Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2018: 0.34

ICV 2017: 161.82

Open Access
See all formats and pricing
More options …
Volume 16, Issue 1


Volume 13 (2015)

The fourth order strongly noncanonical operators

Blanka Baculikova
  • Corresponding author
  • Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, 042 00, Košice, Slovakia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Jozef Dzurina
  • Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, 042 00, Košice, Slovakia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-12-31 | DOI: https://doi.org/10.1515/math-2018-0135


It is shown that the strongly noncanonical fourth order operator


can be written in essentially unique canonical form as


The canonical representation essentially simplifies examination of the fourth order strongly noncanonical equations


Keywords: canonical operator; fourth order differential equations

MSC 2010: 34K11; 34C10

1 Introduction

In the paper, we consider the fourth order delay differential equation


where riC(4–i)(t0, ∞), ri(t) > 0, i = 1, …, 3, p > 0, τ(t) ≤ t, τ′(t) > 0 and τ(t) → ∞ as t → ∞.

Fourth-order differential equations naturally appear in models concerning physical, biological, and chemical phenomena, such as, for instance, problems of elasticity, deformation of structures, or soil settlement, for example, see [1]. In mechanical and engineering problems, questions concerning the existence of oscillatory solutions play an important role. During the past decades, there has been a constant interest in obtaining sufficient conditions for oscillatory properties of different class of fourth order differential equations with deviating argument, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13].

As far as the oscillation theory of fourth-order differential equations is concerned, the problem of investigating ways of factoring disconjugated operators


which are crucial in studying the perturbed differential equations such as (E), has been of special interest. Motivated by the famous work of George Polya, Trench [2] showed that if operator 𝓛y is strongly noncanonical, that is,


then it can be written in an essentially unique canonical form as


so that qiC(4–i)(t0, ∞), qi(t) > 0, i = 0, …, 4 and


However, the computation using the Lemmas 1 and 2 from [2] leading to canonical representation is very complicated and does not provide closed formulas for qi(t). This brings us to the question whether is it possible to establish a closed form formulas for qi. The aim of this paper is to positively answer to this question, showing simultaneously the advantage of the result in the investigation of oscillatory properties of strongly noncanonical equations.

2 Main results

Throughout the paper we assume that (1.2) hold and so we can employ the notation




where i, j, k ∈ {1, 2, 3} are mutually different.

We start with the following auxiliary results which are elementary but very useful.

Lemma 1

Let (1.2) hold. Then





an integration of this equality from t to ∞, yields


To simplify our notation, we denote


The following result provides an alternative formula for Ω(t).

Lemma 2

Let (1.2) hold. Then



Proof of this lemma is similar to that of Lemma 1 and so it can be omitted.□

Now, we are prepared to introduce the main result.

Theorem 1

The strongly noncanonical operator 𝓛 y has the following unique canonical representation



Straightforward evaluation of 𝓛 y, with 𝓛 as defined by (2.2) yields


Employing (2.1), we see that


It follows from Lemma 2 that π12(t) + π21(t) = π1(t)π2(t) and so


On the other hand, by Lemma 1 and 2








which means that the operators (1.1) and (2.2) are equivalent.

Now we shall show that operator (2.2) is canonical. Direct computation shows that


Applying twice the L’Hospital rule, we get


and so



t0t1q3(s)ds=t0tΩ(s)r3(s)π3212(s)ds=π32(t)π321(t)c3 as t.

To evaluate the last one integral it is useful to see that






and we conclude that the operator (2.2) is canonical. By Trench’s result [2] there exists the only one canonical representation of 𝓛 (up to multiplicative constants with product 1) and so our canonical form is unique.□

We support our results with couple of illustrative examples.

Example 1

Let us consider the following operator


By Theorem 1, this operator can be rewritten in canonical form as


Example 2

The operator


can be represented in canonical form as


3 Applications to differential equations

Theorem 1 can be applied to study oscillatory properties of differential equations. We are going to outline one such application based on comparison principle.

Corollary 1

Noncanonical differential equation (E) can be written in canonical form as


Setting z(t) = y(t)/π123(t) we get the following comparison result that reduces oscillation of strongly noncanonical equation to that of canonical equation.

Corollary 2

Noncanonical equation (E) is oscillatory if and only if the canonical equation


is oscillatory.

Example 3

Let us consider the fourth order differential equation


with α > 1, β > 1, γ > 1. By Corollary 2, this equation is oscillatory if and only if the canonical equation


is oscillatory. It is more convenient to study oscillation of (3.12) instead of (3.1).

Now, we are ready to study the properties of (E) with the help of (Ec). Without loss of generality, we can consider only with the positive solutions of (Ec). The following result is a modification of Kiguradze’s lemma [3].

Let us denote




Then (Ec) can be rewritten as


Lemma 3

Assume that z(t) is an eventually positive solution of (Ec), then either




Consequently, the set 𝒩 of all positive solutions of (E) has the decomposition


To obtain oscillation of studied equation (E), we need to eliminate both cases of possible non-oscillatory solutions.

Let us denote




Theorem 2

Let (1.2) hold. Assume that both first-order delay differential equations




are oscillatory. Then (E) is oscillatory.


