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Advances in Difference Equations

, 2010:289340 | Cite as

On the Oscillation of Second-Order Neutral Delay Differential Equations

  • Zhenlai Han
  • Tongxing Li
  • Shurong Sun
  • Weisong Chen
Open Access
Research Article
Part of the following topical collections:
  1. Abstract Differential and Difference Equations

Abstract

Some new oscillation criteria for the second-order neutral delay differential equation Open image in new window , Open image in new window are established, where Open image in new window , Open image in new window , Open image in new window , Open image in new window . These oscillation criteria extend and improve some known results. An example is considered to illustrate the main results.

Keywords

Differential Equation Partial Differential Equation Ordinary Differential Equation Functional Analysis Functional Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

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 recent years, many studies have been made on the oscillatory behavior of solutions of neutral delay differential equations, and we refer to the recent papers [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and the references cited therein.

This paper is concerned with the oscillatory behavior of the second-order neutral delay differential equation

where Open image in new window

In what follows we assume that

(I1) Open image in new window , Open image in new window , Open image in new window

(I2) Open image in new window

(I3) 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.

Some known results are established for (1.1) under the condition Open image in new window Grammatikopoulos et al. [6] obtained that if Open image in new window and, Open image in new window then the second-order neutral delay differential equation

oscillates. In [13], by employing Riccati technique and averaging functions method, Ruan established some general oscillation criteria for second-order neutral delay differential equation

Xu and Meng [18] as well as Zhuang and Li [23] studied the oscillation of the second-order neutral delay differential equation

Motivated by [11], 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 operator Open image in new window and the Riccati technique and averaging technique.

Following [11], we say that a function Open image in new window belongs to the function class 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 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.

Define the operator Open image in new window by

for Open image in new window and Open image in new window The function Open image in new window is defined by

It is easy to see that Open image in new window is a linear operator and that it satisfies

2. Main Results

In this section, we give some new oscillation criteria for (1.1). We start with the following oscillation criteria.

Theorem.

where Open image in new window then (1.1) oscillates.

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 From (1.1), we have
Therefore Open image in new window is a decreasing function. We claim that Open image in new window for Open image in new window Otherwise, there exists Open image in new window such that Open image in new window Then from (2.2) we obtain
and hence,
Taking Open image in new window we get Open image in new window This contradiction proves that Open image in new window for Open image in new window Using definition of Open image in new window and applying (1.1), we get for sufficiently large Open image in new window
Since Open image in new window for Open image in new window we can find a constant Open image in new window such that Open image in new window for Open image in new window Then from (2.8) and the fact that Open image in new window is eventually decreasing, we have

which is a contradiction to (2.1). This completes the proof.

Theorem 2.2.

where Open image in new window is defined as in Theorem 2.1, the operator Open image in new window is defined by (1.5), and Open image in new window is defined by (1.6). 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 , Open image in new window , and Open image in new window for all Open image in new window Define
By (2.2) and the fact Open image in new window we get
From (2.11), (2.12), and (2.13), we have
Similarly, define
By (2.2) and the facting Open image in new window noting that Open image in new window we get
From (2.15), (2.16), and (2.17), we have
Therefore, from (2.14) and (2.18), we get
From (2.6), we obtain
Applying Open image in new window to (2.20), we get
By (1.7) and the above inequality, we obtain
Hence, from (2.22) we have
Taking the super limit in the above inequality, we get

which contradicts (2.10). This completes the proof.

Remark 2.3.

With the different choice of Open image in new window and Open image in new window Theorem 2.2 can be stated with different conditions for oscillation of (1.1). For example, if we choose Open image in new window for Open image in new window , Open image in new window , Open image in new window then

By Theorem 2.2 we can obtain the oscillation criterion for (1.1), the details are left to the reader.

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

Example 2.4.

Consider the following equation:

Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window then by Theorem 2.1 every solution of (2.27) oscillates; for example, Open image in new window is an oscillatory solution of (2.27).

Remark 2.5.

The recent results cannot be applied in (2.27) since Open image in new window so our results are new ones.

Notes

Acknowledgments

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 the Natural Scientific Foundation of Shandong Province (Y2008A28, ZR2009AL003), also supported by University of Jinan Research Funds for Doctors (XBS0843).

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Copyright information

© Zhenlai Han et al. 2010

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.

Authors and Affiliations

  • Zhenlai Han
    • 1
    • 2
  • Tongxing Li
    • 1
  • Shurong Sun
    • 1
    • 3
  • Weisong Chen
    • 1
  1. 1.School of ScienceUniversity of JinanJinanChina
  2. 2.School of Control Science and EngineeringShandong UniversityJinanChina
  3. 3.Department of Mathematics and StatisticsMissouri University of Science and TechnologyRollaChina

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