Abstract
Some sufficient conditions are established for the oscillation of second-order neutral differential equation 
, 
, where 
. The results complement and improve those of Grammatikopoulos et al. Ladas, A. Meimaridou, Oscillation of second-order neutral delay differential equations, Rat. Mat. 1 (1985), Grace and Lalli (1987), Ruan (1993), H. J. Li (1996), H. J. Li (1997), Xu and Xia (2008).
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1. Introduction
In recent years, the oscillatory behavior of differential equations has been the subject of intensive study; we refer to the articles [1–13]; Especially, the study of the oscillation of neutral delay differential equations is of great interest in the last three decades; see for example [14–38] and references cited therein. Second-order neutral delay differential equations have applications in problems dealing with vibrating masses attached to an elastic bar and in some variational problems (see [39]).
This paper is concerned with the oscillatory behavior of the second-order neutral delay differential equation
where 
Throughout this paper, we assume that
-
(a)
, 
and 
is not identically zero on any ray of the form 
for any 
where 
is a constant; -
(b)
for 
is a constant; -
(c)
, 
, 
, 
, 
, 
where 
is a constant.
, 
and 
is not identically zero on any ray of the form 
for any 
where 
is a constant;
for 
is a constant;
, 
, 
, 
, 
, 
where 
is a constant.In the study of oscillation of differential equations, there are two techniques which are used to reduce the higher-order equations to the first-order Riccati equation (or inequality). One of them is the Riccati transformation technique. The other one is called the generalized Riccati technique. This technique can introduce some new sufficient conditions for oscillation and can be applied to different equations which cannot be covered by the results established by the Riccati technique.
Philos [7] examined the oscillation of the second-order linear ordinary differential equation
and used the class of functions as follows. Suppose there exist continuous functions 
such that 
, 
, 
and 
has a continuous and nonpositive partial derivative on 
with respect to the second variable. Moreover, let 
be a continuous function with
The author obtained that if
then every solution of (1.2) oscillates. Li [4] studied the equation
used the generalized Riccati substitution, and established some new sufficient conditions for oscillation. Li utilized the class of functions as in [7] and proved that if there exists a positive function 
such that
where 
and 
then every solution of (1.5) oscillates. Yan [13] used Riccati technique to obtain necessary and sufficient conditions for nonoscillation of (1.5). Applying the results given in [4, 13], every solution of the equation
is oscillatory.
An important tool in the study of oscillation is the integral averaging technique. Just as we can see, most oscillation results in [1, 3, 5, 7, 11, 12] involved the function class 
Say a function 
belongs to a function class 
denoted by 
if 
where 
and 
which satisfies
and has partial derivatives 
and 
on 
such that
In [10], Sun defined another type of function class 
and considered the oscillation of the second-order nonlinear damped differential equation
Say a function 
is said to belong to 
denoted by 
if 
where 
which satisfies 
for 
and has the partial derivative 
on 
such that 
is locally integrable with respect to 
in 
In [8], by employing a class of function 
and a generalized Riccati transformation technique, Rogovchenko and Tuncay studied the oscillation of (1.10). Let 
Say a continuous function 
belongs to the class 
if:
-
(i)
and 
for 
-
(ii)
has a continuous and nonpositive partial derivative 
satisfying, for some 
, 
where 
is nonnegative.
and 
for 
has a continuous and nonpositive partial derivative 
satisfying, for some 
, 
where 
is nonnegative.Meng and Xu [22] considered the even-order neutral differential equations with deviating arguments
where 
, 
The authors introduced a class of functions 
Let 
and 
The function 
is said to belong to the class 
(defined by 
for short) if
, 
, 
for 
has a continuous and nonpositive partial derivative on 
with respect to the second variable;
there exists a nondecreasing function 
such that
Xu and Meng [31] studied the oscillation of the second-order neutral delay differential equation
where 
by using the function class 
an operator 
and a Riccati transformation of the form
the authors established some oscillation criteria for (1.13). In [31], the operator 
is defined by
for 
and 
The function 
is defined by
It is easy to verify that 
is a linear operator and that it satisfies
In 2009, by using the function class 
and defining a new operator 
, Liu and Bai [21] considered the oscillation of the second-order neutral delay differential equation
where 
The authors defined the operator 
by
for 
and 
The function 
is defined by
It is easy to see that 
is a linear operator and that it satisfies
Wang [11] established some results for the oscillation of the second-order differential equation
by using the function class 
and a generalized Riccati transformation of the form
Long and Wang [6] considered (1.22); by using the function class 
and the operator 
which is defined in [31], the authors established some oscillation results for (1.22).
