Oscillation Criteria of Second-Order Dynamic Equations with Damping on Time Scales

Abstract

Using functions in some function classes and a generalized Riccati technique, we establish Kamenev-type oscillation criteria for second-order dynamic equations with damping 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 dynamic equation with damping on a time scale satisfying and .

Throughout this paper we will assume that(C1);(C2), where ;(C3) is a quotient of odd positive integers;(C4) is nondecreasing and for ;(C5) and there exists a function such that ;(C6) is positively regressive, which means and

The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. thesis [1] in 1988 in order to unify continuous and discrete analysis; see also [2]. Preliminaries about time scale calculus can be found in [3–6] and hence we omit them here. Note that, for some typical time scales, we have the following properties, respectively:(1)since , we have (2)since , we have (3)since , we have

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.

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 [1–16] and the references therein. In Došlý and Hilger [9], the authors considered the second-order dynamic equation as and gave necessary and sufficient conditions for the oscillation of all solutions on unbounded time scales. In Del Medico and Kong [7, 8], the authors employed the following Riccati transformation: and gave sufficient conditions for Kamenev-type oscillation criteria of (6) on a measure chain.

In Wang [15], the author considered second-order nonlinear damped differential equation as using the following generalized Riccati transformations: where ,  , and gave new oscillation criteria of (8). In [12], Saker considered second-order nonlinear neutral delay dynamic equation as and improved some well-known oscillation results for second-order neutral delay difference equations. In [13], Saker et al. studied the second-order damped dynamic equation with damping as follows: and gave some new oscillation criteria. In Huang and Wang [10], the authors considered second-order nonlinear dynamic equation as By using a similar generalized Riccati transformation which is more general than (7), where , , the authors extended the results in Del Medico and Kong [7, 8] and established some new Kamenev-type oscillation criteria. In [11], Qiu and Wang considered the second-order nonlinear dynamic equation of a more general form and established some Kamenev-type oscillation criteria.

In [14], Şenel had tried to establish Kamenev-type oscillation criteria for (1). However, it seemed that several mistakes had been made and the obtained theorems and corollaries are incorrect. In this paper, we will correct some mistakes in [14] and establish some Kamenev-type oscillation criteria for (1) by employing functions in some function classes and a similar generalized Riccati transformation as (13) and as used in [15, 16] for nonlinear differential equations. Finally, two examples are included to show the significance of the results.

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

2. Preliminary Results

To establish Kamenev-type criteria for oscillation of (1), we give three lemmas in this section.

Lemma 2. Assume that (C1)–(C6) hold and there exists a sufficiently large such that is a solution of (1) satisfying for . Then, for , one has

Proof. Let such that is a solution of (1) satisfying for ; then we also have . By (1) and (C5), it follows that, for , so we have
Assume that there exists satisfying ; then, for , we also have Integrating (18) from to , we obtain that is, which implies that Integrating (21) from to and letting , by (C6), we obtain which contradicts . Hence , for , which implies from (16) that holds. Lemma 2 is proved.

Remark 3. In [14, (A*)], the key condition that is regressive is missed; then the assumption may not be well presented. In this paper, the condition is added as (C6).

Remark 4. In [14, (2.5)], it seems not to be so obvious to obtain the inequality And in [14, (2.7), (2.8)], the symbol should be . We have improved the proof in Lemma 2.

Lemma 5. Assume that (C1)–(C6) hold and is a solution of (1), for with . Then for , if , one has And, if , one has where

Proof. Since is a solution of (1) satisfying for with , by Lemma 2, we have From (16) it follows that Integrating (29), we obtain Hence, when , we have
Since , , for , we get and we obtain
So Lemma 5 is proved.

Remark 6. In [14, (2.9), (2.10)], when and for any , the integral in must be convergent, which means that The condition should be added to the paper.

