- Research Article
- Open Access
Sturm-Picone Comparison Theorem of Second-Order Linear Equations on Time Scales
© C. Zhang and S. Sun. 2009
- Received: 29 December 2008
- Accepted: 28 May 2009
- Published: 16 June 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.
- Nontrivial Solution
- Riccati Equation
- Product Rule
- Comparison Theorem
- Fundamental Result
First we briefly recall some existing results about differential and difference equations. As we well know, in 1909, Picone  established the following identity.
By (1.4), one can easily obtain the Sturm comparison theorem of second-order linear differential equations (1.3).
Sturm-Picone Comparison Theorem
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.
The following result can be directly derived from (2.4).
The development of the theory uses similar arguments and the definition of the nabla derivative (see [10, Chapter 3]).
In this section, we give and prove the main results of this paper.
can be rewritten as (1.1).
Lemma 3.2 (Picone Identity).
Now, we turn to proving the main result of this paper.
Theorem 3.3 (Sturm-Picone Comparison Theorem).
In the discrete case: . This result is the same as Sturm comparison theorem of second-order difference equations (see [8, Chapter 8]).
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.
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).
- 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
- Kamke E: A new proof of Sturm's comparison theorems. The American Mathematical Monthly 1939, 46: 417-421. 10.2307/2303035MathSciNetView ArticleGoogle Scholar
- Leighton W: Comparison theorems for linear differential equations of second order. Proceedings of the American Mathematical Society 1962, 13: 603-610. 10.1090/S0002-9939-1962-0140759-0MATHMathSciNetView ArticleGoogle Scholar
- Leighton W: Some elementary Sturm theory. Journal of Differential Equations 1968, 4: 187-193. 10.1016/0022-0396(68)90035-1MATHMathSciNetView ArticleGoogle Scholar
- Reid WT: A comparison theorem for self-adjoint differential equations of second order. Annals of Mathematics 1957, 65: 197-202. 10.2307/1969673MATHMathSciNetView ArticleGoogle Scholar
- Zhuang R: Sturm comparison theorem of solution for second order nonlinear differential equations. Annals of Differential Equations 2003,19(3):480-486.MATHMathSciNetGoogle Scholar
- 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.2004.03.004MATHMathSciNetView ArticleGoogle Scholar
- Kelley WG, Peterson AC: Difference Equations: An Introduction with Applications. 2nd edition. Harcourt/Academic Press, San Diego, Calif, USA; 2001:x+403.Google Scholar
- Zhang BG: Sturm comparison theorem of difference equations. Applied Mathematics and Computation 1995,72(2-3):277-284. 10.1016/0096-3003(94)00201-EMATHMathSciNetView ArticleGoogle Scholar
- Bohner M, Peterso A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHView ArticleGoogle Scholar
- Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar
- 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.View ArticleGoogle Scholar
- Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications. Results in Mathematics 1999,35(1-2):3-22.MATHMathSciNetView ArticleGoogle Scholar
- Hilger S: Special functions, Laplace and Fourier transform on measure chains. Dynamic Systems and Applications 1999,8(3-4):471-488.MATHMathSciNetGoogle Scholar
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