Sturm-Picone Comparison Theorem of Second-Order Linear Equations on Time Scales
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Research Article Sturm-Picone Comparison Theorem of Second-Order Linear Equations on Time Scales Chao Zhang and Shurong Sun School of Science, University of Jinan, Jinan, Shandong 250022, China Correspondence should be addressed to Chao Zhang, ss [email protected] Received 29 December 2008; Revised 13 March 2009; Accepted 28 May 2009 Recommended by Alberto Cabada 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. Copyright q 2009 C. Zhang and S. Sun. 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.
1. Introduction In this paper, we consider the following second-order linear equations:
p1 txΔ t
Δ
q1 txσ t 0,
1.1
Δ p2 tyΔ t q2 tyσ t 0,
1.2
where t ∈ α, β ∩ T, p1Δ t, p2Δ t, q1 t, and q2 t are real and rd-continuous functions in α, β ∩ T. Let T be a time scale, σt be the forward jump operator in T, yΔ be the delta derivative, and yσ t : yσt. 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 xt and yt are the nontrivial solutions of p1 tx t q1 txt 0, p2 ty t q2 tyt 0,
1.3
2
Advances in Difference Equations
where t ∈ α, β , p1 t, p2 t, q1 t, and q2 t are real and continuous functions in α, β . If yt / 0 for t ∈ α, β , then
xt p1 tx tyt − p2 ty txt yt
2 xty t − x t . p1 t − p2 t x t q2 t − q1 t x2 t p2 t yt
2
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 xt and yt are the nontrivial solutions of 1.3 and a, b are two consecutive zeros of xt, if p1 t ≥ p2 t > 0,
q2 t ≥ q1 t,
t ∈ a, b ,
1.5
then yt has at least one zero on a, b . 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 Δ p1 t − 1Δxt − 1 q1 txt 0, Δ p2 t − 1Δyt − 1 q2 tyt 0,
1.6
has been investigated in 8, Chapter 8 , where p1 t ≥ p2 t > 0 on α, β 1 , q2 t ≥ q1 t on α 1, β 1 , α, β are integers, and Δ is the forward difference operator: Δxt xt 1 − xt. 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 Stur
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