By using Riccati transformation technique, we will establish some new oscillation criteria for the even order neutral delay differential equations , , where is even, , , and . These oscillation criteria, at least in some sense, complement and improve those of Zafer (1998) and Zhang et al. (2010). An example is considered to illustrate the main results.
1. Introduction
This paper is concerned with the oscillatory behavior of the evenorder neutral delay differential equations
where is even
In what follows we assume that
,
, , , , , , where is a constant,
and for is a constant.
Neutral differential equations find numerous applications in natural science and technology. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines; see Hale [1].
In the last decades, there are many studies that have been made on the oscillatory behavior of solutions of differential equations [2–6] and neutral delay differential equations [7–23].
For instance, Grammatikopoulos et al. [10] examined the oscillation of secondorder neutral delay differential equations
where
Liu and Bai [13] investigated the secondorder neutral differential equations
where
Meng and Xu [14] studied the oscillation of evenorder neutral differential equations
where
Ye and Xu [21] considered the secondorder quasilinear neutral delay differential equations
where
Zafer [22] discussed oscillation criteria for the equations
where
In 2009, Zhang et al. [23] considered the oscillation of the evenorder nonlinear neutral differential equation (1.1) when
To the best of our knowledge, the above oscillation results cannot be applied when and it seems to have few oscillation results for (1.1) when
Xu and Xia [17] established a new oscillation criteria for the secondorder neutral differential equations
Motivated by Liu and Bai [13], we will further the investigation and offer some more general new oscillation criteria for (1.1), by employing a class of function and operator The method used in this paper is different from [17].
Following [13], we say that a function belongs to the function class if where which satisfies
for and has the partial derivative on such that is locally integrable with respect to in
By choosing the special function it is possible to derive several oscillation criteria for a wide range of differential equations.
Define the operator by
for and The function is defined by
It is easy to verify that is a linear operator and that it satisfies
2. Main Results
In this section, we give some new oscillation criteria for (1.1). In order to prove our theorems we will need the following lemmas.
Lemma 2.1 (see [5]).
Let If is eventually of one sign for all large say then there exist a and an integer with even for or odd for such that implies that for and implies that for
Lemma 2.2 (see [5]).
If the function is as in Lemma 2.1 and for then for every there exists a constant such that
Lemma 2.3 (see [14]).
Suppose that is an eventually positive solution of (1.1). Then there exists a number such that for
Theorem 2.4.
Assume that .
Further, there exist functions and , such that for some and for every
where the operator is defined by (1.9), and is defined by (1.10). Then every solution of (1.1) is oscillatory.
Proof.
Let be a nonoscillatory solution of (1.1). Then there exists such that for all Without loss of generality, we assume that for all
By Lemma 2.3, there exists such like that (2.2) for Using definition of and applying (1.1), we get for sufficiently large
thus
where
It is easy to check that we can apply Lemma 2.2 for and conclude that there exist and such that
Next, define
Then, and
From (2.6), (2.7), and (2.8), we have
Similarly, define
Then, and
From (2.6), (2.10), and (2.11), we have
Therefore, from (2.9) and (2.12), we get
From (2.5), note that , and then we obtain
Applying to (2.14), we get
By (1.11) and the above inequality, we obtain
Hence, from (2.16), we have
that is,
Taking the super limit in the above inequality, we get
which contradicts (2.3). This completes the proof.
We can apply Theorem 2.4 to the secondorder neutral delay differential equations
We get the following new result.
Theorem 2.5.
Assume that Further, there exist functions and such that
where is defined as in Theorem 2.4, the operator is defined by (1.9), and is defined by (1.10). Then every solution of (2.20) is oscillatory.
Proof.
Let be a nonoscillatory solution of (2.20). Then there exists such that for all
Without loss of generality, we assume that for all Proceeding as in the proof of Theorem 2.4, we have (2.2) and (2.5) Next, define
Then and
From (2.22) and (2.23), we have
Similarly, define
Then The rest of the proof is similar to that of the proof of Theorem 2.4, hence we omit the details.
Remark 2.6.
With the different choice of and Theorem 2.4 (or Theorem 2.5) can be stated with different conditions for oscillation of (1.1) (or (2.20)). For example, if we choose for then
By Theorem 2.4 (or Theorem 2.5) we can obtain the oscillation criterion for (1.1) (or (2.20)); the details are left to the reader.
For an application, we give the following example to illustrate the main results.
Example 2.7.
Consider the following equations:
Let Take it is easy to verify that
By Theorem 2.5, let one has (2.21). Hence, every solution of (2.27) oscillates. For example, is an oscillatory solution of (2.27).
Remark 2.8.
The recent results cannot be applied in (1.1) and (2.20) when for Therefore, our results are new.
Remark 2.9.
It would be interesting to find another method to study (1.1) and (2.20) when , or for
Remark 2.10.
It would be more interesting to find another method to study (1.1) when is odd.
Acknowledgments
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), and Shandong Postdoctoral funded project (200802018) by the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), and also by University of Jinan Research Funds for Doctors (B0621, XBS0843).
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