Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

989 Angelo B. Mingarelli !E

Ser. y

14~

, "%~

Cat.

Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions ETHICS ETH-BIB

IIIIIIIlUUIIIIIIIIIIIIIIIIIII 00 ] 0 0 0 0 0 3 8 ] 261

Springer-Verlag Berlin Heidelberg New York Tokyo 1983

Author Angelo B. Mingarelli Department of Mathematics, University of Ottawa 585 King-Edward Avenue, Ottawa, Ontario, Canada K1N 9B4

AMS Subject Classifications (1980): Primary: 45 J 05, 45 D 05, 47 A 99 Secondary: 34 B25, 34 C10, 39A10, 39A12, 47 B50 ISBN 3-540-12294-X Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12294-X Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the materia is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

Quest'

opera

ai m i e i Oliviana

cari

~ umilmente genitori

e al m i o

dedicata

Giosafat

fratello

Marco

A.M.D.G.

e

PREFACE

The aim of these notes a d o p t e d by m a n y authors F.V. Atkinson, qualitative

W.T.

is to pursue a line of r e s e a r c h

(W. Feller,

Reid,

M.G.

among others)

Krein,

I.S.

in order to d e v e l o p a

and spectral theory of V o l t e r r a - S t i e l t j e s

e q u a t i o n s with

specific a p p l i c a t i o n s

ential and d i f f e r e n c e

equations

(comparison theorem,

general

setting.

of the second order.

of such e q u a t i o n s

separation

In c h a p t e r and,

theorem)

to this more

3,4,5,

certain g e n e r a l i z e d o r d i n a r y d i f f e r e n t i a l

theory

apply some aspects of

it to the study of the s p e c t r u m of the o p e r a t o r s

g e n e r a t e d by

expressions

associated

integral equations.

In order to make these notes dices have been added w h i c h m a i n text.

results of

2 we study the o s c i l l a t i o n

in C h a p t e r s

with the a b o v e - m e n t i o n e d

integral

to real o r d i n a r y differ-

We begin by an e x t e n s i o n of the c l a s s i c a l Sturm

Kac,

self-contained

include results

some appen-

fundamental

to the

Care has been taken to give due c r e d i t to those

r e s e a r c h e r s who have c o n t r i b u t e d

to the d e v e l o p m e n t of the theory

p r e s e n t e d h e r e i n - any o m i s s i o n s

or errors are the author's

sole

responsibility. I am g r e a t l y w h o s e hands

indebted to P r o f e s s o r F.V. A t k i n s o n at

I learned the s u b j e c t and I also take this o p p o r t u n i t y

to a c k n o w l e d g e w i t h thanks the a s s i s t a n c e of the N a t u r a l

Sciences

and E n g i n e e r i n g R e s e a r c h C o u n c i l of C a n a d a

finan-

cial support.

My sincere

thanks go to Mrs.

for c o n t i n u e d

Frances M i t c h e l l

VI

for her expert typing of the m a n u s c r i p t . Finally,

I am deeply g r a t e f u l

Jean for her c o n s t a n t e n c o u r a g e m e n t

to my wife Leslie

and p a t i e n c e

and I also

wish to thank P r o f e s s o r A. Dold for the p o s s i b i l i t y p u b l i s h the m a n u s c r i p t

to

in the Lecture Note series.

A n g e l o B. M i n g a r e l l i Ottawa, A p r i l

1980.

TABLE

OF

CONTENTS

Page INTRODUCTION

CHAPTER

..........................................

1 Introduction i.i.

CHAPTER

1

oo,o,o,,oooooooo°.oooo,oooo°oooo,°

Comparison Theorems for Differential Equations

i°2.

Separation

1.3.

The

Green's

Theorems

Stieltjes Integro..................

4

.....................

20

....................

25

Function

2 Introduction 2.1.

..................................

Non-Oscillation Criteria for Linear Volterra-Stieltjes Integral Equations

2.1A.

Applications

to

Differential

2.1B.

Applications

to

Difference

2.2.

Oscillation

2.2A.

Applications

to

Differential

2.2B.

Applications

to

Difference

2.3.

An Oscillation Theorem in t h e N o n l i n e a r Case ....................................

Addenda

CHAPTER

x

Criteria

28

...

29

..

52

....

60

Equations Equations

.................... Equations Equations

74 ..

80

....

82

.......................................

87 113

3 Introduction 3.1.

..................................

Generalized

Derivatives

.................

118 120

VIII

Page CHAPTER

3

(continued)

3.2.

Generalized Differential Expressions of the Second Order ........................

123

3.3.

The

129

3.4.

Applications

3.5.

Limit-Point

3.6.

J-Self-Adjointness of G e n e r a l i z e d Differential Operators ..................

156

Dirichlet Integrals Associated with Generalized Differential Expressions

....

180

for Three-Term .....................

183

3.7.

3.8.

CHAPTER

Weyl

and

Limit-Circle

Criteria

143

....

147

4

4.1.

4.2.

...................................

197

Sturm-Liouville Difference Equations with an I n d e f i n i t e W e i g h t - F u n c t i o n ...........

199

Sturm-Liouville Differential Equations with an Indefinite Weight-Function ......

212

5 Introduction 5.1.

5.2.

APPENDIX

.................

............................

Dirichlet Conditions Recurrence Relations

Introduction

CHAPTER

Classification

o

o

o

The Discrete Differential

o

o

o

o

,

o

o

o

°

o

o

o

o

o

o

o

o

o

,

o

o

o

o

°

o

o

o

.

.

o

.

Spectrum of Generalized Operators ..................

The Continuous Spectrum Differential Operators

of Generalized ..................

225

226

242

I I.l.

Functions

1.2.

The

1.3.

G e n e r a l T h e o r y of V o l t e r r a - S t i e l t j e s Integral Equations ......................

264

Construction

273

1.4.

of B o u n d e d

Variation

..........

256

Riemann-Stieltjes

Integral

..........

258

of

the

Green's

Function

....

IX

Page APPENDIX

II II.1.

APPENDIX

in

LP

and

Other

Spaces

..

280

III III.l.

Eigenvalues Equations

of G e n e r a l i z e d Differential ............................

296

....

299

Linear

Operators

in

III.3.

Linear

Operators

in a K r e i n

III.4.

Formally Self-Adjoint Even Order Differential Equations with an Indefinite Weight-Function ...........

Index

a Hilbert

Space

Space

292

..

III.2.

BIBLIOGRAPHY

Subject

Compactness

303

.........................................

309

........................................

318

INTRODUCTION

Let

p,q:

Lebesgue

measure)

Consider

the

I÷

]19, p(t)

and

i/p,

formally

a solution

of

- q(t)y

(i.e.,

py'c

AC(I)

and

Then

a quadrature

p ( t ) y ' (t) where

B =

(PY') (7).

o(t)

=

/t q ( s ) d s a

will

be

a solution

Stieltjes

Since

exists

where

the

sense.

whenever

o e BV(I)

is c o n t i n u o u s used

to d e a l

need

not (2)

discrete term

be

to b e

On

on

be

recurrence

(I)

y: I)

I ÷ C

such

I.

,

that

Let

y e I.

equation

of

and

difference as w e l l

in t h e

variation

on

equations

the

so

equations as

Riemann-

I)

and

y

(2) m a y

Moreover

require

(2) c a n

(2)

a meaning

form

(i).

as w e

a

t c I,

has

of

y

form

also

(as l o n g I)

,

Hence

satisfies

the

say,

(2)

equations

I

on

relations)

if y(t)

hand

integral

~ e AC(I).

only

bounded

differential on

indefinite

ft y ( s ) d o ( s ) Y

other

e.g.,

its

interpreted,

Hence

continuous

problems,

on

t e I and

8 +

the

continuous

on

a.e.

c L(I)

if a n d

(i.e.,

I.

with

=

may

Stieltjes

of

q

integro-differential

integral

c IR

t ~ I,

for

(i)

p ( t ) y ' (t)

[a,b3

t ~ + / y(s)q(s)ds Y

=

of

of

equation

a function

(I)

for

I =

sense

t c I.

continuous

satisfies

gives,

where

differential

mean

absolutely

y(t)

(in t h e

= 0,

(i) w e w i l l

y c AC(I),

a.e.

q c L(I)

symmetric

(p(t)y')' By

> 0

be

a

a solution

used

to

(or t h r e e -

continuous

be

problems

treat

XI

as w e h a v e
m

seen.

= b bea

p,o

More

as f o l l o w s :

I and t h e i r o n l y

jumps,

above with O(tn)

where

p,o

a n d p is d e f i n e d p(t) w h e r e Cn_l,

corresponding polygonal

their ordinates, second-order

step-functions

be r i g h t - c o n t i n u o u s

will

be at the p o i n t s

of o b e i n g

given

on {t.} 1

by

- O ( t n - 0 ) = b n,

on

[a,b]

is a g i v e n upon

these

m

is a g i v e n

solutions

Yn'

linear

of

vertices

satisfying difference

finite

sequence,

t c [ t n _ l , t n)

identifications

whose

real

setting

= Cn_l(tn-tn_l),

real

curves

will

the s a l t u s

n = 0,i,..., With

Define

if any,

b n, n = 0 , 1 , . . . , m - l ,

sequence.

let t _ l = a < t 0 < t I ...
f i x e d partit~n of I

c BV(I)

defined

precisely

positive one

(2) w i l l

finds

real

finite

t h a t the

be c o n t i n u o u s

(tn,Y(tn))

~ (tn,Yn)

the f o r m a l l y

have

symmetric

equation

A ( C n _ i A Y n _ I) - b n Y n = 0, for n = 0,I,..., operator, understand preted

m-l,

and

A

Ay n ~ Y n + l -

Yn"

So use of

that

as c o n t i n u o u s

as the f i n i t e sight

example,

That

of

for

BScher

difference

(2) n o w l e a d s o n e

(3) s h o u l d p e r h a p s defined

to

be i n t e r -

on I a n d n o t

just

y _ l , Y 0 , .... Y m as one m a y a t f i r s t

functions, M.

is the f o r w a r d

functions

sequence

suspect.

as c o n t i n u o u s For

solutions

(3)

(3) s o l u t i o n s has

noted

a r e to be i n t e r p r e t e d

its h i s t o r i c a l in his

survey

precedents. article

[ 6]

Xll

that a Sturmian if

theory

"solutions"

were

could

treated

be n a t u r a l l y

developed

for

as c o n t i n u o u s

functions

(in f a c t ,

the s a m e p o l y g o n a l

curves

advantage

in u s i n g

(2) is t h a t a S t u r m i a n

developed

for

for e a c h of

(2) thus

(i) a n d

If in

that were mentioned

simultaneously

then

If,

in a d d i t i o n ,

yielding

a e C(I)

(2) is a p u r e S t i e l t j e s

once again

theory

The

c a n be

such a theory

(3).

(2) one c h o o s e s

I)

above).

(3)

p e C(I)

to y i e l d

the

say,

(i.e.,

continuous

integro-differential

then

equation.

(2) m a y be i n t e g r a t e d

Volterra-Stieltjes

t

on

integral

equation

t

ds = ~ + B f p(s) + S (t-s)y(s)d~(s) t ci 7 (2) a l s o i n c l u d e s e q u a t i o n s of m i x e d t y p e o b t a i n e d

y(t) Note by,

that say,

points

~ E C 1 (I) e x c e p t

setting

or b y d e f i n i n g

and a s t e p - f u n c t i o n

by Fi ~

the f u n d a m e n t a l of W.T.

Reid

In o r d e r needs

to use

suitable of

s t u d y of e q u a t i o n s

Atkinson

paper

[791,

~ to be a c l - f u n c t i o n

of

o n a p a r t of I

[80]

of the f o r m

[3] in his m o n o g r a p h ,

of K r e i n

[39]

(2) w a s

(See a l s o

and the r e l a t e d

papers

).

to d e r i v e

(2) in o r d e r

space.

number

elsewhere.

An intensive undertaken

at a f i n i t e

a spectral to d e f i n e

To this end,

note

theory

for

an o p e r a t o r that

if

y

(2) o n e on s o m e

is a s o l u t i o n

(2) t h e n t d d--t {p(t)y' (t) - S y ( s ) d o ( s ) }

= 0

(4)

XIII

and c o n v e r s e l y y ~ AC(I)

a s o l u t i o n of t p(t)y' (t) - / y ( s ) d a ( s )

for w h i c h

then recover of

if o n e d e f i n e s

(2) f r o m

(4) d e f i n e s

(4).

a generalized

and s u c h an e x p r e s s i o n on L 2(I)

may

t h e n be u s e d

W e can

the l e f t - s i d e

expression,

equations

with

to d e f i n e

for d o m a i n

to t r e a t b o u n d a r y

viz.

=

a linear

considerations.

problems

a weight-function

-(p(t)y') ' + q ( t ) y consideration

hand

differential

w i t h d u e care

If o n e w i s h e s Sturm-Liouville

c AC(I).

d {p(t)y' (t) - f t y ( s ) d a ( s ) } dt Y

~[y] (t) =

operator

On the o t h e r

(4) as a f u n c t i o n

for r(t)

e L(I),

Ir(t)y,

of the g e n e r a l i z e d

ordinary

differential

expression Z[y](t) m a y be m a d e ,

where

d d~(t)

=

the g e n e r a l i z e d

the r i g h t

is,

c a s e r(t)

>0 c o r r e s p o n d s

in g e n e r a l ,

c a s e of u n r e s t r i c t e d In t h e

former

expression

the latter

r(t)

to

~(t)

symmetric

in t h e w e i g h t e d

space,

signed measure.

t f y(s)do(s)}

(t) -

derivative

appearing

derivative.

non-decreasing

corresponds

case the operator

(Pontrjagin)

y,

a Radon-Nikodym

case the operator

is f o r m a l l y

restrictions)

{p(t)

(under

The

a n d the

by t h e d i f f e r e n t i a l suitable

space

is J - s y m m e t r i c

since the measure

on

to ~(t) e BV(I).

defined

Hilbert

(5)

domain

L 2 (I,d~).

In

in a K r e i n

induced

by ~(t)

is a

XIV

Expressions W.

Feller

o(t~

e

on

BY(I)

monotone,

form

(5) w e r e

[683,[693,[703,[713,[723,[733

- constant

function

of the

I, was

cf.,

on

I, p(t)

(cf.,

also

treated

- i, a n d Langer

by I.S.

[46,p.49].

in t h e

~ a given

[41]). Kac

first

considered case

by

when

non-decreasing

The more

[353,[36],[373

general when

case

~ is

CHAPTER

1

INTRODUCTION: In t h i s Stieltjes of

the

chapter

we

shall

integro-differential

study

equations;

defined valued

on a f i n i t e

> 0

Historical

what

the

of

is,

of

equations

functions

1836.

[a , b]

and

of b o u n d e d

p , o

are

variation

real

on

I

there.

comparison we call

scalar

first

(i.0.0)

y(s)do(s)

I =

and

the

separation

Sturmian

theorems

theory.

of

Sturm

Comparison

com-

theorems

equation

Ip(t)y'(t)l'

were

that

theory

Background:

The

for

interval

right-continuous

p(t)

prise

Sturmian

form

p ( t ) y ' (t) = c +

and

the

obtained In t h a t

by

paper

- q(t)y(t)

Sturm Sturm

[58,

p.

135]

considered

(i.o.i)

= 0

in h i s the

famous

memoir

equations

!

(K~y')

- G~y

= 0

(1.0.2)

2

(K2z')

on a f i n i t e G2 ~ Gi , then

interval equality

between

is at l e a s t result

any one

usually

proof

depended

coefficients to

K2

and

then

as the valid

he

[K2Yz

upon

'

- KlY

'

= 0

that

(1.0.3)

if

0 < K2 ~ K1 ,

everywhere

some

on the

solution

solution

of

of

the

introduction

GI

to the

him

as

location

varied.

This

Theorem.

the

parameter zeros

depended

is the Sturm's

from

was

of the

upon

there

in the

continuously

of the

It a l s o

(1.0.2)

of a p a r a m e t e r

to p a s s

G2 ,

interval,

(1.0.3).

as the Sturm-Comparison

allowed

studied

all

of

of a n y

known

parameter for

zeros

zero

from

showed

not holding

two

which

and

and

- G2z

the

KI

increased, solutions identity

tI , t2 • I ,

t2 z]tl

=

f t2 (G 2 tl

G1)YZ

-

dt + f t2 (K 2 - K l ) y ' z ' tI

dt (1.0.4)

which [13,

c a n be o b t a i n e d p.

case

comparison

[58,

of a t h r e e - t e r m equation

A discrete

analog

[21,

of S t u r m equations.

p.

p.

by

that first

recurrence though

of the

] whose

applied

186]

theorem

difference

Fort

of G r e e n ' s

theorem

291]. It s e e m s

of the

b y an a p p l i c a t i o n

the

to d i f f e r e n c e

having

relation latter

comparison

method

Sturm

came

shown or

result

theorem

of p r o o f

was,

equations

to the c o n c l u s i o n it t r u e

second was was

the

order

not

published.

published

in e s s e n c e ,

instead

for

by

that

of d i f f e r e n t i a ]

In 1909 Picone

[48, p. 18]

proof of the c o m p a r i s o n

theorem

gave by far the simplest

in the continuous

case.

He

made use of the formula

t2

[z

(K2YZ'

KlY'Z)

.t2

t 2

1 f tl

tl

+

dt t 2

It I

(G 2- G1)y2 dt

2

(1.o.5)

-

commonly allows

known

as the Picone

an immediate

[33, p. 226].

One important

a variational Q[y]

y e C 1 (a, b) termed

and

(1.0.2-3) = y(b) For such

Q[y]

acting

had the property

of

(1.0.3) would have

Swanson Q[y]

~ 0

= 0

that

Q[y]

to vanish

the solutions

y 7 0

(such functions

were

y ,

< 0

(1.0.6)

admissible

then every

at some point

Leighton's

reaching

in

functional"

df = fa% (K2Y ,2+ G2y2) dt

[59, p. 3] w e a k e n e d for

the theorem

on functions

The main result was that if some non-trivial y

theorem was

He made use of a "quadratic

y(a)

'admissible').

Theorem

of the c o m p a r i s o n

[42, p. 604] who interpreted

with

(1.0.5)

[74]).

extension

setting:

associated

The use of

proof of the Sturm C o m p a r i s o n

(cf., also

that of Leighton

Identity.

real solution in

(a, b)

condition

Q[y]

the same conclusion

were not constant m u l t i p l e s

of

function

y

.

< 0

provided

to

4

Sturm-Separation

The linearly

independent

separate

one

recurrence known

solutions

another.

relations

in the

of,

A similar and

latter

In s e c t i o n

theorem s t a t e s t h a t the z e r o s of

in f a c t

case. 1 we

say,

(1.0.2)

result

holds

a more

general

(See s e c t i o n

shall

give

an e x t e n s i o n

"Leighton-Swanson

Theorem"

equations

(i.0.0)

as c o r o l l a r i e s ,

continuous

and d i s c r e t e

In s e c t i o n Theorem

for

(i.0.0)

and

and

chapter

a study

with

problem of

the

§l.1

finding

functions Pi(t)

Pi(t)

functions

t e

are

ing a f i n i t e simplicity can

,

of b o u n d e d

> 0 ,

FOR

~i(t)

,

variation

[a, b]

,

number

be o m i t t e d ,

in m o s t We will,

We

(See

its

STIELTJES

over

on

theorems,

integral

corresponding theorem. Separation

this

for b o u n d a r y

for

the

to the solution

3).

INTEGRO-DIFFERENTIAL

,

be

and

real

valued

We

assume

that

there.

without assume

all

with

chapters

in g e n e r a l ,

of

application

[a , b]

following

afore-

to b o t h

[a, b]

i= 1 , 2 ,

of t h e

conclude

section

i : 1 , 2

is

Sturm

function and

result

comparison

of the

of d i s c o n t i n u i t i e s

In the

the

representation

problem.

right-continuous

only.

conclusions.)

Green's

three-term

class

applications

(i.0.0)

COMPARISON THEOREMS EQUATIONS : Let

some

an e x p l i c i t

non-homogeneous

of t h e

equations.

of t h e with

to t h e

a proof

give

difference

associated of

versions

2 we give

differential

problems

give,

for

or

21).

mentioned

and

interlace

(This

possessis for

hypothesis

affecting that

four

each

this

that

all

the these

functions

are

lim

exists

o(t)

continous

Consider

at

a ,b

as

t ÷

the

equations

pl(t)u'(t)

= c +

and

,

i

if

b = ~

,

then

t u(s)dol(s)

(i.i.o)

v(s)do2(s)

(l.i.l)

a

P 2 ( t ) v ' (t)

where

a solution of

by

u(t)

e AC[a,

each

point

b]

with

t c

DQ

=

f2

(I.i.0),

say,

pl(t)u'(t)

we mean

6 BV(a, b)

a function

satisfying

(i.i.0)

at

[a , b]

Associated functional

= c' +

Q[u]

with with

{u : u E A C [ a ,

the

pair

domain

b]

(i.i.0-i)

is

the

quadratic

DQ

, p2 u ' ~ B V ( a

b)

, u(a)

= u(b)

= 0} (i.i.2)

and where,

for

U

e DQ

Q[u]

We

can

Swanson

result.

THEOREM

i.i.0:

Let

Pi

now

,

=

state

' o_ 1

,

'i~

(P2u'2dt+

u2d°2)

and

an

extension

be

defined

prove

i= 1, 2

,

(1.1.3)

of

as

the

above

Leighton-

and

let

u E DQ

,

u ~ 0

,

be

such

that

Q[u]

Then multiple

of

(1.1.4)

every

solution

of

(i.i.i)

u(t)

vanishes

at

least

Proof:

Assume,

(a, b)

and

and

< 0

on

let

the

contrary,

a < s < t < b

once

that .

which

v

is n o t

in

(a , b)

does p 2 v '/v

Then

a constant

not

vanish

in

c B V I o c (a , b)

so

(1.1.5)

exists.

Case

i:

v(a)

~ 0 ,

For

u e DQ

S

u2 d

v(b)

~ 0

satisfying

=

s

u2

=

in p a s s i n g

(i.i.i).

dp2v,

+ p2v' dv -I

s

=

where

(1.1.4),

from

Integrating

dP2V'

(1.1.6)

+ u P2V

u2d~ 9 - 2

p2tT

(1.1.6)

to

(1.1.7)

(1.1.5)

by

parts

we

we

~

used

find

dt

the

that

(1.1.7)

equation

u2d

=

P2

-

s

Combining

(i.i.7),

(i.i.8)

and

i to b o t h

sides

2

fs

v vuu

P2

(i.i.8)

s

we

adding

t

,2 P2 u

s

obtain,

(P2 u ' 2 d t +

u2dd2)

=

EP2 v'

+ S

i

+

t

p2 u

,2

t

ruv, 2

P2[--v--/

S

dt

- 2

It

s

, , u p2 v u v

s 2 t

t

2

= [~ v U I + f ~{u S

=

[p2 v ' u ~ I t

UV~v

S

+ IS" p2 v 21vU--}' 2

S

(1.1.9)

for

a < s < t < b

obtain,

.

Hence

if w e

since

v(a)

let ,

Q[u]

The

hypothesis

on

u

s ÷ a+ v(b)

0

,

t ÷ b-0

in

(1.1.9)

~ 0 ,

(i.i.i0)

u fa~ P 2 V 2 {v) '~=> 0

=

we

implies

Q[u]

= 0

but

since

v 7 0

,

! f

we must [a, b]

~

have

lul

which

we

= 0

or

excluded.

that This

u

is a m u l t i p l e

contradiction

of

shows

v that

on v

must

vanish

Case

2:

at l e a s t

v(a) To

once

= v(b)

settle

in

(a, b)

= 0 .

this

case

it s u f f i c e s

to s h o w

that

in

(1.1.9),

u

2

(t)P2 (t)v' (t)

lim t÷b-0

= 0

(l.l.ll)

= 0

(i.i.12)

v (t)

and u

2

(s)P2

lim s÷a+0

It is p o s s i b l e problem

to s h o w

(i.i.i),

v(a)

See A p p e n d i x

I and

v' (a)

(The p r i m e

~ 0

derivative

which

is c o n t i n u o u s since

v(b)

lim s÷a+0

provided implies Thus

the that

solutions

= cI ,

P2(a)v'(a)

341].

here

point

to the

Thus

usually

(two-sided)

~ 0

latter

limit

exists.

it is c o n t i n u o u s

P2(t)

right-neighborhood

The

in s o m e

a right-

derivative

lim s+a+0

if

°2

[3, p.

2 uv(s) (s)

hypothesis

348],

on

(1.1.13)

°2

right-neighborhood

is c o n t i n u o u s

in some,

Hence

unique:

Hence

in s u c h

a

value

= 0 ,

Similarly

is c o n t i n u o u s

of

are

v(a)

represents

in q u e s t i o n . )

P2(b)v'(b)

initial

= c2

since

(s)P2(S)V' (s) v(s) = P2(a)v' (a)

P2(t)v'(t)

Similarly

that

is an o r d i n a r y

= 0 , 2

v(s)

[3, p.

at the

u

s)v' (s)

v' (t)

of

a neighborhood. possibly

different,

is c o n t i n u o u s

(i.e.

a.

is

an o r d i n a r y

(a, a +

6)

,

In ordinary (a , a + we

can

derivative)

the

,

same

it c a n

be

shown

in

(a, a +

Q)

,

theorem

that

q > 0

2 u ( t ) u ' (t)

L'Hopital's

u' (t)

.

Thus

Since

to

the

u,

limit

v

is a n in

e AC[a,

in t h e

b] ,

right

of

to obtain

2 u (s) v(s)

lim s÷a+0

_

since,

as w e

exists

and

saw

is

above,

we

0

v' (a)

it c a n

(i.i.ii),

(1.1.9)

v' (s)

~ 0

Hence

the

limit

(1.1.12)

zero.

Similarly Combining

2 u ( s ) u ' (s)

lira s÷a+0

=

in

right-neighborhood

way

(u2(t)) ' =

apply

(1.1.13)

some

6 > 0

derivative

q)

in

be

shown

(1.1.12)

obtain

(I.i.i0)

v(a)

= 0

v(b)

This

case

and

that

(i.i.ii)

letting

again

and

holds.

s ÷ a+ thus

0

,

derive

t ÷ ba contra-

diction.

Case

3:

combination and

Associated with

domain

z 0

is e a s i l y

of C a s e s

(1.1.11-12).

Q' [u]

,

1 and

This

with DQ,

or

v(a)

disposed

2 leading

proves

the

(i.i.0)

is

of to

~ 0

,

v(b)

as

it

is

(i.i.i0)

= 0

simply via

a

(1.1.9)

theorem.

the

quadratic

functional

0

10 DQ, = {u : u e A C [ a , b]

, pl u' e B V ( a , b)

, u(a)

= u(b)

= 0}

(1.1.14) and Q' [u] = lab (PlU'2dt + u2dgl )

i.i.0:

COROLLARY

Let u(a)

=

u(b)

u =

(1.1.15)

(Swanson [59, p. 4], L e i g h t o n Cor. i]). be a n o n - t r i v i a l

solution

of

[42, p. 605,

(i.I.0) w i t h

0

Then every s o l u t i o n constant multiple

of

u

v(t)

of

(i.i.i) w h i c h

is not a

must v a n i s h at least once in

(a , b)

provided b ~ {(Pl - P2 )u'2dt + u2d(~l - ~2 ) } ~ 0

Proof:

Let

u

be a s o l u t i o n

of

(i.i.0),

u(a)

(1.1.16)

= u(b)

= 0

Then

ud(PlU')

Using the e q u a t i o n

=

b

[uPlU'] a -

(1.1.0)

plu'

in the l e f t - s i d e

2

dt .

of

(i.i.17)

(1.1.17)

we

find that Q' [u] =

( P l U ' 2 d t + u2dOl )

[UplU,l b a =

o

.

(1.1.18)

11

(1.i.16)

now

says

Q' [u]

that

- Q[u]

or,

> 0

because

of

(1.1.18),

Q[u]

Since

u

applies

is n o t and

Swanson's

a constant

hence

v(t)

extension

obtained

by

[59,

<

0

(1.1.19)

multiple

of

v

Theorem

vanishes

at

p.

Leighton's

Theorem

t E

,

4]

of

least

,

once

in

l.l.0 (a , b) [42]

is

setting

oi(t)

=

qids

,

[a , b]

i= 1 , 2

, (1.1.20)

in

(i.i.0-i)

COROLLARY

in

Pi

> P2(t)

(1.1.16).

(Sturm Comparison

i.i.i:

Let Pl(t)

and

'

> 0

qi ,

E C[a, ql(t)

b]

Theorem) i = 1,

,

2

and

suppose

that

> q2 (t)

If

(plu')'

and

u(a)

for w h i c h which

= u(b) v(c)

is n o t

= 0 = 0

- qlu

= 0

(i.1.21)

(P2V') ' - q 2 v

= 0

(1.1.22)

,

is

then

whenever

a constant

there v

multiple

is of

at

least

a solution u

one of

c

E

(a , b)

(1.1.22)

12

Proof:

Let

o

follows

from

Corollary

1

(t)

is n o n - d e c r e a s i n g We

now

recurrence

a fixed

defined I.i.0

on

as

on

[a, b]

interpret

relation.

t_l

be

be

the

above for

of

the

.-.

interval

Let

be

an arbitrary

b 0 , b I , ..., bm_ 1

for

n=

continuous at

p(t)

= Cn_l(t n-

0 , 1 , 2 ....

the

function

now

- o2(t)

hypothesis. a three-term

[a , b]

a given

p(t)

result

o1(t)

< t m _ I < tm

be

a function

The

Let

c_l , c O , c I , ... , C m _ 1

define

that

results

= a < tO < tI <

partition

(1.1.20).

account

by

these

in

on

[a , b]

t n _ 1)

,m

positive

if

Then

of b o u n d e d

real

(1.i.23)

let sequence.

sequence

and

setting

t e

p(t)

and

real

by

= b

1.1.24)

[tn_ I , t n)

is a p o s i t i v e

variation

with

rlght-

jumps,

if a n y ,

{ti} Now

define

a right-continuous

~(t)

on

[a , b]

by

step-function

with

~ ( t n)

0)

requiring

jumps

at

the

that

it b e

{ti}

of

magnitude

where

With

n=

p(t)

0 , 1 , ... , m - i

, a(t)

- ~(t n-

= -b n

(1.1.25)

.

as d e f i n e d

above

consider

(i.0.0).

On

13

[a, t O ) ,

a(t)

= constant,

la y d a

and

so

(i.i.0)

But

p(t)

= p(a)

function,

hence

fact,

(1.1.27)

[a , t 0)

,

(1.1.26)

[a , t O ) .

(1.1.27)

that

on

[a, t O )

(1.1.27)

letting

t ~

t

= c = p ( a ) y ' (a)

y' (t)

In

--- 0

implies

p ( t ) y ' (t)

hence,

because

implies

= y' (a)

y ( t n)

= Yn

p(t)

is

also

a

step-

that

t

'

c

~

[a , t O )

n=-i

(1.1.28)

, 0 , 1 , ... , m

,

then

gives

y ( t 0) - y ( a ) y' (t)

= tO - a

YO - Y-i t c

[a , t 0) .

t O - t_l (1.1.29)

Hence

p(t)y'

(t)

= p(a)y'

(a) (Y0 - Y - i )

=

c_i(t

O - t_i)

= c_l(y0-Y_l)

•

t O - t_i t ~

[a , t O ) .

(1.1.30)

14 Now

let

from

t e [tn-1' tn)

(1.1.30)

that

'

1 =< n =< m

p(t)y'(t)

When

n = 0 ,

= c_l(y 0 - y_l)

for

we k n o w

t E

[a, t O ) .

Thus

p(t)y' (t) = p(a)y' (a) +

ydo

= p(a)y' (a) + n-i [ i=0 = p(a)y'

(a)

+

t. f i ti_ 1 t.+0

~

i=0

+ ft yda tn_ 1

ydo

yd~

+

ft

ti-0

ydo

tn_l +0

n-i = p(a)y'

(a)

+

Y(ti) (o(t i) - o(t i - 0)]

+ 0

i=0

since

~

constant there,

is c o n s t a n t on

on

[tn_ 1 , t n)

y'(t)

[tn_ 1 , t n) and since

is also c o n s t a n t

y' (t) =

Hence p(t)

p(t)y'(t)

satisfies

is

(1.1.24)

so that

Y n - Yn-I t -t n n-i

t e

[tn_ 1 , t n)

Consequently,

p(t)y' (t) = C n _ l ( y n - Yn_l )

t c [tn_ I , t n)

(1.1.31)

This

is true

for each

t e [t n , tn+ I)

,

n

(i.i.0)

in the range gives

considered.

If

15 t ydo

p(t)y' (t) = p(t n - 0)y' (t n - 0) + t -0 n t +0

ydo = p(t n - 0 y'(t n - 0) + I n t -0 n = Cn_l(Yn-Yn_

I) + y(t n) (o(t n) - O ( t n - 0) ] (1.1.32) (1.1.33)

= C n _ l ( y n - Yn_l ) + yn(-bn)

w h e r e we have used (1.1.33) By

(i.i.31)

and

(1.1.25)

in o b t a i n i n g

(i.1.32),

respectively.

(1.1.31)

we find that

t E [t n , tn+ I)

p(t)y'(t)

Combining

= C n ( Y n + I - yn )

this with

(1.1.33)

if

we o b t a i n

C n ( Y n + I - Yn ) = C n _ l ( y n - Yn_l ) - bnY n

(1.1.34)

CnYn+ 1 + C n _ l Y n _ 1 - (c n + Cn_ I - bn)Y n = 0

(1.1.35

or

which

is e q u i v a l e n t

to

(1.1.36

A ( C n _ i A Y n _ l ) + bnY n = 0

where

A

represents

the f o r w a r d d i f f e r e n c e

operator,

AYn = Yn+l - Yn Summarizing d e f i n e d as in

then, we see that w h e n

(1.1.24-5)

respectively,

p(t)

, o(t)

the S t i e l t j e s

are

integro-

16

differential

equation

gonal

whose

the

curves Yn

satisfy

or

the

n=

0 , 1 , 2,

instead

to

solutions

"vertices"

are

the

second-order

three-term

which

points

recurrence

difference

equation

similar

the

The

poly-

(t n , y n )

relation

(1.1.36)

An

argument

of

(1.1.25),

with

recurrence

we

and

(1.1.35)

for

one

above

shows

that

if,

require

- ~(t n-

the

to

0)

same

= b n - c n - Cn_ 1

p

as

in

(1.i.24),

,

will

(1.1.37)

give

rise

relation

(i.i.38)

CnYn+ 1 + Cn_lYn_ 1 - bnY n = 0

where

are

... , m - i

(1.1.0),

the

has

the

o ( t n)

then

(i.i.0)

n = 0 , 1 , 2 , ... , m - i

initial

conditions

y(a)

= a

p ( a ) y ' (a)

associated

with

(i.i.0)

become,

(1.1.39)

=

in

8

(1.1.40)

the

case

of

a recurrence

relation,

Y_l

(1.1.41)

= ~

C_l(Y 0- Y_l ) =

8

(1.1.42)

17

on

A

account

of

(i.1.30).

fundamental

will

then

solution,

i.e.

one

Y0

the

case

respect

of

see

a recurrence

[3,

With

p.

the

two

arbitrary

and

two

Let

~ =

0

,

B =

1

,

become

Y-I

in

in which

~. (t) 1

,

(1.1.43)

1

(1.1.44)

-i

relation

t

defined

n

finite

(1.1.38).

1,

2

,

as

in

(1.1.23)

sequences

sequences

i=

- c

0

(In

this

97]).

real

positive

=

cn be

bn

, rn

,

' qn n=-l

step-functions

we

suppose

'

n=

given

0, i, ... ,m-]

, 0 , 1 , ... , m - i on

[ a , b]

with

saltus

n=

Let

0 , l,

ol(tn)

- ol(tn-

0)

= bn

(1.1.45)

°2(tn)

- °2 ( t n -

0)

= qn

1.1.46)

... , m - i

Pi(t)

,

i=

1 , 2

,

be

defined

by

Pl(t)

= Cn_ l(t n-

t n _ I)

t e

[tn_ 1 , t n)

1.1.47)

P2(t)

= rn_l(t n-

tn_l)

t

[tn_ 1 , tn)

1.1.48)

6

18

where i=

n

=

1 , 2 ,

continuous

o. 1

,

m-i

and,

with

along

and

of

bounded

Then the

the

variation

now

(i.i.0-i)

with

1 , 2

The

solutions

of

ti

.

will

then

Pi(t)

~. (t) l

Consider i=

points

0 , 1 .....

,

>

i=

for

1 , 2

on

[a , b]

the

above

(i.I.0-i)

0 ,

are

choice

of

evaluated

satisfy

the

recurrence

c n Un+ 1 +Cn_ 1 Un_ 1 -

(c n +

Cn_ I+

r n Vn+ 1 +rn_ 1 Vn_ 1 -

(r n +

rn_ 1 + qn)Vn

b n)u n

right

Pi

at

'

the

relations

=

0

=

0

and

where

n

=

0 , 1 , ... , m - i

equivalent

0 , 1 , ... , m - I

the

Sturm [21,

COROLLARY

latter

are

A(Cn-iAUn-l)

- bn u n

=

0

(1.1.49)

A(rn-iAVn-l)

- qn Vn

=

0

(1.1.50)

We

comparison p.

can

theorem

now

state

one

form

a discrete of

which

analog

was

of

proven

by

].

1.1.2 : Let

equality

The

to

n =

Fort

respectively.

not

c

n

> r = n

holding

> 0

and

for

every

A(Cn_iAUn_

bn n

=> q n .

I) - b n u n

for

If

=

U_l

0

n=

0 , 1 , ....

= um

0

m-i

and

(1.1.51)

,

19

then

there

is at

least

one

node

A(rn-iAVn-l)

in

of

- qn Vn

(1.1.52)

= 0

(a , b)

REMARK:

We lent of

to

note

u(a)

that

the

= u(b)

= 0

when

U_l

= um = 0

is c o n s i d e r e d

u

is e q u i v a a solution

(i.I.0). By

"polygonal the

a node

we mean

a point

curve"

defined

by

the

on

the abscissa where

finite

sequence

vn

the crosses

axis.

Proof:

The

implies

that

we

from

find

condition

c

Pl (t)

> P2 (t)

(1.1.45-6)

ol(tn)

Since n

condition

,

o I , 02 (1.1.53)

for

t e

that

Corollary

(i.i.i) to t h e

[a , b]

has

at

required

Moreover,

> 0

$ ol(t n-

0)

step-functions

implies

with

(1.1.47-8)

since

bn = > qn

that

- o 2 ( t n)

are

along

> r > 0 n = n

that

This,

of(t)

along

is a p p l i c a b l e

least

one

zero

conclusion.

on

in

the and

(a, b)

0) .

