#### Abstract

We investigate the oscillation of the following higher order dynamic equation: , on some time scale , where , and are positive rd-continuous functions on and are the quotient of odd positive integers. We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.

#### 1. Introduction

In this paper, we investigate the oscillation of the following higher order dynamic equation: on some time scale , where , and are positive rd-continuous functions on and are the quotient of odd positive integers. Write then reduces to the following equation:

Since we are interested in the oscillatory behavior of solutions near infinity, we assume that sup and is a constant. For any , we define the time scale interval . By a solution of (2), we mean a nontrivial real-valued function , which has the property that for and satisfies (2) on , where is the space of differentiable functions whose derivative is rd-continuous. The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution of (2) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is called nonoscillatory.

The theory of time scale, which has recently received a lot of attention, was introduced by Hilger's landmark paper [1], a rapidly expanding body of the literature that has sought to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus, where a time scale is an nonempty closed subset of the real numbers, and the cases when this time scale is equal to the real numbers or to the integers represent the classical theories of differential or of difference equations. Many other interesting time scales exist, and they give rise to many applications (see [2]). The new theory of the so-called “dynamic equations” not only unifies the theories of differential equations and difference equations, but also extends these classical cases to cases “in between,” for example, to the so-called -difference equations when , which has important applications in quantum theory (see [3]). In this work, knowledge and understanding of time scales and time scale notation are assumed; for an excellent introduction to the calculus on time scales, see Bohner and Peterson [2, 4]. 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 papers [5–20].

Recently, Erbe et al. in [21–23] considered the third-order dynamic equations respectively, and established some sufficient conditions for oscillation.

Hassan [24] studied the third-order dynamic equations and obtained some oscillation criteria, which improved and extended the results that have been established in [21–23].

#### 2. Main Results

In this section, we investigate the oscillation of (2). To do this, we need the following lemmas.

Lemma 1 (see [25]). *Assume that
**
and . Then,*(1)* implies for ;*(2)* implies for .*

Lemma 2 (see [25]). * Assume that (5) holds. If and for , then there exists an integer with even such that*(1)* for and ;*(2)*if , then there exists such that for and .*

*Remark 3. *Let , and let be the set of integers. Then, Lemmas 1 and 2 are Lemma and Theorem of [26], respectively.

Lemma 4. *Assume that (5) holds. Furthermore, suppose that
**
If is an eventually positive solution of (2), then there exists sufficiently large such that*(1)* for ;*(2)*either for and or .*

*Proof. *Pick so that on . It follows from (2) that
By Lemma 2, we see that there exists an integer with even such that for and , and is eventually monotone.

We claim that implies . If not, then and , and there exist and a constant such that on . Integrating (2) from into , we get that for
Thus,
Again, integrating the above inequality from into , we obtain that for
It follows from (6) that , which is a contradiction to . The proof is completed.

Lemma 5. *Assume that is an eventually positive solution of (2) such that for and for and . Then,
**
and there exist and a constant such that
**
where
*

*Proof. *Since (), it follows that is strictly decreasing on . Then, for ,
On the other hand, we have that for ,
Thus, there exist and a constant such that
Again,
Thus, there exists a constant such that
Again,
Thus, there exists a constant such that
The rest of the proof is by induction. The proof is completed.

Lemma 6 (see [2]). *Let be continuously differentiable and suppose that is delta differentiable. Then, is delta differentiable and the formula
*

Lemma 7 (see [27]). *If are nonnegative and , then
*

Now, we state and prove our main results.

Theorem 8. *Suppose that (5) and (6) hold. If there exists a positive nondecreasing delta differentiable function such that for all sufficiently large and for any positive constants , there is a such that
**
where
**
and is as in Lemma 5. Then, every solution of (2) is either oscillatory or tends to zero.*

*Proof. *Assume that (2) has a nonoscillatory solution on . Then, without loss of generality, there is a , sufficiently large, such that for . Therefore, we get from Lemma 4 that there exists such that(i) for ;(ii)either for and or .Let for and . Consider
It follows from Lemma 6 that
Then,
From (11) and (27), we get
Now, we consider the following three cases.*Case 1*. If , then
*Case 2*. If , then it follows from (12) that there exist and a constant such that
*Case 3*. If , then
Thus,

From (27)–(32), we obtain
Integrating the above inequality from into , we have
which gives a contradiction to (23). The proof is completed.

