Oscillation Criteria for Third-Order Emden–Fowler Differential Equations with Unbounded Neutral Coefficients

Abstract

New sufficient conditions for the oscillation of all solutions to a class of third-order Emden–Fowler differential equations with unbounded neutral coefficients are established. The criteria obtained essentially improve related results in the literature. In particular, as opposed to known results, new criteria can distinguish solutions of third-order differential equations with different behaviors. Examples are also provided to illustrate the results.

1. Introduction

This paper is concerned with the oscillation of solutions of the third-order Emden–Fowler neutral differential equationwhere , is the ratio of odd positive integers. Throughout, the following conditions are assumed to hold: (C1) are continuous functions, , for large , , and is not identically zero for large ;(C2) are continuous functions, , is strictly increasing, and .

The study of (1) is important due to the further development of the oscillation theory and its practical reasons. Emden–Fowler differential equations have numerous applications in physics (mathematical, theoretical, and chemical physics) and engineering; see, e.g., the papers by Agarwal et al. [1], Li and Rogovchenko [2–5], and Wong [6].

By a solution of (1) we mean a continuous function , , such that and satisfies (1) on . We consider only proper solutions of (1) that satisfy for all . Furthermore, we tacitly suppose that (1) possesses such solutions. Such a solution of (1) is said to be oscillatory if it has arbitrarily large zeros on ; i.e., for any , there exists a such that ; otherwise, it is called nonoscillatory, i.e., if it is either eventually positive or eventually negative. Equation (1) is said to be oscillatory if all its proper solutions oscillate.

In recent years, there has been much research activity concerning the oscillation and asymptotic behavior of solutions to various classes of third-order neutral differential equations. We refer the reader to the papers [2, 7–17] and the references contained therein as examples of recent results on this topic. However, the sufficient conditions established in these papers except [10, 13] ensure that every solution of equations either oscillates or converges to zero as . This means that these results cannot distinguish solutions with different behaviors. On the other hand, the papers [2, 7–17] were concerned with the case where is bounded, i.e., the cases where , , and were considered. In view of the observations above, we wish to develop new sufficient conditions which not only ensure oscillation of (1) but also can be applied to the case where is unbounded. We would like to point out that only a few results are known regarding oscillatory and asymptotic behavior of third-order neutral differential equations for unbounded ; see, e.g., the papers [18–20], where the Riccati transformation technique and comparison method were used to obtain the results. A similar observation as above is valid for these papers as well, i.e., the sufficient conditions established in these papers cannot distinguish solutions with different behaviors too.

Consequently, our work is of significance because of the above-mentioned reasons. Moreover, the results obtained in this paper can easily be extended to more general third-order differential equations with unbounded neutral coefficients to derive more general oscillation results. It is our belief that the present paper will contribute significantly to the study of oscillatory behavior of solutions of third-order neutral differential equations. In the sequel, all functional inequalities are supposed to hold eventually.

2. Main Results

We begin with the following lemmas that will play an important role in establishing our main results. For notational purposes, we let where is a constant and is the inverse function of .

Lemma 1 (see [21]). Let the function satisfy , and eventually. Then, for every , eventually.

Lemma 2. Let conditions and be satisfied and assume that is an eventually positive solution of (1). Then for sufficiently large , either (I), , , and ; or(II), , , and .

Proof. The proof is not difficult and so is omitted.

Theorem 3. In addition to conditions and , assume that there exists a function such that for . If for some constants , the two first-order delay differential equationsandoscillate, then (1) oscillates.

Proof. Let be a nonoscillatory solution of (1). Since is also a solution of (1), without loss of generality, we may suppose that there exists a such that, for , , , and . It follows from Lemma 2 that satisfies either case or case .
Assume first that case holds. By virtue of the definition of , we conclude thatTaking into account and Lemma 1 with , we deduce that, for every ,which yields Hence, is nonincreasing for sufficiently large . It follows from and the monotonicities of and thatUsing (8) in (5), we arrive at and thusCombining (1) and (10), we obtainIt follows now from (6) and (11) thatLetting , we haveand inequality (12) can be written asCombining (13) and Lemma 1 with , we get, for every , and soUsing (16) in (14), we deduce that Letting , we see that is a positive solution of the first-order delay differential inequality Therefore, by [22, Theorem 1], we conclude that, for every , (3) has a positive solution, which contradicts the fact that (3) oscillates.
Next, suppose that case holds. Since is strictly decreasing and , we haveUsing (19) in (5), we conclude that and thusSubstitution of (21) into (1) implies thatSince , , , and , for , one can easily arrive atBy virtue of and the fact that is strictly increasing, we deduce that . Substitute and into (23) to obtain Using (24) in (22), we get Letting , we see that is a positive solution of the first-order delay differential inequality The rest of the proof is similar to that of case and hence is omitted. This completes the proof.

