#### Abstract

A class of third-order nonlinear delay dynamic equations on time scales is studied. By using the generalized Riccati transformation and the inequality technique, four new sufficient conditions which ensure that every solution is oscillatory or converges to zero are established. The results obtained essentially improve earlier ones. Some examples are considered to illustrate the main results.

#### 1. Introduction

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various equations on time scales, and we refer the reader to the studies by Bohner and Saker [1] and Erbe et al. [2, 3]. And there are some results dealing with oscillatory behavior of second-order delay dynamic equations on time scales [4–10]. However, there are few results dealing with the oscillation of the solutions of third-order delay dynamic equations on time scales, we refer the reader to the papers [11–14].

In this paper, we consider new oscillatory behavior of all solutions of the third-order nonlinear delay dynamic equation where is the ratio of two positive odd integers.

Throughout this paper, we will assume the following hypotheses.) is a time scale (i.e., a nonempty closed subset of the real numbers ) which is unbounded above, and with , we define the time scale interval of the form by .(), , are positive and real-valued rd-continuous functions defined on , and , satisfy () is a strictly increasing and differentiable function, such that () is a continuous function and there exists some positive constant such that for all .

By a solution of (1), we mean a nontrivial function satisfying (1) which has the properties for , and . Our attention is restricted to those solutions of (1) which satisfy for all , where is the space of -continuous functions. A solution of (1) is said to be oscillatory on if it is neither eventually positive nor eventually negative. Otherwise it is called nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.

If , , then (1) simplifies to the third-order nonlinear dynamic equation

If, furthermore, , , , then (1) reduces to the third-order linear dynamic equation

If, in addition, , then (1) reduces to the nonlinear delay dynamic equation

In 2005, Erbe et al. [11] considered the general third-order nonlinear dynamic equation (4). By employing the generalized Riccati transformation techniques, they established some sufficient conditions which ensure that every solution of (4) is oscillatory or converges to zero. In 2007, Erbe et al. [12] studied the third-order linear dynamic equation (5), and they obtained Hille and Nehari type oscillation criteria for (5). In 2011, Han et al. [13] extended and improved the results of [12], meanwhile obtaining some oscillatory criteria for (6). In 2014, Gao et al. [14] extended some results of [12, 13] to (1). On this basis, we continue to discuss the oscillation of solutions of (1). By using the generalized Riccati transformation and the inequality technique, we obtain some new sufficient conditions which guarantee that every solution of (1) is oscillatory or converges to zero. Our results will improve some results that have been established in [11–14].

Throughout this paper, we will make use of the following product and quotient rules: For and a differentiable function , the Cauchy integral of is defined by The integration by parts formula reads and improper integrals are defined in the usual way by For more details, see [15, 16].

#### 2. Several Lemmas

In this section we present several lemmas that will be needed in the proofs of our results in Section 3.

Lemma 1. *Assume that is an eventually positive solution of (1), then there exists such that either
**
or
*

*Proof. *Assume that is an eventually positive solution of (1), then there exists such that and for all . From (1); we obtain
Hence, is decreasing and therefore eventually of one sign, so is either eventually positive or eventually negative. We assert that for all .

If there exists such that , we get
Let , then
Integrating (16) from to provides
Then there exists such that . Similarly, , we obtain
which contradicts with . So ; this implies that or for all . This completes the proof.

Lemma 2 (see [17]). *Assume that () and the following conditions hold:*(I)*, where for some ;*(II)*, , for .**Then, for each , there exists a constant , such that
*

Lemma 3 (see [15]). *If is differentiable, then
*

Lemma 4 (see [12]). *Assume that satisfies
**
Then
**
where the Taylor monomials are defined recursively by
*

Lemma 5 (see [18]). *Assume that and are nonnegative real numbers. Then
**
where the equality holds if and only if .*

Lemma 6. *Assume that is an eventually positive solution of (1) which satisfies case (II) in Lemma 1, if either
**
or
**
Then .*

*Proof. *Assume that is an eventually positive solution of (1) which satisfies case (II) in Lemma 1. Then is decreasing and . If ; it is easy to see that there exists such that for all . From (14),

If (25) holds, then integrating (27) from to , we get
This is contrary to .

If (26) holds, then integrating (1) from to , we get
and hence,
Again, integrating this inequality from to , we obtain
Finally, integrating the last inequality from to , we get
Hence by (26), we obtain , which contradicts . Thus, we get . This completes the proof.

