 Research
 Open Access
Oscillatory and asymptotic properties of thirdorder quasilinear delay differential equations
 G. E. Chatzarakis^{1},
 J. Džurina^{2} and
 I. Jadlovská^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s1366001919670
© The Author(s) 2019
 Received: 26 October 2018
 Accepted: 15 January 2019
 Published: 25 January 2019
Abstract
In this paper, we consider a class of quasilinear thirdorder differential equations with a delayed argument. We establish new sufficient conditions for all solutions of such equations to be oscillatory or almost oscillatory. Those criteria improve, simplify and complement a number of existing results. The strength of the criteria obtained is tested on Euler type equations.
Keywords
 Quasilinear differential equation
 Delay
 Thirdorder
 Oscillation
MSC
 34C10
 34K11
1 Introduction
 (H_{0}):

α is a quotient of odd positive integers;
 (H_{1}):

\(r\in\mathcal{C}([t_{0},\infty), \mathbb {R})\) is positive and satisfies$$ \int_{t_{0}}^{\infty}\frac{\mathrm {d}t}{r^{1/\alpha}(t)} < \infty; $$
 (H_{2}):

\(q\in\mathcal{C}([t_{0},\infty),\mathbb {R})\) is nonnegative and does not vanish eventually;
 (H_{3}):

the delay function \(\tau\in\mathcal{C}^{1}([t_{0}, \infty),\mathbb {R})\) is strictly increasing, \(\tau(t)\le t\), and \(\lim_{t\to\infty}\tau(t) = \infty\).
A solution y of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is nonoscillatory. The equation itself is termed oscillatory if all its solutions oscillate.
From the early years of the 18th century, differential equations of thirdorder have been used for modeling various phenomena in several areas of the applied sciences. The first step in this direction was taken by J. Bernoulli in 1696 who formulated the famous isoperimetric problem and, five years later, gave the solution that depends upon a thirdorder differential equation [8]. Since then, these equations have shown to be particularly important in the modeling of several physical phenomena, including the interactions between charged particles, in an external electromagnetic field [21], the entryflow phenomenon [11], the propagation of action potentials in squid neurons [16] and others.
Although the importance of thirdorder equations in applications had been realized very early, the majority of the work on the qualitative behavior of those equations has been carried out only relatively recently, in the last three decades. For a review of key results up to 2014, we refer the reader to the recent monographs [17, 18].
For closely related results having in common that the function \(r(t)\) satisfies condition (1.2), we refer the reader to [1, 3–6, 10, 19, 20].
The main objective of this work is to establish results for the solutions of (1.1) to be oscillatory or almost oscillatory under the crucial condition (H_{1}). We postulate new sufficient conditions for oscillations and/or property A (see Definition 1), which improve, simplify and complement some existing results reported in the literature. Finally, we test the strength of our criteria on Euler type equations.
2 Preliminaries, definitions and existing results
At first, we constrain the structure of possible nonoscillatory, let us say positive solutions of (1.1).
Lemma 1
 (I)
\(y > 0\), \(y' >0\), \(y'' > 0\),
 (II)
\(y > 0\), \(y' >0\), \(y'' < 0\),
 (III)
\(y > 0\), \(y' <0\), \(y'' > 0\),
Proof
The proof is straightforward and hence we omit it. □
2.1 Notation and definitions
Remark 1
All functional inequalities considered in the paper are supposed to hold eventually, that is, they are satisfied for all t large enough.
Remark 2
Note that if y is a solution of (1.1), then \(x = y\) is also a solution of (1.1). Thus, regarding nonoscillatory solutions of (1.1), we only need to consider the eventually positive ones.
2.2 Motivation
In the sequel, we state and discuss in detail a triplet of related results for (1.1) under the assumptions (H_{0})–(H_{3}), which are considered to be the primary motivation of the paper.
Grace et al. [9] studying the oscillatory behavior of (1.1) using comparison principles and established the following result, which we present below for the reader’s convenience.
