Abstract and Applied Analysis

Volume 2013 (2013), Article ID 340574, 17 pages

http://dx.doi.org/10.1155/2013/340574

Research Article

Oscillatory and Asymptotic Properties on a Class of Third Nonlinear Dynamic Equations with Damping Term on Time Scales

Qinghua Feng and Huizeng Qin

School of Science, Shandong University of Technology, Zhangzhou Road 12, Zibo 255049, China

Received 4 March 2013; Revised 10 May 2013; Accepted 10 May 2013

Academic Editor: Narcisa C. Apreutesei

Copyright © 2013 Qinghua Feng and Huizeng Qin. 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.

We establish some new oscillatory and asymptotic criteria for a class of third-order nonlinear dynamic equations with damping term on time scales. The established results on one hand extend some known results in the literature on the other hand unify continuous and discrete analysis. For illustrating the validity of the established results, we also present some applications for them.

1. Introduction

The theory of time scale, which was initiated by Hilger [1], trying to treat continuous and discrete analysis in a consistent way, has received a lot of attention in recent years. Various investigations have been done by many authors. Among these investigations, some authors have taken research in the oscillatory and asymptotic properties of dynamic equations on time scales, and there has been increasing interest in obtaining sufficient conditions for the oscillation and asymptotic behavior of solutions of various dynamic equations on time scales (e.g., we refer the reader to [2–20]). But we notice that most of the investigations are concerned with oscillatory and asymptotic properties of solutions of first- or second-order dynamic equations on time scales, while relatively less attention has been paid to oscillatory and asymptotic properties of third-order dynamic equations on time scales. For recent results about the oscillation and asymptotic behavior of solutions of third-order dynamic equations on time scales, we refer the reader to [21–33]. In [34, 35], Saker researched oscillation of the following third-order dynamic equations:

Based on the Riccati substitution and the analysis of the associated Riccati dynamic inequality, some new sufficient oscillatory conditions were presented.

Moreover, to our best knowledge, none of the existing results deal with oscillatory and asymptotic behavior of solutions of third-order nonlinear dynamic equations with damping term on time scales, in which the damping term brings new difficulty in obtaining oscillatory and asymptotic criteria. We now list some important results.

In this paper, we are concerned with oscillatory and asymptotic behavior of solutions of the third-order nonlinear dynamic equation with damping term on time scales of the following form:

where is an arbitrary time scale, , satisfying for , and is a quotient of two odd positive integers.

A solution of (2) is said to be oscillatory if it is neither eventually positive nor eventually negative otherwise it is nonoscillatory. Equation (2) is said to be oscillatory in case all its solutions are oscillatory.

We will establish some new criteria of oscillatory and asymptotic behavior for (2) by a generalized Riccati transformation technique in Section 2 and present some applications for our results in Section 3. Throughout this paper, denotes the set of real numbers and , while denotes the set of integers. denotes an arbitrary time scale and ,. On we define the forward and backward jump operators and such that , . A point with is said to be left-dense if , right-dense if , left-scattered if , and right-scattered if . A function is called rd-continuous if it is continuous in right-dense points and if the left-sided limits exist in left-dense points, while is called regressive if , where . denotes the set of rd-continuous functions, while denotes the set of all regressive and rd-continuous functions, and .

Definition 1. For , the exponential function is defined by

Remark 2. If , then

If , then

The following two theorems include some known properties on the exponential function.

Theorem 3 (see [36, Theorem 5.1]). If and fix , then the exponential function is the unique solution of the following initial value problem

Theorem 4 (see [36, Theorem 5.2]). If , then for  .

For more details about the calculus of time scales, we refer to [37].

2. Main Results

For the sake of convenience, in the rest of this paper, we set .

