Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2010, Article ID 181784, 15 pages
doi:10.1155/2010/181784
Research Article
Oscillation Theorems for Second-Order
Forced Neutral Nonlinear Differential Equations
with Delayed Argument
Jing Shao1, 2 and Fanwei Meng2
1
2
Department of Mathematics, Jining University, Qufu, Shandong 273155, China
School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
Correspondence should be addressed to Jing Shao, shaojing99500@163.com
Received 13 October 2009; Accepted 14 December 2009
Academic Editor: Elena Braverman
Copyright q 2010 J. Shao 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 are concerned with the oscillation of the forced second-order neutral nonlinear differential
equations with delayed argument in the form rtxt atxσt ptfxτt n
λi
i1 qi t|xt| sngxt et. No restriction is imposed on the potentials pt, qi t, andet to
be nonnegative. Our methodology is somewhat different from those of previous authors.
1. Introduction
In this paper, we study the oscillatory behavior of the forced neutral nonlinear functional
differential equation of the form
n
rtxt atxσt ptfxτt qi t|xt|λi sgn xt et,
1.1
i1
where t ≥ t0 . In this paper, we assume that
I1 rt ∈ Ct0 , ∞, 0, ∞, r t ≥ 0,
∞
1/rtdt ∞,
I2 at ∈ Ct0 , ∞, 0, 1,
I3 σt ∈ Ct0 , ∞, R is nondecreasing, σt ≤ t for t ≥ t0 , and limt → ∞ σt ∞,
I4 τt ∈ Ct0 , ∞, R, τt ≤ t for t ≥ t0 and limt → ∞ τt ∞,
I5 pt, qi t, and et are continuous functions defined on 0, ∞, pt > 0, λ1 > · · · >
λm > 1 > λm1 > · · · > λn > 0 n > m ≥ 1,
I6 fx is nondecreasing, fx/x ≥ M > 0, and x /
0.
2
International Journal of Differential Equations
No restriction is imposed on the potentials pt, qi t, and et to be nonnegative. As
usual, a solution of 1.1 is called oscillatory if it is defined on some ray T, ∞ with T ≥ 0 and
has unbounded set of zeros. 1.1 is called oscillatory if all of its solutions on some ray are
oscillatory.
In the last decades, there has been an increasing interest in obtaining sufficient
conditions for the oscillation and/or nonoscillation of second-order linear and nonlinear
delay differential equations see, for example, 1–22 and the references therein. Let us
consider the familiar forced Emden-Fowler equation
x t pt|xt|λ sgn xt et,
1.2
t ≥ t0 .
When λ1 > 1, 1.2 is known as the superlinear equation, and when 0 < λ1 < 1, it is known as
the sublinear equation. The oscillation of 1.2 has been the subject of much attention during
the last 50 years; see the seminal book by Agarwal, et al. 23. Here, we refer to the papers 1–
3 and the references cited therein. In this case, one can usually establish oscillation criteria for
more general nonlinear equations by using a technique introduced by Kartsatos 9 where it is
additionally assumed that f is the second derivative of an oscillatory function. This approach
has been expressed in 5, 6. Sun 4 has extended these results to delay differential equations
of the form of 1.2, where λ ≥ 1 and the potentials pt and et are allowed to change
sign. However, Sun 4 does not say anything else for the oscillation of equation 1.2 with
0 < λ < 1. Later, employing the arguments in 4, Çakmak and Tiryaki 7 have established
similar oscillation criteria for the equation of the form
x t qtfxτt et,
1.3
where fx is assumed to satisfy certain growth conditions.
Very recently, Sun et al. 13, 14 obtained some new oscillation criteria for the equations
in the form
n
rtx t ptxt qi t|xt|λi sgn xt 0,
i1
rtx t ptxt n
1.4
λi
qi t|xt| sgn xt et,
i1
where λ1 > · · · > λm > 1 > λm1 > · · · > λn > 0 n > m ≥ 1. He also established oscillation
theorems when n > 1. When n 1, this approach was initiated by Agarwal and Grace 1,
pages 244–249 for higher-order equations and subsequently developed in papers of Ou and
Wong 15, Q.Yang 18, X.Yang 19, as well as Sun and Agarwal 16, 17.
