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

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.

Copyrightq2009 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:

p1txΔt

_Δ

q1txσt 0, 1.1

p2tyΔt

_Δ

q2tyσt 0, 1.2

where t ∈ α, β ∩T, pΔ₁t, pΔ₂t, q1t, and q2t are real and rd-continuous functions in

α, β ∩T.Let Tbe a time scale, σtbe the forward jump operator in T,yΔ be the delta derivative, andyσt: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

Ifxtandytare the nontrivial solutions of

p1tx_{tq1txt 0,}

wheret ∈ α, β , p₁t, p₂t, q1t,andq2tare real and continuous functions inα, β .If

yt/0 fort∈α, β ,then

_xt

yt

p1tx_{tyt−p2tytxt}

p1t−p2tx2_{t q2t−q1tx}2_{t p2t}

_xtyt

yt −xt

2 .

1.4

By 1.4, one can easily obtain the Sturm comparison theorem of second-order linear differential equations1.3.

Sturm-Picone Comparison Theorem

Assume thatxtandytare the nontrivial solutions of1.3anda, bare two consecutive zeros ofxt,if

p1t≥p2t>0, q2t≥q1t, t∈a, b , 1.5

thenythas at least one zero ona, 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 , wherep1t≥p2t>0 onα, β1 , q2t≥q1ton

α1, β1 , α, βare integers, andΔis the forward difference operator:Δxt xt1−xt. In 1995, Zhang9 extended this result. But we will remark that in8, 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

p1txΔ_t∇_{q1txt 0,}

p2tyΔt

∇

q2tyt 0,

1.7

wherep1t≥ p2t> 0 andq2t≥ q1tfort ∈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 of1.1and1.2.

2. 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 operatorsσ, ρ:T → Tby

σt inf{s∈T:s > t}, ρt sup{s∈T:s < t}, 2.1

where inf∅ supT, sup∅ infT. A pointt ∈ Tis called right-scattered, right-dense, left-scattered, and left-dense ifσt> t, σt t, ρt< t, andρt t,respectively. We putTkT ifTis unbounded above andTkT\ρmaxT,maxT otherwise. The graininess functions

ν, μ:T → 0,∞are defined by

μt σt−t, νt t−ρt. 2.2

Letfbe a function defined onT.fis said to bedeltadifferentiable att∈Tkprovided there exists a constantasuch that for anyε >0, there is a neighborhoodUofti.e.,U t−δ, tδ∩T for someδ >0with

fσt−fs−aσt−s≤ε|σt−s|, ∀s∈U. 2.3

In this case, denotefΔt :a. Iff isdeltadifferentiable for everyt∈Tk, thenfis said to bedeltadifferentiable onT. Iffis differentiable att∈Tk, then

fΔ_t

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

lim s→t s∈T

ft−fs

t−s , ifμt 0, fσt−ft

μt , ifμt>0.

2.4

IfFΔt ftfor allt∈Tk, thenFtis called an antiderivative offonT. In this case, define the delta integral by

t

s

fτΔτ Ft−Fs ∀s, t∈T. 2.5

Moreover, a functionf defined onTis said to be rd-continuous if it is continuous at every right-dense point inTand its left-sided limit exists at every left-dense point inT.

Lemma 2.1. Letf, g:T → Randt∈Tk.

iIffis differentiable att, thenfis continuous att.

iiIffandgare differentiable att, thenfg is differentiable attand

fgΔt fσ_tgΔ_{t f}Δ_{tgt f}Δ_tgσ_{t ftg}Δ_{t. 2.6}

iiiIffandgare differentiable att, andftfσt/0, thenf−1_{gis differentiable attand}

gf−1Δ_{t g}Δ_tft−gtfΔ_tfσ_tft−1_.

2.7

ivIffis rd-continuous onT, then it has an antiderivative onT.

Definition 2.2. A functionf:T → Ris said to be right-increasing att0∈T\{maxT}provided

ifσt0> ft0in the case thatt0is right-scattered;

iithere is a neighborhoodUoft0such thatft> ft0for allt∈Uwitht > t0in the

case thatt0is right-dense.

If the inequalities forfare reversed iniandii,fis said to be right-decreasing att0.

The following result can be directly derived from2.4.

Lemma 2.3. Assume thatf :T → Ris differentiable att0 ∈T\ {maxT}.IffΔt0>0,thenf is

right-increasing att0; and iffΔt0<0, thenfis right-decreasing att0.

Definition 2.4. One says that a solutionxtof1.1has a generalized zero attifxt 0 or, iftis right-scattered andxtxσt<0.Especially, ifxtxσt<0,then we sayxthas a node attσt/2.

A functionp:T → Ris called regressive if

1μtpt/0, ∀t∈T. 2.8

Hilger14 showed that fort0 ∈ Tand rd-continuous and regressivep, the solution of the

initial value problem

yΔ_{t ptyt, yt0 1 2.9}

is given byep·, t0, where

ept, s exp

t

sξμτ

pτΔτ

withξhz

⎧ ⎪ ⎨ ⎪ ⎩

Log1hz

h , if h /0 z, if h0.

