- Research Article
- Open Access

# 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

https://doi.org/10.1155/2010/289340

© Zhenlai Han et al. 2010

**Received:**8 October 2009**Accepted:**10 January 2010**Published:**9 February 2010

## Abstract

Some new oscillation criteria for the second-order neutral delay differential equation , are established, where , , , . These oscillation criteria extend and improve some known results. An example is considered to illustrate the main results.

## Keywords

- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Functional Equation

## 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

where

In what follows we assume that

(*I*_{1})
,
,

(*I*_{2})

(*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.

Define the operator by

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.

where then (1.1) oscillates.

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

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