(tn_ 1 , t n)

- o2(t)

with

i.i.0

- o2(tn-

(1.1.53)

for

each

is n o n - d e c r e a s i n g

above hence

Remark, the

which

shows

equation

is e q u i v a l e n t

20

Note:

In g e n e r a l ,

a comparison

theorem

for

equations

of

the

- bn Yn

= 0

(1.1.54)

r n Z n + l + r n - i Z n - i - q n Zn = 0

(1.1.55)

form

Cn Yn+l + Cn-i Yn-i

under

For

the

assumptions

example

let

rn = qn = c/2 that

in t h i s will

§1.2

on

solutions, case

for

known at t h e

to

this the as

of

If

n=

but

has

large

section

we

separation

all

n

no

,

is n o t

and

b

available.

n

nodes

= 3c We

0 , 1 , ... , m - i

a simple

computation

,

see

shows

then that,

eventually

while

(1.1.55)

classical

Sturm

separation

n

> 0

prove of

p.

186]

the

zeros

of

differences

[58,

of his

n

for ,

b n => q n

,

THEOREMS:

finite

c

n

a consequence

Sturm

end

each

b n > qn

SEPARATION

theorem,

= c > 0

(1.1.54)

nodes

In

the

,

case,

have

n

for

> rn

cn

c

C n => r n

the

one

linearly

results

this

as

of

result can

independent

in

section

was

also

gather

from

i.

In

probably the

remarks

memoir. for

all

n

,

then

the

nodes

of

solutions

of

Cn Y n + l + C n - i

Yn-i - bn Yn

= 0

(1.2.0)

21

separate proof

The case

one

another

of t h i s

Sturm

result

separation

of a g e n e r a l

if t h e s e will

are

follow

theorem

linearly below.)

i~ not v a l i d

three-term

recurrence

[6, p.

solutions

of

176]

points

(1.2.1)

out

holds

P

for all false,

n

in the

in g e n e r a l ,

an e x a m p l e The

nodes

the

initial

Y0 = 6

do not One

if

note

that

the

condition

p(t)

> 0

for

the

n

Pn

p(t)

> 0

then

property

for

(1.2.2)

The

fails. =

= 0 ,

one

result

He g i v e s

1 ,

Qn

=

Rn

is h o w e v e r [6, p.

-i

=

solutions

Y0 = 1

and

177]

for

all

as n .

corresponding Y-I

= -i0

,

another.

separation given

(1.2.2)

property

of

by M o u l t o n

[45,

is the

analog

+ r(t)y

= 0

(1.2.1) p.

of the

under

137].

We

condition

equation

p(t) y" + q(t)y'

If

separation

> 0

independent

was

(1.2.1)

= 0

the

considered.

Y-I

of the

(1.2.2)

R

(1.2.2)

separate

proof

the h y p o t h e s i s

n

linearly

values

in the

if

case where

of the

to the

range

that

in g e n e r a l

relation

Pn Y n + l + Qn Yn + Rn Y n - i

Bocher

(Th e

independent.

the

zeros

of

linearly

.

independent

(1.2.3)

solutions

22

of

(1.2.3)

that

separate

(1.2.3)

can

one

then

another.

(One w a y

be t r a n s f o r m e d

into

of

seeing

this

an e q u a t i o n

is

of the

form

(P(t)y' 1 ' + Q ( t ) y

where

P(t)

property

THEOREM

of

> 0

and

the

the

zeros

result

of

separate

zeros

one

generate and

say,

find stant

the

separation

of

linearly

u

that

has the

two

solutions

= c +

y(s)d~(s)

linearly

independent

solution

Pl = P2 when

v

multiple

must of

u

Sturm

in

space

(1.1.16)

vanishes

vanish

of

(1.2.5)

[3, p. we

in b e t w e e n

348].

can

at t w o

solutions

apply

If w e

v

now

Corollary

consecutive since

u ,v

points

is n o t

set i.i.0 to

a con-

u

In p a r t i c u l a r the c l a s s i c a l

independent

another.

(1.2.5)

°l = °2 to,

from

(1.2.4).)

p(t)y'(t)

which

follows

1.2.0: The

Proof:

(1.2.4)

= 0

if

o E C' (a , b)

separation

theorem.

we

immediately

obtain

23

COROLLARY

1.2.0: If

then

the

o c C'(a,

zeros

of

b)

and

linearly

o'(t)

independent

(p(t)y'] ' - q(t)y

separate

each

other.

Porter

[49,

solutions the

of

limiting

differential

p.

= q(t)

55]

showed

(1.2.0)

generate

process

which

takes

t e

solutions

[a, b]

of

(1.2.6)

= 0

that

the

,

two

solution

linearly space

a difference

independent

and

considered

equation

to a

equation.

Defining

o , p

as

in

(1.1.24-25)

we

obtain

the

discrete

analog

COROLLARY

1.2.1: If

n=

c

n

> 0

,

0 , 1 , ... , m - I

n=

-i,

is a n y

0,

sequence

A ( C n _ i A Y n _ I)

then

the

nodes

of

..., m-I

linearly

and

b

,

n

and

(1.2.7)

- bnY n = 0

independent

solutions

separate

one

another.

As recurrence

an

application

relation

of C o r o l l a r y

(1.2.1)

we

state

1.2.1 the

to

the

following

[45, p.

137].

24

COROLLARY

1.2.2:

Let

P n Yn+l

for

n=

be

Pn ' Qn ' Rn

real

finite

sequences

+ Qn Yn + R n Yn-i

=

and

0

(1.2.8)

, m-i

(1.2.9)

0 , 1 , ... , m - 1

If

P

then

the

nodes

separate

each

n

idea

(1.2.9)

can

be

linearly

is

to

brought

C_l

(1.2.9)

implies

C_l

.

If

>

0,

1,

independent

now

...,

that

into

the

and

that we

show

solutions

(1.2.8) form

consider

Cn_ I ~

bn

n=

0 , 1 ....

of

(1.2.8)

under

(1.2.7)

the after

hypothesis which

we

1.2.1.

0

P n R n

Cn

for

n=

Corollary

Let

> 0

0

other.

The

apply

>

n

of

Proof:

simply

R

c

n

n=

> 0

the

recurrence

relation

0 , 1 , ... , m - i

for

n=

0 , 1 , ....

(1.2.10)

m-i

since

set

=

m-i

-c n

,

-

Cn_ 1

then

a

Cn Qn p n simple

(1.2.11)

computation

shows

that

25

with

the

substitutions

three-term

recurrence

(1.2.10-11), relation

(1.2.7)

(1.2.8).

reduces

Hence

the

to t h e result

followsl

§1.3.

The

GREEN'S

FUNCTION:

In A p p e n d i x tence

of a G r e e n ' s

~(t)

I to t h i s function

= e + B

work

for

we

the

~do

= u2~

=

shown

inhomogeneous

--+

u1~

have

the

exis-

problem

ds +

(1.3.0)

-P

(1.3.1)

0

where

Ui ~ =

2 ~ j=l

i=l,

IMij ~ (j-l) (a) + Nij p ( b ) ~ (j-l) (b)}

2 , (1.3.2)

and the that

M

(1.3.1)

homogeneous the

with

on

~' = q

.

constants,

with

If

(1.3.0)

In t h i s

is c o n t i n u o u s

case

and

p(t) with the

the b o u n d a r y we mean

boundary in

is of the

the h y p o t h e s i s

and

(By t h i s

f = 0

and

under

f = 0)

homogeneous

equation

then

(with

incompatible.

~ • C ' ( a , b)

[a, b]

(i.0.0)

is

solution.)

integral

If

real

problem

equation

zero

resulting

ous

are

the h o m o g e n e o u s

conditions

only

, N.. 13

l]

conditions

(1.21.0)

form

then

reduces

"derivative"

the G r e e n ' s

the has

the

(1.0.0).

is p o s i t i v e f = 0

that

function

and

continu-

to

(1.2.6)

appearing reduces

in to the

26

usual

one.

(See A p p e n d i x

On then

the o t h e r

(1.3.0)

with

f = 0

recurrence

relations.

difference

equations

seems

to h a v e

Another

been

t • same

p(t)

We

in A p p e n d i x

then

given

the

order

constructed

was

showed

case

of h i g h e r

first

,

o(t)

c a n be m a d e

In t h i s

unique

step-functions

to i n c l u d e

and, the

more

Green's

[3, p.

if

three-te@m

generally,

by Bocher

by A t k i n s o n I that

are

function [5, p.

of

83].

148].

(1.3.0-1)

solution

for

with

(1.3.0-1)

f = 0 is

by

~(x)

for

if

treatment

is i n c o m p a t i b l e given

hand

I, p. 278 .)

x •

[a , b]

[a, b] points

=

In the

and

f(t)

where

G ( x , t)df(t)

particular

as usual,

o(t)

the

case when

is a s t e p - f u n c t i o n has

its

fi = f(ti)

where,

(1.3.3)

t.

jumps

with

and

p(t)

= 1 ,

jumps

at t h e

if w e

denote

- f(ti-0)

represent

the

by

(1.3.4)

jump

points

of

f ,

1

then

a simple

computation

~n

- ~(tn)

shows

=

that

G(tn,

t)df(t)

m-i i=0

G ( t n ' ti) • If(t i) - f(t i - 0)) (1.3.5)

27

and if w e w r i t e

Gni

-= G ( t n , t i)

,

0 =< n ,

i =< m - 1

we

find

that

m-.l ~ G .f. i=O nl i

~n =

This

~n

then r e p r e s e n t s

inhomogeneous derived

difference

directly

for e x a m p l e see A p p e n d i x

149]

I, s e c t i o n

We note

the s o l u t i o n boundary

using methods

[3, p.

and

problem.

of f i n i t e

[5, p.

p(t)

of b o u n d e d

variation

appearing

in

is c o n t i n u o u s

Green's

I, the d i s c o n t i n u i t y function

(1.3.6)

For

is

(See

further details

on

are c o n t i n u o u s

[a , b]

t h e n the d e r i v a t i v e

everywhere

in the

and so,

first derivative

from of the

is g i v e n by

Gx(t+

which

Usually

differences.

84].)

, ~(t)

functions

Appendix

to the c o r r e s p o n d i n g

1.4.

that when

(i.0.0)

(1.3.6)

0 , t) - G x ( t -

is the u s u a l m e a s u r e

function

associated

equation

of the f o r m

with

0 , t)

p(t)

of d i s c o n t i n u i t y

a second-order

(1.2.6).

(1.3.7)

of the G r e e n ' s

linear

differential

CHAPTER

2

INTRODUCTION: There subject order

is a v e r y

of o s c i l l a t i o n

differential

[59]).

On

hand

establishing

criteria

behaviour

solutions

particular c a n be p.

found

425].

[12],

case in

[23,

Other

of

the o s c i l l a t o r y

of d i f f e r e n c e

pp.

known

less

second for e x a m p l e ,

non-oscillatory In t h e

relations

and more

some

recently

are m o r e

or

chapter

we

be c o n c e r n e d

the

about

equations.

recurrence

126-128]

and

results

oscillation

be n o t e d

that

y"

will

oscillatory

and

shall

non-oscillation

Stieltjes

the p o t e n t i a l

which

little

(see,

with

in

scattered:

results [32,

[21],

[20] .

non-linear will

is

dealing

of r e a l

on a half-axis

there

for

literature

non-oscillation

of t h r e e - t e r m

In t h i s some

and

equations

the o t h e r

of

extensive

integral

equations

if o n e m a k e s q

for

obtaining linear

on a h a l f - a x i s .

an h y p o t h e s i s

o n the

and It

integral

in

- q(t)y

guarantee

criteria

with

the

solutions,

= 0

existence

then

t e

[a, ~)

(2.0.0)

of o s c i l l a t o r y

a certain

discrete

or n o n -

analog

will

29

exist

for a t h r e e - t e r m In s e c t i o n

Stieltjes order

integral

difference

a result

sufficient

1 we

give

equations

equations.

o n the o s c i l l a t o r y extend

recurrence

condition

some and

their

75]

2 we

and

guarantees

state all

solutions

of a n o n - l i n e a r

equation

corollary

we

the

discrete

analog

Various

examples

are

theorem should

§2.1

shall

[2, p. help

obtain

643].

visualize

the

theorems

NON-OSCILLATION CRITERIA INTEGRAL EQUATIONS: In the

equations

following,

of t h e

we

give

and

that

are

criteria

applications

of s o l u t i o n s

[8, p.

which

non-oscillation

In s e c t i o n

behaviour

of B u t l e r

relation.

to some

for

second results

in s e c t i o n

3 we

a necessary

and

continuable

oscillatory.

As

a

of A t k i n s o n ' s

included

which

stated.

FOR LINEAR

shall

VOLTERRA-STIELTJES

usually

be c o n s i d e r i n g

form

y' (t) = c +

i

t t e

y(s)do(s)

[a , ~)

,

(2.1.0)

a

where

a

variation

is a r i g h t - c o n t i n u o u s

function

on

of

[a , ~)

assume,

in a d d i t i o n ,

remains

finite

can

also

Because that

in f i n i t e

be e x t e n d e d

the

the

number

intervals.

to e q u a t i o n s

p ( t ) y ' (t) = c +

of

locally

of b o u n d e d

applications

we

shall

of d i s c o n t i n u i t i e s The

theorems

the

form

y(s)do(s)

of

proved

here

(2.1.1)

30

in the

case when

p(t)

> 0 ,

(2.1.2) a

p

satisfying

every

the u s u a l

equation

(2.1.2), (2.1.0)

of

can be by

the

conditions

form

will

equation

[a , ~)

A

solution

to the r i g h t

zeros when

then

all

of

the

form

variable

~k

(2.1.3)

P

[0 , ~)

ISee A p p e n d i x

I,

of of

if

(2.1.0) a

,

there

is

said

an i n f i n i t e is

some

oscillatory

to b e

number

tO ~ ~

of

such

zeros

that

if it and

is

it has

no

t a tO

From that

into

satisfies

an equation

=

For

2.1.1:

non-oscillatory

see

T(t)

p

1.

(I.3.14) .1

DEFINITION

has,

take

into

in C h a p t e r

where

of i n d e p e n d e n t

t~-~

which

stated

(2.1.1),

transformed

the c h a n g e

p

the

if o n e

Sturm

solution

solutions Equation

oscillatory)

separation

are

is o s c i l l a t o r y

oscillatory

(2.1.0)

if all o f

theorem,

is

its

said

Theorem

1.2.0,

we

(non-oscillatory)

(non-oscillatory).

to be

solutions

oscillatory (non-

are

oscillatory (non-

oscillatory). Unless

otherwise

stated

we

shall,

in

the

following,

31

assume

that

O(t)

appearing

,

in

(2.1.0),

has

a limit

at

~

,

i.e. l i m o(t) t+~

exists

and

assume

i t is

has

the

(2.1.0)

is

same

zero

(for if w e

result

of H i l l e

tory behaviour

[31,

of

non-linear

THEOREM

2.1.1:

and that

~

the

condition

a solution,

~

for

= o(t)

T(~)

which

~(~)

we

- ~(~)

= 0

can

then

T

Moreover,

is r e p l a c e d

by

T )

of a w e l l - k n o w n

relates

the n o n - o s c i l l a of

solutions

of

a

equation.

and

with

~(~)

(2.1.0)

To

show

(2.1.4)

has

that

+

sufficiently

Proof:

(cf.,

the

a solution that

= o(t)

to the e x i s t e n c e

for

by

locally = 0

of b o u n d e d Then

a necessary

to b e n o n - o s c i l l a t o r y

is

equation

at infinity

implies

and

243]

(2.1.4)

integrable

then

T(t)

be r i g h t - c o n t i n u o u s

integral

limit

is an e x t e n s i o n

p.

v(t)

have

~

integral

satisfying

sufficient

this

let

if

(2.1.0)

certain

variation

as

unchanged

first

Let

Denoting

properties

remains The

theorem

finite.

(2.1.4)

v(t)

(2.1.5a)

v2(s)ds

large

t

,

which

is s q u a r e

[80]).

condition

is

v E L 2 (t O , ~)

sufficient ,

some

is r i g h t - c o n t i n u o u s ,

assume

tO locally

that

(2.1.4) of

32

bounded

variation

and

v(~)

= 0

Put

y(t)

Then

y(t)

is

locally

y' (t)

= exp

v(s)ds

absolutely

= v(t) e x p

(2.1.5b)

continuous

i

and

so

t v(s)ds

(2.1.6

to

everywhere,

as

jump

of

points

Letting

h

> 0

a two-sided v(t) ,

t

derivative,

which

are

the

except

same

as

possibly

those

of

the ~(t)

arbitrary,

exp y (t+h)-y(t) h

= y(t)

v(s) ds

-

[Jt

•

(2.1.7 h

Now

,!exr ){f [

~t

for

each

h

use

Theorem

> 0

v-i

,

= ~

fixed

t

H of Appendix

I to

i I t+h lira ~ h÷0 +

.

~t

v+2![3

Hence

we

find

that

v(s)ds

t

can

= v(t)

v

let

+ "'"

}

h ÷ 0+

(2.1.8

and

(2.1.9)

33

while

the o t h e r

continuity Hence

of

letting

terms

the

are

zero by v i r t u e

h ÷ 0+

the d e r i v a t i v e

derivative

which

(2.1.9)

and

the

integrals. in

(2.1~7)

y' (t)

where

of

is

we

obtain,

from

above,

= y(t) v(t)

is in g e n e r a l locally

(2.1.10)

understood

of b o u n d e d

as a r i g h t -

variation.

Thus

if

t > tO

y' (t)

- y' (to)

=

i

t dy' (s)

to

= ft 0

=

d(y(s)v(s) )

ft

vdy

+

ft

ydv

to

=

where

we h a v e

by

0

v dy +

y do -

to

0

equation

(2.1.10),

=

(vdy-

It 0

vy'ds)

+

ydo

0 Theorem (2.1.11)

K of A p p e n d i x vanishes

for

.

(2.1.11)

0

I now all

yv 2

implies t

that

and h e n c e

the

first

integral

in

34

y' (t) = y' (to)

it y d o

+

t > t =

0

to

so t h a t

y(t)

equation. for

is a p o s i t i v e

This

t ~ tO

and hence

To p r o v e non-oscillatory positive

for

For

implies

that

v(t)

(2.1.0)

of

the

has

above

a positive

the n e c e s s i t y solution

y(t)

we

suppose which

we

that

solution

can

(2.1.0) suppose

has

a

is

t ~ to

t => t O

we

set

is l o c a l l y

=

y' (t) y(t)

of b o u n d e d

(2.1.12)

variation

on

[t O , ~)

is r i g h t - c o n t i n u o u s .

Hence,

integral

is n o n - o s c i l l a t o r y .

v(t)

Then

solution

for

t > tO

v(t)

- v ( t 0) =

dv(s) 0

=

y(s) 0

dy'

(s)

ds

-

to

and

35

ft =

do(s)

-

v 2(s)ds

to

0

Hence

v(t)

= o(t)

- o(t0)

+ v(t0)

-

v

2

(2.1.13)

0

for

t ~ tO

that

the

Suppose,

Since

same

must

a(t)

be

true

if p o s s i b l e ,

square-integrable

has of

that

at

~

there

y' (t) is

a

< 0 t2

for such

v(~)

and

v(t2)

If w e

let

(2.1.13)

t , tO

(2.1.13)

shows

cannot

of

T => t 2

then by

T

be

that

(2.1.14)

because

- d(t2)

replaced

implies

v

= -~

when

+ a(T)

Then

(2.1.13)

t 3 = m a x { t 0 , t I , t2} with

~

= ~ ~ 0

so

t ~ tI that

at

v(t)

l i m v(t) t+~

Hence

a limit

(2.1.12).

Moreover

,

< -i

(2.1.15)

.

using , t3

(2.1.15)

in

respectively,

we

obtain

v(T)

< -i

+ ItT

y'(s) y(s)

v(s) ds

(2.1.16)

3 whenever Exercise

T => t 3 . i]

in

We

(2.1.16)

now

use

Gronwall's

to o b t a i n

inequality

[9, p.

37,

36

v(T)

<

-I

-

exp

ds

t3

-exp

=

ds t$

Y(t 3 ) y(T)

Thus,

by

(2.1.12) ,

y' (T)

for

all

T

positive now

~

t$

which

rewrite

(2.1.17) is

T

that

v(t)

the

2

t a

implies

Now

satisfies

that

y (t) v(~)

Hence

cannot

remain

=

We

0

can

as

- v(t)

t3

(2.1.17)

-Y(t3)

a contradiction.

(2.1.13)

V(T)

where

<

=

~(T)

-

letting

(2.1.4)

for

a(t)

T ÷

-

~

t >

v

in t3

2

(2.1.18)

(2.1.18) This

we

find

completes

proof.

THEOREM

2.1.2:

Let bounded a~ (~)

dI

variation =

0

,

, ~2 on

i = 1 , 2

be

right-continuous

[a , ~)

satisfying

functions (2.1.4)

locally

with

of

37

Assume

that

1oi(t) I __> 1o2(t) I

t__> t o

(2.1.19)

If

v 1(t) = 1o 1(t) l +

has

a solution

for

t >

v(t)

has

a solution

Proof:

We

(Appendix and

for

shall

= -+02 (t)

t ~

make

+

f;

v 2 ds

(2.1.21)

tO

use

of

the

Schauder

2.1.1).

With

the

usual

norm

consider

the

subset

x : {v ~

L2

where

vi(t) For

is v

we

(t 0,-)

as

~ X

(2.1.20)

then

Theorem

the

II,

tO

v21ds

Banach

: Iv(t) l __< v1(t)

in

(2.1.20).

we

define

fixed

an o p e r a t o r

point space

theorem L2(t0

, t__> to}

T

on

X

, ~)

(2.1.22)

by

oo

(Tv) (t)

= O 2 (t)

+

I

v 2 ds

t

If

e c

[ 0 , i]

and

x

, y

c X

,

t >

to

(2.1.23)

38

i~x+(1-~)yi

<__ ~ixl < c~iv I

+ (l-~)]yl + (1-C~)IVll

(2.1.24)

__< ivll and

hence

X

For

is v

convex.

6 X

,

co

i (Tv) (t) I __< 10 2 (t)

+ I v 2 ds t Co

lUl(t)

=

(2.1.25)

implies

We X

Let

IXnl

~ vI

now

that

show

(x n ) c X and

TX

that be

x • L

t vl ds

v i (t)

c X

T

such

t => t O

.

is

a continuous

that

xn ÷ x

map

where

2

fco - x2)ds xn

< =

IXn-X

I IXn + x l d s

to

:< Iix n - xii lix n ÷ xll

by

the

Schwarz

(2.1.25)

inequality.

:< ELxn - xll (liXnlL + Elxll)

from x e X

X

into Since

39

by Minkowski's

inequality.

2.1.26)

=< llxn - xll (llVlll + IIxll)

Letting

now

n ÷ ~

in

I

(2.1.26)

x 2 ds n

tO

The

same

argument

shows

÷

we

I

see

that

x 2 ds

.

2.1.27)

tO

that

oo oo It ~[2ds n -~ ~t x2ds

for

each

2.1.28)

t > t =

0

Hence

oo (Tx n) (t)

= 0 2 (t)

+ I

x 2 ds

t

n

~oo ÷ ~2 (t)

+

I x 2 ds t

df = (Tx) (t)

=

This

implies

n ÷ ~

that

ITx n - Txl 2 ÷ 0

,

each

0

t => t O ,

as

Moreover

I (Tx n) (t) - (Tx)(t) I 2 < 4v21

whenever

(2.1.29)

t > t

t ~ tO

Thus

the

Lebesgue

dominated

(2.1.30)

convergence

40

theorem

[24,

p.

ll0]

I

implies

- Txl 2 ÷ 0

~ITx

t

for

t

> t =

of

as

Hence

T

is

show

Appendix

II.

~ VI

and

that

TX

is

compact

(II.l.4)

is

satisfied

choose

EA

Corollary

since

if

x

II.1.2

e X

=

Vl 2

2.1.32

{t :

tO

-< A

-< t

large

< ~}

so

then

given

6

> 0

will

then

imply

,

we

that

2 vI < E

This

,

tO

sufficiently

A

use

,oo

tO

let

we

so

oo

we

(2.1.31)

continuous.

f Txt2< ] If

n ÷ ~

n

0

To

ITXl

that

(2.1.33)

that

oo

~I for To

all

x

prove

Since

c X

by

virtue

(II.i.6-7) vi

is

a

we

2

V 1 (t)

of

need

solution

(2.1.34)

<

(2.1.32). some

of

=

Txl 2

This

additional

proves

(II.1.5).

information.

(2.1.20),

{

IO 1 (t) I +

f2

V1

t

>

I~ 1(t) 12

t __> t o

(2.1.35)

41

and

so

2 01

e L

(t O , ~)

By

the

same

argument,

v I (t)

and

=>

t > t = 0

vI

so

~oo

VI e L

(2.1.36)

(t O , ~)

t

The

following

If

f

theorem

L P [ t 0 , ~)

[24,

,

p ~

p.

1

,

] will

also

be

useful.

then

llf(x+~)-f(x)ll

÷

as

o

h ÷ 0

(2.1.37)

P

Since

o I e L2[t0

(~2 c L 2 [t O , ~)

, ~)

and

account

of

have

from

(2.1.19)

that

thus

Ilo2(t+h)-

on

we

o 2 ( t ) ll + 0

as

h +

0

(2.1.38)

(2.1.37).

Similarly

if we

set

oo

V(t)

:

i

v~ t

then llV(t+h)

-V(t)

ll ÷ 0

as

h ÷ 0

(2.1.39)

42 because

of

Thus if

(2.1.36).

x ~ X , e > 0

to+h

to+h (2.1.40)

to

if

lhl < 6 ,

=

to

by the c o n t i n u i t y

of the integral.

This proves

(II.l.6). For

x e X ,

e > 0

II(Tx) ( t + h )

- (Tx)(t)ll = lla2(t+h)

- ~ 2 ( t ) + it x2 as II t+h (2.1.41)

+ II it

=< 11~2(t+h)-~2(t)ll

t+h

From

(2.1.38)

we can choose

h

so that if

lla2(t+h )_~2(t)l I < !2 Similarly

there

is a

62 > 0

~211 lhl < 61

"

such that w h e n e v e r

llV(t+h) - V ( t ) ll < _g 2 "

then

(2.1.42)

lh] < 62

(2.1.43)

Thus

t 2 II IIitt+h x211 <__IIIt+h vl

=

llv(t+h)-v(t)ll

< ! 2

(2.1.44)

43

Hence

if

lhl

< 6 = m i n {Id_

, 62}~

II (Tx) ( t + h )

which

proves

quently point

the

(II.1.7) Schauder

v = Tv

COROLLARY

for

and

then,

(2.1.21)

x c X

- (Tx) (t) ll < s

therefore

theorem

for any

TX

implies and

is c o m p a c t .

the e x i s t e n c e

this

completes

Conse-

of a f i x e d

the proof.

2.1.2:

Let

al(t)

~ 0

for

t ~ tO

and

~1(t) => la 2(t) l

t__> t o

(2.1.45)

If

v l(t)

has

a solution

for

= a l(t)

t ~ tO

v(t)

(2.1.46)

+ It~v I 2 ds

then

= ± °2 (t) +

(2.1.47)

v 2 ds t

has

a solution

Proof:

This

THEOREM

2.1.3:

With

for

follows

a l ' °2

t ~ tO

immediately

as a b o v e

from

and

the

theorem.

44

a i(t) suppose

__> I~ 2(t) I

(2.1.48)

t__> t o ,

that

f

y' (t) = c i +

t y ( s ) d a i(s)

(2.1.49)

z(s)da2(s)

(2ol.50)

a

is n o n - o s c i l l a t o r y .

Then

z'(t)

= c 2 -+

f

t

a

is n o n - o s c i l l a t o r y .

Proof:

This

is i m m e d i a t e

from Corollary

2.1.2

and

Theorem

2.1.1.

THEOREM

2.1.4: Let

~ (t)

satisfy

the

conditions

of T h e o r e m

2.1.1.

If

1 tJa(t) J < ~

then

(2.1.0)

is n o n - o s c i l l a t o r y .

Proof:

Let

Theorem

2.1.3.

equivalent

(2.1.51)

t > to > 0

ai(t)

= 1/4 t

This

and

o2(t)

is p e r m i s s i b l e

=- o(t)

since

and

(2.1.49)

apply is

then

to

y"

1 + y = 0 4t 2

(2.1.52)

4~

which

is a n o n - o s c i l l a t o r y

result

now

COROLLARY

Euler

equation

[59, p.

45].

follows.

2.1.3: (2.1.0)

is n o n - o s c i l l a t o r y

if

l i m sup tla(t) I < ! 4 " t÷+~

Proof: t

This

is i m m e d i a t e

is s u f f i c i e n t l y

THEOREM

The

since

(2.1.53)

(2 1.53)

implies

(2.1.51)

if

large.

2.1.5: Let

o I , 02

be as in T h e o r e m

of(t)

>

la2(t) I

If

(2.1.49)

is n o n - o s c i l l a t o r y

of

(2.1.49)

there

corresponds

2.1.2

and

t > tO

then

(2.1.54)

to e v e r y

a solution

solution

z(t)

of

y(t)

(2.1.50)

such t h a t

z(t)

Proof:

We

account

of T h e o r e m

If

y(t)

either

first note

_<_ ly(t) I

that

(2.1.50)

> 0

(2.1.55)

is n o n - o s c i l l a t o r y

on

2.1.3.

is a n o n - o s c i l l a t o r y y(t)

t > t*

or

y(t)

< 0

solution for

of

t > tI

(2.1.49) If

then y(t)

> 0 ,

46

Theorem

2.1.1

implies

that

t ~ t* = m a x { t 0 , tl} for

t ~ t*

Hence

(because

2.1.2

z(t)

> 0

guarantees

w e can r e c o v e r

solution for

has

(2.1.47)

of C o r o l l a r y

some non-oscillatory suppose

(2.1.46)

vl(t)

has a s o l u t i o n

2.1.2)

z(t)

t ~ t*

a solution

which

v(t)

corresponds

of

(2.1.50).

Since

the p r o o f

for

to

We can of T h e o r e m

that

Iv(t) I < vl(t)

t > t*

(2.1.56)

the n o n - o s c i l l a t o r y

solutions

y ,z

to find

that

z(t)

If

z(t)

< 0

other hand the a b o v e

for

if

_< y(t)

t > t*

y(t)

argument

< 0

the l a s t line for

shows

t > t*

t ~ tI

that there

(2.1.57)

is clear.

then

-y(t)

is some

O n the > 0

solution

and z(t)

such that

z(t)

This

completes

THEOREM

< -y(t)

t > t*

(2.1.58)

the proof.

2.1.6: Let

~ (t)

satisfy

the h y p o t h e s e s

of T h e o r e m

2.1.1.

If oo

f

t

o2(s)ds

< llo(t) I

t > tO ,

(2.1.59)

47

then

(2.1.0)

Proof:

By

is n o n - o s c i l l a t o r y .

Theorem

2.1.I

it

a solution

for

use

of

the

Schauder

fixed

Let

X

be

a subset

of

X =

sufficiently

suffices large

point

L2(t0

t

to

show

.

We

that

shall

(2.1.4) again

has

make

theorem.

, ~)

defined

{V £ L 2 (t O , ~) : Iv(t) - O(t) I <

by

lo(t)]

, t > to} (2.1.60)

For

v ~ X

we

define

a map

T

by

oo

(Tv) (t)

= o(t)

+ I

(2.1.61)

v 2 ds t

If

S e

[0 , i]

ISU+

and

u ,v e X

(l-s)V-O

I =

_<

,

IS(U-O)

~lu-ol

+ (i-S)(V-O)

I

+ (1-~)Iv-ol

i ~Io1 + (l-~)lol i lol This

shows

that

Moreover

X if

is

convex.

v ~ X

then

Iv(t) I __< 210(t) I

t > to

(2.1.62)

48 Hence co

i (TV) (t) - O(t) i = I

v 2 ds

t oo

< 4 ]

~2ds

t => t 0 ,

]~(t) l

t => t o ,

t

_-< 4 . ~

1

< i~(t) i

which

implies

in e x a c t l y make

use

that

the

of

TX c X

same w a y

(2.1.62)

instead

of

TX

to the

TX

by m a k i n g

procedure fore

is

can be

similar

omitted. of

a fixed

completes

the proof.

continuity

of

shown

to t h a t

point

2.1.2

by

T

is

shown

wherein

we

now

applying use

of

in T h e o r e m the

of

(2.1.63)

Iv(t) i ~ vl(t)

extensive

Consequently

existence

The

as in T h e o r e m

compactness set

.

t __> t O

v = Tv

Corollary (2.1.62).

2.1.2

Schauder of

The

and

theorem (2.1.61)

is

II.l.2 The there-

implies and

the

this

REMARK : We n o t e (2.1.49-50) priate

THEOREM

and

change

that the

in

~

can be

conclusion

replaced will

be

by

the

-~ same

in with

the

appro

(2.1.0).

2.1.7: Let

o(t)

satisfy

the h y p o t h e s e s

of T h e o r e m

2.1.1

and

49 (2.1.59) .

If

o(t)

> 0

then

z' (t) = c 2 +

will

be n o n - o s c i l l a t o r y

z(s)do2(s)

(2.1.64)

o2(s)ds

(2.1.65)

where

~oo

o2(t)

Proof:

Let

implies

that

o l(t)

= 4 J t

-- o(t)

Since

o2(t)

~i (t) > °2 (t)

Therefore Theorem and

Theorem

2.1.3

this

2.1.6

now implies

completes

COROLLARY

shows that

=> 0 ,

t > tO

that

(2.1.0)

(2.1.64)

(2.1.59)

(2.1.66)

is n o n - o s c i l l a t o r y .

is n o n - o s c i l l a t o r y

the proof.

2.1.4:

Let

o(t)

Then

(2.1.0)

has

for

t => t O ,

~ 0

satisfy

the h y p o t h e s e s

a non-oscillatory

ly(t) I =< exp

solution

{I2 t o(s)ds }

of T h e o r e m y(t)

such

2.1.7. that,

2 .i .67)

to

Proof:

The h y p o t h e s i s

implies

that

(2.1.0

is n o n - o s c i l l a t o r y .

50 The p r o o f solution

of T h e o r e m of

2.1.6

(2.1.4),

then i m p l i e s

such

(2.1.67)

and t h a t

o(t)

that

for

It is p o s s i b l e hold

implies

(2.1.62)

holds.

of This

v(t)

,

a

estimate

y

to r e m o v e

> 0

the e x i s t e n c e

the r e q u i r e m e n t s

in T h e o r e m

2.1.7

that

(2.1.59)

and then s t a t e

the

converse.

THEOREM

2.1.8:

Let With

~2

d(t)

defined

satisfy as in

the h y p o t h e s e s

(2.1.65)

suppose

of T h e o r e m that

2.1.1.

(2.1.64)

is non-

oscillatory. Then

(2.1.0)

solution

z

is n o n - o s c i l l a t o r y of

(2.1.64)

there

and for e a c h n o n - t r i v i a l is a s o l u t i o n

y

of

(2.1.0)

such that

0 < y(t)

< Iz(t)12exp

Id(S) Ids 1

for

t ~ tI

Proof: space

We use S c h a u d e r ' s L 2 ( t 0 , ~)

X = {v e

where

say.

v~ (t)

L2

fixed point

and a s u b s e t

X

theorem.

defined

by

i (t o , ~) : Iv(t) I < ~ vl(t) + Id(t)

is a s o l u t i o n

of the i n t e g r a l

Consider

, t > to}

equation

the

51

v(t)

= 4

c2(s)ds

+

v2(s)ds t

which fine

exists a map

by virtue T

of T h e o r e m

2.1.1.

For

v • X

we de-

by

oo

(Tv) (t) = c(t)

+ I

t > t

v2ds

=

t

As

in T h e o r e m

is c o n v e x .

If

v c X

2.1.2

a simple

0

calculation

shows

that

X

, t => t O ,

0o

f _< lo(t) l + f

l(Tvl (t) l _< l°(t) l +

v2 ds

t oo

1

2

{yv1+lol}

ds

t

=< 1~(t) i + 2 . ~ 1

r

Jt

v i2 d s

+ 2

o2ds

1

__< i~(t) 1 + y v i(t)

Hence TX

TX

c a n be

heavily

c X

The

shown

upon

This

which

means

T

and

as in T h e o r e m

1 Iv(t) l __< yv1(t)

Schauder's

v = Tv

analogously

of

the

compactness

2.1.2

of

relying

the e s t i m a t e

vEX:

Thus

continuity

theorem

implies

necessarily

that

(2.1.0)

+ la(t) i

that

satisfies

T the

has

t => t o

a fixed

latter

is n o n - o s c i l l a t o r y

point

inequality.

and we

can recover

52 an e v e n t u a l l y integral

positive

equation

such

0 < y(t)

solution

y(t)

of

(2.1.0)

f r o m the

that

<

Iz(t)]½exp

lo(s) Ids

}

i

2.1A

APPLICATIONS Let

and s u p p o s e

a(t) that

TO D I F F E R E N T I A L , b(t)

EQUATIONS :

be c o n t i n u o u s

functions

on

[a r ~)

the i n t e g r a l s

oo

exist,

B(t)

=

(2.1.69)

b (s)ds

sense.

results

y" + a ( t ) y = 0

(2.1.70)

z" + b(t) z = 0

(2.1.71)

are all c o n s e q u e n c e s

of s e c t i o n

2.1.

2.1.1A: Let

condition linear

(2.1.68)

the e q u a t i o n s

following

THEOREM

= ft a ( s ) d s

in the l i m i t i n g

Consider

The

A(t)

a(t)

for

integral

be as above.

(2.1.70)

Then

a necessary

to be n o n - o s c i l l a t o r y

equation

and s u f f i c i e n t

is t h a t

the n o n -

5~

i t

i

for s u f f i c i e n t l y

large

v(t) =

have

a solution

Proof:

This

letting

follows

a(t)

This where

it w a s

function

THEOREM

in

assumed

(2.1.4)

extends that

t

from Theorem

2.1.i

and n o t i c i n g

that

the t h e o r e m of H i l l e a(t)

satisfying

[62, p.

(2.1.72)

v 2(s)ds

upon (2.1.0)

is

(2.1.70).

result

v(t)

criterion

to

+

t

immediately

H A(t)

then e q u i v a l e n t

a(s)ds

> 0

[31, p.

The e x i s t e n c e

(2.1.72)

is r e m i n i s c e n t

243]

of a

of W i n t n e r ' s

375].

