Volume 2009, Article ID 496135, 12 pagesdoi:10.1155/2009/496135 Research Article Sturm-Picone Comparison Theorem of Second-Order Linear Equations on Time Scales Chao Zhang and Shurong Su
Trang 1Volume 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 zhangc@ujn.edu.cn
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 Copyrightq 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:
p1txΔtΔ q1tx σ t 0, 1.1
p2tyΔtΔ q2ty σ t 0, 1.2
where t ∈ α, β ∩ T, pΔ
1t, pΔ
2t, q1t, and q2t 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 y σ t : yσt.
First we briefly recall some existing results about differential and difference equations
As we well know, in 1909, Picone1 established the following identity
Picone Identity
If xt and yt are the nontrivial solutions of
p1txt q1txt 0,
p2tyt q2tyt 0, 1.3
Trang 2where t ∈ α, β , p
1t, p
2t, q1t, and q2t are real and continuous functions in α, β If
yt / 0 for t ∈ α, β , then
xt
yt
p1txtyt − p2tytxt
p1t − p2tx2t q2t − q1tx2t p2t xtyt
yt − xt
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
p1t ≥ p2t > 0, q2t ≥ q1t, t ∈ a, b , 1.5
then yt has at least one zero on a, b
Later, many mathematicians, such as Kamke, Leighton, and Reid2 5 developed thier work The investigation of Sturm comparison theorem has involved much interest in the new century6,7 The Sturm comparison theorem of second-order difference equations
Δp1t − 1Δxt − 1 q1txt 0,
Δp2t − 1Δyt − 1 q2tyt 0, 1.6
has been investigated in8, Chapter 8 , where p1t ≥ p2t > 0 on α, β 1 , q2t ≥ q1t on
α1, β1 , α, β are integers, and Δ is the forward difference operator: Δxt xt1−xt.
In 1995, Zhang9 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
p1txΔt∇ q1txt 0,
p2tyΔt∇ q2tyt 0,
1.7
where p1t ≥ p2t > 0 and q2t ≥ q1t for t ∈ ρα, σβ ∩ T, y∇ 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 of1.1 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 inSection 3 InSection 3we first give the Picone identity on time scales, then we will employ this to prove our main result: Sturm-Picone comparison theorem of1.1 and 1.2 on time scales
Trang 32 Preliminaries
In this section, some basic concepts and some fundamental results on time scales are introduced
Let T ⊂ R be a nonempty closed subset Define the forward and backward jump
operators σ, ρ :T → T by
σt inf{s ∈ T : s > t}, ρt sup{s ∈ T : s < t}, 2.1
where inf∅ sup T, sup ∅ inf T A point t ∈ T is called right-scattered, right-dense, left-scattered, and left-dense if σt > t, σt t, ρt < t, and ρt t, respectively We put T k T
ifT is unbounded above and Tk T \ ρmax T, max T otherwise The graininess functions
ν, μ : T → 0, ∞ are defined by
μt σt − t, νt t − ρt. 2.2
Let f be a function defined on T f is said to be delta differentiable at t ∈ T kprovided there
exists a constant a such that for any ε > 0, there is a neighborhood U of t i.e., U t−δ, tδ∩T for some δ > 0 with
2.3
In this case, denote fΔt : a If f is delta differentiable for every t ∈ T k , then f is said to
bedelta differentiable on T If f is differentiable at t ∈ T k, then
fΔt
⎧
⎪
⎪
⎪
⎪
lim
s → t s∈T
ft − fs
t − s , if μt 0, fσt − ft
μt , if μt > 0.
2.4
If FΔt ft 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 fτΔτ Ft − Fs ∀s, t ∈ T. 2.5
Moreover, a function f defined onT is said to be rd-continuous if it is continuous at every right-dense point inT and its left-sided limit exists at every left-dense point in T
For convenience, we introduce the following results11, Chapter 1 , 12, Chapter 1 , and13, Lemma 1 , which are useful in the paper
Trang 4Lemma 2.1 Let f, g : T → R and t ∈ T k
i If f is differentiable at t, then f is continuous at t.
ii If f and g are differentiable at t, then fg is differentiable at t and
fgΔt f σ tgΔt fΔtgt fΔtg σ t ftgΔt. 2.6
iii If f and g are differentiable at t, and ftf σ t / 0, then f−1g is differentiable at t and
gf−1Δ
t gΔtft − gtfΔt
f σ tft−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 t0∈ T\{max T} provided
i fσt0 > ft0 in the case that t0is right-scattered;
ii there is a neighborhood U of t0such that f t > ft0 for all t ∈ U with t > t0in the
case that t0is right-dense
If the inequalities for f are reversed in i and ii, f is said to be right-decreasing at t0 The following result can be directly derived from2.4
Lemma 2.3 Assume that f : T → R is differentiable at t0 ∈ T \ {max T} If fΔt0 > 0, then f is
right-increasing at t0; and if fΔt0 < 0, then f is right-decreasing at t0.
