Abstract

The purpose of this work is to derive sufficient conditions for the oscillation of all solutions of the third-order functional dynamic equation on a time scale . In addition, we present some Hille-type conditions for generalized third-order dynamic equations that improve and extend significant contributions reported in the literature without imposing time-scale restrictions. An example is given to demonstrate the essential results.

1. Introduction

Oscillatory criteria of solutions to dynamic equations on time scales are gaining interest due to their applications in engineering and natural sciences. Eventually, this kind of study aids in comprehending the geometric behavior of solutions. We are interested in the oscillatory and asymptotic behavior of the third-order functional dynamic equation in the form of on an arbitrary time scale with , where , , , , , , , and such that and on and , , such that

By a solution of equation (1), we mean a nontrivial real–valued function for some for a positive constant such that , and satisfies equation (1) on where is the space of right-dense continuous functions. Solutions that disappear near infinity will not be considered. A solution of (1) is oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonnoscillatory. We presume that the reader has a basic understanding of time scales and notation; see [1–4] for providing a great introduction to time scale calculus.

Hille [5] investigated the oscillatory behavior of second-order linear differential equation and shown that if then every solution of (3) is oscillatory. The results in [6–13] generalized the Hille-type criterion for different forms of second-order dynamic equations. Regarding third-order dynamic equations, [14–18, 32] established several Hille-type oscillation criteria for different forms of third-order dynamic equations under some restrictive times, which ensure that the solutions are either oscillatory or nonoscillatory and converge to a finite limit under various restrictive conditions. Recently, Hassan et al. [19] improved the Hille-type criteria in [14–18, 32] for equation (1) with , see ([19], in Discussions and Conclusions) for a good comparison between these results. Some of these results in [19] are as follows.

Theorem 1 (see [19]). Assume that , every solution of equation (1) is either oscillatory or convergent if
(1) (2) where with

The reader is directed to papers [20–33] as well as the sources listed therein.

Contrary to [14–16, 19], we are concerned in this paper in deducing sufficient oscillation criteria that guarantee that all solutions of nonlinear third-order dynamic equation (1) are oscillatory when and without imposing restrictive on the time scales. This solves an open problem posed in [1] (Remark 3.3). Furthermore, we will propose certain Hille-type conditions for generalized third-order dynamic equation (1) for the cases and that improve and extend relevant significant contributions reported in [14–16, 18, 19] without extra imposing time-scale constraints. All functional inequalities reported in this paper are considered to hold eventually, that is, for all sufficiently large .

2. Main Results

Throughout this paper, we let

Before stating the main results, we will present some preliminary lemmas to aid in the proving of the main results.

Lemma 2. Assume that equation (1) has a solution such that

Then, for

Proof. Suppose, without losing generality, that on . From (1) and (10), we have for From (13), we get for and Therefore, So, Integrating the previous inequality with respect to from to yields that Thus, Thus, (12) holds for . The proof is now complete.

Lemma 3. Assume that equation (1) has a solution such that then for and

Proof. Suppose, without losing generality, that on . By dint of (13), on . Therefore, Thus, (21) holds for . From (24), we have Replacing by in (25) and integrating with respect to from to gives Thus, (22) holds for . By virtue of (24), there is a such that Hence, for , Thus, (23) holds for . The proof is now complete.

2.1. Asymptotic Behavior

In this subsection, we debate the asymptotic behavior of the solutions of equation (1) for both cases and

Theorem 4. Assume that and for sufficiently large , If equation (1) has a nonoscillatory solution , then and converge.

