 LMθ g n−2 t, ag t, a 2
w t.
4
ρtrt
Let
Qt ρtM
b
pt, ξf 1 − q gt, ξ dσξ −
a
2
ρ t rt
.
K1 LMθ g n−2 t, ag t, aρt
2.15
That is,
Qt ≤ −w t −
K1 LMθ g n−2 t, ag t, a 2
w t.
4ρtrt
2.16
Integrating by parts for any t > T ≥ t1 , and using properties A7 and A8 , we obtain
t
Ht, sQsds
T
t
t
K1 LMθ g n−2 s, ag s, a 2
w s
4ρsrs
T
T
t
t
K1 LMθ g n−2 s, ag s, a 2
∂Ht, s
Ht, T wT ws
ds − Ht, s
w sds
∂s
4ρsrs
T
T
t
t
K1 LMθ g n−2 s, ag s, a 2
∂Ht, s
ds − Ht, s
w sds
− ws
Ht, T wT −
∂s
4ρsrs
T
T
t K1 LMθ g n−2 s, ag s, a 2
ht, s Ht, swsHt, s
w s ds
Ht, T wT −
4ρsrs
T
≤−
Ht, sw sds −
Ht, s
Ht, T wT −
t
T
⎛
⎝
Ht, sK1 LMθ g n−2 s, ag s, a
wsht, s
4βρsrs
t 2
h t, sρsrs
β
ds
Mθ K1 L T g n−2 s, ag s, a
β − 1 K1 Mθ L t Ht, sg n−2 s, ag s, a 2
−
w sds.
4β
ρsrs
T
⎞2
βρsrs
⎠ ds
K1 LMθ g n−2 s, ag s, a
2.17
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7
We obtain
t βh2 t, sρsrs
Ht, sQs −
ds
K1 LMθ g n−2 s, ag s, a
T
β − 1 K1 Mθ L t Ht, sg n−2 s, ag s, a 2
w sds
≤ Ht, T wT −
4β
ρsrs
T
⎛
⎞2
t n−2
Ht,
sK
LM
g
s,
a
ρsrs
ag
s,
1
θ
⎠ ds.
ws ht, s
− ⎝
4βρsrs
K1 LMθ g n−2 s, ag s, a
T
2.18
From A8 , H t, s ≤ 0, for t1 ≥ t0 , Ht, t1 ≤ Ht, t0 ,
t t1
βh2 t, sρsrs
Ht, sQs −
ds ≤ Ht, t1 wt1 ≤ Ht, t0 wt1 ,
K1 LMθ g n−2 s, ag s, a
2.19
which implies that
t Ht, sQs −
βh2 t, sρsrs
ds
K1 LMθ g n−2 s, ag s, a
t0
t1 h2 t, sρsrs
1
Ht, sQs −
≤ wt1 ds
Ht, t0 t0
K1 LMθ g n−2 s, ag s, a
1
Ht, t0 ≤ wt1 t1
2.20
Qsds < ∞.
t0
Let t → ∞, and taking upper limits, we have
1
lim sup
t→∞
Ht, t0 t Ht, sQs −
t0
h2 t, sρsrsβ
ds < ∞,
K1 LMθ g n−2 s, ag s, a
2.21
which contradicts the assumption 2.1. This complete the proof of Theorem 2.1.
From Theorem 2.1, we have the following oscillation result.
Corollary 2.2. If condition 2.1 of Theorem 2.1 is replaced by
lim sup
t→∞
1
Ht, t0 t
1
lim sup
t→∞
Ht, t0 t
Ht, sQsds ∞,
t0
h2 t, sρsrsβ
ds < ∞,
n−2 s, ag s, a
t0 K1 LMθ g
where Qt is defined by 2.2, then 1.1 is oscillatory.
