#### Abstract

By using the Riccati transformation technique and constructing a class of Philos-type functions on time scales, we establish some new interval oscillation criteria for the second-order damped nonlinear dynamic equations with forced term of the form on a time scale which is unbounded, where is a quotient of odd positive integer. Our results in this paper extend and improve some known results. Some examples are given here to illustrate our main results.

#### 1. Introduction

In this paper, we are concerned with the oscillation criteria for the following forced second-order nonlinear dynamic equations with damping:

on a time scale , where is a quotient of odd positive integer. Throughout this paper and without further mention, we assume that the functions , with , , and .

The theory of time scales, which has recently received a lot of attention, was originally introduced by Stefan Hilger in his Ph.D. thesis in 1988 (see [1]). Since then a rapidly expanding body of the literature has sought to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus, where a time scale is an arbitrary nonempty closed subset of the real numbers , and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. Many other interesting time scales exist, and they give rise to many applications (see [2]). Not only does the new theory of the so-called dynamic equations unify the 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) and can be applied on different types of time scales like , , and , the space of the harmonic numbers. A book on the subject of time scales by Bohner and Peterson [2] summarizes and organizes much of the time scale calculus. For advances of dynamic equations on the time scales we refer the reader to the book [3].

Since we are interested in the oscillatory behavior of solutions near infinity, we make the assumption throughout this paper that the given time scale is unbounded above. We assume and it is convenient to assume . We define the time scale interval of the form by . We assume throughout that has the topology that it inherits from the standard topology on the real numbers .

By a solution of (1), we mean a nontrivial real-valued function satisfying (1) on . A solution of (1) is said to be oscillatory on in case it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation (1) is said to be oscillatory in case all its solutions are oscillatory. Our attention is restricted to those solutions of (1) which exist on some half line and satisfy for all .

In recent years, there has been much research activity concerning the interval oscillation criteria for various second order differential equations; see [4–9]. A great deal of effort has been spent in obtaining criteria for oscillation of dynamic equations on time scales without forcing terms and it is usually assumed that the potential function is positive. We refer the reader to the papers [10–25] and the references cited therein. On the other hand, there has been an increasing interest in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of dynamic equations with forcing terms on time scales, and we refer the reader to the papers [26–35].

In 2004, by using two inequalities due to Hölder and Hardy and Littlewood and Polya as well as averaging functions, Li [4] established several interval oscillation criteria for the second order damped quasilinear differential equation with forced term of the following form: where , are constants. The obtained results were based on the information only on a sequence of subintervals of , rather than on the whole half line, made use of the oscillatory properties of the forcing term, and extended a known result which is obtained by means of a Picone identity.

Erbe et al. [26] studied the forced second-order nonlinear dynamic equation

on a time scale , where . By using the Riccati substitution, the authors established some new interval oscillation criteria, that is, the criteria given by the behavior of and on a sequence of subintervals of .

In [31], by constructing a class of Philos-type functions on time scales, Li et al. established some oscillation criteria for the second order nonlinear dynamic equations with the forced term

on a time scale , where , , and are real-valued rd-continuous functions defined on , with as , and , whenever . The obtained results unified the oscillation of the second order forced differential equation and the second order forced difference equation. An example was considered to illustrate the main results in the end.

Erbe et al. [32] were concerned with the oscillatory behavior of the forced second-order functional dynamic equation with mixed nonlinearities

on an arbitrary time scale , where , , and are nondecreasing rd-continuous functions on , , and , for . Their results in a particular case solved a problem posed by Anderson, and their results in the special cases when the time scale is the set of real numbers and the set of integers involved and improved some oscillation results for second-order differential and difference equations, respectively.

In this paper, we intend to use the Riccati transformation technique to obtain some interval oscillation criteria for (1). Our results do not require that and be of definite sign and are based on the information only on a sequence of subintervals of rather than the whole half line. To the best of our knowledge, nothing is known regarding the oscillation behavior of (1) on time scales until now, and there are few results regarding the interval oscillation criteria for (1) on time scales without the damping term when , so our results expand the known scope of the study.

The paper is organized as follows. In Section 2, we present some basic definitions and useful results from the theory of calculus on time scales on which we rely in the later section. In Section 3, we intend to use the Riccati transformation technique, integral averaging technique, and inequalities to obtain some sufficient conditions for oscillation of every solution of (1). In Section 4, we give two examples to illustrate Theorems 3 and 7, respectively.

#### 2. Some Preliminaries

On any time scale , we define the forward and the backward jump operators by

where and . A point is said to be left-dense if , right-dense if , left-scattered if , and right-scattered if . The graininess function for a time scale is defined by .

For a function , the (delta) derivative is defined by

if is continuous at and is right-scattered. If is right-dense, then the derivative is defined by

provided this limit exists. A function is said to be rd-continuous provided is continuous at right-dense points and there exists a finite left limit at all left-dense points in . The set of all such rd-continuous functions is denoted by . The derivative of and the forward jump operator are related by the formula

Also, we will use which is shorthand for to denote . We will make use of the following product and quotient rules for the derivative of two differentiable functions and :

The integration by parts formula reads

We say that a function is regressive provided

The set of all regressive and rd-continuous functions will be denoted by

If , then we can define the exponential function by where is the cylinder transformation, which is defined by

Next, we give the following lemmas which will be used in the proof of our main results.