In 24, Xu and Meng studied the oscillation of the equation
n
rtxt atxσt qi tfi yτi t 0,
i1
by using the generalized Riccati technique and the function class Y.
t ≥ t0 ,
1.5
International Journal of Differential Equations
3
The purpose of this paper is to give some new oscillation criteria for 1.1, which can
be regarded as further investigation for the 1.1 including the papers of Sun and Wong 13,
Xu and Meng 24. These criteria do not assume that rt, pt, qi t, and et are of definite
sign. Our methodology is somewhat different from those of previous authors, and the results
we obtained are more general than those of Sun and Wong 13.
2. Main Results
We will need the following lemmas that have been proved in 13.
Lemma 2.1 see 13. Let λi , i 1, 2, . . . , n, be n-tuple satisfying λ1 > · · · > λm > 1 > λm1 >
· · · > λn > 0. Then there exists an n-tuple k1 , k2 , . . . , kn satisfying
n
λi ki 1,
a
i1
which also satisfies either
n
ki < 1,
0 < ki < 1,
b
0 < ki < 1.
c
i1
or
n
ki 1,
i1
Lemma 2.2 see 13. Let u, A, B, C, and D be positive real numbers. Then
i Auα B ≥ αα − 11/α−1 A1/α B1−1/α u, α > 1,
ii Cu − Duα ≥ α − 1αα/1−α Cα/α−1 D1/1−α , 0 < α < 1.
Remark 2.3. For a given set of exponents λi satisfying λ1 > · · · > λm > 1 > λm1 > · · · > λn > 0,
Lemma 2.1 ensures the existence of an n-tuple k1 , k2 , . . . , kn such that either a and b hold
or a and c hold. When n 2 and λ1 > 1 > λ2 > 0, in the first case, we have that
k1 1 − λ2 1 − k0 ,
λ1 − λ2
k2 λ1 1 − k0 − 1
,
λ1 − λ2
d
where k0 can be any positive number satisfying 0 < k0 < λ1 − 1/λ1 . This will ensure that
0 < k1 , k2 < 1, and conditions a and b are satisfied. In the second case, we simply solve
a and c and obtain
k1 1 − λ2
,
λ1 − λ2
k2 λ1 − 1
.
λ1 − λ2
e
4
International Journal of Differential Equations
Theorem 2.4. Suppose that, for any T ≥ 0, there exist constants a1 , b1 , a2 , b2 such that T ≤ a1 <
b1 ≤ a2 < b2 , and
qi t ≥ 0,
t ∈ τa1 , b1 ∪ τa2 , b2 , i 1, . . . , n,
et ≤ 0,
t ∈ τa1 , b1 ,
et ≥ 0,
t ∈ τa2 , b2 .
2.1
Let Daj , bj {u ∈ C1 aj , bj : uν1 > 0, ν > 0 is a constant, t ∈ aj , bj , and uaj ubj 0}, for j = 1, 2. Assume that there exists a positive, nondecreasing function ρ ∈ C1 t0 , ∞, R
such that, for some H ∈ Daj , bj and for some θ ≥ 1,
bj θρtrtH ν−1 tA2 t
ν1
H tρtRt −
dt > 0,
4
aj
2.2
for j = 1, 2, then 1.1 is oscillatory, where
n
τt − τ aj
Rt Mpt1 − aτt
qiki t,
a0 1 − at|et|k0
t − τ aj
i1
ρ t
ν 1H t,
At Ht
ρt
a0 n
−ki
i0 ki ,
2.3
and k0 , k1 , . . . , kn are positive constants satisfying a and b of Lemma 2.1.
Proof. Assume to the contrary that there exists a solution xt of 1.1 such that xt >
0, xτt > 0, xσt > 0, when t ≥ t0 > 0, for some t0 depending on the solution xt.
Set
zt xt atxσt.
2.4
By assumption, we have that zt > 0 for t ≥ t0 ≥ 0, and from 2.4 it follows that
n
rtz t et − ptfxτt − qi t|xt|λi sgn xt ≤ 0,
t ≥ t0 ≥ 0.
2.5
i1
It is not difficult to show that z t is eventually positive. In fact, first, we know that z t ≡
/0
for sufficiently large t, since zt is nontrivial. Second, if there exists an t1 ≥ t0 such that
rt1 z t1 C < 0, then rtz t ≤ C for t ≥ t1 ≥ t0 , that is, z t ≤ C/rt, and hence,
t
zt ≤ zt1 t1 C/rtdt → −∞ as t → ∞, which contradicts the fact that zt > 0. Without
loss of generality; say z t > 0, t ≥ t0 ≥ 0. Thus we have that
xt ≥ zt − atzt,
t ≥ t0 ≥ 0.