From [23], it is well known that if then the first-order delay differential equation oscillates, where , , , and . Therefore, by virtue of Theorem 3, we have the following result.

Corollary 4. Let conditions and be satisfied and . Assume that there exists a function such that for . If for some constant ,andthen (1) oscillates.

Corollary 5. Let conditions and be satisfied and . Assume that there exists a function such that for . If for some constant ,andthen (1) oscillates.

Proof. Applications of (31), (32), and [24, Theorem 2] imply that (3) and (4) oscillate. Hence, by Theorem 3, (1) oscillates.

Next, we present the following interesting result for which we need to assume that the function in condition is nondecreasing.

Theorem 6. In addition to conditions and , assume that the function with is nondecreasing on . If for some constant ,andthen (1) oscillates.

Proof. Let be a nonoscillatory solution of (1). Without loss of generality, we may suppose that there exists a such that, for , , , and . It follows from Lemma 2 that satisfies either case or case .
Assume that case holds. Proceeding as in the proof of Theorem 3, we deduce that (13), (14), and (16) hold for every . Integrating (14) from to , and letting , we obtainUsing (16) in (35), we conclude that which yieldsUsing (13) and the fact that , we have and so inequality (37) implies that i.e., Taking as in (40), we obtain a contradiction to (33).
Next, let case hold. Then, we arrive at (22) and (23). For , we see that . Putting and into (23), we getIntegrating (22) from to and using (41), we obtain which can be written as Taking as in (43), we obtain a contradiction to (34). The proof is complete.

We conclude this paper with the following examples and remarks to illustrate the main results. The first example is concerned with the case where as , whereas the second example is concerned with the case where is a bounded function.

Example 7. Consider the sublinear Emden–Fowler neutral differential equationwhere and are constants. Here , , , , , , and . Let . Then and condition (31) becomes Letting , then and condition (32) reduces to Therefore, by Corollary 5, (44) oscillates.

Example 8. Consider the linear differential equationHere , , , , , and is a constant. It is easy to deduce that , , , and for some constant . Using in (33), we have That is, condition (33) with holds.

Next, using in (34), we obtain That is, condition (34) with holds. Therefore, by Theorem 6, (47) oscillates.

Remark 9. For a class of third-order Emden–Fowler delay differential equations with unbounded neutral coefficients (1), we established new oscillation criteria which complement and improve results in the cited papers because these criteria apply also in the case where is unbounded and ensure that all solutions of (1) are oscillatory (that is, these results can distinguish solutions with different behaviors).

Remark 10. Using different methods, we improve results of Li and Rogovchenko [2] by removing restrictive condition , which, in a certain sense, is a significant improvement compared to the results reported in the cited papers.

Remark 11. Combining Theorem 3 and the results obtained in [25], one can derive various oscillation criteria for (1) in the linear case. To study the oscillation of (1) in the superlinear case, it would be of interest to establish oscillation criteria for (3) and (4) assuming that .

Remark 12. In the conclusion of Lemma 1, the existence of the constant is necessary in some cases. For instance, for , if , then , , and , and so the function does not satisfy the conclusion of Lemma 1 provided that there is no . On the basis of Lemma 1, one can easily revisit the results reported in [26–28].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All five authors contributed equally to this work and are listed in alphabetical order. They all read and approved the final version of the manuscript.

Acknowledgments

This research is supported by NNSF of P. R. China (Grant No. 61503171), CPSF (Grant No. 2015M582091), and NSF of Shandong Province (Grant No. ZR2016JL021). The research of the third author is supported by the grant project KEGA 035TUKE-4/2017.