# Oscillation of Even-Order Neutral Delay Differential Equations

- 1.1k Downloads
- 6 Citations

**Part of the following topical collections:**

## Abstract

By using Riccati transformation technique, we will establish some new oscillation criteria for the even order neutral delay differential equations Open image in new window , Open image in new window , where Open image in new window is even, Open image in new window , Open image in new window , and Open image in new window . These oscillation criteria, at least in some sense, complement and improve those of Zafer (1998) and Zhang et al. (2010). An example is considered to illustrate the main results.

### Keywords

Differential Equation Partial Differential Equation Ordinary Differential Equation Functional Analysis Linear Operator## 1. Introduction

where Open image in new window is even Open image in new window

In what follows we assume that

(*I*_{1}) Open image in new window ,

(*I*_{2}) 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,

(*I*_{3}) Open image in new window and Open image in new window for Open image in new window Open image in new window is a constant.

Neutral differential equations find numerous applications in natural science and technology. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines; see Hale [1].

In the last decades, there are many studies that have been made on the oscillatory behavior of solutions of differential equations [2, 3, 4, 5, 6] and neutral delay differential equations [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

where Open image in new window

where Open image in new window

where Open image in new window

where Open image in new window

where Open image in new window

In 2009, Zhang et al. [23] considered the oscillation of the even-order nonlinear neutral differential equation (1.1) when Open image in new window

To the best of our knowledge, the above oscillation results cannot be applied when Open image in new window and it seems to have few oscillation results for (1.1) when Open image in new window

Motivated by Liu and Bai [13], we will further the investigation and offer some more general new oscillation criteria for (1.1), by employing a class of function Open image in new window and operator Open image in new window The method used in this paper is different from [17].

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

By choosing the special function Open image in new window it is possible to derive several oscillation criteria for a wide range of differential equations.

## 2. Main Results

In this section, we give some new oscillation criteria for (1.1). In order to prove our theorems we will need the following lemmas.

Lemma 2.1 (see [5]).

Let Open image in new window If Open image in new window is eventually of one sign for all large Open image in new window say Open image in new window then there exist a Open image in new window and an integer Open image in new window Open image in new window with Open image in new window even for Open image in new window or Open image in new window odd for Open image in new window such that Open image in new window implies that Open image in new window for Open image in new window and Open image in new window implies that Open image in new window for Open image in new window

Lemma 2.2 (see [5]).

Lemma 2.3 (see [14]).

Theorem 2.4.

Assume that Open image in new window .

where Open image in new window the operator Open image in new window is defined by (1.9), and Open image in new window is defined by (1.10). Then every solution Open image in new window of (1.1) is oscillatory.

Proof.

Let Open image in new window be a nonoscillatory solution of (1.1). Then there exists Open image in new window such that Open image in new window for all Open image in new window Without loss of generality, we assume that Open image in new window for all Open image in new window

where Open image in new window

which contradicts (2.3). This completes the proof.

We get the following new result.

Theorem 2.5.

where Open image in new window is defined as in Theorem 2.4, the operator Open image in new window is defined by (1.9), and Open image in new window is defined by (1.10). Then every solution Open image in new window of (2.20) is oscillatory.

Proof.

Let Open image in new window be a nonoscillatory solution of (2.20). Then there exists Open image in new window such that Open image in new window for all Open image in new window

Then Open image in new window The rest of the proof is similar to that of the proof of Theorem 2.4, hence we omit the details.

Remark 2.6.

By Theorem 2.4 (or Theorem 2.5) we can obtain the oscillation criterion for (1.1) (or (2.20)); the details are left to the reader.

For an application, we give the following example to illustrate the main results.

Example 2.7.

By Theorem 2.5, let Open image in new window one has (2.21). Hence, every solution of (2.27) oscillates. For example, Open image in new window is an oscillatory solution of (2.27).

Remark 2.8.

The recent results cannot be applied in (1.1) and (2.20) when Open image in new window for Open image in new window Therefore, our results are new.

Remark 2.9.

It would be interesting to find another method to study (1.1) and (2.20) when Open image in new window , or Open image in new window for Open image in new window

Remark 2.10.

It would be more interesting to find another method to study (1.1) when Open image in new window is odd.

## Notes

### 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 manuscript. This research is supported by the Natural Science Foundation of China (60774004, 60904024), China Postdoctoral Science Foundation funded Project (20080441126, 200902564), and Shandong Postdoctoral funded project (200802018) by the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), and also by University of Jinan Research Funds for Doctors (B0621, XBS0843).