Theorem A
(See [9, Theorem 3])
It is useful to note that conditions (2.1), (2.2) and (2.3) eliminate solutions satisfying cases (I)–(III) of Lemma 1, respectively.
Making further use of comparison principles with firstorder delay equations, Agarwal et al. [2] established the following oscillation result for (1.1) with \(\alpha= 1\).
Theorem B
(See [2, Corollary 1])
In fact, both Theorems A and B strongly depend on the right choice of the auxiliary functions in lim inftype conditions. Since there is no general rule for this choice, the application of such criteria may become difficult.
Using a different technique based on reducing the studied equation into a firstorder Riccatitype inequality, which is generally considered as one of the most valuable tools in the oscillation theory, Li et al. [15] provided the following criterion for property A of (1.1).
Theorem C
(See [15, Theorem 1])
Here, condition (2.7) works to ensure that any solution of type (III) converges to zero as t approaches infinity, while conditions (2.8) and (2.9) eliminate solutions of type (I) and (II), respectively.
3 Main results
3.1 Nonexistence of solutions of type (I) and (II)
We start with a simple condition ensuring the nonexistence of solutions of type (I). As will be shown later, this condition is already included in those eliminating solutions of type (II).
Lemma 2
Proof
Next, we state some useful properties of the type (II) solutions, which are useful when proving the main results.
Lemma 3
 (a)
\(y(t)\ge ty'(t)\) and \(y(t)/t\) is decreasing for \(t\ge t_{2}\), and \(\lim_{t\to\infty}y(t)/t = y' = 0\),
 (b)
\(y'(t)\ge \pi(t) r^{1/\alpha}(t)y''(t)\) and \(y'(t)/\pi(t)\) is increasing for \(t\ge t_{2}\),
Proof
Since y is increasing, by Lemma 1, y satisfies either case (I) or case (II) for \(t\ge t_{1}\), where \(t_{1}\in[t_{0},\infty)\) is such that \(y(\tau(t))>0\) for \(t\ge t_{1}\).
At first, note that because of the assumption (H_{1}), condition (3.4) implies that (3.1) holds. Thus, by Lemma 2, y satisfies case (II) for \(t\ge t_{1}\).
Now, we can proceed to present various simple criteria for property P for (1.1).
Theorem 1
Proof
If we assume that y is of (I)type, then (3.8) contradicts the positivity of \(r(t) (y''(t) )^{\alpha}\).
The next result is based on a comparison with a firstorder delay inequality. This result, in connection with the results from Sect. 3.2, can be viewed as an improved and simplified alternative of Theorem B. In contrast to that theorem, we stress that the next theorem does not require the existence of auxiliary functions (as in condition (2.5)) and, moreover, the nonexistence of solutions of type (I) and (II) is ensured by means of only one condition.
Theorem 2
Proof
A principle like the one we used in the proof of Theorem 2 always requires \(\tau(t)< t\). The results presented in the sequel, however, apply also in the case when \(\tau(t) = t\).
Theorem 3
Proof
Assume for the sake of contradiction that y satisfies case (I) or (II) of Lemma 1 for \(t\ge t_{1}\). At first, note that \(\lim_{t\to\infty}\pi(t) = 0\) holds due to (H_{1}), which together with (3.13) implies (3.1). By Lemma 3, we conclude that y satisfies case (II) and the asymptotic properties (a) and (b) of the lemma for \(t\ge t_{2}\ge t_{1}\).
Theorem 4
Proof
Assume for the sake of contradiction that y satisfies case (I) or (II) of Lemma 1 for \(t\ge t_{1}\). By Lemma 3, we conclude that y satisfies case (II) and the asymptotic properties (a) and (b) of the lemma for \(t\ge t_{2}\ge t_{1}\).
Letting \(\rho(t) = 1/\pi(t)\), the following consequence is immediate.
