1. Introduction

The study of dynamic equations on time-scales, which goes back to its founder Hilger [1], is an area of mathematics that has recently received a lot of attention. It has been created in order to unify the study of differential and difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time-scales reveals such discrepancies, and helps avoid proving results twice—once for differential equations and once again for difference equations.

Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [2], Bohner and Guseinov [3], and references cited therein. A book on the subject of time-scales, by Bohner and Peterson [4], summarizes and organizes much of the time-scale calculus; see also the book by Bohner and Peterson [5] for advances in dynamic equations on time-scales.

In the recent years, there has been increasing interest in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of various equations on time-scales; we refer the reader to the papers [6–38]. To the best of our knowledge, it seems to have few oscillation results for the oscillation of third-order dynamic equations; see, for example, [14–16, 21, 35]. However, the paper which deals with the third-order delay dynamic equation is due to Hassan [21].

Hassan [21] considered the third-order nonlinear delay dynamic equations

where is required, and the author established some oscillation criteria for (1.1) which extended the results given in [16].

To the best of our knowledge, there are no results regarding the oscillation of the solutions of the following third-order nonlinear neutral delay dynamic equations on time-scales up to now:

We assume that is a quotient of odd positive integers, and are positive real-valued rd-continuous functions defined on such that the delay functions are rd-continuous functions such that and

As we are interested in oscillatory behavior, we assume throughout this paper that the given time-scale is unbounded above. We assume and it is convenient to assume We define the time-scale interval of the form by .

For the oscillation of neutral delay dynamic equations on time-scales, Mathsen et al. [26] considered the first-order neutral delay dynamic equations on time-scales

and established some new oscillation criteria of (1.3) which as a special case involve some well-known oscillation results for first-order neutral delay differential equations.

Agarwal et al. [7], Şahíner [28], Saker [31], Saker et al. [33], Wu et al. [34] studied the second-order nonlinear neutral delay dynamic equations on time-scales

by means of Riccati transformation technique, the authors established some oscillation criteria of (1.4).

Saker [32] investigated the second-order neutral Emden-Fowler delay dynamic equations on time-scales

and established some new oscillation for (1.5).

Our purpose in this paper is motivated by the question posed in [26]: What can be said about higher-order neutral dynamic equations on time-scales and the various generalizations? We refer the reader to the articles [23, 24] and we will consider the particular case when the order is 3, that is, (1.2). Set By a solution of (1.2), we mean a nontrivial real-valued function satisfying and and satisfying (1.2) for all

The paper is organized as follows. In Section 2, we apply a simple consequence of Keller’s chain rule, devoted to the proof of the sufficient conditions which guarantee that every solution of (1.2) oscillates or converges to zero. In Section 3, some examples are considered to illustrate the main results.

2. Main Results

In this section we give some new oscillation criteria for (1.2). In order to prove our main results, we will use the formula

where is delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller’s chain rule (see Bohner and Peterson [4, Theorem ]).

Before stating our main results, we begin with the following lemmas which are crucial in the proofs of the main results.

For the sake of convenience, we denote: for Also, we assume that

Lemma 2.1. Assume that holds. Further, assume that is an eventually positive solution of (1.2). If then there are only the following three cases for sufficiently large: , , ,
or
,

Proof. Let be an eventually positive solution of (1.2). Then there exists such that and for all From (1.2) we have Hence is strictly decreasing on We claim that eventually. Assume not, then there exists such that Then we can choose a negative and such that Dividing by and integrating from to we have Letting then by (2.2). Thus, there is a such that for Integrating the previous inequality from to we obtain Therefore, there exist and such that We can choose some positive integer such that for Thus, we obtain The above inequality implies that for sufficiently large which contradicts the fact that eventually. Hence we get It follows from this that either or Since which yields If then there are two possible cases:(1) eventually; or(2) eventually.
If there exists a such that case (2) holds, then exists, and We claim that Otherwise, We can choose some positive integer such that for Thus, we obtain
which implies that and from the definition of we have which contradicts Now, we assert that is bounded. If it is not true, there exists with as such that From which implies that it contradicts that Therefore, we can assume that By we get which implies that so hence,
Assume that We claim that eventually. Otherwise, we have or By there exists we can choose some positive integer such that for and we obtain
which implies that and from the definition of we have which contradicts or Now, we have that here is finite. We assert that is bounded. If it is not true, there exists with as such that From which implies that it contradicts that Therefore, we can assume that By we get which implies that so hence, This completes the proof.

In [4, Section ] the Taylor monomials are defined recursively by

It follows from [4, Section ] that for any time-scale, but simple formulas in general do not hold for

Lemma 2.2 (see [15, Lemma ]). Assume that satisfies case (i) of Lemma 2.1. Then

Lemma 2.3. Assume that is a solution of (1.2) satisfying case (i) of Lemma 2.1. If then satisfies eventually

Proof. Let be a solution of (1.2) such that case of Lemma 2.1 holds for Define Thus We claim that eventually. Otherwise, there exists such that for Therefore, which implies that is strictly increasing on Pick such that for Then we have then for By Lemma 2.2, for any there exists such that Hence there exists so that By the definition of we have that From (1.2), we obtain Integrating both sides of (2.35) from to we get which yields that which contradicts (2.26). Hence and is nonincreasing. The proof is complete.

Lemma 2.4. Assume that holds and is a solution of (1.2) which satisfies case (iii) of Lemma 2.1. If where for then

Proof. Let be a solution of (1.2) such that case of Lemma 2.1 holds for Then , Next we claim that Otherwise, there exists such that for all By the definition of we have that (2.35) holds. Integrating both sides of (2.35) from to we get Integrating again from to we have Integrating again from to we obtain which contradicts (2.38), since by [23, Lemma ] and [3, Remark ], we get Hence and completes the proof.

Theorem 2.5. Assume that (2.2), (2.26), and (2.38) hold, Furthermore, assume that there exists a positive function such that for some and for all constants where Then every solution of (1.2) oscillates or

Proof. Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that and for all Then by Lemma 2.1, satisfies three cases. Assume that satisfies case Define the function by Then Using the product rule, we have By the quotient rule, we get By the definition of and (1.2), we obtain (2.35). From (2.35) and (2.44), we have from (2.25) and (2.27), for any we obtain hence by (2.48), we have