2.10

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:

xΔΔ_{t a}

Proof. We first divide the left part of3.4into two parts

From1.1and the product ruleLemma 2.1ii, we have

Theorem 3.3 Sturm-Picone Comparison Theorem. Suppose that xt and yt are the integrating it fromatobwe get

which is a contradiction. Therefore, in Case1,ythas at least one generalized zero ona, b ∩

T.

Case 2. Supposeais a zero ofxt,bσb/2 is a node ofxt, xb<0,andxσb>0.It follows from the assumption thatythas no generalized zeros ona, b ∩Tand thatyt>0 for allt ∈a, b ∩Tthatyσb >0.Hence by2.4andp2t≥ p1t >0 ona, b ∩T, we

have

xb

yb

p1bxΔbyb−p2byΔbxb

_yxb_bμ1_bp1bxσbyb−p2byσbxb

p2b−p1b

xbyb

<0.

3.12

By integration, it follows from3.12andxa 0 that

_b

a

_xt

yt

p1txΔtyt−p2tyΔtxt

Δ

Δt

_xt

yt

p1txΔ_tyt−p2tyΔ_txtb a

x_yb_bp1bxΔbyb−p2byΔbxb

<0.

3.13

So, from3.9and above argument we obtain that

0>

b

a

⎛ ⎜

⎝p1t−p2txΔ_t2_q2t−q1tx2_σt

⎛ ⎝

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

Case 3. Supposeaσa/2 is a node ofxt, xa >0, xσa< 0,andbis a generalized zero ofxt.Similar to the discussion of3.12, we have

xa

ya

p1axΔaya−p2ayΔaxa

x_ya_aμ1_ap1axσaya−p2ayσaxa p2a−p1axaya

<0,

3.15

which implies

p1axΔaya−p2ayΔaxa

<0. 3.16

iIfbσb/2 is a node ofxt,thenxb<0, xσb>0.Hence, we have3.12, that is,

xb

yb

p1bxΔbyb−p2byΔbxb

<0. 3.17

iiIfbis a zero ofxt,then

xb

yb

p1bxΔbyb−p2byΔbxb

0. 3.18

It follows from3.4andLemma 2.3that

xt

yt

p1txΔtyt−p2tyΔtxt

3.19

is right-increasing ona, b ∩T.Hence, fromiandiithat

xa

ya

p1axΔaya−p2ayΔaxa

< xσa yσa

p1σaxΔσayσa−p2σayΔσaxσa

<0,

3.20

which implies

From3.16,3.21, and2.4, we have

p1xΔy−p2yΔx

_Δ

a 1

μa

p1xΔy−p2yΔx

σa−p1xΔy−p2yΔx

a>0. 3.22

Further, it follows from1.1,1.2, product ruleLemma 2.1ii, and3.22that

p1xΔ_y−p2yΔ_xΔ_{a q2a−q1axσayσa p1a−p2ax}Δ_ayΔ_a>0.

3.23

Ifp1a p2aand fromq2a≥q1a,xσa<0, andyσa>0 we have

q2a−q1a

xσayσa<0. 3.24

This contradicts3.22. Note thatxΔa 1/μaxσa−xa. It follows fromp1a > p2a>0,3.23, and3.24that

yΔ_{a<0. 3.25}

On the other hand, it follows fromxtandytare solutions of1.1and1.2that

yσap1axΔa

_Δ

q1axσa

0,

xσap2ayΔ_aΔ_q2ayσa0.

3.26

Combining the above two equations we obtain

p1axΔa

_Δ

yσa−p2ayΔa

_Δ

xσa

q1a−q2a

xσayσa 0.

It follows from3.27and2.4that

1

μa

p1σaxΔ_σa−p1axΔ_{ayσa−p2σay}Δ_σa−p2ayΔ_axσa

q1a−q2axσayσa

1

μa

p2ayΔaxσa−p1axΔayσa

_μ1_ap1σaxΔσayσa−p2σayΔσaxσa

q1a−q2axσayσa 0.

3.28

Hence, fromq2a≥q1a, xσa<0, yσa>0,and3.21, we get

p2ayΔaxσa−p1axΔayσa<0. 3.29

By referring toxΔa<0 andp1a> p2a>0,it follows that

yΔ_{a>0, 3.30}

which contradictsyΔa<0.

It follows from the above discussion thatyt has at least one generalized zero on

a, b ∩T.This completes the proof.

Remark 3.4. If p1t ≡ p2t ≡ 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 equationsseeSection 1.

Remark 3.6. In the discrete case:μt≡1. This result is the same as Sturm comparison theorem of second-order difference equationssee8, 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 ∩Tqk|k≥0, k∈Z∪ {0}, where 0< q <1.

ByTheorem 3.3, we have ifxtandytare the nontrivial solutions of1.1and1.2,a, bare two consecutive generalized zeros ofxt,andp1t≥p2t>0, q2t≥q1t, t∈a, b ∩T,

then yt has at least one generalized zero ona, b ∩T.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 Remarks3.4–3.6andExample 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.

Acknowledgments

Many thanks to Alberto Cabada the editor and the anonymous reviewers for helpful comments and suggestions. This research was supported by the Natural Scientific Foundation of Shandong ProvinceGrant Y2007A27,Grant Y2008A28, the Fund of Doctoral Program Research of University of Jinan B0621, and the Natural Science Fund Project of Jinan UniversityXKY0704.

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