Corollary 1
Theorem 5
Proof
Assume for the sake of contradiction that y satisfies case (I) or (II) of Lemma 1 for \(t\ge t_{1}\). By Lemma 3, we conclude that y satisfies case (II) and the asymptotic properties (a) and (b) of the lemma for \(t\ge t_{2}\ge t_{1}\).
Finally, we turn our attention to the existing result presented in the introductory section, namely Theorem C. By careful observation, it is easy to show that condition (2.8) is redundant.
Theorem 6
If (2.9) holds for some \(k\in(0,1)\) and for sufficiently large \(t_{3}\ge t_{0}\), then (1.1) has property P.
Proof
The proof is complete. □
Remark 3
We note that in the proof of Theorem C, a weaker version of (a) of Lemma 3 was used for solutions of type (II), namely, \(y(t)\ge k y'(t)\) for \(k\in(0,1)\) and t large enough. Assuming that condition (3.4) holds, one can easily provide a stronger version of the above theorem with \(k = 1\).
3.2 Convergence to zero and/or nonexistence of solutions of type (III)
Lemma 4
Proof
Theorem 7
Proof
3.3 Applications
3.3.1 Property A
Combining Theorems 1–5 with Lemma 4, one can easily provide fundamentally new criteria for property A of (1.1).
3.3.2 Oscillation
We are now interested in the situation in which all solutions of Eq. (1.1) are oscillatory. To attain this goal, we combine Theorems 1–6 with Theorem 7.
4 Examples
Example 1
Among conditions (4.1)–(4.3), we remark that (4.1) is more efficient for small values of λ, while (4.3) for larger ones. Since Lemma 4 is satisfied, we conclude that (\(E_{x}\)) has property A if any of conditions (4.1)–(4.3) hold. Note that Theorem C does not apply due to (2.7).
Finally, by Theorem 13, we conclude that (\(E_{x}\)) is oscillatory if any of conditions (4.1)–(4.3) and (4.4) hold.
5 Conclusions
In the present paper, several new oscillation results for Eq. (1.1) have been presented, which further improve, complement and simplify existing criteria introduced in the paper as Theorems A–C.
In Sect. 3.1, we provided various criteria for the nonexistence of solutions of type (I) and (II). In particular, Theorem 1 serves as a single condition alternative to Theorem A, while Theorem 2 offers a single condition criterion, which is based on similar principles (compared with firstorder delay equations) as Theorem B, but does not require the existence of auxiliary functions. By a simple refinement in the proof of Theorem C, we have shown that (2.8) is unnecessary and can be removed. We have also pointed out how a stronger version with \(k = 1\) can be attained. Using different substitutions as in the proof of Theorem C, we have presented more general results for the nonexistence of solutions of type (I) and (II).
In Sect. 3.2, we were dealing with the asymptotic properties and nonexistence of solutions of type (III) of Lemma 1. In that section, we extended (2.7) from Theorem C to be applied on (2.10). Furthermore, we provided a new criterion for the nonexistence of such solutions.
Finally, we have combined the results from Sects. 3.1 and 3.2 to obtain new results for oscillation and/or property A of (1.1).
Declarations
Acknowledgements
The authors would like to express their gratitude to the editors and three 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. The work on this research has been supported by the grant project KEGA 035TUKE4/2017.
Availability of data and materials
Not applicable.