Lemma 7. Assume that (C1)–(C6) hold and is a solution of (1) satisfying for with . For , define where , . Then satisfies where

Proof. Without loss of generality we may assume that there exists such that for ; then Lemmas 2 and 5 hold. Let be defined by (35). Then, differentiating (35) and using (1), it follows that Using the fact that , we obtain
When , using the Pötzsche chain rule, we have and it follows that By Lemmas 2 and 5, for , we obtain So (39) becomes
When , we have and it follows that By Lemmas 2 and 5, for , we obtain So (39) becomes
By (43) and (47), (36) holds. Lemma 7 is proved.

Remark 8. Note that, in general, . In [14, (3.6)], it seemed that Şenel had been confused with . Some similar mistakes had also been made in [14, (3.8), (3.10)] and, finally, there are some problems about all conclusions of the theorems and corollaries in [14].

3. Main Results

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. Firstly, we give some definitions.

Let and . For any function : , denote by the partial derivative of with respect to . Define These function classes will be used throughout this paper. Now, we are in a position to give our first theorem.

Theorem 9. Assume that (C1)–(C6) hold and that there exist and such that, for any , where Then, (1) is oscillatory.

Proof. Assume that (1) is not oscillatory. Without loss of generality we may assume that there exists such that for . Let be defined by (35). Then, by Lemma 7, (36) holds.
For simplicity, in the following, we let and omit the arguments in the integrals.
Multiplying (36), 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
When , we have
When , we have
When , on the one hand, we have
On the other hand, when , we also have
Using the inequality let , and Then we have So, when , we get while when we get
Therefore, for all , by (52), we have which implies that Hence which contradicts (49) and completes the proof.

When , let , and Theorem 9 can be simplified as the following corollary.

Corollary 10. Assume that (C1)–(C6) hold and that there exists such that, for any , where Then, (1) is oscillatory.

Remark 11. There are some mistakes in [14]. For example, should be in [14, (3.22)], the symbol had not correctly written in [14, (3.29)], should be in [14, (3.30), (3.32)], and the bracket had been missed in [14, (3.34)].

When , (35) is simplified as Now we have the following theorem.

Theorem 12. Assume that (C1)–(C6) hold and that there exists such that, for any , where Then, (1) is oscillatory.

Proof. Assume that (1) is not oscillatory. Without loss of generality we may assume that there exists such that for . Let be defined by (67). Then, by Lemma 7, we have where is simplified as
When , we have
When , we have
When , on the one hand, we have On the other hand, when , we also have Using the inequality let and Then we have So, when , we get while when we get
Therefore, for all , we always have which implies that Let be replaced by , and integrating (82) with respect to from to with and , we obtain which contradicts (68) and completes the proof.

When , let , and Theorem 12 can be simplified as the following corollary.

Corollary 13. Assume that (C1)–(C6) hold and that, for any , where Then, (1) is oscillatory.

Remark 14. It seems that the inequality in [14, (3.19)] is incorrect. From the definition of (see [14, (3.3)]), we can see that is not always positive and we could not obtain the conclusion in [14, Theorem  3.1]. As a result, we simplify as (67) which satisfies in this paper and get the correct results.

4. Examples

In this section, we will show the application of our oscillation criteria in two examples. We first give an example to demonstrate Theorem 9 (or Corollary 10).

Example 15. Consider the equation where , , , and , so we have . Letting , we have(i), and then there exists such that When , we obtain When , we obtain Hence That is, (65) holds. By Corollary 10 we see that (86) is oscillatory.(ii), then there exists such that Similarly, we can get the conclusion that (86) is oscillatory by Corollary 10.

The second example illustrates Theorem 12.

Example 16. Consider the equation where , , , and , so we have . Let , when ; then there exists such that When , we obtain For , we have .
When , we obtain When , we obtain For , we have .
Hence That is, (68) holds. By Theorem 12 we see that (92) is oscillatory. Similarly, we can get the same conclusion when and .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This project was supported by the NNSF of China (no. 11271379).