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

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## Abstract

Some new oscillation criteria for the second-order neutral delay differential equation Open image in new window , Open image in new window are established, where Open image in new window , Open image in new window , Open image in new window , Open image in new window . 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, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and the references cited therein.

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

where Open image in new window

In what follows we assume that

(*I*_{1}) Open image in new window , Open image in new window , Open image in new window

(*I*_{2}) Open image in new window

(*I*_{3}) 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.

Some known results are established for (1.1) under the condition Open image in new window Grammatikopoulos et al. [6] obtained that if Open image in new window and, Open image in new window 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 Open image in new window operator Open image in new window and the Riccati technique and averaging technique.

Following [11], we say that a function Open image in new window belongs to the function class Open image in new window denoted by Open image in new window if Open image in new window where Open image in new window which satisfies Open image in new window 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.

Define the operator Open image in new window by

for Open image in new window and Open image in new window The function Open image in new window is defined by

It is easy to see 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). We start with the following oscillation criteria.

Theorem.

where Open image in new window then (1.1) oscillates.

Proof.

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

Theorem 2.2.

where Open image in new window is defined as in Theorem 2.1, the operator Open image in new window is defined by (1.5), and Open image in new window is defined by (1.6). Then every solution Open image in new window 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 Open image in new window , Open image in new window , Open image in new window , and Open image in new window then by Theorem 2.1 every solution of (2.27) oscillates; for example, Open image in new window is an oscillatory solution of (2.27).

Remark 2.5.

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

## Notes

### 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).

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