1. Introduction

In this paper, we consider the following second-order linear equations:

(1.1)
(1.2)

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

(1.3)

where and are real and continuous functions in If for then

(1.4)

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

(1.5)

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

(1.6)

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

(1.7)

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

(2.1)

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

(2.2)

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

(2.3)

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

(2.4)

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

(2.5)

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 .

  1. (i)

    If is differentiable at , then is continuous at .

  2. (ii)

    If and are differentiable at , then is differentiable at and

    (2.6)
  3. (iii)

    If and are differentiable at , and , then is differentiable at and

    (2.7)
  4. (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

  1. (i)

    in the case that is right-scattered;

  2. (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

(2.8)

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

(2.9)

is given by , where

(2.10)

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:

(3.1)

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

(3.2)

Proof.

Multiplying both sides of (3.1) by , we get

(3.3)

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:

(3.4)

Proof.

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

(3.5)

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

(3.6)

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

(3.7)

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

(3.8)

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

(3.9)

Noting that we have

(3.10)

Hence, by (3.9) and we have

(3.11)

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

(3.12)

By integration, it follows from (3.12) and that

(3.13)

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

(3.14)

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

(3.15)

which implies

(3.16)

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

(3.17)

(ii)If is a zero of then

(3.18)

It follows from (3.4) and Lemma 2.3 that

(3.19)

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

(3.20)

which implies

(3.21)

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

(3.22)

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

(3.23)

If and from , , and we have

(3.24)

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

(3.25)

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

(3.26)

Combining the above two equations we obtain

(3.27)

It follows from (3.27) and (2.4) that

(3.28)

Hence, from and (3.21), we get

(3.29)

By referring to and it follows that

(3.30)

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:

(3.31)

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.