2.6
International Journal of Differential Equations
5
Define
wt ρt
rtz t
,
zt
2.7
t ≥ t0 .
It follows from 2.7 that wt satisfies the following differential equality:
w t ptfxτt
ρ t
et ρt
w2 t
wt − ρt
−
ρt
−
ρt
zt
ρtrt
zt
n
qi txλi t
.
zt
i1
2.8
Using 2.6, we have that
w t ≤
ptf1 − aτtzτt
ρ t
w2 t
wt − ρt
−
ρt
zt
ρtrt
n
et
− ρt qi t1 − atλi zλi −1 t,
ρt
zt
i1
2.9
and by the condition fx/x ≥ M > 0, we have that
w t ≤
ρ t
w2 t
1 − aτtzτt
wt − Mρtpt
−
ρt
zt
ρtrt
n
et
− ρt qi t1 − atλi zλi −1 t.
ρt
zt
i1
2.10
By assumption, we can choose a1 , b1 ≥ t0 such that b1 ≥ τa1 , τ 2 a1 ττa1 ≥ t0 , qi t ≥
0, i 1, 2, . . . , n, for t ∈ τa1 , b1 , and et ≤ 0 for t ∈ τa1 , b1 . Recall the arithmeticgeometric mean inequality see 25
n
i0
k i ui ≥
n
uki i ,
ui ≥ 0,
2.11
i0
where k0 1 − ni1 ki and ki > 0, i 1, 2, . . . , n are chosen to satisfy a and b of Lemma 2.1
for the given λ1 , λ2 , . . . , λn > 0. Now return to 2.10 and identify u0 k0−1 |et|z−1 t and
ui ki−1 qi tzλi −1 t1 − atλi in 2.11 to obtain
w t ≤
ρ t
1 − aτtzτt
wt − Mρtpt
ρt
zt
n
w2 t
− ρt1 − atk0−k0 |et|k0
−
ki−ki qiki t.
ρtrt
i1
2.12
6
International Journal of Differential Equations
From 1.1, we can easily obtain z t ≤ 0, for t ∈ τa1 , b1 . Therefore, we have that, for
t ∈ τa1 , b1 ,
zt − zτa1 z st − τa1 ≥ z tt − τa1 .
2.13
Noting that zt > 0 for t ≥ τa1 , we get by 2.13 that
zt ≥ z tt − τa1 ,
t ∈ τa1 , b1 ,
2.14
that is,
1
z t
≤
,
zt
t − τa1 2.15
t ∈ τa1 , b1 .
Integrating 2.15 from τt to t > a1 , we obtain
zτt τt − τa1 ≥
,
zt
t − τa1 t ∈ a1 , b1 .
2.16
By using 2.16 in 2.12, we have that, for t ∈ a1 , b1 ,
w t ≤
ρ t
τt − τa1 wt − Mρtpt1 − aτt
ρt
t − τa1 −
n
w2 t
− ρt1 − atk0−k0 |et|k0
ki−ki qiki t
ρtrt
i1
2.17
ρ t
w2 t
wt − ρtRt −
.
ρt
ρtrt
Multiplying both sides of 2.17 by H ν1 t as given in the hypothesis of Theorem 2.4 and
integrating 2.17 from a1 to b1 , we obtain
b1
H ν1 tρtRtdt ≤
a1
b1
H ν1 t
a1
ρ t
wtdt −
ρt
b1
H ν1 tw tdt −
a1
b1
H ν1 t
a1
w2 t
dt.
ρtrt
2.18
Using the integration by parts formula, we have that
b1
a1
H ν1 tw tdt H ν1 twt|ba11 −
−
b1
a1
b1
ν 1H ν tH twtdt
a1
ν 1H tH twtdt,
ν
2.19
International Journal of Differential Equations
7
where Ha1 Hb1 0. Substituting 2.19 into 2.18, we obtain
b1
H ν1 tρtRtdt ≤
b1
a1
H ν1 t
a1
b1
ρ t
wtdt ρt
b1
ν 1H ν tH twtdt
a1
w2 t
dt
ρtrt
a1
b1
b1
w2 t
ν
dt.
