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Abstract and Applied Analysis
Volume 2010 (2010), Article ID 294194, 10 pages
Forced Oscillation of Second-Order Half-Linear Dynamic Equations on Time Scales
1Department of Mathematics, Maoming University, Maoming 525000, China
2Department 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.
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.
The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. thesis  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 ). A book on the subject of time scale by Bohner and Peterson  summarizes and organizes much of the time scale calculus (see also ). For the notions used below, we refer to  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 , 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  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 ).
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 , 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.
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 )
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.
This work is supported by the National Natural Science Foundation of China (no.10971232) and by NSF of Maoming University (no. LK201002).
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