Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 618058, 13 pages
doi:10.1155/2012/618058
Research Article
Existence for Eventually Positive Solutions of
High-Order Nonlinear Neutral
Differential Equations with Distributed Delay
Huanhuan Zhao,1 Youjun Liu,1 and Jurang Yan2
1
College of Mathematics and Computer Sciences, Shanxi Datong University, Datong,
Shanxi 037009, China
2
School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China
Correspondence should be addressed to Huanhuan Zhao, zhh9791@126.com
Received 14 February 2012; Accepted 12 March 2012
Academic Editor: Mingshu Peng
Copyright q 2012 Huanhuan Zhao 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 consider the existence for eventually positive solutions of high-order nonlinear neutral
differential equations with distributed delay. We use Lebesgue’s dominated convergence theorem
to obtain new necessary and sufficient condition for the existence of eventually positive solutions.
1. Introduction and Preliminary
In this paper, we consider the high-order nonlinear neutral differential equation:
rtxt −
b
n
pt, τxt − τdτ
a
d
qt, σfxt − σdσ 0,
t ≥ t0 ,
1.1
qt, σfxt − σdσ ≤ 0,
t ≥ t0 ,
1.2
c
and the associated inequality:
rtxt −
b
n
pt, τxt − τdτ
a
d
c
1 where n is a positive integer, 0 < a < b, 0 < c < d,
2 p ∈ Ct0 , ∞ × a, b, R ,
2
Discrete Dynamics in Nature and Society
3 r ∈ Ct0 , ∞, R , rt > 0, q ∈ Ct0 , ∞ × c, d, R ,
4 fu are continuously nondecreasing real function with respect to u defined on R
such that ufu > 0, for u > 0, for n is positive odd integer; fu are continuously
decreasing real function with respect to u defined on R such that ufu > 0, for
u > 0, for n is positive even integer.
Recently, there has been a lot of activities concerning the existence of eventually
positive solutions for nonlinear neutral differential equations. See 1–8. In 1, Liu et al. have
studied the even-order neutral differential equation:
atxt −
m
n
−
bi txt − τi i1
l
pj tfj t, x t − σj 0,
t ≥ 0,
1.3
j1
and the associated differential inequality:
n
m
l
− pj tfj t, x t − σj ≥ 0,
atxt − bi txt − τi i1
t ≥ 0.
1.4
j1
They have obtained that the existences of eventually positive solutions of 1.3 and 1.4 are
equivalent. In 2, Ouyang et al. has studied the odd-order neutral differential equation:
atxt −
m
n
bi txt − τi i1
l
pj tfj x t − σj 0,
t ≥ 0,
1.5
pj tfj x t − σj ≤ 0,
t ≥ 0.
1.6
j1
and the associated differential inequality:
atxt −
m
n
bi txt − τi i1
l
j1
He has obtained that the existences of eventually positive solutions of 1.5 and 1.6 are
equivalent.
As usual, a solution of 1.1 is a continuous function xt defined on −μ, ∞ such that
b
yt : rtxt − a pt, τxt − τdτ is n times differentiable and 1.1 holds for all n ≥ 0. Such
a solution xt is called an eventually positive solution if there is T ≥ t0 , such that xt > 0,
for t ≥ T . Here, μ max{b, d}.
Lemma 1.1. Assume supt≥t0 rt < ∞, and
positive solution of inequality 1.2, and set
b
a
pt, τdτ/rt ≤ M, let xt is an eventually bounded
yt rtxt −
b
a
pt, τxt − τdτ.
1.7
Discrete Dynamics in Nature and Society
3
If n is a positive odd integer, then eventually,
yn t ≤ 0,
−1k1 yk t < 0,
lim yk t 0,
t→∞
k 0, 1, . . . , n − 1,
k 1, 2, . . . , n − 1.
1.8
If n is a positive even integer, then eventually,
yn t ≤ 0,
−1k yk t < 0,
lim yk t 0,
t→∞
k 0, 1, . . . , n − 1,
k 1, 2, . . . , n − 1.
1.9
Proof. We have the following cases.
Case 1 If n is a positive odd integer. Because xt is bounded, and supt≥t0 rt <
b
∞, a pt, τdτ/rt ≤ M, thus yt is bounded. From 1.2, we have yn t ≤
d
− c qt, σfxt − σdσ ≤ 0. Assume yn−1 t ≤ 0, since yn t ≤ 0, yn−1 t decreases, set
limt → ∞ yn−1 t L, then −∞ ≤ L < 0.
