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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 952932, 16 pages
doi:10.1155/2012/952932
Research Article

Interval Oscillation Criteria of Second-Order Nonlinear Dynamic Equations on Time Scales

Yang-Cong Qiu1,2 and Qi-Ru Wang2

1Department of Humanities and Education, Shunde Polytechnic, Foshan, Guangdong 528333, China
2School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China

Received 7 May 2012; Revised 22 July 2012; Accepted 7 August 2012

Academic Editor: Yuriy Rogovchenko

Copyright © 2012 Yang-Cong Qiu 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

Using functions in some function classes and a generalized Riccati technique, we establish interval oscillation criteria for second-order nonlinear dynamic equations on time scales of the form . The obtained interval oscillation criteria can be applied to equations with a forcing term. An example is included to show the significance of the results.

1. Introduction

In this paper, we study the second-order nonlinear dynamic equation on a time scale .

Throughout this paper we will assume that(C1) ; (C2) , where is an arbitrary positive constant;(C3) .

Preliminaries about time scale calculus can be found in [1–3] and hence we omit them here. Without loss of generality, we assume throughout that .

Definition 1.1. A solution of (1.1) is said to have a generalized zero at if , and it is said to be nonoscillatory on if there exists such that for all . Otherwise, it is oscillatory. Equation (1.1) is said to be oscillatory if all solutions of (1.1) are oscillatory. It is well-known that either all solutions of (1.1) are oscillatory or none are, so (1.1) may be classified as oscillatory or nonoscillatory.

The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. thesis [4] in 1988 in order to unify continuous and discrete analysis, see also [5]. In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of dynamic equations on time scales, for example, see [1–27] and the references therein. In Došlý and Hilger’s study [10], the authors considered the second-order dynamic equation and gave necessary and sufficient conditions for the oscillation of all solutions on unbounded time scales. In Del Medico and Kong’s study [8, 9], the authors employed the following Riccati transformation: and gave sufficient conditions for Kamenev-type oscillation criteria of (1.2) on a measure chain. And in Yang’s study [27], the author considered the interval oscillation criteria of solutions of the differential equation In Wang’s study [24], the author considered second-order nonlinear differential equation used the following generalized Riccati transformations: where , and gave new oscillation criteria of (1.5).

In Huang and Wang’s study [16], the authors considered second-order nonlinear dynamic equation on time scales By using a similar generalized Riccati transformation which is more general than (1.3) where , , the authors extended the results in Del Medico and Kong [8, 9] and Yang [27], and established some new Kamenev-type oscillation criteria and interval oscillation criteria for equations with a forcing term.

In this paper, we will use functions in some function classes and a similar generalized Riccati transformation as (1.8) and was used in [24, 25] for nonlinear differential equations, and establish interval oscillation criteria for (1.1) in Section 2. Finally in Section 3, an example is included to show the significance of the results.

For simplicity, throughout this paper, we denote , where , and are denoted similarly.

2. Main Results

In this section, we establish interval criteria for oscillation of (1.1). Our approach to oscillation problems of (1.1) is based largely on the application of the Riccati transformation.

Let and . For any function : , denote by and the partial derivatives of with respect to and , respectively. For , denote by the space of functions which are integrable on any compact subset of . Define These function classes will be used throughout this paper. Now, we are in a position to give our first lemma.

Lemma 2.1. Assume that (C1)–(C3) hold and that there exist , functions such that for , for all and . If is a solution of (1.1) such that on (or on ), for any one defines on , and , . Then for any , and , one has where for , for , and

Proof. Suppose that is a solution of (1.1) such that on . First, Hence, we always have Then differentiating (2.4) and using (1.1), it follows that . Noting that on , from (2.10), we have That is, for , (ii) For , from (2.10), we have Then (2.12) also holds.
From (i) and (ii) above, we see that (2.12) holds for . For simplicity in the following, we let , and omit the arguments in the integrals. For , Since on , we see that . Multiplying (2.12), where is replaced by , by , and integrating it with respect to from to , we obtain Noting that , by the integration by parts formula, we have Since on , from (2.8), we see that For , and , we have For , and , we have Therefore, for , we have Then from (2.16), (2.17), and (2.20), we obtain that (2.5) holds for .
If on , then we see that on and Following the steps above, we have that (2.5) holds for . The proof is complete.

