Oscillation Criteria for Functional Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integral

Abstract

We present new oscillation criteria for the second order nonlinear dynamic equation under mild assumptions. Our results generalize and improve some known results for oscillation of second order nonlinear dynamic equations. Several examples are worked out to illustrate the main results.

1. Introduction

In this paper, we are concerned with the oscillatory behavior of the second order nonlinear functional dynamic equation with -Laplacian and nonlinearities given by Riemann-Stieltjes integral where the time scale is unbounded above; , ; with is strictly increasing; is a time scale; is a positive rd-continuous function on ; and are nonnegative rd-continuous functions on and with ; the functions and are rd-continuous functionssuch that lim and for and .

Both of the following two cases: are considered. We define the time scale interval by . By a solution of (1) we mean a nontrivial real-valued function , , which has the property that and satisfies (1) on , where is the space of rd-continuous functions. The solutions vanishing identically in some neighborhood of infinity will be excluded from our consideration. A solution of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is nonoscillatory.

Not only does the theory of the so-called “dynamic equations” unify theories of differential equations and difference equations, but also it 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 [1])) and can be applied in different types of time scales like , , and the set of harmonic numbers. In this work knowledge and understanding of time scales and time scale notation is assumed; for an excellent introduction to the calculus on time scales, see Bohner and Peterson [2–4].

In the last few years, there has been increasing interest in obtaining sufficient conditions for the oscillation/nonoscillation of solutions of different classes of dynamic equations; we refer the reader to [5–25] and the references cited therein. Recently, Erbe et al. [26] considered on an arbitrary time scale , where is a quotient of odd positive integers and sgn with and , is a positive rd-continuous function on , , , are nonnegative rd-continuous functions on , and , , satisfy . In [26], some oscillation criteria have been established when , , and is nondecreasing and delta differentiable with on . In this paper, we will establish oscillation criteria for the more general equation (1) under mild assumptions on the time scale and the time delay. Note that (1) not only contains a -Laplacian term and the advanced/delayed function , but also allows an infinite number of nonlinear terms and even continuous nonlinearities determined by the function .

2. Main Results

Throughout this paper, we denote

Lemma 1. Assume that or where If (1) has a positive solution on , then there exists a , sufficiently large, so that

Proof. Pick sufficiently large such that , , and on . From (1), we have, for , Then is nonincreasing on , and is of definite sign eventually. We claim that is eventually positive. If not, is eventually negative; that is, there exists such that for .
First, we assume (5) holds. Using the fact that is nonincreasing, we obtain, for , Hence, by (5), we have , which contradicts the fact that is a positive solution of (1).
Second, we assume that (6) holds. Using the fact that is nonincreasing, we obtain, for , where . By choosing sufficiently large such that and , for and , we get, for and , where . From (1) and (12) we find that Integrating this last inequality from to , we see that which implies Again, integrating this last inequality from to , we get From (6), we have , which contradicts the fact that is a positive solution of (1). This completes the proof.

Lemma 2. Assume that there exists sufficiently large such that Then where

Proof. Since is strictly decreasing on . If , then by the fact that is strictly increasing. Now we consider the case when . We first have which implies On the other hand, we have It implies that Therefore, (22) and (24) yield that and hence Let so that and for and . Thus, we have that, for , This completes the proof.

We denote by the set of Riemann-Stieltjes integrable functions on with respect to . Let such that . We further assume that such that

We start with the following two lemmas cited from [25] which will play an important role in the proofs of our results.

Lemma 3. Let Then there exists such that on ,

Lemma 4. Let and satisfying , on and . Then where we use the convention that and .

Theorem 5. Assume that one of conditions (5) and (6) holds. Furthermore, suppose that there exists a positive -differentiable function such that, for all sufficiently large , where with and being defined by (19) and (20), respectively. Then every solution of (1) is oscillatory.

Proof. Assume (1) has a nonoscillatory solution on . Then, without loss of generality, there is , sufficiently large, so that and on. By Lemma 1, we have, for , Define By the product rule and the quotient rule, we have that From (1) and the definition of , we have By the Pötzsche chain rule [3, Theorem 1.90], we obtain If , we have that whereas if , we have that Using the fact that is strictly increasing and is nonincreasing, we get that From (40), (41), and (42), we obtain where . By (18) and the definition of , we have that, for and , where and . We let be defined as in Lemma 3. Then satisfies (31). This follows the fact that From Lemma 4 we get This together with (44) shows that, for , Define and by Then, using the inequality [27] we get that From this last inequality and (47) we get, for , Integrating both sides from to , we get which leads to a contradiction to (33).

In the following examples, for , , and , we assume that such that , , and for .

Example 6. Consider the nonlinear dynamic equation where are rd-continuous functions with on , and , , are positive constants, and , , are nonnegative rd-continuous functions on . Here, Choose an -tuple with satisfying (31). By Example 5.60 in [4], condition (5) holds since Also, by choosing , we have Then, by Theorem 5, every solution of (55) is oscillatory.

Example 7. Consider the nonlinear dynamic equation where is a positive real number, , , , are positive constants, , , are nonnegative rd-continuous functions on , and , are rd-continuous functions with on . Assume It is clear that satisfies This holds for many time scales, for example, when . To see that (6) holds note that Since we can find such that for Therefore, we get To apply Theorem 5, it remains to prove that condition (33) holds. By putting , we get We conclude that if , , is a time scale, where , then every solution of (59) is oscillatory by Theorem 5.

We are now ready to state and prove Philos-type oscillation criteria for (1). Its proof can be similarly done as [28] and hence is omitted.

Theorem 8. Assume that one of conditions (5) and (6) holds. Furthermore, suppose that there exist functions , where such that and has a nonpositive continuous -partial derivative with respect to the second variable and satisfies and, for all sufficiently large , where is a positive -differentiable function. Then every solution of (1) is oscillatory on .

Example 9. Consider the following dynamic equation: where , , , , are rd-continuous functions with and on , and and , , are positive constants. It is easy to see that (5) holds. Choose an -tuple with satisfying (31). By the definition of , we know . On the other hand, let and . From (67), we obtain We have that for and hence for . Therefore, By Theorem 8, we can say that every solution of (69) is oscillatory if

Theorem 10. Assume that one of conditions (5) and (6) holds and Then every solution of (18) is oscillatory.

Proof. Assume (1) has a nonoscillatory solution on . Then, without loss of generality, there is a , sufficiently large, so that and on. Then, by Lemma 1, we have, for , Integrating both sides of the dynamic equation (18) from to , we obtain As shown in the proof of Theorem 5, we have Then, from (75) and (76), we get Since and , we have Also, by using the fact that is nonincreasing, we have or In view of (78) and (80), we get which gives us the contradiction This completes the proof.

Example 11. For , we consider the following dynamic equation: where , , , , are rd-continuous functions with on , , , are nonnegative rd-continuous functions on , , and , , are positive constants. It is obvious that (5) holds. Choose an -tuple with satisfying (31). On the other hand, noting that and , we can easily verify that By Theorem 10, every solution of (83) is oscillatory.

The last theorem is under the assumption that . Its proof can be similarly done as in [28] and hence is omitted.

Theorem 12. Assume that one of conditions (5) and (6) holds and is a (delta) differentiable function with . Furthermore, assume that and Then every solution of (1) is oscillatory.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the associate editor and the reviewers for their valuable comments on this paper. This work was supported by the Natural Science Foundation of Shandong Province under Grant nos. JQ201119 and ZR2010AL002 and the National Natural Science Foundation of China under Grant no. 61174217.