We study asymptotic behavior of solutions to a class of odd-order delay differential equations. Our theorems extend and complement a number of related results reported in the literature. An illustrative example is provided.
Keywords:asymptotic behavior; odd-order; delay differential equation; oscillation
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  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 odd-order delay differential equation
By a solution of (1.1) we mean a function , , such that and satisfies (1.1) on . We consider only those extendable solutions of (1.1) that do not vanish eventually, that is, condition holds for all . We tacitly assume that (1.1) possesses such solutions. As customary, a solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros on the ray ; otherwise, we call it non-oscillatory.
Analysis of the oscillatory and non-oscillatory behavior of solutions to different classes of differential and functional differential equations has always attracted interest of researchers; see, for instance, [1-19] 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 . In particular, (1.1) may be viewed as a special case of a more general class of higher-order differential equations with a one-dimensional p-Laplacian, which, as mentioned by Agarwal et al., have applications in continuum mechanics.
which was studied by Zhang et al. who established the following result.
Theorem 1.1 ([, Corollary 2.1])
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  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 is a solution of (1.1), so is .
2 Main results
We need the following auxiliary lemmas.
which means that the function
Inequality (2.3) yields
Lemma 2.2 (Agarwal et al.)
Lemma 2.3 (Agarwal et al.)
For a compact presentation of our results, we introduce the following notation:
Theorem 2.4Assume that
provided that either
(i) (1.2) holds or
However, it follows from the result due to Werbowski [, 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.
where is sufficiently large. Assume first that (2.1) holds. As in the proof of the part (i), one arrives at a contradiction with the condition (2.4). Suppose now that (2.8) holds. For , define a new function by
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
(see Zhang and Wang [, 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. [, Theorem 5.3].
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 Philos-type condition.
Proof Assuming that is an eventually positive solution of (1.1) that satisfies (2.7) and proceeding as in the proof of Theorem 2.4, we arrive at the inequality (2.12) which holds for all . Multiplying (2.12) by and integrating the resulting inequality from to t, we obtain
Using the inequality
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.
Proof Assuming again that is an eventually positive solution of (1.1) that satisfies (2.7) and proceeding as in the proof of Theorem 2.4, we obtain (2.12) which holds for all . By virtue of Lemma 2.3, we conclude that (2.14) is non-oscillatory, which is a contradiction. The proof is complete. □
The following example illustrates possible applications of theoretical results obtained in the previous section.
It is not difficult to verify that (1.4) holds and
Remark 3.2 Note that Theorems 2.4, 2.7, and 2.8 ensure that every solution of (1.1) is either oscillatory or satisfies (2.5) and, unfortunately, these results cannot distinguish solutions with different behaviors. Since the sign of the derivative is not known, it is difficult to establish sufficient conditions which guarantee that all solutions of (1.1) are just oscillatory and do not satisfy (2.5). Neither is it possible to use the technique exploited in this paper for proving that all solutions of (1.1) satisfy (2.5). Therefore, these two interesting problems remain for future research.
The authors declare that they have no competing interests.
Both authors contributed equally to this work and are listed in alphabetical order. They both read and approved the final version of the manuscript.
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.
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, 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
Liu, S, Zhang, Q, Yu, Y: Oscillation of even-order half-linear 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 second-order 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 first-order 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 higher-order delay differential equations with p-Laplacian 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 higher-order half-linear delay differential equations. Appl. Math. Lett.. 24, 1618–1621 (2011). Publisher Full Text
Zhang, Q, Liu, S, Gao, L: Oscillation criteria for even-order half-linear functional differential equations with damping. Appl. Math. Lett.. 24, 1709–1715 (2011). Publisher Full Text
Zhang, S, Wang, Q: Oscillation of second-order nonlinear neutral dynamic equations on time scales. Appl. Math. Comput.. 216, 2837–2848 (2010). Publisher Full Text