We obtain several oscillation criteria for a class of second-order nonlinear neutral differential equations. New theorems extend a number of related results reported in the literature and can be used in cases where known theorems fail to apply. Two illustrative examples are provided.
Keywords:oscillation; second-order; neutral differential equation; integral averaging
In this paper, we are concerned with the oscillation of a class of nonlinear second-order neutral differential equations
By a solution of equation (1) we mean a continuous function defined on an interval such that is continuously differentiable and satisfies (1) for . We consider only solutions satisfying and tacitly assume that equation (1) possesses such solutions. A solution of (1) is called oscillatory if it has arbitrarily large zeros on ; otherwise, it is called nonoscillatory. We say that equation (1) is oscillatory if all its continuable solutions are oscillatory.
During the past decades, a great deal of interest in oscillatory and nonoscillatory behavior of various classes of differential and functional differential equations has been shown. Many papers deal with the oscillation of neutral differential equations which are often encountered in applied problems in science and technology; see, for instance, Hale . It is known that analysis of neutral differential equations is more difficult in comparison with that of ordinary differential equations, although certain similarities in the behavior of solutions of these two classes of equations are observed; see, for instance, the monographs [2-4], the papers [5-22] and the references cited there.
although several authors were concerned with the oscillation of equation (1) in the case where
In particular, Xu and Meng [, Theorem 2.3] established sufficient conditions for the oscillation of (1) assuming that
Further results in this direction were obtained by Ye and Xu  under the assumptions that
see also the paper by Han et al. where inaccuracies in  were corrected and new oscillation criteria for (1) were obtained [, Theorems 2.1 and 2.2]. We conclude this brief review of the literature by mentioning that Li et al. and Sun et al. extended the results obtained in  to Emden-Fowler neutral differential equations and neutral differential equations with mixed nonlinearities.
Our principal goal in this paper is to derive new oscillation criteria for equation (1) without requiring restrictive conditions (4) and (5). Developing further ideas from the paper by Hasanbulli and Rogovchenko  concerned with a particular case of equation (2) with , we study the oscillation of (1) in the case where .
2 Oscillation criteria
(ii) H has a nonpositive continuous partial derivative with respect to the second variable satisfying
In order to establish our main theorems, we need the following auxiliary result. The first inequality is extracted from the paper by Jiang and Li [, Lemma 5], whereas the second one is a variation of the well-known Young inequality .
Then equation (1) is oscillatory.
Proof Let be a nonoscillatory solution of (1). Since γ is a quotient of two odd positive integers, is also a solution of (1). Hence, without loss of generality, we may assume that there exists a such that , , and for all . Then , and by virtue of
It follows from (14) and (15) that
Define a generalized Riccati substitution by
Differentiating (18) and using (16) and (17), one arrives at
By virtue of Lemma 1, part (i), we have the following estimate:
It follows now from (19) and (20) that
Application of Lemma 1, part (ii), yields
Hence, by the latter inequality and (22), we have
which contradicts (8).
Hence, we have
Differentiating (25), we have
Letting in Lemma 1, part (i),
It follows from (24) and (26) that
Letting in Lemma 1, part (ii),
we conclude that
Using the latter inequality and (28), we have
Proceeding as in the proof of Case 1, we obtain contradiction with our assumption (9). Therefore, equation (1) is oscillatory. □
equation (1) is oscillatory.
Proof Without loss of generality, assume again that (1) possesses a nonoscillatory solution such that , , and on for some . From the proof of Theorem 2, we know that there exists a such that either or for all .
Assume now that
which implies that
Since η is an arbitrary positive constant,
but the latter contradicts (36). Consequently,
and, by virtue of (35),
which contradicts (33).
As an immediate consequence of Theorem 3, we have the following result.
Theorem 4Let, , , andbe as in Theorem 3, and assume that conditions (H1)-(H4), (3), and (6) are satisfied. Suppose also that there exist functions, , such that (30), (33), and (34) hold. If, for alland for some,
equation (1) is oscillatory.
Most oscillation results reported in the literature for neutral differential equation (1) and its particular cases have been obtained under the assumption (2) which significantly simplifies the analysis of the behavior of for a nonoscillatory solution of (1). In this paper, using a refinement of the integral averaging technique, we have established new oscillation criteria for second-order neutral delay differential equation (1) assuming that (3) holds.
We stress that the study of oscillatory properties of equation (1) in the case (3) brings additional difficulties. In particular, in order to deal with the case when (which is simply eliminated if condition (2) holds), we have to impose an additional assumption . In fact, it is well known (see, e.g., [6,14]) that if is an eventually positive solution of (1), then
One of the principal difficulties one encounters lies in the fact that (44) does not hold when (3) is satisfied, cf.. Since the sign of the derivative is not known, our criteria for the oscillation of (1) include a pair of assumptions as, for instance, (8) and (9). On the other hand, we point out that, contrary to [8,13,18,19], we do not need in our oscillation theorems quite restrictive conditions (4) and (5), which, in a certain sense, is a significant improvement compared to the results in the cited papers. However, this improvement has been achieved at the cost of imposing condition (6).
Therefore, two interesting problems for future research can be formulated as follows.
(P1) Is it possible to establish oscillation criteria for (1) without requiring conditions (4), (5), and (6)?
The authors declare that they have no competing interests.
All three authors contributed equally to this work and are listed in alphabetical order. They all read and approved the final version of the manuscript.
The research of TL and CZ was supported in part by the National Basic Research Program of PR China (2013CB035604) and the NNSF of PR China (Grants 61034007, 51277116, and 51107069). YR acknowledges research grants from the Faculty of Science and Technology of Umeå University, Sweden and from the Faculty of Engineering and Science of the University of Agder, Norway. TL would like to express his gratitude to Professors Ravi P. Agarwal and Martin Bohner for support and useful advices. Last but not least, the authors are grateful to two anonymous referees for a very thorough reading of the manuscript and for pointing out several inaccuracies.
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