Assume that y(t) is an eventually positive solution of (E), say for tt1. Then by Corollary 2, z(t) = y(t)/π123(t) is a solution of (Ec).

It follows from Lemma 3 that either z(t) ∈ 𝒩1 or z(t) ∈ 𝒩3. At first, we admit that z(t) ∈ 𝒩1. Noting that q1(t)z′(t) is decreasing, we see that


Integrating (Ec) from t to ∞, we have


Taking into account that z(τ(t)) is increasing, the last inequality yields


Integrating once more, we are led to


Combining the last inequality with (3.5), one gets


Thus, the function x(t) = q1(t)z′(t) is a positive solution of the differential inequality


Hence, by Philos theorem [4], we conclude that the corresponding differential equation (3.3) also has a positive solution, which contradicts the assumptions of the theorem.

Now, we shall assume that z(t) ∈ 𝒩3. Since q3q2q1z is decreasing, we are led to


Integrating the above inequality, one can verify that


Integrating once more, we see that x(t) = q3q2q1z(t) satisfies


Setting the last estimate into (Ec), we see that x(t) is a positive solution of the differential inequality


which in view of Philos theorem in [4] guarantees that the corresponding differential equation (3.4) has also a positive solution. This is a contradiction and the proof is complete now.□

Applying suitable criteria for oscillation of (3.3) and (3.4), we immediately obtain the criteria for oscillation of (E). We use the one which is due to Ladde et al. [5].

Corollary 3

Let (1.2) hold. Assume that for i = 1, 2

lim inftτ(t)tQi(s)ds>1e(3.10)

hold. Then (E) is oscillatory.

We support our results by another example.

Example 4

Let us consider the general Euler delay differential equation


with α > 1, β > 1, γ > 1, a > 0, and 0 < λ < 1. By Corollary 2 and Example 3, this equation is oscillatory if and only if the canonical equation


is oscillatory. The straightforward computation yields that




By Corollary 4 considered equation is oscillatory, provided that both conditions




are satisfied.

4 Summary

In this paper we provided canonical representation for strongly noncanonical operator. This canonical transformation is easy and immediate. Moreover, we point out its application in the oscillation theory.


The paper has been supported by the grant project KEGA 035TUKE-4/2017.


  • [1]

    Bartusek M., Dosla Z., Asymptotic problems for fourth-order nonlinear differential equations. Boundary Value Problems 2013, 1–15 Web of ScienceGoogle Scholar

  • [2]

    Trench W., Canonical forms and principal systems for general disconjugate equations, Trans. Amer. Math. Soc. 184, 1974, 319–327 Google Scholar

  • [3]

    Kiguradze I. T., Chanturia T. A., Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equation, Kluwer Acad. Publ., Dordrecht 1993 Google Scholar

  • [4]

    Philos Ch. G., Ot the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delay, Arch. Math., 36, 1981, 168–178 CrossrefGoogle Scholar

  • [5]

    Ladde G. S., Lakshmikantham V., Zhang B. G., Oscillation Theory of Differential Equations with Deviating Argument, Marcel Dekker, New York, 1987 Google Scholar

  • [6]

    Dzurina J., Comparison theorems for nonlinear ODE’s, Math. Slovaca 42, 199), 299–315 Google Scholar

  • [7]

    Dzurina J., Kotorova R., Zero points of the solutions of a differential equation, Acta Electrotechnica et Informatica 7, 2007, 26–29 Google Scholar

  • [8]

    Jadlovska, I., Application of Lambert W function in oscillation theory, Acta Electrotechnica et Informatica 14, 2014, 9–17 CrossrefGoogle Scholar

  • [9]

    Koplatadze R., Kvinkadze G., Stavroulakis I. P., Properties A and B of n-th order linear differential equations with deviating argument, Georgian Math. J. 6, 1999, 553–566 CrossrefGoogle Scholar

  • [10]

    Kitamura Y., Kusano T., Oscillations of first-order nonlinear differential equations with deviating arguments, Proc. Amer. Math. Soc., 78, 1980, 64–68 CrossrefGoogle Scholar

  • [11]

    Kusano T., Naito M., Comparison theorems for functional differential equations with deviating arguments, J. Math. Soc. Japan 3, 1981, 509–533 Google Scholar

  • [12]

    Zhang T. Li, Ch., Thandapani E., Asymptotic behavior of fourth-order neutral dynamic equations with noncanonical operators, Taiwanese J. Math. 18, 2014, 1003–1019 Web of ScienceCrossrefGoogle Scholar

  • [13]

    Mahfoud W. E., Comparison theorems for delay differential equations, Pacific J. Math. 83, 1979, No. 83, 187–197 CrossrefGoogle Scholar

About the article

Received: 2018-05-22

Accepted: 2018-11-26

Published Online: 2018-12-31

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1667–1674, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0135.

Export Citation

© 2018 Baculikova and Dzurina, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

B. Baculikova and J. Dzurina
Advances in Difference Equations, 2019, Volume 2019, Number 1

Comments (0)

Please log in or register to comment.
Log in