Abstract
We study the oscillatory properties of the following even order delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals: where , a time scale which is unbounded above, is even, , is a constant, and is a strictly increasing right-dense continuous function; , , , and are right-dense continuous functions; is strictly increasing. Our results extend and supplement some known results in the literature.
1. Introduction
In this paper, we consider the following even order delay dynamic equations with nonlinearities of the form: where is even, , and , is a time scale which is unbounded above, and the following are satisfied:) is another time scale, denotes the collection of all functions which are right-dense continuous on ;(), , , and is a strictly increasing and satisfying , ;(), , for , , is right-dense continuous on , and is a time scale, for all , where is the forward jump operator on ;(), , and , for any ;() is a continuous function such that , for all and there exist a positive right-dense continuous function defined on and a constant , such that for all and for all ;() is strictly increasing. denotes the Riemann-Stieltjes integral of the function on with respect to .
By 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 . A nontrivial solution of (1) is called oscillatory if it has arbitrary large zeros; otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.
If is a quotient of odd positive integers, , , then (1) simplifies to the even order dynamic equation
If , , , for , and ; satisfying ; ; , then (1) reduces to
In recent years, more and more people have been interested in studying the oscillatory behavior of higher order dynamic equations on time scales, see [1–13] and references therein. For an introduction to time scale calculus and dynamic equations, we refer the reader to the landmark paper of Hilger [14] and the seminal book by Bohner and Peterson [15] for a comprehensive treatment of the subject. In particular, Grace [9] studied the even order dynamic equation (2). By employing generalized Riccati techniques, he established some new criteria which ensure that (2) is oscillatory. Chen and Qu [8] investigated the even order advanced type dynamic equation (3). They got some new oscillation criteria for (3) by introducing parameter functions.
In the present paper, we will establish several oscillation criteria for the more general equation (1). Our work is of significance because (1) allows an infinite number of nonlinear terms and even a continuum of nonlinearities determined by the function . Our results extend and supplement a number of other existing results and handle the cases which are not covered by known criteria.
2. Preliminaries
In the sequel, we denote by the set of Riemann-Stieltjes integrable functions on with respect to , and we use the convention that .
Lemma 1 (Kiguarde's Lemma [16, Theorem 5]). Let , , and . Suppose that is either positive or negative and is not identically zero and is either nonnegative or nonpositive on for some . Then there exist , such that holds for all with(i) holds for all and all ;(ii) holds for all and all .
In order to present the next lemma, we use the Taylor monomials (see [15, section 1.6]) which are defined recursively by where .
Lemma 2 (see [1]). Let and . Moreover, suppose that Kigurade's theorem holds with and on . Then there exists a sufficiently large such that
Lemma 3 (see [1]). Assume that the conditions of Lemma 2 hold. Then
Lemma 4 (see [17]). Suppose that () holds. Let . If exists for all sufficiently large , then for all sufficiently large .
Lemma 5 (see [15]). Assume that is -differentiable and eventually positive or eventually negative; then
Lemma 6 (see [18]). Suppose and are nonnegative; then where equality holds if and only if .
Lemma 7 (see [19]). Let and satisfy (0), on , and Then
3. Main Results
Theorem 8. Assume that ()–() hold. If there exist a function and a function satisfying on and for all , 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 . From (1) and , we have
Therefore is a nonincreasing function and is eventually of one sign on .
We claim that
Otherwise, if there exists a such that for , then from (16), for some positive constant , we have
that is
integrating the above inequality from to , we have
Letting , from (), we get . Analogously, we have , which contradicts the fact that for . Thus, we have proved (17).
So from (16) and (17) and Lemma 5, we obtain
Therefore, it follows from the fact , we have , and from Lemma 1, there exists an integer such that (i) and (ii) hold on . Thus, we have and then we conclude
For the case , from Lemmas 1 and 2, we get . For the case , we claim that . Otherwise, we obtain , Therefore, it follows from (ii) of Lemma 1 that , on . From (1), we have
where
Integrating (23) from to and from (17) we obtain
Letting , we have
Integrating both sides of the last inequality from to and from , we get
Letting , we get
which contradicts (13). Thus, we have , so from Lemma 2, there exists a sufficiently large such that
Define
Obviously, . From (1), , (30) and , it follows that
Now we consider the following two cases.
In the first case . By and Lemmas 4 and 5, we have
From , (29)–(32), , , and the fact that
we obtain
In the second case . By and Lemmas 4 and 5, we get
From , (29)–(31), (35), , , , and the fact that
we have
Therefore, for , from (34) and (37), we get
On the other hand, by (11) and (12), we have
Therefore, by , Lemma 7, (39), and , we have that for
Substituting (40) into (38) we obtain
where and are defined by (15).
Taking , , by Lemma 6 and (41), we obtain
Integrating above inequality (42) from to , we have
Since for , we have
Taking upper limit of both sides of the inequality (44) as , the right-hand side is always bounded, which contradicts condition (14). This completes the proof of Theorem 8.
Theorem 9. Assume that ()–() and (13) hold for all . If there exist a function and a function such that on and (11), (12) hold, where , then (1) is oscillatory.
Proof. The proof is in fact a simpler version of the proof of Theorem 8. We need only to note that is positive in (37).
Theorem 10. Assume that ()–() and (13) hold for all . If there exist a function and a function such that on and (11) and (12) hold, where then (1) is oscillatory.
Proof. We proceed as in the proof of Theorem 8 to have (30). From (1) and , (30), Lemmas 3–5, , , and , it follows that From (40), we obtain Integrating this inequality from to , we have Taking upper limit of both sides of the inequality (50) as and using (46) we obtain a contradiction to the fact that on . This completes the proof of Theorem 9.
Theorem 11. Assume that ()–() and (13) hold for all . If there exist a function and a function such that on and (11), (12) hold, where , then (1) is oscillatory.
Proof. The proof is in fact a simpler version of the proof of Theorem 10. We need only to note that is positive in (49).
Remark 12. If we let be the ratio of positive odd integers, , , and and use the convention that , , then Theorem 8 reduces to [9, Theorems 2.3] and Theorem 10 reduces to [9, Theorems 2.2].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by the National Natural Science Foundations of China (nos. 11171178 and 61104136), the Natural Science Foundation of Shandong Province of China (no. ZR2010FQ002), and the Foundation of Qufu Normal University (no. XJ201014).