- Research Article
- Open Access
Oscillation of Even-Order Neutral Delay Differential Equations
https://doi.org/10.1155/2010/184180
© Tongxing Li et al. 2010
- Received: 28 November 2009
- Accepted: 29 March 2010
- Published: 19 May 2010
Abstract
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.
Keywords
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Linear Operator
1. Introduction
where
is even
In what follows we assume that
(I1)
,
(I2)
,
,
,
,
,
, where
is a constant,
(I3)
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].
where
where
where
where
where
In 2009, Zhang et al. [23] considered the oscillation of the even-order 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
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].
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.
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]).
Lemma 2.3 (see [14]).



Theorem 2.4.
Assume that
.
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




where



which contradicts (2.3). This completes the proof.
We get the following new result.
Theorem 2.5.
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



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.




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.
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.
Declarations
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).
Authors’ Affiliations
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