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An improved approach for studying oscillation of secondorder neutral delay differential equations
 Said R. Grace^{1},
 Jozef Džurina^{2},
 Irena Jadlovská^{2} and
 Tongxing Li^{3, 4}Email author
https://doi.org/10.1186/s136600181767y
© The Author(s) 2018
 Received: 10 February 2018
 Accepted: 30 June 2018
 Published: 27 July 2018
Abstract
The paper is devoted to the study of oscillation of solutions to a class of secondorder halflinear neutral differential equations with delayed arguments. New oscillation criteria are established, and they essentially improve the wellknown results reported in the literature, including those for nonneutral differential equations. The adopted approach refines the classical Riccati transformation technique by taking into account such part of the overall impact of the delay that has been neglected in the earlier results. The effectiveness of the obtained criteria is illustrated via examples.
Keywords
 Neutral differential equation
 Delayed argument
 Secondorder
 Riccati substitution
 Oscillation
MSC
 34C10
 34K11
1 Introduction
 (H_{1}):

α is a quotient of odd positive integers;
 (H_{2}):

\(r\in \mathrm{C}^{1}([t_{0},\infty),(0,\infty))\), \(p, q \in\mathrm{C}([t_{0},\infty),[0,\infty))\), \(0\le p(t)< 1\), and \(q(t)\) does not vanish identically on any halfline of the form \([t_{*},\infty )\), \(t_{*}\ge t_{0}\);
 (H_{3}):

\(\tau, \sigma\in\mathrm{C}([t_{0},\infty),\mathbb {R})\) satisfy \(\tau(t)\le t\), \(\sigma(t )< t\), and \(\lim_{t\to\infty}\tau(t) =\lim_{t\to\infty}\sigma(t) = \infty\).
The oscillation theory of differential equations with deviating arguments was initiated in a pioneering paper [2] of Fite, which appeared in the first quarter of the twentieth century. Since then, there has been much research activity concerning the oscillation of solutions of various classes of differential and functional differential equations. The interest in this subject has been reflected by extensive references in monographs [3–7]. We also refer the reader to the papers [8–10] and the references cited therein regarding similar discrete analogues of (E) and its particular cases and modifications.
A neutral delay differential equation is a differential equation in which the highest order derivative of the unknown function appears both with and without delay. During the last three decades, oscillation of neutral differential equations has become an important area of research; see, e.g., [11–27]. This is due to the fact that such equations arise from a variety of applications including population dynamics, automatic control, mixing liquids, and vibrating masses attached to an elastic bar; see Hale [28]. Especially, secondorder neutral delay differential equations are of great interest in biology in explaining selfbalancing of the human body and in robotics in constructing biped robots [29].
The objective of this paper is to establish new oscillation results for (E), which would improve the abovementioned ones. The paper is organized as follows. First, motivated by [35], we generalize conditions (\(1_{a}'\)) and (\(1_{b}'\)) for linear equation (2) to be applicable to the halflinear neutral equation (E). Second, we refine classical Riccati transformation techniques to obtain new oscillation criteria, which, to the best of our knowledge, essentially improve a large number of related results reported in the literature, including those for secondorder delay differential equations. The adopted approach lies in establishing sharper estimates relating a nonoscillatory solution with its derivatives in the case when conditions analogous to (\(1_{a}'\))–(\(1_{b}'\)) fail to apply. We illustrate the effectiveness of the obtained criteria via a series of examples and comparison with other known oscillation results.
In what follows, all occurring functional inequalities are assumed to hold eventually, that is, they are satisfied for all t large enough. As usual and without loss of generality, we can deal only with eventually positive solutions of (E).
2 Preliminaries
To prove our oscillation criteria, we need the following auxiliary results.
Lemma 1
(see [12, Lemma 3])
Lemma 2
(see [27, Lemma 2.3])
3 Main results
Now, we state and prove our first oscillation result, which extends [35, Theorem 3] obtained for the linear delay differential equation (2) to the halflinear neutral delay differential equation (E).
Theorem 3
Proof
Letting \(p(t) = 0\) in (E), the following result is an immediate consequence.
Corollary 1
Remark 1
Note that for \(\alpha= 1\), \(r(t) = 1\), and \(p(t)=0\), Theorem 3 reduces to [35, Theorem 3].
Example 1
Lemma 4
Proof
Assume that (E) has a nonoscillatory solution \(x(t)\) on \([t_{0},\infty)\). Without loss of generality, we can suppose that there exists a \(t_{1}\ge t_{0}\) such that \(x(t)>0\), \(x(\tau(t))>0\), and \(x(\sigma (t))>0\) for \(t\ge t_{1}\). As in the proof of Theorem 3, we deduce that \(y(t): = r(t) (z'(t) )^{\alpha}\) is a positive solution of the firstorder delay differential inequality (18). Proceeding in a similar manner as in the proof of [41, Lemma 1], we see that estimate (29) holds. □
In what follows, we employ the Riccati substitution technique to obtain new oscillation criteria for (E), which are especially effective in the case when Theorem 3 fails to apply.
Theorem 5
Proof
Remark 2
Theorem 5 is new because of the constant \(f_{n}(\rho)\) (for some \(n\ge 0\)) appearing in (30). So far, all results obtained in a similar manner have been formulated for \(n=0\); see, e.g., [16, 17, 20–23, 26, 27, 36, 37]. Thus, for any given \(n>0\), our result essentially improves the previous ones.
Letting \(\varphi(t)= R^{\alpha}(\sigma(t)) \) in (30), Theorem 5 yields the following result.
Corollary 2
Example 2
The following theorem serves as an alternative to Theorem 5.
Theorem 6
Proof
Example 3
4 Conclusions
In the present paper, we have studied the oscillatory behavior of the secondorder halflinear neutral delay differential equation (E). As it has been illustrated through several examples, the results obtained improve a large number of the existing ones. Our technique lies in establishing some sharper estimates relating a nonoscillatory solution with its derivatives in the case when criteria analogous to (\(1_{a}'\))–(\(1_{b}'\)) fail to apply.
The results presented in this paper strongly depend on the properties of firstorder delay differential equations. An interesting problem for further research is to establish different iterative techniques for testing oscillations in (E) independently on the constant \(1/\mathrm {e}\).
Declarations
Acknowledgements
The authors express their sincere gratitude to the editors and two anonymous referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details.
Funding
This research is supported by NNSF of P.R. China (Grant No. 61503171), CPSF (Grant No. 2015M582091), NSF of Shandong Province (Grant No. ZR2016JL021), DSRF of Linyi University (Grant No. LYDX2015BS001), and the AMEP of Linyi University, P.R. China. The research of the second and third authors is supported by the grant project KEGA 035TUKE4/2017.
Authors’ contributions
All four authors contributed equally to this work. They all read and approved the final version of the 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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