Definition 2.4 One says that a solution x t of 1.1 has a generalized zero at t if xt 0 or,
if t is right-scattered and xtxσt < 0 Especially, if xtxσt < 0, then we say xthas a
node att σt/2.
A function p :T → R is called regressive if
1 μtpt / 0, ∀t ∈ T. 2.8
Hilger14 showed that for t0 ∈ T and rd-continuous and regressive p, the solution of the
initial value problem
yΔt ptyt, yt0 1 2.9
is given by e p ·, t0, where
e p t, s exp
t
s ξ μτ
pτΔτ
with ξ h z
⎧
⎪
⎪
Log1 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 derivativesee 10, Chapter 3
Trang 53 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:
xΔΔt a1tx Δσ t a2tx σ t 0 3.1 can be rewritten as1.1
Theorem 3.1 If 1 μta1t / 0 and a2t is continuous, then 3.1 can be written in the form of
1.1, with
p1t e a1t, t0, q1t e a1t, t0a2t. 3.2
Proof Multiplying both sides of3.1 by e a1t, t0, we get
0 e a1t, t0xΔΔt e a1t, t0a1tx Δσ t e a1t, t0a2tx σ t
e a1t, t0xΔΔt e a1t, t0 Δx Δσ t e a1t, t0a2tx σ t
e a1t, t0xΔtΔ e a1t, t0a2tx σ t,
3.3
where we usedLemma 2.1 This equation is in the form of1.1 with p1t and q1t as desired.
Lemma 3.2 Picone Identity Let xt and yt be the nontrivial solutions of 1.1 and 1.2 with
p1t ≥ p2t > 0 and q2t ≥ q1t for t ∈ α, β ∩ T If yt has no generalized zeros on α, β ∩ T,
then the following identity holds:
xt
yt
p1txΔtyt − p2tyΔtxtΔ
p1t − p2txΔt2q2t − q1tx2σt
⎛
⎝
yt
p2tyσt
p2tyΔt
yt xσt −
p2tyσt
yt xΔt
⎞
⎠
2
.
3.4
Proof We first divide the left part of3.4 into two parts
xt
yt
p1txΔtyt − p2tyΔtxtΔ
p1txΔtxt − p2tyΔt
yt x2t
Δ
p1txΔtxtΔ−
p2tyΔt
yt x2t
Δ
.
3.5
Trang 6From1.1 and the product rule Lemma 2.1ii, we have
p1txΔtxtΔp1txΔtΔxσt p1txΔtxΔt
p1txΔt2− q1tx2σt ∀t ∈α, β
∩ T.
3.6
It follows from 1.2, 2.4, product and quotient rules Lemma 2.1ii, iii and the
assumption that yt has no generalized zeros on α, β ∩ T that
p2tyΔt
yt x2t
Δ
x2σt
p2tyΔt
yt
Δ
xσtxΔt p2tyΔt
yt xΔtxt
p2tyΔt
yt
x2σt
−q2t − p2t
yΔt2
ytyσt
xσtxΔt p2tyΔt
yt
xΔtxσt − μtxΔt p2tyΔt
yt
p2txΔt2− q2tx2σt − p2t
yΔt2x2σt
ytyσt
2xσtxΔt p2tyΔt
yt −
p2t μt p2tyΔt
yt
xΔt2
p2txΔt2− q2tx2σt − yt
p2tyσt
p2tyΔt
yt
2
x2σt
2xσtxΔt p2tyΔt
yt −
p2tyσt
yt
xΔt2
p2txΔt2− q2tx2σt
−
⎛
⎝
yt
p2tyσt
p2tyΔt
yt xσt −
p2tyσt
yt xΔt
⎞
⎠
2
∀t ∈α, β
∩ T.
3.7
Combiningp1txΔtxtΔand−p2tyΔt/ytx2tΔ, we get3.4 This completes the proof
Now, we turn to proving the main result of this paper
Trang 7Theorem 3.3 Sturm-Picone Comparison Theorem Suppose that xt and yt are the
nontrivial solutions of1.1 and 1.2, and a, b are two consecutive generalized zeros of xt, if
p1t ≥ p2t > 0, q2t ≥ q1t, t ∈ a, b ∩ T, 3.8
then yt has at least one generalized zero on a, b ∩ T.
Proof Suppose to the contrary, y t 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 x t Then byLemma 3.2,3.4 holds and
integrating it from a to b we get
b
a
xt
yt
p1txΔtyt − p2tyΔtxtΔΔt
b
a
⎛
⎜p
1t − p2txΔt2q2t − q1tx2σt
⎛
⎝
yt
p2tyσt
p2tyΔt
yt −
p2tyσt
yt xΔt
⎞
⎠
2⎞
⎟
⎠Δt.
3.9
Noting that xa xb 0, we have
b
a
xt
yt
p1txΔtyt − p2tyΔtxtΔΔt
xt
yt
p1txΔtyt − p2tyΔtxt b
a
0.
