- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2011 (2011), Article ID 387483, 9 pages

http://dx.doi.org/10.1155/2011/387483

## Oscillation Criteria for Certain Second-Order Nonlinear Neutral Differential Equations of Mixed Type

^{1}School of Science, University of Jinan, Jinan, Shandong 250022, China^{2}School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China

Received 19 September 2010; Accepted 19 January 2011

Academic Editor: Josef Diblík

Copyright © 2011 Zhenlai Han 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

Some oscillation criteria are established for the second-order nonlinear neutral differential equations of mixed type , , where is a quotient of odd positive integers. Our results generalize the results given in the literature.

#### 1. Introduction

This paper is concerned with the oscillatory behavior of the second-order nonlinear neutral differential equation of mixed type

Throughout this paper, we will assume the following conditions hold.(A_{1}), , and , , are positive constants;(A_{2}),
.

By a solution of (1.1), we mean a function for some which has the property that and satisfies (1.1) on . As is customary, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros on , otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

Neutral functional differential equations have numerous applications in electric networks. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines which rise in high speed computers where the lossless transmission lines are used to interconnect switching circuits; see [1].

Recently, many results have been obtained on oscillation of nonneutral continuous and discrete equations and neutral functional differential equations, we refer the reader to the papers [2–35], and the references cited therein.

Philos [2] established some Philos-type oscillation criteria for the second-order linear differential equation

In [3–5], the authors gave some sufficient conditions for oscillation of all solutions of second-order half-linear differential equation by employing a Riccati substitution technique.

Zhang et al. [15] examined the oscillation of even-order neutral differential equation

Some oscillation criteria for the following second-order quasilinear neutral differential equation were obtained by [12–17].

However, there are few results regarding the oscillatory properties of neutral differential equations with mixed arguments, see the papers [20–24]. In [25], the authors established some oscillation criteria for the following mixed neutral equation: here and are nonnegative real-valued functions. Grace [26] obtained some oscillation theorems for the odd order neutral differential equation where is odd. Grace [27] and Yan [28] obtained several sufficient conditions for the oscillation of solutions of higher-order neutral functional differential equation of the form where and are nonnegative real constants.

Clearly, (1.6) is a special case of (1.1). The purpose of this paper is to study the oscillation behavior of (1.1).

In the sequel, when we write a functional inequality without specifying its domain of validity we assume that it holds for all sufficiently large .

#### 2. Main Results

In the following, we give our results.

Theorem 2.1. *Assume that , . If
**
where
**
for , then every solution of (1.1) oscillates.*

*Proof. *Let be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists such that , ,, , and for all . Setting
Thus , , and
Then, is of constant sign, eventually. On the other hand,
Note that , , is a convex function. Hence, by the definition of convex function, we obtain
Using inequality (2.7), we get
Similarly, we obtain
Thus, from (2.6), we have

In the following, we consider two cases.*Case 1. *Assume that . Then, . In view of (2.10), we see that
Applying the monotonicity of , we find
Combining the last two inequalities, we obtain the inequality
Therefore, is a positive increasing solution of the differential inequality
However, by [11], condition (2.1) contradicts the existence of a positive increasing solution of inequality (2.14).*Case 2. *Assume that . Then, . In view of (2.10), we see that
Applying the monotonicity of , we find
Combining the last two inequalities, we obtain the inequality
Therefore, is a positive decreasing solution of the differential inequality
However, by [11], condition (2.2) contradicts the existence of a positive decreasing solution of inequality (2.18).

*Remark 2.2. *When , Theorem 2.1 involves results of [25, Theorem 1].

Theorem 2.3. *Let , . Suppose that, for , there exist functions
**
such that
**
where are as in (2.3) for . If the first-order differential inequality
**
has no eventually negative solution for and no eventually positive solution for , then (1.1) is oscillatory.*

*Proof. *Let be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists such that , , , , and for all . Define and as in Theorem 2.1. Proceeding as in the proof of Theorem 2.1, we get (2.10).

In the following, we consider two cases.*Case 1. *Assume that . Clearly, . Then, just as in Case 1 of Theorem 2.1, we find that is a positive increasing solution of inequality (2.14). Let . Then . Using (2.19) and (2.20), we obtain
Define . Then, is a positive solution of (2.21) for , which is a contradiction.*Case 2. *Assume that . Clearly, . Then, just as in Case 2 of Theorem 2.1, we find that is a positive decreasing solution of inequality (2.18). Let . Then . Using (2.19) and (2.20), we obtain
Define . Then, is a negative solution of (2.21) for . This contradiction completes the proof of the theorem.

