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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 962590, 16 pages

http://dx.doi.org/10.1155/2013/962590

## Oscillation Criteria for Some New Generalized Emden-Fowler Dynamic Equations on Time Scales

^{1}School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China^{2}Department of Mathematics, University of Houston, Houston, TX 77204, USA

Received 4 September 2012; Revised 8 January 2013; Accepted 11 January 2013

Academic Editor: Patricia J. Y. Wong

Copyright © 2013 Haidong Liu and Puchen Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By means of novel analytical techniques, we have established several new oscillation criteria for the generalized Emden-Fowler dynamic equation on a time scale , that is, , with respect to the case and the case , where , is a constant, , is a constant satisfying , and , , and are real valued right-dense continuous nonnegative functions defined on . Noting the parameter value probably unequal to , our equation factually includes the existing models as special cases; our results are more general and have wider adaptive range than others' work in the literature.

#### 1. Introduction

In the past two decades, the theory of time scales proposed by Hilger [1] in 1990 has received extensive attention because of its advantage to unify continuous model and discrete model into one case under the scholars’ investigation. Numerous authors have considered many aspects of this new theory. Many of those results focus on oscillation and nonoscillation of some equations on time scales. Reader can refer to articles [2–25] and there references cited therein.

In this paper, we consider the oscillatory behavior of the solutions of second-order generalized Emden-Fowler dynamic equation of the form
with , parameter constant , and conditions (H_{1})–(H_{6}):(H_{1}) is a time scale which is unbounded above. , where with , denotes the collection of all functions which are right-dense continuous on ;(H_{2}), ; (H_{3});
(H_{4}), , for , , , , for , ;(H_{5}) is right-dense continuous on , and for all, where is the forward jump operator on ;(H_{6}) is a continuous function such that , for all and there exists a positive right-dense continuous function defined on such that for all and for all , where is a constant satisfying .

As a solution of (1), we mean a function such that and , and satisfying (1) for all , where denotes the set of right-dense continuously -differentiable functions on . In the sequel, we restrict our attention to those solutions of (1) which exist on the half-line and satisfy for any . We say that a nontrivial solution of (1) is oscillatory if it has arbitrary large zeros, otherwise we say that it is nonoscillatory. We say that (1) is oscillatory if all its solutions are oscillatory.

Among researchers in the oscillation of functional equations with time scales, Agarwal et al. [2] studied a special case of (1), which is where and are positive constants and is a quotient of odd positive integers. They got some oscillation criteria of (2) for the case when under the condition , and the case when under the condition . Subsequently, for the case when is an odd positive integer, Saker [7] did not require the conditions and and obtained some new oscillation results for (2) under the conditions (3).

Very Recently, in [10–13], Saker et al. have considered the oscillation of several equations with time scales. For example in paper [13], the author is concerned with the quasilinear equation of the form: where , , and are ratios of odd positive integers.

However the value range of the equation parameters in our work is wider than those in [2, 7, 10–13] and the equation itself is also different from those in [2, 7, 10–13]. In fact, our approach in constructing the criteria is different from those of Saker and his coauthors’ work.

For (2) with being a quotient of odd positive integers and without the restrictive conditions and without , Wu et al. [21] obtained several oscillation criteria for the equation: under the conditions (3).

Chen [25] investigated the following second-order Emden-Fowler neutral delay dynamic equation with , under the conditions (3). He obtained some oscillation criteria when is a constant and without assuming the conditions and .

All the above results cannot apply to our model (1) since our model (1) is more general than (2), (6) and those in [10–13], and the function in (1) satisfies (H_{6}) which makes our model (1) distinguished from all the existing cases. To the best of our knowledge, nothing is known regarding the necessary and sufficient conditions for the qualitative behavior of (1) with in (H_{6}) on time scales.

In this paper, even if in (H_{6}) and there is no assumptions and , we have established several new oscillation criteria of (1) for the both cases

Factually, we have employed new analytical techniques to present and construct our criteria in Section 3 after reciting two useful lemmas in Section 2. Our results have extended and unified a number of other existing results and handled the cases which are not covered by current criteria. Finally, in Section 4 two examples are demonstrated to illustrate the efficiency of our work with relevant remark.

