Asymptotic behavior of an odd-order delay differential equation
© Li and Rogovchenko; licensee Springer. 2014
Received: 29 January 2014
Accepted: 22 April 2014
Published: 9 May 2014
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.
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.
where and is an odd natural number, is a ratio of odd natural numbers, , , , , , , , and .
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])
Then every solution of (1.3) is either oscillatory or converges to zero as .
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.
is decreasing for . Therefore, does not change sign eventually, that is, there exists a such that either or for all .
which implies that . This completes the proof. □
Lemma 2.2 (Agarwal et al. )
holds on .
Lemma 2.3 (Agarwal et al. )
(1.2) holds or
- (ii)(1.4) is satisfied and, for some ,(2.6)
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.
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].
Remark 2.6 For , Theorem 2.4 includes Theorem 1.1.
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.
for some function .
for all and for some . Then the conclusion of Theorem 2.4 remains intact.
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.
is oscillatory for some , then the conclusion of Theorem 2.4 remains intact.
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.
for some . Hence, by Theorem 2.4, every solution of (3.1) is either oscillatory or satisfies (2.5). As a matter of fact, is a solution of this equation satisfying condition (2.5).
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 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.
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