Abstract

This paper is concerned with oscillation of second-order forced functional dynamic equations of the form on time scales. By using a generalized Riccati technique and integral averaging techniques, we establish new oscillation criteria which handle some cases not covered by known criteria.

1. Introduction

The theory of time scales was introduced by Stefan Hilger in his Ph.D. thesis in 1988 in order to unify the continuous and discrete analysis. Not only can this theory of the so-called “dynamic equations” unify theories of differential equations and difference equations but also it can extend these classical cases to cases “in between,” for example, to the so-called -difference equations. A time scale is an arbitrary nonempty closed subset of the real numbers with the topology and ordering inherited form , and the cases when this time scale is equal to or to the integers represent the classical theories of differential and difference equations. Of course many other interesting time scales exist, and they give rise to plenty of applications. In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations on time scales, and we refer the readers to [1–20].

In 2006, Saker [1] provided sufficient conditions for the boundedness of the solutions of forced dynamic equations of the form: In 2007, Sun and Wong [2] considered interval oscillation of second-order forced ordinary differential equations with mixed nonlinearities: where .

In 2008, Erbe et al. [3] established oscillation criteria for the forced second-order nonlinear dynamic equation:

In 2009, Li and Chen [4] considered oscillation of second-order functional differential equations with mixed nonlinearities:

In 2010, Lin et al. [5] considered forced oscillation of second-order half-linear dynamic equations on time scales: where is a quotient of odd positive integers. Also, Erbe et al. [6] obtained some interval oscillation criteria for forced second order nonlinear delay dynamic equations with oscillatory potential of the form:

In 2011, Hassan et al. [7] discussed oscillation of the following forced second-order differential equations with mixed nonlinearities: where is a quotient of odd positive integers, , and with , , is positive and nondecreasing continuous and for . The authors established some sufficient conditions for the oscillation of (7), which did not assume that and are of definite sign.

In 2013, Anderson and Saker [8] establish some oscillation criteria for the second-order nonlinear Emden-Fowler functional dynamic equation with oscillatory potential and forcing term on time scales of the form: where is a time scale unbounded above, , the potentials and and the forcing function are right dense continuous with , and satisfies . They also did not assume that and are of definite sign.

In this paper, motivated by [1–8] and others, we study the second-order nonlinear dynamic equation: on a time scale , where , is a quotient of odd positive integers, , and is a real -tuple satisfying , .

This paper is organized as follows. In the next section, we give some preliminaries and lemmas. In Sections 3 and 4, we will use the Riccati transformation technique to prove our main results. In Section 5, we present two examples to illustrate our results.

2. Preliminaries and Lemmas

For convenience, we recall some concepts related to time scales. More details can be found in [9].

Definition 1. Let be a time scale; for the forward jump operator is defined by , the backward jump operator by , and the graininess function by , where and . If , is said to be right scattered; otherwise, it is right dense. If , is said to be left scattered; otherwise, it is left dense. The set is defined as follows. If has a left-scattered maximum , then ; otherwise, .

Definition 2. For a function and , we define the delta-derivative of to be the number (provided that it exists) with the property that, given any , there is a neighborhood of (i.e., for some ) such that

We say that is delta differentiable (or in short: differentiable) on provided that exists for all .

It is easily seen that if is continuous at and is right scattered, and then is differentiable at with Moreover, if is right dense, then is differential at iff the limit exists as a finite number. In this case In addition, if , then is nondecreasing. A useful formula is as follows:

We will make use of the following product and quotient rules for the derivative of the product and the quotient (where ) of two differentiable functions and :

Definition 3. Let be a function, is called right dense continuous (rd-continuous) if it is continuous at right-dense points in , and its left-sided limits exist (finite) at left-dense points in . A function is called an antiderivative of provided that holds for all . By the antiderivative, the Cauchy integral of is defined as , and .

In (9), we assume that is a time scale satisfying and , and(h1), and , ;(h2), and , .

By a solution of (9), we mean a nontrivial real-valued function satisfying and (9). Our attention is restricted to those solutions of (9) that exist on some half-line and satisfy for any . A solution of (9) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is called nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.

For convenience, we use the notations ,   , , and, for and , Now, we give the first lemma.

Lemma 4. Let conditions ( and ) hold. Furthermore, assume that, for any , there exist constants such that where , , .
If is a nonoscillatory solution of (9), then, for , we have , , .

Proof. If is an eventually positive solution of (9), then by ( there exists a such that By assumption, we can choose such that and on . By (9) we have which implies that is decreasing on .
Case 1. and . In this case, we get It follows that Also, since , we see for . So, we get which implies that It follows that
Case 2. and . In this case, we obtain Therefore, Also, since , we see for , and we have which implies that It follows that
Case 3. and . It is easy to get Combing (24), (29), and (30), we have When is an eventually negative solution of (9), its proof follows the similar argument using the interval instead of .
The proof is complete.

Lemma 5 (see [2]). Let , , be the n-tuple satisfying . Then there exists an n-tuple with satisfying and which also satisfies either or

Lemma 6 (see[7]). Let , , , , and be positive real numbers and let be a quotient of odd positive integers. Then

Lemma 7 (Yong’s Inequality). If and are conjugate numbers , then and equality holds if and only if .
Let , , , and . It follows from Lemma 7 that for all . Rewriting the above inequality we also have for all .

