# Sturm-Picone Comparison Theorem of Second-Order Linear Equations on Time Scales

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## Abstract

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.

## Keywords

Nontrivial Solution Riccati Equation Product Rule Comparison Theorem Fundamental Result## 1. Introduction

where Open image in new window and Open image in new window are real and rd-continuous functions in Open image in new window Let Open image in new window be a time scale, Open image in new window be the forward jump operator in Open image in new window , Open image in new window be the delta derivative, and Open image in new window .

First we briefly recall some existing results about differential and difference equations. As we well know, in 1909, Picone [1] established the following identity.

Picone 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 Open image in new window has at least one zero on Open image in new window

where Open image in new window and Open image in new window for Open image in new window 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.

## 2. Preliminaries

In this section, some basic concepts and some fundamental results on time scales are introduced.

Moreover, a function Open image in new window defined on Open image in new window is said to be rd-continuous if it is continuous at every right-dense point in Open image in new window and its left-sided limit exists at every left-dense point in Open image in new window .

For convenience, we introduce the following results ([11, Chapter 1], [12, Chapter 1], and [13, Lemma Open image in new window ]), which are useful in the paper.

Lemma 2.1.

- (i)
If Open image in new window is differentiable at Open image in new window , then Open image in new window is continuous at Open image in new window .

- (ii)If Open image in new window and Open image in new window are differentiable at Open image in new window , then Open image in new window is differentiable at Open image in new window and(2.6)
- (iii)If Open image in new window and Open image in new window are differentiable at Open image in new window , and Open image in new window , then Open image in new window is differentiable at Open image in new window and(2.7)
- (iv)
If Open image in new window is rd-continuous on Open image in new window , then it has an antiderivative on Open image in new window .

Definition 2.2.

- (i)
Open image in new window in the case that Open image in new window is right-scattered;

- (ii)
there is a neighborhood Open image in new window of Open image in new window such that Open image in new window for all Open image in new window with Open image in new window in the case that Open image in new window is right-dense.

If the inequalities for Open image in new window are reversed in (i) and (ii), Open image in new window is said to be right-decreasing at Open image in new window .

The following result can be directly derived from (2.4).

Lemma 2.3.

Assume that Open image in new window is differentiable at Open image in new window If Open image in new window then Open image in new window is right-increasing at Open image in new window ; and if Open image in new window , then Open image in new window is right-decreasing at Open image in new window .

Definition 2.4.

One says that a solution Open image in new window of (1.1) has a generalized zero at Open image in new window if Open image in new window or, if Open image in new window is right-scattered and Open image in new window Especially, if Open image in new window then we say Open image in new window has a node at Open image in new window

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).

Theorem 3.1.

Proof.

where we used Lemma 2.1. This equation is in the form of (1.1) with Open image in new window and Open image in new window as desired.

Lemma 3.2 (Picone Identity).

Proof.

Combining Open image in new window and Open image in new window , 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 Open image in new window has at least one generalized zero on Open image in new window

Proof.

Suppose to the contrary, Open image in new window has no generalized zeros on Open image in new window and Open image in new window for all Open image in new window

Case 1.

which is a contradiction. Therefore, in Case 1, Open image in new window has at least one generalized zero on Open image in new window

Case 2.

which is a contradiction, too. Hence, in Case 2, Open image in new window has at least one generalized zero on Open image in new window .

Case 3.

which contradicts Open image in new window

It follows from the above discussion that Open image in new window has at least one generalized zero on Open image in new window This completes the proof.

Remark 3.4.

If Open image in new window then Theorem 3.3 reduces to classical Sturm comparison theorem.

Remark 3.5.

In the continuous case: Open image in new window . This result is the same as Sturm-Picone comparison theorem of second-order differential equations (see Section 1).

Remark 3.6.

In the discrete case: Open image in new window . This result is the same as Sturm comparison theorem of second-order difference equations (see [8, Chapter 8]).

Example 3.7.

By Theorem 3.3, we have if Open image in new window and Open image in new window are the nontrivial solutions of (1.1) and (1.2), Open image in new window are two consecutive generalized zeros of Open image in new window and Open image in new window then Open image in new window has at least one generalized zero on Open image in new window 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.

## Notes

### Acknowledgments

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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