Abstract and Applied Analysis

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

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

Research Article

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

Haidong Liu¹ and Puchen Liu²

¹School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China

²Department 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 (H₁)–(H₆):

(H₁) is a time scale which is unbounded above. , where with , denotes the collection of all functions which are right-dense continuous on ;(H₂), ; (H₃); (H₄),  , for ,  , , , for , ;(H₅) is right-dense continuous on , and for all, where is the forward jump operator on ;(H₆) 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 (H₆) 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 (H₆) on time scales.

In this paper, even if in (H₆) 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 (H₅) 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 (H₁)–(H₆) 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), (H₃), and (H₆):

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), (H₆), 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 (H₅), (20), (23)–(25), and the fact that is nondecreasing, we obtain

Case  2. Let . By (15), Lemmas 1 and 2, we get

From (H₄), (H₅), (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 (H₁)–(H₆) 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 (H₁)–(H₆) 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 (H₃), we deduce

and from for , we can get is nonincreasing. Hence, we have

which implies

Choosing such that for , we get

From (H₆), (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 (H₁)–(H₆) 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 (H₁)–(H₆) 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

Then

which contradicts (76). This completes the proof of Theorem 8.

The case

Theorem 9. Assume that (H₁)–(H₆) and (8) hold and there exists a such that ,   for , and suppose that there exists a function such that (12) holds for any positive number , and there exists a function satisfying ,   for such that for any positive number and for every

where

then (1) is oscillatory.

Proof. Suppose to the contrary that is an eventually positive solution of (1), then there exists a such that , , for all , (the case of is negative and can be considered by the same method). It follows form (H₃) that for . From (14) it is easy to conclude that there exist two possible cases of the sign of .

Case  1. Suppose for sufficiently large , then we are back to the case of Theorem 4. Thus the proof of Theorem 4 goes through, and we may get contradiction by (12).

Case  2. Suppose for . Define

Then for . From the fact that is positive and nonincreasing, we get that

holds for some positive constant .

Noting that , so we have

Integrating the above inequality (88) with respect to from to , we have

Letting in the above inequality, we obtain

From (86) and (90), we have

If . From , Lemmas 1 and 2, we have

From (1), (H₆), (85), and (92), we get

If . From , Lemmas 1 and 2, we have

From (1), (H₆), (85) and (94), we get

Therefore, for , from (93) and (95), we get

Noticing that and , from , we see that for , and from we can get

Thus from (86), (87), (96), (97) and the fact that , we have

where .

That is

Multiplying (99) (with replaced by ) by , integrating it with respect to from to , we have

Next, we consider the following two cases.

Case  (i)  (let ). From Lemma 2 and , we have

From (100) and (101), we get

That is

Taking , , by Lemma 3 and (103), we obtain

That is

By (83), we get a contradiction with (91).

Case  (ii)  (let ). From Lemma 2 and , we get

From (100) and (106), we obtain

That is

Taking , , by Lemma 3 and (108), we obtain

That is

By (83), we get a contradiction with (91). This completes the proof of Theorem 9.

4. Examples

Example 10. Consider the following dynamic equation:

where are constants. In (111), , ,  .

If , and , where and , then . It is easy to get that . Choosing , therefore,

Hence, by Theorem 6, (111) is oscillatory.

Example 11. Consider the following dynamic equation:

where . In (113), , , .

If , and , where and , then . It is easy to get that . Choosing ,  , therefore, ,

Hence, by Theorem 7, (111) is oscillatory.

Acknowledgments

This research was supported by the National Natural Science Foundations of China (nos. 11171178, 61104136), Natural Science Foundation of Shandong Province of China (no. ZR2010FQ002), and Foundation of Qufu Normal University (no. XJ201014).

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