In 1985, Grammatikopoulos et al. [16] obtained that if 
and 
then equation
is oscillatory. Li [18] studied (1.1) when 
and established some oscillation criteria for (1.1). In [15, 19, 25], the authors established some general oscillation criteria for second-order neutral delay differential equation
where 
In 2002, Tanaka [27] studied the even-order neutral delay differential equation
where 
or 
The author established some comparison theorems for the oscillation of (1.26). Xu and Xia [28] investigated the second-order neutral differential equation
and obtained that if 
for 
and 
then (1.27) is oscillatory. We note that the result given in [28] fails to apply the cases 
or 
for 
To the best of our knowledge nothing is known regarding the qualitative behavior of (1.1) when 
Motivated by [10, 21], for the sake of convenience, we give the following definitions.
Definition 1.1.
Assume that 
. The operator is defined by 
by
for 
and 
Definition 1.2.
The function 
is defined by
It is easy to verify that 
is a linear operator and that it satisfies
In this paper, we obtain some new oscillation criteria for (1.1). The paper is organized as follows. In the next section, we will use the generalized Riccati transformation technique to give some sufficient conditions for the oscillation of (1.1), and we will give two examples to illustrate the main results. The key idea in the proofs makes use of the idea used in [23]. The method used in this paper is different from that of [27].
2. Main Results
In this section, we give some new oscillation criteria for (1.1). We start with the following oscillation result.
Theorem 2.1.
Assume that 
for 
Further, suppose that there exists a function 
such that for some 
and some 
one has
where 
Then every solution of (1.1) is oscillatory.
Proof.
Let 
be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists 
such that 
, 
, 
, for all 
Define 
for 
then 
for 
From (1.1), we have
It is obvious that 
and 
for 
imply 
for 
Using (2.2) and condition 
there exists 
such that for 
we get
We introduce a generalized Riccati transformation
Differentiating (2.4) from (2.2), we have 
Thus, there exists 
such that for all 
,
Similarly, we introduce another generalized Riccati transformation
Differentiating (2.6), note that 
by (2.2), we have 
then for all sufficiently large 
one has
From (2.5) and (2.7), we have
By (2.3) and the above inequality, we obtain
Multiplying (2.9) by 
and integrating from 
to 
we have, for any 
and for all 
From the above inequality and using monotonicity of 
for all 
we obtain
and, for all 
By (2.12),
which contradicts (2.1). This completes the proof.
Remark 2.2.
We note that it suffices to satisfy (2.1) in Theorem 2.1 for any 
which ensures a certain flexibility in applications. Obviously, if (2.1) is satisfied for some 
it well also hold for any 
Parameter 
introduced in Theorem 2.1 plays an important role in the results that follow, and it is particularly important in the sequel that 
With an appropriate choice of the functions 
and 
one can derive from Theorem 2.1 a number of oscillation criteria for (1.1). For example, consider a Kamenev-type function 
defined by
where 
is an integer. It is easy to see that 
and
As a consequence of Theorem 2.1, we have the following result.
Corollary 2.3.
Suppose that 
for 
Furthermore, assume that there exists a function 
such that for some integer 
and some 
where 
and 
are as in Theorem 2.1. Then every solution of (1.1) is oscillatory.
For an application of Corollary 2.3, we give the following example.
Example 2.4.
Consider the second-order neutral differential equation
where 
, 
Let 
, 
, and 
Then 
, 
Take 
Applying Corollary 2.3 with 
for any 
for 
Hence, (2.17) is oscillatory for 
Remark 2.5.