2.1.2A: Let

a(t)

, b(t)

IA(t) I >

be d e f i n e d

IB(t) I

as a b o v e

and s u p p o s e

that

t > tO

(2.1.73)

V~ ds

(2.1.74)

If co

V l(t)

=

IA(t) I + I t

has

a solution

for

t => t O

v(t)

a l s o has

a solution

for

then

= B(t)

+

t ~ tn

I

~ v2 ds t

(2.1.75)

54 Proof:

This

o i ( t ) E A(t)

COROLLARY

is i m m e d i a t e ,

~2(t)

from Theorem

2.1.2

upon

setting

~ B(t)

2.1.2A:

Let

A(t)

=> 0

A(t)

for

t => t O

=> IB(t) I

and

t => t O

(2.1.76)

v~ ds

(2.1.77)

If

v l(t)

has a s o l u t i o n

Proof:

Immediate

When by Hille

THEOREM (those

then

2.1.3A: stated

(2.1.75)

f r o m the

a(t)

[31, p.

> 0 ,

(2.1.70)

oscillatory.

+

has a s o l u t i o n .

theorem.

b(t)

Theorem

> 0

2 . 1 . 2 A was p r o v e n

245].

Let

a(t)

, b(t)

in the b e g i n n i n g

A(t)

and

= A(t)

__> [B(t) I

is n o n - o s c i l l a t o r y

satisfy

of this

the u s u a l

subsection).

t > to

then

(2.1.71)

conditions If

(2.1.78)

is a l s o n o n -

55

Proof: o2(t)

This H B(t)

As p.

he

(See a l s o

stated

~ B(t)

Hille's with

[31,

A(t)

p.

H A(t)

and

of

369,

theorem

exercise

general the

7.9].)

theorem

form

to T a a m

[60,

[63, p.

a(t)

~ 0

,

In fact,

than

(i.0.i).

to W i n t n e r

criterion

is due

the

The

257]

above

as

case

who

b(t)

~ 0

conditions.

If

extended along

~ B(t)

Let

a(t)

t

(2.1.70)

Proof:

245]

of(t)

2.1.4A:

THEOREM

then

more

is due

setting

2.1.3.

[25, p.

equations

~ 0

by

the p r e v i o u s

a slightly

considered

A(t)

verified

in T h e o r e m

it s t a n d s ,

495].

Taam

is r e a d i l y

This

setting

COROLLARY

o(t)

iit

satisfy

the

a(s)ds

< ~

usual

1

t > to > 0 ,

(2.1.79)

is n o n - o s c i l l a t o r y .

follows

immediately

from

Theorem

2.1.4

upon

- A(t)

2.1.3A : (2.1.70)

is n o n - o s c i l l a t o r y

l i m sup t÷~

t

a(s)ds t

if

< 1 4 "

(2.1.80)

56 Proof:

Set

o(t)

Wintner 2.1.4A

= A(t)

[62, p.

by r e p l a c i n g

Thus

he 3 -7

to

The

latter

number

2.1.4A

was

Corollary shown

that

[31, pp.

THEOREM

~ a(s)ds

lower

bound

seems

is b e s t

possible

by H i l l e can be

the b o u n d

t > to

(2.1.81)

1 -4

of

to be o p e n or not.

[31,

found

p.

in

appearing

in

as to w h e t h e r When

246].

[31,

in

appearing

p.

a(t)

For

a(t)

246]

where

(2.1.80)

is b e s t

(2.1.79) the

> 0

Theorem

> 0 , it is a l s o possible

2.1.5A:

the

a(t)

, b(t)

integrals

conditionally).

be

continuous

(2.1.68-69)

Suppose

A(t)

>

are

further

(2.1.70)

is n o n - o s c i l l a t o r y

of

(2.1.70)

there

corresponds

on

[a , ~)

convergent

and

suppose

(possibly

that

IB(t) I

If

such

< ~i

248-49].

Let that

theorem

with

question

2.1.3A

extends

t

the

proven

2.1.3.

essentially

I

=

improved

370]

(2.1.79)

3 < t 4

in C o r o l l a r y

t > tO

then

(2.1.82)

to e v e r y

a solution

solution

z(t)

of

y(t)

(2.1.71)

that

z(t)

=< ly(t) I

t > t*

(2.1.83)

57

Proof:

This

follows

in the p r o o f

The only

of T h e o r e m

conclusion

assume

sign.

The

Note

that

need

not be

(2.1.71)

latter

is due

the above where

a(t)

A(t) Thus

can be o b t a i n e d whenever

via

from

theorem

a(t)

to H a r t m a n

requires

so for

can be estimated.

the s u b s t i t u t i o n s

and W i n t n e r

(2.1.83), of

t

635].

b u t this

for solutions

under

of

the above

(2.1.70)

[26, p.

to

[26, p.

for large

estimates

(See for e x a m p l e

if we

is u n r e s t r i c t e d

> 0

the s o l u t i o n s

also holds

are k n o w n

or

636]).

2.1.6A: Let

the i n t e g r a l for

of

~ b(t)

(2.1.82)

2.1.5

2.1.3A.

a(t)

hypotheses,

THEOREM

from T h e o r e m

a(t)

be c o n t i n u o u s converges

A (t)

on

[a , ~)

(possibly

and suppose

conditionally).

that If

t => t O oo

I A21s)ds ¼1A(t l

(2.1.84)

t

then

(2.1.70)

Proof:

This

setting

o(t) When

[47, p. which

312]

states

is n o n - o s c i l l a t o r y .

follows

immediately

from T h e o r e m

2.1.6

upon

E A(t) A(t)

~ 0

the above

and e x t e n d e d that

(2.1.70)

a result

t h e o r e m was p r o v e n of W i n t n e r

is n o n - o s c i l l a t o r y

[62, p. if

by Opial 371]

58

Thus

in T h e o r e m

non-negative. the E u l e r

THEOREM

A 2(t)

=< ~1 a(t)

2.1.6A

A(t)

Equality

equation

in

t => t 0

is no l o n g e r

(2.1.84)

required

is a t t a i n e d

to be

in the case of

(2.1.52).

2.1.7A: Let

along with

a(t)

satisfy

(2.1.84).

If

the h y p o t h e s e s A(t)

> 0

of T h e o r e m

for l a r g e

t

2.1.6A

then

y" + 4A2(t) y = 0

(2.1.85)

is n o n - o s c i l l a t o r y .

Proof:

Refer

to T h e o r e m

Whether (2.1.85)

is,

we shall

discuss

COROLLARY

(2.1.68)

appears

2.1.7 w i t h

being

~ t)

-- A(t)

non-oscillatory

to be an o p e n q u e s t i o n

in s e c t i o n

implies

[59, p.

that

93] w h i c h

2.2.

2.1.4A:

Let

A(t)

Then

(2.1.68)

for

t => t O ,

has

> 0

and s u p p o s e

a non-oscillatory

ly(t) I __< exp

2

that

(2.1.84)

solution

A(s)ds

.

is s a t i s f i e d .

y(t)

such

that,

(2.1.86)

59 Proof:

This follows

from C o r o l l a r y

2.1.4 with

a(t)

H A(t)

T H E O R E M 2.1.8A: Let

A(t)

be d e f i n e d

as in

(2.1.68)

and suppose

z" + 4A2(t) z = 0

is n o n - o s c i l l a t o r y .

(2.1.87)

Then

(2.1.88)

y" + a(t)y = 0

is n o n - o s c i l l a t o r y (2.1.87)

that

and for each n o n - t r i v i a l

there is a solution

0 < y(t)

y(t)

of

< [z(t)12exp

solution

(2.1.88)

z(t)

of

such that

(2.1.89

[A(s) Ids l

for

t

sufficiently

Proof:

large,

say,

t -> t I

This is an a p p l i c a t i o n of T h e o r e m 2.1.8. The first part of the theorem is i d e n t i c a l w i t h a

t h e o r e m of H a r t m a n and W i n t n e r (2.1.89) where

is stronger

than the c o r r e s p o n d i n g

the absolute value

not appear.

[27, p. 216]

sign about

A(t)

though the e s t i m a t e estimate in

in

(2.1.89)

[27] does

Thus the first part of T h e o r e m 2.1.8 extends

Hartman-Wintner (2.1.0), while

r e s u l t cited

above

to e q u a t i o n s

the second p a r t extends

result only w h e n

o(t)

~ 0

of the type

the c o r r e s p o n d i n g

in T h e o r e m 2.1.8.

the

60 2.1B

APPLICATIONS

In 2.1

to

this

TO

DIFFERENCE

subsection

recurrence

relations

CnYn+ 1 +

where

cn

> 0

sequence,

n We

, =

we

n

=

EQUATIONS:

apply

of

the

the

Cn_lYn_

theorems

of

section

form

1 + bnY n

-1 , 0 , 1 , ....

=

(2.1.90)

0

(b n)

is

any

given

specified,

that

real

0 , 1 , ....

shall

assume,unless

otherwise

co

(2.1.91) 0 Cn-i

be

satisfied We

with

saw

jumps,

where

of

as

in

at

t_l

=

an

a

extra Chapter

fixed

a

and

tn

-

of

1

that

increasing

1 Cn_l

tn_ 1 -

upon if

the

o(t)

sequence

n=0

c

n

is

a

of

points

, 1 , ...

step-function t n)

,

2.1.92)

magnitude

~(tn)

for

condition

n= some

resulting [a , ~)

~(t n

0 , 1 , 2 , ... "extended" solution which

has

,

0)

then

recurrence is the

-b n

(2.1.0)

in

curve

that

if

2.1.93)

Cn-i

gives

relation

a polygonal property

n

rise the

y(t) we

write

to

sense

solutions that

defined Yn

the

on

H Y(tn)

'

61

then

the

sequence

currence

relation We

for m

given

> 0

(yn)

note

(2.1.90) that,

=

a solution for

n=

whenever

sequences

cn

, bn

the

three-term

re-

0 , 1 , ....

o(t)

is d e f i n e d

then

,

to

for

t c

by

(2.1.93)

[tm , t m + I)

,

,

(t) =

This

is

follows

from

o(a)

(2.1.93)

m - ~ 0

(2.1.94)

(b n + c n + Cn_ 1 )

and

the

relation

o(t n-

0)

O ( t n _ I)

THEOREM

2.1.1B: Let

~(~)

o(t)

exists

condition

for

and

be is

defined

zero.

(2.1.90)

as

Then

to b e

in

(2.1.94)

and

a necessary

and

non-oscillatory

is

assume

that

sufficient that

oo

v(t)

f

v 2 ds

(2.1.94),

have

= o(t)

+

(2.1.95)

t

where

o(t)

L2

infinity.

at

is

Proof:

This

results

of

REMARK

given

by

follows

Chapter

immediately

from

a solution

which

is

in

Theorem

2.1.1

and

the

is

to b e

oscillatory

i.

:

A solution

(yn)

of

(2.1.90)

said

62

if

the

and

sequence

exhibits

non-oscillatory

constant

sign.

theorem

shows

if,

The that

if

Moreover

the

transition

when

is

(2.1.0)

for

all

by

number

> N

,

the

version

solutions from

of

is

to

shows

(2.1.90)

with

defined

sequence Sturm

the

that

a

property.

a given and

retains

(non-

same

if

changes

separation

(2.1.90),

(non-oscillatory) of

sign

oscillatory

(2.1.0)

solution

of

the

inherit

(2.1.94),

is o s c i l l a t o r y

corresponding

n

a solution

then

given

infinite

discrete

oscillatory)

o

an

in

case

solution

only

is o s c i l l a t o r y

the

if

of

the

(non-

oscillatory). Thus,

y' (t)

~

= c +

f

as

in

(2.1.94),

t t c

y(s)do(s)

[a , ~)

,

(2.1.96)

a

is o s c i l l a t o r y

(non-oscillatory)

if a n d

only

CnYn+ 1 + en_lYn_ 1 + bnY n = 0

is o s c i l l a t o r y The Hille's

functions

latter

(2.1.97)

theorem

thus

gives

the

discrete

version

of

' gn

define

step

[31].

given °1

n = 0 , 1 , ...

(non-oscillatory).

theorem For

if

' ~2

sequences on

c

[a , ~)

n

> 0 by

,

b

n

we

setting

m 01 (t)

= a I (a)

- [

(b n + c n

÷

Cn_ 1

)

(2.1.98)

63

if

t e

[tm , tin+I)

,

m => 0

,

and

m ~2(t)

if

t ~

=

(~2(a)

[tm , tm+ l)

(2.1.92)

We

afortiori, With

°i

of T h e o r e m

Theorem

2.1.2B

latter

two

former

two,

omit

refer

to e i t h e r

corollary

we

shall

mean

THEOREM

2.1.3B: Let

both

exist

c

by

n

and

conditionally

and

denoted

respectively.

in the

same way

it s h a l l

2.1.2

satisfy

the d i s c r e t e

2.1.2

of T h e o r e m

lim m÷~

m [ 0

lim

m [

satisfy

be

2.1.2B

or its

by

Since as

the

the

understood or its

corollary

with

finite

further

(2.1.91).

Suppose

that

(b n + c n + Cn_ 1 )

(2.1.100)

(gn + c n + C n - i )

(2.1.101)

(so t h a t

convergent).

Suppose

also

n +

we o b t a i n

and

Theorem

cn

(2.1.98-99).

> 0

are

as

2.1.2B

them

we

given

÷ ~

n

stated

that when

o I , 02

the

and C o r o l l a r y

c a n be

shall

t

(2.1.99)

Cn_ 1 )

+

that

so d e f i n e d

2.1.2

results

(gn + c n

recall

and C o r o l l a r y

we

~ 0

so t h a t

' ~2

analogs

~

that

the

series

need

only

be

64 oo

oo

(c n + Cn_ 1 + bn)

m

for

m => m 0

>

(2.1.102)

[m (c n + Cn_ 1 + gn)

If

(2.1.103)

CnYn+ 1 + C n _ l Y n _ 1 + bnY n = 0

is n o n - o s c i l l a t o r y

then

(2.1.104)

CnZn+ 1 + C n _ l Z n _ 1 + gnZn = 0

is n o n - o s c i l l a t o r y .

Proof:

Define

(2.1.100-101) ~i(~)

~I

by

(2.1.98),

are then e q u i v a l e n t

, ~2 (~)

by an a d d i t i v e This

' ~2

exist

then i m p l i e s

to r e q u i r i n g

and be finite.

factor,

(2.1.99)

we can assume

respectively. that b o t h

Since we can alter that

a1(~)

these

= ~2(~) = 0

that

c~

Ol (a) = ~ (c n + C n _ 1 + b n)

(2.1.105)

0 oo

O2 (a) = ~ (c n + Cn_ 1 + g n )

(2.1.106)

0

Hence,

for

t e

[tm_ 1 , t m)

,

oo

Ol(t)

= [ (c n + C n _ 1 + b n) m

(2.1.107)

oo

02 (t) = ~ (C n + C n _ 1 + g n ) m

(2.1.108)

65

Thus

the r e q u i r e m e n t

is e q u i v a l e n t m

.

From

that

to the

the r e m a r k

oscillatory.

Hence

(2.1.48)

be

satisfied

for

large

t

requirement

that

(2.1.102)

hold

for

large

we

that

(2.1.49)

2.1.3

applies

see

Theorem

is n o n - o s c i l l a t o r y .

Consequently

oscillatory

completes

of

the

this

The

latter

theorem

Taam

result

[60].

extension [63]

and

of

the d i s c r e t e

and H i l l e

[31]

is

must

be n o n -

and hence

(2.1.104)

is a l s o

(2.1.50) non-

the proof. therefore

the d i s c r e t e

Simultaneously

it p r o v i d e s

version

theorem

(see T h e o r e m

of

the

2.1.3A).

Thus

analog an

of W i n t n e r for e x a m p l e ,

if

b

> 0 n =

n = 0 , 1 , ...

(2.1.109)

gn => 0

n = 0 , 1 ....

(2.1.ii0)

and

~bn= m

then be

(2.1.i04)

the

is n o n - o s c i l l a t o r y

formulation

of

(2.1.ili)

m > m=0 '

>~gn m

the d i s c r e t e

if

(2.1.103)

analog

is.

of H i l l e ' s

This

theorem

[31].

THEOREM

2.1.4B: Let

sequence

c

(b n)

n

> 0

and

assume

satisfy

that

(2.1.91).

(2.1.100)

For

exists.

a given If

would

66

1

li then

(2.1.90)

Proof: we

1 (c n + Cn_ 1 + b n)

We

shall

m > m0

<

(2.1.112)

is n o n - o s c i l l a t o r y .

define

o

by

(2.1.93).

Then,

for

t e

[tm_ 1 , t m) ,

have oo

(t) = ~

(2.1.113)

(C n + C n _ 1 + b n)

m

For

(2.1.51)

to h o l d

for

large

t

it

is

necessary

that

oo

t

for we

all let

t e

(c n + C n _ 1 + b n)

[tm_ 1 , tm)

t ~ t

in

m

when

(2.1.114)

a +

1

[ m

0

for

m ~ m0

t_l

= a = 0

satisfied

Without Then

for by

oscillatory

which

In t_l

b

n

= -i in

the we

Theorem

and

use

of

is w h a t

tn b

n

hence

- 2

we

(2.1.92)

+bn )

imply that

wished

when

n

for

all

we

obtain

Thus

large.

to o b t a i n

(2.1.115)

=<

we

can

that

assume

(2.1.51)

(2.1.96)

Consequently

case ----

(2.1.114)

sufficiently

generality will

2.1.4.

particular

by

and

(c n + C n _ l

loss

t

obtain

(2.1.90)

is

(2.1.115)

large

oscillatory

m

J =< l

that is

is n o n -

(2.1.90)

is

non-

n

and

to p r o v e .

c

n

= 1

n > 0

for By

all

replacing

a non-oscillation

criterion

for

the

equation

A2

The

latter

Yn-i

theorem

i:

Let

b

n

n=

that

li

bn

=

= 0

bnYn

shows

m

Example

+

(2.1.116)

< ~1

.

(2.1.116)

is n o n - o s c i l l a t o r y

m => m 0

-2

y(n+l)

0 , 1 , ...

if

(2.1.117)

n=0,1

....

in

(2.1.116).

Then ¥ A2yn_ 1 +

2 Yn

(2.1.118)

= 0

(n + i)

is n o n - o s c i l l a t o r y

if

1 y < ~

.

For

= m

mibn

~ m

Y (n+l) oo

< m y • Im x-2 dx

< =

my m+l

1 < -= 4

Consequently is

(2.1.117)

non-oscillatory

equation.

holds

where

m

with

1 y ~ ~

.

> 0 =

.

m0 = 0

and

thus

This

the

discrete

is

(2.1.118) Euler

88 Example

2:

Let

b n = y(-l)n/(n+l)

,

n=0

,1

Then A2 Yn-i

is n o n - o s c i l l a t o r y

V (-i) n ( n + l ) Yn = 0

+

(2 .i.119)

[y[ < ~1 .

if

co

For

[ bn

is c o n d i t i o n a l l y

m ~m bn

convergent

and

~m (n+-li (-i) n

=

mly]

<

IV]

<

I~l m + l

{ ( m +il )

Im +i2

I}

i

m+3

m

1 < -=

Consequently

if

m > 0

4

2.1.i17)

=

applies

and so

(2.1.119)

is non-

oscillatory.

COROLLARY

2.1.3B:

If limm÷~sup

then

Proof:

(2.1.90)

1

(Cn + Cn-Z + bn)

< 14

(2.1.120)

is n o n - o s c i l l a t o r y .

Follows

from C o r o l l a r y

2.1.3.

i.e.

(2.1.120)

implies

69 that

(2.1.i12)

holds

for l a r g e

m

In p a r t i c u l a r

if

oo

limm÷~Sup m m~ b n

then

(2.1.116)

(2.1.121) which

is n o n - o s c i l l a t o r y

extends

the s e r i e s

bn

Because

appear

t h a t the u p p e r b o u n d

but

there

exists

(2.1.116)

THEOREM

such

the s e q u e n c e s

2.1.3B

along with

c

n

in

conver-

it w o u l d

is b e s t p o s s i b l e ,

that e q u a l i t y

holds

, b

the h y p o t h e s e s

Define

solution

a solution

The r e s u l t

of(t)

then

z

of

n

£ zn ,

similarly

of

in

(2.1.121)

y

,

(2.1.103)

such

is

that

(2.1.122)

of T h e o r e m

2.1.5 w h e r e w e

and use

of

(2.1.103)

.

as in the p r o o f

from Theorem for

Yn

If

(2.1.104)

n => N

, o2(t)

follows

satisfy

n

z n =< lynl

z(t n)

(2.1.121)

(2.1.100-101-102).

then to e v e r y

corresponds

Proof:

section

427]

is o s c i l l a t o r y .

non-oscillatory there

(bn)

in

[32, p.

to be a b s o l u t e l y

in the n e x t 1 ~

(2.1.117)).

2.1.5B: Let

Theorem

the r e s u l t s

of

and L e w i s

is r e q u i r e d

gent.

i.e.

of

[because

a r e s u l t of H i n t o n for

(2.1.121)

< 14

(2.1.55)

2.1.3B.

set to o b t a i n

(2.1.122) .

THEOREM

2.1.6B: Let

the s e q u e n c e s

c

n

, b

n

satisfy

the h y p o t h e s e s

of

70 Theorem 2.1.4B. If, for

i=m

m => m 0 ,

i

j= +i

Co-1

= ~

(c i + ci_ 1 + b i)

i=m+l

(2.1.123) then

(2.1.103)

Proof: for

is non-oscillatory.

We define

o

t • [tm_ 1 , t m)

as in the proof of Theorem 2.1.4B. ,

~

is given by

o2(s)ds = t

(2.1.113).

o (s)dsi= -i

Then

Consequently

o (s)ds

i

tm-i (2.1.124)

Since

o

is constant on each

[tm_ 1 , t m)

,

m = 0 , 1 , ...

we

obtain oo

I

oo

2 (S) ds =

t

i=m-1

(ti+l- ti){ j= ~ +i

- ( t - t m _ I)

~ j m

(c J + cj-i +bj)

(cj + C j _ l + b

j)

(2.1.125)

Co

=

I i=m-1

G i2+ l 1

-

(t-

t m - i ) G m2

where oo

G i = j=i ~

(cj + cj_ 1 + b j)

(2.1.126)

71

Since

t e

neglected

[tm_ 1 , t m) and

o

the

(s)ds

=<

t

so

and

. 1

i=m-i 1 =< ~

t c

term

in

(2.1.126)

c a n be

so

t

since

second

IGml

[tm_ 1 , t m) (2.1.0)

i+l

1 = ~

Thus

t e

[tm_ 1 , t m)

(2.1.127)

(2.1.128)

[°(t) I

(2.1.59)

is n o n - o s c i l l a t o r y .

is s a t i s f i e d

for

large

Consequently

(2.1.103)

is n o n - o s c i l l a t o r y .

From 2.1.4B We

we

the d i s c u s s i o n

can o b t a i n

therefore

find

following

the p r o o f

a non-oscillation

that

of T h e o r e m

criterion

for

(2.1.116).

if

(2.1.129) i=m

for

m ~ m0 ,

follows

from

(2.1.129) be d e f i n e d

= +i

then the

is b e s t

3

(2.1.116)

above

is n o n - o s c i l l a t o r y .

theorem.

possible.

To see

b

1 n

0

then

numerical this

let

bound

the

in

sequence

b

n

as

0

We

The

This

notice

that we have

n=0 (2.1.130)

n=l n>2

equality

in

(2.1.129)

when

m = 0 .

72

A simple

computation

corresponding the

lower

now

to the

shows

initial

that

values

(2.1.116)

THEOREM assume

solution

Y-I

= 0

,

of

(2.1.116)

Y0 = 1

admits

bound

Yn => n - i

Hence

the

is n o n - o s c i l l a t o r y .

2.1.7B: that

n => i

With

the

(2.1.123)

c

n

, b

n

is s a t i s f i e d

as in T h e o r e m for

large

m

2.1.4B

.

If

co

(t)

-

[ i=m

(C i + C i _ 1 + b i) > 0

,

t c

[tm_ I , t m)

(2.1.131) then

the

differential

equation

z" + 4~ 9 (t) z = 0

(2.1.132)

is n o n - o s c i l l a t o r y .

Proof:

This

COROLLARY

suppose

oscillatory

immediately

from

Theorem

2.1.7.

2.1.4B:

Let and

follows

c

n

that

, b

n

satisfy

(2.1.131)

solution

(yn)

the h y p o t h e s e s

holds. such

Then that

of T h e o r e m

(2.1.90)

for

n ~ N

has

2.1.7B

a non-

,

t lynl

< e x p { 2 IT n o ( s ) d s }

(2.1.133)

73 where

o

is as in

Proof:

Follows

THEOREM

2.1.8B: Let

(2.1.131).

from Corollary

o(t)

be as in T h e o r e m

n e e d n o t be n o n - n e g a t i v e .

Assume

z" + 4o

is n O n - o s c i l l a t o r y .

2.1.4.

2

2.1.7B

except

that

that

(t) z = 0

(2.1.134)

Then

(2.1.135)

CnYn+ 1 + Cn_lYn_ 1 + bnY n = 0

is n o n - o s c i l l a t o r y (2.1.134)

there

and for e a c h n o n - t r i v i a l

is a s o l u t i o n

(yn)

of

solution

(2.1.135)

z

such

of that

t 0 < Yn <

JZnJ2exp ~ If

n Jo(s) Ids } T

where

Proof:

z n ~ z(t n)

Follows

and

and 2.1.8B For g i v e n

(2.1.135)

will

is g i v e n

from Theorem

W e can t h e r e f o r e 2.1.7B

~

by

(2.1.131)

2.1.8.

summarize

the r e s u l t s

of T h e o r e m

as follows: sequences

c

n

be n o n - o s c i l l a t o r y

, b

n

satisfying

(Theorem

(2.1.123),

2.1.6B).

If,

in

74

addition, will

we

that

be n o n - o s c i l l a t o r y

2.1.7B

gives

differential recurrence Theorem

relation

2.1.8B

recurrence

equations

relation

of

oscillation

THEOREM

the

the

will

ensure

converse.

That

is n o n - o s c i l l a t o r y

will

also

then

(2.1.132)

Thus

Theorem

that

a certain

if a c e r t a i n

related

On

hand,

the

is, then

other

if a c e r t a i n a related

be n o n - o s c i l l a t o r y .

CRITERIA: section form

theorems

we

continue

(2.1.0)

and

in s e c t i o n

the

investigation

complement

2.1 w i t h

some

of

oscillation

of the n o n theorems.

2.2.1: Let

suppose

2.1.7B).

is n o n - o s c i l l a t o r y

equation

In this

which

is s a t i s f i e d

is n o n - o s c i l l a t o r y .

gives

OSCILLATION

(2.1.131) (Theorem

a criterion equation

differential

§2.2

assume

satisfy

the

conditions

of T h e o r e m

2.1.1

and

that

o(t)

=> 0

t => t O

(2.2.1)

to(t)

1 < ~

t > tO

(2.2.2)

If

then

If

(2.1.0)

e > 0

is n o n - o s c i l l a t o r y .

and

to(t)

1 > ~ + e

t > tO

(2.2.3)

75 then

(2.1.0)

Proof:

The

is o s c i l l a t o r y .

first part

p a r t we w r i t e

~i (t) -- ~(t)

then i m p l i e s

2.1.4.

To p r o v e

the s e c o n d

and

ill 1

a2(t)

(2.2.3)

is T h e o r e m

- ~ + ~

(2.2.4)

t __> t o

that

a 1(t)

> a 2 (t)

z'(t)

= c +

t > tO

(2.2.5)

z(s)dJ2(s)

(2.2.6)

Furthermore t

s

a

is o s c i l l a t o r y

since

it is e q u i v a l e n t

to

(2.2.7)

and

the l a t t e r

Since

(2.1.0)

equation must

is o s c i l l a t o r y

either

c a n n o t be n o n - o s c i l l a t o r y that

(2.2.6)

is o s c i l l a t o r y

THEOREM

2.2.2:

Let

be o s c i l l a t o r y

a

satisfy

which

e > 0

[59, p.

45].

or n o n - o s c i l l a t o r y

for then T h e o r e m

is n o n - o s c i l l a t o r y

(2.1.0)

for

2.1.3 w o u l d

is i m p o s s i b l e .

a n d the t h e o r e m

is p r o v e d .

the h y p o t h e s e s

of T h e o r e m

imply Thus

2.1.1

and

it

76

assume

that

(2.2.1)

holds.

If

o2(s)ds

__< ~ o(t)

t > tO

(2.2.8)

t

then

If

(2.1.0)

~ > 0

is n o n - o s c i l l a t o r y .

and

It 2

then

(2.1.0)

Proof:

The

(s)ds

>

first

part

under

is T h e o r e m

(2.2.9)

it is n o n - o s c i l l a t o r y . such

Then

then

we

2.1.6.

To p r o v e

assume,

there

on

exists

the

that

contrary,

a solution

(2.1.0) that

y(t)

of

that

y(t)

This

(2.2.9)

is o s c i l l a t o r y .

is o s c i l l a t o r y

(2.1.0)

t > tO ,

+ c o(t)

implies

that

> 0

the

t __> T

integral

.

(2.2.10)

equation

~oo

v(t)

admits

a solution

non-negative

for

= ~(t)

+ I t

v 2 ds

v(t)

for

t > T

t > T

and

v(t)

v(t)

=> o(t)

t > T

This + 0

t __> T

as

solution t ÷ ~

(2.2.11)

v(t)

is

Moreover

(2.2.12)

77 Set

(2.2.9)

then implies that

oo

I

g2(s)ds > ~ ( t )

t => t O

(2.2.13)

t NOW,

oo

v(t) => g(t) + I

°2(s)ds

t => T

(2.2.14)

t

> (l+e)g(t)

- elo(t)

t > T .

(2.2.15)

We may take it that oo

I 2(s)d s < t since

v E L9

and

Using

(2.2.15)

in

v > g

for large

t

(2.2.11) we obtain

oo

v(t)

> g(t) + e~ I

gg(s)ds

(2.2.16)

t

2 > g(t) + 6~i~g(t) 2 > (i +a1~)O(t)

~2o(t)

t ~ T

(2.2.17)

Repeating the above process we find that

v(t)

> e d(t) =

n

t > T =

(2.2.18)

78

where 2 i + aen_ I

an

is i n d e p e n d e n t (en) must

For

of

t

is i n c r e a s i n g be

if

A

n => 2

simple

and hence

induction

tends

B < ~

then,

1 e > ~

since

letting

n ÷ ~

bounded

at every

by

solution

shows

can

such

B

(2.2.18)

point of

v

we

limit

shows ~

that

which

t

where

on

finite

must

theorems for

of T h e o r e m

v(t)

B = ~ must

~ 0

This This

Thus

be

un-

contradicts contra-

a non-oscillatory

be oscillatory•

therefore

we

hold

in T h e o r e m

2.1.7

therefore

complement

Theorem

in T h e o r e m

that

have

(2.1.0)

2.2.2

Hence

intervals.

(2.1.0)

conditions

(2.2.20)

o(t)

and hence

previous

2

find

cannot

because

THEOREM

to s o m e

can e x i s t .

(2.1.0)

sufficient"

(2.1.59)

argument

(2.2.19),

that

The

Thus

no in

the b o u n d e d n e s s diction

(2.2 19)

infinity.

= 1 + aB

and

,

give

"e-necessary

to b e n o n - o s c i l l a t o r y .

see

that

is n o t

too

2.1.8

the

condition

restrictive.

with

the

result

that We stated

2.1.7.

2.2.3: Let

o

be

right-continuous

and

and

locally

of bounded

79

variation

on

[a , ~)

If

lim t+~

then

(2.1.0)

Proof:

t => t O 2.1.1

on

the

solution If

we

(2.2.21)

= -~

is o s c i l l a t o r y .

Suppose,

oscillatory

o(t)

v(t)

contrary,

of

(2.1.0)

= y' ( t ) / y ( t )

that

with ,

y(t)

say

t >

is

y(t)

tO

,

a non-

> 0

then

for from

Theorem

have

v(t)

= o(t)

- ~ ( t 0)

+ v ( t 0)

.t v 2 ds

-

t > tO

to (2.2.22)

< ~(t)

We

can

then

obtain

that

- o ( t 0)

proceed

to

the

v(t)

Arguing

then

diction

(2.1.17).

and

this

as

proves

During

in

the

+ v ( t 0)

limit

t > tO

as

÷ -~

t + ~

in

(2.2.23)

t + ~

Theorem

2.1.i,

Thus

non-oscillatory

no

(2.2.23)

(2.2.24)

to

(2.2.24)

leads

to

solution

the

contra

can

exist

theorem.

the writing

of these

notes

there

appeared

a

80 paper of Reid

[50, p. 801] who also,

independently,

proved

T h e o r e m 2.2.3.

2.2A

APPLICATIONS

TO D I F F E R E N T I A L EQUATIONS:

T H E O R E M 2.2.1A: Let and suppose

a(t) that

satisfy the c o n d i t i o n s A(t)

~ 0

(where

A(t)

of T h e o r e m 2.1.6A is d e f i n e d in

(2.1.68)>. If tA(t)

then

If

(2.1.70)

E > 0

(2.2.25)

is fixed and

(2.1.70)

Proof:

t => t O

is n o n - o s c i l l a t o r y .

tA(t)

then

1 =< ~

1 => ~ + s

t > tO

(2.2.26)

is oscillatory.

This is a c o n s e q u e n c e

of T h e o r e m 2.2.1 where

o(t)

A(t) The first part of this t h e o r e m is due to W i n t n e r p. 260]

and the second part follows a l m o s t i m m e d i a t e l y

this r e s u l t

(see

[44, p. 131],

[63, p. 259]).

[63,

from

81

THEOREM

2.2.2A: Let

and

suppose

a(t) that

satisfy A(t)

the

conditions

of T h e o r e m

2.1.6A

=> 0

If A 2 (s)ds

< ~ A(t)

t > to

(2.2.27)

t

then

If

(2.1.70)

is n o n - o s c i l l a t o r y .

~ > 0 ,

5

A2(s)ds

=>

+~

t

then

(2.1.70)

Proof:

This The

THEOREM

A(t)

t > tO

(2.2.28)

is o s c i l l a t o r y .

follows above

from Theorem

theorem

is d u e

2.2.2. to O p i a l

[47,

p.

309].

2.2.3A: Let

(2.1.68)

1

a (t)

be c o n t i n u o u s

on

[a i ~)

and

suppose

that

exists.

If co

I

a(s)ds

a

then

(2.1.70)

is o s c i l l a t o r y .

= ~

(2.2.29)

82

Proof:

We

let

The the p.

case 115]

§2.2B

latter

when for

= -

a(s)ds

theorem

a(t)

> 0

general

APPLICATIONS

THEOREM

was

in T h e o r e m

proven

and w a s

2.2.3.

by F i t e

extended

[19, p.

by Wintner

347]

in

[61,

a(t)

TO D I F F E R E N C E

EQUATIONS:

2.2.1B: Let

2.1.4B

o(t)

and

of T h e o r e m

the

c

assume

, b

n

satisfy

n

further

2.1.6B,

that

G

the h y p o t h e s e s ,

m

is n o n - n e g a t i v e

defined

for

of T h e o r e m

in the p r o o f

m ~ m0

If

i }Z

(C n + C

0 Cn-i

then

(2.1.90)

1 <= ~

+ b n)

n-i

m => m 0

(2.2.30)

is n o n - o s c i l l a t o r y .

If

1 0 Cn-I

• m

1 => ~ + e

(c n + en_ 1 + b n)

m >m =

0

(2.2.31)

where

Proof: An

e > 0

The

argument

latter

is

first

fixed,

part

similar

theorem

shows

then

(2.1.90)

is a c o n s e q u e n c e

to the one that

used

(2.2.3)

is o s c i l l a t o r y .

of T h e o r e m

in the p r o o f

is e q u i v a l e n t

of to

2.1.4B. the (2.2.31).

83

As

COROLLARY

a consequence

this w e

obtain

in p a r t i c u l a r ,

2.2.1B:

Let convergent

of

be a n y

(bn)

sequence

whose

series

is c o n d i t i o n a l l y

and co

b n => 0

(2.2.32)

m => m 0

m

If co

m

< 1 [ bn = ~

m > m0

(2.2.33)

m

then

If

(2.1.116)

s > 0

is

is n o n - o s c i l l a t o r y .

fixed

and

CO

m

[ b n => ~ +1s

m>

m0

(2.2.34)

m

then

(2.1.116)

Proof: 2.1.4B

Example

This

is o s c i l l a t o r y .

follows

as a p p l i e d

l:

from

the

to T h e o r e m

The d i s c r e t e

discussion

Euler

equation,

Y

+

=

2 Yn

0

(n + i)

is o s c i l l a t o r y

whenever

Theorem

2.2.1B.

Z~2

Yn-i

following

y > ! 4

This

is b e c a u s e

(2.2.35)

84 oo

oo

m [ m

Y (n+l)

>my 2 =

> =

x

m m+l

y

1 > ~-+

for

e > 0

some

Consequently

if

the

m

is

above

2 dx

+1

e

(2.2.36)

sufficiently

corollary

large

implies

that

1 y > ~

since

(2.2.35)

.

is

oscillatory. Using comparison (b n)

is

the

discrete

equation

we

a positive

(2.1.116)

2b

b n => gn

=> 0

to a c o n t r a d i c t i o n Similarly

(2.1.116)

deduce such

and if it

the

=

(2.2.35)

following

that,

> 1 n = 4 + s

for

n => n O

For

if w e

as

(2.1.116) can be

2.1.3B was

shown

< 1 = ~

is n o n - o s c i l l a t o r y .

If

fixed

s > 0

,

(2.2.37)

,

let

(2.2.38)

would

assumed

that

a

result.

[i + e] ( n + l ) - 2

Theorem

(n+l)2bn

then

equation

is o s c i l l a t o r y .

gn

then

can

sequence

(n+l)

then

Euler

if

n > nO

lead

immediately

non-oscillatory. b n =>

0

and

(2.2.39)

85 THEOREM

2.2.2B: Let

(c n)

satisfy

, (b n)

the

hypotheses

of

Theorem

2.1.4B.

If

for

m => m 0

, co

1 G2 c-- i+l i=m 1

then If

(2.1.103)

for

(2.2.40)

is n o n - o s c i l l a t o r y .

s > 0

fixed

and

(2.1.103)

m ~ m0 ,

I----G2 c. i+l z

i=m

then

1 < 4 Gm+l

is o s c i l l a t o r y

> =

+ s Gm

,

(2.2.41)

where

co

Gm = i=m[

Proof:

The

follows

from

holds

first the

whenever

(2.1.126)

we

part proof

t ~

(ci + c i - I

is T h e o r e m of

the

+ bi)

2.1.6B

latter

[tm_ 1 , t m)

(2.2.42)

while

theorem.

Thus

letting

find

co

oo

I 21slds > t

= m-i

i

i+l

m oo

=fiG

c.

m

1

2

i+l

i

G2

Cm_ 1

m

the

second

For

(2.1.126)

t ÷ tm - 0

part

in

86

since

o(t)

= Gm

when

t e

[tm_ 1 , tm)

Thus co

t E [tm_ 1 , t m) t and thus T h e o r e m

2.2.2

applies.

