#### Abstract

In this work, we use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation , on a time scale where is the quotient of odd positive integers and *p(t)* and *q(t)* are positive right-dense continuous (rd-continuous) functions on . Our results improve and extend some results established by Sun et al. 2009. Also our results unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our main results.

#### 1. Introduction

The theory of time scales was introduced by Hilger [1] in order to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus. Many authors have expounded on various aspects of this new theory, see [2β4]. A time scale is a nonempty closed subset of the real numbers, If the time scale equals the real numbers or integer numbers, it represents the classical theories of the differential and difference equations. Many other interesting time scales exist and give rise to many applications. The new theory of the so-called βdynamic equationβ not only unify the theories of differential equations and difference equations, but also extends these classical cases to the so-called -difference equations (when for or ) which have important applications in quantum theory (see [5]). Also it can be applied on different types of time scales like , and the space of the harmonic numbers . In the last two decades, there has been increasing interest in obtaining sufficient conditions for oscillation (nonoscillation) of the solutions of different classes of dynamic equations on time scales, see [6β9]. In this paper, we deal with the oscillation behavior of all solutions of the second-order nonlinear delay dynamic equation
subject to the hypotheses (H_{1}) is a time scale which is unbounded above, and with . We define the time scale interval by .(H_{2}) is the quotient of odd positive integers.(H_{3}) and are positive rd-continuous functions on an arbitrary time scale , and(H_{4}) is a strictly increasing and differentiable function such that , .(H_{5}) is a continuous function such that for some positive constant , it satisfies for all .

By a solution of (1.1), we mean that a nontrivial real valued function satisfies (1.1) for . A solution of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. (1.1) is said to be oscillatory if all of its solutions are oscillatory. We concentrate our study to those solutions of (1.1) which are not identically vanishing eventually.

It is easy to see that (1.1) can be transformed into a half linear dynamic equation where , . If , then (1.1) is transformed into the equation If , then (1.4) has the form If , then (1.5) becomes Recently, Zhang et al. [10] have considered the nonlinear delay (1.1) and established some sufficient conditions for oscillation of (1.1) when . Also Grace et al. [11] introduced some new sufficient conditions for oscillation of the half linear dynamic equation (1.3). In 2009, Sun et al. [12] extended and improved the results of [6, 13, 14] to (1.1) when , but their results can not be applied for . In 2008, Hassan [15] considered the half linear dynamic equation (1.3) and established some sufficient conditions for oscillation of (1.3). In 2007, Erbe et al. [13] considered the nonlinear delay dynamic equation (1.4) and obtained some new oscillation criteria which improve the results of Εahiner [14]. In 2005, Agarwal et al. [6] studied the linear delay dynamic equation (1.6), also Εahiner [14] considered the nonlinear delay dynamic equation (1.5) and gave some sufficient conditions for oscillation of (1.6) and (1.5). In this work, we give some new oscillation criteria of (1.1) by using the generalized Riccati transformation and the inequality technique. Our results are general cases for some results of [12, 15].

This paper is organized as follows. In Section 2, we present some preliminaries on time scales. In Section 3, we give several lemmas. In Section 4, we establish some new sufficient conditions for oscillation of (1.1). Finally, in Section 5, we present some examples to illustrate our results.

#### 2. Some Preliminaries on Time Scales

A time scale is an arbitrary nonempty closed subset of the real numbers . On any time scale , we define the forward and backward jump operators by A point , inf is said to be left dense if , right dense if and , left scattered if , and right scattered if . The graininess function for a time scale is defined by .

A function is called rd-continuous provided that it is continuous at right-dense points of , and its left-sided limits exist (finite) at left-dense points of . The set of rd-continuous functions is denoted by . By , we mean the set of functions whose delta derivative belongs to .

For a function (the range of may be actually replaced with any Banach space), the delta derivative is defined by provided that is continuous at , and is right scattered. If is not right scattered, then the derivative is defined by provided that this limit exists.

A function is said to be differentiable if its derivative exists. The derivative and the shift of a function are related by the equation The derivative rules of the product and the quotient (where ) of two differentiable functions and are given by An integration by parts formula reads or and the infinite integral is defined by Note that in case , we have

and in case , we have

Throughout this paper, we use where

#### 3. Several Lemmas

In this section, we present some lemmas that we need in the proofs of our results in Section 4.

