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Sturm-Picone Comparison Theorem of Second-Order Linear Equations on Time Scales
Advances in Difference Equations volume 2009, Article number: 496135 (2009)
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.
1. Introduction
In this paper, we consider the following second-order linear 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 [1] established the following identity.
Picone Identity
If and
are the nontrivial solutions of

where and
are real and continuous functions in
If
for
then

By (1.4), one can easily obtain the Sturm comparison theorem of second-order linear differential equations (1.3).
Sturm-Picone Comparison Theorem
Assume that and
are the nontrivial solutions of (1.3) and
are two consecutive zeros of
if

then has at least one zero on
Later, many mathematicians, such as Kamke, Leighton, and Reid [2–5] developed thier work. The investigation of Sturm comparison theorem has involved much interest in the new century [6, 7]. The Sturm comparison theorem of second-order difference equations

has been investigated in [8, Chapter 8], where on
on
are integers, and
is the forward difference operator:
In 1995, Zhang [9] extended this result. But we will remark that in [8, Chapter 8] the authors employed the Riccati equation and a positive definite quadratic functional in their proof. Recently, the Sturm comparison theorem on time scales has received a lot of attentions. In [10, Chapter 4], the mathematicians studied

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.
2. Preliminaries
In this section, some basic concepts and some fundamental results on time scales are introduced.
Let be a nonempty closed subset. Define the forward and backward jump operators
by

where ,
. A point
is called right-scattered, right-dense, left-scattered, and left-dense if
, and
respectively. We put
if
is unbounded above and
otherwise. The graininess functions
are defined by

Let be a function defined on
.
is said to be (delta) differentiable at
provided there exists a constant
such that for any
, there is a neighborhood
of
(i.e.,
for some
) with

In this case, denote . If
is (delta) differentiable for every
, then
is said to be (delta) differentiable on
. If
is differentiable at
, then

If for all
, then
is called an antiderivative of
on
. In this case, define the delta integral by

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
.
For convenience, we introduce the following results ([11, Chapter 1], [12, Chapter 1], and [13, Lemma ]), which are useful in the paper.
Lemma 2.1.
Let and
.
-
(i)
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) -
(iv)
If
is rd-continuous on
, then it has an antiderivative on
.
Definition 2.2.
A function is said to be right-increasing at
provided
-
(i)
in the case that
is right-scattered;
-
(ii)
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).
Lemma 2.3.
Assume that is differentiable at
If
then
is right-increasing at
; and if
, then
is right-decreasing at
.
Definition 2.4.
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
A function is called regressive if

Hilger [14] showed that for and rd-continuous and regressive
, the solution of the initial value problem

is given by , where

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.
First, we will show that the following second-order linear equation:

can be rewritten as (1.1).
Theorem 3.1.
If and
is continuous, then (3.1) can be written in the form of (1.1), with

Proof.
Multiplying both sides of (3.1) by , we get

where we used Lemma 2.1. This equation is in the form of (1.1) with and
as desired.
Lemma 3.2 (Picone Identity).
Let and
be the nontrivial solutions of (1.1) and (1.2) with
and
for
If
has no generalized zeros on
then the following identity holds:

Proof.
We first divide the left part of (3.4) into two parts

From (1.1) and the product rule (Lemma 2.1(ii), we have

It follows from (1.2), (2.4), product and quotient rules (Lemma 2.1(ii), (iii) and the assumption that has no generalized zeros on
that

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).
Suppose that and
are the nontrivial solutions of (1.1) and (1.2), and
are two consecutive generalized zeros of
if

then has at least one generalized zero on
Proof.
Suppose to the contrary, has no generalized zeros on
and
for all
Case 1.
Suppose are two consecutive zeros of
. Then by Lemma 3.2, (3.4) holds and integrating it from
to
we get

Noting that we have

Hence, by (3.9) and we have

which is a contradiction. Therefore, in Case 1, has at least one generalized zero on
Case 2.
Suppose is a zero of
is a node of
and
It follows from the assumption that
has no generalized zeros on
and that
for all
that
Hence by (2.4) and
on
, we have

By integration, it follows from (3.12) and that

So, from (3.9) and above argument we obtain that

which is a contradiction, too. Hence, in Case 2, has at least one generalized zero on
.
Case 3.
Suppose is a node of
and
is a generalized zero of
Similar to the discussion of (3.12), we have

which implies

(i)If is a node of
then
Hence, we have (3.12), that is,

(ii)If is a zero of
then

It follows from (3.4) and Lemma 2.3 that

is right-increasing on Hence, from (i) and (ii) that

which implies

From (3.16), (3.21), and (2.4), we have

Further, it follows from (1.1), (1.2), product rule (Lemma 2.1(ii), and (3.22) that

If and from
,
, and
we have

This contradicts (3.22). Note that . It follows from
, (3.23), and (3.24) that

On the other hand, it follows from and
are solutions of (1.1) and (1.2) that

Combining the above two equations we obtain

It follows from (3.27) and (2.4) that

Hence, from and (3.21), we get

By referring to and
it follows that

which contradicts
It follows from the above discussion that has at least one generalized zero on
This completes the proof.
Remark 3.4.
If then Theorem 3.3 reduces to classical Sturm comparison theorem.
Remark 3.5.
In the continuous case: . 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: . This result is the same as Sturm comparison theorem of second-order difference equations (see [8, Chapter 8]).
Example 3.7.
Consider the following three specific cases:

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.
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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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Zhang, C., Sun, S. Sturm-Picone Comparison Theorem of Second-Order Linear Equations on Time Scales. Adv Differ Equ 2009, 496135 (2009). https://doi.org/10.1155/2009/496135
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DOI: https://doi.org/10.1155/2009/496135
Keywords
- Nontrivial Solution
- Riccati Equation
- Product Rule
- Comparison Theorem
- Fundamental Result