Advances in Difference Equations

, 2010:184180 | Cite as

Oscillation of Even-Order Neutral Delay Differential Equations

Open Access
Research Article
Part of the following topical collections:
  1. Recent Trends in Differential and Difference Equations

Abstract

By using Riccati transformation technique, we will establish some new oscillation criteria for the even order neutral delay differential equations Open image in new window , Open image in new window , where Open image in new window is even, Open image in new window , Open image in new window , and Open image in new window . 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

This paper is concerned with the oscillatory behavior of the even-order neutral delay differential equations

where Open image in new window is even Open image in new window

In what follows we assume that

(I1) Open image in new window ,

(I2) Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , where Open image in new window is a constant,

(I3) Open image in new window and Open image in new window for Open image in new window Open image in new window 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, 3, 4, 5, 6] and neutral delay differential equations [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

For instance, Grammatikopoulos et al. [10] examined the oscillation of second-order neutral delay differential equations

where Open image in new window

Liu and Bai [13] investigated the second-order neutral differential equations

where Open image in new window

Meng and Xu [14] studied the oscillation of even-order neutral differential equations

where Open image in new window

Ye and Xu [21] considered the second-order quasilinear neutral delay differential equations

where Open image in new window

Zafer [22] discussed oscillation criteria for the equations

where Open image in new window

In 2009, Zhang et al. [23] considered the oscillation of the even-order nonlinear neutral differential equation (1.1) when Open image in new window

To the best of our knowledge, the above oscillation results cannot be applied when Open image in new window and it seems to have few oscillation results for (1.1) when Open image in new window

Xu and Xia [17] established a new oscillation criteria for the second-order 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 Open image in new window and operator Open image in new window The method used in this paper is different from [17].

Following [13], we say that a function Open image in new window belongs to the function class Open image in new window if Open image in new window where Open image in new window which satisfies

for Open image in new window and has the partial derivative Open image in new window on Open image in new window such that Open image in new window is locally integrable with respect to Open image in new window in Open image in new window

By choosing the special function Open image in new window it is possible to derive several oscillation criteria for a wide range of differential equations.

It is easy to verify that Open image in new window 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 Open image in new window If Open image in new window is eventually of one sign for all large Open image in new window say Open image in new window then there exist a Open image in new window and an integer Open image in new window Open image in new window with Open image in new window even for Open image in new window or Open image in new window odd for Open image in new window such that Open image in new window implies that Open image in new window for Open image in new window and Open image in new window implies that Open image in new window for Open image in new window

Lemma 2.2 (see [5]).

If the function Open image in new window is as in Lemma 2.1 and Open image in new window for Open image in new window then for every Open image in new window there exists a constant Open image in new window such that

Lemma 2.3 (see [14]).

Suppose that Open image in new window is an eventually positive solution of (1.1). Then there exists a number Open image in new window such that for Open image in new window

Theorem 2.4.

Assume that Open image in new window .

Further, there exist functions Open image in new window and Open image in new window , such that for some Open image in new window and for every Open image in new window

where Open image in new window the operator Open image in new window is defined by (1.9), and Open image in new window is defined by (1.10). Then every solution Open image in new window of (1.1) is oscillatory.

Proof.

Let Open image in new window be a nonoscillatory solution of (1.1). Then there exists Open image in new window such that Open image in new window for all Open image in new window Without loss of generality, we assume that Open image in new window for all Open image in new window

By Lemma 2.3, there exists Open image in new window such like that (2.2) for Open image in new window Using definition of Open image in new window and applying (1.1), we get for sufficiently large Open image in new window

where Open image in new window

It is easy to check that we can apply Lemma 2.2 for Open image in new window and conclude that there exist Open image in new window and Open image in new window such that
Next, define
From (2.6), (2.7), and (2.8), we have
Similarly, define
From (2.6), (2.10), and (2.11), we have
Therefore, from (2.9) and (2.12), we get
From (2.5), note that Open image in new window , and Open image in new window then we obtain
Applying Open image in new window 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 second-order neutral delay differential equations

We get the following new result.

Theorem 2.5.

Assume that Open image in new window Further, there exist functions Open image in new window and Open image in new window such that

where Open image in new window is defined as in Theorem 2.4, the operator Open image in new window is defined by (1.9), and Open image in new window is defined by (1.10). Then every solution Open image in new window of (2.20) is oscillatory.

Proof.

Let Open image in new window be a nonoscillatory solution of (2.20). Then there exists Open image in new window such that Open image in new window for all Open image in new window

Without loss of generality, we assume that Open image in new window for all Open image in new window Proceeding as in the proof of Theorem 2.4, we have (2.2) and (2.5) Open image in new window Next, define
From (2.22) and (2.23), we have
Similarly, define

Then Open image in new window 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 Open image in new window and Open image in new window 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 Open image in new window for Open image in new window 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:

By Theorem 2.5, let Open image in new window one has (2.21). Hence, every solution of (2.27) oscillates. For example, Open image in new window is an oscillatory solution of (2.27).

Remark 2.8.

The recent results cannot be applied in (1.1) and (2.20) when Open image in new window for Open image in new window Therefore, our results are new.

Remark 2.9.

It would be interesting to find another method to study (1.1) and (2.20) when Open image in new window , or Open image in new window for Open image in new window

Remark 2.10.

It would be more interesting to find another method to study (1.1) when Open image in new window is odd.

Notes

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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Copyright information

© Tongxing Li et al. 2010

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  • Tongxing Li
    • 1
  • Zhenlai Han
    • 1
    • 2
  • Ping Zhao
    • 3
  • Shurong Sun
    • 1
    • 4
  1. 1.School of ScienceUniversity of JinanJinanChina
  2. 2.School of Control Science and EngineeringShandong UniversityJinanChina
  3. 3.School of Control Science and EngineeringUniversity of JinanJinanChina
  4. 4.Department of Mathematics and StatisticsMissouri University of Science and TechnologyRollaUSA

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