2.22
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International Journal of Differential Equations
Remark 2.3. By introducing various Ht, s from Theorem 2.1 or Corollary 2.2, we can obtain
some oscillatory criteria of 1.1. For example, let Ht, s t − sm−1 , t ≥ s ≥ t0 , in which
m > 2 is a integer. By choosing
ht, s t − sm−3/2 m − 1,
2.23
it is clear that the conditions of A7 and A8 hold; then, from Theorem 2.1 and Corollary 2.2,
we have the following.
Corollary 2.4. Assume that there exists a function ρt ∈ C t0 , ∞, 0, ∞ such that
lim sup
t→∞
t t − sm−1 Qs −
m−1
1
t
t0
ρsrsβ
t − sm−3 m − 12 ds ∞,
K1 LMθ g n−2 s, ag s, a
2.24
where Qt is defined by 2.2, then 1.1 is oscillatory.
Corollary 2.5. Assume that there exists a function ρt ∈ C t0 , ∞, 0, ∞ such that
lim sup
t→∞
lim sup
t→∞
1
tm−1
1
tm−1
t
t − sm−1 Qsds ∞,
t0
t
ρsrsβ
t − sm−3 m − 12 ds < ∞,
n−2 s, ag s, a
t0 K1 LMθ g
2.25
where Qt is defined by 2.2, then 1.1 is oscillatory.
Theorem 2.6. Assume that the conditions of Theorem 2.1 hold, and
Ht, s
≤ ∞.
0 < inf lim inf
s≥t0 t → ∞
Ht, t0 2.26
If there exists a function ϕt ∈ Ct0 , ∞, R satisfying
1
lim sup
t→∞
Ht, u
t Ht, sQs −
u
h2 t, sρsrsβ
ds ≥ ϕu,
K1 LMθ g n−2 s, ag s, a
t
lim sup
t→∞
g n−2 u, ag u, a 2
ϕ udu ∞,
ρuru
t0
where Qt is defined by 2.2, then 1.1 is oscillatory.
u ≥ t0 ,
ϕ u max ϕu, 0 ,
u≥t0
2.27
2.28
International Journal of Differential Equations
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Proof. Assume that there exists a nonoscillatory solution xt of 1.1 on t0 , ∞, such that
xt /
0 on t0 , ∞. Without loss of generality, assume that xt > 0, t ≥ t0 . Then, proceeding as
in the proof of Theorem 2.1, for t > u ≥ t1 ≥ t0 , we have
t Ht, sQs −
βh2 t, sρsrs
ds
K1 LMθ g n−2 s, ag s, a
u
β − 1 K1 Mθ L t Ht, sg n−2 s, ag s, a 2
1
w sds.
≤ wu −
Ht, u
4β
ρsrs
u
1
Ht, u
2.29
Let t → ∞, and taking upper limits, we have
t Ht, sQs −
βh2 t, sρsrs
ds
K1 LMθ g n−2 s, ag s, a
u
β − 1 K1 Mθ L t Ht, sg n−2 s, ag s, a 2
1
w sds,
≤ wu − lim inf
t→∞
Ht, u
4β
ρsrs
u
1
lim sup
t→∞
Ht, u
2.30
thus, from 2.27, we have
β − 1 K1 Mθ L t Ht, sg n−2 s, ag s, a 2
1
w sds,
wu ≥ ϕu lim inf
t→∞
Ht, u
4β
ρsrs
u
2.31
then wu ≥ ϕu, and
1
lim inf
t→∞
Ht, u
t
Ht, sg n−2 s, ag s, a 2
4β
w sds < wu − ϕu < ∞.
ρsrs
β − 1 K1 Mθ L
u
2.32
Now we can claim that
∞
t1
g n−2 s, ag s, a 2
w sds < ∞,
ρsrs
t < t1 .
2.33
t < t1 .
2.34
In fact, assume the contrary, that
∞
t1
g n−2 s, ag s, a 2
w sds ∞,
ρsrs
From 2.26, there exists a constant ρ > 0 such that
Ht, s
inf lim inf
> ρ > 0,
s≥t0 t → ∞
Ht, t0 2.35
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International Journal of Differential Equations
this is
lim inf
t→∞
Ht, s
> ρ > 0,
Ht, t0 2.36
and there exists a T2 ≥ t1 such that Ht, T /Ht, t0 ≥ ρ, for all t ≥ T2 . On the other hand, by
virtue of 2.34, for any positive number α, there exists a T1 ≥ t1 , such that, for all t ≥ T1
t
g n−2 s, ag s, a 2
α
w sds > .