Lemma 5. Suppose , and assume that

and (2) has an eventually positive solution . Then there exists a sufficiently large such that

Proof. By , we have . Since is eventually a positive solution of (2), there exists a sufficiently large such that on , and for , we obtain that

Then is strictly decreasing on , and together with , we deduce that is eventually of one sign. We claim on . Otherwise, assume there exists a sufficiently large such that on . Then

By (7), we have , and thus there exists a sufficiently large such that on . By the assumption one can see is strictly decreasing on , and then

Using (8), we have , which leads to a contradiction. So on , and the proof is complete with taking .

Lemma 6. Under the conditions of Lemma 5, furthermore, assume that

Then either there exists a sufficiently large such that on or .

Proof. By Lemma 5, we deduce that is eventually of one sign. So there exists a sufficiently large such that either or on , where is defined as in Lemma 5. If , together with is an eventually positive solution of (2), we obtain and . We claim . Otherwise, assume . Then on , and for , an integration for (10) from to yields

which is followed by

Substituting with in (15), an integration for (15) with respect to from to yields

which implies

Substituting with in (17), an integration for (17) with respect to from to yields

By (18) and (13) we have , which leads to a contradiction. So one has , and the proof is complete.

Lemma 7. Suppose , and assume that is a positive solution of (2) such that

where is sufficiently large. Then for , we have

Proof. Take , where , are defined as in Lemmas 5 and 6, respectively. By Lemma 5 we have strictly decreasing on . So

and then

Furthermore,

which is the desired result.

Lemma 8 (see [38, Theorem 41]). Assume that and are nonnegative real numbers. Then

Theorem 9. Suppose , and assume that (7), (8), and (13) hold, and for all sufficiently large ,

where are two given nonnegative functions on with . Then every solution of (2) is oscillatory or tends to zero.

Proof. Assume (2) has a nonoscillatory solution on . Without loss of generality, we may assume on , where is sufficiently large. By Lemmas 5 and 6, there exists sufficiently large such that on , and either on or . Now we assume on . Define the generalized Riccati function:

Then for , we have

By [37, Theorem 1.93], we have . Then

Using the following inequality (see [25, ]):

where are constants and is a quotient of two odd positive integers, we obtain

A combination of (28) and (30) yields

Setting

Using Lemma 8 in (31) we get that

Substituting with in (33), an integration for (33) with respect to from to yields

which contradicts (25). So the proof is complete.

In Theorem 9, if we take for some special cases, then we can obtain the following corollaries.

Corollary 10. Let . Assume that

and for all sufficiently large ,

 Then every solution of (2) is oscillatory or tends to zero.

Corollary 11. Let and . Assume that

and for all sufficiently large ,

Then every solution of (2) is oscillatory or tends to zero.

Theorem 12. Suppose , and assume that (7), (8), and (13) hold, and for all sufficiently large ,

where are defined as in Theorem 9, then every solution of (2) is oscillatory or tends to zero.

Proof. Assume (2) has a nonoscillatory solution on . Similar to Theorem 9, we may assume on , where is sufficiently large. By Lemmas 5 and 6, there exists sufficiently large such that on , and either on or . Now we assume on . Let be defined as in Theorem 9. By Lemma 7, we have the following observation:

Using (40) in (28) we get that

Substituting with in (41), an integration for (41) with respect to from to yields

which contradicts (39). So the proof is complete.

Based on Theorems 9 and 12, we will establish some Philos-type oscillation criteria for (2).

Theorem 13. Suppose , and assume that (7), (8), and (13) hold, and define . If there exists a function such that

and has a nonpositive continuous partial derivative with respect to the second variable, and for all sufficiently large ,

where are defined as in Theorem 9. Then every solution of (2) is oscillatory or tends to zero.

Proof. Assume (2) has a nonoscillatory solution on . Without loss of generality, we may assume on , where is sufficiently large. By Lemmas 5 and 6, there exists sufficiently large such that on , and either on or . Now we assume on . Let be defined as in Theorem 9. By (33) we have

Substituting with in (45) and multiplying both sides by and then integrating with respect to from to yield

Then

So

which contradicts (44). So the proof is complete.