Next, we have the second lemma.

Lemma 2.2. Assume that (C1)–(C3) hold, and that there exist , , functions such that for and and (2.3) holds for all and . If is a solution of (1.1) such that on on ), define as in (2.4) on . Then for any , one has where is defined as before, and

Proof. Suppose that is a solution of (1.1) such that on . For simplicity in the following, we let , and omit the arguments in the integrals. Multiplying (2.12), where is replaced by , by , and integrating it with respect to from to and then using the integration by parts formula we have that For , Hence,
Furthermore, for , and , For , and , Hence, for , we have
From (2.25), (2.27), and (2.30), we have that (2.23) holds for .
If on , then we see that on . Following the steps above, we have that (2.23) holds for . The proof is complete.

Theorem 2.3. Assume that (C1)–(C3) and the following two conditions hold:(C4) For any , there exist , , functions such that for , and (2.3) holds for all and .(C5) There exist , , , , , such that for , where , and are defined as before.

Then (1.1) is oscillatory.

Proof. Suppose that is a nonoscillatory solution of (1.1) which is eventually positive, say when for some depending on the solution . From the assumption (C4), we can choose so that on the interval with . From Lemmas 2.1 and 2.2, we see that (2.5) and (2.23) hold for . By dividing (2.5) and (2.23) by and , respectively, and then adding them, we obtain a contradiction to assumption (2.32) with .
When is eventually negative, we choose so that on to reach a similar contradiction. Hence, every solution of (1.1) has at least one generalized zero in or .
Pick a sequence such that and as . By assumption, for each there exists such that and (2.32) holds, where are replaced by , respectively. Hence, every solution has at least one generalized zero . Noting that , we see that every solution has arbitrarily large generalized zeros. Thus, (1.1) is oscillatory. The proof is complete.

Corollary 2.4. Assume that (C1)–(C4) hold and that
(C6) there exist , , , , , such that for , where , and are defined as before. Then (1.1) is oscillatory.

Proof. By (2.33) and (2.34), we get (2.32). Therefore, (1.1) is oscillatory by Theorem 2.3. The proof is complete.

When , we have the following corollary.

Corollary 2.5. Assume that (C1)–(C3) hold and that there exists a function such that . Also, suppose that there exist , , , such that for any Then (1.1) is oscillatory.

Proof. When (C3) holds and there exists a function such that , it follows that (C4) holds for and . Now . For any , let . In (2.35), we choose . Then there exists such that In (2.36), we choose . Then there exists such that Combining (2.37) and (2.38) we obtain (2.32) with .
Next, in (2.35) we choose . Then there exists such that In (2.36), we choose . Then there exists such that Combining (2.39) and (2.40) we obtain (2.32) with . The conclusion thus follows from Theorem 2.3. The proof is complete.

3. Example

In this section, we will show the application of our oscillation criteria in an example. The example is to demonstrate Theorem 2.3.

Example 3.1. Consider the equation where , , , and . So we have .
For any , there exists such that . Let , we have (i)Consider , So for , we have

When , we have , so (2.32) holds, which means that (C5) holds. By Theorem 2.3, we have that (3.1) is oscillatory. However, when , we do not know whether (3.1) is oscillatory. (2) Consider ,

So we have

When   , we have   , so (2.32) holds, which means that (C5) holds. By Theorem 2.3, we have that (3.1) is oscillatory. However, when   , we do not know whether (3.1) is oscillatory.

Acknowledgments

The authors sincerely thank the referees for their valuable comments and useful suggestions that have led to the present improved version of the original paper. This project was supported by the NNSF of China (nos. 10971231, 11071238, 11271379) and the NSF of Guangdong Province of China (no. S2012010010552).

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