#### 2. Some Lemmas

Lemma 1 (see [25]). *Suppose that (H _{5}) holds. Let . If exists for all sufficiently large , then for all sufficiently large .*

Lemma 2 (Bohner and Peterson [26, Theorem 1.90]). *Assume that is -differentiable and eventually positive or eventually negative, then
*

Lemma 3 (see [27]). *Let , where , , are constants, , , , and . Then attains its maximum value on at , and
*

#### 3. Main Results

The case

Theorem 4. *Assume that (H _{1})–(H_{6}) and (7) hold. If there exists a function such that for any positive number ,
*

*where*

*then (1) is oscillatory.*

* Proof. *Suppose that (1) has a nonoscillatory solution , then there exists such that for all . Without loss of generality, we assume that , and for , because a similar analysis holds for , and . Then the following are deduced from (1), (H_{3}), and (H_{6}):
Therefore is a nonincreasing function and is eventually of one sign.

We claim that
Otherwise, if there exists a such that for , then from (14), for some positive constant , we have
that is,
integrating the above inequality from to , we have
Letting , from (7), we get , which contradicts (14). Thus, we have proved (15).

We choose some such that for . Therefore from (14), (15), and the fact , we have that
which follows that
On the other hand, from (1), (H_{6}), and (15), we have
Noticing (15) and the fact , we get
where .

Define
Obviously, . By (22), (23) and the product rule and the quotient rule, we obtain
Now we consider the following two cases.*Case 1.* Let . By (15), Lemmas 1 and 2, we have
From (H_{5}), (20), (23)–(25), and the fact that is nondecreasing, we obtain
*Case 2.* Let . By (15), Lemmas 1 and 2, we get
From (H_{4}), (H_{5}), (20), (23)–(25), and the fact that is nondecreasing, we have
Therefore, for , from (26) and (28), we get
From (14) and (15), there exists a constant such that
that is
integrating the above inequality from to , we have
Thus, there exist a constant , and such that
so we have
where .

From (29) and (34), we obtain
Let
then . So from (35) and (36) we get
where .

Taking , , by Lemma 3 and (37), we obtain
where .

Integrating the above inequality (38) from to , we have
Since for , we have
which contradicts (12). This completes the proof of Theorem 4.

Next, we use the general weighted functions from the class which will be extensively used in the sequel.

Letting , we say that a continuous function belongs to the class if(i) for and for ,(ii) has a nonpositive right-dense continuous -partial derivative with respect to the second variable.

Theorem 5. *Assume that (H _{1})–(H_{6}) and (7) hold. If there exist a function and a function such that for any positive number ,
*

*where*

*then (1) is oscillatory.*

* Proof. *We proceed as in the proof of Theorem 4 to have (37). From (37) we obtain
Multiplying (46) (with replaced by ) by , integrating it with respect to from to for , using integration by parts and (i)-(ii), we get
where is defined as in (44).

Taking , , by Lemma 3 and (47), we obtain
where ,
Then it follows that
Thus we get
Then
which contradicts (41). This completes the proof of Theorem 5.

Theorem 6. *Assume that (H _{1})–(H_{6}) and (7) hold and . Furthermore, assume that . If there exists a function such that for any positive number ,
*

*where*

*then (1) is oscillatory.*

* Proof. *We proceed as in the proof of Theorem 4 to have (24). On the other hand, from (22) and (H_{3}), we deduce
and from for , we can get is nonincreasing. Hence, we have
which implies
Choosing such that for , we get
From (H_{6}), (15), (20), (24), (25), (58), and as is nonincreasing, we obtain
Now, from the fact that is nonnegative and nonincreasing, there exists a sufficiently large such that
holds for some positive constant and therefore
Combining (59) and (61), we obtain that
Letting
then . So
Integrating the above inequality from to , we have
Since for , we have
which contradicts (53). This completes the proof of Theorem 6.

Theorem 7. *Assume that (H _{1})–(H_{6}) and (7) hold and . Furthermore, assume that . If there exist a function and a function such that
*

*where*

*then (1) is oscillatory.*

* Proof. *We proceed as in the proof of Theorem 6 to have (64). From (64) we obtain
Multiplying (70) (with replaced by ) by , integrating it with respect to from to for , using integration by parts and (i)-(ii), we get
Using (67) in the above inequality (71), we get
Then it follows that
Thus we get
Then
which contradicts (68). This completes the proof of Theorem 7.

Theorem 8. *Assume that (H _{1})–(H_{6}) and (7) hold and . Furthermore, assume that . If there exist a function and a function such that for any positive number ,
*

*where*

*then (1) is oscillatory.*

* Proof. *We proceed as those in the proof of Theorem 7 to have (71), that is,
Then it follows that
Thus we get