3.10
Hence, by3.9 and p1t ≥ p2t > 0, q2t ≥ q1t, for all t ∈ a, b ∩ T we have
0
b
a
⎛
⎜p
1t − p2txΔt2q2t − q1tx2σt
⎛
⎝
yt
p2tyσt
p2tyΔt
yt −
p2tyσt
yt xΔt
⎞
⎠
2⎞
⎟
⎠Δt
> 0,
3.11
Trang 8which is a contradiction Therefore, in Case1, yt has at least one generalized zero on a, b ∩ T.
Case 2 Suppose a is a zero of x t, b σb/2 is a node of xt, 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 p2t ≥ p1t > 0 on a, b ∩ T, we
have
xb
yb
p1bxΔbyb − p2byΔbxb
xb
yb
1
μb
p1bxσbyb − p2byσbxb p2b − p1bxbyb
< 0.
3.12
By integration, it follows from3.12 and xa 0 that
b
a
xt
yt
p1txΔtyt − p2tyΔtxtΔΔt
xt
yt
p1txΔtyt − p2tyΔtxt b
a
xb
yb
p1bxΔbyb − p2byΔbxb
< 0.
3.13
So, from3.9 and above argument we obtain that
0 >
b
a
⎛
⎜p
1t − p2txΔt2q2t − q1tx2σt
⎛
⎝
yt
p2tyσt
p2tyΔt
yt −
p2tyσt
yt xΔt
⎞
⎠
2⎞
⎟
⎠Δt
> 0,
3.14
which is a contradiction, too Hence, in Case 2, yt has at least one generalized zero on
a, b ∩ T.
Trang 9Case 3 Suppose a σa/2 is a node of xt, xa > 0, xσa < 0, and b is a generalized zero of xt Similar to the discussion of 3.12, we have
xa
ya
p1axΔaya − p2ayΔaxa
xa
ya
1
μa
p1axσaya − p2ayσaxa p2a − p1axaya
< 0,
3.15
which implies
p1axΔaya − p2ayΔaxa< 0. 3.16
i If b σb/2 is a node of xt, then xb < 0, xσb > 0 Hence, we have 3.12, that is,
xb
yb
p1bxΔbyb − p2byΔbxb< 0. 3.17
ii If b is a zero of xt, then
xb
yb
p1bxΔbyb − p2byΔbxb 0. 3.18
It follows from3.4 andLemma 2.3that
xt
yt
p1txΔtyt − p2tyΔtxt 3.19
is right-increasing ona, b ∩ T Hence, from i and ii that
xa
ya
p1axΔaya − p2ayΔaxa
< xσa
yσa
p1σaxΔσayσa − p2σayΔσaxσa
< 0,
3.20
which implies
p1σaxΔσayσa − p2σayΔσaxσa > 0. 3.21
Trang 10From3.16, 3.21, and 2.4, we have
p1xΔy − p2yΔxΔ
a 1
μa
p1xΔy − p2yΔx
σa −p1xΔy − p2yΔx
a> 0 3.22
Further, it follows from1.1, 1.2, product rule Lemma 2.1ii, and 3.22 that
p1xΔy − p2yΔxΔ
a q2a − q1axσayσa p1a − p2axΔayΔa > 0.
3.23
If p1a p2a and from q2a ≥ q1a, xσa < 0, and yσa > 0 we have
q2a − q1axσayσa < 0. 3.24
This contradicts3.22 Note that xΔa 1/μaxσa − xa It follows from p1a >
p2a > 0, 3.23, and 3.24 that
On the other hand, it follows from xt and yt are solutions of 1.1 and 1.2 that
yσa
p1axΔaΔ q1axσa
0,
xσa
p2ayΔaΔ q2ayσa
0.
3.26
Combining the above two equations we obtain
p1axΔaΔyσa −p2ayΔaΔxσa
q1a − q2axσayσa 0.
3.27
Trang 11It follows from3.27 and 2.4 that
1
μa
p1σaxΔσa − p1axΔayσa −p2σayΔσa − p2ayΔaxσa
q1a − q2axσayσa
1
μa
p2ayΔaxσa − p1axΔayσa
1
μa
p1σaxΔσayσa − p2σayΔσaxσa
q1a − q2axσayσa
0.
3.28
Hence, from q2a ≥ q1a, xσa < 0, yσa > 0, and 3.21, we get
p2ayΔaxσa − p1axΔayσa < 0. 3.29
By referring to xΔa < 0 and p1a > p2a > 0, it follows that
which contradicts yΔa < 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 p1t ≡ p2t ≡ 1, thenTheorem 3.3 reduces to classical Sturm comparison theorem
Remark 3.5 In the continuous case: μ t ≡ 0 This result is the same as Sturm-Picone
comparison theorem of second-order differential equations seeSection 1
Remark 3.6 In the discrete case: μ t ≡ 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
0,1
2 ∪
2
3, 1 ,
0, 1 ∩ T
0,1
2 ∪
! 1 2N − 1,
1
N − 1 ,
3 2N − 1, , 1
"
, N > 2,
0, 1 ∩ T q k | k ≥ 0, k ∈ Z ∪ {0}, where 0 < q < 1.
3.31