About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 281581, 5 pages
http://dx.doi.org/10.1155/2013/281581
Research Article

Oscillation Criteria for Linear Neutral Delay Differential Equations of First Order

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 10 May 2013; Revised 11 August 2013; Accepted 12 August 2013

Academic Editor: Aref Jeribi

Copyright © 2013 Fatima N. Ahmed 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 new sufficient conditions for oscillation of all solutions of the first-order linear neutral delay differential equations are obtained. Our new results improve many well-known results in the literature. Some examples are inserted to illustrate our results.

1. Introduction

A neutral delay differential equation (NDDE) is a differential equation in which the highest-order derivative of the unknown function is evaluated both at the present state at time and at the past state at time for some positive constant .

In the last two decades, there has been an increasing interest in obtaining sufficient conditions for the oscillation and/or nonoscillation of solutions of neutral delay differential equations. Particularly, we mention the papers by Ladas and Sficas [1], Chuanxi and Ladas [2], Ruan [3], Elabbasy and Saker [4], Kulenović et al. [5], and Karpuz and Öcalan [6] who investigated NDDEs with variable coefficients. To a large extent, this is due to its theoretical interest as well as to its importance in applications. It suffices to note that NDDEs appear in the study of networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits) in population dynamics and also in many applications in epidemics and infection diseases. We refer reader to [118] for relevant studies on this subject.

In this paper, we consider the linear first-order NDDE of the type where   and . When   and   is a constant, Jaroš [9] established some new oscillation conditions for all solutions of (1), and his technique was based on the study of the characteristic equation

Zhang [19], Ladas and Sficas [1], Grammatikopoulos et al. [10], and Yu et al. [8] considered (1) when , and they obtained some sufficient conditions for oscillation of (1). The purpose of this work is to present some new sufficient conditions under which all solutions of (1) are oscillatory. In order to achieve this object, we are first concerned with NDDE (1) with constant coefficients (when   is a constant). That is,

Some illustrating examples are given. In some sense, the established results extend and improve some previous investigations such as [1, 810, 19].

As usual, a solution of (1) is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory if it is eventually positive or eventually negative. A function     is called eventually positive (or negative) if there exists    such that     (or  ) for all. Equation (1) is called oscillatory if all its solutions are oscillatory; otherwise, it is called nonoscillatory.

2. Main Results

In this section, we give some new sufficient conditions for the oscillation of all solutions of (1) and (3). This is done by using the following well-known lemmas which are from [11, 12].

Lemma 1. Consider the NDDE where  ,   and .
Let   be a positive solution of (4). Set
If , then is a positive and decreasing solution of (4); that is,

Lemma 2. Let and   be positive constants. Let    be an eventually positive solution of the delay differential inequality
Then for sufficiently large, where

Our main results can now be given as follows.

Theorem 3. Consider NDDE (3). Assume that (i), and(ii),
where is the unique real root of the equation
Then all solutions of (3) are oscillatory.

Proof. Assume, for the sake of a contradiction, that (3) has a nonoscillatory solution  . Without loss of generality, assume that  . Let So that is also a positive solution of (3).
That is, where
Set for
Thus it follows from Lemma 1 that    is a positive and decreasing solution of and in particular (as implies that .), it follows that
But we have
This implies that
Applying Lemma 2 with (18) we get
Then    is bounded.
Dividing (16) by   and integrating from    to  , we get
Let .
Then, it follows from (20) that for and sufficiently small,
As is arbitrary, so we have
Let
Then
Let be the unique real root of the equation
Then
Hence
This contradicts condition (ii) and then completes the proof.

Example 4. Consider the NDDE
We note that
Then we have (i)  ,  (ii)  where    is the unique real root of the equation Then all the hypotheses of Theorem 3 are satisfied, and therefore every solution of (28) oscillates. (Indeed    is such a solution.)

Theorem 5. Consider the NDDE (1). Assume that (iii)  ,  and   is periodic with period  , (iv)  ,where    is defined as in Theorem 3. Then all solutions of (1) are oscillatory.

Proof. Assume, for the sake of contradiction, that (1) has a nonoscillatory solution . Without loss of generality, assume that . Let which is oscillation invariant transformation. Then is a positive solution of the equation where     is periodic with period .
Let
Then   is decreasing positive solution of the equation
Set
This implies that  , since  .
Dividing both sides of (33) by and then integrating from to  , we obtain that
Hence
Since is periodic with period  , then we obtain
Substituting in (38) we find, for all ,
Now, we want to prove that   is bounded.
Applying the assumption (iv), we can find such that where is similar as in the proof of Theorem 3.
Integrating (33) from to we obtain
Using Bonnet’s Theorem and in particular (as ), we get
Integrating (33) from to , we get
Using Bonnet’s Theorem and in particular (as ), we get
Combining (43) and (45), we conclude or
Then    is bounded.
Now, let
But we have proved that   is bounded; that is,   is finite.
From (40), we obtain
Therefore, we get
Hence
This contradicts our assumption (iv) and then completes the proof.