Theorem 14. Suppose , and assume that (7), (8), and (13) hold. Let be defined as in Theorem 13, and for all sufficiently large ,

where are defined as in Theorem 9. Then every solution of (2) is oscillatory or tends to zero.

Proof. Assume (2) has a nonoscillatory solution on . Without loss of generality, we may assume on , where is sufficiently large. By Lemmas 5 and 6, there exists sufficiently large such that on , and either on or . Now we assume on . Let be defined as in Theorem 9. By (41) we have

Substituting with in (45), multiplying both sides by , and then integrating with respect to from to yield

Then similar to Theorem 13, we obtain

which contradicts (44). So the proof is complete.

In Theorems 13 and 14, if we take for some special functions such as or , then we can obtain some corollaries. For example, if we take , then we have the following corollaries.

Corollary 15. Suppose , and assume that (7), (8), and (13) hold, and for all sufficiently large ,

Then every solution of (2) is oscillatory or tends to zero.

Corollary 16. Suppose , and assume that (7), (8), and (13) hold, and for all sufficiently large ,

Then every solution of (2) is oscillatory or tends to zero.

Remark 17. The established results above extend the main results in [25, Theorems ] except that the latter is related to time delay.

Remark 18. In Theorems 12–14, if we take for some special time scales, we can obtain similar results as in Corollaries 10 and 11, which are omitted here.

In Theorems 9, 12, 13, and 14, if we let , then , and subsequently we obtain the following four corollaries concerning oscillatory criteria of the following equation:

Corollary 19. Assume (8) holds. If

and for all sufficiently large ,

where are defined as in Theorem 9. Then every solution of (2) is oscillatory or tends to zero.

Corollary 20. Assume that (8), (56), and (57) hold, and for all sufficiently large ,

where are defined as in Theorem 9. Then every solution of (2) is oscillatory or tends to zero.

Corollary 21. Assume that (8), (56), and (57) hold. If for all sufficiently large ,

where are defined as in Theorems 9 and 13, respectively, then every solution of (2) is oscillatory or tends to zero.

Corollary 22. Assume that (8), (56), and (57) hold. If for all sufficiently large ,

where are defined as in Theorems 9 and 13, respectively, then every solution of (2) is oscillatory or tends to zero.

Remark 23. In [34, Theorems ] and [35, Theorems ], Saker established some new oscillatory criteria for the equation

under the condition . We note that the conditions (8) and (56) in Corollaries 19–22 are consistent with those in [34, ] and [35, ], which were used in [34, Theorems ] and [35, Theorems ], respectively, while is assumed in [34, 35]. Moreover, in the results established above, the Riccati substitution function is defined by (see Theorem 9), which is different form that in [34, Theorem 3.3] and [35, Theorem 2.7], where the Riccati substitution function is defined by . Since the Riccati substitution function is the most important fact in establishing sufficient oscillatory conditions, so our results in Corollaries 19–22 are essentially different from Saker's results in [34, 35].

3. Applications

In this section, we will present some applications for the established results above. First we consider the following third-order nonlinear differential equation with damping term.

Example 24. Consider

where is a quotient of two odd positive integers.

We have in (2) ,. Then , and . So =. Moreover, we have

Then we have

Furthermore,

On the other hand, for a sufficiently large , we have

So there exists a sufficiently large such that for . Taking in (36), we get that

So (35)–(36) all hold, and by Corollary 10 we deduce that every solution of (63) is oscillatory or tends to zero.

Next we consider the following third-order difference equation:

Example 25. Consider

where denotes the difference operator, is a constant, and is a quotient of two odd positive integers.

We have in (2) ,, . Then , and

which implies . So by [2, Lemma 2] we obtain

Then we have

Furthermore,

On the other hand, for a sufficiently large , we have

So there exists such that for . Let in (38). Then by the inequality , we obtain

provided that . So (37) and (38) all hold, and by Corollary 11 we obtain that every solution of (69) is oscillatory or tends to zero under the condition .

Acknowledgments

The authors would like to thank the reviewers very much for their valuable suggestions on this paper.

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