Abstract
We study asymptotic behavior of solutions to a class of oddorder delay differential equations. Our theorems extend and complement a number of related results reported in the literature. An illustrative example is provided.
MSC: 34K11.
Keywords:
asymptotic behavior; oddorder; delay differential equation; oscillation1 Introduction
Professor Ivan Kiguradze is widely recognized as one of the leading contemporary experts in the qualitative theory of ordinary differential equations. His research has been partly summarized in the monograph written jointly with Professor Chanturia [1] where many fundamental results on the asymptotic behavior of solutions to important classes of nonlinear differential equations were collected. In particular, the Kiguradze lemma and Kiguradze classes of solutions are well known to researchers working in the area and are extensively used to advance the knowledge further.
In this tribute to Professor Kiguradze, we are concerned with the asymptotic behavior of solutions to an oddorder delay differential equation
where
By a solution of (1.1) we mean a function
Analysis of the oscillatory and nonoscillatory behavior of solutions to different classes of differential and functional differential equations has always attracted interest of researchers; see, for instance, [119] and the references cited therein. One of the main reasons for this lies in the fact that delay differential equations arise in many applied problems in natural sciences, technology, and automatic control, cf., for instance, Hale [20]. In particular, (1.1) may be viewed as a special case of a more general class of higherorder differential equations with a onedimensional pLaplacian, which, as mentioned by Agarwal et al.[4], have applications in continuum mechanics.
Let us briefly comment on a number of closely related results which motivated our study. In [2,58,14], the authors investigated asymptotic properties of a thirdorder delay differential equation
Using a Riccati substitution, Liu et al.[11], Zhang et al.[16], and Zhang et al.[18] studied oscillation of (1.1) assuming that
In the special case when
which was studied by Zhang et al.[17] who established the following result.
Theorem 1.1 ([[17], Corollary 2.1])
Let
and assume that
and, for some
Then every solution of (1.3) is either oscillatory or converges to zero as
To the best of our knowledge, only a few results are known regarding oscillation of
(1.1) for n odd. Furthermore, in this case the methods in [11,18] which employ Riccati substitutions cannot be applied to the analysis of (1.1). Therefore,
the objective of this paper is to extend the techniques exploited in [17] to the study of (1.1) in the case when the integral in (1.2) is finite, that is,
for all
As usual, all functional inequalities considered in this paper are supposed to hold
for all t large enough. Without loss of generality, we may deal only with positive solutions
of (1.1), because under our assumption that γ is a ratio of odd natural numbers, if
2 Main results
We need the following auxiliary lemmas.
Lemma 2.1Assume that (1.2) is satisfied and let
Proof Let
Thus,
which means that the function
is decreasing for
We claim that
and, for all
where
Inequality (2.3) yields
Integrating this inequality from
Passing to the limit as
It follows now from the inequalities
which implies that
Lemma 2.2 (Agarwal et al.[3])
Assume that
holds on
Lemma 2.3 (Agarwal et al.[4])
The equation
where
For a compact presentation of our results, we introduce the following notation:
Theorem 2.4Assume that
Then every solution
provided that either
(i) (1.2) holds or
(ii) (1.4) is satisfied and, for some
Proof Assume that (1.1) has a nonoscillatory solution
Case (i) By Lemma 2.1, we conclude that (2.1) holds for all
for every
By virtue of (1.1), we conclude that
However, it follows from the result due to Werbowski [[15], Corollary 1] that the latter inequality does not have positive solutions under the assumption (2.4), which is a contradiction. The proof of part (i) is complete.
Case (ii) Similar analysis to that in Lemma 2.1 leads to the conclusion that a nonoscillatory
positive solution with the property (2.7) satisfies, for
where
Then
we deduce that the function
Dividing both sides of (2.10) by
Letting
Hence,
which yields
Thus, by (2.9), we conclude that
Differentiation of (2.9) yields
It follows now from (1.1) and (2.9) that
On the other hand, it follows from Lemma 2.2 that
for every
Multiplying (2.12) by
Let
(see Zhang and Wang [[19], Lemma 2.3] for details) and the definition of φ, we derive from (2.11) that
which contradicts (2.6). This completes the proof for the part (ii). □
Remark 2.5 For a result similar to the one established in part (i) in Theorem 2.4, see also Zhang et al. [[16], Theorem 5.3].
Remark 2.6 For
In the remainder of this section, we use different approaches to arrive at the conclusion of Theorem 2.4. First, we employ the integral averaging technique to replace assumption (2.6) with a Philostype condition.
To this end, let
and H has a nonpositive continuous partial derivative
for some function
Theorem 2.7Let
for all
Proof Assuming that
Let
and
Using the inequality
we obtain
which contradicts assumption (2.13). This completes the proof. □
Finally, we formulate also a comparison result for (1.1) that leads to the conclusion of Theorem 2.4.
Theorem 2.8Let
is oscillatory for some
Proof Assuming again that
3 Example
The following example illustrates possible applications of theoretical results obtained in the previous section.
Example 3.1 For
It is not difficult to verify that (1.4) holds and
Let
for some
Remark 3.2 Note that Theorems 2.4, 2.7, and 2.8 ensure that every solution
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to this work and are listed in alphabetical order. They both read and approved the final version of the manuscript.
Acknowledgements
The authors express their sincere gratitude to both 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.
References