Thus,
yn−2 t − yn−2 t0 t
yn−1 tdt ≤ L εt − t0 −→ −∞,
t −→ ∞,
1.10
t0
that is, limt → ∞ yn−2 t −∞. Simile, limt → ∞ yk t −∞, k 0, 1, . . . , n − 2, this is
a contradiction and yt is bounded, therefore, yn−1 t > 0.
Again, since it decreases, set limt → ∞ yn−1 t L, thus L ≥ 0. Next, we proof L 0. Assume
L > 0, then,
y
n−2
t − y
n−2
t0 t
yn−1 tdt ≥ L − εt − t0 −→ ∞,
t −→ ∞,
1.11
t0
then, limt → ∞ yn−2 t ∞. Simile, limt → ∞ yk t ∞, k 0, 1, . . . , n − 2, this is a
contradiction and yt is bounded, therefore, L 0. That is, limt → ∞ yn−1 t 0. Similarly,
we obtain
−1k1 yk t < 0,
k 0, 1, . . . , n − 1,
lim yk t 0,
t→∞
k 1, 2, . . . , n − 1.
1.12
Case 2 If n is a positive even integer. The proof of Case 2 is similar to that of part Case 1,
therefore, it is omitted. We obtain
−1k yk t < 0,
k 0, 1, . . . , n − 1,
The proof is complete.
lim yk t 0,
t→∞
k 1, 2, . . . , n − 1.
1.13
4
Discrete Dynamics in Nature and Society
Lemma 1.2 see 3, page 21. Let η ∈ −∞, 0, τ ∈ 0, ∞, t0 ∈ R, and suppose that a function
xt ∈ Ct0 − τ, ∞, R satisfies the inequality:
xt ≤ η max xs,
t−τ≤s≤t
t ≥ t0 .
1.14
Then, xt cannot be a nonnegative function.
Lemma 1.3. Suppose that supt≥t0 rt < ∞, and
positive solution of 1.2 and set
yt rtxt −
b
a
b
pt, τdτ/at ≤ 1. Let xt be an eventually
pt, τxt − τdτ,
1.15
a
then eventually
yt > 0.
1.16
Proof. From 1.2 and 1.7, eventually, we have
y
n
t ≤ −
d
qt, σfxt − σdσ ≤ 0,
1.17
c
and the hypotheses on qt, σ, fx, and xt yield that yn is not eventually zero. Thus,
yi t i 0, l, . . . , n − 1 is eventually nonzero. Hence, if yt > 0 does not hold, then
eventually yt < 0, y t < 0, or yt < 0, y t > 0.
Case 1. yt < 0, y t < 0, then there exists t1 > 0 such that yt ≤ yt1 < 0 for t > t1 . Then,
b
α : −yt1 /supt≥t0 rt > 0. In view of 1.3 and a pt, τdτ/at ≤ 1, we obtain
b
yt
1
pt, τxt − τdτ
rt rt a
b
1
pt, τdτ max xs
≤ −α rt a
t−b≤s≤t
xt 1.18
≤ −α max xs.
t−b≤s≤t
According to Lemma 1.2, we obtain xt < 0. This is a contradiction and so yt is eventually
positive.
Case 2. yt < 0, y t > 0, one argues that y t < 0 and repeats the argument to obtain that
yn t > 0 which contradicts the offset after yn t < 0.
The proof is complete.
Discrete Dynamics in Nature and Society
5
2. Comparison Theory of Existence for Eventually Positive Solution
Theorem 2.1. Assume all conditions of Lemma 1.1 hold, n is a positive odd integer. And
H1 pt, τ qt, σ > 0, for sufficiently large t.
Then 1.1 has an eventually bounded positive solution if and only if inequality 1.2 has an eventually
bounded positive solution.
Proof. It is clear that an eventually bounded positive solution of 1.1 is also an eventually
bounded positive solution of 1.2. So, it suffices to prove that if 1.2 has an eventually
bounded positive solution xt, for t > t0 , then so does 1.1. Set
yt rtxt −
b
2.1
pt, τxt − τdτ.
a
It follows from Lemma 1.1 and 1.2 that eventually,
yn t ≤ 0,
−1k1 yk t > 0,
lim yk t 0,
k 0, 1, . . . , n − 1,
2.2
k 1, 2, . . . , n − 1.
t→∞
By using 1.8 and integrating 1.2 from t to ∞, we obtain
y
n−1
∞ − y
n−1
t ≤ −
∞ d
t
yn−1 t ≥
qt, σfxt − σdσ ds,
c
∞ d
t
2.3
qs, σfxs − σdσ ds.
c
By repeating the same procedure n times and by using 1.8, we are led to the inequality:
yt ≥
∞
∞
dtn
t
∞
dtn−1
tn
dtn−2 · · ·
∞ d
tn−1
t2
qs, σfxs − σdσ ds,
t ≥ t0 .