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
References
 Agarwal, R., Grace, S., Smith, T.: Oscillation of certain thirdorder functional differential equations. Adv. Math. Sci. Appl. 16(1), 69–94 (2006) MathSciNetMATHGoogle Scholar
 Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: Oscillation of thirdorder nonlinear delay differential equations. Taiwan. J. Math. 17(2), 545–558 (2013) MathSciNetView ArticleGoogle Scholar
 Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: A philostype theorem for thirdorder nonlinear retarded dynamic equations. Appl. Math. Comput. 249, 527–531 (2014) MathSciNetMATHGoogle Scholar
 Agarwal, R.P., Grace, S.R., O’Regan, D.: On the oscillation of certain functional differential equations via comparison methods. J. Math. Anal. Appl. 286(2), 577–600 (2003) MathSciNetView ArticleGoogle Scholar
 Baculíková, B., Džurina, J.: Oscillation of thirdorder nonlinear differential equations. Appl. Math. Lett. 24(4), 466–470 (2011) MathSciNetView ArticleGoogle Scholar
 Chatzarakis, G.E., Grace, S.R., Jadlovská, I.: Oscillation criteria for thirdorder delay differential equations. Adv. Differ. Equ. 2017(1), 330 (2017) MathSciNetView ArticleGoogle Scholar
 Džurina, J., Jadlovská, I.: Oscillation of thirdorder differential equations with noncanonical operators. Appl. Math. Comput. 336, 394–402 (2018) MathSciNetView ArticleGoogle Scholar
 Fraser, C.G.: Isoperimetric problems in the variational calculus of Euler and Lagrange. Hist. Math. 19(1), 4–23 (1992) MathSciNetView ArticleGoogle Scholar
 Grace, S.R., Agarwal, R.P., Pavani, R., Thandapani, E.: On the oscillation of certain thirdorder nonlinear functional differential equations. Appl. Math. Comput. 202(1), 102–112 (2008) MathSciNetMATHGoogle Scholar
 Hassan, T.S.: Oscillation of thirdorder nonlinear delay dynamic equations on time scales. Math. Comput. Model. 49(7–8), 1573–1586 (2009) MathSciNetView ArticleGoogle Scholar
 Jayaraman, G., Padmanabhan, N., Mehrotra, R.: Entry flow into a circular tube of slowly varying crosssection. Fluid Dyn. Res. 1(2), 131–144 (1986) View ArticleGoogle Scholar
 Kiguradze, I.T., Chanturia, T.A.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Mathematics and Its Applications (Soviet Series), vol. 89. Kluwer Academic, Dordrecht (1993) Translated from the 1985 Russian original Google Scholar
 Kusano, T., Naito, M.: Comparison theorems for functionaldifferential equations with deviating arguments. J. Math. Soc. Jpn. 33(3), 509–532 (1981) MathSciNetView ArticleGoogle Scholar
 Ladde, G.S., Lakshmikantham, V., Zhang, B.G.: Oscillation Theory of Differential Equations with Deviating Arguments. Monographs and Textbooks in Pure and Applied Mathematics, vol. 110. Dekker, New York (1987) MATHGoogle Scholar
 Li, T., Zhang, C., Baculíková, B., Džurina, J.: On the oscillation of thirdorder quasilinear delay differential equations. Tatra Mt. Math. Publ. 48, 117–123 (2011) MathSciNetMATHGoogle Scholar
 McKean, H.P.: Nagumo’s equation. Adv. Math. 4(3), 209–223 (1970) MathSciNetView ArticleGoogle Scholar
 Padhi, S., Pati, S.: Theory of ThirdOrder Differential Equations. Springer, New Delhi (2014) View ArticleGoogle Scholar
 Saker, S.: Oscillation Theory of Delay Differential and Difference Equations: Second and Third Orders. LAP Lambert Academic Publishing (2010) Google Scholar
 Saker, S., Džurina, J.: On the oscillation of certain class of thirdorder nonlinear delay differential equations. Math. Bohem. 135(3), 225–237 (2010) MathSciNetMATHGoogle Scholar
 Şenel, M.T., Utku, N.: Oscillation criteria for thirdorder neutral dynamic equations with continuously distributed delay. Adv. Differ. Equ. 2014(1), 220 (2014) MathSciNetView ArticleGoogle Scholar
 Vreeke, S.A., Sandquist, G.M.: Phase space analysis of reactor kinetics. Nucl. Sci. Eng. 42(3), 295–305 (1970) View ArticleGoogle Scholar
 Wu, H., Erbe, L., Peterson, A.: Oscillation of solution to secondorder halflinear delay dynamic equations on time scales. Electron. J. Differ. Equ. 2016, 71 (2016) MathSciNetView ArticleGoogle Scholar