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- Open Access
Oscillation criteria for third-order delay differential equations
- George E Chatzarakis^{1},
- Said R Grace^{2} and
- Irena Jadlovská^{3}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-017-1384-y
© The Author(s) 2017
- Received: 24 August 2017
- Accepted: 2 October 2017
- Published: 13 October 2017
Abstract
Keywords
- third-order
- differential equation
- delay
- Grönwall inequality
- oscillation
MSC
- 34C10
- 34K11
1 Introduction
We restrict our attention to those solutions of (1.1) which exist on I and satisfy the condition \(\sup\{\vert x(t)\vert : t\ge t _{1}\}>0\) for any \(t_{1}\geq t_{y}\). We assume that (1.1) possesses such a solution.
A solution \(y(t)\) of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be nonoscillatory. The equation itself is termed oscillatory if all of its solutions oscillate.
As is well known, differential equations of third-order have long been considered as valuable tools in the modeling of many phenomena in different areas of applied mathematics and physics. Indeed, it is worthwhile to mention their use in the study of entry-flow phenomenon [1], the propagation of electrical pulses in the nerve of a squid approximated by the famous Nagumo’s equation [2], the feedback nuclear reactor problem [3] and so on.
Hence, a great amount of work has been done over the last three decades on the oscillation theory of third-order differential equations with variable coefficients. The most significant results published up to 2014 have been collected and summarized in recent monographs [4, 5].
It follows from the generalized Kiguradze lemma (cf. Lemma 1) that the first derivative of any positive solution y of (1.1) is of one sign eventually, i.e., y is either increasing or decreasing. Since in the ordinary case (when \(\tau(t)\equiv t\)) there always exists a decreasing solution of (1.1), see [6, Lemma 1], authors have used various techniques to present sufficient conditions guaranteeing that any solution of (1.1) oscillates or converges to zero eventually. For such results, we refer the reader to [6–14] and the references cited therein.
However, it is interesting to note that the delay argument can cause that (1.1) becomes oscillatory. As an example of this property, we can consider the third-order differential equation \(y'''(t)+y(t- \tau) = 0\), \(\tau>0\), which is oscillatory if and only if \(\tau \mathrm {e}>3\) (see [15, Theorem 1]). But the corresponding third-order ordinary differential equation \(y'''(t)+y(t) = 0\) has a nonoscillatory solution \(y(t) = \mathrm {e}^{-t}\). Therefore, it is of special interest to establish new criteria ensuring oscillation of all solutions of (1.1) when \(\tau(t)< t\).
An interesting method in oscillation theory is to use some comparison principles based on which the oscillatory behavior of the solutions of the studied differential equation is inherited from the oscillations in a first-order delay differential equations, resulting in conditions involving \(1/\mathrm {e}\). The results concerned with this problem for (1.1) and its various generalizations were presented in [7, 16–22].
Theorem A
(See [17, Theorem 2])
Theorem B
(See [17, Theorem 3])
On the other hand, oscillations in (1.5) eliminate the existence of positive decreasing solutions in (1.2). Since there is no general rule as to how to choose a function \(\xi(t)\) satisfying the imposed conditions, an interesting problem is how to establish a corresponding result without requiring the existence of the unknown function \(\xi(t)\). Here, we will also address this problem.
The main objective in this paper is to study the asymptotic and oscillatory behavior of the solutions of (1.1). Our method is essentially based on establishing sharper estimates for increasing positive solutions of (1.1) than (1.6), using an iterative technique; and to obtain analogous iterative estimates for decreasing positive solutions of (1.1). Similar ideas as those presented here have been successively employed in investigating oscillatory behavior of second-order advanced differential equations, see a recent work [27] for details. The results obtained are new even in the case of (1.1) when \(r_{i}(t) = 1\), \(i = 1,2\). Furthermore, the iterative nature of the results enables us to test for oscillations, even when the previously known results fail to apply. We demonstrate the improvement we achieve with these results by applying them to Euler-type delay differential equations.
Remark 1
In the sequel, all functional inequalities are assumed to hold eventually, that is, they are satisfied for all t large enough.
Remark 2
Without loss of generality, we can deal only with the positive solutions of (1.1).
2 Some lemmas and auxiliary results
In this section, we state and prove some lemmas that will be useful in establishing our main results. For completeness, we start by recalling the adaptation of the generalized Kiguradze lemma. Next, we will provide some important monotonic properties of a positive solution y of (1.1) in cases (2.1) and (2.2), respectively.
Lemma 1
(See [28, Lemma 2])
Lemma 2
Proof
Lemma 3
Proof
Lemma 4
Proof
Since the proof is similar to that of [17, Lemma 2], we omit it. □
3 Main results
We are prepared to provide the main results of the paper. At first, we improve Theorem A by establishing a new sufficient condition for all nonoscillatory solutions of (1.1) to converge to zero as \(t\to\infty\). Second, we relax condition (2.17) and employ another one in order to eliminate (2.2)-type solutions in (1.1) and attain oscillation of (1.1).
Theorem 1
Proof
Let \(y(t)\) be a nonoscillatory solution of (1.1), say \(y(t)>0\) and \(y(\tau(t))>0\) for \(t\ge T_{1}\) for some \(T_{1}\ge t_{0}\). By Lemma 1, \(y(t)\) satisfies either (2.1) or (2.2) for \(t\ge t_{1}\ge T_{1}\).
Now, assume that \(y(t)\) satisfies (2.2) for \(t\ge t_{1}\). By Lemma 4, we have \(\lim_{t\to\infty}y(t) = 0\). The proof is complete. □
Applying the known oscillation criteria for (3.1), one immediately gets an oscillation criterion for (1.1). The following is due to Ladde et al. [30, Theorem 2.1.1].
Corollary 1
Theorem 2
Proof
Let \(y(t)\) be a nonoscillatory solution of (1.1), say \(y(t)>0\) and \(y(\tau(t))>0\) for \(t\ge T_{1}\) for some \(T_{1}\ge t_{0}\). By Lemma 1, \(y(t)\) satisfies either (2.1) or (2.2) for \(t\ge t_{1}\ge T_{1}\).
Assume first that \(y(t)\) satisfies (2.1) for \(t\ge t_{1}\). Proceeding as in the proof of Theorem 1, we arrive at a contradiction.
4 Examples
The following examples are provided to show the improvement we achieve by our results over those presented earlier. All numerical calculations can be easily performed in MATLAB.
Example 1
We remark that Theorem A, as well as [7, Theorem 1], [22, Theorem 12], [25, Theorem 2.9], fail due to (4.2).
Example 2
5 Summary
In this paper, we have obtained new oscillation criteria for (1.1), which, to the best of our knowledge, essentially improve a number of related results reported in the literature, even in the case when \(r_{1} = r_{2} \equiv1\).
Our approach is based on refining classical techniques, in which a desired property of the studied equation is deduced from oscillation of first-order equations by taking into account such part of the overall impact of the delay which has been neglected in earlier results.
An interesting problem for further research is to establish different iterative techniques for testing oscillations in (1.1) independently on the constant \(1/\mathrm {e}\).
Declarations
Acknowledgements
We are grateful to the editors and three anonymous referees for a very careful reading of the manuscript and for pointing out several inaccuracies. The work on this research has been supported by the internal grant project No. FEI-2015-22.
Availability of data and materials
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Funding
Not applicable.
Authors’ contributions
The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
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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