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On the Oscillation of Second-Order Neutral Delay Differential Equations

Abstract

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

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 [223] and the references cited therein.

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

(11)

where

In what follows we assume that

(I1), ,

(I2)

(I3), , , , , where is a constant.

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

(12)

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

(13)

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

(14)

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 operator and the Riccati technique and averaging technique.

Following [11], we say that a function belongs to the function class denoted by if where which satisfies for and has the partial derivative on such that is locally integrable with respect to in By choosing the special function it is possible to derive several oscillation criteria for a wide range of differential equations.

Define the operator by

(15)

for and The function is defined by

(16)

It is easy to see that is a linear operator and that it satisfies

(17)

2. Main Results

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

Theorem.

If

(21)

where then (1.1) oscillates.

Proof.

Let be a nonoscillatory solution of (1.1). Then there exists such that for all Without loss of generality, we assume that for all From (1.1), we have

(22)

Therefore is a decreasing function. We claim that for Otherwise, there exists such that Then from (2.2) we obtain

(23)

and hence,

(24)

Taking we get This contradiction proves that for Using definition of and applying (1.1), we get for sufficiently large

(25)

and thus,

(26)

Integrating (2.6) from to we obtain

(27)

Noting that we have

(28)

Since for we can find a constant such that for Then from (2.8) and the fact that is eventually decreasing, we have

(29)

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

Theorem 2.2.

Assume that and there exist functions and such that

(210)

where is defined as in Theorem 2.1, the operator is defined by (1.5), and is defined by (1.6). Then every solution of (1.1) is oscillatory.

Proof.

Let be a nonoscillatory solution of (1.1). Then there exists such that for all Without loss of generality, we assume that , , and for all Define

(211)

Then and

(212)

By (2.2) and the fact we get

(213)

From (2.11), (2.12), and (2.13), we have

(214)

Similarly, define

(215)

Then and

(216)

By (2.2) and the facting noting that we get

(217)

From (2.15), (2.16), and (2.17), we have

(218)

Therefore, from (2.14) and (2.18), we get

(219)

From (2.6), we obtain

(220)

Applying to (2.20), we get

(221)

By (1.7) and the above inequality, we obtain

(222)

Hence, from (2.22) we have

(223)

that is,

(224)

Taking the super limit in the above inequality, we get

(225)

which contradicts (2.10). This completes the proof.

Remark 2.3.

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

(226)

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:

(227)

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

Remark 2.5.

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

References

  1. 1.

    Hale J: Theory of Functional Differential Equations. 2nd edition. Springer, New York, NY, USA; 1977:x+365. Applied Mathematical Sciences

    Book  MATH  Google Scholar 

  2. 2.

    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)00280-1

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Berezansky L, Diblik J, Šmarda Z: On connection between second-order delay differential equations and integrodifferential equations with delay. Advances in Difference Equations 2010, 2010:-8.

    Google Scholar 

  4. 4.

    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)00243-6

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    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-R

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Grammatikopoulos MK, Ladas G, Meimaridou A: Oscillations of second order neutral delay differential equations. Radovi Matematički 1985,1(2):267-274.

    MathSciNet  MATH  Google Scholar 

  7. 7.

    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.2009.12.006

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    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.2008.12.075

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Li H-J, Yeh C-C: Oscillation criteria for second-order neutral delay difference equations. Computers & Mathematics with Applications 1998,36(10-12):123-132. 10.1016/S0898-1221(98)80015-1

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    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.2006.11.006

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    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.2009.04.009

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    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/s12175-007-0006-7

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Ruan SG: Oscillations of second order neutral differential equations. Canadian Mathematical Bulletin 1993,36(4):485-496. 10.4153/CMB-1993-064-4

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Sun YG, Meng F: Note on the paper of Dourina and Stavroulakis. Applied Mathematics and Computation 2006,174(2):1634-1641. 10.1016/j.amc.2005.07.008

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    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-X

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Xu R, Meng F: Some new oscillation criteria for second order quasi-linear neutral delay differential equations. Applied Mathematics and Computation 2006,182(1):797-803. 10.1016/j.amc.2006.04.042

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    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.2007.01.108

    MathSciNet  Article  MATH  Google Scholar 

  18. 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.2006.11.004

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    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.2006.09.012

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    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.2008.10.051

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Zafer A: Oscillation criteria for even order neutral differential equations. Applied Mathematics Letters 1998,11(3):21-25. 10.1016/S0893-9659(98)00028-7

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    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.2009.06.027

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Zhuang R-K, Li W-T: Interval oscillation criteria for second order neutral nonlinear differential equations. Applied Mathematics and Computation 2004,157(1):39-51. 10.1016/j.amc.2003.06.016

    MathSciNet  Article  MATH  Google Scholar 

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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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Correspondence to Zhenlai Han.

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Han, Z., Li, T., Sun, S. et al. On the Oscillation of Second-Order Neutral Delay Differential Equations. Adv Differ Equ 2010, 289340 (2010). https://doi.org/10.1155/2010/289340

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation