Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 496135, 12 pages
doi:10.1155/2009/496135
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:
p
1
t
x
Δ
t
t
y
σ
t
0, 1.2
where t ∈ α, β ∩ T,p
Δ
1
t,p
Δ
2
t,q
1
t, and q
2
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, y
Δ
be the delta
derivative, and y
σ
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 t he nontrivial solutions of
t
q
2
t
y
t
0,
1.3
2 Advances in Difference Equations
where t ∈ α, β,p
1
t,p
2
t,q
1
t, and q
2
t are real and continuous functions in α, β. If
yt
y
t
x
t
p
1
t
− p
2
t
x
2
t
t
y
t
− x
t
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
p
1
t
≥ p
2
t − 1
q
1
t
x
t
0,
Δ
p
2
t − 1
Δy
t − 1
q
2
t
∇
q
1
t
x
t
0,
p
2
t
y
Δ
t
∇
q
2
t
σ
t
inf
{
s ∈ T : s>t
}
,ρ
t
sup
{
s ∈ T : s<t
}
, 2.1
where inf ∅ sup T,sup∅ inf T.Apointt ∈ T is called right-scattered, right-dense, left-
scattered, and left-dense if σt >t,σtt, ρt <t,andρtt, respectively. We put T
k
T
if T is unbounded above and T
k
T \ ρmax T, max T otherwise. The graininess functions
ν, μ : T → 0, ∞ are defined by
μ
t
σ
σ
t
− s
≤ ε
|
σ
t
− s
|
, ∀s ∈ U. 2.3
In this case, denote f
Δ
t : a.Iff is delta differentiable for every t ∈ T
k
, then f is said to
be delta differentiable on T.Iff is differentiable at t ∈ T
k
, then
f
Δ
t
f
σ
t
− f
t
μ
t
, if μ
t
> 0.
2.4
If F
Δ
tft for all t ∈ T
k
, then Ft is called an antiderivative of f on T. In this case, define
the delta integral by
t
s
f
t
g
Δ
t
f
Δ
t
g
t
f
Δ
t
g
σ
t
f
t
t
− g
t
f
Δ
t
f
σ
t
f
t
−1
. 2.7
iv If f is rd-continuous on T, then it has an antiderivative on T.
Definition 2.2. A function f : T → R is said to be right-increasing at t
0
∈ T\{max T} provided
i fσt
Δ
t
0
< 0,thenf is right-decreasing at t
0
.
Definition 2.4. One says that a solution xt of 1.1 has a generalized zero at t if xt0or,
if t is right-scattered and xtxσt < 0. Especially, if xtxσt < 0, then we say xthas a
node at t σt/2.
A function p : T → R is called regressive if
1 μ
t
p
t
/
0, ∀t ∈ T. 2.8
Hilger 14 showed that for t
0
∈ T and rd-continuous and regressive p, the solution of the
initial value problem
y
Δ
t
p
p
τ
Δτ
with ξ
h
z
⎧
⎪
⎨
⎪
⎩
Log
1 hz
h
, if h
/
0
z, if h 0.
2.10
The development of the theory uses similar arguments and the definition of the nabla
0 3.1
can be rewritten as 1.1.
Theorem 3.1. If 1 μta
1
t
/
0 and a
2
t is continuous, then 3.1 can b e written in the form of
1.1,with
p
1
t
e
a
1
t, t
0
,q
1
t
e
a
1
1
t, t
0
a
1
t
x
Δσ
t
e
a
1
t, t
0
a
2
t
x
σ
a
1
t, t
0
a
2
t
x
σ
t
e
a
1
t, t
0
x
Δ
t
2
t > 0 and q
2
t ≥ q
1
t for t ∈ α, β ∩ T. If yt has no generalized zeros on α, β ∩ T,
then the following identity holds:
x
t
y
t
p
1
t
x
Δ
t
y
t
x
Δ
t
2
q
2
t
− q
1
t
x
2
σ
t
t
y
t
p
1
t
x
Δ
t
y
t
− p
2
t
y
Δ
1
t
x
Δ
t
x
t
Δ
−
p
2
ty
Δ
t
yt
x
2
t
Δ
.