2.4
c
Using Tonelli s theorem, we reverse the order of integration and obtain
∞
t
s − tn−1
n − 1!
d
qs, σfxs − σdσ ds,
t ≥ t0 .
2.5
c
That is,
1
xt ≥
rt
b
1
pt, τxt − τdτ rt
a
∞
t
s − tn−1
n − 1!
d
qs, σfxs − σdσ ds,
t ≥ t0 .
c
2.6
6
Discrete Dynamics in Nature and Society
Let T ≥ t0 be such that 2.6 hold, and xt−μ > 0, t ≥ T . Now, we consider the set of functions
Ω z ∈ C T − μ, ∞ , R : 0 ≤ zt ≤ 1, t ≥ T − μ
2.7
and define an operator S on Ω as follows:
⎧
t−T μ
t−T μ
⎪
⎪
1
−
,
T − μ ≤ t < T,
SzT
⎪
⎪
⎪
μ
μ
⎪
⎪
⎪
⎪
⎪
⎨ 1
1 b
Szt xt rt pt, τzt − τxt − τdτ
a
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1 ∞ s − tn−1 d
⎪
⎪
⎪
qs, σfzs − σxs − σdσ ds , t ≥ T.
⎩
rt t n − 1!
c
2.8
Then, it follows from Lebesgue s dominated convergence theorem that S is continuous. By
using 2.6, it is easy to see that S maps Ω into itself, and for any z ∈ Ω, we have Szt > 0,
for T − μ ≤ t < T . Next, we define the sequence zk t ∈ Ω
z0 t ≡ 1,
zk1 t Szk t,
t ≥ T − μ,
2.9
t ≥ T − μ, k 0, 1, . . . .
Then, by using 2.6 and a simple induction, we can easily see that
0 ≤ zk1 t ≤ zk t ≤ 1,
t ≥ T − μ.
2.10
Set limk → ∞ zk t zt, t ≥ T − μ then zt satisfies
⎧
t−T μ
t−T μ
⎪
⎪
,
1
−
SzT
⎪
⎪
⎪
μ
μ
⎪
⎪
⎪
⎨ 1
1 b
zt xt rt pt, τzt − τxt − τdτ
⎪
a
⎪
⎪
∞
⎪
⎪
1
s − tn−1 d
⎪
⎪
⎪
qs, σfzs − σxs − σdσ ds ,
⎩
rt t n − 1!
c
T − μ ≤ t < T,
t ≥ T.
2.11
Again, set ωt ztxt, then ωt satisfies ωt > 0, T − μ ≤ t ≤ T , and
1
ωt rt
b
1
pt, τωt − τdτ rt
a
∞
t
s − tn−1
n − 1!
d
qs, σfωs − σdσ ds,
t ≥ T.
c
2.12
Thus, ωt is a positive solution of 1.1 for t ≥ T .
Discrete Dynamics in Nature and Society
7
The following:
t ≥ T − μ.
ωt > 0,
2.13
Assume that there exists t∗ ≥ T − μ, such that ωt > 0, for T − μ ≤ t ≤ t∗ , and ωt∗ 0. Then,
1
0 ωt rt∗ ∗
b
pt∗ , τωt∗ − τdτ
a
1
rt∗ ∞
t∗
s − t∗ n−1
n − 1!
d
2.14
qs, σfωs − σdσ ds,
t∗ ≥ T,
c
which implies
pt∗ , τ 0,
qs, σfωs − σ ≡ 0,
2.15
which contradicts H1 . Thus, ωt is an eventually bounded positive solution of 1.1.
The proof is complete.
Theorem 2.2. Assume all conditions of Lemma 1.3 hold, n is a positive odd integer. And
H1 pt, τ qt, σ > 0, for sufficiently large t.
Then, 1.1 has an eventually positive solution if and only if inequality 1.2 has an eventually positive
solution.
Proof. It is clear that an eventually positive solution of 1.1 is also an eventually positive
solution of 1.2. So, it suffices to prove that if 1.2 has an eventually positive solution xt,
for t > t0 , then so does 1.1. Set
yt rtxt −
b
pt, τxt − τdτ.
