Abstract and Applied Analysis

Volume 2012 (2012), Article ID 743469, 20 pages

doi:10.1155/2012/743469

Research Article

Oscillation Criteria for Second-Order Nonlinear Dynamic Equations on Time Scales

Shao-Yan Zhang¹ and Qi-Ru Wang²

¹Department of Mathematics, Guangdong University of Finance, 527 Yingfu Lu, Guangdong, Guangzhou 510520, China

²School of Mathematics & Computational Science, Sun Yat-Sen University, 135 Xinguang Xi Lu, Guangdong, Guangzhou 510275, China

Received 31 August 2012; Accepted 15 October 2012

Academic Editor: Zhanbing Bai

Copyright © 2012 Shao-Yan Zhang and Qi-Ru Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerned with oscillation of second-order nonlinear 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 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 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, we refer the reader to [1–14].

In 2006, Wu et al. [1] considered the second-order nonlinear neutral dynamic equation with variable delays

where is a quotient of odd positive integers. In 2007, Saker et al. [2] also discussed (1.1) for an odd positive integer . In 2010, Zhang and Wang [3] extended and complemented some results in [1, 2] for and gave some new results for . In 2011, Saker [4] considered (1.1) in different conditions. In 2010, Sun et al. [5] considered the second-order quasiliner neutral delay dynamic equation

where , , and are quotients of odd positive integers with .

In this paper, we study the second-order nonlinear dynamic equation

on a time scale , where , is a quotient of odd positive integers.

The paper is organized as follows. In the next section, we give some preliminaries and lemmas. In Section 3, we will use the Riccati transformation technique to prove our main results. In Section 4, 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 [6].

Definition 2.1. 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 2.2. For a function and , one defines the delta-derivative of to be the number (provided 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 exists, for all .

It is easily seen that if is continuous at and is right-scattered, then is differentiable at with

Moreover, if is right-dense then is differential at if the limit

exists as a finite number. In this case

In addition, if , then is nondecreasing. A useful formula is

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 2.3. 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 holds for all . By the antiderivative, the Cauchy integral of is defined as , and .

Let denote the set of all rd-continuous functions mapping to . It is shown in [6] that every rd-continuous function has an antiderivative. An integration by parts formula is

In (1.3), we assume that is a time scale and

, , , , and , ,, , , where , is continuous function such that for all , there exist (), quotients of odd positive integers and such that , , and .

Since we are interested in the oscillatory and asymptotic behavior of solutions near infinity, we assume throughout that the time scale under consideration satisfies and . For , let . Throughout this paper, these assumptions will be supposed to hold. Let , and . Clearly for , is nondecreasing and coincides with the inverse of when the latter exists.

By a solution of (1.3), we mean a nontrivial real-valued function which has the properties and . Our attention is restricted to those solutions of (1.3) that exist on some half line and satisfy for any . A solution of (1.3) 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 notation , and , and set

Then (1.3) becomes

Now, we give the first lemma. Set

Lemma 2.4. Let conditions hold. If is an eventually positive solution of (1.3), then there exists sufficiently large such that , , , and () for .

Proof. If is an eventually positive solution of (1.3), then by there exists a such that

From (2.8), (1.3), and , we see that . Also by (1.3) and , we have

which implies that is decreasing on .

We claim that on . Assume not, there is a such that . Since for , we have

Integrating the inequality above form to , by we get

and this contradicts the fact that , for all . Thus we have on and so on .

Note that

we have

Since and , we get

The proof is complete.

Remark 2.5. By on , , (1.3), (2.8) and , we get

Lemma 2.6 (see [3]). Let , where and B are constants, is a quotient of odd positive integers. Then attains its maximum value on at , and

Lemma 2.7 (see [11]). and are delta-differentiable on . For and any , one has

3. Main Results

In this section, by employing the Riccati transformation technique we will establish oscillation criteria for (1.3) in two cases: and . Set

.

Theorem 3.1. Assume that hold and . Furthermore, assume that there exists a positive rd-continuous -differentiable function such that for all sufficiently large ,

then (1.3) is oscillatory.

Proof. Suppose to the contrary that is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that is eventually positive (note that in the case when is eventually negative, the proof is similar, since the substitution transforms (1.3) into the same form). Then, by there exists sufficiently large such that , , , and Lemma 2.4 holds for , where is defined by (2.8). Define the function by the Riccati substitution

then and

By (1.3), , and (2.18), we obtain

Noting that , we have

By Young’s inequality

with

we have

By and Lemma 2.4, we get

In view of and (3.5)–(3.10), for all , we obtain

Using , Lemma 2.4 and the Keller’s chain rule, we get

Also from Lemma 2.4 and , we have

By (3.11)–(3.13), we get

Setting

then by Lemma 2.6, from (3.14) we obtain that for all ,

Integrating the above inequality from to , we get

Taking lim sup on both sides of the above inequality as , we obtain a contradiction to condition (3.2). The proof is complete.

The following theorem gives new oscillation criteria for (1.3) which can be considered as the extension of Philos-type oscillation criterion. Define and

Theorem 3.2. Assume that hold and . Furthermore, assume that there exist a positive rd-continuous -differentiable function and a function such that for all sufficiently large ,

then (1.3) is oscillatory.

Proof. Suppose to the contrary that is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that is eventually positive. Then, by there exists sufficiently large such that , , , and Lemma 2.4 holds for , where is defined by (2.8). Define as in (3.3). Proceeding as in the proof of Theorem 3.1, we can get (3.14). From (3.14), for function and all we have

Using the integration by parts formula (2.7), we obtain

It follows that

Setting

by Lemma 2.6 we obtain that for all ,

That is,

Taking lim sup on both sides of the above inequality as , we obtain a contradiction to condition (3.19). The proof is complete.