AtH twtdt −
H ν1 t
ρtrt
a1
a1
−
H ν1 t
2.20
Then
b1
H
a1
ν1
b1 w2 t
ν
ν1
−AtH twt H t
dt
tρtRtdt ≤ −
ρtrt
a1
⎤2
⎡
b1 ν1
θρtrt
H t
− ⎣
wt −
H ν tAt⎦ dt
ν1 t
θρtrt
4H
a1
⎤2
⎡
b1 b1
θρtrt ν
θ − 1H ν1 t 2
⎦
⎣
w tdt.
dt
−
H
tAt
θρtrt
4H ν1 t
a1
a1
2.21
From the hypothesis of Theorem 2.4 and 2.21, we have that
⎞2 ⎤
⎛
θρtrt
⎢ ν1
⎥
H ν tAt⎠ ⎦dt
⎣H tρtRt − ⎝
4H ν1 t
ai
bi
⎡
bi ai
H ν1 tρtRt −
ν−1
θρtrtH
4
tA t
2
2.22
dt ≤ 0,
which contradicts 2.2. When xt is eventually negative, we can obtain similar contradiction
using the interval τa2 , b2 instead of τa1 , b1 . This completes the proof.
Remark 2.5. Let rt 1, qi t 0, and at 0 for i 1, 2, . . . , n. It is easy to see that
Theorem 2.4 reduces to Theorem 1 of 7.
In Theorem 2.6, we do not impose any restriction on signs of those coefficients
corresponding to sublinear terms of 1.1, that is, ql t for l m 1, . . . , n. If it is nonpositive,
we can easily see that Theorem 2.4 is invalid. However, the following theorem is valid for this
case.
8
International Journal of Differential Equations
Theorem 2.6. Suppose that, for any T ≥ 0, there exist constants a1 , b1 , a2 , b2 such that T ≤ a1 <
b1 ≤ a2 < b2 and
qi t ≥ 0,
t ∈ τa1 , b1 ∪ τa2 , b2 , i 1, . . . , m,
et < 0,
t ∈ τa1 , b1 ,
et > 0,
t ∈ τa2 , b2 .
2.23
Assume that there exists a positive, nondecreasing function ρ ∈ C1 t0 , ∞, R such that, for some
H ∈ Daj , bj and for some θ ≥ 1,
bj θρtrtH ν−1 tA2 t
ν1
H tρtRt −
dt > 0,
4
aj
2.24
for j = 1, 2, then 1.1 is oscillatory, where
ρ t
ν 1H t,
ρt
m
τt − τ aj
1−1/λi 1/λi
Rt ptM1 − aτt
qi t
μi βi |et|
t − τ aj
i1
At Ht
−
n
2.25
l
γl δl |et|1−1/λl q1/λ
t ,
l
lm1
n
1/λi −1
with m
, i 1, . . . , m, γl λl 1 −
i1 βi lm1 δl 1 for βi > 0, δl > 0, μi λi λi − 1
λl 1/λl −1 , and ql t max{−ql t, 0}, l m 1, . . . , n.
Proof. Assume to the contrary that there exists a solution xt of 1.1 such that xt >
0, xτt > 0, when t ≥ t0 > 0, for some t0 depending on the solution xt. When xt
is eventually negative, the proof follows the same argument using the interval τa2 , b2 instead of τa1 , b1 . Apply the assumption of βi and δl , then 1.1 is rearranged as
rtxt atxσt ptfxτt
m n qi t|xt|λi − βi et ql t|xt|λl − δl et 0.
i1
2.26
lm1
Noting the assumption 2.23 and applying Lemma 2.2i to the first summation term in
2.26, we get that
m
1−1/λi 1/λi
rtxt atxσt ptfxτt xt μi βi |et|
qi t
i1
n
lm1
λl
ql t|xt| − δl et ≤ 0.
2.27
International Journal of Differential Equations
9
Introduce the Riccati substitution as 2.4 and apply Lemma 2.2ii to each of the nonlinear
terms in the last sum in 2.27. Here u xt, α λl , D ql t, and
C α1 − α1/α−1 δl |et|1−1/α q1/α
l t.
2.28
We can obtain from 2.27 the following Riccati inequality:
w t ≤
ρ t
τt − τa1 w2 t
wt − ρtptM1 − aτt
−
ρt
t − τa1 ρtrt
m
n
1−1/λi 1/λi
l
qi t ρt
γl δl |et|1−1/λl q1/λ
− ρt μi βi |et|
t
l
i1
2.29
lm1
ρ t
w2 t
wt − ρtRt −
,
ρt
ρtrt
where ql t max{−ql t, 0}. The remaining argument is the same as that in Theorem 2.4 and
this completes the proof.
Remark 2.7. Let fx x, at 0, and τt t, n 1. Theorems 2.4 and 2.6 reduce to
Theorem 1 of 8, 11.