Theorem 9. *Suppose that (5) and (6) hold. If there exists a positive nondecreasing delta differentiable function such that for all sufficiently large and for any positive constants , there is a such that
**
where
**
and are as in Lemma 5. Then, every solution of (2) is either oscillatory or tends to zero. *

*Proof. *Assume that (2) has a nonoscillatory solution on . Then, without loss of generality, there is a , sufficiently large, such that for . Therefore, we get from Lemma 4 that there exists such that(i) for ;(ii)either for and or .Let for and . Note that
From (11), we have
Then it follows from (27) that for ,
Now, we consider the following three cases.*Case 1*. If , then
*Case 2*. If , then it follows from (12) that there exist and a constant such that
Thus,
with .*Case 3*. If , then
Thus,

From (39)–(44), we obtain that for ,
Let
with . By Lemma 7, we have
Integrating the above inequality from into , it follows that
which gives a contradiction to (35). The proof is completed.

*Remark 10. *The trick used in the proofs of Theorems 8 and 9 is from [16].

Theorem 11. *Suppose that (5) and (6) hold. If for all sufficiently large ,
**
then every solution of (2) is either oscillatory or tends to zero. *

*Proof. *Assume that (2) has a nonoscillatory solution on . Then, without loss of generality, there is a , sufficiently large, such that for . Therefore, we get from Lemma 4 that there exists such that(i) for ;(ii)either for and or .Let for and . Then, for ,
It follows from (2) that
Integrating the above inequality from into , we have
which gives a contradiction to (49). The proof is completed.

Theorem 12. *Suppose that (5) and (6) hold. If for all sufficiently large ,
**
where
**
and is as in Lemma 5, then every solution of (2) is either oscillatory or tends to zero. *

*Proof. *Assume that (2) has a nonoscillatory solution on . Then, without loss of generality, there is a , sufficiently large, such that for . Therefore, we get from Lemma 4 that there exists such that(i) for ;(ii)either for and or .Let for and . Then, it follows from (2) and (11) that for ,
Using the fact that is strictly increasing on , we obtain
Thus,
Now, we consider the following three cases.*Case 1*. If , then
*Case 2*. If , then it follows from (12) that there exist and a constant such that
Thus,
with .*Case 3.* If , then
Thus,

From (57)–(62), we obtain that for ,
which gives a contradiction to (53). The proof is completed.

#### 3. Examples

In this section, we give some examples to illustrate our main results.

*Example 1. *Consider the following higher order dynamic equation:
on an arbitrary time scale with , where , and are as in (2) with , and . Then, every solution of (64) is either oscillatory or tends to zero.

*Proof. *Note that
by Example 5.60 in [4]. Pick such that
Then,
Let , sufficiently large, and such that , then
Thus, conditions (5), (6), and (49) are satisfied. By Theorem 11, every solution of (64) is either oscillatory or tends to zero.

*Example 2. *Consider the following higher order dynamic equation:
on an arbitrary time scale with , where , are as in (2) with , , and are the quotient of odd positive integers with . Then, every solution of (69) is either oscillatory or tends to zero.

*Proof. *Note that
Pick such that , then

Let with being positive constants, , and . Pick such that
Let , then
Thus,
So conditions (5), (6), and (23) are satisfied. Then, by Theorem 8, every solution of (69) is either oscillatory or tends to zero.

#### Acknowledgments

This project is supported by NNSF of China (11261005 and 51267001), NSF of Guangxi (2011GXNSFA018135 and 2012GXNSFDA276040), and SF of ED of Guangxi (2013ZD061).