ρsrs
ρ
t1
2.37
Using integration by parts, we conclude that, for all t ≥ T > t1 ,
t
Ht, sg n−2 s, ag s, a 2
w sds
ρsrs
t1
s
t
g n−2 u, ag u, a 2
1
w sdu
Ht, sd
Ht, t1 t1
ρuru
t1
t s n−2
g u, ag u, a 2
1
∂H
w sdu
ds
−
Ht, t1 t1
ρuru
∂s
t1
1
Ht, t1 ≥
2.38
α Ht, T ≥ α.
ρ Ht, t1 Since α is an arbitrary positive constant,
lim inf
t→∞
1
Ht, t1 t
Ht, sg n−2 s, ag s, a 2
w sds ∞,
ρsrs
t1
2.39
which contradicts 2.32, consequently, 2.33 holds, and, by virtue of ωu ≥ ϕu for u ≥
t 1 ≥ t0 ,
t
g u, ag n−2 u, a 2
lim sup
ϕ udu ≤ lim sup
t→∞
t→∞
ρuru
t0
t
g u, ag n−2 u, a 2
ω udu < ∞,
ρuru
t0
2.40
which contradicts 2.28, and therefore, 1.1 is oscillatory.
Remark 2.7. Choosing H as in Remark 2.3, it is not difficult to see that condition 2.26 is
satisfied because, for any s ≥ t0 ,
Ht, s
t − sn−1
lim
1.
t → ∞ Ht, t0 t → ∞ t − t n−1
0
lim
2.41
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11
Consequently, one immediately derives from Theorem 2.6 the following useful
corollary for the oscillation of 1.1.
Corollary 2.8. Assume that there exist functions ρt ∈ Ct0 , ∞, 0, ∞ and ϕt ∈ Ct0 , ∞, R
satisfying
lim sup
t→∞
t 1
tm−1
t− sm−1 Qs−
t0
ρsrsβ
m−3
2
ds ≥ ϕu,
t−s
m−1
K1 LMθ g n−2 s, ag s, a
t
g n−2 u, ag u, a 2
ϕ udu 0,
ρuru
t0
lim sup
t→∞
u ≥ 0,
ϕ u max ϕu, 0 ,
u≥t0
2.42
where Qt is defined by 2.2, then 1.1 is oscillatory.
Theorem 2.9. Assume that the conditions of Theorem 2.1 and 2.26 hold, and
1
lim inf
t→∞
Ht, u
t Ht, sQs −
u
h2 t, sρsrsβ
ds ≥ ϕu,
K1 LMθ g n−2 s, ag s, a
t
g n−2 u, ag u, a 2
ϕ udu ∞,
lim sup
t→∞
ρuru
t0
u ≥ t0 ,
ϕ u max ϕu, 0 ,
2.43
u≥t0
then 1.1 is oscillatory.
Proof. Assume that there exists a nonoscillatory solution xt of 1.1 on t0 , ∞, such that
xt /
0 on t0 , ∞. Without loss of generality, assume that xt > 0, t ≥ t0 . Then, proceeding as
in the proof of Theorem 2.1, for t > u ≥ t1 ≥ t0 , we have
t Ht, sQs −
βh2 t, sρsrs
ds
K1 LMθ g n−2 s, ag s, a
T
β − 1 K1 Mθ L t Ht, sg n−2 s, ag s, a 2
1
w sds.
≤ wT −
Ht, T 4β
ρsrs
T
1
Ht, T 2.44
Let t → ∞, and taking lower limits, we have
t Ht, sQs −
βh2 t, sρsrs
ds
K1 LMθ g n−2 s, ag s, a
u
β − 1 K1 Mθ L t Ht, sg n−2 s, ag s, a 2
1
w sds.