We will use the function class Y to study the oscillatory of 1.1. We say that a function
Φ Φt, s, l belongs to the function class Y, denoted by Φ ∈ Y if Φ ∈ CE, R, where E 0 for
{t, s, l : t0 ≤ l ≤ s ≤ t < ∞}, which satisfies Φt, t, l 0, and Φt, l, l 0, Φt, s, l /
l < s < t, and has the partial derivative ∂Φ/∂s on E such that ∂Φ/∂s is locally integrable with
respect to s in E.
We defined the operator B·; l, t by
B g; l, t t
Φ2 t, s, lgsds,
for t ≥ s ≥ l ≥ t0 , gs ∈ Ct0 , ∞,
2.30
l
and the function ϕ ϕt, s, l is defined by
∂Φt, s, l
ϕt, s, lΦt, s, l.
∂s
2.31
It is easy to verify that B·; l, t is a linear operator and satisfies
B g ; l, t −2B gϕ; l, t ,
for g ∈ C1 t0 , ∞.
2.32
Theorem 2.8. Suppose that, for any T ≥ 0, there exist constants a1 , b1 , a2 , b2 such that T ≤ a1 <
b1 ≤ a2 < b2 and satisfies 2.1. Assume that there exists a function Φ ∈ Y, such that, for each t ≥ t0
and for some θ ≥ 1,
ρ s 2
θρsrs
2ϕ −
; t0 , t > 0,
lim supB ρsRs −
4
ρs
t→∞
2.33
10
International Journal of Differential Equations
for j=1, 2, then 1.1 is oscillatory, where the operator B is defined by 2.30, ϕ ϕt, s, t0 is defined
by 2.31,
n
τs − τ aj
qiki s,
Rs Mps1 − aτs
a0 1 − as|es|k0
s − τ aj
i1
a0 n
−ki
i0 ki ,
2.34
and k0 , k1 , . . . , kn are positive constants satisfying a and b of Lemma 2.1.
Proof. We proceed as in Theorem 2.4. Assume to the contrary that there exists a solution xt
of 1.1 such that xt > 0, xτt > 0, when t t0 > 0, for some t0 depending on the
solution xt. From the proof of Theorem 2.4, we obtain 2.16 for all t ∈ τa1 , b1 . Applying
B·; t0 , t, t ≥ t0 to 2.17, we have that
ρ s
w2 s
ws; t0 , t − B ρsRs; t0 , t − B
; t0 , t .
B w s; t0 , t ≤ −B
ρs
ρsrs
2.35
By 2.32 and above inequality, we have, for t ≥ t0 , that
ρ s
w2 s
ws; t0 , t − B
; t0 , t
B ρsRs; t0 , t ≤ 2B wsϕ; t0 , t − B
ρs
ρsrs
⎡ !
$2
#
ρ s
ws
⎣
− θρsrs ϕ −
B − "
2ρs
θρsrs
⎤
ρ s 2 θ − 1 w2 s
θρsrs
2ϕ −
; t0 , t⎦
−
4
ρs
θ ρsrs
2.36
ρ s 2
θρsrs
2ϕ −
; t0 , t .
≤B
4
ρs
That is,
θρsrs
ρ s 2
B ρsRs −
2ϕ −
; t0 , t ≤ 0.
4
ρs
2.37
Taking the super limit in the above inequality, we have that
θρsrs
ρ s 2
lim supB ρsRs −
; t0 , t ≤ 0,
2ϕ −
4
ρs
t→∞
2.38
which contradicts assumption 2.33. When xt is eventually negative, the proof follows
the same argument using the interval τa2 , b2 instead of τa1 , b1 . This completes the
proof.
International Journal of Differential Equations
11
Theorem 2.9. Suppose that, for any T ≥ 0, there exist constants a1 , b1 , a2 , b2 such that T ≤ a1 <
b1 ≤ a2 < b2 and satisfies 2.23. Assume that there exists a function Φ ∈ Y such that, for each t ≥ t0
and for some θ ≥ 1,
ρ s 2
θρsrs
2ϕ −
; t0 , t > 0,
lim supB ρsRs −
4
ρs
t→∞
2.39
for j = 1, 2, then 1.1 is oscillatory, where the operator B is defined by 2.30, ϕ ϕt, s, t0 is defined
by 2.31, and Rt is defined by 2.25.
The proof of Theorem 2.9 can be completed by following the proofs of Theorems 2.6
and 2.8 with suitable changes, we omit it here.
3. Corollaries
As Theorems 2.4–2.9 are rather general, it is convenient for applications to derive a number
of oscillation criteria with the appropriate choice of the functions H, ρ, and Φt, s, l.