Kiguradze, IT, Chanturia, TA: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Academic, Dordrecht (1993)

Agarwal, RP, Aktas, MF, Tiryaki, A: On oscillation criteria for third order nonlinear delay differential equations. Arch. Math.. 45, 1–18 (2009)

Agarwal, RP, Grace, SR, O’Regan, D: Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic, Dordrecht (2000)

Agarwal, RP, Grace, SR, O’Regan, D: Oscillation Theory for Second Order Linear, HalfLinear, Superlinear and Sublinear Dynamic Equations, Kluwer Academic, Dordrecht (2002)

Baculíková, B, Džurina, J: Comparison theorems for the thirdorder delay trinomial differential equations. Adv. Differ. Equ.. 2010, (2010) Article ID 160761

Baculíková, B, Džurina, J, Rogovchenko, YuV: Oscillation of third order trinomial delay differential equations. Appl. Math. Comput.. 218, 7023–7033 (2012). Publisher Full Text

Džurina, J, Komariková, R: Asymptotic properties of thirdorder delay trinomial differential equations. Abstr. Appl. Anal.. 2011, (2011) Article ID 730128

Džurina, J, Kotorová, R: Properties of the third order trinomial differential equations with delay argument. Nonlinear Anal. TMA. 71, 1995–2002 (2009). Publisher Full Text

Grace, SR, Agarwal, RP, Pavani, R, Thandapani, E: On the oscillation of certain third order nonlinear functional differential equations. Appl. Math. Comput.. 202, 102–112 (2008). Publisher Full Text

Li, T, Rogovchenko, YuV, Tang, S: Oscillation of secondorder nonlinear differential equations with damping. Math. Slovaca (2014, in press)

Liu, S, Zhang, Q, Yu, Y: Oscillation of evenorder halflinear functional differential equations with damping. Comput. Math. Appl.. 61, 2191–2196 (2011). Publisher Full Text

Philos, ChG: Oscillation theorems for linear differential equations of second order. Arch. Math.. 53, 482–492 (1989). Publisher Full Text

Rogovchenko, YuV, Tuncay, F: Oscillation criteria for secondorder nonlinear differential equations with damping. Nonlinear Anal.. 69, 208–221 (2008). Publisher Full Text

Tiryaki, A, Aktas, MF: Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping. J. Math. Anal. Appl.. 325, 54–68 (2007). Publisher Full Text

Werbowski, J: Oscillations of firstorder differential inequalities with deviating arguments. Ann. Mat. Pura Appl.. 140, 383–392 (1985). Publisher Full Text

Zhang, C, Agarwal, RP, Li, T: Oscillation and asymptotic behavior of higherorder delay differential equations with pLaplacian like operators. J. Math. Anal. Appl.. 409, 1093–1106 (2014). Publisher Full Text

Zhang, C, Li, T, Sun, B, Thandapani, E: On the oscillation of higherorder halflinear delay differential equations. Appl. Math. Lett.. 24, 1618–1621 (2011). Publisher Full Text

Zhang, Q, Liu, S, Gao, L: Oscillation criteria for evenorder halflinear functional differential equations with damping. Appl. Math. Lett.. 24, 1709–1715 (2011). Publisher Full Text

Zhang, S, Wang, Q: Oscillation of secondorder nonlinear neutral dynamic equations on time scales. Appl. Math. Comput.. 216, 2837–2848 (2010). Publisher Full Text

Hale, JK: Theory of Functional Differential Equations, Springer, New York (1977)