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Abstract and Applied Analysis

Volume 2011 (2011), Article ID 927690, 15 pages

http://dx.doi.org/10.1155/2011/927690

## Oscillation of Second-Order Neutral Functional Differential Equations with Mixed Nonlinearities

^{1}School of Science, University of Jinan, Jinan, Shandong 250022, China^{2}Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA^{3}School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China

Received 2 September 2010; Revised 26 November 2010; Accepted 23 December 2010

Academic Editor: Miroslava Ružicková

Copyright © 2011 Shurong Sun et al. 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.

#### Abstract

We study the following second-order neutral functional differential equation with mixed nonlinearities , where , . Oscillation results for the equation are established which improve the results obtained by Sun and Meng (2006), Xu and Meng (2006), Sun and Meng (2009), and Han et al. (2010).

#### 1. Introduction

This paper is concerned with the oscillatory behavior of the second-order neutral functional differential equation with mixed nonlinearities where are constants, , , , , are nonnegative, is a constant. Here, we assume that there exists such that , , , and for .

One of our motivations for studying (1.1) is the application of this type of equations in real word life problems. For instance, neutral delay equations appear in modeling of networks containing lossless transmission lines, in the study of vibrating masses attached to an elastic bar; see the Euler equation in some variational problems, in the theory of automatic control and in neuromechanical systems in which inertia plays an important role. We refer the reader to Hale [1] and Driver [2], and references cited therein.

Recently, there has been much research activity concerning the oscillation of second-order differential equations [3–8] and neutral delay differential equations [9–20]. For the particular case when , (1.1) reduces to the following equation:

Sun and Meng [6] established some oscillation criteria for (1.2), under the condition they only obtained the sufficient condition [6, Theorem 5], which guarantees that every solution of (1.2) oscillates or tends to zero.

Sun and Meng [7] considered the oscillation of second-order nonlinear delay differential equation and obtained some results for oscillation of (1.4), for example, under the case (1.3), they obtained some results which guarantee that every solution of (1.4) oscillates or tends to zero, see [7, Theorem 2.2].

Xu and Meng [10] discussed the oscillation of the second-order neutral delay differential equation and established the sufficient condition [10, Theorem 2.3], which guarantees that every solution of (1.5) oscillates or tends to zero.

Han et al. [11] examined the oscillation of second-order neutral delay differential equation and established some sufficient conditions for oscillation of (1.6) under the conditions (1.3) and The condition (1.7) can be restrictive condition, since the results cannot be applied on the equation

The aim of this paper is to derive some sufficient conditions for the oscillation of solutions of (1.1). The paper is organized as follows. In Section 2, we establish some oscillation criteria for (1.1) under the assumption (1.3). In Section 3, we will give three examples to illustrate the main results. In Section 4, we give some conclusions for this paper.

#### 2. Main Results

In this section, we give some new oscillation criteria for (1.1).

Below, for the sake of convenience, we denote

Theorem 2.1. *Assume that (1.3) holds, , and there exists , such that , , ,. If for all sufficiently large ,
**
then (1.1) is oscillatory.*

*Proof. *Suppose to the contrary that is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that for all large . The case of can be considered by the same method. From (1.1) and (1.3), we can easily obtain that there exists a such that
or
If (2.4) holds, we have
From the definition of , we obtain
Define
Then, for . Noting that , we get for . Thus, from (1.1), (2.7), and (2.8), it follows that
By (2.4), (2.9), and , we get
In view of (2.4), (2.6), and (2.10), we have
By (2.4), we obtain
that is,
Set
Using Young’s inequality
we have
Hence, by (2.11), (2.13), and (2.16), we obtain
Integrating (2.17) from to , we get
Letting in (2.19), we get a contradiction to (2.2). If (2.5) holds, we define the function by
Then, for . It follows from that is nonincreasing. Thus, we have
Dividing (2.21) by and integrating it from to , we obtain
Letting in the above inequality, we obtain
that is,
Hence, by (2.20), we have
Differentiating (2.20), we get
by the above equality and (1.1), we obtain
Noticing that , from [10, Theorem 2.3], we see that for , so by , , we have
On the other hand, from , , we obtain
Thus, by (2.20) and (2.27), we get
Set
Using Young’s inequality (2.15), we obtain
Hence, from (2.30), we have
that is,
Multiplying (2.34) by and integrating it from to implies that
Set , , and
Using Young's inequality (2.15), we get
Thus,
Therefore, (2.35) yields
Letting in the above inequality, by (2.3), we get a contradiction with (2.25). This completes the proof of Theorem 2.1.

