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.
where and are real and rd-continuous functions in Let be a time scale, be the forward jump operator in , be the delta derivative, and .
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
then has at least one zero on
where and for is the nabla derivative, and they get the Sturm comparison theorem. We will make use of Picone identity on time scales to prove the Sturm-Picone comparison theorem of (1.1) and (1.2).
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.
Moreover, a function defined on is said to be rd-continuous if it is continuous at every right-dense point in and its left-sided limit exists at every left-dense point in .
If is differentiable at , then is continuous at .
- (ii)If and are differentiable at , then is differentiable at and(2.6)
- (iii)If and are differentiable at , and , then is differentiable at and(2.7)
If is rd-continuous on , then it has an antiderivative on .
in the case that is right-scattered;
there is a neighborhood of such that for all with in the case that is right-dense.
If the inequalities for are reversed in (i) and (ii), is said to be right-decreasing at .
The following result can be directly derived from (2.4).
Assume that is differentiable at If then is right-increasing at ; and if , then is right-decreasing at .
One says that a solution of (1.1) has a generalized zero at if or, if is right-scattered and Especially, if then we say has a node at
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).
where we used Lemma 2.1. This equation is in the form of (1.1) with and as desired.
Lemma 3.2 (Picone Identity).
Combining and , we get (3.4). This completes the proof.
Now, we turn to proving the main result of this paper.
Theorem 3.3 (Sturm-Picone Comparison Theorem).
then has at least one generalized zero on
Suppose to the contrary, has no generalized zeros on and for all
which is a contradiction. Therefore, in Case 1, has at least one generalized zero on
which is a contradiction, too. Hence, in Case 2, has at least one generalized zero on .
It follows from the above discussion that has at least one generalized zero on This completes the proof.
If then Theorem 3.3 reduces to classical Sturm comparison theorem.
In the continuous case: . This result is the same as Sturm-Picone comparison theorem of second-order differential equations (see Section 1).
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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