Example 6. Consider the NDDE where
Then we have(1); (2) is periodic with period and satisfies
where is the unique real root of the equation
Therefore (52) satisfies all the hypotheses of Theorem 5. Hence every solution of this equation is oscillatory.

Theorem 7. Suppose that condition (iii) holds. If (v)  ,
then every solution of (1) is oscillatory.

Proof. Proceeding as in the proof of Theorem 5, we get (49) which implies that
Hence
But this is a contradiction of assumption (v), and then the proof is complete.

Example 8. Consider the NDDE
Here we have
Note that   is positive and periodic with period , and also(1), (2)
Then (58) satisfies hypotheses of Theorem 7, and so all its solutions are oscillatory.

Funding

This research has been completed with the support of these Grants: ukm-DLP-2011-049, DIP-2012-31 and FRGS/1/2012/SG04/ukm/01/1.

References

  1. G. Ladas and Y. G. Sficas, “Oscillations of neutral delay differential equations,” Canadian Mathematical Bulletin, vol. 29, no. 4, pp. 438–445, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  2. Q. Chuanxi and G. Ladas, “Oscillations of first-order neutral equations with variable coefficients,” Monatshefte für Mathematik, vol. 109, no. 2, pp. 103–111, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  3. S. G. Ruan, “Oscillations for first order neutral differential equations with variable coefficients,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 147–152, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  4. E. M. Elabbasy and S. H. Saker, “Oscillation of delay differential equation with several positive and negative coefficients,” Discussiones Mathematicae, vol. 23, pp. 39–52, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  5. M. R. S. Kulenović, G. Ladas, and A. Meimaridou, “Necessary and sufficient condition for oscillations of neutral differential equations,” Australian Mathematical Society Journal B, vol. 28, no. 3, pp. 362–375, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  6. B. Karpuz and O. Öcalan, “Oscillation criteria for some classes of linear delay differential equations of first-order,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 3, no. 2, pp. 293–314, 2008. View at MathSciNet
  7. I. R. Al-Amri, “On the oscillation of first-order neutral delay differential equations with real coefficients,” International Journal of Mathematics and Mathematical Sciences, vol. 29, no. 4, pp. 245–249, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. S. Yu, Z. Wang, and C. X. Qian, “Oscillation of neutral delay differential equations,” Bulletin of the Australian Mathematical Society, vol. 45, no. 2, pp. 195–200, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J. Jaroš, “On characterization of oscillations in first-order linear neutral differential equations,” Funkcialaj Ekvacioj, vol. 34, no. 2, pp. 331–342, 1991. View at MathSciNet
  10. M. K. Grammatikopoulos, E. A. Grove, and G. Ladas, “Oscillations of first-order neutral delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 120, no. 2, pp. 510–520, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  11. I. Győri and G. Ladas, Oscillation Theory of Delay Differential Equations, Oxford Mathematical Monographs, The Clarendon Press, New York, NY, USA, 1991. View at MathSciNet
  12. I. Kubiaczyk and S. H. Saker, “Oscillation of solutions to neutral delay differential equations,” Mathematica Slovaca, vol. 52, no. 3, pp. 343–359, 2002. View at MathSciNet
  13. R. D. Driver, “A mixed neutral system,” Nonlinear Analysis. Theory, Methods & Applications, vol. 8, no. 2, pp. 155–158, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  14. E. M. Elabbasy and S. H. Saker, “Oscillation of first order neutral delay differential equations,” Kyungpook Mathematical Journal, vol. 41, no. 2, pp. 311–321, 2001. View at MathSciNet
  15. K. Farrell, E. A. Grove, and G. Ladas, “Neutral delay differential equations with positive and negative coefficients,” Applicable Analysis, vol. 27, no. 1–3, pp. 181–197, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  16. M. K. Grammatikopoulos, E. A. Grove, and G. Ladas, “Oscillation and asymptotic behavior of neutral differential equations with deviating arguments,” Applicable Analysis, vol. 22, no. 1, pp. 1–19, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  17. J. K. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 1977. View at MathSciNet
  18. J. S. Yu, M.-P. Chen, and H. Zhang, “Oscillation and nonoscillation in neutral equations with integrable coefficients,” Computers & Mathematics with Applications, vol. 35, no. 6, pp. 65–71, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  19. B. G. Zhang, “Oscillation of first order neutral functional-differential equations,” Journal of Mathematical Analysis and Applications, vol. 139, no. 2, pp. 311–318, 1989. View at Publisher · View at Google Scholar · View at MathSciNet