2.16
a
It follows from Lemma 1.3 and 1.2 that eventually, yn t ≤ 0, yt > 0, which
implies that there exists a nonnegative even integer n∗ ≤ n − 1, such that eventually
yk t > 0,
−1k yk t > 0,
k 0, 1, . . . , n∗ ,
k n∗ , . . . , n − 1.
2.17
We consider the following possible cases.
Case 1 n∗ 0. Since n is a positive integer, n − 1 is an even integer, we can easily see that
there exists a T > 0, such yn−1 t > 0, and yn t ≤ 0, t ≥ T , yn−1 ∞ ≥ 0.
8
Discrete Dynamics in Nature and Society
By using 2.17 and integrating 1.2 from t to ∞, we obtain
y
n−1
∞ − y
n−1
t ≤ −
∞ d
t
y
n−1
t
qs, σfxs − σdσ ds,
c
∞ d
t ≥
2.18
qs, σfxs − σdσ ds.
c
By repeating the same procedure n times and by using 2.31, we are led to the inequality:
yt ≥
∞
∞
dtn
t
∞
dtn−1
tn
dtn−2 · · ·
∞ d
tn−1
t2
qs, σfxs − σdσ ds,
t ≥ t0 .
2.19
c
Using Tonelli s theorem, we reverse the order of integration and obtain
∞
t
s − tn−1
n − 1!
d
qs, σfxs − σdσ ds,
t ≥ t0 .
2.20
c
That is,
1
xt ≥
rt
b
1
pt, τxt − τdτ rt
a
∞
t
s − tn−1
n − 1!
d
qt, σfxt − σdσ ds,
t ≥ t0 .
c
2.21
Let T ≥ t0 be such that 2.21 hold, and xt − μ > 0, t ≥ T . Now, we consider the set of
functions
Ω z ∈ C T − μ, ∞ , R : 0 ≤ zt ≤ 1, t ≥ T − μ ,
2.22
and define an operator S on Ω as follows:
⎧
t−T μ
t−T μ
⎪
⎪
1
−
,
T − μ ≤ t < T,
SzT
⎪
⎪
μ
μ
⎪
⎪
⎪
⎪
⎪
⎨
1 b
1
pt, τzt − τxt − τdτ
Szt ⎪ xt rt a
⎪
⎪
⎪
⎪
⎪
⎪
1 ∞ s − tn−1 d
⎪
⎪
qs, σfzs − σxs − σdσ ds , t ≥ T.
⎩
rt t n − 1!
c
2.23
Discrete Dynamics in Nature and Society
9
Then, it follows from Lebesgue s dominated convergence theorem that S is continuous. By
using 2.21, it is easy to see that S maps Ω into itself, and for any z ∈ Ω, we have Szt > 0,
for T − μ ≤ t < T . Next, we define the sequence zk t ∈ Ω
z0 t ≡ 1,
zk1 t Szk t,
t ≥ T − μ,
2.24
t ≥ T − μ, k 0, 1, . . . .
Then, by using 2.21 and a simple induction, we can easily see that
0 ≤ zk1 t ≤ zk t ≤ 1,
t ≥ T − μ.
2.25
Set limk → ∞ zk t zt, t ≥ T − μ, then zt satisfies:
⎧
t−T μ
t−T μ
⎪
⎪
1
−
,
SzT
⎪
⎪
⎪
μ
μ
⎪
⎪
⎪
⎨ 1
1 b
zt xt rt pt, τzt − τxt − τdτ
⎪
a
⎪
⎪
∞
⎪
⎪
1
s − tn−1 d
⎪
⎪
⎪
qs, σfzs − σxs − σdσ ds ,
⎩
rt t n − 1!
c
T − μ ≤ t < T,
t ≥ T.
2.26
Again, set ωt ztxt, then ωt satisfies ωt > 0, T − μ ≤ t ≤ T , and
1
ωt rt
b
1
pt, τωt − τdτ rt
a
∞
t
s − tn−1
n − 1!
d
qs, σfωs − σdσ ds,
t ≥ T.
c
2.27
Thus, ωt is a positive solution of 1.1 for t ≥ T .
Consider the following:
t ≥ T − μ.