3.5
Δ
x
σ
t
p
1
t
x
Δ
t
x
Δ
t
p
1
t
ty
Δ
t
yt
x
2
t
Δ
x
2
σ
t
p
2
ty
Δ
t
yt
Δ
x
σ
t
2
t
y
Δ
t
y
t
x
2
σ
t
−q
2
t
− p
2
t
p
2
t
y
Δ
t
y
t
x
Δ
t
x
σ
t
− μ
t
t
2
− q
2
t
x
2
σ
t
− p
2
t
y
Δ
t
2
x
t
y
Δ
t
y
t
−
p
2
t
μ
t
p
2
t
y
Δ
t
x
2
σ
t
−
y
t
p
2
t
y
σ
t
p
2
ty
t
y
t
−
p
2
t
y
σ
t
y
t
x
Δ
t
yt
p
2
tyσt
p
2
ty
Δ
t
yt
xσt −
p
2
tyσt
yt
x
Δ
t
⎞
⎠
2
∀t ∈
α, β
∩ T.
3.7
Combining p
1
t
≥ q
1
t
,t∈
a, b
∩ T, 3.8
then yt has at least one generalized zero on a, b ∩ T.
Proof. Suppose to the contrary, yt has no generalized zeros on a, b ∩ T and yt > 0 for all
t ∈ a, b ∩ T.
Case 1. Suppose a, b are two consecutive zeros of xt. Then by Lemma 3.2, 3.4 holds and
integrating it from a to b we get
b
a
x
t
y
t
Δ
Δt
b
a
⎛
⎜
⎝
p
1
t
− p
2
t
x
Δ
t
2
ty
Δ
t
yt
−
p
2
tyσt
yt
x
Δ
t
⎞
⎠
2
⎞
⎟
⎠
Δt.
3.9
Noting that xaxb0, we have
b
a
x
t
t
Δ
Δt
x
t
y
t
p
1
t
x
Δ
t
y
t > 0,q
2
t ≥ q
1
t, for all t ∈ a, b ∩ T we have
0
b
a
⎛
⎜
⎝
p
1
t
− p
2
t
x
Δ
t
p
2
ty
Δ
t
yt
−
p
2
tyσt
yt
x
Δ
t
⎞
⎠
2
⎞
⎟
⎠
Δt
> 0,
3.11
8 Advances in Difference Equations
which is a contradiction. Therefore, in Case 1, yt has at least one generalized zero on a, b∩
T.
Case 2. Suppose a is a zero of xt, b σb/2isanodeofxt,xb < 0, and xσb > 0. It
follows from the assumption that yt has no generalized zeros on a, b ∩ T and that yt > 0
for all t ∈ a, b ∩ T that yσb > 0. Hence by 2.4 and p
b
y
Δ
b
x
b
x
b
y
b
1
μ
b
p
1
p
2
b
− p
1
b
x
b
y
b
< 0.
