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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

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 [220]). 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 [2133]. 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 yieldswhich 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 thatSubstituting with in (41), an integration for (41) with respect to from to yieldswhich 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 <