# On the Oscillation of Second-Order Neutral Delay Differential Equations

- Zhenlai Han
^{1, 2}Email author, - Tongxing Li
^{1}, - Shurong Sun
^{1, 3}and - Weisong Chen
^{1}

**2010**:289340

**DOI: **10.1155/2010/289340

© Zhenlai Han et al. 2010

**Received: **8 October 2009

**Accepted: **10 January 2010

**Published: **9 February 2010

## Abstract

## 1. Introduction

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 recent years, many studies have been made on the oscillatory behavior of solutions of neutral delay differential equations, and we refer to the recent papers [2–23] and the references cited therein.

This paper is concerned with the oscillatory behavior of the second-order neutral delay differential equation

In what follows we assume that

(*I*_{3})
,
,
,
,
,
where
is a constant.

Some known results are established for (1.1) under the condition Grammatikopoulos et al. [6] obtained that if and, then the second-order neutral delay differential equation

oscillates. In [13], by employing Riccati technique and averaging functions method, Ruan established some general oscillation criteria for second-order neutral delay differential equation

Xu and Meng [18] as well as Zhuang and Li [23] studied the oscillation of the second-order neutral delay differential equation

Motivated by [11], we will further the investigation and offer some more general new oscillation criteria for (1.1), by employing a class of function operator and the Riccati technique and averaging technique.

Following [11], we say that a function belongs to the function class denoted by 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.

for and The function is defined by

It is easy to see that is a linear operator and that it satisfies

## 2. Main Results

In this section, we give some new oscillation criteria for (1.1). We start with the following oscillation criteria.

Theorem.

Proof.

which is a contradiction to (2.1). This completes the proof.

Theorem 2.2.

where is defined as in Theorem 2.1, the operator is defined by (1.5), and is defined by (1.6). Then every solution of (1.1) is oscillatory.

Proof.

which contradicts (2.10). This completes the proof.

Remark 2.3.

By Theorem 2.2 we can obtain the oscillation criterion for (1.1), the details are left to the reader.

For an application, we give the following example to illustrate the main results.

Example 2.4.

Let , , , and then by Theorem 2.1 every solution of (2.27) oscillates; for example, is an oscillatory solution of (2.27).

Remark 2.5.

The recent results cannot be applied in (2.27) since so our results are new ones.

## Declarations

### Acknowledgments

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 the Natural Scientific Foundation of Shandong Province (Y2008A28, ZR2009AL003), also supported by University of Jinan Research Funds for Doctors (XBS0843).

## Authors’ Affiliations

## References

- Hale J:
*Theory of Functional Differential Equations*. 2nd edition. Springer, New York, NY, USA; 1977:x+365. Applied Mathematical SciencesView ArticleMATHGoogle 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)00280-1MathSciNetView ArticleMATHGoogle Scholar - Berezansky L, Diblik J, Šmarda Z:
**On connection between second-order delay differential equations and integrodifferential equations with delay.***Advances in Difference Equations*2010,**2010:**-8.Google 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)00243-6MathSciNetView ArticleMATHGoogle 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-RMathSciNetView ArticleMATHGoogle Scholar - Grammatikopoulos MK, Ladas G, Meimaridou A:
**Oscillations of second order neutral delay differential equations.***Radovi Matematički*1985,**1**(2):267-274.MathSciNetMATHGoogle 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.2009.12.006MathSciNetView ArticleMATHGoogle 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.2008.12.075MathSciNetView ArticleMATHGoogle Scholar - Li H-J, Yeh C-C:
**Oscillation criteria for second-order neutral delay difference equations.***Computers & Mathematics with Applications*1998,**36**(10-12):123-132. 10.1016/S0898-1221(98)80015-1MathSciNetView ArticleMATHGoogle 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.2006.11.006MathSciNetView ArticleMATHGoogle 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.2009.04.009MathSciNetView ArticleMATHGoogle 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/s12175-007-0006-7MathSciNetView ArticleMATHGoogle Scholar - Ruan SG:
**Oscillations of second order neutral differential equations.***Canadian Mathematical Bulletin*1993,**36**(4):485-496. 10.4153/CMB-1993-064-4MathSciNetView ArticleMATHGoogle Scholar - Sun YG, Meng F:
**Note on the paper of Dourina and Stavroulakis.***Applied Mathematics and Computation*2006,**174**(2):1634-1641. 10.1016/j.amc.2005.07.008MathSciNetView ArticleMATHGoogle 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-XMathSciNetView ArticleMATHGoogle Scholar - Xu R, Meng F:
**Some new oscillation criteria for second order quasi-linear neutral delay differential equations.***Applied Mathematics and Computation*2006,**182**(1):797-803. 10.1016/j.amc.2006.04.042MathSciNetView ArticleMATHGoogle 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.2007.01.108MathSciNetView ArticleMATHGoogle 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.2006.11.004MathSciNetView ArticleMATHGoogle 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.2006.09.012MathSciNetView ArticleMATHGoogle 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.2008.10.051MathSciNetView ArticleMATHGoogle Scholar - Zafer A:
**Oscillation criteria for even order neutral differential equations.***Applied Mathematics Letters*1998,**11**(3):21-25. 10.1016/S0893-9659(98)00028-7MathSciNetView ArticleMATHGoogle 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.2009.06.027MathSciNetView ArticleMATHGoogle Scholar - Zhuang R-K, Li W-T:
**Interval oscillation criteria for second order neutral nonlinear differential equations.***Applied Mathematics and Computation*2004,**157**(1):39-51. 10.1016/j.amc.2003.06.016MathSciNetView ArticleMATHGoogle 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.