Consequently

(2.1.103)

is

the discrete

analog of Opial's

oscillatory. The latter

theorem gives

theorem

(see Theorem

THEOREM

2.2.3B: Let the

Theorem

(cn)

2.2.2A).

, (bn)

satisfy

the hypotheses

of

2.1.4B.

If oo

(2.2.43)

(Cn + C n _ 1 + b n) = co 0

then

(2.1.103)

Proof:

is oscillatory.

We define

o(t)

as in

is a s t e p - f u n c t i o n

with

find that

is then e q u i v a l e n t

Theorem

(2.2.43)

2.2.3 applies

oscillatory.

jumps

(2.1.93) at the

and c o n s e q u e n t l y

remembering (tn)

to

From

(2.2.21).

(2.1.103)

is

that

o(t)

(2.1.94) Hence

we

87

The p.

426]

theorem

and extended,

by R e i d

§2.3

latter

in the

shown

same

In this gives

equation

THEOREM

section

we

a necessary

continuable

solutions

extend

and

of

t e

class

[a , ~) tion

of a s e c o n d

the p r o o f

the

[a , ~) of all

whose

on

[32, 2.2.3,

in

[8]

first

for

nonlinear shall

to e q u a t i o n s

[8, p.

differential

be m a i n l y of

of

the

an

form

derivatives

(2.3.1)

(2.3.1)

absolutely are

75]

all

f(y(s))dq(s)

A solution locally

of B u t l e r

condition

order

Our proof

CASE:

is a g a i n

continuous locally

sought

functions

of b o u n d e d

in

on

varia-

[a , ~) We

case

& Lewis

as T h e o r e m

a result

sufficient

y' (t) = c -

the

direction

IN THE N O N L I N E A R

to be o s c i l l a t o r y .

adaptation

where

by H i n t o n

[50].

AN OSCILLATION

which

was

of

shall

(2.3.1) :

be m a i n l y

concerned

It is c h a r a c t e r i z e d

with by

the the

"superlinear" convergence

of

integral

I Equations

of

"Emden-Fowler

the

form

type",

-+~

(2.3.1) i.e.

dt f (t)

include

those

(2.3.2)

those

equations

which

with

are of

88

In

the

in

f(y)

= y

case

of

given

as

(2.3.3)

first

characterization

2n+l

n=

ordinary

the

result of

1 , 2 .....

differential

equations

of A t k i n s o n

oscillatory

(2.3.3)

[2, p. 643]

solutions

of

with gave

f the

the

equation

y"

when

p(t)

tion [8]

on

> 0

the

+ p(t)y 2n+l

and

= 0

continuous,

coefficient.

This

n=l

in has

, 2 , ...

terms

of

,

an

recently

(2.3.4)

integral

been

condi-

generalized

to e q u a t i o n s

y"

where into prove

p(t)

is u n r e s t r i c t e d

a "superlinear" later

sufficient

give,

condition

for

to be

oscillatory.

crete

analog

the

+ bnf(Yn)

As

for

42 Yn-i

k

sign The

(2.3.5)

and

= 0

theorem

f

result

difference

a corollary

> 1

= 0

in p a r t i c u l a r ,

of A t k i n s o n ' s

then,

to

equation.

on will

2 A Yn-i

positive

+ p(t) f(y)

which

a necessary

= 0

shall and

0 , 1 , ...

shall

[2],

i.e.

obtain If

,

~ 2k+l + mnYn

we

(2.3.5)

equation

n=

we

turns

n=0

, 1 , ...

(2.3.6)

the (bn)

disis

89

has

a non-oscillatory

solution

if a n d o n l y

if

oo

< co [nb n 0

In the so

that

f oy

of

(2.3.1),

absolutely

following

continuous

has meaning

THEOREM

2.3.1: o

such

y

is a s o l u t i o n

that

case

the

f

be o n l y

integral

integral.

function,

locally

yf(y)

exists,

further

> 0

that

for all

y ~ 0

f' (y) > 0

i

dt f(t)

-i

dt (t----~ f < ~

--co

b)

l i r a inf T+co

c)

I

T | P(s)ds Jt

P2(r)drdst

s

>-~

< ~

of

(2.3.8)

and

i

in

that

Suppose

and

f ~ C' (-~ , co)

when

In a n y

f

infinite.

y ~ 0

76].

T - lira do(s) T÷~ t

f e C' (-co , ~) if

[8, p.

that

to r e q u i r e

as a S t i e l t j e s

and

P(t)

a)

continuous

be a right-continuous

variation

and may be

assume

it is p o s s i b l e

(2.3.1)

bounded

shall

is a b s o l u t e l y

though

Let

we

(2.3.7)

for all

where

P_(t) +

t

= max{~P(t)

, 0}

90

Then a necessary (2.3.1),

and sufficient

continuable

over

condition

the half-axis,

for all solutions to be o s c i l l a t o r y

of is

that P(s)

+

P2(r)dr

t

Note:

We shall prove

showing

that

ds

= +~

(2.3.9)

s

(2.3.9)

the sufficiency with

P2(r)

of

(2.3.9)

replaced

by

by first

P+2(r)

will

F

imply

that all solutions

oscillate

Isince

(2.3.9)

and

(c)

k

along with

the relation

I t Proof:

p

+

p2 (t)

2(r)dr ds = + ~

Assume,

some n o n - o s c i l l a t o r y Isince

+

imply

(2.3.10)

@

s

(Sufficiency)

positive

2 = P+(t)

P2(t)

solution

-y(t)

on the contrary, y(t)

that there is

which we can take to be

is also a solution I .

Thus y(t)

> 0

t > tO

(2.3.11)

We let !

g(t)

where

the prime

Then

g(t)

variation

= f~y(t) (t)]

represents

t > tO

in general

shall be r i g h t - c o n t i n u o u s on

(2.3.12)

a right-derivative. and locally

of b o u n d e d

(t o , ~)

An integration

by parts

shows

that,

for

tO ~ t ~ T ,

91

IT i tfl.y,s,, [ ~ J)l dy' (s) = g(T)

ds (y')2 + ]t f' (y) f(y) 2

- g(t)

(2.3.13)

where we have omitted simplicity. (2.3.1)

the v a r i a b l e s

Moreover,

shows

an a p p l i c a t i o n

T 1 t f(Y)

combining

g(t)

of the i n t e g r a l

for equation

that

I Hence

in the i n t e g r a n d

(2.3.13-14)

= g(T)

+ O(T)

IT dy'

=-

we

t

d~

(2.3.14)

.

find

- o(t)

+

j.T f'

< (y)if(y)!

2 ds

(2.3.15)

t t O =< t < T .

whenever

Our basic

assumption

leads

us to two

cases:

Case t > t =

I)

lim sup

P(s)ds

= +~

some

II)

lim sup

P(s)ds

< ~

all

I:

(2.3.16) If there

implies

that

is a s e q u e n c e

g(Tn)

then for

n

sufficiently

t => t O

the r e l a t i o n

0

T

n

+ ~

t => t O

is v a l i d

such

(2.3.17)

for all

that

> 0

l a r g e we

(2.3.16)

(2.3.18)

s h a l l have,

for

g2

t O <- t -< T n T g(t)

= g ( T n)

-

+ O ( T n)

o(t)

+

I n --

f' (y)g

2

ds .

(2.3.19)

Jt

Hence T

g(t)

in

>

T

do

in

+

t

so lim on

that

letting

sup

account

of

If n o

and

n ÷ ~

of both

we

sides

ds

of

obtain

g(t)

> P(t)

the

latter

we

Taking

obtain

the

a contradiction

(a). such

sequence

exists

then

we must

have

g(t)

< 0

t => t I

(2.3.21)

y' (t)

< 0

t => t I

(2.3.22)

-

do

(2.3.23)

that

P(t)

> 0

all

such)

so

(2.3.15)

now

implies

g(T)

Moreover large of

(2.3.20)

f' (y)

t

(2.3.16) t

some

=< g(t)

implies

(not n e c e s s a r i l y t2 ~ ti

such

for

which

arbitrarily

shows

the

existence

that

.T do

> 0

T >

t2

(2.3.24)

t 2

if w e

assume

that

P(t)

< ~

for

large

t

.

We

note

that

93

P(t)

= ~

for

some

P(t)

= ~

for

all

g(t)

÷ -~

proceed

as

t

if a n d

larger

t ÷ ~

t

only

.

if

This

because

of

~(~)

would

= ~

then

(2.3.15).

and

thus

imply

that

And

we

can

then

as b e l o w . Hence

by

(2.3.23)

g(T)

Replacing

t

by

t2

for

T > t2 ,

< g(t2)

in

- -K

(2.3.15)

and

(2.3.25)

< 0

using

( 2 . 3 . 2 4 - 25)

we

obtain

g(T)

< -K

+ IT

=

f' (y) IY' I g ds .

t

(2.3.26)

f(Y)

2 If w e

write

t => t 9

and

~(t)

=

f' [y(t)] IY' (t) [ flY(t)]

~(t)

> 0

for

so

g(T)

|,T + I ~(s)g(s)ds

<-K

2

An

then

application

of

the

Gronwall

(2.3.27)

.

t 2

inequality

to

(2.3.27)

then

gives

g(T)

<-K

f(y(T)]

T > t2

(2.3.28)

T __> t 2

(2.3.29)

i.e.

y' (T)

and

the

latter

<

implies

-K f l Y ( t 2 ) )

that

y(t)

cannot

remain

positive

for

94

large

Case

t

which

II:

In

contradicts

this

case

(2.3.11).

it is n e c e s s a r y

oo

oo

t

s

that

(2.3.30)

and

that

there

be

an

Mt > 0

I T P(s)ds t

We

now

such

proceed

that

as

holds

that

[ < Mt

in C a s e

(2.3.18)

such

I. for

If

T > t

there

large

n

.

(2.3.31)

is

a sequence

we

find

T

from

n

+

(2.3.20)

that

g

Now

either

i)

ii)

i)

Let

y(t)

2

2 > P+(t)

(t)

y(t)

=> 6 > 0

there

is

y ( t n)

+ 0

=> ~ > 0

t > t3 .

,

t => t 3 ,

(tn)

such

t => t 3

that

then

implies

or

tn

co

f

Since

C = i n f { f ' (u) : 6 < u < ~}

(2.3.20)

(2.3.32)

> 0

(2.3.33)

that

oo

g(t)

> P(t)

+ c I

g2 ds t

(2.3.34)

95

g(t)

Integrating T ÷ ~

sides

+ c It P+2 (r) dr

over

[t, T)

w e get a c o n t r a d i c t i o n

integral

ii)

both

> P(t)

of the r i g h t

Let

t

For large

n

+ ~

n

side of

be s u c h

and

t

because

of

(t)

(a)

y(t

n

lim sups

s i n c e by h y p o t h e s i s

as the

is d i v e r g e n t .

) + 0

t => t 3 ,

t

>

f(s)

(2.3.201),

and t a k i n g

(2.3.35)

that

fixed,

Y (t n) 0 > ly - ds =

to

(2.3.35)

e

ftn[ f P(s)

+

f' ( y ) P

dr ds

s

in the limit,

and

(2.3.32).

Thus Y (t n ) fy

(t)

tn ds f(s)

> ft =

(2.3.36)

P(s)ds

so t h a t

n lim inf n+~

as

f(s)

}

P(s)ds

=

(t)

n +~

and so

0 <

because

But

of

(b).

f

y(t) 0

ds f(s)

< oo

(2.3.37)

96

0 <_

l

y(t)

f, (s) f(s) ds <

sup [0,y(t)]

0

f' (s) •

I y(t)

ds f(s)

0

(2.3.38)

Because

of

(2.3.37)

m u s t be

finite

the e x p r e s s i o n

l

=

is i m p o s s i b l e

settles

this

y(t)

f, (s)

0

< 0

since

f(0)

imply

P+(t)

(2.3.30). (2.3.24)

set o u t

t .

large

t

H 0

for

T

and so o b t a i n

contradiction

We s h a l l

exists with

n

for e l s e t ~ T

P(t)

fixed point

P(t)

~ 0

> 0

for

the a r g u m e n t

then

for

t ~ T

would

contradict

beginning

at

a contradiction.

suppose

that

(2.3.9)

solution

theorem.

First

= P(t)

+

r

p2(s)ds

~t Then co

I t

R(s)ds

<

is f a l s e of

(2.3.1)

of all w e

co

R(t)

(2.3.18)

and this w o u l d

to find a n o n - o s c i l l a t o r y

Schauder's

This

Consequently

We can then r e p e a t

(Necessity).

= 0

case.

for l a r g e

arbitrarily

(2.3.39)

ds < ~

f(s)

If no such s e q u e n c e g(t)

(2.3.38)

and thus

0 <

which

on the r i g h t of

let

and so using

97 and so

are

oo

oo

t

t

finite.

Moreover

if

g ~ 0

the l a t t e r we see

then

(f + g ) +

~ f+

for any

f .

Using

that co

oo

ft R+ds ~ ;t P+ oo

oo

It~ ~s~It ~ and so

P

is a b s o l u t e l y

I

integrable.

ip(s) i + t

We d e f i n e

Q (t)

p2

Thus

ds

<

s

by

Q(t)

= alP(t) I + 2ab

p2 ds

(2.3.40)

t where

W e let

B

functions

a =

sup 0!y~2

f(y)

b =

sup 0sys2

f'(y)

be the s p a c e on

[T , ~)

,

of all b o u n d e d where

T

absolutely

is to be c h o s e n

continuous later,

98

which some

have

a right-derivative

multiple

of

I

with

point

that

is b o u n d e d

by

Q(t)

yl

B = {y c BAC[T, ~) :y+ -

Associate

at each

B

the

=

exists and

norm

IY' (t)] < (const)Q(t)

defined,

for

y e B

,

for

=

t > T}

by

JIy[EB : llyli +

where

I II~

is

the

usual

uniform

norm.

Then

B

a subset

B

is

a Banach

space. For

Bn =

n = 1 , 2 , ...

y c B :

0 =< y(t)

we

define

=< 2

,

t =>T

I]y ' Q -i][

,

of

n

=< 1

B

by

,

I Y'(t 2) -Y'(ti) l < alo(t 2) -o(ti) l + It2 - t i] , t i , t 2 ~ i T , T + n ) " and

For

each

fact

n

Bn

by

B

= constant

n

is

compact.

continue

II, as

on

a closed The

in A p p e n d i x We

n

,

is

presented

B

y(t)

in

[T+n

convex

proof Lemma [8]

of

, ~)

subset

the

of

latter

B

and

result

in

is

II.l.l.

and

define

an o p e r a t o r

A

n

on

*

99 oo

oo

t e t

(2.3.41)

[T , T + n]

s

(ANY) (t) = 1 -

For

y e Bn

each

t

,

t c

t > T+n

f (y) do ds T+n

.

(2.3.42)

s

[T , T +n)

,

AnY

has

a right-derivative

at

g i v e n by

oo

(ANY) '(t) = f

f(y(s))do(s)

t

If

y e B

,

n

then

f(y(s))do(s)

integration

= f(y(t)]

by p a r t s

do +

t

shows

do

that

f' (y)y'ds

.

t

Hence

f t

Thus

f(y)do

= f(y(t))P(t)

+

i

P(s)f' (y(s))y' (s)ds

t

for

T -< t -< T + n

,

I (ANY)' (t) I < alP(t) ] + b It IP(s)l{aIP(s) ] + 2ab

and p r o c e e d

to show as in

[8] that

] (ANY)' (t) I < Q(t)

if

T

is so large

that

s p2 ds

t e

[T , T + n )

(2.3.43)

100 oo

2b I

IP(s) lds

< 1

t => T

t

If

t => T + n

,

(ANY) ' (t) : 0

If, in addition,

we require

hence

An(Bn)

c B

T so large that

oo

I

Q(s)ds

< 1

t > T

(2.3.44)

t

then 0 =< (AnY) (t) =< 2

t => T ,

since we can estimate

the inner integrals

(2.3.43)

then gives

and

also follows

(2.3.44) from

(2.3.42)

T =< t I < t 2 < T + n

in (2.3.41 -42)

(2.3.45).

For

y c B

n

by it

that

t > T+n

(ANY) (t) = constant

If

(2.3.45)

.

, t2

I (ANY)' (t 2) - (ANY) ' (t l) I =

It

f (Y(S) ]d°(s) 1 1

t2 <

f(Y(tl)) t2

=< a

It

It d<~ 1

+

It i

t2 do I + It

i If necessary

t2

t2 f' (Y)Y'

Is

do ds

t 2

b,Q(s) ,

Is

do Ids

1

we can restrict

T

further by requiring

that

101

rt2 | da

bQ(s)

< 1

t 2 => s > T

Js Substituting

this

I ( A N Y ) ' (t 2)

Thus

A n (Bn)

be

tions

done in

the

-(ANY)'

former

equation

we

(tl) ] =< a l o ( t 2) - o

obtain

tl) I +

It2 - tll

c Bn

There can

in

remains

as

the

in

to

[8, p.

definitions

show

that

A

is

n

82]

with

the

of

a(6)

, b(6)

continuous:

appropriate

This

modifica-

there.

i.e.

a(6)

: suP{if(y)

b(6)

= sup{if'

From

the

b(~)

÷ 0

above

given

s ,

6

we

c(6)

lY-xl

< 6}

0 < x , y < 2 , lY-x]

see

that

as

6 ÷ 0

< 6}

both

f E C' is

chosen

sufficiently

small

so

that,

fix - Yll < 6 ,

2 c(~)

where

0 < x ,y < 2 ,

(y) - f'(x) I :

definitions

since If

- f(x) I :

= max{a(6)

, 2ab(6)

IIAn y

<

+6b(a+l)}

- A nxll

<

, then

for

a(6)

,

102

which

shows

theorem

that

A

therefore

is c o n t i n u o u s .

n

implies

that there

A

and this

x

n

x

is a s o l u t i o n

n

The S c h a u d e r

= x

n

of

is some

x

n

fixed point

e B

such

n

n

(2.3.41 - 4 2 ) .

Since t2

,

IXn(t 2) - X n ( t l ) I =< It

t2

IXn(t) Idt < It i

the f a m i l y

{Xn}

hence

is a s u b s e q u e n c e

there

compact

the i n t e g r a l

Note:

is e q u i c o n t i n u o u s

intervals

which

and u n i f o r m l y converges

to a n o n - n e g a t i v e

equation

Q(t) dt i

uniformly

function which

and is e v e n t u a l l y

bounded on

satisfies

non-oscillatory.

Since

I

~ Q(s)ds t

then

l i m o(t)

assume

(exists

it is zero.

and)

Thus

If we a s s u m e

m u s t be

Q(t)

that

<

÷ 0

finite as

and so w e c a n

t ÷

g ~ C' (a , ~)

,

t

o(t)

s

=

p(s)ds

t > a

(2.3.46)

a

then

(2.3.1)

becomes,

upon differentiation,

y"(t)

+ p(t) f(y(t)J

= 0

(2.3.47)

103

and

P(t)

Thus For

Theorem

the v a r i o u s

differential mainly

equations with

difference

Defining -b

and

n

specify can

then

the the be

equation Y(tn)

3.1.i

(2.3.1)

E Yn

formulating

o(t)

using will

defined

if

t e

by

[8].

and

We

m-i [ 0

[tm_ 1 , t m)

bn

with

to shall

be

criteria

for

replaced

by

(2.1.91)

we

t_l

y(t)

1 that

t c

for

can It

the

values

relation

n + b n f ( y n)

that,

= a

whose

recurrence

(Cn+Cn_l)y

see

n

of C h a p t e r

a solution

we

b

satisfies

(2.1.92)

nonlinear

above

+

with

the m e t h o d s

have

the

(t) = o(a)

so t h a t

to

of B u t l e r .

theorem

oscillation

(1.i.25)

(t n)

CnYn+ 1 + Cn_lYn_ 1 -

With

theorem

latter

refer

is p o s i t i v e

satisfy

the

the

shall

as in

sequence shown

of

.

equations.

~

(c n)

we

p(s)ds

includes

applications

concerned

nonlinear

= lim T+co

m

= 0.

(2.3.48)

> 1 ,

[tm_ 1 , t m)

(2.3.49)

,

co

P(t)

= [ b n --- P m - i m

(2.3.50)

104

THEOREM

2.3.2: Let

Assume

a)

(Cn)

be as above.

, (bn)

the f o l l o w i n g :

That

f

should

satisfy

the a s s u m p t i o n

(a) of T h e o r e m

3.1.1.

N b)

lim inf

i

X n=m

c)

Let

Qj-I

p

~

n

> -co

n

~ max{-Pj-i'

0}

and

co

Rj-I

the a b o v e h y p o t h e s e s

sufficient

condition

1 m = n Cm-i

Proof:

[ c ! Q2 i:j -i i i

i R. < oo j=n cj 3

Then

Under

:

This

follows

able i n t e r p r e t a t i o n condition

(2.3.9)

COROLLARY

2.3.1:

Let

P

n

for

(2.3.48)

Pm-I

+

that a necessary to be o s c i l l a t o r y

[ i=m-i

p

of the h y p o t h e s e s

for

becomes

n > N =

and is t h a t

=+oo

(2.3.51)

i

f r o m the p r e c e d i n g

which

> 0

we have

theorem with

a suit-

and the i n t e g r a l

(2.3.51).

.

Then

a necessary

and

105

sufficient

condition

for

1 m=n

Proof:

[

Pm_l

follows

(2.3.51 - 5 2 )

THEOREM

+

Cm_l

This

(2.3.48)

to be o s c i l l a t o r y

i

1p

m=n Cm-1 i = mX - 1

immediately

are e q u i v a l e n t

is t h a t

from

since

P

n

~i

(2.3.52)

l

Theorem

2.3.2

because

> 0 .

2.3.3: Let

satisfy

the b a s i c

hypotheses

of T h e o r e m

2.3.1.

a)

If l i m o(t) t÷co

then

b)

If

(2.3.1)

o(t)

= ~

is o s c i l l a t o r y .

is n o n - d e c r e a s i n g

sufficient

(2.3.53)

condition

for

then

a necessary

(2.3.1)

and

to be o s c i l l a t o r y

is

that

oo

I tda (t) = ~ to Proof: P(t)

a) - ~

To p r o v e

follows for

all

b) we m u s t

immediately t .

from

(2.3.54)

Theorem

2.3.1

since

Hence

(2.3.9)

is i d e n t i c a l l y

that

(2.3.9)

is e q u i v a l e n t

show

satisfied. to

(2.3.54) . If

o

is n o n - d e c r e a s i n g

and

(2.3.9)

is

finite

then

co

It P ( s ) d s

< co

(2.3.55)

106

Consequently assume

o(t)

is zero.

must tend to a finite

Thus

P(t)

= -o(t)

limit,

which we can

and the latter

is non-

increasing. On the other hand if

It

P(t)

+

p2 dt <

P(t)

0

since

(2.3.55)

P

t

+ P(t)

then

P(s)ds

dt

to

is non-increasing,

< fj P(t) 't 0 For

holds

sufficiently

< ft 1 +

P(s)ds

}

dt .

large

co

1 + It P(s)ds

Hence

(2.3.9)

equivalent

to

is finite.

(2.3.56)

= O(i)

Thus we have shown

that

(2.3.9)

is

(2.3.55).

Now oo

oo

to

oo

0

i ft = I

(2.3.57)

(t - to)d~(t) t o

with

the interchange

justified

by the Fubini

theorem.

Thus

107

(2.3.55) latter

is f i n i t e

is f i n i t e

(2.3.9)

and

if and o n l y

if and o n l y

(2.3.54)

if

if

(2.3.57)

(2.3.54)

must diverge

is f i n i t e

is finite.

together

and

and the Thus

this c o m p l e t e s

the proof.

COROLLARY

2.3.2:

Let sequence

(c n)

(tn)

be p o s i t i v e

satisfy

t h a t the s e q u e n c e

a)

The n e c e s s a r y t i o n s of

(2.1.92)

(b n)

with

and s u f f i c i e n t

(2.3.48)

(2.1.91).

t_l = a

Let

the

Suppose

is a l s o p o s i t i v e .

condition

to be o s c i l l a t o r y

{

m i}

a

m=n 0 b)

and s a t i s f y

bm

for all

the s o l u -

is t h a t

= ~

(2.3.58)

If oo

bm

=

(2.3.59)

m=n 0 then all

solutions

is n o t n e c e s s a r i l y

Proof:

Part

satisfies

(2.3.48)

are o s c i l l a t o r y

(here

(b n)

o(t)

positive I .

(b) f o l l o w s

from

(a) of T h e o r e m

2.3.3 w h e r e

from

(b) of T h e o r e m

2.3.3 w h e r e we

(2.3.49).

Part only need

of

(a) f o l l o w s

to n o t e

tm

that

o(t)

m t_l + ~

is as in

1 Ci_ 1

(2.3.49)

m > 0 =

t

and

108

so

that

(2.3.58)

In all

n

when

= b

is

equivalent

particular

we

can

-i , 0 , 1 , .... > 0

n

We

to

(2.3.54).

choose

a =

then

obtain

-i

and

from

c

=

n

(2.3.48)

1

for

that,

,

2 A

is

oscillatory

if

Yn-i

and

+ bnf(Yn)

only

=

0

(2.3.60)

if

oo

=

m bm

(2.3.61)

m=n 0

This

is

the

follows

from

Example

i:

b

n

and

=

i/(n if

we

discrete the

Let + i)

,

choose

[ 0

analog

previous

c

n

=

n = a =

1 0 ci-i

of

Atkinson's

,

n

=

- i , 0 , 1 , ...

0 , 1 , ....

bm

[2]

which

corollary.

n +2

0

theorem

,

Then

(2.1.91)

and is

let

satisfied

then

1 i+2+

=

1 "'" + m - - - ~

1 m+l

oo

>

Hence

Corollary

(2.3.48) (a)

of

2.3.2(a)

are

oscillatory

Theorem

2.3.1.

[ m+l 0

implies where

f

that

all

solutions

is

any

function

of satisfying

109

Example

2:

If w e

let

b

(Un)

be as in E x a m p l e

1

-

and

6>0

(n + i) 1+6

n

1 above

then

so t h a t

(2.3.48)

I

i_i_

0

0 ci-i

b

< co m

has at l e a s t one n o n t r i v i a l

non-oscillatory

solution.

For m

I m=O

since

the s e r i e s

oo

i=0

b

~

~fo m ,=

i

_

terms

(m+l)

i+6

I m=i

b m

and

co

co

1 [ m=i

i

i=O C i _ l

have positive

oo

1 i+i

f

el_ 1

1 < i=l i + l

li (x + i) -1-6 dx -i

co

+

[ m=O

(m + i) -1-6

i.e. co

< 6-1

+ 0 (i) ~ i6 i i=l (i + i) oo

< 6-1

and

since

6 > 0

the

~ i=l

i---l--6+1 + 0(i) i

last t e r m is finite.

(2.3.62)

110

Example

3:

Let

cn

Pk-i

=

1

=

and

let

bn

=

(-l)n/n

,

n

= 1

.....

, 2

Then oo

and

thus

Pk-i

is odd.

By

the

verified

that

[ n=k

(-l)n n

=

(-i)

is p o s i t i v e

if

alternating

series

N lim ~ N+~ k=n

and

is

finite.

(Its

(b)

of

Theorem

2.3.2

k

can

IO

tk-I 1 +----td t

is e v e n

and

theorem

it

(2.3.63)

negative is

be

computed

via

2.3.63.)

is v e r i f i e d .

I

s k2

I

I

I

k=n-i

m=n

=

k=m-i

sk2

where

S k = m a x { P k , 0} Hence

m=n

A look

at

(2.3.63)

~ k=m-i

2 Sk

shows

=

that

k

readily

Now

m=n

if

exists

Pk-i

value

k

~ k=n-i

2 (k - n + 2) S k .

X S k2 < =

Moreover,

Thus

111

~[ k=n-i

ks k =

~[ k=n-i

k {I/

} tk +---i-~ d t

~

at

--

k=n-i Hence

2 [ Sk k=m - 1

[ m=n

and

so T h e o r e m

2.3.2

implies

A2

+

=

that

(-i) n n f (yn)

Yn-i

= 0

is o s c i l l a t o r y .

Example

4:

Let

c

n

= 1

,

b

n

=

( - l ) n /In+ 6 -

,

n = 1 , 2 , ....

Then oo

Pn-i

[ m=n

=

and

the

alternating

series

(-l)m m

+------~i

theorem

<

implies

co

Pm-i m

As

in

the

previous

example

n

< ~

that

6

> 0

112

[ m=n

[ k=m-i

Pk

=

[ k=n-i

(k-n+2)P

and

k=n-i

k=n-I

m=k-i

m

co

X

1

k-

k=n-i

(k - i)

2+26

oo

k=n-1

(k - 1) 1+2~

<

Thus

2 [ (k - n + 2)P k < ~

corresponding oscillatory

equation solution.

and has

so at

(2.3.51) least

one

is

finite.

nontrivial

Thus non-

the

113

ADDENDA:

The arise

when

ON A RELATION BETWEEN NON-OSCILLATORY DIFFERENTIAL AND DIFFERENCE EQUATIONS

preceding one

chapter

studies

of d i f f e r e n t i a l

plausible

that

tial

a general

of determining

similarities

theorem all

which

and non-oscillatory

and difference

when

exists

equations. which

solutions

reduces

to a g i v e n

It s e e m s the differen-

equation

y"

where,

say,

to the

same

f

+ f(t)y

is c o n t i n u o u s

problem

but

for

A2 Y n - i

One

the

the o s c i l l a t o r y

behaviour

problem

shows

such

THEOREM

theorem

is the

(2.3.64)

= 0

on

[a , ~)

a difference

,

are o s c i l l a t o r y ,

equation

+ fnYn = 0

(2.3.65)

following

2.3.4: be

Let

f

Let

t_l

a continuous

non-increasing

function

on

[a , ~)

tn

Denote Then

f (tn)

by

a necessary

> a

,

tn_ 1

and

1

n=0

(2.3.66)

, 1 , ....

fn and

sufficient

condition

that

(2.3.64)

be

114

non-oscillatory

is t h a t

the

difference

equation

(2.3.65)

be

non-oscillatory.

Proof:

The

equations there

idea

and

is no

[a , ~)

is to r e w r i t e

use

loss

Let

Theorem

2.1.1.

of g e n e r a l i t y

~i

, o2

both

as S t i e l t j e s

Since

f

in a s s u m i n g

be d e f i n e d

integral

is n o n - i n c r e a s i n g that

f ~ 0

on

by

oo

Of(t)

=

[ m-i

f

t e

[tm 1 tm) - '

(2.3.67)

t c

[tm 1 tm) - '

(2.3.68)

n

0o

o2(t)

If we

then

let

y(t)

y ( t n)

~ Yn

=

[ m+l

, z(t)

'

f n

be

solutions

of

y' (t) = c I +

ydo I

(2.3.69)

z' (t) = c 2 +

zdo 2

(2.3.70)

z(t n)

~ zn

will

satisfy

the

difference

equations

Now

a change

if a n d o n l y

42 Y n - i

+ fn-i Yn = 0

(2.3.71)

42 Zn-I

+ fn+l z n = 0

(2.3

of v a r i a b l e if e i t h e r

of

shows

that

(2.3.71)

or

(2.3.65) (2.3.72)

72)

is n o n - o s c i l l a t o r y is n o n -

115

oscillatory. Suppose Theorem

2.1.1

that

(2.3.64)

implies

v(t)

=

t h a t the i n t e g r a l

i

Then

is n o n - o s c i l l a t o r y .

f(s)ds

+

equation

v2(s)ds

(2.3.73)

t

has

a solution

integrable finite. upper

for s u f f i c i e n t l y

at

~

Let

[

f

t

and so

(2.3.67 - 6 8 )

[tm_ 1 , t m)

Estimating

f

m u s t be

are b o t h

the i n t e g r a l

by

sums w e o b t a i n

< ]~ f ( s ) d s t

=

n

m+l

Consequently

t e

and l o w e r

large

< =

~ m-I

f

t e n

[tm 1 , tm ) (2.3.74)

In p a r t i c u l a r co

It f ( s ) d s

and so this

Since of

is true

(2.3.73)

(2.3.75),

> o2(t)

has

for all

a solution

t e

(2.3.75)

[tm_ 1 , t m)

t

Corollary

2.1.2

applies,

because

and so

v(t)

has a s o l u t i o n non-oscillatory (2.3.65)

> 0

= o 2 (t) +

for s u f f i c i e n t l y and c o n s e q u e n t l y

is n o n - o s c i l l a t o r y .

v 2 ds

large so is

t .

Hence

(2.3.72).

(2.3.70) Hence

is

116

If n o w w e a s s u m e the same m u s t be implies

that

true of

(2.3.65)

(2.3.71).

is n o n - o s c i l l a t o r y

Theorem

2.1.1

then

therefore

that oo

v(t)

has a s o l u t i o n

= ~l(t)

for l a r g e

+ It v 2 ds

t .

But

(2.3.74)

implies

that

co

Of(t)

Hence

the l a t t e r

2.1.2

again we

(2.3.64)

=> It f(s)ds

holds

find

for all

that

t

(2.3.73)

has a n o n - o s c i l l a t o r y

is n o n - o s c i l l a t o r y .

This

As a c o n s e q u e n c e crete

t c

Euler

a n d thus has

applying

a solution

solution

completes

(2.3.76)

[tm_ 1 , t m)

which

and

implies

Corollary thus t h a t it

the proof.

we i m m e d i a t e l y

obtain

that the d i s -

equation

A2

Yn-i

Y

+

2 Yn

=

0

(2.3.77)

(n + I)

is o s c i l l a t o r y (see e x a m p l e

when

1 y > ~

i, s e c t i o n

Furthermore

A2 Y n - i

non-oscillatory

when

we

and n o n - o s c i l l a t o r y

2.1,

and E x a m p l e

shall have

+ lYn = 0

I ! 0

[30, p.

when

1 of s e c t i o n

Y < 2B).

30],

n=0,1,

and o s c i l l a t o r y

...

when

I > 0

1

117

because

of

the

analogous

property

y"

for

+ ly = 0

CHAPTER

3

INTRODUCTION: The p u r p o s e framework

for

the theory

Volterra-Stieltjes preceding these

of this

generalized undertaken

Kac

to d i f f e r e n t i a l

been

used by H. L a n g e r defined

operators. [35]

though

equations.

shall

extension

of that used by Kac

[35]

Liouville

problems

difference

with

a formalism

the n o t i o n equation

use here

was

there was

formalism

has

of an

of the

and its a p p l i c a t i o n s

weight

form

is a n a t u r a l

and in p a r t i c u l a r ,

indefinite

that

will

Sturm-

functions

and

equations.

In s e c t i o n differential

equations

show

the a p p l i c a t i o n

integral

we

differential

Such

A different

The m e t h o d w h i c h

in the

of as d e f i n i n g

[41] to deal w i t h

by a Stieltjes

a basic

by the

encountered

used here will

(2.1.0).

include

generated

can be t h o u g h t

differential

only

operator

equations

The m e t h o d

equations

by I.S.

is to p r o v i d e

of o p e r a t o r s

integral

chapters.

integral

chapter

2 we

shall

proceed

to define

the g e n e r a l i z e d

fx

1

(3.0.0)

expression

[f] =

d~)d(x)

I

f' (x) -

a

f(s)do(s)

119

where

v ,o

variation, section

are

real

after

having

(limit-point,

of b o u n d e d

given

the b a c k g r o u n d

material

3 we

shall

study

classification

limit-circle)

operators

with

-y"

where

functions

in

i. In s e c t i o n

tial

right-continuous

of

the Weyl

singular

an a p p l i c a t i o n

interval.

Other

recurrence

relation

t E

r(t)

applications

differen-

to the p a r t i c u l a r

+ q(t) y = l r ( t ) y

the w e i g h t - f u n c t i o n

generalized

[a , ~)

vanishes

will

case

(3.0.1)

identically

include

the

on

three-term

(3.0.2)

-CnYn+ 1 - Cn_lYn_ 1 + bnY n = lanY n

where

c

n

> 0

These will

In s e c t i o n to d e t e r m i n e or the

5 we

whether

limit-circle

adjointness

of

In s e c t i o n integrals

generalized 7 we

associated

implications

more

discuss

with

6 we

which

is in shall

generally,

c a n be u s e d

the be

the

4.

limit-point

considering J-self-

operators. the

finiteness

(3.0.0) and

consider

[39]

DI => CD =>

in s e c t i o n

criteria

equation

In s e c t i o n and,

such

some

a certain

case.

self-adjointness

give

be discussed

some

SLP => LP

of Dirichlet the

chain

of

120

where

these

Dirichlet,

abbreviations Strong

Finally, three-term

§3.1

in

GENERALIZED

section

of

at each

interior

Limit-Point

8 we

relation

be

bounded

two

real

variation

point

and

are

Associated defined

on

on

~(x)

function decreasing

with

finite

if

the

for

a

examples.

[a , ~)

,

functions

a > -~

Then

,

,

lim

9(x)

~

(or

~)

(~ , 8]

and

is a s e t [e , ~]

function in

m

[a , ~)

by

( e , B]

= ~(8)

- ~(e)

e =< B

(3.1.0)

m

[~,

=

-

~

(3.1.1)

~

is

B]

measure

measure

~(B)

~(~-0)

non-decreasing

functions

Borel

some

notions

X÷yi0

of bounded

signed

these

m

Borel ~

respectively.

finite.

intervals

When o-finite

give

Conditionally

right-continuous

Y c ~

lim

exist

define

and

X+y±O both

Dirichlet,

DERIVATIVES:

D ,~

locally

for

Limit-Point,

recurrence

Let

stand

on

such on

original

then

~

[55,

p.

[a , ~)

variation

is

a function [a , ~)

function

<

[55,

B

induces 262].

Since

a difference will p.

induce 264,

is b o u n d e d

on

a

of a

ex.

two

every non-

o-finite ii]

which

[a , ~)

We

is

121

will

denote

set

E

such

a measure

where

m+ P

, m-

obtained

are

by

its

function

variation

p

(E) = m~(E)

where

Jordan

a

Borel

m

, m

are

(or t o t a l

absolutely

p(x)

valued

is

as in

appearing

in

with

set

E

E

to

measurable

measures

[24,

such

p.

is c a l l e d ~

~

if

[a , ~)

by

the

total

that

p

Im

varia-

is

I (E) = 0

I (E) = 0 there

[24, p.

exists

(3.1.4)

131,

with

125].

a finite-

~dm

the

for

that

ex.

4].

The

Radon-Nikodym

It is u n i q u e function

on

(3.1.3)

say

]m

continuous

f (E) = JE

Thus

of b o u n d e d

defined

we

to

for w h i c h

~

of

of

respect

function

set

I

123].

measure

is c a l l e d

measure)

with

(3.1.4)

respect

is a n o t h e r

(3.1.2)

signed

m

Borel

Borel

+ m-(E) P

~-absolutely

for e v e r y

locally

I (E) = m+(E) P

are

measurable

and

signed

variations

[24, p.

