Advertisement

Advances in Difference Equations

, 2009:496135 | Cite as

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

  • Chao ZhangEmail author
  • Shurong Sun
Open Access
Research Article
  • 1.8k Downloads
Part of the following topical collections:
  1. Boundary Value Problems on Time Scales

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

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

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

Assume that Open image in new window and Open image in new window are the nontrivial solutions of (1.3) and Open image in new window are two consecutive zeros of Open image in new window if

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

Later, many mathematicians, such as Kamke, Leighton, and Reid [2, 3, 4, 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 Open image in new window on Open image in new window on Open image in new window are integers, and Open image in new window is the forward difference operator: Open image in new window 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 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.

Let Open image in new window be a nonempty closed subset. Define the forward and backward jump operators Open image in new window by
where Open image in new window , Open image in new window . A point Open image in new window is called right-scattered, right-dense, left-scattered, and left-dense if Open image in new window , and Open image in new window respectively. We put Open image in new window if Open image in new window is unbounded above and Open image in new window otherwise. The graininess functions Open image in new window are defined by
Let Open image in new window be a function defined on Open image in new window . Open image in new window is said to be (delta) differentiable at Open image in new window provided there exists a constant Open image in new window such that for any Open image in new window , there is a neighborhood Open image in new window of Open image in new window (i.e., Open image in new window for some Open image in new window ) with
In this case, denote Open image in new window . If Open image in new window is (delta) differentiable for every Open image in new window , then Open image in new window is said to be (delta) differentiable on Open image in new window . If Open image in new window is differentiable at Open image in new window , then
If Open image in new window for all Open image in new window , then Open image in new window is called an antiderivative of Open image in new window on Open image in new window . In this case, define the delta integral by

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.

Definition 2.2.

A function Open image in new window is said to be right-increasing at Open image in new window provided
  1. (i)

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

     
  2. (ii)
     

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

A function Open image in new window is called regressive if
Hilger [14] showed that for Open image in new window and rd-continuous and regressive Open image in new window , the solution of the initial value problem

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 Open image in new window and Open image in new window is continuous, then (3.1) can be written in the form of (1.1), with

Proof.

Multiplying both sides of (3.1) by Open image in new window , we get

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).

Let Open image in new window and Open image in new window be the nontrivial solutions of (1.1) and (1.2) with Open image in new window and Open image in new window for Open image in new window If Open image in new window has no generalized zeros on Open image in new window 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 Open image in new window has no generalized zeros on Open image in new window that

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).

Suppose that Open image in new window and Open image in new window are the nontrivial solutions of (1.1) and (1.2), and Open image in new window are two consecutive generalized zeros of Open image in new window if

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.

Suppose Open image in new window are two consecutive zeros of Open image in new window . Then by Lemma 3.2, (3.4) holds and integrating it from Open image in new window to Open image in new window we get
Hence, by (3.9) and Open image in new window we have

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.

By integration, it follows from (3.12) and Open image in new window that
So, from (3.9) and above argument we obtain that

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.

Suppose Open image in new window is a node of Open image in new window and Open image in new window is a generalized zero of Open image in new window Similar to the discussion of (3.12), we have
which implies
(i)If Open image in new window is a node of Open image in new window then Open image in new window Hence, we have (3.12), that is,
It follows from (3.4) and Lemma 2.3 that
is right-increasing on Open image in new window 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
This contradicts (3.22). Note that Open image in new window . It follows from Open image in new window , (3.23), and (3.24) that
On the other hand, it follows from Open image in new window and Open image in new window 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 Open image in new window and (3.21), we get

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.

Consider the following three specific cases:

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).

References

  1. 1.
    Picone M: Sui valori eccezionali di un parametro da cui dipend unèquazione differenziale linear ordinaria del second ordine. JMPA 1909, 11: 1-141.MathSciNetGoogle Scholar
  2. 2.
    Kamke E: A new proof of Sturm's comparison theorems. The American Mathematical Monthly 1939, 46: 417-421. 10.2307/2303035MathSciNetCrossRefGoogle Scholar
  3. 3.
    Leighton W: Comparison theorems for linear differential equations of second order. Proceedings of the American Mathematical Society 1962, 13: 603-610. 10.1090/S0002-9939-1962-0140759-0zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Leighton W: Some elementary Sturm theory. Journal of Differential Equations 1968, 4: 187-193. 10.1016/0022-0396(68)90035-1zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Reid WT: A comparison theorem for self-adjoint differential equations of second order. Annals of Mathematics 1957, 65: 197-202. 10.2307/1969673zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Zhuang R: Sturm comparison theorem of solution for second order nonlinear differential equations. Annals of Differential Equations 2003,19(3):480-486.zbMATHMathSciNetGoogle Scholar
  7. 7.
    Zhuang R-K, Wu H-W: Sturm comparison theorem of solution for second order nonlinear differential equations. Applied Mathematics and Computation 2005,162(3):1227-1235. 10.1016/j.amc.2004.03.004zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Kelley WG, Peterson AC: Difference Equations: An Introduction with Applications. 2nd edition. Harcourt/Academic Press, San Diego, Calif, USA; 2001:x+403.Google Scholar
  9. 9.
    Zhang BG: Sturm comparison theorem of difference equations. Applied Mathematics and Computation 1995,72(2-3):277-284. 10.1016/0096-3003(94)00201-EzbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Bohner M, Peterso A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.zbMATHCrossRefGoogle Scholar
  11. 11.
    Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.CrossRefGoogle Scholar
  12. 12.
    Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic Systems on Measure Chains, Mathematics and Its Applications. Volume 370. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:x+285.CrossRefGoogle Scholar
  13. 13.
    Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications. Results in Mathematics 1999,35(1-2):3-22.zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Hilger S: Special functions, Laplace and Fourier transform on measure chains. Dynamic Systems and Applications 1999,8(3-4):471-488.zbMATHMathSciNetGoogle Scholar

Copyright information

© C. Zhang and S. Sun. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.School of ScienceUniversity of JinanJinanChina

Personalised recommendations