Lemma 7 (see [19]). *Let . Then for positive rd-continuous functions , one has
**
where and .*

#### 3. Main Results

New we state and prove the main results of this paper.

Theorem 8. *Assume that ()–(), (26), and hold. Furthermore, suppose that there exists a positive function with , and for all sufficiently large , there exists , such that
**
where . Then every solution of (1) is either oscillatory or converges to zero.*

*Proof. *Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that there exists sufficiently large , such that and for all . In the case when is eventually negative, the proof is similar. By Lemma 1, we see that satisfies either case (I) or case (II).

If case (I) holds, then , . Define the function by
Then . By the product rule (7) and the quotient rule (8), we have
Let , from case (I) in Lemma 1, we get , . In view of that
and , it is not difficult to see that . Thus, by Lemma 2, for every , there exists with , such that for all ; this implies that
By Lemma 3, we get , that is,
Using (38) and (39),
Let for all ; it is easy to see that , , , . Thus, by Lemma 4, there exists such that for all . Then, we get
From , we obtain . By (41), we get
Therefore, from (38) and (42), there exists with such that
Using (43), we get
that is,
Now, set
where , and . Using the equality (24), we obtain
From (47), we obtain
Integrating (48) from to , we get
consequently,
This is contrary to (34).

If case (II) holds, from (26), we get . This completes the proof.

Theorem 9. *Assume that ()–(), (26), and hold. Furthermore, suppose that there exist functions , where such that
**
and has a nonpositive continuous -partial derivative on with respect to the second variable and satisfies, for all sufficiently large , that there exists , such that
**
where and are defined in Theorem 8. . Then every solution of (1) is either oscillatory or converges to zero.*

*Proof. *Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that there exists sufficiently large , such that and for all . In the case when is eventually negative, the proof is similar. By Lemma 1, we see that satisfies either case (I) or case (II). If case (I) holds, we proceed as in the proof of Theorem 8 and get (45). In (45), replace by and multiply both sides by and integrate with respect to from to , ; we get
Integrating by parts using (51) and (52), we obtain
and so
Now, set
where , and . Using inequality (24), we obtain
Combining (56) and (58), we get
which contradicts (53).

If case (II) holds, from (26), we get . This completes the proof.

If (53) is not held, then we get the following result.

Theorem 10. *Assume that ()–(), (26), and hold. Furthermore, suppose that there exist functions , where , such that (51) holds, has a nonpositive continuous -partial derivative on with respect to the second variable and satisfies (52). Assume that
**
and a real rd-continuous function such that
**
for , where and are defined in Theorem 8, . Then every solution of (1) is either oscillatory or converges to zero.*

*Proof. *Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that there exists sufficiently large , such that and for all . In the case when is eventually negative, the proof is similar. By Lemma 1, we see that satisfies either case (I) or case (II).

If case (I) holds, proceeding as in the proof of Theorem 9, we get that (56) and (58) hold. Then we conclude that
From (63), we obtain
By (56), we get
We denote
meanwhile noting that (63), we obtain
Now we assert that
holds. Suppose to the contrary that
by (60), there exists a constant such that
from (71); there exists for arbitrary real number such that
By (10), we obtain
From (72), there exists , we get for , so that . Since is arbitrary, we obtain
Selecting a sequence : with satisfying
then there exists a constant such that
for sufficiently large positive integer . From (75), we can easily see
(77) implies that
From (77) and (78), we obtain
that is,
for sufficiently large positive integer , which together with (79) implies
On the other hand, by Lemma 7, we obtain
The above inequality shows that
Hence, (82) implies
This contradicts (61). Therefore (70) holds. Noting for , by using (70), we obtain
This contradicts (62). This completes the proof.

If case (II) holds, from (26), we get . This completes the proof.

Theorem 11. *Assume that ()–(), (26), (52), (60), (62), and hold, where , , and are defined in Theorem 10. Furthermore suppose that there is a real rd-continuous function such that
**
for , where is defined in Theorem 8, . Then every solution of (1) is either oscillatory or converges to zero.*

*Proof. *Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that there exists sufficiently large , such that and for all . In the case when is eventually negative, the proof is similar. By Lemma 1, we see that satisfies either case (I) or case (II).

If case (I) holds, proceeding as in the proof of Theorem 9, we get that (56) and (58) hold. We conclude that
From (88), we obtain
Using (87) and (92), we get
Thus, there exists a sequence : with such that