≤ wu − lim sup
t→∞
Ht, u
4β
ρsrs
u
1
lim inf
t→∞
Ht, u
2.45
The following proof is similar to Theorem 2.6, so we omit the details. This completes
the proof of Theorem 2.9.
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3. When fx Is Not Monotone
In this section, we will deal with the oscillation for 1.1 under the assumptions A1 –A8 and the following assumption:
A10 fx/x ≥ K2 and ψx ≤ L−1 for x /
0.
Theorem 3.1. Let A1 –A8 and A10 hold. Equation 1.1 is oscillatory provided that ρt ∈
C1 t0 , ∞, R such that
1
lim sup
t→∞
Ht, t0 t Ht, sQ2 s −
t0
h2 t, sρsrsβ
ds ∞,
LMθ g n−2 s, ag s, a
3.1
where
Q2 t ρtK2
b
pt, ξf 1 − q gt, ξ dσξ −
a
2
ρ t rt
,
LMθ g n−2 t, ag t, aρt
3.2
then 1.1 is oscillatory.
Proof. Let xt be an eventually positive solution of 1.1. As in the proof of Theorem 2.1, there
exists t1 t0 , such that 2.3, 2.4, and 2.7 hold. Thus, from 1.1 and A10 , we get
0 rtψxtZ
n−1
t
b
pt, ξf x gt, ξ dσξ
a
b
≥ rtψxtZn−1 t K2 pt, ξx gt, ξ dσξ
3.3
a
rtψxtZ
n−1
t
K2
b
pt, ξ Z gt, ξ − q gt, ξ x gt, ξ − τ dσξ.
a
Noting that
Z gt, ξ ≥ Z gt, ξ − τ ≥ x gt, ξ − τ .
3.4
Thus, 3.3 implies that
b
pt, ξ 1 − q gt, ξ Z gt, ξ dσξ ≤ 0,
t ≥ t1 .
3.5
b
n−1
rtψxtz
t K2 Z gt, a
pt, ξ 1 − q gt, ξ dσξ ≤ 0,
t ≥ t1 .
3.6
rtψxtZ
n−1
t
K2
a
From 2.10 and 3.5 we get
a
International Journal of Differential Equations
13
Define
wt ρt
rtψxtZn−1 t
,
Z gt, a/2
3.7
t t1 .
Differentiating 3.7 and using 3.6, Lemma 1.1, and 1.3 we get
b
Mθ Lg n−2 t, ag t, a 2
ρ t
w t ≤
wt − ρt K2 pt, ξ 1 − q gt, ξ dσξ −
w t
ρt
2rtρt
a
≤ −K2 ρt
b
pt, ξ 1 − q gt, ξ dσξ a
−
2
ρ t rt
Mθ Lg n−2 t, ag t, aρt
Mθ Lg n−2 t, ag t, a 2
w t
4ρtrt
−Q2 t −
Mθ Lg n−2 t, ag t, a 2
w t.
4ρtrt
3.8
The rest proof is similar to that of Theorem 2.1 and hence is omitted. This completes the proof
of Theorem 3.1.
Theorem 3.2. Assume that the conditions of Theorem 2.1 and 2.26 hold; if there exists a function
ϕt ∈ Ct0 , ∞, R satisfying
1
lim sup
t→∞
Ht, u
t t
u
h2 t, sρsrsβ
Ht, sQ2 s −
ds ≥ ϕu,
LMθ g n−2 s, ag s, a
g n−2 u, ag u, a 2
lim sup
ϕ udu ∞,
t→∞
ρuru
t0
u ≥ t0 ,
ϕ u max ϕu, 0 ,
3.9
u≥t0
then 1.1 is oscillatory.
Theorem 3.3. Let all assumptions of Theorem 2.6 be satisfied except that lim sup in condition
Theorem 3.2 is replaced with lim inf, then 1.1 is oscillatory.
Acknowledgment
This research was partial supported by the NNSF of China 10771118.
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