### References

- 1.Hale J:
*Theory of Functional Differential Equations, Applied Mathematical Sciences, vol. 3*. 2nd edition. Springer, New York, NY, USA; 1977:x+365.CrossRefGoogle Scholar - 2.Džurina J, Stavroulakis IP:
**Oscillation criteria for second-order delay differential equations.***Applied Mathematics and Computation*2003,**140**(2-3):445-453. 10.1016/S0096-3003(02)00243-6MATHMathSciNetCrossRefGoogle Scholar - 3.Grace SR:
**Oscillation theorems for nonlinear differential equations of second order.***Journal of Mathematical Analysis and Applications*1992,**171**(1):220-241. 10.1016/0022-247X(92)90386-RMATHMathSciNetCrossRefGoogle Scholar - 4.Koplatadze R:
**On oscillatory properties of solutions of functional differential equations.***Memoirs on Differential Equations and Mathematical Physics*1994,**3:**1-180.MathSciNetGoogle Scholar - 5.Philos ChG:
**A new criterion for the oscillatory and asymptotic behavior of delay differential equations.***Bulletin de l'Académie Polonaise des Science, Série des Sciences Mathématiques*1981,**39:**61-64.Google Scholar - 6.Sun YG, Meng FW:
**Note on the paper of J. Džurina and I. P. Stavroulakis.***Applied Mathematics and Computation*2006,**174**(2):1634-1641. 10.1016/j.amc.2005.07.008MATHMathSciNetCrossRefGoogle Scholar - 7.Agarwal RP, Grace SR:
**Oscillation theorems for certain neutral functional-differential equations.***Computers & Mathematics with Applications*1999,**38**(11-12):1-11. 10.1016/S0898-1221(99)00280-1MATHMathSciNetCrossRefGoogle Scholar - 8.Han Z, Li T, Sun S, Sun Y:
**Remarks on the paper [Appl. Math. Comput. 207 (2009) 388–396].***Applied Mathematics and Computation*2010,**215**(11):3998-4007. 10.1016/j.amc.2009.12.006MATHMathSciNetCrossRefGoogle Scholar - 9.Han Z, Li T, Sun S, Chen W:
**On the oscillation of second order neutral delay differential equations.***Advances in Difference Equations*2010,**2010:**-8.Google Scholar - 10.Grammatikopoulos MK, Ladas G, Meimaridou A:
**Oscillations of second order neutral delay differential equations.***Radovi Matematički*1985,**1**(2):267-274.MATHMathSciNetGoogle Scholar - 11.Karpuz B, Manojlović JV, Öcalan Ö, Shoukaku Y:
**Oscillation criteria for a class of second-order neutral delay differential equations.***Applied Mathematics and Computation*2009,**210**(2):303-312. 10.1016/j.amc.2008.12.075MATHMathSciNetCrossRefGoogle Scholar - 12.Lin X, Tang XH:
**Oscillation of solutions of neutral differential equations with a superlinear neutral term.***Applied Mathematics Letters*2007,**20**(9):1016-1022. 10.1016/j.aml.2006.11.006MATHMathSciNetCrossRefGoogle Scholar - 13.Liu L, Bai Y:
**New oscillation criteria for second-order nonlinear neutral delay differential equations.***Journal of Computational and Applied Mathematics*2009,**231**(2):657-663. 10.1016/j.cam.2009.04.009MATHMathSciNetCrossRefGoogle Scholar - 14.Meng F, Xu R:
**Oscillation criteria for certain even order quasi-linear neutral differential equations with deviating arguments.***Applied Mathematics and Computation*2007,**190**(1):458-464. 10.1016/j.amc.2007.01.040MATHMathSciNetCrossRefGoogle Scholar - 15.Rath RN, Misra N, Padhy LN:
**Oscillatory and asymptotic behaviour of a nonlinear second order neutral differential equation.***Mathematica Slovaca*2007,**57**(2):157-170. 10.2478/s12175-007-0006-7MATHMathSciNetCrossRefGoogle Scholar - 16.Şahiner Y:
**On oscillation of second order neutral type delay differential equations.***Applied Mathematics and Computation*2004,**150**(3):697-706. 10.1016/S0096-3003(03)00300-XMATHMathSciNetCrossRefGoogle Scholar - 17.Xu R, Xia Y:
**A note on the oscillation of second-order nonlinear neutral functional differential equations.***International Journal of Contemporary Mathematical Sciences*2008,**3**(29–32):1441-1450.MATHMathSciNetGoogle Scholar - 18.Xu R, Meng F:
**New Kamenev-type oscillation criteria for second order neutral nonlinear differential equations.***Applied Mathematics and Computation*2007,**188**(2):1364-1370. 10.1016/j.amc.2006.11.004MATHMathSciNetCrossRefGoogle Scholar - 19.Xu R, Meng F:
**Oscillation criteria for second order quasi-linear neutral delay differential equations.***Applied Mathematics and Computation*2007,**192**(1):216-222. 10.1016/j.amc.2007.01.108MATHMathSciNetCrossRefGoogle Scholar - 20.Xu Z, Liu X:
**Philos-type oscillation criteria for Emden-Fowler neutral delay differential equations.***Journal of Computational and Applied Mathematics*2007,**206**(2):1116-1126. 10.1016/j.cam.2006.09.012MATHMathSciNetCrossRefGoogle Scholar - 21.Ye L, Xu Z:
**Oscillation criteria for second order quasilinear neutral delay differential equations.***Applied Mathematics and Computation*2009,**207**(2):388-396. 10.1016/j.amc.2008.10.051MATHMathSciNetCrossRefGoogle Scholar - 22.Zafer A:
**Oscillation criteria for even order neutral differential equations.***Applied Mathematics Letters*1998,**11**(3):21-25. 10.1016/S0893-9659(98)00028-7MATHMathSciNetCrossRefGoogle Scholar - 23.Zhang Q, Yan J, Gao L:
**Oscillation behavior of even-order nonlinear neutral differential equations with variable coefficients.***Computers and Mathematics with Applications*2010,**59**(1):426-430. 10.1016/j.camwa.2009.06.027MATHMathSciNetCrossRefGoogle Scholar

## Copyright information

This article is published under license to BioMed Central Ltd. 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.