Corollary 2.3 can be applied to the second-order Euler differential equation
where 
Let 
, and 
Then 
, 
Take 
, 
Applying Corollary 2.3 with 
for any 
for 
Hence, (2.19) is oscillatory for 
It may happen that assumption (2.1) is not satisfied, or it is not easy to verify, consequently, that Theorem 2.1 does not apply or is difficult to apply. The following results provide some essentially new oscillation criteria for (1.1).
Theorem 2.6.
Assume that 
for 
and for some 
Further, suppose that there exist functions 
and 
such that for all 
and for some 
where 
, 
are as in Theorem 2.1. Suppose further that
where 
Then every solution of (1.1) is oscillatory.
Proof.
We proceed as in the proof of Theorem 2.1, assuming, without loss of generality, that there exists a solution 
of (1.1) such that 
, and 
for all 
We define the functions 
and 
as in Theorem 2.1; we arrive at inequality (2.10), which yields for 
sufficiently large
Therefore, for 
sufficiently large
It follows from (2.22) that
for all 
and for any 
Consequently, for all 
we obtain
In order to prove that
suppose the contrary, that is,
Assumption (2.21) implies the existence of a 
such that
By (2.30), we have
and there exists a 
such that 
for all 
On the other hand, by virtue of (2.29), for any positive number 
there exists a 
such that, for all 
Using integration by parts, we conclude that, for all 
It follows from (2.33) that, for all 
Since 
is an arbitrary positive constant, we get
which contradicts (2.17). Consequently, (2.28) holds, so
and, by virtue of (2.27),
which contradicts (2.23). This completes the proof.
Choosing 
as in Corollary 2.3, it is easy to verify that condition (2.21) is satisfied because, for any 
Consequently, we have the following result.
Corollary 2.7.
Suppose that 
for 
Furthermore, assume that there exist functions 
and 
such that for all 
for some integer 
and some 
where 
and 
are as in Theorem 2.1. Suppose further that (2.23) holds, where 
is as in Theorem 2.6. Then every solution of (1.1) is oscillatory.
From Theorem 2.6, we have the following result.
Theorem 2.8.
Assume that 
for 
Further, suppose that 
such that (2.21) holds, there exist functions 
and 
such that for all 
and for some 
where 
and 
are as in Theorem 2.1. Suppose further that (2.23) holds, where 
is as in Theorem 2.6. Then every solution of (1.1) is oscillatory.
Theorem 2.9.
Assume that 
for 
Further, assume that there exists a function 
such that for each 
for some 
where 
, 
are defined as in Theorem 2.1, the operator 
is defined by (1.28), and 
is defined by (1.29). Then every solution of (1.1) is oscillatory.
Proof.
We proceed as in the proof of Theorem 2.1, assuming, without loss of generality, that there exists a solution 
of (1.1) such that 
, 
, and 
for all 
We define the functions 
and 
as in Theorem 2.1; we arrive at inequality (2.9). Applying 
to (2.9), we get
By (1.30) and the above inequality, we obtain
Hence, from (2.43) we have
that is,
Taking the super limit in the above inequality, we get
which contradicts (2.41). This completes the proof.
If we choose
for 
and 
then we have
Thus by Theorem 2.9, we have the following oscillation result.
Corollary 2.10.
Suppose that 
for 
Further, assume that for each 
there exist a function 
and two constants 
such that for some 
where 
, 
are as in Theorem 2.1. Then every solution of (1.1) is oscillatory.
If we choose
where 
then we have
where 
, 
are defined as the following:
According to Theorem 2.9, we have the following oscillation result.
Corollary 2.11.
Suppose that 
for 
Further, assume that for each 
there exist two functions 
such that for some 
where 
, 
are as in Theorem 2.1. Then every solution of (1.1) is oscillatory.
In the following, we give some new oscillation results for (1.1) when 
for 
Theorem 2.12.
Assume that 
for 
Suppose that there exists a function 
such that for some 
and for some 
one has
where 
and 
is as in Theorem 2.1. Then every solution of (1.1) is oscillatory.
Proof.