Theorem 3.3. Assume that hold and . Then (1.3) is oscillatory if for all sufficiently large ,

Proof. Suppose to the contrary that is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that is eventually positive. Then, by there exists sufficiently large such that , , , and Lemma 2.4 holds for , where is defined by (2.8). Set . By Lemma 2.4, we get . Using and the keller’s chain rule, we get

So we have and there is a constant such that for . Then (1.3) becomes . By (2.18), we have

Similar to the proof of (3.10), we get

Using the Keller’s chain rule and , we get and

From and , it follows that

By Lemma 2.4, we get

It follows that

Integrating the above inequality from to , we obtain

Taking lim sup on both sides of the above inequality as , we obtain a contradiction to condition (3.26). The proof is complete.

Theorem 3.4. Assume that hold and . Furthermore, assume that there exists a positive rd-continuous -differentiable function such that for all sufficiently large ,

then (1.3) is oscillatory.

Proof. Suppose to the contrary that is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that is eventually positive. Then, by there exists sufficiently large such that , , , and Lemma 2.4 holds for , where is defined by (2.8). Define as in (3.3). By (2.6), we obtain

By Lemma 2.4, , and (3.5)–(3.13), for all we obtain

It follows that

By completing the square, we have

Integrating the above inequality from to , we get

Taking lim sup on both sides of the above inequality as , we obtain a contradiction to condition (3.35). The proof is complete.

Theorem 3.5. Assume that and hold. Furthermore, assume that there exist a positive rd-continuous -differentiable function and a function such that for all sufficiently large ,

then (1.3) is oscillatory.

Proof. By (3.39), the proof is similar to Theorems 3.2 and 3.4, so we omit it.

Theorem 3.6. Assume that hold and . Furthermore, assume that there exists a rd-continuous -differentiable function such that for all sufficiently large ,

then (1.3) is oscillatory.

Proof. Suppose to the contrary that is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that is eventually positive. Then, by there exists sufficiently large such that , , , and Lemma 2.4 holds for , where is defined by (2.8). Define the function by the Riccati substitution

then . From (3.5) and (3.10), it follows that for all

By (3.12) and Lemma 2.7, we obtain

It follows from Lemma 2.4 that

Integrating the above inequality from to , we get

Taking lim sup on both sides of the above inequality as , we obtain a contradiction to condition (3.42). The proof is complete.

From Theorem 3.6, we can establish different sufficient conditions for the oscillation of (1.3) by using different choices of . For instance, if or , we have the following results.

Corollary 3.7. Assume that hold and . Then (1.3) is oscillatory if for all sufficiently large ,

Corollary 3.8. Assume that hold and . Then (1.3) is oscillatory if for all sufficiently large ,

where .

.

Theorem  . Assume that hold and . Furthermore, assume that there exists a positive rd-continuous -differentiable function such that for all sufficiently large ,

then (1.3) is oscillatory.

Proof. Suppose to the contrary that is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that is eventually positive. Then, by there exists sufficiently large such that , , and Lemma 2.4 holds for , where is defined by (2.8). Define as in (3.3). Proceeding as in the proof of Theorem 3.1, we get

Using , Lemma 2.4 and the Keller’s chain rule, we get

By (3.10), (3.51), and (3.52), we get

Since

it follows that

It is easy to see (3.55) is of the same form as (3.14). The following is similar to the proof of Theorem 3.1 and hence omitted.

For , Theorem 3.2 also holds. Its proof is similar to those of Theorems   and 3.2.

Theorem  . Assume that hold and . Furthermore, assume that there exist a positive rd-continuous -differentiable function and a function such that for all sufficiently large ,

then (1.3) is oscillatory.

Theorem  . Assume that hold and . Then (1.3) is oscillatory if for all sufficiently large ,

Proof. Suppose to the contrary that is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that is eventually positive. Then, by there exists sufficiently large such that , , , and Lemma 2.4 holds for , where is defined by (2.8), is defined as in Theorem 3.3. Similar to the proof of Theorem 3.3, we get

Using the Keller’s chain rule, , and , we get

From and , it follows that

It is easy to see (3.31) is of the same form as (3.60). The following is similar to the proof of Theorem 3.3 and hence omitted.

Last, we give a theorem which holds for all , a quotient of add positive integers.

Theorem 3.9. Assume that hold. Furthermore, assume that there exists a positive rd-continuous -differentiable function such that for all sufficiently large ,

then (1.3) is oscillatory.

Proof. Suppose to the contrary that is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that is eventually positive. Then, by there exists sufficiently large such that , , , and Lemma 2.4 holds for , where is defined by (2.8). Define as in (3.3). Then, and because we get

From (3.5), (3.10), and (3.13), we obtain

Integrating the above inequality from to , we get

Taking lim sup on both sides of the above inequality as , we obtain a contradiction to condition (3.61). The proof is complete.

4. Examples

In this section, we give two examples to illustrate our main results. To obtain the conditions for oscillation we will use the following facts:

We first give an example to show Theorems 3.1 and  .

Example 4.1. Consider the equation

where is a time scale, satisfies , and satisfy , , and is a quotient of odd positive integers.

We choose , , and , then , , and . For any sufficiently large and , there exists a constant sufficiently large such that

Hence, by Theorems 3.1 and  , (4.2) is oscillatory.

The second example illustrates Corollary 3.7.

Example 4.2. Consider the equation

where is a time scale, , , , , and satisfy , has an inverse function , and satisfies .

We choose and . For any sufficiently large and , there exists a constant sufficiently large such that

Hence, by Corollary 3.7, (4.4) is oscillatory.

Acknowledgment

This project was supported by the NNSF of China (nos. 10971231, 11071238, and 11271379).

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