From Theorem 2.1, when , we have the following result.

Corollary 2.2. *Assume that (1.3) holds, , and , . If for all sufficiently large such that (2.2) holds and
**
then (1.1) is oscillatory.*

Theorem 2.3. *Assume that (1.3) holds, , and there exists , such that , , , . If for all sufficiently large such that (2.2) holds and
**
then (1.1) is oscillatory. *

*Proof. *Suppose to the contrary that is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that for all large . The case of can be considered by the same method. From (1.1) and (1.3), we can easily obtain that there exists a such that (2.4) or (2.5) holds.

If (2.4) holds, proceeding as in the proof of Theorem 2.1, we obtain a contradiction with (2.2).

If (2.5) holds, we proceed as in the proof of Theorem 2.1, then we get (2.25) and (2.34). Multiplying (2.34) by and integrating it from to implies that
In view of (2.25), we have , . From (1.3), we get
Letting in (2.42) and using the last inequalities, we obtain
which contradicts (2.41). This completes the proof of Theorem 2.3.

From Theorem 2.3, when , we have the following result.

Corollary 2.4. *Assume that (1.3) holds, , , . If for all sufficiently large such that (2.2) holds and
**
then (1.1) is oscillatory.*

Theorem 2.5. *Assume that (1.3) holds, , and there exists , such that , , , . If for all sufficiently large such that (2.2) holds and
**
then (1.1) is oscillatory. *

*Proof. *Suppose to the contrary that is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that for all large . The case of can be considered by the same method. From (1.1) and (1.3), we can easily obtain that there exists a such that (2.4) or (2.5) holds.

If (2.4) holds, proceeding as in the proof of Theorem 2.1, we obtain a contradiction with (2.2).

If (2.5) holds, we proceed as in the proof of Theorem 2.1, and we get (2.21). Dividing (2.21) by and integrating it from to , letting , yields
By (1.1), we have
Noticing that , from [10, Theorem 2.3], we see that for , so by , , we get
Hence, we obtain
where . Integrating the above inequality from to , we have
Integrating the above inequality from to , we obtain
which contradicts (2.46). This completes the proof of Theorem 2.5.

#### 3. Examples

In this section, three examples are worked out to illustrate the main results.

*Example 3.1. *Consider the second-order neutral delay differential equation (1.8), where is a constant.

Let , , , , , , , and , then

Setting , we have , . Therefore, for all sufficiently large ,
if . Hence, by Theorem 2.1, (1.8) is oscillatory when .

Note that [11, Theorem 2.1] and [11, Theorem 2.2] cannot be applied in (1.8), since . On the other hand, applying [11, Theorem 3.2] to that (1.8), we obtain that (1.8) is oscillatory if . So our results improve the results in [11].

*Example 3.2. *Consider the second-order neutral delay differential equation
Let , , , , ,, , and , then
Therefore, for all sufficiently large ,
Hence, by Theorem 2.1, (3.3) oscillates. For example, is a solution of (3.3).

*Example 3.3. *Consider the second-order neutral differential equation
where , for , are constants, , for .

Let , , , , , , , , , , and , then ,
It is easy to see that (2.2) and (2.41) hold for all sufficiently large . Hence, by Theorem 2.3, (3.6) is oscillatory.

#### 4. Conclusions

In this paper, we consider the oscillatory behavior of second-order neutral functional differential equation (1.1). Our results can be applied to the case when , ; these results improve the results given in [6, 7, 10, 11].

#### Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. This research is supported by the Natural Science Foundation of China (nos. 11071143, 60904024, 11026112), China Postdoctoral Science Foundation funded Project (no. 200902564), the Natural Science Foundation of Shandong (nos. ZR2010AL002, ZR2009AL003, Y2008A28), and also the University of Jinan Research Funds for Doctors (no. XBS0843).

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