- Research Article
- Open Access

# Oscillation of Even-Order Neutral Delay Differential Equations

- Tongxing Li
^{1}, - Zhenlai Han
^{1, 2}Email author, - Ping Zhao
^{3}and - Shurong Sun
^{1, 4}

**2010**:184180

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

In what follows we assume that

(*I*_{2})
,
,
,
,
,
, where
is a constant,

(*I*_{3})
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].

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.

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

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

## References

- Hale J:
*Theory of Functional Differential Equations, Applied Mathematical Sciences, vol. 3*. 2nd edition. Springer, New York, NY, USA; 1977:x+365.View ArticleGoogle Scholar - Džurina J, Stavroulakis IP:
**Oscillation criteria for second-order delay differential equations.***Applied Mathematics and Computation*2003,**140**(2-3):445-453. 10.1016/S0096-3003(02)00243-6MATHMathSciNetView ArticleGoogle Scholar - Grace SR:
**Oscillation theorems for nonlinear differential equations of second order.***Journal of Mathematical Analysis and Applications*1992,**171**(1):220-241. 10.1016/0022-247X(92)90386-RMATHMathSciNetView ArticleGoogle Scholar - Koplatadze R:
**On oscillatory properties of solutions of functional differential equations.***Memoirs on Differential Equations and Mathematical Physics*1994,**3:**1-180.MathSciNetGoogle Scholar - Philos ChG:
**A new criterion for the oscillatory and asymptotic behavior of delay differential equations.***Bulletin de l'Académie Polonaise des Science, Série des Sciences Mathématiques*1981,**39:**61-64.Google Scholar - Sun YG, Meng FW:
**Note on the paper of J. Džurina and I. P. Stavroulakis.***Applied Mathematics and Computation*2006,**174**(2):1634-1641. 10.1016/j.amc.2005.07.008MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP, Grace SR:
**Oscillation theorems for certain neutral functional-differential equations.***Computers & Mathematics with Applications*1999,**38**(11-12):1-11. 10.1016/S0898-1221(99)00280-1MATHMathSciNetView ArticleGoogle Scholar - Han Z, Li T, Sun S, Sun Y:
**Remarks on the paper [Appl. Math. Comput. 207 (2009) 388–396].***Applied Mathematics and Computation*2010,**215**(11):3998-4007. 10.1016/j.amc.2009.12.006MATHMathSciNetView ArticleGoogle Scholar - Han Z, Li T, Sun S, Chen W:
**On the oscillation of second order neutral delay differential equations.***Advances in Difference Equations*2010,**2010:**-8.Google Scholar - Grammatikopoulos MK, Ladas G, Meimaridou A:
**Oscillations of second order neutral delay differential equations.***Radovi Matematički*1985,**1**(2):267-274.MATHMathSciNetGoogle Scholar - Karpuz B, Manojlović JV, Öcalan Ö, Shoukaku Y:
**Oscillation criteria for a class of second-order neutral delay differential equations.***Applied Mathematics and Computation*2009,**210**(2):303-312. 10.1016/j.amc.2008.12.075MATHMathSciNetView ArticleGoogle Scholar - Lin X, Tang XH:
**Oscillation of solutions of neutral differential equations with a superlinear neutral term.***Applied Mathematics Letters*2007,**20**(9):1016-1022. 10.1016/j.aml.2006.11.006MATHMathSciNetView ArticleGoogle Scholar - Liu L, Bai Y:
**New oscillation criteria for second-order nonlinear neutral delay differential equations.***Journal of Computational and Applied Mathematics*2009,**231**(2):657-663. 10.1016/j.cam.2009.04.009MATHMathSciNetView ArticleGoogle Scholar - Meng F, Xu R:
**Oscillation criteria for certain even order quasi-linear neutral differential equations with deviating arguments.***Applied Mathematics and Computation*2007,**190**(1):458-464. 10.1016/j.amc.2007.01.040MATHMathSciNetView ArticleGoogle Scholar - Rath RN, Misra N, Padhy LN:
**Oscillatory and asymptotic behaviour of a nonlinear second order neutral differential equation.***Mathematica Slovaca*2007,**57**(2):157-170. 10.2478/s12175-007-0006-7MATHMathSciNetView ArticleGoogle Scholar - Şahiner Y:
**On oscillation of second order neutral type delay differential equations.***Applied Mathematics and Computation*2004,**150**(3):697-706. 10.1016/S0096-3003(03)00300-XMATHMathSciNetView ArticleGoogle Scholar - Xu R, Xia Y:
**A note on the oscillation of second-order nonlinear neutral functional differential equations.***International Journal of Contemporary Mathematical Sciences*2008,**3**(29–32):1441-1450.MATHMathSciNetGoogle Scholar - Xu R, Meng F:
**New Kamenev-type oscillation criteria for second order neutral nonlinear differential equations.***Applied Mathematics and Computation*2007,**188**(2):1364-1370. 10.1016/j.amc.2006.11.004MATHMathSciNetView ArticleGoogle Scholar - Xu R, Meng F:
**Oscillation criteria for second order quasi-linear neutral delay differential equations.***Applied Mathematics and Computation*2007,**192**(1):216-222. 10.1016/j.amc.2007.01.108MATHMathSciNetView ArticleGoogle Scholar - Xu Z, Liu X:
**Philos-type oscillation criteria for Emden-Fowler neutral delay differential equations.***Journal of Computational and Applied Mathematics*2007,**206**(2):1116-1126. 10.1016/j.cam.2006.09.012MATHMathSciNetView ArticleGoogle Scholar - Ye L, Xu Z:
**Oscillation criteria for second order quasilinear neutral delay differential equations.***Applied Mathematics and Computation*2009,**207**(2):388-396. 10.1016/j.amc.2008.10.051MATHMathSciNetView ArticleGoogle Scholar - Zafer A:
**Oscillation criteria for even order neutral differential equations.***Applied Mathematics Letters*1998,**11**(3):21-25. 10.1016/S0893-9659(98)00028-7MATHMathSciNetView ArticleGoogle Scholar - Zhang Q, Yan J, Gao L:
**Oscillation behavior of even-order nonlinear neutral differential equations with variable coefficients.***Computers and Mathematics with Applications*2010,**59**(1):426-430. 10.1016/j.camwa.2009.06.027MATHMathSciNetView ArticleGoogle Scholar

## Copyright

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.