# Oscillation Criteria for Second-Order Nonlinear Neutral Delay Differential Equations

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

**2010**:763278

**DOI: **10.1155/2010/763278

© Zhenlai Han et al. 2010

**Received: **10 December 2009

**Accepted: **1 July 2010

**Published: **19 July 2010

## Abstract

Some sufficient conditions are established for the oscillation of second-order neutral differential equation , , where . The results complement and improve those of Grammatikopoulos et al. Ladas, A. Meimaridou, Oscillation of second-order neutral delay differential equations, Rat. Mat. 1 (1985), Grace and Lalli (1987), Ruan (1993), H. J. Li (1996), H. J. Li (1997), Xu and Xia (2008).

## 1. Introduction

In recent years, the oscillatory behavior of differential equations has been the subject of intensive study; we refer to the articles [1–13]; Especially, the study of the oscillation of neutral delay differential equations is of great interest in the last three decades; see for example [14–38] and references cited therein. Second-order neutral delay differential equations have applications in problems dealing with vibrating masses attached to an elastic bar and in some variational problems (see [39]).

- (a)
, and is not identically zero on any ray of the form for any where is a constant;

- (b)
for is a constant;

- (c)
, , , , , where is a constant.

In the study of oscillation of differential equations, there are two techniques which are used to reduce the higher-order equations to the first-order Riccati equation (or inequality). One of them is the Riccati transformation technique. The other one is called the generalized Riccati technique. This technique can introduce some new sufficient conditions for oscillation and can be applied to different equations which cannot be covered by the results established by the Riccati technique.

is oscillatory.

Say a function is said to belong to denoted by if where which satisfies for and has the partial derivative on such that is locally integrable with respect to in

- (i)
and for

- (ii)
has a continuous and nonpositive partial derivative satisfying, for some , where is nonnegative.

where , The authors introduced a class of functions Let and The function is said to belong to the class (defined by for short) if

, , for

has a continuous and nonpositive partial derivative on with respect to the second variable;

there exists a nondecreasing function such that

Long and Wang [6] considered (1.22); by using the function class and the operator which is defined in [31], the authors established some oscillation results for (1.22).

and obtained that if for and then (1.27) is oscillatory. We note that the result given in [28] fails to apply the cases or for To the best of our knowledge nothing is known regarding the qualitative behavior of (1.1) when

Motivated by [10, 21], for the sake of convenience, we give the following definitions.

Definition 1.1.

for and

Definition 1.2.

In this paper, we obtain some new oscillation criteria for (1.1). The paper is organized as follows. In the next section, we will use the generalized Riccati transformation technique to give some sufficient conditions for the oscillation of (1.1), and we will give two examples to illustrate the main results. The key idea in the proofs makes use of the idea used in [23]. The method used in this paper is different from that of [27].

## 2. Main Results

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

Theorem 2.1.

where Then every solution of (1.1) is oscillatory.

Proof.

which contradicts (2.1). This completes the proof.

Remark 2.2.

We note that it suffices to satisfy (2.1) in Theorem 2.1 for any which ensures a certain flexibility in applications. Obviously, if (2.1) is satisfied for some it well also hold for any Parameter introduced in Theorem 2.1 plays an important role in the results that follow, and it is particularly important in the sequel that

As a consequence of Theorem 2.1, we have the following result.

Corollary 2.3.

where and are as in Theorem 2.1. Then every solution of (1.1) is oscillatory.

For an application of Corollary 2.3, we give the following example.

Example 2.4.

for Hence, (2.17) is oscillatory for

Remark 2.5.

for Hence, (2.19) is oscillatory for

It may happen that assumption (2.1) is not satisfied, or it is not easy to verify, consequently, that Theorem 2.1 does not apply or is difficult to apply. The following results provide some essentially new oscillation criteria for (1.1).

Theorem 2.6.

where Then every solution of (1.1) is oscillatory.

Proof.

which contradicts (2.23). This completes the proof.

Consequently, we have the following result.

Corollary 2.7.

where and are as in Theorem 2.1. Suppose further that (2.23) holds, where is as in Theorem 2.6. Then every solution of (1.1) is oscillatory.

From Theorem 2.6, we have the following result.

Theorem 2.8.

where and are as in Theorem 2.1. Suppose further that (2.23) holds, where is as in Theorem 2.6. Then every solution of (1.1) is oscillatory.

Theorem 2.9.

where , are defined as in Theorem 2.1, the operator is defined by (1.28), and is defined by (1.29). Then every solution of (1.1) is oscillatory.

Proof.

which contradicts (2.41). This completes the proof.

Thus by Theorem 2.9, we have the following oscillation result.

Corollary 2.10.

where , are as in Theorem 2.1. Then every solution of (1.1) is oscillatory.

According to Theorem 2.9, we have the following oscillation result.

Corollary 2.11.

where , are as in Theorem 2.1. Then every solution of (1.1) is oscillatory.

In the following, we give some new oscillation results for (1.1) when for

Theorem 2.12.

where and is as in Theorem 2.1. Then every solution of (1.1) is oscillatory.

Proof.

The rest of the proof is similar to that of Theorem 2.1, we omit the details. This completes the proof.

Take where is an integer. As a consequence of Theorem 2.12, we have the following result.

Corollary 2.13.

where and are as in Theorem 2.12. Then every solution of (1.1) is oscillatory.

For an application of Corollary 2.13, we give the following example.

Example 2.14.

for Hence, (2.63) is oscillatory for

By (2.61), similar to the proof of Theorem 2.6, we have the following result.

Theorem 2.15.

where is defined as in Theorem 2.6. Then every solution of (1.1) is oscillatory.

Choosing , where is an integer. By Theorem 2.15, we have the following result.

Corollary 2.16.

where and are as in Theorem 2.12. Suppose further that (2.66) holds, where is defined as in Theorem 2.6. Then every solution of (1.1) is oscillatory.

From Theorem 2.15, we have the following result.

Theorem 2.17.

where and are as in Theorem 2.12. Suppose further that (2.66) holds, where is as in Theorem 2.6. Then every solution of (1.1) is oscillatory.

Next, by (2.60), similar to the proof of Theorem 2.9, we have the following result.

Theorem 2.18.

where are defined as in Theorem 2.12, the operator is defined by (1.28), and is defined by (1.29). Then every solution of (1.1) is oscillatory.

If we choose as (2.47), then from Theorem 2.18, we have the following oscillation result.

Corollary 2.19.

where , are as in Theorem 2.12. Then every solution of (1.1) is oscillatory.

If we choose as (2.50), then from Theorem 2.18, we have the following oscillation result.

Corollary 2.20.

where , are as in Theorem 2.12. Then every solution of (1.1) is oscillatory.

Remark 2.21.

The statement and the formulation of the results are left to the interested reader.

Remark 2.22.

One can easily see that the results obtained in [15, 16, 18, 19, 25, 28] cannot be applied to (2.17), (2.63), so our results are new.

## Declarations

### Acknowledgment

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), Shandong Postdoctoral Funded Project (200802018) and supported by the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), also supported by University of Jinan Research Funds for Doctors (XBS0843).

## Authors’ Affiliations

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