*Remark 2.4. *When , Theorem 2.3 involves results of [25, Theorem 2].

From Theorem 2.3 and the results given in [12], we have the following oscillation criterion for (1.1).

Corollary 2.5. *Let , . Assume that (2.19) and (2.20) hold for . If
**
then (1.1) is oscillatory.*

*Proof. *It is known (see [12]) that condition (2.24) is sufficient for inequality (2.21) (for ) to have no eventually negative solution. On the other hand, condition (2.25) is sufficient for inequality (2.21) (for ) to have no eventually positive solution.

For an application of our results, we give the following example.

*Example 2.6. *Consider the second-order differential equation
where are constants and for .

It is easy to see that , . Assume that . Let , . Clearly, (2.19) holds. If
for , then (2.20) holds. Moreover, we see that
Hence by applying Corollary 2.5, we find that (2.26) is oscillatory.

#### Acknowledgments

The authors sincerely thank the referees for their constructive suggestions which improved the content of the paper. This research is supported by the Natural Science Foundation of China (11071143, 60904024, 11026112), China Postdoctoral Science Foundation funded project (200902564), and Shandong Provincial Natural Science Foundation (ZR2010AL002, ZR2009AL003, Y2008A28); it was also supported by University of Jinan Research Funds for Doctors (XBS0843).