3. Main Results

In this section, by employing the Riccati transformation technique we will establish oscillation criteria for (9). Set We define for with the admissible set By employing (32) and (33) in Lemma 5, we have the following theorem.

Theorem 8. Let conditions ( and ) hold. Furthermore, assume that, for any , there exist constants such that (17) holds. If there exists a function such that then (9) is oscillatory, where satisfy (32)-(33) and .

Proof. Suppose to the contrary that is a nonoscillatory solution of (9). Without loss of generality, we may assume that is eventually positive. Then, there exists sufficiently large such that , , for all . By assumption, we can choose , such that and on the interval . From Lemma 4 and (9), we obtain, for , Defining the function by the Riccati substitution and then By (43), we obtain Corresponding to the exponents in (9), let be chosen to satisfy (32) and (33) in Lemma 5 and set .
Setting and by the arithmetic geometric mean inequality in [21], we get for For (32), we have By (45)–(50), we get Set . Since is a quotient of odd positive integers, it is easy to prove . Multiplying by on (51) and then using the identity we obtain where As demonstrated in [10], we know that and that if and only if where stands for the inverse function. In our case, sine , and dynamic equation (55) has a unique solution satisfying . Clearly, the unique solution is . Therefore, on . So, we get Integrating from to and using , we find which leads to a contradiction to (42).
The proof when is eventually negative follows the similar arguments using the interval instead of
The proof is complete.

By employing (32) and (34) in Lemma 5, we have the following theorem.

Theorem 9. Let conditions ( and ) hold. Furthermore, assume that, for any , there exist constants such that (17) holds. If there exists a function such that then (9) is oscillatory, where satisfy (32) and (34).

Proof. Suppose to the contrary that is a nonoscillatory solution of (9). Without loss of generality, we may assume that is eventually positive. Then, there exists sufficiently large such that , , , for all . By assumption, we can choose , such that and on the interval . Define the Riccati substitution as (44). Let be chosen to satisfy (32) and (34) in Lemma 5. By (46), we can get Setting using again the arithmetic-geometric mean inequality in [21], and similar to (50), we have By (45) and (62), we get The reminder of the proof is similar to that of Theorem 8. The proof is complete.

By employing (35) and (36) in Lemma 6, we have the following theorem.

Theorem 10. Let conditions ( and ) hold. Furthermore, assume that, for any , there exist constants such that (17) holds. If there exists a function such that then (9) is oscillatory, where are positive numbers with and , .

Proof. Suppose to the contrary that is a nonoscillatory solution of (9). Without loss of generality, we may assume that is eventually positive. Then, there exists sufficiently large such that , , , for all . By assumption, we can choose , such that and on the interval . Define the Riccati substitution as (44). Let be chosen to satisfy . Similar to the proof of Theorem 8, we can get From (36), we get, for and , From (35), we get, for and , By (45), (65), (66), and (67), we get The reminder of the proof is similar to that of Theorem 8. The proof is complete.

By employing (38) and (39), we have the following theorem.

Theorem 11. Let conditions ( and ) hold. Furthermore, assume that, for any , there exist constants such that (17) holds. If there exists a function such that then (9) is oscillatory, where are positive numbers with , , ,

Proof. Suppose to the contrary that is a nonoscillatory solution of (9). Without loss of generality, we may assume that is eventually positive. Then, there exists sufficiently large such that , , , for all . By assumption, we can choose , such that and on the interval . Define the Riccati substitution as (44). By (43), we have Applying (38) and setting we have On the other hand, we can get Applying (39) and setting we obtain that From (71), (73), and (76), we have By (45) and (77), we have The reminder of the proof is similar to that of Theorem 8. The proof is complete.

Related to (9) are the dynamic equations with mixed delta and nabla derivatives: It is not difficult to see that time scale modifications of the previous arguments give rise to completely parallel results for the above dynamic equations. For an illustrative example we provide below the version of Theorem 8 for (79), for which a lemma analogous to Lemma 4 can be stated easily. The other theorems for (79) and (80) can be obtained by employing arguments developed for (9).

Set

Theorem 12. Let conditions ( and ) hold. Furthermore, assume that, for any , there exist constants such that (17) holds. If there exists a function such that then (79) is oscillatory, where denotes the backward jump operator, and satisfy (32)-(33) and .

5. Examples

In this section, we give two examples to illustrate our main results. We first give an example to show Theorems 8.

Example 1. Consider on the differential equation: We set Also, let , and such that (32) and (33) hold Setting , we get . By (42), we have Hence, by Theorems 8, (83) is oscillatory.

Example 2. Consider on the equation: where , , We set , , , , , , , , .
Setting , , we can get , .
So, satisfies the assumption in Theorem 9. Let such that (32) and (34) hold. Setting , we get . By (58), we get Consequently, if the constant coefficients satisfy the relation then the above sum is nonnegative and hence (88) is oscillatory by Theorem 9.

Conflict of Interests

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

Acknowledgments

The authors sincerely thank the referees for their valuable comments and suggestions. This project is supported by the NNSF of China (no. 11271379), the Science Foundation of Guangdong University of Finance (no. 13XJ03-05), and the Foundation for Technology Innovation in Higher Education of Guangdong, China (no. 2013KJCX0136).