Lemma 1 (Bohner and Peterson [3, Theorem 1.90]). * βIf x(t) is delta differentiable and eventually positive or negative, then
*

Lemma 2 (Hardy et al. [16, Theorem 41]). * If and are nonnegative real numbers, then
**
where the equality holds if and only if .*

Lemma 3. *If (H _{1})β(H_{3}) and (1.2) hold and (1.1) has a positive solution on , then
*

*Proof. *The proof is similar to the proof of Lemma 2.1 in [15] and, hence, is omitted.

#### 4. Main Results

Theorem 1. *Assume that (H _{1})β(H_{5}), (1.2), Lemma 3 hold and , . Furthermore, assume that there exists a positive -differentiable function such that
*

*Then every solution of (1.1) is oscillatory on .*

*Proof. *Assume that (1.1) has a nonoscillatory solution on . Then, without loss of generality, we assume that for all , and there is such that satisfies the conclusion of Lemma 3 on . Consider the generalized Riccati substitution
Using the delta derivative rules of the product and quotient of two functions, we have
using the fact and , we have
If , then using the chain rule and the fact that is strictly increasing on , we obtain
which implies
since , then by integrating from to , we get
that is,

If , then using the chain rule and the fact that is strictly increasing on , we obtain
From (4.4), (4.7), and (4.10), we have
By (4.9), (4.11), and the definition of , we have for
where . Defining and by
then using Lemma 2, we get
From this last inequality and (4.12), we get
Integrating both sides from to , we get
which contradicts the assumption (4.1). This contradiction completes the proof.

Theorem 2. *Assume that (H _{1})β(H_{5}), (1.2), Lemma 3 hold and , . Furthermore, assume that there exist functions (where ) such that
*

*and has a nonpositive continuous -partial derivative with respect to the second variable which satisfies*

*where is positive -differentiable function and*

*Then every solution of (1.1) is oscillatory on .*

*Proof. *Assume that (1.1) has a nonoscillatory solution on . Then, without loss of generality, we assume that for all , and there is such that satisfies the conclusion of Lemma 3 on . Define as in the proof of Theorem 1. Replacing with () in (4.12), we have
Multiplying (4.21) by , and integrating with respect to from to , we get
Integrating by parts and using (4.17) and (4.18), we obtain
Defining and by
then using Lemma 2, we get
therefore,
By the definition of , we get
and this implies that
which contradicts the assumption (4.19). This contradiction completes the proof.

Theorem 3. *Assume that (H _{1})β(H_{5}), (1.2), Lemma 3 hold and , . Furthermore, assume that there exists a positive -differentiable function such that for *

*Then every solution of (1.1) is oscillatory on .*

*Proof. *Assume that (1.1) has a nonoscillatory solution on . Then, without loss of generality, we assume that for all , and there is such that satisfies the conclusion of Lemma 3 on . Proceeding as in the proof of Theorem 1, we get (4.15) from which we have
therefore,
The right hand side of the above inequality gives
Since for , , then we have
then,
which contradicts (4.29). This contradiction completes the proof.

Theorem 4. *Assume that and
**
Then every solution of (1.1) is oscillatory on .*

*Proof. *Assume that (1.1) has a nonoscillatory solution on . Then, without loss of generality, we assume that for all , and there is such that satisfies the conclusion of Lemma 3 on . From (1.1), we have
Integrating last equation from to , we obtain
Since decreasing and , then we have
Since , then , and consequently
but
Since and are strictly increasing, then we get that
therefore,
This contradiction completes the proof.

#### 5. Examples

In this section, we give some examples to illustrate our main results.

*Example 1. *Consider the second-order nonlinear delay dynamic equation
where is a positive constant,l and is the quotient of odd positive integers.

Here,
If , then
Therefore, by Theorem 1, every solution of (5.1) is oscillatory.

*Example 2. *Consider the second-order nonlinear delay dynamic equation
where is a positive constant, and is the quotient of odd positive integers, that is, .

Here,
It is clear that (1.2) holds.

Since , then we can find such that
If , then
if . Then by Theorem 1, every solution of (5.4) is oscillatory if .

*Example 3. *Consider the second-order nonlinear delay dynamic equation
where is a positive constant and is the quotient of odd positive integers.

Here,
It is clear that , for , (i.e., (1.2) holds) and for , and .

Then,
if . Then by Theorem 4, every solution of (5.8) is oscillatory if .

*Remarks 1. *(1) The recent results due to Hassan [15], Grace et al. [11] and Agarwal et al. [7] cannot be applied to (5.1), (5.4), and (5.8) as they deal with ordinary equations without delay.

(2) If , the results of Sun et al. [12] cannot be applied to (5.1) and (5.4).