Im

continuous

measurable

and negative

The m e a s u r e

variation

p , ~

(3.1.2)

decomposition

a-finite

(3.1.2).

If

If

for e v e r y

- m~(E)

the p o s i t i v e

]m

every

Then,

right-continuous

induces

satisfying

tion

m

,

m

each

by

in the

this

function derivative

sense

property

that then

if # =

of

122

v-almost everywhere possibly on a set When and

~

(that is they are equal everywhere except E

p(x)

with

Im I (E) = 0 I -

is n o n - d e c r e a s i n g and right-continuous

is a non-negative Borel measurable

Lebesgue-Stieltjes

integral of

~

function the

with respect to

~

is

defined by

I ~(x)d~(x)

If

is both positive and negative

respect to p(x)

- [ %dm

p

(3.1.5)

it is integrable with m

if it is integrable with respect to

When

is of bounded variation

i

b ~ (x) d~ (x) a

agrees with the ordinary Riemann-Stieltjes the latter is defined When

p ,v

integral whenever

[55, p. 261].

are of bounded variation and

p

is

v-absolutely continuous,

(x)

exists

+h) - ~ ( x - h ) } lim ]p(x [~ + h) v (x h) h+0

m -almost everywhere, v

point of d i s c o n t i n u i t y of

u

in particular when it exists and

p(x+O)

d~(X) = v ( x + O ) = ~ (x)

-p(x-O) -v(x-O)

(3.1.6)

x

is a

123

where

~

to

defined

v

is the

derivatives,

Radon-Nikodym

in

(3.1.4).

see

From

[24,

p.

(3.1.4-5),

p(B

+ O)

derivative

of

(For g e n e r a l

for

- p(a±

a , B e

=

O)

are of b o u n d e d

of d i s c o n t i n u i t y

p(x)dD(x)

IJa-+O B±O

[a, ~)

is the

variation

[18,

p.

134],

dD(x)

and have

B±0 [p(x)~(x) ]e±0

=

general point

p{x}

(or

§3.2

GENERALIZED

no c o m m o n

points

[ B-+0 -

D(x)dp(x)

for

It is d e f i n e d

DIFFERENTIAL

[35]

x ~ I ,

-

we

by

of

the

a ~ I

d(

d~(x)

y+(x)

shall

EXPRESSIONS

shall

by p a r t s . be

When

denoted

by

(3.1.1).

essentially

in the d e f i n i t i o n

expression

£[y] (x)

for i n t e g r a t i o n

~ ( or ~ ) m e a s u r e

section

of K a c

differential

formula

its

~{x}).

In this

i.e.

,

J~i 0

is a s i n g l e

I ,

these

¢(x)d~(x)

Je~±0

approach

on

then

B-+0

This

respect

132].)

$-+0 dp(x) Ja+-0 dv (x)

~ ,~

with

information

i When

p

second

OF T H E

SECOND

pursue

ORDER:

the

of a g e n e r a l i z e d order

on

some

interval

fixed,

rxo Ja+0

y(s)do(s)

1

(3.2.0)

124

where and

o ~

was was

assumed

bounded

Basic

variation. in

the

We

both

finite

If

I

~

that

o

an a r b i t r a r y

see b e l o w

case,

that

to d e f i n e

remainder

at most

intervals

is a f i n i t e

infinite at

the

have

continuous

of

is,

in

function

i t is

of

still

(3.2.0).

this

the

a finite

chapter we

interval

finite

end,

interval

should

both

hypotheses

a), of

then both

b)

of

then both and

I

shall

of d i s c o n t i n u i t i e s

~ ,o

of

§i.i) .

~ ,o If

I

shall

be

that neither

has

s h a l l be is a s e m i continuous

a discontinuity

i.e.

lim ~(x)

both

number

(see the h y p o t h e s e s

a t the e n d - p o i n t s

at infinity,

~ ,O

on

that:

~ ,~ in

ThUs

variation

assumptions:

assume

b)

of bounded

to a s s u m e

and

shall

latter

Throughout

a)

convenient

right-continuous

possible,

locally

non-decreasing.

It s h a l l be addition,

to be

exist

(may be

shall be

,

lim

infinite).

assumed

right-continuity

o(x)

in a d d i t i o n

and bounded

to the u s u a l

variation

for

125

Let

~ ,v

be

variation. induces

right-continuous

As w e

a

saw

o-finite

addition,

we

respect

two

to

the

signed

assume m

in

that

then

previous

Borel m

the

functions section

measure

is

of b o u n d e d

on

absolutely

Radon-Nikodym

each I

of

these

If,

in

continuous

with

derivative

am ¢ --- d m

exists

~-almost

Moreover

(3.2.1) Let

bounded

~ N

agreees

~ ,o

be

variation A

D

everywhere

on

[a , b]

have

relation

(3.1.6)

(3.1.4).

~-almost

right-continuous

everywhere.

functions

of

[a , b ]

f

function

and we

with

two

(3.2.1) l)

is

said

to b e l o n g

to

continuous

on

the

class

if

i)

f

is

absolutely

ii)

f

has

at each

point

x e

[a , b]

[a , b]

,

a right-

!

derivative

f+(x)

the

function

fx

,

]J(x)

and

- f+(x)

-

f(s)do(s) a

is

~-absolutely

continuous

on

[a , b]

I

We

note

that

(ii)

necessitates

that

f+ (x) I

variation an

on

"associated

[a , b] number"

The [35,

quantity p.

212].

f+(b)

be

of b o u n d e d

can

be

termed

126

Thus

the p r e c e d i n g

discussion

shows

that

if

f c ~

the

quantity

Z[f] (x)

exists

w-almost

everywhere tion

of

is

possibly

m

is

(x)-

on on

c a s e of

(2.1.0).

is L e b e s g u e

(2.1.0).

f

fx

f(s)do(s)

a

[a , b]

(i.e.,

a s e t on w h i c h

}

(3.2.2)

it h a s

the

meaning

total

varia-

zero).

A particular expression

{+

everywhere

except

v

d dw(x)

-

_

To

measure

It is c l e a r

and

a generalized

see

this w e

let

differential

let

y(x)

be

v(x) any

= x

so

solution

that

of

that

fx y' (x)

-

}

y(s)d~(s)

= 0

a

so t h a t form

equations

of

the

form

(2.1.0)

can be brought

into

the

(3.2.2). B y a solution of

d d~) (x)

is m e a n t

purpose

f' (x) -

a function

everywhere,

which

we

for d e f i n i n g

contained

in

the

the g e n e r a l i z e d

f c V shall

f(s)do(s)

which

followinq

of

theorem.

equation

= ~(x)

satisfies

abbreviate

expressions

differential

as

the

[~] form

(3.2.3)

(3.2.3) The (3.2.2)

v-almost real is

127

THEOREM

3.2.0: In

(3.2.3) it

order

on

[a , b]

satisfy

f(x)

=

th~it a

the

e +

,

function

say,

integral

B(x-a)

it

is

f(x)

be

necessary

a

solution

and

of

sufficient

that

equation

+

(x-s)f(s)do(s)

-

(x - s) ~ (s) d ~ (s) (3.2.4)

for

some

Proof: f

be

ous

This a

on

given

e , B

,

x

is

not

solution

of

[a , b]

e

[a , b]

very

different

(3.2.4).

and

f

from

Then

has

f

[35, is

p.

215].

absolutely

a right-derivative

for

For

let

continueach

x

,

by

f' (x)

=

B +

f(s)do(s)

-

#(s)dv(s)

(3.2.5)

i.e.

~(x)

=

B -

i

x ~(s)d~(s)

x

:

[a , b]

latter

shows

a

where

~

was

~-absolutely

defined

in

continuous

~(x)

= -

(ii). and

. (x)

=

The

that

~

is

hence

d9]-x)

[~]

f' (x)

(3.2.6)

Consequently

f

e ~

and

f

satisfies

(3.2.3)

[v]

128

On x

¢

the

[a , b ]

other

x

~(s)d~(s)

a result

in

= -

fa

= -

d

[24,

x

is

(3.2.4).

The

THEOREM

p.

the

proof

3.2.1: If

[35,

shown

is

the

This there

through

of

(3.2.3),

f' (s) -

f' (s) -

fdo

dw(s)

fdq

134],

+

f(s)do(s)

+ f' (a)

p.

[a , x]

over

we

obtain

complete.

250] [~ , B]

a I

,

- g(s)Z[f] (s)}dw(s)

:

[f'g - f g ' ] a

Lagrange identity.

is e x a c t l y when

without

variation

latter

is n o w

f ,g E D

{ f ( s ) £ [ g ] (s)

Proof:

a solution

(3.2.5). Integrating

This

is

d dw(s)

= -f' (x)

which

f

if

,

fa from

hand

and

so

w any

Lemma

5,

p.

250

of

is n o n - d e c r e a s i n g . essential

is o m i t t e d .

change

when

[35] The ~

though

proof

it

is

carries

is o f b o u n d e d

129

Thus

solutions

to the

d<

d~ (x)

initial

f' (.x) -

value

fdo

problem

}

(3.2.7)

= ¢

f (a) =

f' (a) = B

exist

on

[a , b]

in A p p e n d i x

Note:

If

locally

because

(3.2.8-9)

solutions

to

(3.2.4)

exist

as w e

saw

I.

I

[a ,,~)

=

of b o u n d e d

and

p(x)

"variation

>

is r i g h t - c o n t i n u o u s ,

0

and

oo

f

the

theory

which

would

dx p(x)

arise

_

by

consideration

of

the m o r e

general

d~(x)

would

be

similar

p(x) f' (x)

to the

variable

(Appendix

(3.2.10)

to an e x p r e s s i o n

§3.3

THE W E Y L Let

variation

on

~

I)

one

and

use of

form

(cf.,

right-continuous

[a , ~)

As we

here

of T h e o r e m the

saw

(3.2.10)

f(s)do(s)

presented

CLASSIFICATION be

-

since

3.2.0

a change

would

of

reduce

(3.2.2).

[81]) and

locally

in s e c t i o n

of b o u n d e d

3.1,

~

induces

130

a

o-finite

Im I

signed

denote

equivalent

Borel

the total

measure

variation

[24, p. 126]

m

on

measure

to the m e a s u r e

If w e

[a , ~) then

of mV

induced

let

Im[

is

by the

function

V(x)

where

the i n t e g r a l

general,

imposed

[a , ~)

is a L e b e s g u e - S t i e l t j e s

integral

since

can be t a k e n the l a t t e r

(3.3.0)

integral,

the L e b e s g u e

of all e q u i v a l e n c e

square-integrable functions

to be a R i e m a n n -

exists

by the h y p o t h e s e s

f ~ L211m

classes

with respect

are e q u i v a l e n t I ; I1

space

L211mvl

of f u n c t i o n s

; I1

to the m e a s u r e

if they are e q u a l

if and o n l y

latter

functions

where

V

Stieltjes

space f

is also e q u i v a l e n t

such

Im I

are Two

[v]

if

(3.3.1)

to the s p a c e

L 2 (V ; I)

of

that

is d e f i n e d integral.

as the

which

I E Ix) J2dlml < The

in

upon We d e f i n e

Thus

x E

Ida(s) I

b u t in our c a s e s

Stieltjes

space

---

i If(x)12dV(x)

<

in

(3.3.2)

(3.3.0)

and

(3.3.2)

is a L e b e s g u e -

131

f ~ L 2 (V ; I)

If

we

define

the n o r m

of

f

as

(3.3.3)

llfll- { I IIf(x) 1 2 d V ( x ) } ½

The

norm

Hilbert

defined space Let

b < ~ We

let

DL

which I

We

(3.3.3)

shall - D

use

be

then

we d e n o t e

be one

D =

For

by

of

by

defined

L 2 (V ; I)

into

a

H

[a , b]

(3.2.2)

{f ~ H :

turns

or

[a , ~)

,

a > -~

to d e f i n e

an o p e r a t o r

(I)

£[f]

,

L

on

by

f E D

and

e H} .

(3.3.4

f c D

Lf = i[f]

where

Z[f]

is as in

The manifold define

domain

on w h i c h

valued. operator

(For L

i

arise

can be

hereafter the is

as

by m e a n s

the m a x i m a l of

(3.3.5),

linear to

H concerning L

sufficient

of

assume

c a n be c o n s i d e r e d

operator

non-decreasing,

shall

in

problems

the r e s u l t i n g

valuedness

D

(3.3.5)

(3.2.2).

it is p o s s i b l e ,

an o p e r a t o r Some

of

H

in

H

conditions found

that

i

applications

single-valued.)

the

in

In the

case

guaranteeing

[35,

defined that we

single-valuedness

p.

260]

and

when the

~

is

single-

[36].

We

by

(3.3.5)

is s i n g l e -

are

dealing

with,

the

132

DEFINITION

3.3.1:

Let solution

I =

[a , ~)

If for a p a r t i c u l a r

10 e •

every

of the e q u a t i o n

iy = loy

(3.3.6)

satisfies o0

la y ( x ) 1 2 d V ( x )

then

i

is said

the c o n t r a r y

< ~

to be of l i m i t - c i r c l e

case

[

(3.3.7)

type at i n f i n i t y .

is said to be of l i m i t - p o i n t

In

type at

infinity.

We now p r o c e e d o n l y on

THEOREM

i

and n o t on the

iy = ly

10 ~ C

solution ,

then

is of c l a s s

Proof:

Let

(3.3.6)

satisfying

depends

chosen.

% , 9

solutions

of

(3.3.6)

for any

is of c l a s s

I ~ {

every

L 2 ( V ; I)

solution

of

L 2 ( V ; I)

be l i n e a r l y

(x)~'(x)

Such

1

that the c l a s s i f i c a t i o n

3.3.1: If e v e r y

for some

to s h o w

-

independent

}'(x)~(x)

e x i s t by T h e o r e m

3.2.0

=

and

solutions

of

(3.3.8)

i

[3, p.

348]

(see

133

also

[35,

(3.3.~)

p.

220]).

is,

We

in g e n e r a l ,

Let

I ~ I0

remark

that

the

(')

appearing

in

a right-derivative.

and

y

be

a solution

[y = ly

of

.

Then

ly = 10y

We

now

apply

use it

the

to

variation

(3.3.10)

when

of

+

constants

it

(3.3.10)

(I - l o ) y

formula

is r e w r i t t e n

in

[3, p. the

351]

and

equivalent

form

y(x)

= y(a)

+ y'(a)(x-a)

-(t

to o b t a i n

y(x)

a representation

= ~(x)

+

where

- t O)

e ,

+

2

(x-s)y(s)d(~(s)

(x - s ) y ( s ) d ~ ( s )

of

y

-toU(S)

,

)

(3.3.11)

as

+ B~(x)

(X - I 0 )

are

{~(x)~(t)-~(x)*(t)

constants

and

c < x

}y(t)d~(t)

is

to be

chosen

(3.3.12)

later.

If w e w r i t e

ITyTJc

]y(x) 12dv(x)

(3.3.13)

134 then since, such

q5 , ~ c

1, 2

(V ; I)

,

it is p o s s i b l e

to c h o o s e

R

that

R > max{flail c for all

(3.3.14)

, II~lIc}

t > c .

Then

i

t {¢(x)~(t) - ¢(x) C(t) }y(t) dv(t) c

[¢ (x)~2 (t)y (t)I dV (t) +

=<

I~ (x)q5 (t)y (t)IdV(t) (3.3.15)

and use of

the S c h w a r z

i~(x) I

with

a similar

inequality

I~(t)y(t)

inequality

gives

IdV(t) < i~(x)[ll~llc[lYll c

holding

for the o t h e r

integral

in

(3.3.15) . Hence,

by

(3.3.14) ,

I¢(~) I

I~yldV + I~<x) I

ICyldV __< (L¢(x) l + I~(x)Ilalyllc (3.3.16)

Consequently

iiylic _< i~lil~ilc + i~lli~ilc + Ix-~01aLyilc{ti~11c+ll~lic}

=< (I~1 + IBI)R + IX-Xol • 2R21IYIIc

135

We

now

choose

c

so

large

that,

for

all

t > c

,

lx - ~0 IR2 < !4 from which

it w i l l

follow

that

ityEic =< 2R([~I + IBII for

all

y

T,2(v: (e oo))

•

t > c

It most

one

THEOREM

.

Thus and

follows

linearly

letting

t ÷ ~

hence

in

from

this

independent

is

that

L2(V; in t h e

solution

~

be

non-decreasing

and

I~ Iy(x ' I) 12d ~ ( x ) all

1

(3.3.9). solution

Proof: , ~

find

that

I) limit-point

can

be

in

case

at

L 2 ( V ; I)

3.3.2: Let

for

we

real If

of

two

is

complex

I ~ 0

(3.3.9)

This be

Im

or

the

then

in

there

that

for

> 0

b

> a

(3.3.17)

Y I 0

is

a solution

exists

at

least

of

one

L 2 ( V ; I)

standard

linearly

where

suppose

"nesting

independent

circles"

solutions

of

analysis.

Let

(3.3.9)

satisfying ~(a,

k)

= sin

~ ( a , k)

= cos

~

~ ' ( a , I)

= -cos

~ ' ( a , l)

=

sine

(3.3.18-19)

136

where

~ c

functions

[0 , z) of

1

Then for

, 4

fixed

x

, ~'

, 4'

(Appendix

are

III,

entire

section

i) .

If w e w r i t e

[}~](x)

then

it

follows

-

from

~(x)4'(x)

(3.3.18-19)

[~] The

solutions

Every of

~, 4

are

(x)

real

(3.3.20)

4(x)~'(x)

that

:

for

(3.3.21)

1

real

1

and

~(a,

l)cos

e + ~' (a , l ) s i n

e = 0

4(a,

l)sin

~ - 4' (a , l ) c o s

e = 0

solution

the

-

~

of

(3.3.9)

is,

up

to

satisfy

a constant

multiple,

form

8 = ~ + m4

where

m

a < b < ~ requiring

is

some

and

zeros must

8 c of

introduce

which

depends

a real

on

boundary

~

Now

condition

let at

b

by

that

g(1)

for

number

(3.3.22)

-: y ( b ,

[0 , z) the

The

entire

necessarily

be

l)cos

B + y' (b, l ) s i n

eigenvalues

function real

g(1)

(Appendix

of

8 = 0

(3.3.9) Since

III,

are

these

Theorem

(3.3.23)

then

the

eigenvalues

III.l.2),

137

(3.3.23) have

does

no

of

accumulation.

We

seek

m

such

now the

boundary

as

1

are

, b

functions in

meromorphic

If

we

let

B

varies

z =

image

D =

A

lies

on

of

=

~' ( b ,

m-plane.

cot

B

0

to

from the

and

Thus Cb

8 From

m

that

is

given

solution

8

A

the

zeros

above

simple

computa-

the by

circle

and

real

z

under

~'(b,

AD

- BC

z

0

satisfy

(3.3.24)

,

I

the

is

B+

Dm Cm

is

Theorem

, ~ ,

III.l.0),

the

moment, (-~,

then

~)

so

as

that

transformation

(3

C

=

~(b,

a circle

(3.3.23 have

which

, ~'

l

over

,

we

A+

%

'

I) ,

III,

for

Az + B Cz+D

=

Since

real

varies

=

B

for

b,

m

,

, b , 8) (Appendix

fix

axis

Cb

= m(l

,

will

, I) + %' (b , I) , l) + ~ ' ( b , I)

1

z =

so

consequently

(3.3.23).

of

and

real

~ ( b , I) I)

8~(b 8~(b

vary

entire

is

where

the

condition

cot cot

--

, 8

m

the

that

and

that

m

~'

identically

point

shows

Thus

vanish

finite

satisfies tion

not

if

I)

,

Cb and

3.24)

in

only

the if

m

(3.3.25)

the

image

of

Im

z =

0

,

138

( A + Cm) ( B + Dm)

Since

every

described

circle

see

mb

of

( A + Cm) ( B + Dm)

center

y

on

2

-

IYI

comparing

Cb

2

+ Y~

+ {~

coefficients

is g i v e n

radius

rb

- ~o

of

r

can

= 0

be

(3.3.27)

(3.3.26-27)

A5- B~ CD-

its

radius

that

center

by

mb -

and

and

(3.3.26)

= 0

by

r

we

with

-

is g i v e n

(3.3.28)

DC

by

I A D - BC 1 (3.3.29) r b = i ~ D _ De I

Substituting obtain

the

the

values

equivalent

for

A

same

way

we

find

1 =

Hence,

, C

, D

into

(3.3.26)

we

equation

[88] (b)

In t h e

, B

= 0

(3.3.30)

that

[~]

(b)

= AD

- BC

[~]

(b)

= DC

- CD

[~]

(b)

= AD

- BC

139 mb =

[#~] (b) [ ~ ] (b)

(3.3.31)

and _

rb

The c o e f f i c i e n t interior

of

of

Cb

mm

in

in the

1

(3.3.32)

(3.3.26)

m-plane

[80] (b)

is

[ ~ ] (b)

and so the

is g i v e n by

<

0

(3.3.33)

[ ~ ] (b)

Since

~ , ~'

t a k e on real v a l u e s

[ ~ ] (b)

-

[~]

(a)

=

at

[~]

a

(b)

b (~ Lt~

by T h e o r e m

we have

t~L~) d'o

3.2.1, : 2i

Im(~L~) dv

= 2i(Im

X)

[bl~12 d~ .

(3.3.34)

10

(3.3.35)

& Similarly

we can show

[88] (b) =

[88] (a) + 2 i ( I m

X)

dv

where [08] (a) = -2i

Im(m)

(3.3.36)

140 Combining

(3.3.34 - 36)

into

b Im I I a

(3.3.33)

we o b t a i n

]6) 12dr - Im(m) < 0

(3.3.37)

Im I I b l @ 1 2 d ~

and so if w e

take

it t h a t

Im I > 0 ,

Ifl The p o i n t s

m

are on

b la

then

Im(m) Im(1)

Ol2d

Cb

say,

if and o n l y

(3.3.38)

if

el2d.~ _ Ira(m)

(3.3.39)

Im(1)

The r a d i u s

rb

of

Cb

is also g i v e n b y

(3.3.40)

rb = 2 Im I ib I~ 12 d~) a

If,

say,

~

is c o n s t a n t

on some

rb = constant

and so the c i r c l e s

lies o u t s i d e

I

If

a < y < b < ~ ,

interval remain

£

f~

then for

the same u n t i l

then

f lel2d a

I

lel2d < Im(m) zm(x)

b e I b

141 and

so

C b ~ Cy

Thus

the

sequence

(C b)

if

b > y

of c i r c l e s

is

.

(3.3.41)

"nested"

in the

sense

(3.3.41) . Assuming sequence

(C b)

or a p o i n t then

its

~

shall

as

If the

which

is p o s i t i v e

is n o t

therefore

b ÷ ~

radius

(3.3.40)

that

eventually converge Cb

constant

to e i t h e r

converge

is n e c e s s a r i l y

given

the

a circle

to a c i r c l e

by

the

lim r b

C in

and hence

b

/a I.~12dv < ~o In this

case

within

Cb

,

if

m

is a n y p o i n t

(3.3.42)

on

C

then

m

lies

b > 0

Thus

ib

1¢ + ~ 1 2

dv <

a and

Im(m

)

(3.3.43)

Im(1)

so

+ moo~ E L 2 (V ; [a, co))

since

we

can The

m

since

let same

in this

b ÷ ~ argument case

in

(3.3.43).

applies

if

C

reduces

to a p o i n t

142

+ m ~ e L 2 (V ; [a, ~1)

In the

latter

case

8

is a l w a y s

a solution

the

" l l"m l t - c i r c l e "

terms

expressed

is u n i q u e .

Hence

(3.3.9)

L2

of

and

in the p r e c e d i n g

in

(3.3.44)

if

Im(1)

~ 0

there

The m o t i v a t i o n

"limit-point"

for

is c l e a r l y

discussion.

REMARK: Difficulties bounded

variation

the e i g e n v a l u e eigenvalues function

problem

g(1)

equations,

Another ness

of

the

so t h a t

not vanish that,

the n u m b e r cases

(3.3.9),

4)

does

chapter

in these

if one

assumes

theorem.

(3.3.23) one

has

g(1)

arises

vanish because

is of is t h a t

non-real

to g u a r a n t e e

that

(We shall and

eigenvalues

cannot

~

such

admit

identically.

of n o n - r e a l

that

One

may

for d i f f e r e n t i a l

difficulty

sign

only

in the p r e c e d i n g

(Chapter

in the n e x t

Thus,

arise

the

show

difference must

be

finite.

identically.) of

the

indefinite-

of

I

b l y ( x , I)12d~(x) a

when For

y

is a n o n - t r i v i a l

varying

and n e g a t i v e (3.3.37),

b

the

values

say,

In any

for case,

solution

latter thus b > a

integral

making

of

(3.3.9)

may

take

it d i f f i c u l t

and

b > a

on b o t h

positive

to i n t e r p r e t

.

even when

~

is of b o u n d e d

:

variation

143

(3.3.9) hence and

can

for

have

all

solutions

1

by

"limit-circle" this

see

§3.4

APPLICATIONS: Let

on

[a , ~)

suppose q(x)

in

~

the

be

such

= o' (x)

L2(V

can

both

following

that

r(x)

locally

; I)

3.3.1).

a locally

is

that

Theorem

cases

shall

in

for

Thus

occur

for

some

the

1

(and

"limit-point"

general

~

We

section.

absolutely = v'(x)

continuous

> 0

absolutely

a.e.

function

Furthermore

continuous

and

a.e.

Then

~(x)

= ~(a)

f

+

x r(s)ds

(3.4.1)

q(s)ds

(3.4.2)

a

and

o(x)

= o(a)

f

+

x

a

If w e

consider

the

problem

d~(x)

on

[a, ~)

then

(3.4.3)

r(x)dx

and

the

latter

y' ( x ) -

fx } a

reduces

y' (x) -

is e q u i v a l e n t

ydo

to

yqds

to

(3.4.3)

= ly

= ly

a.e.

144

-y"(x)

on

[a , ~)

ment as

allows

to

when on

Thus one

(3.4.3)

to i n c l u d e

sign.

Also

v

intervals

has

some

(3.4.4)

+ q(x)y(x)

of

interval.

A

when

r(x)

> 0

Since

(3.4.3)

y(x)

= a +

Ir(x)y(x)

includes (3.4.4)

interest of

=

is

(3.4.4). when

that

constancy,

treatment

of

can be

found

+

f

A

r(x)

e.g.

similar is

(3.4.3)

is

when

the W e y l

is e q u i v a l e n t

B(x-a)

(3.4.4)

the

indefinite also

r(x)

defined = 0

classification

in H e l l w i g to

treat-

a.e. for

[28].

integral

equation

x (x-s)y(s)d{o(s)

-Iv(s)}

(3.4.5)

a

then

using

include (t n)

the

methods

three-term

be

a given

outlined

recurrence

relations

1 we

in

can

then

(3.4.3).

For

let

sequence,

t 1 = a < tO < tI <

and

in C h a p t e r

..-

< tn

<

"'"

+ ~

let

_

tn

where

jumps

C_l

> 0

We

let

at

the

o ( t n)

tn_ 1

o

be

points

- o(t n-

_

1 Cn_ 1

0 , 1 , ...

a right-continuous (tn)

0)

n =

step-function

(3.4.6)

with

where

= cn + Cn_ 1 - b n

n=0,1,... (3.4.7)

145

where

(bn)

is a g i v e n

a right-continuous

sequence.

step-function

Moreover with

by d e f i n i n g

jumps

at the

as

(tn)

where

v ( t n)

where

(a n )

is a n o t h e r

a solution

y(t)

recurrence

whose

- ~ ( t n - 0) = a n

given

sequence

values

Yn

(3.4.8)

then

~ Y(tn)

(3.4.5)

will

satisfy

the

relation

- C n Y n + 1 - C n _ l Y n _ 1 + b n Y n = lanY n

We n o t e

have

here

that

(3.4.6)

(3.4.9)

•

implies

oo

(3.4.10) 0 Cn-i

though a new

this

requirement

function

(1.1.24)

with

p(t) p(t)

c a n be o m i t t e d

which > 0

.

by

the

introduction

is r i g h t - c o n t i n u o u s We

can

then

consider

of

and d e f i n e d

by

the m o r e

general

d~(x)

which

would The

case given

then

reduce

construction

(3.4.9). by

p(x)y' (x) -

to

in s e c t i o n

For example

(3.3.40)

thus

(3.4.9)

the

if we

ydo

without 3.3

radius

let

of

(3.4.11)

= ly

(3.4.10).

also the

applies

in the

circle

Cb

is

146 a = t_l

< t o < t I < '''

< tm_ 1 = b

then i~

(since

we

always

I 12d

assume

m-i

=

that

X0

a

~

is

n

I ~n12

(3.4.12)

continuous

at

a ) .

Consequently

1 m-i I) ~ 0

rb = 2(Im

For

the

latter

result

(5.4.6)].

The

nesting

recurrence

relations

Moreover the

space

weight

of

see

the

be

space

i.e.

(yn)

125-26

analysis

found

in

L 2 ( V ; I)

square-summable

an " '

a n l ~ n 12

[3, pp.

circle

can

(3.4.13)

~

sequences ~2(V ; I)

and

for

equation

three-term

[3, pp.

125-29].

is

equivalent

then

"with

respect

if a n d

only

to

to the

if

co

2 la n Y n I

< co

0

Thus

if

[3, p.

c

n

129,

> 0

,

Theorem

b

n

is

5.4.2]

any for

sequence any

1

and

a

> 0 n =

Im

I ~ 0

,

then

,

-CnYn+ 1 - Cn_lYn_ 1 + bnY n = lanY n

has

at

least

one

nontrivial

solution

~ =

(~n)

in

£ 2 ( V ; I) .

147

§3.5

LIMIT-POINT In

and

~

this

which

limit-circle

AND

LIMIT-CIRCLE

section

will

we

enable

0{

space

L 2 ( V ; I)

over

the

interval

This

space

was

defined

~ , 9

bounded

variation

10

such

~ ~

x > a

conditions the

on

limit-point

or

of

(x) -

in

some

establish

ydo

V(x)

[a , x]

,

be

=

is

and

(3.5.0)

ly

the

I =

total

variation

[a, ~)

,

of

a >-~

(3.3.3).

right-continuous

on

that

[a, ~)

o(x)

that

locally

there

of

exists

is n o n - d e c r e a s i n g

for

a solution

with

say.

(3.5.0),

with

Proof:

Let

initial

conditions

I = 10

y(x)

be

Theorem

3.2.0,

,

has

> 1

the

y(a)

by

functions

Suppose

- 10~(x)

y(x)

Then,

give

3.5.0: Let

Then

y

to

where

~(x)

LEMMA

us

classification

d~(x)

in t h e

shall

CRITERIA:

x

solution

= 1

,

y(x)

> a

of

y' (a)

is

y(x)

(3.5.1)

(3.5.0)

satisfying

= 0

a solution

the

(3.5.2)

of

the

integral

148

equation

y(x)

= 1 +

f

X

(x- s)y(s)d(o(s)

(3.5.3)

- Xo~(S))

a

for an

x ~ a interval

for and

Since

x e

y(a)

[a , a +

[a , a +

6]

6]

,

the

then

6 > 0

by

continuity

in w h i c h

integral

in

y(x)

(3.5.3)

there > 0

exists Then,

is n o n - n e g a t i v e

so

y(x)

Since in

= 1

y(a+

6)

~ 1

[ a + 6 , 61 ]

y(x)

> 1

x

there

6)

[a , a +

exists

Consequently

= y(a+

e

+

6]

61 > 0

for

x

in

(3.5.4)

such

that

such

an

(x- s)y(s)d(o(s)

y(x)

> 0

interval

- lO~)(s) )

+6 and

so

y(x)

Repeating of

real

this

> 1

process

numbers.

It

x e

we

obtain

is t h e n

lim

otherwise so w e

if

could

diction

lim

6

repeat

proves

that

n

=

the

6*

(3.5.5)

an

increasing

necessary

6

n

sequence

(3.5.5)

y(6*)

process holds

(6 n)

that

= ~

then

above

[a+ 6 , 6 1 ]

and

> 1 =

past thus

by 6"

continuity This

and

contra-

149

y(x)

THEOREM

3.5.1:

Let

3.5.0.

Suppose

further

> 1

~ , v

satisfy

(3.5.0)

Proof: solution

of

where

the

latter

there

exists

which

exists

hypotheses

of L e m m a

(3.5.6)

at

to s h o w

(3.5.0)

the

d~(t) I =

is l i m i t - p o i n t

It s u f f i c e s

all

that

la] Then

x > a

that,

is n o t

for in

by h y p o t h e s i s .

a solution

y(t)

of

some

I ,

L2(V;

I)

From

(3.5.0)

there Let

Lemma

such

that

is a I = 10

3.5.0 (3.5.1)

holds. Then

for

such

a solution,

ly(x) I2 Idv(t) I =>

hence

y

is not

COROLLARY

3.5.1:

Let

(a n )

in

L 2 ( V ; I)

be a s e q u e n c e

0 Let

(b) n

sequence.

be any

Ida(t) I =

given

such

that

and

(c n)

fan I =

sequence

another

positive

150 If

there

bn

cn

_

exists

a real

number

I

Cn_ 1 + 10 a n > 0

_

such

0

n=

that

0 , 1 , ...

(3.5.7)

then

CnYn+l

is

limit-point

solution

+ Cn-lYn-i

at

(yn)

co ,

such

i.e.

- bnYn

for

(3.5.8)

= lanYn

some

I

there

corresponds

a

that

co

X I anllYn 12

= ~

(3.5.9)

0

Proof:

We

equations

note of

continuous locally

(t n)

and

L(a,

of

form

(3.4.11)

bounded

co)

define

in p a s s i n g

The

that when

p(x)

variation

proof

Lemma

is

3.5.0 > 0

a step-function

~(t)

to

right-

satisfying

similar

extends

p(t)

-I

with

minor

changes.

with

jumps

at

the

by

require

n = 0, 1, has

the

and

We

here

....

be

constant

We y(t)

relation

(3.5.10) satisfied.

- v(t n-

that

solutions

recurrence

~ ( t n)

and

Moreover

define such

0)

on

o(t) that

= -a n

(3.5.10)

[tn_ 1 , t n)

as

in

Y(tn)

(3.4.7).

= Yn

, Then

satisfies

(3.5.0) the

(3.5.8). the

hypothesis

for

I = In ,

imply

that

(3.5.7)

(3.5.6)

implies

is

that

151

a - ~09

is n o n - d e c r e a s i n g .

Thus

Theorem

3.5..1 a p p l i e s

and

so

f

~Jy(t) 12 Ida(t) [ = a

which

implies

(3.5.9)

In this

form,

p.

Theorem

135,

THEOREM

3.5.1

5.8.2]

where

is a m i n o r the c a s e

a

extension n

> 0

of

is c o n s i d e r e d .

a , ~

be

variation

right-continuous

on

[0 , ~)

functions

locally

a necessary

limit-circle

and

of

and

(3.5.11)

I tJda(t) I < 0

Then

[3,

3.5.2: Let

bounded

Corollary

sufficient

condition

for

(3.5.0)

to be

is t h a t

(3.5.12) 0

Proof:

We

rewrite

y(x)

the

solution

= ~ + 8x +

f

of

(3.5.0)

as

x (x-s)y(s)d(o(s)-

(3.5.13)

l~(s))

0

Then only

(3.5.0)

L 2 ( V ; I)

Theorem of

is l i m i t - c i r c l e solutions.

12.5.2]

solutions

we

y , z

find such

if say Using

now

that (3.5.13) that

(3.5.13)

with

a result with

in

I = 0

I = 0

has

[3, p.

389,

has

a pair

152

The

solutions

limit

circle

Since

z(x)

are

÷ 1

x ÷ ~

(3.5.14)

z(x)

~ x

x + ~

(3.5.15)

then

then ~ x

y(x)

linearly

these for

independent.

solutions

large

x

we

IxlZ(sl12Id(s)l

Jx I

If

must

belong

will

have

(3.5.0)

to

is

L 2 ( V ; I)

zls112s21d (s)l s I

oo

> C IxS21dv(s)

Hence

(3.5.12)

is

since

y ÷ 1

then

over

satisfied. this

A

similar

forces

v

(3.5.16)

I

calculation

to be

of

shows

bounded

that

variation

[0, ~) Next,

bounded

if

(3.5.11-12)

variation

over

are

[0, ~)

I

°°lY(x

satisfied and

then

~

must

be

of

hence

2 idv(x) I <

0

on

account

of

to

the

leading

one

(3.5.14). to

Since

z(x)

(3.5.16)

shows

Iz(x) 12 Ida(x) I = O 0

and

thus

3.3.1

z

implies

~ x

an

argument

similar

that

t21dg(t) I 0

is

square-integrable.

that

(3.5.0)

is

limit

Consequently circle.

This

Theorem completes

153

the proof. If w e 3.5.11)

take

it t h a t

is t r i v i a l l y

satisfied

-

is l i m i t - c i r c l e The

latter

Theorem (3.5.17)

and We

the

conclusion For

proved ~

note

negation

extends

of

E constant and

~ d~(x)

at i n f i n i t y

result

i] w h o

o(t)

[0 , ~)

find

a theorem 3.5.2

then

that

= ly

if a n d

Theorem

thus we

on

(3.5.17)

only

if

(3.5.12)

of M.G. for

Krein

equations

holds. [39,

p.

of t h e

882, form

non-decreasing. here

that

(3.5.11)

of t h e

(3.5.11) does

not,

is n o t

superfluous,

in g e n e r a l ,

produce

i.e. the

theorem.

let t

-t

s i

(t) =

e-xdx

+ 1 = -e

0 t

o(t)

=

idx

= t

0

Then

(3.5.0)

is e q u i v a l e n t

-y"

to

+ y = le-ty

Here ~oo

J

tld°(t) [ = 0

while

t e

[0 , ~)

(3.5.18)

154 oo

~[

t 2 Id~ (t)

<

oo

0

However

(3.5.18)

solution

y(t)

is l i m i t - p o i n t

= exp

is n o n - d e c r e a s i n g . we

shall

bounded

t

which

Thus

see p r e s e n t l y , variation

over

slnce is n o t

(3.5.11)

for in

I = 0

it h a s

L 2 ( V : I)

cannot

since

be o m i t t e d

it is n o t

sufficient

[0, ~)

In fact,

that if w e

the

and ~

only

as

b e of assume

that

I~ ~Id~(t) I < ~ then

(3.5.12)

(3.5.0) For

is no l o n g e r

to be

limit

both

<

6 <

necessary

1

(3.5.19)

and

sufficient

for

circle.

let

~(t)

=

(x+ l)-4dx

(t) = ~

Then

0

(3.5.0)

reduces

_y,, +

A computation (3.5.19). it a d m i t s

shows

dx

•

to

3 2 y = i 4 ( x + I)

However the

( x + i)-2

that

Y 4 ( x + i)

(3.5.12)

(3.5.20)

is l i m i t - p o i n t

=

[0, ~) .

is s a t i s f i e d

solution

y(t)

on

( t + i) 3/2

along

since

for

(3.5.20)

with I = 0

155

which

is n o t

COROLLARY

in

such

b

, c

n

the

be

n

real

sequences

c

n

> 0

for

all

n

that

i-/--

n=0

Then

I)

3.5.2: Let

and

L2(V;

Tc +

0 cj-I

a necessary

and

limit-circle

n

Cn-i

sufficient

case

is

-bnl

< ~

condition

(3.5.21)

that

(3.5.8)

be

in

that

[

1

2 (3.5.22)

n=0

Proof: implies result

We

define

(3.5.11) follows

j=0

cj-i

~ , o and

as

we

< ~

in C o r o l l a r y

(3.5.22)

after

]an]

note

3.5.1.

is e q u i v a l e n t

to

Then

(3.5.21)

(3.5.12).

