`Abstract and Applied AnalysisVolumeΒ 2011Β (2011), Article IDΒ 927690, 15 pagesdoi:10.1155/2011/927690`
Research Article

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

Shurong Sun,1,2Β Tongxing Li,1,3Β Zhenlai Han,1,3Β and Yibing Sun1

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

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

Copyright Β© 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 [38] and neutral delay differential equations [920]. 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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