With an appropriate choice of the functions Ht, ρt, one can derive from Theorems
2.4 and 2.6 a number of oscillation criteria for 1.1. For example, we
# consider the simple case
√
n 2; hence ,λ1 > 1 > λ2 > 0. Let bj aj π/ cj , bj aj π/4 dj , j 1, 2, where aj , bj
√
are given in Theorem 2.4. This determines cj , dj , j 1, 2. Choose Ht sin cj t − aj ,
#
ρt sink dj t − aj , and k being natural number. We obtain from Theorems 2.4 and 2.6
the following corollaries.
Corollary 3.1. Let k0 , k1 , k2 be chosen to satisfy a, b of Lemma 2.1 for λ1 > 1 > λ2 > 0. If for
any T ≥ t0 , there exist constants a1 , b1 , a2 , b2 such that T ≤ a1 < b1 ≤ a2 < b2 , then it holds that
q1 t ≥ 0,
q2 t ≥ 0,
t ∈ τa1 , b1 ∪ τa2 , b2 ,
et ≤ 0,
t ∈ τa1 , b1 ,
et ≥ 0,
t ∈ τa2 , b2 ,
and for cj π 2 /bj − aj 2 , 16dj π 2 /bj − aj 2 , and θ ≥ 1, one has that
$
τt − τ aj
−k0 −k1 −k2
k2
k0 k1
ptM1 − aτt
k0 k1 k2 |et| q1 tq2 t1 − at
t − τ aj
√ k # θrtcj sinν−1 cj t − aj
ν1 "
× sin
cj t − aj sin
dj t − aj −
# 4sink dj t − aj
bj !
aj
3.1
12
International Journal of Differential Equations
# # " #
× k dj sin cj t − aj sink−1 dj t − aj cos dj t − aj
" # 2
"
ν 1 cj sink dj t − aj cos cj t − aj
dt > 0,
3.2
then all solutions of the equation
rtxt atxσt ptfxτt
q1 t|xt|λ1 sgn xt q2 t|xt|λ2 sgn xt et
3.3
are oscillatory.
Corollary 3.2. Assume that, for any T ≥ 0, there exist constants a1 , b1 , a2 , b2 such that T ≤ a1 <
b1 ≤ a2 < b2 and
q1 t ≥ 0,
t ∈ τa1 , b1 ∪ τa2 , b2 ,
et < 0,
t ∈ τa1 , b1 ,
et > 0,
t ∈ τa2 , b2 .
3.4
Also assume that, for λ1 > 1 > λ2 > 0, and that there exists some λ ∈ 0, 1 such that
bj !
aj
τt − τ aj
ptM1 − aτt
λ1 λ1 − 11/λ1 −1 λ|et|1−1/λ1 q11/λ1 t1 − at
t − τ aj
$
2
−λ2 1 − λ2 1/λ2 −1 1 − λ|et|1−1/λ2 q1/λ
t1 − at
2
√ " # θrtcj sinν−1 cj t − aj
cj t − aj sink dj t − aj −
# 4sink dj t − aj
# # #
" × k dj sin cj t − aj sink−1 dj t − aj cos dj t − aj
" # 2
"
k
dt > 0,
ν 1 cj sin
dj t − aj cos cj t − aj
× sinν1
3.5
where q2 t max{−q2 t, 0}, then all solutions of 3.3 are oscillatory.
If we choose Φt, s, l ρst − sα s − lβ for α, β > 1/2 and ρs ∈ C1 t0 , 0, ∞,
then we have that
ϕt, s, l ρ s βt − α β s αl
.
ρs
t − ss − l
Thus by Theorems 2.8 and 2.9, we have two new oscillation results.
3.6
International Journal of Differential Equations
13
Corollary 3.3. Suppose that, for any T ≥ 0, there exist constants a1 , b1 , a2 , b2 such that T ≤ a1 <
b1 ≤ a2 < b2 and satisfies 2.1. For each t ≥ t0 and for some θ ≥ 1,
⎡
!
$2 ⎤
t
βt
−
α
β
s
αt
θρsrs
ρ
s
0
2α
2β
⎦ds > 0,
2
lim sup ρ2 tt − s s − t0 ⎣ρsRs −
4
ρs
t − ss − t0 t→∞
t0
3.7
for j = 1, 2, then 1.1 is oscillatory, where
n
τs − τ aj
qiki s,
Rs Mps1 − aτs
a0 1 − as|es|k0
s − τ aj
i1
a0 n
−ki
i0 ki ,
3.8
and k0 , k1 , . . . , kn are positive constants satisfying a and b of Lemma 2.1.
