Sturm-Picone Comparison Theorem of Second-Order Linear Equations on Time Scales
© C. Zhang and S. Sun. 2009
Received: 29 December 2008
Accepted: 28 May 2009
Published: 16 June 2009
This paper studies Sturm-Picone comparison theorem of second-order linear equations on time scales. We first establish Picone identity on time scales and obtain our main result by using it. Also, our result unifies the existing ones of second-order differential and difference equations.
First we briefly recall some existing results about differential and difference equations. As we well know, in 1909, Picone  established the following identity.
By (1.4), one can easily obtain the Sturm comparison theorem of second-order linear differential equations (1.3).
Sturm-Picone Comparison Theorem
This paper is organized as follows. Section 2 introduces some basic concepts and fundamental results about time scales, which will be used in Section 3. In Section 3 we first give the Picone identity on time scales, then we will employ this to prove our main result: Sturm-Picone comparison theorem of (1.1) and (1.2) on time scales.
In this section, some basic concepts and some fundamental results on time scales are introduced.
The following result can be directly derived from (2.4).
The development of the theory uses similar arguments and the definition of the nabla derivative (see [10, Chapter 3]).
3. Main Results
In this section, we give and prove the main results of this paper.
can be rewritten as (1.1).
Lemma 3.2 (Picone Identity).
Now, we turn to proving the main result of this paper.
Theorem 3.3 (Sturm-Picone Comparison Theorem).
In the discrete case: . This result is the same as Sturm comparison theorem of second-order difference equations (see [8, Chapter 8]).
By Theorem 3.3, we have if and are the nontrivial solutions of (1.1) and (1.2), are two consecutive generalized zeros of and then has at least one generalized zero on Obviously, the above three cases are not continuous and not discrete. So the existing results for the differential and difference equations are not available now.
By Remarks 3.4–3.6 and Example 3.7, the Sturm comparison theorem on time scales not only unifies the results in both the continuous and the discrete cases but also contains more complicated time scales.
Many thanks to Alberto Cabada (the editor) and the anonymous reviewer(s) for helpful comments and suggestions. This research was supported by the Natural Scientific Foundation of Shandong Province (Grant Y2007A27), (Grant Y2008A28), the Fund of Doctoral Program Research of University of Jinan (B0621), and the Natural Science Fund Project of Jinan University (XKY0704).
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