Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 315158, 12 pages

http://dx.doi.org/10.1155/2013/315158

Research Article

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

Yang-Cong Qiu1,2 and Qi-Ru Wang²

¹Department of Humanities & Education, Shunde Polytechnic, Foshan, Guangdong 528333, China

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

Received 20 November 2012; Accepted 14 December 2012

Academic Editor: Elena Braverman

Copyright © 2013 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 from some function classes and a generalized Riccati technique, we establish Kamenev-type oscillation criteria for second-order nonlinear dynamic equations on time scales of the form . Two examples are 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 a fixed positive constant,(C3), and there exist such that for all ,(C4).

Preliminaries about time scale calculus can be found in [1–3], and hence we omit them here. Note that for some typical time scales, we have the following properties, respectively:

(1) , we have

(2) , we have

(3) , we have

(4) , we have

Without loss of generality, we assume throughout that since we are interested in extending oscillation criteria for the typical time scales above.

Definition 1. A solution of (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) is said to be oscillatory if all solutions of (1) are oscillatory.

The theory of time scales, which has recently received a lot of attention, was introduced by Stefan 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 [1–28] and the references therein. In Došlý and Hilger [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 [8, 9], the authors employed the following Riccati transformation

and gave sufficient conditions for Kamenev-type oscillation criteria of (6) on a measure chain.

In Wang [25], the author considered second-order nonlinear damped differential equation

used the following generalized Riccati transformations

where , and gave a new oscillation criteria of (8). In Huang and Wang [16], the authors considered second-order nonlinear dynamic equation on time scales

By using a similar generalized Riccati transformation which is more general than (7)

where , , the authors extended the results in Del Medico and Kong [8, 9] and established some new Kamenev-type oscillation criteria.

In this paper, we will use functions in some function classes and a similar generalized Riccati transformation as (11) and was used in [25, 26] for nonlinear differential equations, and establish Kamenev-type oscillation criteria for (1) in Section 2. Finally, in Section 3, two examples are included to show the significance of the results.

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

2. Kamenev-Type Criteria

In this section we establish Kamenev-type criteria for oscillation of (1). Our approach to oscillation problems of (1) is based largely on the application of the Riccati transformation. Now, we give the first lemma.

Lemma 2. Assume that (C1)–(C4) hold and that there exists a function such that . Also, suppose that is a solution of (1) satisfies for with . For , define

where , , and for . Then, satisfies

where = + ,   = , = .

Proof. By (C3), we see that and are both positive, both negative, or both zero. When , which implies that , it follows that

When , which implies that , it follows that

When , which implies that and , it follows that

Hence, we always have

that is, (13) holds. Then, differentiating (12) and using (1), it follows that

that is, (14) holds. Lemma 2 is proved.

Remark 3. In Lemma 2, the condition ensures that the coefficient of in (14) is always negative. The condition is obvious and easy to be fulfilled. For example, when for all , we have , by (C3), we see that , and when , the condition is also fulfilled.

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 theorem.

Theorem 4. Assume that (C1)–(C4) hold and that there exists a function such that . Also, suppose that there exist and such that and for any ,

where is defined as before, and

Then, (1) is oscillatory.

Proof. Assume that (1) is not oscillatory. Without loss of generality, we may assume there exists such that for . Let be defined by (12). Then, by Lemma 2, (13) and (14) hold.

For simplicity in the following, we let , and and omit the arguments in the integrals. For ,

Since on , we see that . From and (C3), we have

Multiplying (14), where is replaced by , by and integrating it with respect to from to with and , we obtain

Noting that , by the integration by parts formula, we have

Since on , from (13) we see that for ,

For , , and , from (24), we have

For , , and , from (24), we have

Therefore, for all , , we have

Then, from (26), (27), and (30), we obtain that for and ,

Hence,

which contradicts (21) and completes the proof.

Remark 5. If we change the condition in the definition of with a stronger one , (24) in the proof of Theorem 4 will be changed with

Then, the definition of can be simplified as

In the sequel, we define

When , by (C3), we see that and (1) is simplified as

Now, we have the following theorem, but we should note that this result does not apply to the case where all points in are right dense.

Theorem 6. Assume that (C1)–(C4) with hold and that there exists a function such that . Let , and be defined by (35) and (36). Then, (37) is oscillatory provided there exists , , such that for any , one of the following holds

(i) and

(ii) and

(iii) and

 where , is defined as before, and

Proof. Assume that (37) is not oscillatory. Without loss of generality, we may assume there exists such that for . Let be defined by (12) with . Then, by Lemma 2, (13) and (14) hold for . So, we have

where and are defined as in Lemma 2.

For simplicity in the following, we let , and and omit the arguments in the integrals. Multiplying , 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 all , , we have

From (42), (44), and (47), we have

For , implies that

Hence,

Assume that condition (i) holds. Let in (50). Then, we obtain

Taking the as on both sides, we have

which contradicts (38).

The conclusions with conditions (ii) and (iii) can be proved similarly. We omit the details. The proof is complete.

When , Theorems 4 and 6 can be simplified as the following corollaries, respectively.

Corollary 7. Assume that (C1)–(C4) hold and that there exists a function such that . Also, suppose that there exists such that for any ,

Then, (1) is oscillatory.

Corollary 8. Assume that (C1)–(C4) with hold and that there exists a function such that . Let , and be defined by (35) and (36). Then, (37) is oscillatory provided that there exists , , such that for any , one of the following holds

(i) and

(ii) and

(iii) and

Remark 9. When and , Theorems 4 and 6 reduce to [16, Theorems 2.1 and 2.2], respectively. When , , , and , Theorems 4 and 6 reduce to [8, Theorems 2.1 and 2.2], respectively.

3. Examples

In this section, we will show the application of our oscillation criteria in two examples. We first give an example to demonstrate Theorem 4 (or Corollary 7).

Example 10. Consider the equation

where , , , and , so we have , , and . Let and , we have

(1) ,

That is, (53) holds. By Corollary 7, we see that (57) is oscillatory;

(2) ,

that is, (53) holds. By Corollary 7, we see that (57) is oscillatory;

(3) , ,

that is, (53) holds. By Corollary 7, we see that (57) is oscillatory;

(4) ,

that is, (53) holds. By Corollary 7, we see that (57) is oscillatory.

The second example illustrates Theorem 6.

Example 11. Consider the equation

where , , , and , so we have , . Let , we have

(1) , let . When is sufficiently large, there exists such that

that is, in Theorem 6, (i) and (38) hold. Then, (62) is oscillatory;

(2) , , let . When is sufficiently large, there exists such that

that is, in Theorem 6, (ii) and (39) hold. Then, (62) is oscillatory;

(3) , let . When is sufficiently large, there exists such that

that is, in Theorem 6, (iii) and (40) hold. Then, (62) is oscillatory.

Acknowledgments

The authors sincerely thank the referees for their valuable comments and suggestions. This Project was supported by the NNSF of China (nos. 10971231, 11071238, and 11271379).

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