#### References

- J. Hale,
*Theory of Functional Differential Equations*, vol. 3 of*Applied Mathematical Sciences*, Springer, New York, NY, USA, 2nd edition, 1977. View at Zentralblatt MATH - Ch. G. Philos, “Oscillation theorems for linear differential equations of second order,”
*Archiv der Mathematik*, vol. 53, no. 5, pp. 482–492, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. P. Agarwal, S.-L. Shieh, and C.-C. Yeh, “Oscillation criteria for second-order retarded differential equations,”
*Mathematical and Computer Modelling*, vol. 26, no. 4, pp. 1–11, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Džurina and I. P. Stavroulakis, “Oscillation criteria for second-order delay differential equations,”
*Applied Mathematics and Computation*, vol. 140, no. 2-3, pp. 445–453, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Baštinec, L. Berezansky, J. Diblík, and Z. Šmarda, “On the critical case in oscillation for differential equations with a single delay and with several delays,”
*Abstract and Applied Analysis*, vol. 2010, Article ID 417869, 20 pages, 2010. - J. Baštinec, J. Diblík, and Z. Šmarda, “Oscillation of solutions of a linear second-order discrete-delayed equation,”
*Advances in Difference Equations*, vol. 2010, Article ID 693867, 12 pages, 2010. View at Zentralblatt MATH - J. Diblík, Z. Svoboda, and Z. Šmarda, “Explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case,”
*Computers & Mathematics with Applications*, vol. 56, no. 2, pp. 556–564, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Berezansky, J. Diblík, and Z. Šmarda, “Positive solutions of second-order delay differential equations with a damping term,”
*Computers & Mathematics with Applications*, vol. 60, no. 5, pp. 1332–1342, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. G. Sun and F. W. Meng, “Note on the paper of J. Džurina and I. P. Stavroulakis,”
*Applied Mathematics and Computation*, vol. 174, no. 2, pp. 1634–1641, 2006. View at Publisher · View at Google Scholar - B. Baculíková, “Oscillation criteria for second order nonlinear differential equations,”
*Archivum Mathematicum*, vol. 42, no. 2, pp. 141–149, 2006. View at Zentralblatt MATH - R. G. Koplatadze and T. A. Chanturiya,
*Ob ostsillyatsionnykh svoistvakh differentsialnykh uravnenii s otklonyayushchimsya argumentom (Oscillatory Properties of Differential Equations with Deviating Argument)*, Izdat. Tbilis. Univ., Tbilisi, Georgia, 1977. - G. S. Ladde, V. Lakshmikantham, and B. G. Zhang,
*Oscillation Theory of Differential Equations with Deviating Arguments*, vol. 110 of*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 1987. - J. Diblík, Z. Svoboda, and Z. Šmarda, “Retract principle for neutral functional differential equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 12, pp. e1393–e1400, 2009. View at Publisher · View at Google Scholar - L. H. Erbe and Q. Kong, “Oscillation results for second order neutral differential equations,”
*Funkcialaj Ekvacioj*, vol. 35, no. 3, pp. 545–555, 1992. View at Zentralblatt MATH - Q. Zhang, J. Yan, and L. Gao, “Oscillation behavior of even-order nonlinear neutral differential equations with variable coefficients,”
*Computers & Mathematics with Applications*, vol. 59, no. 1, pp. 426–430, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Q. Wang, “Oscillation theorems for first-order nonlinear neutral functional differential equations,”
*Computers & Mathematics with Applications*, vol. 39, no. 5-6, pp. 19–28, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Han, T. Li, S. Sun, and Y. Sun, “Remarks on the paper [Appl. Math. Comput. 207 (2009) 388–396],”
*Applied Mathematics and Computation*, vol. 215, no. 11, pp. 3998–4007, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Liu and Y. Bai, “New oscillation criteria for second-order nonlinear neutral delay differential equations,”
*Journal of Computational and Applied Mathematics*, vol. 231, no. 2, pp. 657–663, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Xu and F. Meng, “Oscillation criteria for second order quasi-linear neutral delay differential equations,”
*Applied Mathematics and Computation*, vol. 192, no. 1, pp. 216–222, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J.-G. Dong, “Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments,”
*Computers & Mathematics with Applications*, vol. 59, no. 12, pp. 3710–3717, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Džurina and D. Hudáková, “Oscillation of second order neutral delay differential equations,”
*Mathematica Bohemica*, vol. 134, no. 1, pp. 31–38, 2009. - M. Hasanbulli and Y. V. Rogovchenko, “Oscillation criteria for second order nonlinear neutral differential equations,”
*Applied Mathematics and Computation*, vol. 215, no. 12, pp. 4392–4399, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Baculíková and J. Džurina, “Oscillation of third-order neutral differential equations,”
*Mathematical and Computer Modelling*, vol. 52, no. 1-2, pp. 215–226, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. H. Saker, “Oscillation of second order neutral delay differential equations of Emden-Fowler type,”
*Acta Mathematica Hungarica*, vol. 100, no. 1-2, pp. 37–62, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Dzurina, J. Busha, and E. A. Airyan, “Oscillation criteria for second-order differential equations of neutral type with mixed arguments,”
*Differential Equations*, vol. 38, no. 1, pp. 137–140, 2002. View at Publisher · View at Google Scholar - S. R. Grace, “On the oscillations of mixed neutral equations,”
*Journal of Mathematical Analysis and Applications*, vol. 194, no. 2, pp. 377–388, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. R. Grace, “Oscillations of mixed neutral functional-differential equations,”
*Applied Mathematics and Computation*, vol. 68, no. 1, pp. 1–13, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Yan, “Oscillations of higher order neutral differential equations of mixed type,”
*Israel Journal of Mathematics*, vol. 115, pp. 125–136, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Wang, “A necessary and sufficient condition for the oscillation of higher-order neutral equations,”
*The Tôhoku Mathematical Journal*, vol. 41, no. 4, pp. 575–588, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Han, T. Li, S. Sun, and W. Chen, “On the oscillation of second-order neutral delay differential equations,”
*Advances in Difference Equations*, vol. 2010, Article ID 289340, 8 pages, 2010. View at Zentralblatt MATH - Z. Han, T. Li, S. Sun, C. Zhang, and B. Han, “Oscillation criteria for a class of second order neutral delay dynamic equations of Emden-Fowler type,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 653689, 26 pages, 2011. View at Publisher · View at Google Scholar - T. Li, Z. Han, P. Zhao, and S. Sun, “Oscillation of even-order neutral delay differential equations,”
*Advances in Difference Equations*, vol. 2010, Article ID 184180, 9 pages, 2010. - Z. Han, T. Li, S. Sun, and C. Zhang, “An oscillation criteria for third order neutral delay differential equations,”
*Journal of Applied Analysis*, vol. 16, no. 2, pp. 295–303, 2010. - Z. Han, T. Li, S. Sun, and W. Chen, “Oscillation criteria for second-order nonlinear neutral delay differential equations,”
*Advances in Difference Equations*, vol. 2010, Article ID 763278, 23 pages, 2010. - S. Sun, T. Li, Z. Han, and Y. Sun, “Oscillation of second-order neutral functional differential equations with mixed nonlinearities,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 927690, 15 pages, 2011. View at Publisher · View at Google Scholar