- Research Article
- Open Access
Sturm-Picone Comparison Theorem of Second-Order Linear Equations on Time Scales
Advances in Difference Equations volume 2009, Article number: 496135 (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.
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  established the following 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  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.
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 .
Let and .
If is differentiable at , then is continuous at .
If and are differentiable at , then is differentiable at and(2.6)
If and are differentiable at , and , then is differentiable at and(2.7)
If is rd-continuous on , then it has an antiderivative on .
A function is said to be right-increasing at provided
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
A function is called regressive if
Hilger  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).
If and is continuous, then (3.1) can be written in the form of (1.1), with
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:
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
Suppose to the contrary, has no generalized zeros on and for all
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
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 .
Suppose is a node of and is a generalized zero of Similar to the discussion of (3.12), we have
(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
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
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]).
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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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).