Lemma 1 (see [2, Chapter 2]). * If ; that is, is rd-continuous and such that for all , then the initial value problem , has a unique and positive solution on , denoted by . This “exponential function” satisfies the semigroup property .*

Lemma 2 (see [36]). * If and are conjugate numbers , then for any ,
*

#### 3. Main Results

Now, we are in a position to state and prove some new results which guarantee that every solution of (1) oscillates. In the sequel, we say that a function belongs to a function class

denoted by .

Theorem 3. *Assume that and for any , there exist constants and , such that , , with
**
where . Furthermore, assume that there exist functions , , and , , such that
**
where
**
Then (1) is oscillatory on .*

*Proof. *Assume that is a nonoscillatory solution of (1) on . Without loss of generality, we may assume that there exists a , such that , for all . By assumption, we can choose , then and on the interval . From (1), we have

Using Lemma 1 and the above inequality, we get

Hence is nonincreasing on . So for ,

Therefore,

Define the function by

Using the product rule and the quotient rule, we obtain

In view of (1), (26), and (27), we have

From (19), (25), and (28), we get

Set

From Lemma 2, it is easy to see that

Since , we obtain

Thus, combining (29)–(32) and noticing that , we have

where is defined as in Theorem 3. Multiplying (33) by and integrating from to , we get

Using integration by parts on the first integral, we obtain

Rearranging and using , we have

Adding and subtracting the term and using (20), we get

It follows that

This implies that

Solving for , we get that solves the IVP

Since , we obtain from [2, Theorem ] that on , which is a contradiction. The proof when is eventually negative follows the same arguments using the interval instead of , where we use , on , and . The proof is complete.

*Remark 4. *When and , Theorem 3 contains Theorem 3.2 in [26].

Theorem 5. *Assume that and for any , there exist constants and , such that , , with
**
Furthermore, assume that there exist functions , , and , , such that
**where
**
and is defined as in Theorem 3. Then (1) is oscillatory on .*

*Proof. * Assume that is a nonoscillatory solution of (1) on . Without loss of generality, we may assume that there exists a , such that , for all . By assumption, we can choose , then and on the interval . We define as in Theorem 3. Proceeding as in the proof of Theorem 3 and from (25) and (32), we get

where is defined as in Theorem 5. Multiplying (44) by and integrating from to , we get

The rest of the argument proceeds as in Theorem 3 to get a contradiction to (42). The proof is complete.

*Remark 6. *When and , Theorem 5 contains Theorem 2.1 in [26].

Theorem 7. *Assume that and for any , there exist constants and , such that , , with
**
where . Furthermore, assume that there exist functions , , and , , such that
**
where
**
and is defined as in Theorem 3. Then (1) is oscillatory on .*

*Proof. * Assume that is a nonoscillatory solution of (1) on . Without loss of generality, we may assume that there exists a , such that , for all . By assumption, we can choose , then and on the interval . We define as in Theorem 3. Proceeding as in the proof of Theorem 3, we have (28). Hence, from (25), (28), and (47), we get

Set

From Lemma 2, it is easy to see that

Thus, combining (32), (50), and (52) and noticing that , we have
where is defined as in Theorem 7. Multiplying (53) by and integrating from to , we get

The rest of the argument proceeds as in Theorem 3 to get a contradiction to (47). The proof is complete.

Next, let us introduce the class of functions , which will be extensively used in the sequel.

Let and . We say that the function belongs to the class , if(i), , on ;(ii) has continuous -partial derivatives and on such that

where .

Theorem 8. *Assume that and for any , there exist constants and , such that , , with
**
where . Furthermore, assume that there exists a function such that for some and ,
**
where
**
and and are defined as in Theorem 3. Then (1) is oscillatory on .*

*Proof. * Assume that is a nonoscillatory solution of (1) on . Without loss of generality, we may assume that there exists a , such that , for all . By assumption, we can choose , then and on the interval . We define the function as in Theorem 3. Proceeding as in the proof of Theorem 3 and from (25) and (31), we get
where is defined as in Theorem 7. Since is nonincreasing on , we obtain

hence

From (25), we have

Therefore, from (61) and (62), we get

Combining (59) and (63), we obtain

Multiplying both sides of (64) by and integrating with respect to from to for , we have

In view of (i) and (ii), we see that

Using (66) in (65) leads to

Letting in the above inequality, we get

Similarly, multiplying both sides of (64) by and integrating with respect to from to for , we obtain

Letting in the above inequality, we get

Adding (68) and (70), we get a contradiction to (57). This completes the proof.

Theorem 9. *Assume that and for any , there exist constants and , such that , , with
**where . Furthermore, assume that there exists a function such that for some and ,
**
where
** is defined as in Theorem 3, is defined as in Theorem 7, and , , and are defined as in Theorem 8. Then (1) is oscillatory on .*

The proof of Theorem 9 is similar to that of Theorem 8, so we omit the proof.

*Remark 10. *The main results in this paper can also be extended to the following second order damped dynamic equations with mixed nonlinearities:
where , , , or the more general equation

on any arbitrary time scale , where , and are nondecreasing rd-continuous functions on with , . Due to the limited space, we omit it here and leave it to the readers who are interested in this problem.

#### 4. Examples

In this section, we will show the applications of our interval oscillation criteria in two examples. Firstly, we will give an example to illustrate Theorem 3.

*Example 1. * Consider the following second order forced difference equations with damping: