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Oscillation Criteria for Some New Generalized Emden-Fowler Dynamic Equations on Time Scales
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.
In the past two decades, the theory of time scales proposed by Hilger  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.  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  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 , 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.  obtained several oscillation criteria for the equation: under the conditions (3).
Chen  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 ). 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 ). Let , where , , are constants, , , , and . Then attains its maximum value on at , and
3. Main Results
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.
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.
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