The

that

n I Cj_l

tn =

since

t_l

for

defining

similar

differential

-y"

by

t

1

l

Q

•

•

= 0

Results stated

n = 0

~,

as

those

equations

+ q(t)y

o

to

of of

= lr(t)y

the

indefinite

Corollary

3.5.1-2

the

form

t {

[0, ~)

integrals

of

can

r, q

be

156

respectively

§3.6

and

limit-point,

turn

to the Krein

sequel

that

with

space

(Appendix

Hilbert course

with

space. of t h e

be a d a p t i n g

See proof

[a, ~)

is e q u i v a l e n t

operator ~

in

absolutely known,

in a

continuous,

since we

are

operator

in a

for e x a m p l e

[46,

§18.3].

In the

of

general

the more

of E v e r i t t

shall both

a > -~

As

, v

we

functions

variation

of

"formally

self-adjoint"

by

~[y] (x)

i

It

differential

following

right-continuous

is d e f i n e d

order

a property

function

are

is w e l l

of

of

§17 a n d

[15,

equivalence, p.

42]

we

shall

to g e n e r a l i z e d

expressions.

In the

I =

~ , ~

OPERATORS:

expression.

limit-point

the

case.

the n o t i o n

in t e r m s by t h i s

When

and

a second

DIFFERENTIAL

of a p a r t i c u l a r

equivalence

an a r g u m e n t

differential

defined of

in t h i s

interpreting

(3.5.0),

III.3).

is n o n - d e c r e a s i n g

now dealing

be

the c o n c e p t

above-mentioned

of

shall

"J-self-adjointness"

(3.5.0)

on

we

of an o p e r a t o r

out

(3.5.11-12)

OF G E N E R A L I Z E D

associated

the d o m a i n

will

the

interpreting

J-SELF-ADJOINTNESS In the

of

then

=

V

.

locally usual

d~(x)

of all

we

operator

generalized

as in s e c t i o n consists

The

assume

y' (x) -

3.2 of t h i s functions

that

9 , ~

are

of b o u n d e d

variation

denote

total

i

the

generated

differential

by the

expression

(3.6.0)

y(s)do(s)

chapter.

Thus

f E L 2 (V ; I)

the such

domain that

157

i)

f

ii)

f

is l o c a l l y has

absolutely

at e a c h

point

continuous

x e

[a, ~)

on

I

a right-derivative

!

f+(x) iii)

The

{ f'(x)

function

~(x)

E f'

(x)

-

fx

f(s)d~(s)

a

is iv) For

V-absolutely

~[f](x)

f ~ P

continuous

locally

on

I

.

e L 2 ( V ; I)

i

is d e f i n e d

by

if = i[f]

The

notions

expression when

~

of

"regularity"

(3.6.0)

are

if the

set

values

is b o u n d e d

growth

of

is n o t

of

o

in

in s o m e

then

In o u r

3.2

that

if

are due

definitions

and

below

[35,

p.

regularity"

of

249]

case

I = and

the

of

V(x)

if t h e

and,

in the

end

a

and

the

a

belongs

to K a c [a, ~)

a

is r e g u l a r the

set of

set of p o i n t s

in a d d i t i o n

to be

the b a s i c

the e n d

that

right-neighborhood

it is s a i d

regular

case

say

of g r o w t h "

from below

definitions

section

shall

"points

variation regular

latter

we

is b o u n d e d

is c o m p l e t e l y These

defined

"complete

is n o n - d e c r e a s i n g . In g e n e r a l

bounded

and

(3.6.1)

of

singular. to the

o a

its

of

is of .

The

If end

interval

a a

concerned.

[35]. It is t h e n assumptions

is c o m p l e t e l y

clear on

from

~, ~

regular.

of If

the

158

I =

[a, b]

in t h e

then

be

a, b

note

case

of

a finite

of

a solution

equal. by

that

This

setting

respectively

THEOREM

on

are

since

can

each some

both

completely

regular.

9 , ~

are

continuous

interval,

the

left

y(t)

also

equal

be to

interval

of

and

(3.5.0)

seen

by

~(a)

, 0(a)

at

,

right-

will

exist

extending

containing

a , b

~ , ~

and

~(b)

there past , ~(b)

[a , b]

3.6.1: Let

£[.]

ends

We

derivatives and

the

on

I =

I

[a, b]

Let

g

be be

a finite

interval

any

function

[y]

= g

in

and

consider

L 2 ( V , I)

The

equation

has

a solution

y(x)

satisfying

y(a)

y' (a)

if a n d

only

solutions

Proof:

We

if

of

the

the

note

function

homogeneous

that

(3.6.2)

f

is

= y(b)

= y' (b)

g(x)

0

(3.6.3)

= 0

(3.6.4)

=

is

equation

J-orthogonal i[y]

J-orthogonal

to

to

all

= 0

g

if a n d

only

if b f(x)g(x)dv(x) a

= 0

(3.6.5)

159 (The

J-orthogonality

L 2 ( V ; I)

,

see A p p e n d i x

This Lemma

I].

equation y(a)

theorem

(3.6.2)

= 0 ,

= 0

can be ~ r o v e d

has

y'(a)

3.2.0,

a unique

Theorem

conditions

solution

= I

,

z 2(b)

= 0

,

z 2(b)

y

and

=

I Ixl d

~[3.6.4)

p.

[46,

1.3.1

which

62,

the

satisfies

system

of

solutions

of

= 0

we

3

[ y' zj -yzj]b',

Ixl-

j = 1, 2 ,

reduces

and using

the b o u n d a r y

j: 1 (3.6.7)

(x) d9 (x)

z

(3.6.6)

to

[ -y' (b) g (x)

find

a "

= 0 ,

(3.6.6)

= 1

J

y (b)

Thus

in

Z[y] (x) zj (x) dg (x)

b a

z

J

£[zj]

above,

to

zj (x)d~(x)

ib y(x) a

I

in

!

3.2.1

g(x)

that

as

and Theorem

!

By n o t i n g

product

satisfy

z l(b)

:

exactly

be a f u n d a m e n t a l

z l(b)

Applying

J-inner

= 0

zi , z2

which

f r o m the

III.3.)

F o r by T h e o r e m

Let ~[z]

stems

is s a t i s f i e d

if and

only

if

j= 2

(3.6.7)

vanishes

for

160

j= 1 , 2 ,

i.e.

solutions

of

f

if

£[z]

J-orthogonal

= 0

and thus

N o w s i n c e the m e a s u r e absolutely V

continuous

the q u a n t i t y

with

(section

dV(x)

the c o n c l u s i o n

induced

respect

by

~ ,

s y s t e m of

follows. in

to the m e a s u r e

(3.6.0),

is

induced by

3.1)

d~ (x) dV(x)

Consequently,

to a f u n d a m e n t a l

exists

[V]

(3.6.8)

the e x p r e s s i o n

y' (x) -

yd~

= -dV(x----~" d~(x-----~ y' (x) -

yd~

a

(3.6.9)

V-almost

Thus

everywhere

if w e d e n o t e

(section

by

by

[24, p.

£%[y]

w e see t h a t

i, and T h e o r e m

the e x p r e s s i o n

to

£% £

dV(x)

is a n o t h e r by

(3.6.9),

£%[y] (x)

and b o t h of t h e s e (3.6.10)

Ex.

defined

by

A].

y ~ D

3.2),

£%[y] (x) -

related

135,

gives

rise

generalized i.e.

=

d~

~(x)

are d e f i n e d to

an

y' (x) -

for

yda

differential

expression

y c D

• Z[y] (x)

(3.6.11)

on the same d o m a i n

operator

(3.6.10)

Lt

on

D ,

D where

Thus D

is

161 the

domain

of

L

defined

earlier,

in t e r m s

of

the

Gram

that

for

y

e D

Ly = d___~_~. dV

LtY

or,

such

operator

,

(3.6.12)

J

defined

in A p p e n d i x

III.3,

L % = JL

If,

in T h e o r e m

then

we

can

conclusion usual

3.6.1,

replace will

then

orthogonality

we

assume

i[y]

in

follow in

(3.6.13)

that

v

(3.6.2)

with

is n o n - d e c r e a s i n g

by

£%[y]

and

J-orthogonality

L 2 ( V ; I)

,

since

the

being

v { V

the

in t h i s

case. !

We

now

define

a new

operator,

denoted

by

L

o

,

with

l

domain

DO

defined

by

[46,

p.

60]

!

P0 -- {f c P : f - 0

outside

a finite

interval

[e , 8] c (a , b) } (3.6.14)

!

The

restriction

of

the

operator

L

to

!

DO

defines

L0

!

Thus

for

y

e

DO

' L0Y

=

Ly

!

Similarly

we

can

define

(L~)

(3.6.15)

= Z [y]

%

(L o )

= Lfy

, %

by

y

~

:

Zt[y]

(D 0)

= DO

(3.6.16)

162

THEOREM

3.6.2: !

a)

If

y • 90

z c D

,

then

!

[/0y , z] =

where

[ , ]

side of

(3.6.17)

[y, Lz]

J - i n n e r p r o d u c t d e f i n e d by the left hand

is the

(3.6.5). !

Moreover,

the o p e r a t o r

L0

J-hermitian,

is

!

i.e.

!

!

[L0Y , z] =

y , z e 90

[y , [0 z]

(3.6.18)

!

b)

If

y

e

90

,

z

•

D

then

writing

LI

E

(/O)f

we

have (ily , z) = (y , Lfz)

where

( ,

)

is the inner p r o d u c t

Again,

the o p e r a t o r

(L1y, z) =

Proof:

iI

in

L 2 ( V ; I)

is hermitian,

(y, Llz)

i.e.

y, z E P

Both a), b) can be shown as in

(3.6.19)

[46, p. 61] m a k i n g use

of T h e o r e m 3.2.1 so we omit the details.

We now p r o c e e d as in operators

L0

Suppose £, £f

and

[46, §17]

in d e f i n i n g the

L%0

that the interval

are both r e g u l a r on

[a, b]

[a, b] .)

is finite.

(Then

163

We

define

DO =

and,

for

y

the

{y • D :

domain

DO

i.e.

similar.

any

y

operator

for

Proof:

L0

by

(3.6.20)

= ty

L% y

(3.6.21)

.

(3.6.22)

3.6.3:

For

the

operator

~ DO ,

% i0y =

and

the

y(a) = y(b) = y' (a) = y' (b) = 0}

toY

THEOREM

of

We

any

c DO ,

L0 y,

refer

z e D

[i0y , z]

=

[y , iz]

(3.6.23)

(L0%Y, z)

=

(y,

(3.6.24)

is

L+z)

J-hermitian

while

L %0

is h e r m i t i a n ,

z e DO ,

to

[L0Y , z]

=

[y , L0 z]

(3.6.25)

(L0%Y, z)

=

(y,

(3.6.26)

62,

I,

[46,

p.

L0%z)

II]

since

the

proofs

are

164

LEMMA

3.6.1: Rt 0

Let solutions

of

~ range

the

of

L0%

and = 0

~[z]

equation

M

let

be

the

set

of

all

.

Then H : Lg(V;

Proof:

Since

continuous so

M

all

solutions

functions

c H

It

is a l s o

readily

in

solution

(3.6.2),

with

in

DO

Hence

implies lies H

R% 0

in

is

if a n d

a Hilbert

THEOREM

i0

have

to

i%0

3.6.3,

the it

DO

then

solution

Z

if

the

they

all

that

replaced Thus

it

belong

the

H

y

,

and

y

of

y

states to

is

that

g

.

Since

M

(3.6.27)

is a

then

existence

is o r t h o g o n a l

decomposition

to

If i%

then

are

is a f i n i t e

2

by

3.6.1

equation

M

dimension

i~y

only

homogeneous

seen

Theorem

space

domain

Since

and

gonal

=

of

,

(3.6.27)

follows.

3.6.4: The

Proof:

Z%[y]

H

g c R %@

that

the

[a , b]

subspace

of

of

on

dimensional

= R %0 + M

I)

of

the

domain

D

suffices

to

is

zero.

(h, y) of

DO

~%[z]

= 0 = h

operator

is t h e

0

show

for

h

all For

y y

same

that

Letting

f0

be

~ DO

for

every such

E DO

is d e n s e

the

Let we

have,

h

element, z

be by

H

operators

element an

in

orthowe

any

Theorem

165

(z , L%0Y) =

and

so

z

previous

is o r t h o g o n a l lemma,

We is d e n s e theorem

expresses

the

THEOREM

3.6.5:

The symmetric

R %0

a set norm fact

is d e n s e

operators

0 ,

i.e.

is d e n s e

= 0

the

Thus

are

space

the

the

if

it

latter

90

space

% i0

and

by

h = 0

domain

Krein

z • M

in a K r e i n

topology.

in t h e

(h , y)

Consequently

that

i0

=

of

the

H

J-symmetric

and

respectively.

This

Note:

The

shown

to be

follows

rest

in

case

whose

Theorem

of

true

regular

operator

from

the

Theorems

results

this the

more

3.6.3-

in

[46,

general

operator

J-adjoint

ix 0

i0

4.

§17]

can

setting.

be

Thus,

is a c l o s e d

is e q u a l

to

i

similarly e.g.,

J-symmetric

[46,

p.

66,

i].

In a > -~

(Z%[z] , y)

£%[z] =

that

Hilbert

=

to

so

in t h e

i0

in t h e

and

recall

operator

Proof:

(L%z , y)

,

the we

singular

follow

the

case,

i.e.

approach

when

outlined

I = in

[a , ~) [46,

,

§17.4]

!

where

we

begin

with

the

operator

i0

defined

in

(3.6.14-15).

both

defined

on

the

!

We

recall

domain

~

that

L0

and

il

are

same

166

THEOREM

3.6.6: I

The

domain

of

definition

!

00

of

i 0

is

dense

in

!

and

L0

Proof:

is t h e r e f o r e

An argument

a

J-symmetric

similar

operator.

to t h a t of

[46, p.

68]

shows

that

in

.

I

00

,

when

viewed

as

the

domain

of

[1

'

is

dense

H

!

Thus is

LI

is a s y m m e t r i c

operator,

by T h e o r e m

3.6.2,

and

L0

J-symmetric.

W e n o w take space

topology

the c l o s u r e

of

L0

, L0

in the H i l b e r t

and d e f i n e

L0 = L0

it t h e n

follows

f r o m the p r e c e d i n g

closed

J-symmetric

i+

(and so

L)

that

i0

is a

operator.

We n o w p r o c e e d of

theorem

to find a p r o p e r t y

when

£%

of the d o m a i n

is in the l i m i t - p o i n t

0

case

in

L 2 (V ; I)

LEMMA

3.6.2:

[14].

F o r any set of six f u n c t i o n s {gq : 1 ~ q ~ 3} [0, ~) point

each being

and each having x • [0 , =)

,

locally

a finite

{fp : 1 ~ p ~ 3} absolutely

right-derivative

continuous at e a c h

on

167

d e t { [fp qq] (x) } = 0

x~

[0,~)

where !

[fg] (x)

Proof:

LEMMA

See

[14, p.

--- f(x)g+(x)

- g(x)f+(x)

374].

3.6.3: Let

be the

p

be a B o r e l m e a s u r e

s p a c e of s q u a r e

Suppose functions

that

which

integrable

f, g

X > 0

[0 , ~)

"functions"

and let

L2(~)

with respect

are c o m p l e x - v a l u e d

to

p-measurable

satisfy

f ~ L 2 <~; [0,~)I for all

on

,

g ~ T2(~; [0,x) 1

and that

g 4 =2(~; [0,~)) Then

fx {fx

lim x +~

Proof: in

fg d~

0

This result

[15, p.

Let

42] w i t h

i%

T h e n by T h e o r e m

Ig

d~

= 0

0

c a n be p r o v e n

in e x a c t l y

the n e c e s s a r y

modifications.

be the o p e r a t o r 3.3.2,

for

defined

Im I ~ 0 ,

by

t h e same w a y as

(3.6.12),

the p r o b l e m

(3.6.10).

168 £f[y]

has

at

least

I =

[0, ~)

Lemmas

one nontrivial Using

3.6.1-2

Everitt

[15,

(3.6.28)

= ty

we

pp.

this

can

result

4 2 - 45]

[0 , ~)

solution

show,

is l i m i t - p o i n t

on

in

and

L2(V ; I)

Theorem

by a d a p t i n g

to our there

(3.6.28)

3.3.1

where along

the a r g u m e n t

situation,

that

with

of

whenever

follows

!

lim

{f(x)g+(x)-

f+(x)g(x)}

: 0

f, g E ?

(3.6.29)

X-~OO

Conversely I = 0

if

(3.6.28)

In this

independent

case

is l i m i t - c i r c l e , it is p o s s i b l e

solutions

~ , ~

satisfy

L 2 ( V ; I)

and

(3.3.18-19)

say.

consequently

in

!

it m u s t

find

two

be

real

so for

linearly

of

£%[y]

which

to

then

= 0

(3.6.30)

By h y p o t h e s i s D

.

Moreover,

these by

are

in

(3.3.18-19)

!

(x)~+(x)

-

}+(x)~(x)

=

1

hence

lim

Summarizing,

THEOREM

we

[%~] (x)

~ 0

obtain

3.6.7: A necessary

and

sufficient

condition

for

(3.6.28)

to

169

be limit-point is that for all

f,g

c P ,

(3.6.29) be

satisfied.

THEOREM 3.6.8: Let

~

satisfy

(3.5.11).

Then

(3.6.28)

is limit-

point if and only if

i[y] = ly

is limit-point related by

Proof:

fin the

L2(V ; I)

(3.6.31)

sense I

where i

and

i+

are

(3.6.11).

For Theorem 3.5.2 implies that

(3.6.28)

is limit-

circle if and only if

I

where

V(t)

=

Id~(s) ]

I

t2dV(t)=

~

(3.6.32)

However the latter is equivalent to

t 2 Id~(t) I = ~

and Theorem 3.5.2 again implies that if and only if (3.6.33)

and so

(3.6.33)

(3.6.31)

(3.6.32)

is limit circle

is satisfied.

The

result now follows.

COROLLARY 3.6.1: In order that infinity

fin the space

(3.6.31) be in the limit-point case at L2(V ; I) I

it is necessary and

170

sufficient

that

lim

[fg]

x)

= 0

f,g

c D

(3.6.34)

X+~

where

[

of

L%

L,

]

defined

We

D

of

is d e f i n e d

now

L

D

where

:

e ~

the

defined

{f e D :

[0, z)

L

f

,

in t h e

First

is

f(0)cos

for

is d e f i n e d

the

same,

limit-point

of

all

we

the

f E D

f = Lf

on

the

f,g

e D

,

X > 0

the

Let

domain

the

iT = Ji

that

identity

a : 0}

domain

(3.6.35)

,

.

(3.6.36)

same

domain

D

and

f E D

= Lff

case,

note

Lagrange

is

, Lf

~- f~(0)sin

We

now

f Le

is

if

f, g c D

[fg] (0)

Next

D

by

and

Lff

or w h a t

and

L

operators

L

Similarly

3.6.2

earlier.

define

be

in L e m m a

(3.6.37)

proceed

to

show

that,

self-adjoint.

= 0

(Theorem

then

.

3.2.1)

(3.6.38)

shows

that,

for

171

I o { f ( x ),%9[g] (x) - ~[f] ( x ) g ( x ) } d V ( x )

= -[fg] (x)

Consequently X + ~

in

if

£%

(3.6.39)

is and

(L f , g]

and

so

i% e

it c o n t a i n s the

singular

to t h a t shown

in

symmetric

the

domain

case [46,

by

p.

the

3.6.7

of

the

L0

VII).

and

, we

(3.6.38)

In

in

The

proof

the

same

L0

of

find

(3.6.40)

L 2 ( V ; I)

operator

let

to

f ' g ~ De

is d e n 3 e

e

f , g c D

since

defined

this

fashion

is it

in

similar can

be

that

so t h a t

domain

ID

L0 =

[ief , g]

THEOREM

Theorem

DO

71,

and

t [f, L e g )

:

(3.6.39)

[fg] (0)

limit-point use

is

+

L

e

is

3.6.9: of

=

[f , Leg]

J-symmetric.

In t h e

limit-point

self-adjointness

following

i)

ii)

(3.6.41)

f , g e De

of

case,

L% e

the

if a n d

domain

only

if

D D

e

e

properties,

For

all

If

g c D

f e D

e

f, g e D

,

e

,

satisfies then

g e 0

e

[fg] (0)

= 0

[fg] (0)

= 0

,

for

all

is has

a

172

Proof: We theorem

note

that

of N a i m a r k

this

[46,

result

p.

73,

is a p a r t i c u l a r

Theorem

l] and

case

of a

c a n be p r o v e n

similarly. With shows and

that

D

defined

both

(i) a n d

consequently,

adjoint.

On the

the d e f i c i e n c y Consequently

(3.6.35)

are

[46,

if p.

a simple

satisfied

limit-point

hand,

indices the

in

(ii)

in the other

as

case

L% 26]

computation

in T h e o r e m L%

3.6.9

is s e l f -

is s e l f - a d j o i n t of the o p e r a t o r

then are

equation

(3.6.41)

(i~)* z = Iz

has

no n o n - t r i v i a l

self-adjoint

(0 , 0) .

solution

(3.6.41)

in

implies

L 2 ( V ; I)

that

Since

i%

is

the problem

L%z = Iz

z(0)cos

has

no

solutions

point.

Hence

THEOREM

3.6.10: The

L2(V;

I)

in

we have

equation

if a n d o n l y

self-adjoint.

e - z'(0)sin

L 2 ( V ; I)

Thus

e = 0

(3.6.28)

is l i m i t -

proved

(3.6.28)

is in the

if t h e o p e r a t o r

limit-point L%

, e c

case

[0, ~)

,

in is

173

In the

the

Hilbert

following

space

operators

L% e

self-adjoint. limit its

point

adjoint i

t

J-adjoint

by

and Lx e

Let

f

e D

and

Krein

Theorem

so

i

space

exists.

e

, g

[Lf

Let 3.6.10,

is

e

Lx

(i~)

respectively.

e

Then case

discussion

a

us

'

e

adjoint

of

suppose

that

(3.6.28)

J-symmetric

We d e n o t e

will

'

its

is

denote

the

in

i% e the

operator

domain

by

is

and Dx e

e Dx e

, g]

=

[f , Leg] x (3.6.43)

NOW

L % = JL

since

e

e

L% =

,

=

(L~)*

e

JL x = e

=

(JLe)*

L*J e

.

Moreover

L*J

(3.6.44)

e

"k

Substituting

(3.6.44)

into

(3.6.43)

we

find

g

that

E D

LeJ)

But

hence

g

e De

J-symmetric

Consequently D

c Dx e

e

.

Hence

Dx

c D

D

=

e

and

since

Dx e

and

so

be

J-self-adjoint

e

L i

e

e

1s

is

J-self-adjoint. On each

e

E

the

other

[0 , ~)

,

hand, so

that

let

i

we

have

L

=

Lx e

or

for

•

174

[f , Leg]

Using f,g

the E ~

Lagrange

=

[ief , g]

identity

in

(3.6.45)

we

find

that

for

e fx 0 = x+~lim 0 { f i - - ~ - g / e f } d v

Since

(3.6.45)

f , g E P

f, g ~ D

,

e

lim

g ( 0 ) f ' (0)

{f(x)g'

= x÷~lim [gf' _ ~,f]x0

- g' (0)f(0)

(x) - f' ( x ) g ( x ) }

= 0

=

"

so t h a t

(3.6.46)

0

X-~OO

for

all

f, g e

equality suffice

in to

fact show

We

e holds that

functions,

f e 0e

holds.

For

if

[fg] (x)

- f ( x ) g ' (x)

=

where

now

for

if

wish

all

show

f, g

f , g

' g e ~B

to

are

'

that

E ~

any

e , B c

the

For

two

latter

this

it w o u l d

real-valued

[0 , ~)

then

(3.6.46)

f , g c D

- f' (x)g(x)

[fRgR ] (x) + [fIgi ] (x) + i{ [figR ] (x) + [gifR ] (x) }

f = fR + ifI

'

g = gR

+ igI

Hence

for

given !

f c ~ are

,

fRc

real.

B ~ e

,

for

A similar B ~

Thus f E ~e

~e

, g

some result

[0, ~) let

e D8 ,

The

f , g ~ e

e e

[0 , ~)

holds

for

result

now

be

two

.

We

real will

since fI

fR(0)

e ~B

,

fR(O)

where

follows.

valued show

functions

that

under

with certain

175

hypotheses

on

a , ~

we

can

find

a

function

g*

E D

such

that

[fg] (x)

One

such

=

[fg*] (x)

condition

continuous

so

is

that

the

for

the

g*(x )

sense be

of

defined

the

some

a , b

absolutely

are

also

=

to

x

o

.

be

~-absolutely

function

~

,

~d~

measures

defined

to

(g(x) ~ ax+b

=

be

continuous

wish

Let

~-measurable

by

}

,

x

-> 1

[

,

0

< x

o

Let

and

have

(3.6.47)

determined. so

g*

We

< 1

need

g*(x)

to

be

that

g*(1)

We

large

by

g*(x)

where

all

following:

do

in

for

e D

= g(1)

(3.6.48)

where

~

is

as

above.

Consequently

g*(0)cos

(3.6.48-49) function

then

g*(x)

e

determine has

the

- g*' ( 0 ) s i n

a , b following

in

~

=

(3.6.49)

0

(3.6.47).

properties:

The

resulting

176

g*

{ A C I o c (0 , oo)

g +*'

and

(x)

-= g +*, (x)

G(x)

exists

Sx

-

for

each

x

> 0

g*(s)d~(s)

(3.6.50)

0

is

then

defined

for

G(x)

for

x

> 0

Since

~-absolutely more 0

< x

> 0

a

-

g*

<

implies 1

.

since Thus g

ig

given

~ g

for

is.

(s)~(s)d~(s)

for x

(3.6.51)

> 1

,

G(x)

since

,

g

is e D6

Further-

9-absolutely

is

m-absolutely

continuous

continuous

L2(v; t0 ~)) (3.6.49)

there

large

and

From

> i

is

implies

exists

g*

therefore

[fg] (x)

=

lira

that

~ D

for

[fg*]

f

g*

~ D

such

that

E D

,

it

then

(x)

X-WCO

=

(3.6.46).

x

G(x)

X-~OO

by

fact

0

G(x)

~

in

g*

g

g e D8 ,

lim

x

that

Finally

x

and

if

Hence

Lg*

i =

continuous

(3.6.51)

if

=

x

the

0

preceding

discussion

that

[fg] (x)

:

0(i)

f , g

{ D

.

follows

and

177

Theorem case

3.6.7

now

and Theorem

self-adjoint. the

3.6.10

Hence

with

jumps o

on

with

difference

then

i%

.

same

forces

under

When

points

then

operator

is

J-self-adjoint

differential

since

hypothesis

on

~, ~

than

those

THEOREM

3.6.11:

other

to be implies

step-functions continuity

thus

the

result

in t h i s

in the o r d i n a r y

operators

i

in the K r e i n

continuous

general

of

and

A similar

operators

[0 , ~)

absolute

is s a t i s f i e d

ordinary

limit-point

above-mentioned

the

9

this

~ E

are b o t h

to

if it is s e l f - a d j o i n t .

weakening

the

respect

absolutely

is in the

i% ,

o , ~

and only

are

(3.6.28)

J-self-adjointness

of

o , ~

at the

that

the

self-adjointness

restrictions

of

implies

case

sense.

we may

space

if

holds

for

both

~,

In f a c t

include

previously

resulting

by

more

mentioned.

Suppose

(3.6.52)

So tido(t) I <

Then

a necessary

to be

and

J-self-adjoint

sufficient

condition

for

is t h a t

I/ t21d~(t) I = Proof:

For

the

assumption

limit-point

if and

The

being

latter,

L

the o p e r a t o r

only

if

(3.6.53)

(3.6.52)

implies

(3.6.53)

holds,

limit-point,

is n e c e s s a r y

that

(3.6.28)

by T h e o r e m and

is

3.5.2.

sufficient

178

for

the

operator

to

is

being

The quoted these

theorem [4, p.

As w e

saw

be w a i v e d ,

theorems,

for the

sake

and

to g i v e

~

the

and

~

to the

[38,

Thus,

75].

equations

Using

by means

the

therein.

existence a similar

case

with

the m e t h o d s analogs

we

between

the

the

here notion

functions,

refer

result

of a u n i q u e

usual

of C h a p t e r s of all

to

In p a r t i c u l a r ,

result

(3.6.52)

n o t be u n d e r t a k e n

spectral

the

of

a(t)

requirement

relation of

discussion.

of a r e s u l t

particular

task will

existence

of the

extension

discrete

For

is e q u i v a l e n t

preceding

the

is n o n - d e c r e a s i n g

is e q u i v a l e n t p.

3.5

the

this

which

expressions

is n o n - d e c r e a s i n g ,

the bibliographies

constant

in the

in g e n e r a l .

and

the

is a m i n o r

in s e c t i o n

of b r e v i t y .

when

from

122],

however

limit-point

the c a s e

self-adjoint

differential

it is p o s s i b l e

above

of

above

generalized

cannot

to be

J-self-adjoint

by L a n g e r

= const.

1-2

i%

[35],

[38]

where

d

is

in T h e o r e m

spectral

applies

in

3.6.11

function

to d i f f e r e n c e

construction

(cf.,[64], [65], [661

REMARKS:

i.

In T h e o r e m (3.5.11) This is

3.6.8

the h y p o t h e s i s

c a n be o m i t t e d

follows

essentially

limit-circle

then

can only

have

However,

(3.6.11)

without from

it m u s t

solutions states

that

in

should

affecting

the

(3.6.11). be

so for

i 2 ( V ; I)

that

~

~

and

conclusion.

Since ~ = 0 for Z%

satisfy

if

.

such have

(3.6.28)

Thus a

it

k the

same

179

solutions

2.

to either

homogeneous

limit-circle

if and only

Consequently

the result

The

latter

that

discussion

(3.6.31)

holds.

if

operator.

For p r o b l e m s

Daho

of g e n e r a l i z e d

treated,

when

of Kac

~

[37].

In these

in the

of the

J-self-adjoint

chapter

(3.6.31),

that,

resolvent

sets of s e c o n d - o r d e r

operators

with

"indefinite

empty.

In the

not

resolvent

the

generated

of such finite

set of the

in

weight [ii,

p.

an o p e r a t o r number

171]).

into

a homogeneous

the paper

of

and c o m p l e t e n e s s operators

the

were

also

in a paper to

resolvent

set

In the

in the r e g u l a r difference

case,

the

and d i f f e r e n t i a l

functions"

are non-

it is not k n o w n w h e t h e r J-self-adjoint

need be empty It w o u l d

necessarily

of n o n - r e a l

with

Sturm-Liouville

function

is

J-self-adjoint

be non-empty.

weight

case

by an o r d i n a r y

indefinite footnote

singular

(3.6.31)

we cite

that

operators

show

for

it is n e c e s s a r y

case,

3.6.7)

(3.6.29)

non-decreasing,

arguments

limit-point

we w i l l

3.3.1.

to the e x p a n s i o n

differential

is a s s u m e d

assume,

next

related

The e x p a n s i o n

theorems

is

if

of a r e l a t e d

at the origin,

[II].

Z%

(Theorem

if and only

of the p r o b l e m

and L a n g e r

implies

of l i m i t - p o i n t

to the e x i s t e n c e

condition

is, by T h e o r e m

therefore

the n o t i o n

eigenfunctions

Thus

follows.

equivalent

boundary

Z

is l i m i t - p o i n t

Hence

equation.

seem

consists

eigenvalues

operator

problem or not that

with (see the

the

of at m o s t

because

or

spectrum a

of the result

180

in t h e

§3.7

regular

case.

DIRICHLET INTEGRALS ASSOCIATED DIFFERENTIAL EXPRESSIONS:

In t h i s "maximal

domain"

basic

material

shall

extend

generalized three-term

DEFINTION i)

~

for

the

we

examine

defined this

differential

properties

earlier

c a n be

Notions

GENERALIZED

various

in t h e

section

various

recurrence

found

contained

expressions

section.

and

in

The

[17].

therein

thus,

of t h e

We

to c o v e r

in p a r t i c u l a r ,

relations.

3.7.1:

The

is s a i d

section

WITH

operator

to h a v e

the

~oo

L ,

defined

in

(3.3.5),

Dirichlet property

with

domain

at infinity

(DI)

if

!

] If+(x) 12dx <

,

f • P

(3.7.1)

a and

I

~If(x) 121do(x) I < ~

,

f • D

(3.7.2)

a

ii)

k

is Conditionally

lim

Dirichlet

fg d~

(CD)

at infinity if

exists

(3.7.3)

X-~Oo

and

is

f e D

iii)

finite

for all

f, g • ~

and

if

(3.7.1)

holds

for all

.

k

is

said

to be Strong

Limit-Point

(SLP)

at

infinity

181

if

l i m f(x) g+(x)

= 0

f , g c D

(3.7.4)

X-~Oo

iv)

is Limit-Point

i

(LP)

at i n f i n i t y

if

!

lim

{f(X)gx(X)

- f+(x)g(x)}

f,g

: 0

c ~

(3.7.5)

X-~OO

This

definition

section

is c o n s i s t e n t

3.6 b e c a u s e

It f o l l o w s is s t r o n g

limit

with

of T h e o r e m

immediately

point

then

i

the u s u a l

3.6.7

from

and

definition

Remark

i.

(iii) - (iv)

is l i m i t - p o i n t ,

of

that

if

i.e.

SLP => LP

The

converse

given if

is n o t v a l i d

later).

i

Similarly

is D i r i c h l e t

then

(3.7.6)

in g e n e r a l

(an e x a m p l e

it f o l l o w s

from

will

the d e f i n i t i o n s

it is c o n d i t i o n a l l y

Dirichlet,

DI => CD

with tion

the

CD =>

313-14] case

converse SLP

false was

for o r d i n a r y

the p r o o f For

is n o t v e r y

suppose

that

[17,

to be v a l i d

differential

that i.e.

(3.7.7)

in g e n e r a l

shown

be

p.

313].

The

by E v e r i t t

expressions.

[17,

In the

implicapp. general

different. i

is

CD

at

infinity.

We

find

182

upon

integrating

IX

__

f(Ig)d~

by

=

parts

that,

(fg') (a)

-

for

all

(fg') (x)

f, g

IX

+

a

i

limit

as

for

,

(f'g' + fgd~)

a

Since

hand

e ~

is

CD

the

x ÷ ~

integral every

Moreover

tends

to

f , g ~ ~

~ lim

If p o s s i b l e ,

right-hand

tends

to a f i n i t e

f , [g c L 2 (V ; I)

since

a finite

limit

as

x + ~

the

left-

Consequently,

,

(fg')(x)

let

integral

exists

us

assume

that

lim

If(x)g'

(x) I =

and

is

e ~ 0

finite.

for

some

(3.7.8)

f,g

~ P

Then

I~I

> 0

X-WOO Thus

for

x > X

,

1 I f ( x ) g ' (~) I > ~

If

f

is u n i f o r m l y

inequality zero.

implies

bounded that

Consequently If

increasing

f

is n o t

sequence

g'

above

g'(x) { L2(X,

uniformly {Xn}

with

(3.7.9)

I~

on

[X, ~)

is u n i f o r m l y ~)

which

bounded xn ÷ ~

then

the

bounded

latter

away

from

is a c o n t r a d i c t i o n .

then

there

along

exists

which

an

183

f ( x n)

(3.7.9) and

then

implies

÷ co

that

,

n ÷

If(x) I > 0

for

x ~ X1 ,

say,

hence

1 If' (x)g' (x) I > ~

Integrating

the

latter

f' (x) f(x)

Isl

over

x => X 1

[X 1 , x n]

and

letting

n ÷ ~

we

find

If'(x)~'(x)

dx

=

X1 a contradiction,

by

f' , g'

( L 2 (a, ~)

Again,

in g e n e r a l ,

p.

313].

the

Hence this

now

interpret

relations, ordinary

§3.8

the

n

> 0

conclusion

implication

=>

these theory

differential

CD

for

(c n) all

having

is

since is

both

that

CD

irreversible

=>

SLP

[17,

SLP

=>

for

been

LP

.

(3.7.10)

three-term developed

recurrence in

the

case

of

expressions.

, (bn) n

=>

results

DIRICHLET CONDITIONS RELATIONS: Let

c

the

inequality,

Thus

DI

We

Schwarz

Let

FOR

be

THREE-TERM

real

(a n )

sequences be

RECURRENCE

and

a sequence

suppose of

real

that numbers

184

where

an

~ 0

sequence

where t

n

÷

both that

of

C_l ~

for

numbers

tn

tn_ 1 -

-

and

as

n ÷

Now

define be

these

o (t n)

Let defined

=

n=

a

an

increasing

by

1 Cn-i

t_l

be

(t n)

is

(3.8.1)

0 , 1 , ...

fixed.

We

also

assume

by

requiring

that

~ step-functions

constant

have

_

n

real

> 0

v , ~

all

on

v , o

[tn_ 1 , t n)

discontinuities

0 (t n

0)

_

cn

=

at

Cn_ 1

+

-

n=

l

the

b n

0,

(t n)

1,

...

only,

n=0,1,

,

that and

given

by

...

and

9 ( t n)

We that

also

-

suppose

neither Let

~(t n-

0)

that

~ , o

have ~

a

be

jump

=

at

-a n

are

,

n=

both

0 , 1 .....

continuous

(3.8.2-3)

at

a

and

infinity.

summable

and

y

y do

consider

the

differential

equation

£[y] (x)

=

d~(x)

(x) -

=

~(x)

x

e

[a , ~) (3.8.4)

where above

~ , o as

the

are

defined

solution

of

above.

