Abstract and Applied Analysis

Volume 2010 (2010), Article ID 294194, 10 pages

http://dx.doi.org/10.1155/2010/294194

## Forced Oscillation of Second-Order Half-Linear Dynamic Equations on Time Scales

^{1}Department of Mathematics, Maoming University, Maoming 525000, China^{2}Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

Received 14 March 2010; Revised 19 June 2010; Accepted 12 July 2010

Academic Editor: Allan C. Peterson

Copyright © 2010 Quanwen Lin et al. 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 will establish a new interval oscillation criterion for second-order half-linear dynamic equation on a time scale which is unbounded, which is a extension of the oscillation result for second order linear dynamic equation established by Erbe et al. (2008). As an application, we obtain a sufficient condition of oscillation of the second-order half-linear differential equation , where , , are odd positive integers.

#### 1. Introduction

The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. thesis [1] in order to unify continuous and discrete analysis. Not only can this theory of so-called “dynamic equations” unify the theories of differential equations and of difference equations, but also it is able to extend these classical cases to cases “in between,” for example, to so-called q-difference equations. A time scale is an arbitrary closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. Many other interesting time scales exist, and they give rise to plenty of applications, among them the study of population dynamic models (see [2]). A book on the subject of time scale by Bohner and Peterson [2] summarizes and organizes much of the time scale calculus (see also [3]). For the notions used below, we refer to [2] and to the next section, where we recall some of the main tools used in the subsequent sections of this paper.

In the last years, there has been much research activity concerning the oscillation and nonoscillation of solutions of some dynamic equations on time scales, and we refer the reader to the paper [4–6]. Following this trend, in this paper we will provide some sufficient conditions for oscillation of second-order half-linear dynamic equation.

Consider the second-order half-linear dynamic equation on a time scale which is unbounded above, , , and are rd-continuous functions. is a quotient of odd positive integer. When , (1.1) is the second-order linear dynamic equation

In [7], by using the Riccati substitution the authors established a interval oscillation criterion, that is, a criterion given by the behavior of and on a sequence of subintervals of . In this paper, we extend the result of [7] to the second-order half-linear dynamic (1.1). As a application, we prove the equation is oscillatory, if , where , are odd positive integers and is defined in Example 3.1.

For completeness, (see [2, 3] for elementary results for the time scale calculus), we recall some basic results for dynamic equations and the calculus on time scales. Let be a time scale (i.e., a closed nonempty subset of ) with . The forward jump operator is defined by and the backward jump operator is defined by where , where denotes the empty set. If , we say is right-scattered, while if , we say is left-scattered. If , we say is right-dense, while if and we say is left-dense. Given a time scale interval in the notation denotes the interval in case and denotes the interval in case . The graininess function for a time scale is defined by , and for any function the notation denotes . We say that is differentiable at provided that exists, when (here by it is understood that approaches in the time scale) and when is continuous at and , Note that if , then the delta derivative is just the standard derivative, and when the delta derivative is just the forward difference operator. Hence, our results contain the discrete and continuous cases as special cases and generalize these results to arbitrary time scales (e.g., the time scale which is very important in quantum theory [8]).

#### 2. Main Theorem

Theorem 2.1. *Assume that given any there exists points such that
**
Further assume that there exists a function such that for , one has
**
satisfies and on , with Then the dynamic (1.1) is oscillatory on *

*Remark 2.2. *When , the above theorem becomes [7, Theorem ].

*Proof. *Assume that (1.1) is nonoscillatory. Then there is a solution of (1.1) and a such that is of one sign on . We consider the case on . Make the substitution
Then, for (note that on ),
If we define
then we have

(i) Suppose that is right-dense. Then,
so we have
We use Young's inequality [9], which says that
with equality if and only if

So if we let
then we have and

(ii) Suppose next that is right-scattered and . Then,

Let us put , Then we have
where

Note that and
It follows that if , then , and so . Likewise, if , then , and so .

In other words, and

(iii) Suppose next that is right-scattered but . It is easy to get that
So and

From (i), (ii), and (iii), we get that and

Integrating (2.6) from to (using ), we get that
Since , we obtain that
From (2.18) and , , we get that , . That is,
So,
Hence From , we get that , which is a contradiction.

#### 3. Example

*Example 3.1. *Consider the second-order half-linear differential equation
where , are odd positive integers. Let
and . We have
Noticing that is even number and
using the following formula (see [10])
we get that
It is easy to see that , so from (3.6), we obtain that when
where denotes , equation (3.1) is oscillatory.

In particular, take . From (3.7), we get that when
the second-order half-linear equation
is oscillatory.

*Example 3.2. *Consider the second-order half-linear difference equation
where , is a quotient of odd positive integers,
Let , and note that

Furthermore, we have (note that )
Therefore, if , then (3.10) is oscillatory.

*Example 3.3. *Consider the second-order half-linear q-difference equation
where , . is a quotient of odd positive integers.

Let . We have
Similarly, we have
Therefore, if , then , for large . So (3.14) is oscillatory.

Many other interesting examples can be similarly given.

#### Acknowledgment

This work is supported by the National Natural Science Foundation of China (no.10971232) and by NSF of Maoming University (no. LK201002).

#### References

- S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,”
*Results in Mathematics*, vol. 18, no. 1-2, pp. 18–56, 1990. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Bohner and A. Peterson,
*Dynamic Equations on Time Scales: An Introduction with Applications*, Birkhäuser, Boston, Mass, USA, 2001. View at MathSciNet - M. Bohner and A. Peterson, Eds.,
*Advances in Dynamic Equations on Time Scales*, Birkhäuser, Boston, Mass, USA, 2003. View at MathSciNet - M. Bohner and S. H. Saker, “Oscillation criteria for perturbed nonlinear dynamic equations,”
*Mathematical and Computer Modelling*, vol. 40, no. 3-4, pp. 249–260, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Bohner and T. S. Hassan, “Oscillation and boundedness of solutions to first and second order forced functional dynamic equations with mixed nonlinearities,”
*Applicable Analysis and Discrete Mathematics*, vol. 3, no. 2, pp. 242–252, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - R. P. Agarwal and A. Zafer, “Oscillation criteria for second-order forced dynamic equations with mixed nonlinearities,”
*Advances in Difference Equations*, vol. 2009, Article ID 938706, 20 pages, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Erbe, A. Peterson, and S. H. Saker, “Oscillation criteria for a forced second-order nonlinear dynamic equation,”
*Journal of Difference Equations and Applications*, vol. 14, no. 10-11, pp. 997–1009, 2008. View at Publisher · View at Google Scholar · View at Scopus - V. Kac and P. Cheung,
*Quantum Calculus*, Universitext, Springer, New York, NY, USA, 2002. View at MathSciNet - O. Došlý and P. Řehák, “Half-Linear Differential Equations,” North-Holland Mathematics Studies 202, Elsevier, Amsterdam, The Netherlands, 2005. View at Google Scholar
- E. T. Wittaker and G. N. Watson,
*Modern Analysis*, Cambridge University Press, New York, NY, USA, American edition, 1945.