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Notes on oscillation of linear delay differential equations
- Božena Dorociaková^{1}Email authorView ORCID ID profile,
- Radoslav Chupáč^{1} and
- Rudolf Olach^{1}
https://doi.org/10.1186/s13662-018-1608-9
© The Author(s) 2018
Received: 17 January 2018
Accepted: 21 April 2018
Published: 4 May 2018
Abstract
This paper deals with the oscillation criteria for the linear delay differential equations. We present new sufficient conditions for the oscillation of all solutions of such equations. The results improve and complement some earlier ones in the literature.
Keywords
- Oscillation
- Delay differential equation
- Sufficient conditions
MSC
- 34K06
- 34K11
1 Introduction
Our aim is to establish new sufficient conditions for the oscillation of all solutions of Eq. (1). This problem has been recently investigated by many authors. See, for example, [1–14] and the references cited therein.
A continuously differentiable function defined on \([\tau (T _{0}),\infty)\) for some \(T_{0}\ge t_{0}\) and satisfying Eq. (1) for \(t\ge T_{0}\) is called a solution of Eq. (1). Such a solution is called oscillatory if it has arbitrarily large zeros. Otherwise, it is called nonoscillatory.
In the case \(0< k\le 1/e\), all conditions in the papers [4–6, 8, 9, 12–14] are dependent on \(0< L<1\). The aim of this article is to establish such conditions for oscillation of solutions of Eq. (1) which are independent of L.
In the second section we will use the next lemma by Jaroš and Stavroulakis [5].
Lemma 1.1
([5])
2 Oscillatory properties
In this section we will study the oscillatory properties of Eq. (1).
Lemma 2.1
Proof
Theorem 2.1
Proof
Theorem 2.2
Proof
In the next example we observe the case \(k=1/e\). Then \(\lambda_{1}= \lambda_{2}=e\).
Example
We point out that the results in [6, 13] are dependent on the constant \(0< L<1\), while our results do not depend on the constant L.
Declarations
Acknowledgements
The authors gratefully acknowledge the Scientific Grant Agency VEGA of the Ministry of Education of Slovak Republic and the Slovak Academy of Sciences for supporting this work under the Grant No. 1/0812/17. The authors would like to thank the anonymous referees for their valuable comments.
Authors’ contributions
The authors have made the same contribution. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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