ωt > 0,
2.28
Assume that there exists t∗ ≥ T − μ, such that ωt > 0, for T − μ ≤ t ≤ t∗ , and ωt∗ 0. Then,
0 ωt∗ 1
rt∗ b
pt∗ , τωt∗ − τdτ
a
1
rt∗ ∞
t∗
s − t∗ n−1
n − 1!
d
c
2.29
qs, σfωs − σdσ ds,
∗
t ≥ T,
10
Discrete Dynamics in Nature and Society
which implies
pt∗ , τ 0,
2.30
qs, σfωs − σ ≡ 0,
which contradicts H1 . Thus, ωt is an eventually positive solution of equation.
Case 2 2 ≤ n∗ ≤ n − 1. By using 2.17 and integrating 1.2 from t to ∞, we obtain
y
n∗ t ≥
∞
t
s − tn−1
n − 1!
d
qs, σfxs − σdσ ds,
t ≥ t0 .
2.31
c
Let T ≥ 0 be such that 2.17 and H1 hold. Integrating 2.31 from T to t and using 2.17,
we have
xt ≥
1
rt
b
pt, τxt − τdτ
a
1
rt
t
T
∗
t − sn −1
n∗ − 1!
∞
t
∗
u − sn−n −1
n − n∗ − 1!
d
2.32
qs, σfxs − σdσ du ds,
t ≥ t0 .
c
Using a method similar to the proof of Case 1 yields that 1.1 also has an eventually positive
solution.
The proof is complete.
Theorem 2.3. Assume all conditions of Lemma 1.1 hold, n is a positive even integer, and
H1 pt, τ qt, σ > 0, for sufficiently large t.
Then, 1.1 has an eventually positive solution if and only if inequality 1.2 has an eventually positive
solution.
Proof. It is clear that an eventually bounded positive solution of 1.1 is also an eventually
bounded positive solution of 1.2. So, it suffices to prove that if 1.2 has an eventually
bounded positive solution xt, for t > t0 , then so does 1.1. Set
yt rtxt −
b
pt, τxt − τdτ.
2.33
a
It follows from Lemma 1.1 and 1.2 that eventually,
yn t ≤ 0, −1k yk t < 0,
lim yk t 0,
t→∞
k 0, 1, . . . , n − 1,
k 1, 2, . . . , n − 1.
2.34
Discrete Dynamics in Nature and Society
11
By using 1.9 and integrating 1.2 from t to ∞, we obtain
y
n−1
∞ − y
n−1
t ≤ −
∞ d
t
yn−1 t ≥
∞ d
t
qs, σfxs − σdσ ds,
c
2.35
qs, σfxs − σdσ ds.
c
By repeating the same procedure n times and by using 1.9, we are led to the inequality:
yt ≤ −
∞
∞
dtn
t
∞
dtn−1
tn
dtn−2 · · ·
∞ d
tn−1
t2
qs, σfxs − σdσ ds,
t ≥ t0 .
2.36
c
Using Tonelli s theorem, we reverse the order of integration and obtain
−
∞
t
s − tn−1
n − 1!
d
qs, σfxs − σdσ ds,
t ≥ t0 .
2.37
c
That is,
1
xt ≤
rt
b
a
pt, τωt − τdτ −
∞
t
s − tn−1
n − 1!
d
qs, σfxs − σdσ ds,
t ≥ t0 .
c
2.38
Let T ≥ 0 be such that 2.38 hold, and xt − μ > 0, t ≥ T . Now, we consider the set of
functions:
Ω z ∈ C T − μ, ∞ , R : 0 < zt ≤ 1, t ≥ T − μ ,
2.39
and define an operator S on Ω as follows:
Szt ⎧t − T μ
t−T μ
⎪
⎪
1
−
,
SzT
⎪
⎪
μ
μ
⎪
⎪
⎪
⎪
⎪
⎪
T − μ ≤ t < T,
⎪
⎪
⎨
xt
⎪
⎪
⎪
,
⎪
⎪
⎪
xt − τ
1 ∞ s − tn−1 d
1 b
xs − σ
⎪
⎪
dτ −
dσ ds
pt, τ
qs, σf
⎪
⎪
rt a
zt − τ
rt t n − 1! c
zs − σ
⎪
⎪
⎩
t ≥ T.
2.40
12
Discrete Dynamics in Nature and Society
Then, it follows from Lebesgue s dominated convergence theorem that S is continuous. By
using 2.38, it is easy to see that S maps Ω into itself and, for any z ∈ Ω, we have Szt < 0,
for T − μ ≤ t < T . Next, we define the sequence zk t ∈ Ω:
z0 t ≡ 1,
zk1 t Szk t,
t ≥ T − μ,
2.41
t ≥ T − μ, k 0, 1, . . . .