Corollary 3.4. Suppose that, for any T ≥ 0, there exist constants a1 , b1 , a2 , b2 such that T ≤ a1 <
b1 ≤ a2 < b2 and satisfies 2.23. Assume that there exists a function Φ ∈ Y such that, for each t ≥ t0
and for some θ ≥ 1,
⎡
!
$2 ⎤
t
βt
−
α
β
s
αt
ρ
θρsrs
s
0
⎦ds > 0,
2
lim sup ρ2 tt − s2α s − t0 2β⎣ρsRs −
4
ρs
t − ss − t0 t→∞
t0
3.9
for j = 1, 2, then 1.1 is oscillatory, where Rt is defined by 2.25.
We say that a function H Ht, s belongs to the function class X, if H ∈ CD, R ,
where D {t, s : t0 ≤ s ≤ t < ∞}, which satisfies Ht, t 0, Ht, s > 0 for t > s and has
partial derivatives ∂H/∂s and ∂H/∂t on D such that
#
∂H
h1 t, s Ht, s,
∂t
#
∂H
−h2 t, s Ht, s,
∂s
3.10
where h1 t, s, h2 t, s are locally
" integrable with respect to t and s, respectively, in D.
If we choose Φt, s, l H1 s, lH2 t, s, for H1 , H2 ∈ X, then we have that
1
ϕt, s, l 2
1
!
1
2
h s, l
h t, s
− "2
"1
H2 t, s
H1 s, l
$
,
3.11
2
where h1 s, l, h2 t, s are defined as the following:
#
∂H1 s, l
1
h1 s, l H1 s, l,
∂s
#
∂H2 t, s
2
−h2 t, s H2 t, s.
∂s
Thus by Theorems 2.8 and 2.9, we also have the two following oscillation results.
3.12
14
International Journal of Differential Equations
Corollary 3.5. Suppose that, for any T ≥ 0, there exist constants a1 , b1 , a2 , b2 such that T ≤ a1 <
b1 ≤ a2 < b2 and satisfies 2.1. For each t ≥ t0 and for some θ ≥ 1,
⎡
! 1
$2 ⎤
t
2
h2 t, s
θρsrs h1 s, t0 ρ s
⎦ds > 0,
lim sup H1 s, t0 H2 t, s⎣ρsRs−
−"
−
"
4
ρs
H1 s, t0 H2 t, s
t→∞
t0
3.13
1
2
for j = 1, 2, then 1.1 is oscillatory, where h1 s, t0 , h2 t, s are defined by 3.12,
n
τs − τ aj
Rs Mps1 − aτs
qiki s,
a0 1 − as|es|k0
s − τ aj
i1
a0 n
−ki
i0 ki ,
3.14
and k0 , k1 , . . . , kn are positive constants satisfying a and b of Lemma 2.1.
Corollary 3.6. Suppose that, for any T ≥ 0, there exist constants a1 , b1 , a2 , b2 such that T ≤ a1 <
b1 ≤ a2 < b2 and satisfies 2.23. Assume that there exists a function Φ ∈ Y such that, for each t ≥ t0
and for some θ ≥ 1,
⎡
! 1
$2 ⎤
t
2
h2 t, s
θρsrs h1 s, t0 ρ s
⎦ds > 0,
lim sup H1 s, t0 H2 t, s⎣ρsRs−
−
−"
"
4
H2 t, s ρs
H1 s, t0 t→∞
t0
3.15
1
2
for j = 1, 2, then 1.1 is oscillatory, where h1 s, t0 , h2 t, s is defined by 3.12 and Rt are
defined by 2.25.
Acknowledgments
The aouthors thank the referees for their helpful comments to improve this paper. This
research was supported by the NNSF of China Grant no. 10771118 and NSF of Shandong
Grants no. ZR2009AM011 and no. ZR2009AQ010.
References
1 M. S. Keener, “On the solutions of certain linear nonhomogeneous second-order differential
equations,” Applicable Analysis, vol. 1, no. 1, pp. 57–63, 1971.
2 S. M. Rankin III, “Oscillation theorems for second-order nonhomogeneous linear differential
equations,” Journal of Mathematical Analysis and Applications, vol. 53, no. 3, pp. 550–553, 1976.
3 H. Teufel Jr., “Forced second order nonlinear oscillation,” Journal of Mathematical Analysis and
Applications, vol. 40, pp. 148–152, 1972.