3.12
By integration, it follows from 3.12 and xa0that
b
a
x
t
x
t
Δ
Δt
x
t
y
t
p
1
t
x
Δ
t
y
y
b
p
1
b
x
Δ
b
y
b
− p
2
b
y
Δ
b
x
t
2
q
2
t
− q
1
t
x
2
σ
t
⎛
⎝
yt
p
zero of xt. Similar to the discussion of 3.12, we have
x
a
y
a
p
1
a
x
Δ
a
y
a
− p
2
a
y
σ
a
y
a
− p
2
a
y
σ
a
x
a
p
2
a
y
a
− p
2
a
y
Δ
a
x
a
< 0. 3.16
i If b σb/2isanodeofxt, then xb < 0,xσb > 0. Hence, we have 3.12,
that is,
x
b
y
b
< 0. 3.17
ii If b is a zero of xt, then
x
b
y
b
p
1
b
x
Δ
b
y
b
− p
2
b
y
t
− p
2
t
y
Δ
t
x
t
3.19
is right-increasing on a, b ∩ T. Hence, from i and ii that
x
a
y
a
<
x
σ
a
y
σ
a
p
1
σ
a
x
Δ
σ
a
y
σ
a
x
Δ
σ
a
y
σ
a
− p
2
σ
a
y
Δ
σ
a
p
1
x
Δ
y − p
2
y
Δ
x
σ
a
−
p
1
x
Δ
y − p
2
y
Δ
x
a
x
σ
a
y
σ
a
p
1
a
− p
2
a
x
x
σ
a
y
σ
a
< 0. 3.24
This contradicts 3.22.Notethatx
Δ
a1/μaxσa − xa. It follows from p
1
a >
p
2
a > 0, 3.23,and3.24 that
y
Δ
a
< 0. 3.25
On the other hand, it follows from xt and yt are solutions of 1.1 and 1.2 that
0,
x
σ
a
p
2
a
y
Δ
a
Δ
q
2
a
y
p
2
a
y
Δ
a
Δ
x
σ
a
q
1
a
− q
2
a
x
Δ
σ
a
− p
1
a
x
Δ
a
y
σ
a
−
p
2
q
1
a
− q
2
a
x
σ
a
y
σ
a
1
μ
a
a
1
μ
a
p
1
σ
a
x
Δ
σ
a
y
σ
a
2
a
x
σ
a
y
σ
a
0.
3.28
Hence, from q
2
a ≥ q
1
a,xσa < 0,yσa > 0, and 3.21,weget
p
2
a
y
1
a >p
2
a > 0, it follows that
y
Δ
a
> 0, 3.30
which contradicts y
Δ
a < 0.
It follows from the above discussion that yt has at least one generalized zero on
a, b ∩ T. This completes the proof.
Remark 3.4. If p
1
t ≡ p
2
t ≡ 1, then Theorem 3.3 reduces to classical Sturm comparison
theorem.
Remark 3.5. In the continuous case: μt ≡ 0. 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: μt ≡ 1. 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:
0, 1
∩ T
N − 1
,
3
2
N − 1
, ,1
,N>2,
0, 1
∩ T
q
k
| k ≥ 0,k∈ Z
∪
{
0
}
, where 0 <q<1.
3.31
12 Advances in Difference Equations
By Theorem 3.3, we have if xt and yt are the nontrivial solutions of 1.1 and 1.2, a, b are
two consecutive generalized zeros of xt, and p
1968.
5 W. T. Reid, “A comparison theorem for self-adjoint differential equations of second order,” Annals of
Mathematics, vol. 65, pp. 197–202, 1957.
6 R. Zhuang, “Sturm comparison theorem of solution for second order nonlinear differential
equations,” Annals of Differential Equations, vol. 19, no. 3, pp. 480–486, 2003.
7 R K. Zhuang and H W. Wu, “Sturm comparison theorem of solution for second order nonlinear
differential equations,” Applied Mathematics and Computation, vol. 162, no. 3, pp. 1227–1235, 2005.
8 W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications,Har-
court/Academic Press, San Diego, Calif, USA, 2nd edition, 2001.
9 B. G. Zhang, “Sturm comparison theorem of difference equations,” Applied Mathematics and
Computation, vol. 72, no. 2-3, pp. 277–284, 1995.
10 M. Bohner and A. Peterso, Advances in Dynamic Equations on Time Scales,Birkh
¨
auser, Boston, Mass,
USA, 2003.
11 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications,
Birkh
¨
auser, Boston, Mass, USA, 2001.
12 V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan, Dynamic Systems on Measure Chains, vol.
370 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands,
1996.
13 R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,” Results in
Mathematics, vol. 35, no. 1-2, pp. 3–22, 1999.
14 S. Hilger, “Special functions, Laplace and Fourier transform on measure chains,” Dynamic Systems and
Applications, vol. 8, no. 3-4, pp. 471–488, 1999.
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