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Oscillation Criteria for SecondOrder Nonlinear Neutral Delay Differential Equations
Advances in Difference Equations volume 2010, Article number: 763278 (2010)
Abstract
Some sufficient conditions are established for the oscillation of secondorder neutral differential equation , , where . The results complement and improve those of Grammatikopoulos et al. Ladas, A. Meimaridou, Oscillation of secondorder 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).
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. Secondorder 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 secondorder 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.
In the study of oscillation of differential equations, there are two techniques which are used to reduce the higherorder equations to the firstorder 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 secondorder 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 secondorder 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.
Meng and Xu [22] considered the evenorder 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 secondorder 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 secondorder 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 secondorder 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 secondorder neutral delay differential equation
where In 2002, Tanaka [27] studied the evenorder neutral delay differential equation
where or The author established some comparison theorems for the oscillation of (1.26). Xu and Xia [28] investigated the secondorder 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 Kamenevtype 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 secondorder 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 secondorder 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 secondorder 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 SecondOrder Nonlinear Neutral Delay Differential Equations. Adv Differ Equ 2010, 763278 (2010). https://doi.org/10.1155/2010/763278
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Keywords
 Function Class
 Nonoscillatory Solution
 Oscillation Criterion
 Oscillation Result
 Neutral Delay Differential Equation