4 Y. G. Sun, “A note on Nasr’s and Wong’s papers,” Journal of Mathematical Analysis and Applications,
vol. 286, no. 1, pp. 363–367, 2003.
5 A. H. Nasr, “Necessary and sufficient conditions for the oscillation of forced nonlinear second order
differential equations with delayed argument,” Journal of Mathematical Analysis and Applications, vol.
212, no. 1, pp. 51–59, 1997.
International Journal of Differential Equations
15
6 J. S. W. Wong, “Second order nonlinear forced oscillations,” SIAM Journal on Mathematical Analysis,
vol. 19, no. 3, pp. 667–675, 1988.
7 D. Çakmak and A. Tiryaki, “Oscillation criteria for certain forced second-order nonlinear differential
equations with delayed argument,” Computers & Mathematics with Applications, vol. 49, no. 11-12, pp.
1647–1653, 2005.
8 A. H. Nasr, “Sufficient conditions for the oscillation of forced super-linear second order differential
equations with oscillatory potential,” Proceedings of the American Mathematical Society, vol. 126, no. 1,
pp. 123–125, 1998.
9 A. G. Kartsatos, “On the maintenance of oscillations of nth order equations under the effect of a small
forcing term,” Journal of Differential Equations, vol. 10, pp. 355–363, 1971.
10 Q. Kong, “Interval criteria for oscillation of second-order linear ordinary differential equations,”
Journal of Mathematical Analysis and Applications, vol. 229, no. 1, pp. 258–270, 1999.
11 J. S. W. Wong, “Oscillation criteria for a forced second-order linear differential equation,” Journal of
Mathematical Analysis and Applications, vol. 231, no. 1, pp. 235–240, 1999.
12 D. Çakmak and A. Tiryaki, “Oscillation criteria for certain forced second-order nonlinear differential
equations,” Applied Mathematics Letters, vol. 17, no. 3, pp. 275–279, 2004.
13 Y. G. Sun and J. S. W. Wong, “Oscillation criteria for second order forced ordinary differential
equations with mixed nonlinearities,” Journal of Mathematical Analysis and Applications, vol. 334, no.
1, pp. 549–560, 2007.
14 Y. G. Sun and F. Meng, “Interval criteria for oscillation of second-order differential equations with
mixed nonlinearities,” Applied Mathematics and Computation, vol. 198, no. 1, pp. 375–381, 2008.
15 C. H. Ou and J. S. W. Wong, “Forced oscillation of nth-order functional differential equations,” Journal
of Mathematical Analysis and Applications, vol. 262, no. 2, pp. 722–732, 2001.
16 Y. G. Sun and R. P. Agarwal, “Forced oscillation of nth-order nonlinear differential equations,”
Functional Differential Equations, vol. 11, no. 3-4, pp. 587–596, 2004.
17 Y. G. Sun and R. P. Agarwal, “Interval oscillation criteria for higher-order forced nonlinear differential
equations,” Nonlinear Functional Analysis and Applications, vol. 9, no. 3, pp. 441–449, 2004.
18 Q. Yang, “Interval oscillation criteria for a forced second order nonlinear ordinary differential
equations with oscillatory potential,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 49–64,
2003.
19 X. Yang, “Forced oscillation of nth-order nonlinear differential equations,” Applied Mathematics and
Computation, vol. 134, no. 2-3, pp. 301–305, 2003.
20 J. S. W. Wong, “On Kamenev-type oscillation theorems for second-order differential equations with
damping,” Journal of Mathematical Analysis and Applications, vol. 258, no. 1, pp. 244–257, 2001.
21 A. Tiryaki, D. Çakmak, and B. Ayanlar, “On the oscillation of certain second-order nonlinear
differential equations,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 565–574,
2003.
22 Y. V. Rogovchenko and F. Tuncay, “Oscillation criteria for second-order nonlinear differential
equations with damping,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 1, pp. 208–
221, 2008.
23 R. P. Agarwal, S. R. Grace, and D. O’Regan, Oscillation Theory for Second Order Linear, HalfLinear, Superlinear and Sublinear Dynamic Equations, Kluwer Academic Publishers, Dordrecht, The
Netherlands, 2002.
24 R. Xu and F. Meng, “New Kamenev-type oscillation criteria for second order neutral nonlinear
differential equations,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1364–1370, 2007.
25 E. F. Beckenbach and R. Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete,
Springer, Berlin, Germany, 1961.