Then, by using 2.38 and a simple induction, we can easily see that
0 < zk1 t ≤ zk t ≤ 1,
t ≥ T − μ.
2.42
Set limk → ∞ zk t zt, t ≥ T − μ, then zt satisfies
⎧
t−T μ
t−T μ
⎪
⎪
,
1
−
SzT
⎪
⎪
⎪
μ
μ
⎪
⎪
⎪
⎪
T − μ ≤ t < T,
⎪
⎪
⎨
xt
zt ⎪
,
⎪
n−1 ⎪
⎪
xs − σ
d
⎪ 1 b pt, τ xt − τ dτ − 1 ∞ s − t
⎪
dσ ds
qs, σf
⎪
⎪
⎪ rt a
zt − τ
rt t n − 1! c
zs − σ
⎪
⎪
⎩
t ≥ T.
2.43
Again, set ωt xt/zt, then ωt satisfies ωt > 0, T − μ ≤ t ≤ T , and
ωt 1
rt
b
pt, τωt − τdτ
a
1
−
rt
∞
t
s − tn−1
n − 1!
d
2.44
qs, σfωs − σdσ ds,
t ≥ T.
c
Thus, ωt is a positive solution of 1.1 for t ≥ T .
Consider the following
t ≥ T − μ.
ωt > 0,
2.45
Assume that there exists t∗ ≥ T − μ, such that ωt > 0, for T − μ ≤ t ≤ t∗ , and ωt∗ 0. Then,
0 ωt∗ 1
rt∗ b
pt∗ , τωt∗ − τdτ
a
1
−
rt∗ ∞
t∗
s − t∗ n−1
n − 1!
d
c
2.46
qs, σfωs − σdσ ds,
∗
t ≥ T,
Discrete Dynamics in Nature and Society
13
which implies
pt∗ , τ 0,
2.47
qs, σfωs − σ ≡ 0,
which contradicts H1 . Thus, ωt is an eventually positive solution of 1.1.
The proof is complete.
Example 2.4. Consider high-order neutral differential equation with distributed delay
xt −
2
n
−t
e xt − τdτ
1
1
−1
2e − 3e−2
2
σxt − σdσ 0.
2.48
1
Here, rt ≡ 1, pt − τ e−t , qt, σ σ/2e−1 − 3e−2 , a c 1, b d 2. It is easy to see
that
2
rt ≡ 1 < ∞,
1
e−t dτ
e−t < 1,
1
t > 0.
2.49
From Theorem 2.2, we have that 1.1 has an eventually positive solution if and only if
inequality 1.2 has an eventually positive solution. In fact, xt et is a positive solution
of 1.1.
Acknowledgments
This research is supported by the Natural Sciences Foundation of Shanxi Province and
Scientific Research Project of Shanxi Datong University. The authors are greatly indebted to
the referees for their valuable suggestions and comments.
References
1 Y. Liu, H. Zhao, and J. Yan, “Eventually positive and bounded solutions of even-order nonlinear
neutral differential equations,” Applied Mathematics Letters, vol. 21, no. 11, pp. 1118–1123, 2008.
2 Z. Ouyang, Y. Li, and M. Qing, “Eventually positive solutions of odd-order neutral differential
equations,” Applied Mathematics Letters, vol. 17, no. 2, pp. 159–166, 2004.
3 I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations With Applications, Clarendon
Presss, Oxford, UK, 1991.
4 T. Candan and R. S. Dahiya, “On the oscillation of certain mixed neutral equations,” Applied
Mathematics Letters, vol. 21, no. 3, pp. 222–226, 2008.
5 T. Candan and R. S. Dahiya, “Existence of nonoscillatory solutions of first and second order neutral
differential equations with distributed deviating arguments,” Journal of the Franklin Institute, vol. 347,
no. 7, pp. 1309–1316, 2010.
6 J. G. Dong, “Oscillation of solutions for first order neutral differential equations with distributed
deviating arguments,” Computers and Mathematics with Applications, vol. 58, no. 4, pp. 784–790, 2009.
7 X. Lin, W. Liu, and Y. Yu, “Oscillation criteria for even-order half-linear distributed delay differential
equations with damping,” Computers and Mathematics with Applications, vol. 60, no. 8, pp. 2206–2211,
2010.
8 L. Berezansky and E. Braverman, “Oscillation of equations with an infinite distributed delay,”
Computers and Mathematics with Applications, vol. 60, no. 9, pp. 2583–2593, 2010.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014