Rewriting

a Volterra-Stieltjes

the

solution integral

of

the

185

equation

we

see that,

solution

y(t)

then

satisfies

Yn

-Cn Yn+l

using

is linear

the m e t h o d s

on

[tn_ 1 , tn)

the r e c u r r e n c e

- Cn-i Yn-i

of C h a p t e r

+ b n Yn =

and

if

i, the Yn

~ Y(tn)

relation

n=0,1,

an %n

... (3.8.5)

where Thus

~n H ~(tn) the d o m a i n

generalized gonal

Moreover space

of the o p e r a t o r

differential

curves,

continuous

D

i.e.

and

the

space

£2(Ia I)

,

expression

each

linear

function

on

generated

above

in

~

[tn_ 1 , tn)

L 2 ( V ; I) i.e.

i

f e i2(la I)

of p o l y -

is a b s o l u t e l y

for

becomes,

consists

by the

n= 0 , 1 , ....

in this

case,

the

if

oo

[. lanIIfn12

<

(3.8.6)

co

0

where Since

fn z f(tn) the d o m a i n

D

is e s s e n t i a l l y

D = {f ~ L 2 (V ; I) :

we

see t h e r e f o r e

sequence

(fn)

that

Z[f] (x)

a function

satisfies

la n] Ifn 12 < 0

( L 2 (V ; I) j1

f e D

if and only

(3.8.7)

if the

186 and

if

£[f]

--- - C n f n + l - C n - i a n

n

fn-l+bnfn

n=0

, 1 , ...

(3.8.8)

then

c i2

£[f]n

Hence

the

given

by

resulting

"difference

= {f = (fn) •

where that

i[f]n each

Thus

we

f c D

is d e f i n e d

such

[tn_ 1 , t n)

~,2

sequence

and

belongs

identify then

the

f' (t)

(la]) operator"

(la]) : £[f]n

in

(3.8.8).

defines

We

a function

to t h e

domain

•

domain

(3.8.7)

is c o n s t a n t

on

D

with

has

,£2

domain

D

(]al)}

have

(3.8.9)

to k e e p

which

is

defined

[tn_ 1 , t n)

linear

by

(3.8.9).

in m i n d

So

on

(3.8.7). if

and

f

f' (t)

-f n n-i tn - t n - i

-

= Cn-l(fn-

Thus

if

fed

fn-I )

t •

[tn_ 1 , t n)

, t !

If+(x) I 2dx a

If+ (X) I2 dx

= o

tn_ 1

(3.8.10)

187 =

c2

~

n-i ]fn

_ f

12

n-i

• (t n - t n _ I)

O oo

(3.8.11)

: [ C n _ i I A f n _ I] 0

Similarly

for

f e ~

, t

f(x) 121do(xll

-- [ 0

If(x)

IdoCxll

tn_ 1 t +0

= ~

ftn

0

since

d

is c o n s t a n t

on

n

If (x)

12Ido (x) I

-0

[tn_ 1 , t n)

,

n =

0 , 1 .....

oo

: [

IfnI2 ICn + Cn_ 1

(3.8.12)

bnl

0

DEFINITION

3.8.1:

The

difference

operator

L

defined

on

(3.8.9)

if = £[f]

by

(3.8.13)

where £[f]

is s a i d f =

(fn)

= £[f]n

to h a v e ,

g =

= a n l { b n f n - C n - i fn- 1 - c n f n+l }

(3 " 8 " 14)

Dirichlet property at infinity if for all

the (gn)

in

~

we h a v e

Cn_iIAfn_112 0

<

(3.8.15)

188 and

(3.8.16)

Ic n + en_ 1 - bnl If n 12 < 0

Next

let

x ~ [t n , tn+ I)

Then

for

f ,g E D

,

t .+0 f(t)g(t)da(t)

=

fgdc~ + 0

ix

-0 ]

fg do

t +0 n

n

: [0 fj %j (c j + cj -i - bj)

since

a

is c o n s t a n t

DEFINITION The

on

(t n , x)

This

motivates

3.8.2: difference

at infinity if for all

operator

L

is Conditionally Dirichlet

f ,g ~ D ,

(3.8.15)

holds

and

m

lim m÷~

exists Let

and

[ 0

fn g n ( C n + C n - l -

(3.8.17)

bn)

is f i n i t e .

f, g e D

Then

!

lira

(3.8.18)

X+OO

= lim n÷~

{f(tn)g'+(tn) - f + ( t n ) g ( t n ) }

= lim

{fnCn

n+oo

by

(3.8.10).

Agn - ~n C n A f n

}

189

= lim C n {fn g n + l

- fn+l gn }

(3.8.19)

n+~

whenever stems

either

of

(3.8.18-19)

exists.

lim f(x)g'(x)

= lim C n f n A g

n

f, g c 9 .

(3.8.20)

n+~

DEFINITION

3.8.3:

The d i f f e r e n c e

operator

L

is said to be in the

Strong Limit-Point case at i n f i n i t y

lim c n fn Agn n÷~

is said

all

to be

if for all

exists

f ,g e D

(= 0)

in the Limit-Point c a s e

(3.8.21)

at i n f i n i t y

if for

f, g c

lim c n {fn g n + l - fn+l gn } = 0

The

also

the r e l a t i o n

X~

L

F r o m the l a t t e r

latter

point

is c o n s i s t e n t

for a t h r e e - t e r m

[3, pp.

with

[32, p.

We note,

in p a s s i n g ,

section

3.6 a l s o

includes

cases.

Thus

a certain

(3.8.22)

difference

the u s u a l d e f i n i t i o n

recurrence

498-99],

425,

relation.

Theorem

(See for e x a m p l e ,

that the t h e o r y

for all

operator

[

of l i m i t -

2].)

the d i f f e r e n c e

holds

(3.8.22)

developed

operators

f, g e V defined

by

in

as s p e c i a l

if and o n l y

if

190

i f :

£[f]

f E ~

(3.8.23)

and f_icos

is

J-self-adjoint

for

all

n

,

~ - C_l(f 0 -f

in

then

symmetric

extension

statement

is a l s o

(3.8.22)

is

"maximal" L

=

Krein

i

is

of

i

true

in

satisfied

operator

it

L

space

£2(Iai)'

self-adjoint must the

implies

[3, p.

and

coincide

Kreln

499],

If

an > 0

consequently

with

space

the

(3.8.24)

~ = 0

i

(This

setting.)

When

self-adjointness it

then

every

follows

of

the

that

L Moreover

generally

EXAMPLE

the

in

(3.7.10)

are

valid

and

~ Z2

and

3.8.1: c

n

= n

I1 Yn

b

a

n

= 1

and

if n = 2 m

let

some

m

> 0

if

n

2 TM

by

n

b

=

CnYn+l

n

Y0

,

= 1 n

Define

implications

irreversible.

Let

where

the

1)sin

= 0

+ Cn-I Yn-i

n = 1,

2,

...

that

Yn

Yn

say.

A computation

shows

191

if

zn

is a l i n e a r l y

Yn Zn+l

- Yn+l

by

the d i s c r e t e

z

c Z2

n

would

the

analog

LP

-

n

the

n

=

of the W r o n s k i a n

inequality

right

Cn Y n + l

is

-

a contradiction

while

solution

const

=

Zn

Schwarz

produce

finite

independent

side

+cn-I

applied since

the

diverges.

Yn-1

then

we m u s t

have

1 , 2 , ...

identity. to the left

Thus

latter

if

identity

side would

be

Thus

- bnYn

= 0

(3.8.25)

However

lim n Yn AYn

does

not

even

exist.

lim inf n +~

Thus

(3.8.25)

show

that,

irreversible next

result

THEOREM

In

fact,

n Y n A Y n = -i

is n o t

SLP.

in g e n e r a l , even

for

follows

,

lim sup n Y n n÷~

Other

the

implications

three-term

from

examples

remarks

AYn = 0

may

(3.7.10)

recurrence i,

be

2 of the

and

sufficient

condition

preceding

for

to

are

relations.

3.8.1:

A necessary

found

The section.

192

-Cn Yn+l

to be for

LP

all

in t h e

(here

a

all

n

lanl

is

that

(3.8.26)

(3.8.22)

should

hold

O)

> 6 > 0

for

(3.8.26)

independently

of

Proof:

let

If we

n

the

is l i m i t - p o i n t coefficient

f ~ D

,

and

that

0 < cn < M

fn ÷ 0

as

(3.8.22)

n ÷ ~ holds

[

for for

b

in the

£2(lal)-sense

n

then

]fn ]2 < 6 -I

bounded

all

.

Then

Thus

~

n

= la n y n

3.8.1: Let

for

+ bnYn

£2(lai)-sense

f, g c V

COROLLARY

- Cn-lYn-i

lanl [fn 12 <

every

f { D

all

f , g c D

= 1

in

Since .

The

the

Cn

result

now

follows. If we

let

c

n

= a

n

(3.8.26)

we

find

that

equation

A2yn_l

is a l w a y s p.

LP

+ b n Y n = ly n

in the

i2-sense

n=0,

(see

i,

[32,

...

p.

436]

and

[3,

499]). Unlike

the

results

in C h a p t e r

2,

the

limit-point,

the

are

193

limit-circle

theory

substantially equations. general

of d i f f e r e n c e

from

One

equations

the a n a l o g o u s

reason

limit-point

for

this

criterion

theory appears

for r e c u r r e n c e

relations.

and

f =

then

let

should

(fn)

exist

and

automatically

be

need

unbounded. thus that,

Hence

say,

if

and

to s t r e n g t h e n same

hand

to the

interpretation set

c

that

(3.8.22)

other

= ly

of

to b e

f ~ D

then

= a

n

lim

fn

is

if w e

consider

is

operator ~)

,

far

generated Thus

if

upon

and

by

the

f e L2 ,

f'

satisfied

b(x) have

and

to e n s u r e limits

at

theorem

conclusion

we

show

of C o r o l l a r y

that 3.8.1

it is p o s s i b l e under

the

3.8.2: Let

Corollary

a

n

3.8.1.

~ 0

,

f

and can be essentially

from being

imposed f

[0, oo)

set of h y p o t h e s e s .

THEOREM

= ]

n

satisfied.

following the

the

at i n f i n i t y

have

be

x c

L2(0

(3.6.29)

(3.6.29)

In the

if w e

Consequently

of

to a l i m i t

conditions

infinity

domain

is a s u b s e t

tend

For

its

it is n e c e s s a r y

On t h e

+ b(x)y

the m a x i m a l

not

and

related

equation

y"

expression

zero.

satisfied.

the differential

then

c £2

for d i f f e r e n t i a l to be

(3.6.29)

(3.8.22)

differs

and

c

n

satisfy

the h y p o t h e s e s

of

194

Then

(LY)n

- a-1 n {-c n Y n + l

- Cn-lYn-i

n = 0 , 1 , ...

+ bnYn}

(3.8.27)

has

the

Proof:

Dirichlet

property

According

to

at

infinity.

Definition

3.8.1

it

is n e c e s s a r y

to

show

(3.8.15-16). Let Since

[anl

in

be

(lal)

> 6 > 0

square-summable £2

£2

f {

that

f c z2

then the

such

usual

,

sense.

(Lf)n

i.e. Since

_

the

(Afn)

hypothesis

c

n

co

12 < M

0

sequence

f =

(fn)

f is

n

is

in

same

is n o w

=

£2

(fn+l

< M

fn ) c

implies

,

then

that

[

IAf n

2

< ~

for

f ~

0

argument

shows

[ Icn + C n l

It

(lal)

co

[ Cn_iI&fn

The

~2

then

Af =

and

the

c

sufficient

to

that

I Ifn 12 < ~

show

that

f { D

(3.8.28)

195 co

7.

Ibnl Ifn[

2

co

f E ~

imply

(3.8.16).

<

(3.8.29)

0

since

(3.8.28-29)

multiply

(3.8.27)

bnlYn12 = Thus

for

both will by

an Yn

an(LY)nYn

To this end

Then

+ Cn Yn+l Yn + Cn-i Yn-i Yn

y c ~ ,

oo

oo

oo

7. IbnllYn 12 < Y. I(LY) n '~n Ila n 0

+ M X Yn+lll~nl

0

0

oo

19nl

+ M ~. l Y n _ l l 0

Since

y

is

£2

the last two serles on the right

are finite by the Schwarz y, Ly ~ £2(Ia I)

{

finite

since

}{

[ lanl lYn 12 ½

0

0

inequality

again.

it now follows

The series on the right

that

oo

Y. IbnI lYn 12 0

and so the conclusion

}

~ lanII(LY)n 12 ½ .

0

being

Moreover

then

7. I(LY) n Ynl lanl <

by the Schwarz

inequality.

follows.

< co

196

COROLLARY

3.8.2: Set

by

b

n

+ 2

c

n

= 1

Then

,

the

a

n

= -i

in

bn

in t h e

Dirichlet

In p a r t i c u l a r

infinity.

(cf.,

also

and

replace

b

n

operator

(Ly) n = A 2 Y n - 1

is

(3.8.27)

condition (3.8.30) [75]).

at

+ bnY n

infinity

is a l w a y s

(3.8

independently

limit-point

at

of

30)

CHAPTER 4

INTRODUCTION: The study of s e c o n d - o r d e r with an indefinite w e i g h t - f u n c t i o n the century.

Sturm-Liouville

dates back to the turn of

During the past ten years or so,

topic of current research.

problems

it has been a

We shall not a t t e m p t to give a

d e t a i l e d h i s t o r y of the subject here though we shall m e n t i o n some aspects of the theory, eigenvalues

e s p e c i a l l y those d e a l i n g w i t h

and their d i s t r i b u t i o n

the complex plane.

along the real axis and in

We shall be d e a l i n g e x c l u s i v e l y with the

h i s t o r y of d i f f e r e n t i a l

equations

since n o t h i n g appears

known about t h r e e - t e r m r e c u r r e n c e relations with weight-functions;

indefinite

though we shall prove a t h e o r e m in section

4.1 d e a l i n g with this topic.

The p r o b l e m

(4.0.0)

-y" + q(x)y = Ik(x)y

y(0)

where

q(x)

~ 0

and

c o n s i d e r e d by H i l b e r t (4.0.0-1)

to be

= y(1)

k(x)

(4.0.1)

: 0

has both

signs in

[0 , I]

[29] who proved in this case that

admits an infinite

sequence of e i g e n v a l u e s

was

198

• -- I_2

with

no f i n i t e , y_i(x)

This

r e s u l t was

vanish

[52]. tion

q

theorem

[53], w h i c h

of

are

(4.0.1), paper

lil

an i n d e f i n i t e

more

detailed

what happens

to the

[51].

it s e e m s

that

indefinite,

these

dubious.*

on the o s c i l l a t i o n

corresponding

* See the n o t e

to n o n - r e a l

in A p p e n d i x

that

the

the o s c i l l a -

by a n o t h e r

In this

paper,

order

problems

same p a p e r

theorem

the q u e s t i o n

as to

numbers".

are r e l a t e d

to the n o n - r e a l

his a t t e m p t

eigenvalues

III,

of n o n - r e a l is p r o v e d .

p. 307.

In a n s w e r

at p r o v i n g

the f o l l o w i n g

properties

a

in the

oscillation

Still,

that

for s u f f i c i e n t l y

a s u r v e y of s e c o n d

though

by

by R i c h a r d s o n

is v a l i d

followed

(c.f., [67])

k(x)

case,

along with

shows

thus r a i s i n g

of the p r o b l e m

appears

For a r b i t r a r y

The g e n e r a l

of the o s c i l l a t i o n

"missing

general

the o s c i l l a t i o n

was c o n s i d e r e d

gave

(0 , i)

and s i m u l t a n e o u s l y

Richardson

c a s e w a s given,

eigenvalues

[48]

weight-function.

proof

to m o r e

~ 0 :

This was

essentially

with

indefinite

k

the e i g e n f u n c t i o n s

though

for the e i g e n f u n c t i o n s

large v a l u e s

theorem

by P i c o n e

and

In the l a t t e r

existence

k(x)

[56] a n d R i c h a r d s o n

conditions

to this,

[43]

(4.0.2)

in the i n t e r v a l

extended,

shown only when

is w h e n b o t h boundary

(i-l)-times

by M a s o n

the r e s u l t w a s p r o v e n Sanlievici

and such t h a t

subsequently

conditions,

theorem was

< 0 < 10 < 11 < -'"

limit point

Yi(X)

boundary

< I_1

their

interesting eigenfunctions

199

THEOREM

A:

[53,

Let of

the

once

q(x)

interval

only

(4.0.0-1)

Then

the

v(x,

X)

that

let

X)

of

one

section

finding

an

of

the

In

the

best

sequel of

X

be

= u(x,

where

and

that

in a s u b i n t e r v a l

k(x)

a non-real X)

changes

sign

eigenvalue

+ iv(x,

imaginary

shall

bound

X)

be

and

for

shall

show

of

the

u(x,

X)

,

let

> 0

of

a three-term

the

It

turns

recurrence

these

upper

of

non-real

case.

b

EQUATIONS

n

numbers

is

number

results

theorems

bound

obtained

(cf., [76],[77],[78]).

DIFFERENCE

we

the

illustrate

that

possible.

the

indefinite

Furthermore C_l

parts,

complement

for

in t h e

We

real

0 , 1 , ... , m - i

formally

negative

suppose

theorem

true.

examples also

we

(4.0.0-1)

analogous

sequence

..., m-i

be

another.

upper

STURM-LIOUVILLE WEIGHT-FUNCTION:

finite

real

In t h i s

of

the

the

separate

by means

n=

y(x,

X].

eigenfunction.

zeros

is

and

and

is a l s o

4.1

(4.0.0),

[0, i]

relation

therein

in

,

Let

eigenvalues out

Theorem

[0 , i]

,

by

302~

in

corresponding

[53]

p.

, and

second

-A(c n - lAY n-i ) + bnYn

= lanYn

a

let

fixed.

self-adjoint

n=

AN

INDEFINITE

0 , 1 , ... , m - i ~ 0

n c

n

We

order

WITH

any

,

> 0 shall

, be

difference

n=0

be

n=

-i , 0 , 1 ,

dealing

with

equation

' 1 ' .... m-i

(4.1.0)

200

where

A

is t h e

meter

and

m

forward

introduce

Y-I

(4.1.0-i)

(4.1.0-1)

define

is t h e n

y(l)

where

y_l(1) For

define

a

=

where

we

m-vectors

The also

following be

we

,

1

is a p a r a -

conditions

Ym = 0

an e i g e n v a l u e

(4.1.1)

problem.

complex)

, yl(1)

solution

m-vector

.....

y(1)

of ,

(4.1.2)

Ym_l(l)]

f =

(f0 ' fl . . . . .

f-1

their

[f , g]

=

m-i [ 0

summation

(4.1.3)

+ bnfn}

= fm = 0

define

of

we

fm-i )

by

= ani{-A(Cn-iAfn-1)

formula

A

= 0

i[f]

it t h a t

f ,g

= 0

m-vector

m-vector

take

boundary

(yo(1)

= ym(1)

£[f]n

the

(possibly

a given

the

operator,

> 2

If we

then

difference

by

definition.

"J-inner-product"

For by

(4.1.4)

fngnan

by

parts

given

[30,

p.

17]

should

useful, N

N+I

uk v k : [Uk_iVk] M

N

- [ vkAuk_ 1 M

M

(4.1.5)

201

We now define the c o l l e c t i o n

Q(f)

a quadratic

of all c o m p l e x

:

functional

m-vectors

f

Q(f)

with domain

by

(4.1.6)

Cf , ]~[f]] m-i

= c_11f012

+

I {CnJAfnl2+bnlfn12}

(4.1.7)

0

w h e r e we o b t a i n

(4.1.7)

from

(4.1.6)

upon

the a p p l i c a t i o n

of

(4.1.5).

THEOREM

4.1.i: Let

tion of 0

<

n

<

y(1)

(4.1.0) m-1

=

(y0(1) , yi(1) . . . . .

satisfying

y_l(1)

Proof: 4.2.1]

THEOREM

= 0

Then

be a s o l u -

for

,

n

([-I)

ym_l(1)]

[ arYr(1) r= 0

This result

yr(p)

= c

Y n + l (I)

Yn+l (p)

Yn(1)

Yn(P)

n

can be p r o v e n

as in

[3, p.

98, T h e o r e m

a n d so we o m i t the d e t a i l s .

4.1.2: If

1

is n o n - r e a l ,

0 < n < m-i

,

then

202

n [ r=O

ar lyr (i) 12

Proof:

We

refer

THEOREM

4.1.3: Let

to

Tr =

eigenvalues

of

-

2i

1I m

[3, p.

Y n + l (I)

Yn (I)

Yn (I)

Cn

I

99,

Theorem

[y(Ir) , y(Ir) ]

(4.1.0-1)

Y n + l (~)

and

[r

[y(l s) , y(Ir)]

4.2.3].

If 1s

Ir

' 1s

are

non-real

then

(4.1.8a)

= 0

Hence

[y(Is) , y(Ir) ] = T r 6rs

where

is the

rs

that

1

r

,

1

are

s

Proof:

Similar

THEOREM

4.1.4:

Let the

to

Kronecker

delta

r ~ s

and when

[3, p.

104,

Theorem

4.4.1].

; f(10) ' "'" ' f(Im-l)

and corresponding

eigenvectors

the

boundary

conditions

(4.1.1).

Let

of

denote

the p r o b l e m

(4.1.9)

- & ( C n _ I A Y n _ 1) + b n Y n = ly n

with

we mean

not conjugates.

I0 ' II ' "'" ' l m - 1

eigenvalues

(4.1.8b)

y

be an a r b i t r a r y

203

m-vector.

Then

Yn :

m-i [ r=O

V(Ir) p r l f n ( I r )

n = 0 , 1 , ... , m - i

(4.1.10)

r= 0 , 1 .....

(4.1.11)

where

v(l r) =

m-i [ Ysfs(Ir ) s=O

m-i

and

m-i Pr = ~ n=O

Proof: make

This

use

THEOREM

follows

of T h e o r e m

from 4.4.2

components)

y

results

(with

a

with

be an a r b i t r a r y Y-I

Q(y)

Proof:

the

n

in

§4.4

of

[3] w h e r e

we

= 1 ).

4.1.5: Let

where

(4.1.12)

Ifn(Ir) 12

I ,v , p

Let

are

us w r i t e

m-vector

= Ym = 0

=

m-i [ n=0

(with

real

Then

I n l V ( I n ) 12pn I

as in T h e o r e m

k I E v(li)

or c o m p l e x

(4.1.13)

4.1.4.

p;1

for b r e v i t y .

Note

also

that

£[f(Is) ]

= I s a n l f n (Is) n

(4.1.14)

204 Hence m-i Q(y) = n=0

Yn £[Y]n an

m-i {m[l }{mil l[sf n } [ k f (In) Isa n (l s) a n n=O r=O r n s 0

where we have used the expansion

(4.1.10) along with

(4.1.14),

m-i m-i m-i [ t k ~. V(Ir) [ prlfn(lr)fn(ls) s= 0 s s r=0 n=0 m-i m-i [ 1 k [ v(l r) 6 s= 0 S S r=0 rs

by Theorem 4.1.3 with

an = 1

(see also

[3, p. 105,

(4.4.2)]],

m-i =

y.

S=0

XsksV(Xs)

m-i

~.

S= 0

x

S

Iv(xs) l2

-1

QS

which is what we set out to prove.

COROLLARY

4.1.1:

Let

c_iIy012+

y

be any

m-vector.

Then

m-1 { } m-i -i ~ Cn]AYn 12 + bnlYn 12 = ~ Inlv(In) [2P n 0 0 (4.1.15)

205

Proof:

LEMMA

This

follows

(4.1.7)

and

the

preceding

theorem.

4.1.1: Let

values

from

of

N

the

> 0

be

the

problem

(4.1.9),

10 , I i , ... , IN_ 1

with

the

corresponding

N

Let

there

be

collection

an

of

number

of

distinct

(4.1.1)

and

negative

denote

f(l O) , ... , f ( I N _ I)

eigen-

these

by

representing

eigenvectors.

m-vector

y

eigenvectors,

which

is o r t h o g o n a l

to t h e

above

i.e.

m-i s = 0 , 1 , ... , N - I

Yr f r ( I s ) = 0

.

(4.1.16)

r=O Then

Q (y)

if

(4.1.9),

Proof:

(4.1.1)

(4.1.16)

has

at

implies

(4.1.17)

> 0

least

that

one

v(l s)

positive

= 0 ,

eigenvalue.

s=

Hence

N-I

Q(Y)

=

[ n=0 m-i

m-i

+

[

} lnlV(ln)

n=N

Z Xnlv(X n) 12p

n=N

> 0

12 -i Pn

0 , ... , N - 1

.

206

since

at l e a s t

completes

of the

is p o s i t i v e .

1 N , ... , Im_ I

note

only

that

0

is an e i g e n v a l u e

if it is an e i g e n v a l u e

same multiplicity

in b o t h

of

cases.

On

of

the o t h e r

(4.1.1)

has

no p o s i t i v e

eigenvalues

of p a i r s

of n o n - r e a l

eigenvalues

of

than

(4.1.9), values we

(4.1.1)

are,

shall

one

or e q u a l

Let eigenvalues and

of

its

since

in b o t h

always

positive

to the

assume

(4.1.0-1)

of n e g a t i v e

the d e f i n i n g

cases,

relations

polynomials

that

(4.1.9),

having hand

then

the

if

the

number

is n e c e s s a r i l y eigenvalues for

of d e g r e e (4.1.1)

(4.1.1)

have

the m at

of

eigenThus least

eigenvalue. M = the of

number

(4.1.0-1).

of d i s t i n c t (By a p a i r

pairs

we mean

of n o n - r e a l an e i g e n v a l u e

conjugate.) Let

N = the

(4.1.9),

(4.1.1).

THEOREM

number

(4.1.9),

(4.1.0-1)

(4.1.9),

less

This

the proof.

We if a n d

one

number

of d i s t i n c t

negative

eigenvalues

4.1.6: Let

M, N

be d e f i n e d

as

in the p r e c e d i n g

Remark.

Then

M

The

upper

bound

is b e s t

<

N

possible.

.

(4.1.18)

207

Proof:

Let

non-real

~0 ' ~1 ' "'" ' ZM-1

eigenvalues

y(~i ) =

of

; [0 ' [i ' "'" ' ~M-I

(4.1.0-1)

(y0(~i)

eigenvector

corresponding

where

0 ~ i ~ M-I

[ we note

corresponding

to

We define

[i)

.

an

m-vector

the

~i

'

and

, Yl(~i ) .....

be the

be

Y m _ l ( ~ i )]

to the

that

eigenvalue

y(zi )

z =

is the

eigenvector

(z 0 , z i , ... , Zm_ I)

where M-I [ e n yi(~n ) n=0

zi =

and the of the

e

n

are

to be c h o s e n

eigenvalues

later.

(4.1.19)

m-i

Because

of the

indexing

we have

~i

Consequently

i= 0 .....

~ ~j

Theorem

4.1.3

0 =< i , j =< M - I

implies

[y(~i ) , y ( ~ j ) ]

= 0 ,

that

(4.1.20)

i ~ j

and

[y(~i ) , y ( ~ i ) ] = 0

since

the

eigenvalues

Suppose,

are

non-real.

if p o s s i b l e ,

that

M > N

.

We

shall

proceed

208

to

show

that

orthogonal

the

to

Thus

e

n

in

(4.1.19)

must

e

in

M

(4.1.19)

must

n

satisfy

=

into

0 , ... , N - I

the

latter

equation

following

ej

~ n=0

fn(Ir)Yn(~j)

r =

0 , 1 , ... , N - I

always

has

a non-trivial

system

a set

of

property

(e i)

Since solution

The

Lemma

implies

}

M

that

z

is

.

we

find

that

of

N

equations

=

> N

the

last

0

equation

e 0 , e I , ... , e M _ 1

resulting

m-vector

z

then

Fix has

that

Q(z)

by

r=

0

the

where

the

so

unknowns,

[ j =0

such

chosen

have

m-i [ z fn(Ir) n=0 n

the

be

f0 ' fl ' "'" ' f N - i we

Substituting

can

4.1.1.

Moreover

> 0

Z_l

=

(4.1.21)

zm

=

0

that

Q(z)

=

[z , £ [ z ] ]

m-i =

[

n=0

z

£[z]

n

a

n

n

Hence

(4.1.7)

209

m-i

M-I

}rM[l

{r~0

n=0

erYn(Ur)

{s:0

es~sYn(Us)}an

M-I

Z

eres~s{~il

r,s=0

anYn(Ur)Yn(Us)}

M-I e r e s ~ s [ Y ( U r) , Y(U s) ]

r, s¼0

Now

since

(4.1.20)

[r ~ U s

for all

r, s ,

0 < r , s < M-1

,

implies

[y(u r) , y(u s) ] = 0

for all

r, s ,

0 < r , s < M-I

.

Hence

Q(z)

This, and

however,

the

EXAMPLE

is in c o n t r a d i c t i o n

theorem

is c o m p l e t e l y

(4.1.22)

• 0 .

with

M < N

(4.1.21).

Thus

=

Consider

proved.

4.1.1: Let

c

n

= 1 ,

b

n

= -2

and

a

n

(-i) n

the

problem

-A 2 Y n - i

- 2Yn = I (-i) n Yn

Y-I

= 0 = Ym

n= 0 ....

, m-i

(4.1.23a)

(4.1.23b)

210

where

m > 2

The c o r r e s p o n d i n g

2 -A Y n - i

with

the same b o u n d a r y

values

and c o n s e q u e n t l y If w e put

that,

if

m = 2k

the

zeros

negative

Y2k(1)

of

of

number,

are p r e c i s e l y

are all real. computation

shows

problem. 1

In fact

while

table

illustrates

k ,

of p o s i t i v e

and

the e i g e n v a l u e s

Y2k+l(1)

of The f o l l o w i n g

ym(1)

eigen-

= Y2k(-l)

has an e q u a l

definite

only even powers

m u s t h a v e o n l y real

then

zeros which

associated

(4.1.24)

a straightforward

Y2k(l)

and thus

problem

- 2Yn = lYn

conditions

Y0 = 1

"definite"

this:

Y2k(1)

of the

consists

of

has o n l y odd p o w e r s

211

TABLE

I

~0(~)

i

"'"

{&(x)

0

-1

~2(I)

-i

0

i

~3(I)

0

2

0

-i

~4(I)

1

0

-3

0

1

~5(i)

0

-3

0

4

0

-i

y6(l)

-i

0

6

0

-5

0

1

y7(1)

0

4

0

-i0

0

6

0

-i

~S(1)

1

0

-i0

0

15

0

-7

0

1

yg(1)

0

-5

0

17

0

-21

0

8

0

"'" "'" ''" "'" -'' "'" ''" .'' -i

---

o o .

The t a b l e power of

change

of the polynomial

(.4.1.22-23)

zeros must there and

table then

corresponding

coefficients

to

solutions

shows

even

be n o n - r e a l

of the

if

Z2k(1)

then

the

former

powers

and o c c u r

of

eigenvalues

indefinite we

simply signs.

is t h e p o l y n o has p o s i t i v e

I ; consequently

in c o n j u g a t e

of

the

t a b l e to p o s i t i v e

that

pairs of non-real

negative

To o b t a i n

sign change)

in t h e a b o v e

Y2k(1)

and o n l y

are M = k N = k

signs

of the corresponding

concerned.

(up to a p o s s i b l e

the negative

The r e s u l t i n g mial

the coefficients

I and o f t h e p o l y n o m i a l

coefficients problem

gives

pairs.

eigenvalues, (4.1.24-23).

its Thus

if m = 2k

,

212

§4.2

STURM-LIOUVILLE WEIGHT-FUNCTION The main

problems regular

DIFFERENTIAL

difference

for d i f f e r e n t i a l case,

is t h a t

number

of e i g e n v a l u e s

number

of e i g e n v a l u e s ,

the

"eigenfunction"

is m o r e is,

in b o t h

the

p(x)

cases,

> 0

latter

while

the

under

on

former

proper

number

and bounded shall ,

case it

eigenvalues constants.

following

~ L l o c ( a , b)

assumptions:

is s u c h

that

problem

-(p(x)y') ' + q(x)y

y(a)

admits point that

infinite

in the o t h e r

similar

the

an

a finite

In o n e

of n o n - r e a l

make

q(x)

has

while

by

in the

at m o s t

always

is t r i v i a l

finite

[a , b]

has

INDEFINITE

of e i g e n v a l u e

conditions.

the

AN

equations,

always

Still,

we

WITH

the h a n d l i n g

and difference

the

In t h i s s e c t i o n That

between

expansion

involved.

EQUATIONS

a denumerable

number

of a c c u m u l a t i o n . both

8.4.6].]

p

-i

set

in

Associated

case

(4.2.2)

= 0

(Conditions

the

(4.2.1)

of e i g e n v a l u e s

, q ~ L ( a , b)

In t h i s

orthonormal

: y(b)

: ly

,

see

having

which

guarantee

[3, p.

eigenfunctions

no

215, form

finite this

are

Theorem a complete

L 2 ( a , b) with

(4.2.1-2)

is the

"indefinite"

boundary

problem

-(p(x) z')'

+ q(x) z = Ir(x) z

(4.2.3)

213

z(a)

where such

r(x) that

measure. that p.

288]

is a r e a l - v a l u e d r(x)

takes

both

(4.2.4)

= 0

function signs

on

defined

some

on

subsets

"Indefinite ease" is c h a r a c t e r i z e d

The

both

= z(b)

q, r

have

and not being

a variable equal

sign

in

[a , b]

[a , b]

of p o s i t i v e by the (see

fact [53,

a.e.

4.2.1:

LEMMA

f

Let eigenvalue

be an e i g e n f u n c t i o n of

1

(4.2.3-4).

corresponding

to s o m e

Then

(plf'12+qlfl2)dx

= ~

fbrIfl 2 d x

(4.2.5)

a

Proof: over

We multiply

the

interval

fa

(pf')

Integrating boundary

LEMMA

and

(4.2.3) [a , b]

f dx + I

by to

F

f

and

first

conditions

rlf] 2 dx =

integral

(4.2.4)

both

sides

find

a

the

integrate

by p a r t s

the

result

la

qlf[ 2 dx .

and

applying

the

follows.

4.2.2:

Let eigenfunctions

I , H , f, g

I ~ ~

be

two

respectively.

non-real Then

eigenvalues

with

214

b r(x) f (x)g(x)dx

i.e.

f,g

l

are

J-orthogonal

= 0

(4.2.6)

in the K r e i n

space

L2

Irl)

and

f

b

g' {p(x) f'(x)

(x) + q ( x ) f ( x ) g ( x )

}dx = 0

(4.2.7)

a

Proof:

We have

along with

f(a)

(4.2.8)

g

by

we obtain,

upon

(l-i)

-(pf')'

+ qf = Irf

(4.2.8)

-(pg')

+ qg : urg

(4.2.9)

= f(b)

and

= g(a)

(4.2.7)

integration

the

latter

~ab

f

over

:

of the b o u n d a r y

Multiplylng

[a , b]

,

{(pg')

f-

integral

g}dx

= 0

and s u b t r a c t i n g

' {(pg')' f - ( p f ' )

because

by

r(x)f(x)g(x)dx

and i n t e g r a t i n g

= g(b)

by parts

(pf')

b [P(g'f- gf')]a

=

0

This

g}dx

we find

=

conditions.

the r e s u l t s

proves

(4.2.6)

since

215

Multiplying

(4.2.8)

by

g

and integrating

over

[a , b]

we obtain b b la {-(pf')' g + qfg}dx = X la rfg dx

Integrating

the first term in the left by parts we see that b -

la (pf,), ~

=

-

[pf,_g ] a b

+

dx

pf'g

= 'lab pf'q' dx

Thus

b{pf,~, + qfg}dx = ~

=

by

(4.2.6).

This completes

Associated A

defined

in

(A) = {y E L 2(a

where

for

with

L2(a , b)

,

~(A)

is the differential

with domain

'

c ACIo c(a

D(A)

,

,

Ay = -(py')'

If we let

0

the proof.

(4.2.1)

b) : y , py

y E ~(A)

rfgdx

be defined

by

+ qy

b)

defined

and

operator by

Ayc L 2 (a , b) }

216

D(A)

= {y • O(A)

: y(a)

= y(b)

(4.2.10)

= 0}

and let

Ay = Ay

then

A

is a r e s t r i c t i o n

a symmetric The regular

operator

[34,

following

y • D(A)

of

A

§4.11,

lemmas

Sturm-Liouville

to

(4.2.11)

D(A)

Theorem

is,

and i].

are p a r t of the t h e o r y

equation

in fact,

and can be f o u n d

of the

in

[34],

thus w e o m i t the proofs.

LEMMA

4.2.3: a)

above,

The r e g u l a r

is b o u n d e d

below,

Sturm-Liouville i.e.

there

operator

exists

A ,

defined y •

a constant

such t h a t

(Af , f) > y(f , f)

where

(

,

b) negative

Proof:

)

is the u s u a l

The o p e r a t o r

A

,

f • D(A)

inner product

has at m o s t

in

(4.2.12)

L 2 ( a , b)

a finite

number

of

eigenvalues.

For p a r t a)

Corollary].

For

see

P a r t b)

f • D(A)

[34,

§5.17,

is p r o v e d

,

in

Ex.

[34,

the e x p r e s s i o n

5.3 ° and §5.8,

§6.7,

Theorem

(Af, f)

2].

defines

a

217

quadratic

f u n c t i o n a l with v a l u e s

(Af, f) =

{plf'I2 + q l f I g } d x

This is i m m e d i a t e

LEMMA

f ~ 0(A)

if we f o l l o w the a r g u m e n t

leading

to

(4.2.5).

4.2.4:

We d e f i n e

D (Q)

by

oo

: {y ( L 2 (a , b)

D(Q)

: [ Iljl I (y , ~j)I 9 < oo} 0

oo

Q(y)

where

(lj)

(4.2.1-2)

= [ I ] (y , ~j) 12 0 J

, (~j)

Y { D(Q)

are the e i g e n v a l u e s

and e i g e n f u n c t i o n s

of

respectively.

Henceforth L 2 ( a , b)

( ,

)

will denote

Then the q u a d r a t i c

an e x t e n s i o n

of the f u n c t i o n a l

Q(y)

=

the inner p r o d u c t

functional (Ay , y)

Q(y) in

D(A)

in

D(Q)

,

i.e.

in is

y e N(A)

(Ay , Y)

or

~a~{PLY' for

f ~

Proof:

I2 + qlY

12}dx

: [ ljl (y , qbj)12 0

(4.2.13)

D(A) The proof of this t h e o r e m

is c o n t a i n e d

in

[34,

§6,

218

Theorem

i, pp.

6.1-6.5].

We now d e f i n e domain

D(Q')

D(Q')

given

another

quadratic

y(a)

and

for

Q' (y)

is d e f i n e d

The crux of the m a t t e r

LEMMA

~

and

L (a , b)

= y(b)

(4.2.14)

= O}

{plf'

+qlfl 2}

and e x t e n d s is the

(Ay , y)

following

[34, p.

6.6].

