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

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## Abstract

Some sufficient conditions are established for the oscillation of second-order neutral differential equation Open image in new window , Open image in new window , where Open image in new window . 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).

### Keywords

Function Class Nonoscillatory Solution Oscillation Criterion Oscillation Result Neutral Delay Differential Equation## 1. Introduction

In recent years, the oscillatory behavior of differential equations has been the subject of intensive study; we refer to the articles [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 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, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 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)
Open image in new window , Open image in new window and Open image in new window is not identically zero on any ray of the form Open image in new window for any Open image in new window where Open image in new window is a constant;

- (b)
Open image in new window for Open image in new window Open image in new window is a constant;

- (c)
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.

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 Open image in new window is said to belong to 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

- (i)
- (ii)
Open image in new window has a continuous and nonpositive partial derivative Open image in new window satisfying, for some Open image in new window , Open image in new window where Open image in new window is nonnegative.

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

Open image in new window , Open image in new window , Open image in new window for Open image in new window

Open image in new window has a continuous and nonpositive partial derivative on Open image in new window with respect to the second variable;

there exists a nondecreasing function Open image in new window such that

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

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

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

Definition 1.1.

for Open image in new window and Open image in new window

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

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

Corollary 2.3.

where Open image in new window and Open image in new window 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 Open image in new window Hence, (2.17) is oscillatory for Open image in new window

Remark 2.5.

for Open image in new window Hence, (2.19) is oscillatory for Open image in new window

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

Theorem 2.9.

where Open image in new window , Open image in new window are defined as in Theorem 2.1, the operator Open image in new window is defined by (1.28), and Open image in new window 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 Open image in new window , Open image in new window 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 Open image in new window , Open image in new window 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 Open image in new window for Open image in new window

Theorem 2.12.

where Open image in new window Open image in new window and Open image in new window 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 Open image in new window where Open image in new window is an integer. As a consequence of Theorem 2.12, we have the following result.

Corollary 2.13.

where Open image in new window and Open image in new window 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 Open image in new window Hence, (2.63) is oscillatory for Open image in new window

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

Theorem 2.15.

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

Choosing Open image in new window , Open image in new window where Open image in new window is an integer. By Theorem 2.15, we have the following result.

Corollary 2.16.

where Open image in new window and Open image in new window are as in Theorem 2.12. Suppose further that (2.66) holds, where Open image in new window 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 Open image in new window and Open image in new window are as in Theorem 2.12. Suppose further that (2.66) holds, where Open image in new window 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 Open image in new window are defined as in Theorem 2.12, the operator Open image in new window is defined by (1.28), and Open image in new window is defined by (1.29). Then every solution of (1.1) is oscillatory.

If we choose Open image in new window as (2.47), then from Theorem 2.18, we have the following oscillation result.

Corollary 2.19.

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

If we choose Open image in new window as (2.50), then from Theorem 2.18, we have the following oscillation result.

Corollary 2.20.

where Open image in new window , Open image in new window 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.

## Notes

### 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).

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