Let 
be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a solution 
of (1.1) such that 
, 
, and 
for all 
Proceeding as in the proof of Theorem 2.1, we obtain (2.2) and (2.3). In view of (2.2), we have 
for 
We introduce a generalized Riccati transformation
Differentiating (2.55) from (2.2), we have 
Thus, there exists 
such that for all 
,
Similarly, we introduce another generalized Riccati transformation
Differentiating (2.57), then for all sufficiently large 
one has
From (2.56) and (2.58), we have
Note that 
then we have 
By (2.3) and the above inequality, we obtain
Multiplying (2.60) by 
and integrating from 
to 
we have, for any 
and for all 
The rest of the proof is similar to that of Theorem 2.1, we omit the details. This completes the proof.
Take 
where 
is an integer. As a consequence of Theorem 2.12, we have the following result.
Corollary 2.13.
Suppose that 
for 
Furthermore, assume that there exists a function 
such that for some integer 
and some 
where 
and 
are as in Theorem 2.12. Then every solution of (1.1) is oscillatory.
For an application of Corollary 2.13, we give the following example.
Example 2.14.
Consider the second-order neutral differential equation
where 
, 
, 
, 
, 
, 
, and 
, for 
Let 
, 
, and 
. Then 
, 
Applying Corollary 2.13 with 
for any 
for 
Hence, (2.63) is oscillatory for 
By (2.61), similar to the proof of Theorem 2.6, we have the following result.
Theorem 2.15.
Assume that 
for 
Assume also that 
such that (2.21) holds. Moreover, suppose that there exist functions 
and 
such that for all 
and for some 
where 
and 
are as in Theorem 2.12. Suppose further that
where 
is defined as in Theorem 2.6. Then every solution of (1.1) is oscillatory.
Choosing 
, 
where 
is an integer. By Theorem 2.15, we have the following result.
Corollary 2.16.
Suppose that 
for 
Furthermore, assume that there exist functions 
and 
such that for all 
some integer 
and some 
where 
and 
are as in Theorem 2.12. Suppose further that (2.66) holds, where 
is defined as in Theorem 2.6. Then every solution of (1.1) is oscillatory.
From Theorem 2.15, we have the following result.
Theorem 2.17.
Assume that 
for 
Assume also that 
such that (2.21) holds. Moreover, suppose that there exist functions 
and 
such that for all 
and for some 
where 
and 
are as in Theorem 2.12. Suppose further that (2.66) holds, where 
is as in Theorem 2.6. Then every solution of (1.1) is oscillatory.
Next, by (2.60), similar to the proof of Theorem 2.9, we have the following result.
Theorem 2.18.
Assume that 
for 
Further, assume that there exists a function 
such that for each 
for some 
where 
are defined as in Theorem 2.12, the operator 
is defined by (1.28), and 
is defined by (1.29). Then every solution of (1.1) is oscillatory.
If we choose 
as (2.47), then from Theorem 2.18, we have the following oscillation result.
Corollary 2.19.
Suppose that 
for 
Further, assume that for each 
there exist a function 
and two constants 
such that for some 
where 
, 
are as in Theorem 2.12. Then every solution of (1.1) is oscillatory.
If we choose 
as (2.50), then from Theorem 2.18, we have the following oscillation result.
Corollary 2.20.
Suppose that 
for 
Further, assume that for each 
there exist two functions 
such that for some 
where 
, 
are as in Theorem 2.12. Then every solution of (1.1) is oscillatory.
Remark 2.21.
The results of this paper can be extended to the more general equation of the form
The statement and the formulation of the results are left to the interested reader.
Remark 2.22.
One can easily see that the results obtained in [15, 16, 18, 19, 25, 28] cannot be applied to (2.17), (2.63), so our results are new.
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Acknowledgment
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (60774004, 60904024), China Postdoctoral Science Foundation Funded Project (20080441126, 200902564), Shandong Postdoctoral Funded Project (200802018) and supported by the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), also supported by University of Jinan Research Funds for Doctors (XBS0843).
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Han, Z., Li, T., Sun, S. et al. Oscillation Criteria for Second-Order Nonlinear Neutral Delay Differential Equations. Adv Differ Equ 2010, 763278 (2010). https://doi.org/10.1155/2010/763278
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DOI: https://doi.org/10.1155/2010/763278