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
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.
In the special case when , (1.1) reduces to a two-term differential equation
which was studied by Zhang et al. who established the following result.
Theorem 1.1 ([, Corollary 2.1])
and assume that . Suppose also 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  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.
Lemma 2.1Assume that (1.2) is satisfied and let be an eventually positive solution of (1.1). Then there exists a sufficiently large such that, for all ,
Proof Let be an eventually positive solution of (1.1). Then there exists a such that and for all . By virtue of (1.1),
which means that the function
is decreasing for . Therefore, does not change sign eventually, that is, there exists a such that either or for all .
We claim that for all . Otherwise, there should exist a such that
and, for all ,
Inequality (2.3) yields
Integrating this inequality from to t, , we conclude that
Passing to the limit as and using (1.2), we deduce that
It follows now from the inequalities and that , which contradicts our assumption that . Finally, write (2.2) in the form
which implies that . This completes the proof. □
Lemma 2.2 (Agarwal et al.)
Assume that , is non-positive for all largetand not identically zero on . If , then for every , there exists a such that
holds on .
Lemma 2.3 (Agarwal et al.)
where is a quotient of odd natural numbers, , and is non-oscillatory if and only if there exist a number and a function such that, for all ,
For a compact presentation of our results, we introduce the following notation:
Theorem 2.4Assume that
Then every solution of (1.1) is either oscillatory or satisfies
provided that either
(i) (1.2) holds or
(ii) (1.4) is satisfied and, for some ,
Proof Assume that (1.1) has a non-oscillatory solution which is eventually positive and such that
Case (i) By Lemma 2.1, we conclude that (2.1) holds for all , where is sufficiently large. It follows from Lemma 2.2 that
for every and for all sufficiently large t. Let
By virtue of (1.1), we conclude that is a positive solution of a differential inequality
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.
Case (ii) Similar analysis to that in Lemma 2.1 leads to the conclusion that a non-oscillatory positive solution with the property (2.7) satisfies, for , either conditions (2.1) or
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
Then for . Since
we deduce that the function is decreasing. Thus, for ,
Dividing both sides of (2.10) by and integrating the resulting inequality from t to T, we obtain
Letting and taking into account that and , we conclude that
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 and for all sufficiently large t. Therefore, (2.11) yields
Multiplying (2.12) by and integrating the resulting inequality from to t, we have
Let and . Using the fact that and the inequality
(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].
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.
To this end, let . We say that a function belongs to the class if
and H has a non-positive continuous partial derivative with respect to the second variable satisfying the condition
for some function .
Theorem 2.7Let be as in Theorem 2.4 and suppose that (1.4) and (2.4) hold. Assume that there exists a function such that
for all and for some . Then the conclusion of Theorem 2.4 remains intact.
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.
Theorem 2.8Let be as above, and assume that (1.4) and (2.4) hold. If a second-order half-linear ordinary differential equation
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.
Example 3.1 For , consider the third-order differential equation
It is not difficult to verify that (1.4) holds and
Let . Then , , , and thus
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 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.
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