Oscillation Criteria of Third-Order Nonlinear Damped Dynamic Equations on Time Scales

Abstract

We establish oscillation criteria of third-order nonlinear damped dynamic equations on time scales of the form by employing functions in some function classes and the generalized Riccati transformation. Two examples are given to show the significance of the conclusions.

1. Introduction

In this paper, we study third-order nonlinear damped dynamic equation on a time scale satisfying and .

Throughout this paper we shall assume that(C1) such that (C2) is a quotient of odd positive integers;(C3), and for any , (C4) and there exists a function such that for with a same sign, (C5)when , it always satisfies

In 1988, Hilger introduced the theory of time scales in his Ph.D. thesis [1] in order to unify continuous and discrete analysis; see also [2]. Preliminaries about time scale calculus can be found in [3–6] and omitted here.

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.

There has been much research achievement about the oscillation of dynamic equations on time scales in the last few years; see the papers [1–14] and the references therein.

Wang [14] discussed the even order nonlinear damped differential equation and obtained some oscillation criteria for (6). Hassan [8] gave some oscillation criteria for the third order nonlinear delay dynamic equation Afterwards, Erbe et al. [7] established some new oscillation criteria for (7). Saker et al. [11] studied the second-order damped dynamic equation and they gave some oscillation criteria.

Qiu and Wang [9] considered the second-order nonlinear dynamic equation By using a generalized Riccati transformation the authors established some Kamenev-type oscillation criteria. Şenel [12] had tried to establish Kamenev-type oscillation criteria for the second-order nonlinear dynamic equation of the form However, it seemed that the obtained theorems and corollaries are incorrect. Qiu and Wang [10] corrected some mistakes in [12] and established correct oscillation criteria for (11) by employing functions in some function classes and the generalized Riccati transformation. Şenel [13] considered the third-order nonlinear dynamic equation and established some sufficient conditions which guarantee that every solution of (12) oscillates or converges to zero on an arbitrary time scale . In this paper, we shall establish new oscillation criteria of (1), which is more general than (12), and give two examples to show the significance of the results.

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

2. Preliminary Results

To establish oscillation criteria of (1), we give four lemmas in this section.

Lemma 2. Assume that (C1)–(C5) hold and there exists a sufficiently large such that is a solution of (1) satisfying for . Then there exists such that for , we have

Proof. Let such that is a solution of (1) satisfying for ; then we also have . By (1) and (C4), it follows that, for , Hence, is strictly decreasing on . We claim that Assume not, then there exists such that for . So there exists a constant and such that for , which means that Integrating (16) from to , we obtain Letting , by (C1) we have . Then there exists such that for , which implies that Integrating (18) from to , we obtain Letting , by (C1) we have , which contradicts . So (15) holds, which implies that Therefore, is strictly increasing on . It follows that is either eventually positive or eventually negative. Then, there exists such that, for , we have (13) holding. Lemma 2 is proved.

Lemma 3. For , assume that (C1)–(C5) hold and is a solution of (1) satisfying , for with . Then, we have where

Proof. Since is a solution of (1) satisfying for with , by Lemma 2 we have By , it follows that, for , Integrating (24) from to , by (23) we obtain As is strictly decreasing on , for we have Hence when , we have which implies that Lemma 3 is proved.

Lemma 4. For , assume that (C1)–(C4) hold and is a solution of (1) satisfying , for with , and Then .

Proof. Since is a solution of (1) satisfying for with , by Lemma 2 we have By , there exists such that . Assume , by (24) and , we obtain Letting , , we have , and Integrating (32) from to , we obtain By (29), there exists a sufficiently large such that , , which contradicts . So . Lemma 4 is proved.

Lemma 5. Assume that (C1)–(C5) hold and is a solution of (1) satisfying , for with . For , define where and . Then, satisfies where

Proof. Without loss of generality we may assume there exists such that , for , then Lemmas 2 and 3 hold. Let be defined by (34). Then, differentiating (34) 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 3, for , we obtain So (39) becomes When , we have and it follows that By Lemmas 2 and 3, for , we obtain which implies that So (39) becomes By (43) and (48), (35) holds. Lemma 5 is proved.

3. Main Results

In this section, we establish oscillation criteria of (1) by generalized 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 give our first theorem.

Theorem 6. Assume that (C1)–(C5) hold and that there exist and such that, for any , where Then, (1) is oscillatory or exists.

Proof. Assume that (1) is not oscillatory. Without loss of generality we may assume there exists such that for . By Lemma 2, for , either or holds. Assume , . Let be defined by (34). Then by Lemma 5, (35) holds.
For simplicity in the following, we let , , and we omit the arguments in the integrals.
Multiplying (35), where is replaced by , by and integrating it with respect to from to with , we obtain Noting that , by the integration by parts formula we have When , we have When , we have Using the inequality let , and then we have Therefore, for all , by (53) we have which implies that Hence, which contradicts (50). So , , and it is clearly that exists. The proof is completed.

When , if (29) holds, we have the following corollary on the basis of Lemma 4 and Theorem 6.

Corollary 7. When , assume that (C1)–(C4) and (29) hold. If there exist and such that, for any , where Then, (1) is oscillatory or .

Remark 8. In Corollary 7, letting , we can simplify (62) as When , (34) is simplified as Now we have the following theorem.

Theorem 9. Assume that (C1)–(C5) hold and that there exists such that, for any , where Then, (1) is oscillatory or exists.

Proof. Assume that (1) is not oscillatory. Without loss of generality we may assume that there exists such that for . By Lemma 2, for , either or holds. Assume , . Let be defined by (65). Then by Lemma 5, we have where is simplified as When , we have When , we have Using the inequality let , and then we have Therefore, for all , we always have which implies that Letting be replaced by , and integrating (76) with respect to from to , we obtain which is a contradiction of (66). So , , and, as before, exists. The proof is completed.

When , if (29) holds, from Lemma 4 and Theorem 9, we have the following result.

Corollary 10. When , assume that (C1)–(C4) and (29) hold. If there exists such that, for any , then (1) is oscillatory or .

Remark 11. Actually, letting , by (29) we have (78). Hence, it is easy to satisfy the conditions in Corollary 10.

Remark 12. When , for with a same sign, if we assume that or instead of (4) in (C4), by (31) in Lemma 4, the conclusions above are also applicable. Furthermore, if , , and by (79) it is clear that the conclusions above include the results in Şenel [13].

Remark 13. When , if , the conclusions above are not applicable. Similarly, the assumption that in Şenel [12] and Qiu and Wang [10] should be changed to (C3) in this paper.

4. Examples

In this section, the application of our oscillation criteria will be shown in two examples. The first example is given to demonstrate Theorem 6.

Example 1. Consider the equation where , is a constant and , , , , , and , by (C4) we get . Letting , , we have Since , when , so we have where . Hence, When , where . Hence, That is, (50) holds. By Theorem 6 we see that (82) is oscillatory or exists.
The second example illustrates Corollary 10.

Example 2. Consider the equation (see Şenel [13, Example 3.4]) where , , , and . Letting , by (79) we have . Then, letting , we obtain Therefore,

That is, (78) holds. By Corollary 10 we see that (89) is oscillatory or . This result is consistent with the conclusion in Şenel [13, Example  3.4].

Conflict of Interests

The author declares 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).