Proof. Suppose, without losing generality, that and are eventually positive, by virtue of (1), we deduce that and are eventually of one sign and by (2), and we can easily see that is eventually positive, see [34], part of the proof of Theorem 4]. Therefore, we consider the following two cases:
is eventually positive. In this case, there is a such that Consider Hence, (1)If , by means of the Pötzsche chain rule and the definitions of and , we obtain for Using the fact that for , we get Therefore, (33) becomes Now, for any , there exists a such that where Substituting (36) into (35), we get for , (2)If , by means of the Potzsche chain rule and the definitions of and , we obtain for Substituting (36) into (39), we get for , Combining (38) with (40), we conclude that for and , From (1), we see that If by the fact that for then whereas if by the fact that for then Now, consider the case when . From (23) and using the fact that , we deduce that Next consider the case when . From (23) and using the fact that , we conclude that By combining (45) with (46), we have From (43), (44), and (47), we get for where Substituting (47) into (42), we obtain for , Integrating (49) from to , we have Taking into consideration that and passing to the limit as , we conclude that Multiplying both sides of (51) by , we get for , Taking the of both sides of the last inequality (52) as , we obtain By dint of the fact that is arbitrary, we deduce that Setting and . Then, using inequality (see [35]) we achieve Thus, (54) becomes as a result of which there is a contradiction with (29).
is eventually positive. In this case, there is a such that By virtue of (59), it is easy to see that and converge. This completes the proof.

Remark 5. [1] The deduction of Theorem 4 keeps intact if assumption (29) is replaced by 2) If either or then nonoscillatory solutions of the investigated equation (1) are convergent to zero, see [35], [Theorem 4].

2.2. Oscillation Criteria

In this subsection, we establish oscillation criteria of the solutions of equation (1) when and on . This solves an open problem posed in [1].

Theorem 6. Assume that (29) and either or hold. Then, all solutions to equation (1) are oscillatory.

Proof. Suppose, without losing generality, that and are eventually positive, by virtue of (1), we deduce that and are eventually of one sign and by (2), we can easily see that is eventually positive. Therefore, we consider the following two cases:
is eventually positive. The same proof as in part of the proof of Theorem 4 hence is omitted.
is eventually positive. In this case, there is a such that (59) for
Let (63) hold. Integrating (1) form to gives By using Lemma 2 and setting and we obtain Substituting (66) into (65) yields We get a contradiction with (63) by taking the on both sides of the last inequality as .
Let (64) hold. Integrating (1) form to , we get Subsequently, Integrating the last inequality form to , we conclude that As a result, Integrating again the last inequality form to , we achieve We get a contradiction with (64) by taking the on both sides of the last inequality as . The proof is complete.

Example 7. Consider the third-order delay dynamic equation

It is obvious that condition (2) is fulfilled. Now,

Therefore, the conditions (63) and (29) are satisfied. Then, all solutions to equation (73) are oscillatory, according to Theorem 6.

3. Conclusions

(1)The important point to note here is that the proposed results in Theorem 6 are new for third-order dynamic equation (1) and also, these results, in a special case, have answered an open problem stated by [1] (Remark 3.3), which is concerned with studying the sufficient conditions that guarantee that all solutions of third-order delay dynamic equations oscillate(2)In comparison to the results in the cited papers [14–16, 18], Hille-type criteria for dynamic equation (1) in the cases , and have been developed, and the results in this study are a significant improvement; for more details, see ([19], in Discussions and Conclusions). Moreover, our results improve and expand upon those described in [19], see the following details(i)If and , then condition (29) becomes

By dint of

Hille-type criterion (75) improves (5); (ii)Condition (29) reduces to (6) in the case where and (3)The asymptotic behavior of solutions is viable to dynamic equation (1) for both and , whereas the oscillation conditions are viable to dynamic equation (1) for . As a result, oscillations can be ensured by a delay in equations(4)The results presented here are for equation (1) on an unbounded above arbitrary time scale; therefore, they are applicable to different of time scales(5)It would be interesting to define Hille-type criteria for the third-order dynamic equation (1) under noncanonical assumptions

Data Availability

The numerical data used to support the findings of this study are included in the article.

Conflicts of Interest

The authors declare that they have no competing interests. There are no any nonfinancial competing interests (political, personal, religious, ideological, academic, intellectual, commercial, or any other) to declare in relation to this manuscript.

Authors’ Contributions

Hassan oversaw the study and help inspection. All the authors carried out the main results of this article and drafted the manuscript and read and approved the final manuscript.

Acknowledgments

This research has been funded by Scientific Research Deanship at University of Ha’il, Saudi Arabia through project number RG-21008.