Q(y)

Q'(y)

lemma,

4.2.5: When

p(x)

of the q u a d r a t i c D(Q)

with

y e D(Q')

Q'(y) --

Then

Q'(y)

by

{y c A C ( a , b) : ply'l 2

=

functional

= D(Q')

> 0

a.e.

functional

Proof:

(Ay , y)

are

identical,

,

i.e.

and

{ply'f2 + q l y

where

the e x t e n s i o n s

}dx : [ ljl (y, %j) I2 0

y e D(Q')

This

important

result

is p r o v e d

in

[34, p.

6.8,

Theorem

3].

219

THEOREM 4.2.1: Let

p(x)

r(x)

as

value

problem

non-real

(cf.,

in the

[76])

> 0

hypotheses

(4.2.3-4)

N = the

following

If we

number

values

of

[a , b]

possesses

eigenvalues.

M = the

on

a.e.

,

q • L ( a , b)

(4.2.3-4).

at m o s t

The

a finite

and

eigen-

number

of

let

of p a i r s

of

distinct

non-real

eigen-

negative

eigenvalues

(4.2.3-4),

number

(4.2.1-2)

of

distinct

(which

we

know

is

finite

by

of

Lemma

4.2.3(b)),

then

M

Proof: of

We

let

(4.2.1-2)

Let

~0'

for

some

"'"'

< N

.

10 , 11 , ... , IN_ 1

arranged ~N-I

be

the

in an

increasing

the

corresponding

be

f • D(Q')

(4.2.15)

we

have

(f , ~j)

negative

order

eigenvalues

of m a g n i t u d e .

eigenfunctions.

= 0 ,

j = 0 .....

If N-I

,

then

Q' (f)

> 0

since co

Q' (f)

= [ ljJ (f , ~j) I 2 N

(4.2.16)

220

and

the

I. > 0 3 We

non-real

now

let

Z0 ' ~1 ' "'" ' ~ M - I

eigenvalues

[i

We

write

the

of

z.(x) 1

replacing of

M

0

~ Zj

corresponding

z 0(x)

Then

(4.2.3-4

g

by

mutually

arranged

i

.....

i= 0 ..... f

,

j

<

Thus

M-I

(4.2.17)

the

.

as

(4.2.17)

because

of

constitutes

fin the

be

that

ZM_ l(x)

M-I

J-orthogonal

~ 0 )

such

eigenfunctions

, z l(x)

e D(Q')

<

( Im ~i

Kreln

(4.2.7),

upon

a collection

space

L2(Irl)

]

eigenfunctions. Let

e. e C 3

and

form

the

sum

M-I

f(x)

where

the

e.'s 3 > N .

that

M

Then is o r t h o g o n a l For

it

it

shall

so

be

is p o s s i b l e

(in t h e

is n e c e s s a r y

(f , }j) and

=

[ 0

e.z. (x) 3 3

chosen

(4.2.18)

Assume,

later.

to c h o o s e

L 2 (a , b ) - s e n s e )

the to

(ej)

so t h a t

~0 . . . . .

that

: 0

if p o s s i b l e

j = 0 , 1 , ... , N - I

~N-I

f

221

M-I ei(z i , ~j)

j = 0 , 1 , ... , N-1

= 0 ,

i=0

The l a t t e r

constitutes M > N

a set of .

Thus

N

unknowns

where

this

solution

(ej)

not all

necessary

that,

for such a c h o i c e

zero w h i c h

linear system

equations

(ej)

It is then ,

Q' (f) > 0

because

of a p r e c e d i n g

Q'(f)

remark.

M

has a n o n - t r i v i a l

we fix. of

in

(4.2.19)

Moreover,

= Q'{Zejzj}

=

i

n

i

_

-!

_

_

a p(Tejzj) (7eiz ~) + a(Zejzj)~ (Zeiz i) M-I

I~ {P z'i' j i + q zj~i}dx

ejei i,j=0

But since applies

Pi ~ ~j

for all

Lemma

4.2.2

and so ~ab {p

for all

0 =< i , j =< M-I

i , j ,

zI jzi + q z j ~ i } d x

0 < i , j < M-I

.

= 0

Consequently

Q' (f) = 0

which

contradicts

completely

proved.

(4.2.19).

Thus

M < N =

and the t h e o r e m

is

222

The Richardson make

the

preceding [53]

theorem

mentioned

following

where ous

p(x)

and

at

L2(Ir[

and

is n e g a t i v e

continuous sign

> 0

on

least

once

functions

f

[a , b]

space

,

III

discussion

these

Let Appendix

1.4,

those

the

chapter

equation

,

q(x)

that

is

and r(x)

case

the

continur(x)

f]

and

-

ifl 2 Irldx

the

indefinite

space

(equivalence

classes

-

If I2

the

references

r

dx

<

inner

product

given

by

f c H

therein

for m o r e

spaces).

Ul(y ) , equation

U2(Y )

be

the

linear

forms

(I.4.1).

We

denote

the

u~ (y)

= 0

U o (y)

= 0

is

changes

that

(see A p p e n d i x on

by

this

[a , b]

In t h i s

of

(4.2.20)

of

property

[a , b] defined

with

[f

on

the

to

Consider

continuous

(f , f)

is a K r e i n

plausible.

with

results

introduction

in a s u b i n t e r v a l

= H

the

= fry

in

such

with

z -(py') ' + q y

[a , b]

; [a , b])

in t h e

notions

Ly

along

defined

in

relationships

of)

223

by

U(y)

= 0

(4.2.21)

The problem

:

then

defines

I £ C ing

such

ly = fry

an e i g e n v a l u e that

,

Uy = 0

problem,

(4.2.22)

has

i.e.,

(4.2.22)

we

a non-trivial

seek

values

solution

of

satisfy-

(4.2.21). With

eigenvalue

some

loss

problem

~

of g e n e r a l i t y ,

f , g e C 2 (a , b)

a

J-self-adjoint

may

not exist

number also,

but,

appears because

results

in

formulate

[53,

the

problem,

finite

the

if

satisfy

= 0

.

non-real if t h e y

for g e n e r a l

preceding

§4] w i t h

say t h a t

[if , g]

= U(g)

in a n y case,

to b e of

=

which

U(f)

For

shall

formally J-self-adjoint

is

[f , [g]

for all

we

theorem.

the preceding

eigenvalues do exist, boundary

or

their conditions

Combining theorem

may

the

we can

224

THEOREM

4.2.2:

The for.~ally

Ly = hry

has a f i n i t e

number

none

and on

at

all,

no f i n i t e and p l u s

point

y(a)

of n o n - r e a l Ill > A

= y(b)

= 0

eigenvalues,

in

some

cases

has o n l y real e i g e n v a l u e s ,

of a c c u m u l a t i o n ,

clustering

at m i n u s

with

infinity

infinity.

The s e c o n d [53, p.

301,

Theorem

4.2.2

boundary

problem

J-self-adjoint

p a r t of this

Theorem

VII].

remains

conditions

true

time.

changes

in the a r g u m e n t ,

self-adjoint

It w o u l d

Theorem

4.2.1

differential

is due to R i c h a r d s o n

seem plausible

for a r b i t r a r y

t h o u g h we

present

theorem

shall

"J-self-adjoint"

not go into

extends,

with

to the g e n e r a l

that

this

at the

appropriate

even order

formally

equation

(_i) n[poy(n) ) (n) + (-i) n - l ( p l y ( n - l ) ] (n-l) + ... + Pn y = fry

y(J) (a) = y(J) (b) = 0

where [a , b] that

P0 > 0 , Pk

and

where e C

(n-k)

i

Pi(X)

changes

is in the r a n g e

(a, b)

j = 0 .....

n-i

s i g n at l e a s t 0 < i < n

(see A p p e n d i x

III.4).

once

in

and w e a s s u m e

CHAPTER

5

INTRODUCTION: In t h i s differential

operators

i[y] (x)

where

I =

variation Chapter

=

d~(x)

[a , ~)

,

3.

We

we

y(s)do(s)

satisfying

x

o

a lemma

[23]

non-oscillatory

and

~

I

(5.0.0)

of b o u n d e d -

the b a s i c

by proving

of G l a z m a n

of g e n e r a l i z e d

expression

is n o n - d e c r e a s i n g ,

begin

the

spectrum

y' (x) -

to a t h e o r e m

is to r e l a t e

the

by the

~ , o

shall

study

generated

~

locally,

in c o n t e n t which

chapter

assumptions which

is s i m i l a r

the purpose

behaviour

of

of

of

solutions

of

Z[y] (x) = ly(x)

to t h e then

finiteness

use

some

for the

spectrum

of a n y

which

shall

and

then

the

spectrum

non-oscillation

criteria

we

of

obtain

discreteness self-adjoint

define

later.

an e x t e n s i o n

to t h e

results of

the

left

of

I .

from Chapter negative

extension We

(5.0.1)

x e I

shall

part

of a t h e o r e m

the

of M.

of

the operator,

latter Sh.

can

2 to o b t a i n

of t h e m i n i m a l apply

We

result

Birman

226

[23,

p.

93]

gives

discreteness operators ous

of the

in the

spectrum

generalized bounded show

spectrum

polar

(also

the

We

note

that

always

the p r o o f s , will

be o m i t t e d .

operators in

[35],

are

are

generality

§5.1

in a s s u m i n g

shall

[y] (x) =

I =

general we

SPECTRUM

be d e a l i n g

d dr(x)

[a, ~)

,

can

case

o

is of

we

shall

is b o u n d e d

extends

a result

setting,

shall

property

the

we

found

shall in

adaptable

that

the

this

theory

[23] and

not or so

resulting

c a n be

found

operators

thus

defined

is no g r e a t

loss

of

in g e n e r a l .

OF G E N E R A L I Z E D

with

be

for

space

general

though

are

assume

the

the

the

either

so t h e r e

y'+ (x) -

v

which

continu-

when

extension

Conditions

and this

the

of t h e s e

latter

be u s e d

applications

single-valued

THE DISCRETE OPERATORS: We

where

In the

case

consequently

These

Moreover

study

is n o n - d e c r e a s i n g

shall

single-valued.

[36].

indeed

~

spaces

in the m o r e

bound

the

p = 1

space,

the d e t a i l s :

in the

for

differential

spectrum)

In the

lower

since

in t h e s e

then

essential

I

case

is a H i l b e r t

give

shall

self-adjoint

an e x p l i c i t in the

of o p e r a t o r s

of

condition

second-order

We

the

on all

[16]

I)

sufficient

particularly

of E v e r i t t

L2(v,

of

corresponding

and give

and

case.

called

operators

variation

that

below

a necessary

DIFFERENTIAL

expression

y(s)do(s)

is a r i g h t - c o n t i n u o u s

x • I

(5.1.0)

non-decreasing

227

function locally as

and

~

on

usual,

I

In c a s e

that

both

In t h e for

the

compact we

space

of

mean

finite

the

v , ~

in

[a , b])

a function

of

variation

is

finite

continuous

at

the

, b]

on

functions

[a , b]

By

vanishes

we

assume,

end-points.

( A C 0 [ a , b])

continuous

which

bounded

I

AC[a

absolutely

and

interval

are

following,

support

shall

some

is r i g h t - c o n t i n u o u s

will

stand

(having

a finite

identically

function

outside

interval.

We

define

the

spaces

S(~ , 8)

and

S ( ~ , 8) =

{z(x) e A C [ e , 8] : z' e T, 2 (~ , 8)

T(e,

{y(x) e A C [ e ,

T ( ~ , 8)

by

z(~) = z(8) = 0}

and

!

8) =

8] : y + ( x )

exists

everywhere

on

I

[a, 8]

,

y+(x)

is

[~ , B]

,

dy ~./ d r

c L 2 (~ ; [e , B])

y(~)

Note

= Y(8)

v-absolutely

= y' (~)

= y' (8)

continuous

on

and

= 0}

that

T ( ~ , 8) c s ( ~ , 8)

The

first

(see

the

LEMMA

result book

similar

Glazman

to

[23,

an p.

important 35,

Lemma

result

of

Krein

5]).

5.1.0:

Let [~ , 8]

of

is

z(x)

Then

for

e S(e, any

B) e > 0

be ,

a finite there

function

exists

with

a finite

support function

228

y(x)

having

e T(~ , B)

the

same

support

as

z(x)

and

such

that

(5.1.1)

IQ[ z] - Q[Y] I < s where

Q[z]

Proof:

We

This

said,

case

where

Q'[z]

shall

{Iz'12dx+ IzI2d (x)} .

adapt

it w i l l

be

the

sufficient

g { constant,

defined

is b e c a u s e

or

in

for the

the

-

the

situation.

lemma

quadratic

in the

functional

5.1.3)

IZ' 12 dx .

fundamental

nature

c a n be r e a l i z e d

by

i.e.

{yj(x) }

is any

sequence

for all

j ,

y- (~) = 0 ]

to t h i s

to p r o v e

"Q-metric" If

[23]

by

Q' [z]

This

ideas

(5.1.2)

of any

its n a t u r e

[yj (x) - Yk(X) I <

in

sequence

in the

in the

"Q'-metric",

A C [ ~ , B]

with

IYj - Y k Idx

I Y j - Yk Idx

<

B - c~)½

IY

- Yk 12dx

"

(5.1.4)

229 !

Thus

e.g.

If

uniformly

to

inner

product

which

are

yj

÷ y

some on

in t h e

function the

finite

then

[e, 6]

on

space

with

Q'-metric

F

of

support

all

[~,

We

yj

can

define

functions

6]

converges an

z c S(~,

8)

by

0o

(f ' g)

- I

f, g c F

f' (x)g' (x)dx

.

(5.1.5)

a

This

is w e l l - d e f i n e d

and

then

induces

II IIF

that

the

let

{zj(x)}

a norm

resulting be

Q' [f]

=

(f , f)

defined

metric

space

a Cauchy

by

The

IlfllF =

is c o m p l e t e .

sequence

in

F

inner

product

(f, f)½

such

To

this

prove

Then

if w e w r i t e

!

z. E w. 3 3

we

find

that

Ilwj - wkll 2 = II j - zkll F <

for The

j, k

large,

completeness

a function

of

h(x)

112

II

where

L 2 ( ~ , 6)

such

is t h e then

L2-norm

implies

the

on

(~,

B)

existence

of

that

!

lim j÷~

in t h e

norm We

of now

z. (x) ]

L 2 (c~ , 6) set

z(x)

Then

= h(x)

z c A C [ ~ , 8]

,

z(~)

=

i

x

= 0

h(s)ds

and

z'

E L 2 (~ , 6)

Using

230

(5.1.4), z. 3

and

with

y

converges

replaced on

lim j÷=

= z(x)

z.(x) ]

[a , 8]

see

that

the

sequence

so t h a t

x

[~, 8]

~

thus

Hence

z c F

II IIF , that

and

the

finite this

it

z ~ F

so

F

F

and

have

be

to

is

such

G

an

of

[~ , 8]

show

that

by

=

-

f

for

shows

Z' (x)y' (x)dx

-

if t h e r e

= 0

f8 =

in

=

in

is

of

some G

,

18

y' (x)dz(x)

z(x)dy'

(x)

8 z(x) (x

y e G

that

z(x)dy'

-

dy' (x) d~) (x)

the

(x)

d~(x)

norm

proceed

T ( a , 8)

all

8 18

to

now

Then

parts

= [~'z]~

We

is d e n s e

to a l l

a function.

relative

space.

functions

F-orthogonal

integration

= 0

is c o m p l e t e

support

suffices

z.(8) 3

is a H i l b e r t

collection

which z

Thus

= lim j+~

I~ z' (x)y' (x)dx

Now

we

z ,

uniformly

z(8)

Let

by

F

to

which To

are

prove

function then

show

z E 0

231

because

y'

is

v-absolutely

continuous.

8 { d~' (X)~dV(x) c~ z(x) - dr(x) J

for

all

y

domain

c G

G

We

defined

now

in

tial

fact

the

that

we

have

= 0

the

that

(5.1.6)

operator

L0

with

by

dy' = - dv

L0Y

is

note

Hence

minimal

Y

operator

~ G

,

associated

with

the

differen-

expression

i[y] (x)

(see C h a p t e r deficiency

for

all

i0

which

3.6)

and

indices

y

= _dy' (x) dv

(L0Y , z) v

=

is e q u a l

to

z

c

a closed Thus

Thus

(see

is

(2, 2)

e G .

L 2 ( v ; [~ , 8])

so

x

is

[~ , 8]

symmetric (5.1.6)

operator

implies

that

(y , 0) v

in the

the

"maximal"

Chapter

3.6,

(5.1.7)

domain

of

operator

and

with

[46,

the

adjoint

i

defined

in

§17.3,

Theorem

i]).

!

Hence

z

is a b s o l u t e l y

continuous

exists

everywhere

on

[e , 8]

and

there.

Moreover

from

(5.1.7)

we

i0 z =

iz =

on is see

dz' (x) dv (x)

[~,

8]

,

v-absolutely

z+(x) continuous

that

- 0

(5.1.8)

232

for

all

that

x

~

[~ , 8]

(5.1.8)

Integrating

implies

so

z(e)

=

z(x) z(B)

=

must

be

0

Hence

=

so

there

G

is

exists

dense y

to

~

we

see

constant

linear

on

z (x)

and

respect

that

z' (x)

and

with

• G

in

F

such

[e,

But

8]

since

z

c

E

e >

F ,

- 0

Thus

for

given

z

z

there

F ,

0

that

IIz- yllr < i.e.

IQ' [z]

Let y

e >

• G

0

such

be

given,

thus

-

Q' [y] I < e

for

given

• F

exists

that

IQ'[z]

- Q'[y]

I < £2

"

Hence,

IQ[z]-Q[y][

__< I Q ' [ z ] - Q ' [ y ] I < ~- +

Izl 2 -

+

(Izl 2-

lyl 2 lao(x) l

lyl 2) do-

a

,

233

Since

Iz(x) -y(x) l =< ellz-yllF ,

by

(5.1.4),

Y(X) I = O ( I z ( x ) I )

uniformly

for

x c

[~ , B]

Thus

Isllzl2 -lyl 2

Hence

we may

right

hand

follows

Izl-lyl

of

this

J

I ~ - yl 2 do

cc'"

llz- y]l F •

Is2

further,

if n e c e s s a r y ,

y

the

inequality

completes

f

f'

8]

e BY[e,

consist with

functions

Proof:

Idol

small.

the

proof.

of

the

collection

support

and

f(~)

[~, B]

= f(B)

G c

so

Izl+lY]

The

to m a k e

result

the

now

5.1.0: Let

and

Idol

c'

restrict

side

and

COROLLARY

Idol ~

J

is d e n s e

If

f c G

,

in

J

= 0

c

such

of

all

that

finite

f c AC[~,

B]

,

Then

(5.1.9)

F

F

then

f'

is

~-absolutely

continuous

and

234

so

of

bounded

other The

hand,

for

result

ceding

now

f

c

on

J

,

follows

[e , 8] f'

~ BV[~

since

O

is

Thus , B]

f

c

J

.

On

the

and

so

f'

~ L 2 (e , B) •

in

F

by

the

pre-

y

J

dense

theorem.

COROLLARY

5.1.1: Let

such

variation

that,

z

c

for

F

Then

e >

0

there

exists

This

function

c

,

Q[z] - Q[Y] I <

Proof:

a

(5.1.10)

£

follows

immediately

from

the

preceding

i[y] (x)

=

non-oscillatory

dis-

cussion.

LEMMA

5.1.1: Let

Then

for

z

c O

Xy(x)

be

for

I =

l0 •

,

Q[z]

>

fooIzl2dv"

~0

(5.i.ii)

a

Proof:

We

can

assume

non-oscillatory,

then

y'(x)

=

c

that

+

f

0

=

0

Since

[y] (x)

=

0

is

x y(s)do(s)

x

c

[a , ~)

(5.1.12)

a

and

so

exists

the a

latter

solution

equation y(x)

is of

non-oscillatory.

(5.1.12)

such

that

Thus

there

y(x)

~

0

for

235

all

x ~ x0

X => x 0 , u'

if

there

(x 0 , X) u

can

by

.] be

for

COROLLARY

is n o

function

,

u ( x 0)

were

one

Thus

all

u

= 0

then

such

every

by

then

the minimal

that

of

Q[u]

would Q[u]

, ]

< 0

> 0

a zero

B

all

,

.

Since e,

for

~ 0

any

e A C [ x 0 , X]

have

with

,

Q[u]

u

for

in

such

replaced u

~ J

and

(5.1.9).

the

(5.1.11)

spectrum

operator

Proof:

This

THEOREM

5.1.0:

L0

follows

A necessary

is

from

and

to b e o s c i l l a t o r y spectrum

of

lying

the

The

any

for

of

argument

any

finite

[23,

pp.

sufficient

10

is

be

for

all

z e G

self-adjoint to

the

34-35,

left

is t h a t extension an

similar

of

that

of

I = 0

28].

for

(5.1.13)

the of

infinite

to

for

extension

Theorem

condition

and

= ly(x)

I = l0

self-adjoint

left

holds

of

[y] (x)

Proof:

and

,

u

then

that

y(x)

such

that,

5.1.2:

Then

to

such

= 0

as e l e m e n t s

E G

implies

u

= u(X)

respectively,

Suppose 10

for

regarded

x0 , X

thus

I.i.0

there

c B V ( x 0 , X)

[For

Theorem

part

of

the

the minimal set.

in

[28,

p.

40,

operator,

236

Theorem

31].

oscillatory

We for

solution

assume

10 = 0

yl(x)

B > ~ > tI an

can

of

Then

eigenfunction

that

then

(5.1.12) on

[~,

for

8]

If

x = tI

which

to

at

can

I = 0

(5.1.13)

there

vanishes

yi(x)

corresponding

£ [y]

I0 = 0

be of

is

exists e,

B

a

where

regarded the

as

problem

= 0 (5.1.14)

y(~)

If we

let

el

= e

and

principles

it

follows

81

the

have

a negative

corresponding

(

,

)9

Thus

applying

such

that

> B ,

then

(5.1.14)

the

Lemma

inner we

(L0~ I , ~i)

Choosing eventually

t 2 > 81 obtain

we an

11 we

can

with

product can

infinite

find

= Q[~I]

iterate

Writing

see

= l l ( f I , fl)

5.1.0

variational

= y ( B 1) = 0

eigenfunction

is

from

along

eigenvalue

Q[fl]

where

= 0

that

y(el)

should

= y(B)

the

sequence

for

fl(x)

that

< 0

in the

space

a function

L2(v) ¢i

in

< 0 .

construction of

finite

and

we

functions

237

~k ~ D(Lo)

with

disjoint

(L0} k , ~k )

Applying tive

now Theorem

part

of the

Conversely for imply This

that

completes

As

just

spectrum the

32 of

of C h a p t e r

proved,

that

it is o s c i l l a t o r y must

p.

that

2 e ....

15]

we

is an i n f i n i t e

that

(5.1.13)

5.1.1

along left

find

that

the n e g a -

set.

is n o n - o s c i l l a t o r y

with

of

Corollary

0

must

be

5.1.2 finite.

proof.

[23, 3.) if

we o b t a i n

§2.14].

for all

Theorem

(For t h i s

It a l s o

(5.1.13)

follows,

31 of

reduction from

is o s c i l l a t o r y

I > 10

[23,

Hence

the for

one

§2.12]

see

the

theorem ~ = l0

of t h r e e

then cases

occur:

i)

It is n o n - o s c i l l a t o r y

for all

2)

It is o s c i l l a t o r y

all

3)

There

is some

oscillatory

This

[23,

to the

an a p p l i c a t i o n

and Theorem methods

Lemma

such

k=lt

,

13 of

suppose

Then the

< 0

spectrum

let us

10 = 0

supports

also

THEOREM

applies

and

for

~0

such

for

to t h r e e - t e r m

l

1

that < l0

recurrence

for

~ > 10

it

is

it is n o n - o s c i l l a t o r y .

relations.

5.1.2: Let

~

satisfy

the u s u a l

hypotheses

and

suppose

that

238

o(t)

tends

is zero.

to a f i n i t e

Suppose

limit

furthermore

lim t÷~

Let

~

for

the

when

~(~)

Proof:

The

can

assume

of

t(~(~)

(5.1.15)

a necessary

(5.1.13)

- v(t))

and

sufficient

to be d i s c r e t e

= 0

is t h a t

(5.1.16)

<

spectrum

is n o n - o s c i l l a t o r y and only

we

that

Then

spectrum

lim t÷~

which

tlo(t) I = 0

be n o n - d e c r e a s i n g .

condition

at i n f i n i t y

if the

will

for

be d i s c r e t e

all

1

if a n d

Moreover

integro-differential

only

the

if

(5.1.13)

latter

holds

if

equation

t

y'(t)

f

= c +

y(s)dIo(s)-

l~(s) 1

1

assume

(5.1.17)

a

is n o n - o s c i l l a t o r y

for all

since

Thus

~(~)

< ~

we n e e d

We

can

only

show

that

that

~(~)

under

(5.1.15),

lim tI~(t) I = 0

if and For

only

given

if

(5.1.17)

1 ,

choose

rio(t) I +

is n o n - o s c i l l a t o r y t

so l a r g e

1 llItlg(t) I =< ~

= 0 ,

(5.1.18)

for

that

t > T

all

1

239

Then 1 19(t) ] < ~

tlo(t)-

and

consequently

Theorem

oscillatory

for

and

that

suppose

Suppose

that,

such

on

1

the

our

hypothesis

We

now

choose

t

the

that

other

hand

is n o n - o s c i l l a t o r y

(5.1.17)

is n o n -

let

(5.1.15)

for

all

hold

1

contrary,

~(t)

t(o(t)

implies

On

(5.1.17)

lim t÷~

By

2.1.4

t => T

tI~(t) I -= ~ ~ 0

< 0

and

- l~(t)]

so

: to(t)

large

t O(t)

so

+

Itlv(t) I

that

> -y

t > T

and

ti~)(t) i > C~

Then,

for

t > T

,

I > 0

,

tIo(t) - l~(t))

Since

by

hypothesis

t > T

(5.1.17)

> y(l-

i)

is n o n - o s c i l l a t o r y

for

all

1

we

240

can choose

l

so

large

that

1 > ~+

(l-l)

where

e > 0

is s o m e

fixed

number.

Thus

for

such

a choice

such

I

of

l

f

.

1 > ~ + e

.

t[o~t) -l~(t)l_ -,

Thus

o(t)

- l~(t)

application for

such

of T h e o r e m

1

completes

if p o s i t i v e

This

the

s e e p.

superfluous. Birman

[23,

97,

The p.

93,

sign

case

that

,

(5.1.17) and

> 0

Glazman

though

5.1.0,

[38].]

result

and

other

E 0

7]

An

is o s c i l l a t o r y

thus

we

~ = 0

This

when

the

latter

of

p(x)

can ~(t)

x c

[a , ~)

§29]

calls

is u s u a l l y in for

(5.1.19). the

the

result

(For t h e

(5.1.15)

extended

since we

[23,

obtain

11.9°].

Again

had

then,

criteria

,

Proposition

= Ip(x)y

indefiniteness

Theorem

of

Theorem

to

-y"

o(t)

78,

latter

equivalent

"polar"

p.

(2),

continuous

p(x)

when

[38,

absolutely

where

shows

t > T

g

proof.

and Kre[n

original

for

is a c o n t r a d i c t i o n

In p a r t i c u l a r of K a c

2.2.1

t => T

is n o t

a theorem

let

~

of

be

~ 0 ,

(5.1.13)

is

(5.1.19)

this

case

connected Because

finiteness

of

the with of the

the

241

negative

part

oscillation

Example and

to

criteria

spectrum

of

i:

let

(5.1.0)

c

the

for

theorem if a n d

the

operators.

If w e

o(t) all

includes

the

difference

an

> 0

by

(above) only

Za

(5.1.16)

Example

z 0

2:

n

is

t

that

the

negative

= n

n

part

by

all

of

the

obtain

v(t)

for

non-

because

therefore

define

n=

The

n

of

the

(3.8.3)

,

0 , 1 , ...

discrete

spectrum

then

of

(5.1.20)

analog

(5.1.20)

of

Birman's

is d i s c r e t e

if

n

<

and

is

In t h e

lim n ~ n÷ ~ j=n

a

The

follows

proof

therefore

addenda

is n o n - o s c i l l a t o r y implies

that

whenever the

=

0

3

the

usual

substitutions

omitted.

to C h a p t e r

2 -A -±Vn-- = ly n

theory

the

the

equation

= lanY n

hypothesis.

we

and

or

via

Moreover

of

difference

for

n

obtained

2.1.

finiteness

let

be

relations,

= 1

whenever in

can

of C h a p t e r

recurrence

-A2yn_l

where

spectrum

theorems

applidations some

of

2 we

n = 0 , 1

I < 0

spectrum

of

i

saw

i

that

(5.1.21)

o

Consequently, (5.1.21)

is

the

finite

above below

242

zero.

Thus

it s h a r e s

the

-y"

Other

criteria

difference

§5.2

for

THE CONTINUOUS OPERATORS:

generalized

"almost

We

shall

over

all

SPECTRUM

we

differential

I =

[0, ~)

,

everywhere"

respect,

as

of d i f f e r e n t i a l from

Theorem

OF G E N E R A L I Z E D

study

the

and 2.3.4.

DIFFERENTIAL

continuous

spectrum

of the

equation

d { , /x } -d-~ y ( x ) yd~ a

and

~(x)

assume

[0 , ~)

in t h i s

[a , co)

= x

in the u s u a l

hereafter of

on

c a n be o b t a i n e d

section

£[y] (x) =

where

= ly

properties,

the d i s c r e t e n e s s

operators

In this

same

that

x c I

so t h a t

(5.2.0)

equality

is

sense.

0

is of b o u n d e d

variation

Thus

I

(5.2.1)

~Ido(x) [ < 0

If

[y] (x) = ly(x)

then

y

satisfies

(5.2.2)

x E I

the V o l t e r r a - S t i e l t j e s

integral

equation

243

y(x)

= y(0)

+ xy' (0) +

(x- s)y(s)d(a(s)

x e I

- Is 1

0 (5.2.3) (by the r e s u l t s

in C h a p t e r

THEOREM

(Atkinson

5.2.1: Let

a(x)

any p o s i t i v e

3).

[3, p.

satisfy

392],

(5.2.1).

eigenvalues.

Hence

Theorem

Then

(5.2.2)

12.6.1)

(5.2.3)

cannot

is l i m i t - p o i n t

have at

infinity.

Proof:

For w h e n

I > 0 ,

by v i r t u e

of T h e o r e m

can e x i s t

and thus

(5.2.3)

12.6.1

in

the d i s c r e t e

has no s o l u t i o n

[3].

Hence

spectrum

in

L2(0 , ~)

no e i g e n f u n c t i o n

is c o n t a i n e d

in

(-~, 0]

Defining

the m i n i m a l

(5.2.0)

as in C h a p t e r

L0

L0

of

boundary if

which

implies

in

that

at

0

(5.2.2),

(5.2.3)

ting the a s y m p t o t i c

It follows

3, there

is d e t e r m i n e d

conditions

I < 0 ,

operator

f r o m this

has

is t h e n

i0

corresponding

a self-adjoint

to

extension

by a set of h o m o g e n e o u s

(see for e x a m p l e then T h e o r e m

12.5.1

a p a i r of s o l u t i o n s

[46, of Yl

§17.5]). [3, p. ' Y2

384] admit-

representations,

yl(x)

~ exp(-x/ilJ

Y2(X)

~ exp( x /I I )

that

Yl

' Y2

)

are b o t h

eventually

Now

of

244

constant

sign

and

so,

Consequently

the

5.1.0.

there

in

Thus

(-~, 0)

THEOREM

~

is f i n i t e

be no p o i n t

and

satisfy

is p r e c i s e l y

From

minimal

can

(5.2.2) on

is n o n - o s c i l l a t o r y .

(-~, 0]

of the

by T h e o r e m

continuous

spectrum

5.2.2:

(5.2.2)

below

spectrum

I < 0 ,

Hence

Let of

when

this

self-adjoint

the

any

the

that

the

below.

generated

by

continuous

spectrum

[0 , ~)

spectrum

self-adjoint

is b o u n d e d

operator

Then

semi-axis

it f o l l o w s

consequently

operator

(5.2.1).

is b o u n d e d

extension

Thus, the

for

of the

example,

differential

the

expression

Z [y] : ly

and

y(0)cos

is b o u n d e d

below,

~ - y' (0)sin

If we d e n o t e

i.e.

(5.2.4)

~ = 0

such

an e x t e n s i o n

by

i

then

(i f , f) > - y ( f ,

where We will (5.2.5)

y £ ~

and

now

proceed

in t e r m s

(

of

,

)

is the

to g i v e ~

f)

f ~ D

usual

an e x p l i c i t

The

approach

(5.2.5)

L2

inner-product.

lower used

bound

here

y

for

is e s s e n t i a l l y

245

an e x t e n s i o n lemma w i l l

LEMMA

of an a r g u m e n t

of E v e r i t t

[16].

The

following

be useful.

5.2.1: Let

(Ganelius f ~ 0

on the c l o s e d

[22])

and

interval

I fdg J

g

be f u n c t i o n s

J

of b o u n d e d

variation

Then

=< linf < J

f + var J

(5.2.6)

f} sup IK dg KcJ

where

var J

and the recall

sup

the f o l l o w i n g

(5.2.0)

in

J

is t a k e n over

The m a x i m a l by

f - S Idf(x) I

domain

= {fe L2(0 , ~)

and

subsets

on

~

of the o p e r a t o r

is d e f i n e d

:

f c A C l o c ( 0 , ~)

[0 , ~)

,

F(x)

' -= f+(x)

- Sx f(s)do(s) a

f E D ,

Lf = ~ [ f ]

L

We

generated

, f'(x)+

~ A C I o c ( 0 , ~) ,

where

For

J

by

F'(x) E L 2 (0 , ~) }

F(x)

of

notions:

L 2 ( 0 , ~)

exists

all c o m p a c t

246

and

i

is a s i n g l e - v a l u e d

the

self-adjoint

by

TO

operator

operator

in

generated

by

We

L 2 (0 , ~) (5.2.2)

denote

y(O)

and

= 0

Thus

~ ( T 0) =

{f

e ~ : f(0)

Tof

= ~ [f]

(5.2.7)

= 0}

and

LEMMA

5.2.2: Let

exists

a

o

satisfy

C = C(g)

fx

If(t)

> 0

Ida(t)

Then

(5.2.1). such

for

every

e > 0

there

that

I <= c(~) •

0

If

dt + s

0

f7

I f ' 12 d t (5.2.8)

for

all

Proof: the

f c ~

We

use

variation

,

and

Lemma of

o

x c

5.2.1 over

[i, ~)

with

f , g

[0, x]

replaced

by

respectively.

x

''Ifl2

and

Thus

}Sx

[ If(t)12 Ido(t) I < ~ inf J0 = [0,x]

Ifl 2 +

var

Ifl 2

[0,x]

Ido(t) I .

0 (5.2.9)

Now

if

x e

[i, ~)

then

inf

[O,x]

if]2

<

ix o

If

12 dt

.

(5.2.10)

247

Moreover,

Var [0,x]

=

dlfl 2

=

21re(ff' ) Idt

0 < 2{fO 1f ,2dtld {f~' f ' [2 dt} ½

by the Schwarz

inequality.

A(x)

(5.2.11)

Let us write

Ifl2dt

=

B(X) = {f~,f',2dt} ½

Inserting

(5.2.10-ii)

into

(5.2.9) we find

fx Ifl 2 Idol __< {A2(x) + 2A(x)B(x) } •

Idc~(t) ] 0

for all

f e D

For

e > 0 ,

1 A(x) - /-{B(x)

> 0 .

Hence 2A(x)B(x)

< 1 A2(x) =

Inserting

this in (5.2.12)

+

£B2(x)

S

we obtain

f~f20o ~ {il+~IA2 x +~B2 x } c

(5.2.12)

248

where

C

is

the

quantity

(5.2.1).

Replacing

~

e/C

by

we

find

If

Idol

__< C ( s ) A 2 ( x )

+ eB2(x)

where C(~)

This

completes

When be Our

found

in

proof

o [18,

C

.

proof.

is

absolutely

continuous

p.

339,

I]

appears

Consequently, f ~ ~

the

= 1 +-

to be

if w e

Lemma simpler

above

lemma p=

in

the

case

when

than

the

case

p=

1 E = ~

choose

the

we

find

that

1

for

1

of

can .

[18].

each

,

,fx

If[ 2 Idol =< 2

where

LEMMA

c

+ C'

(5.2.13)

Ifl2dt

c<½1

5.2.3: For

every

f c ?

lim

Proof:

0 If'12dt

This

can

be

f(x)

,

f'

= lim

shown

as

e L 2 (0 , ~)

f(x) f' (x)

in L e m m a

and

= 0

2 of

[18].

For

if

249

f' / L 2 (0 , ~)

{If'] 2dt+ 1~12ao} => 0

If'

dt -

're,

-_> y

by

(5.2.13).

A simple

Ifl21dol

0

f'l

Since

f e L 2 (0 , ~)

lim ix x÷~ 0

{I~' [2 dt+

calculation

ix

(Lf) ( t ) f ( t ) d t

also

=

0

dt

-

fo,

f

12 dt

we must have

Ill 2 do(t)}

shows

C'

(5.2.14)

= ~

that

fx

{if,12dt+

ifl2do(t)}

_

0

[~f,]x 0 (5.2.15a)

However,

since

f , Lf e L 2 ( 0 , ~)

we must

therefore

have

lim f (x) f' (x) = X-~Oo

B u t by t a k i n g that This

o

real

is real,

contradiction

implies

and imaginary the l a t t e r proves

parts

equation

that

f'

in

(5.2.15a), and n o t i n g

is c l e a r l y

~ L 2 ( 0 , ~)

impossible. Hence

(5.2.13)

that

I

~Ifl21dol

< ~

(5.2.15b)

0

Thus

(5.2.15a) i m p l i e s

that

f(x) f'(x)

+ ~ ,

as

x ÷ ~

But

250

since

ff'

that

• L(0,

If(x) 12 ÷

B

~)

,

,

as

e =

0

The

x ÷ ~

2

f • L 2 (0 , ~)

Since

We

also

=

,

If(0) I

8 =

obtain

0

relation

implies

•

2

If(x) I

following

~x J 2 re(ff')dt 0

+

.

The

If(x)f'(x)

lemma

I ~

is

(5.2.16)

.

proved.

llff'II~

for

all

large

x .

Thus

I

(5.2.17)

~ I f ( x ) f' (x) I Id o ( x ) I < Cll ~f'lloo < ~o

0

for

all

f • D

.

Moreover,

I f ' ( x ) I2 =

=

for

Since Thus y =

of

(5.2.17)

f' , L f

0

,

+

{~'df'+ f'd~'}

If(O)[

+

{f'f + f'f}d~(t)

the

2

first

• L 2 (0 , ~)

I f ' ( x ) I2 ÷ y

f

If(O) [ 2

-

Because

such

as

f2

{f'Lf-f'Lf}

integral

the

second

x ÷ ~

But

in

dt .

(5.2.18)

integral since

is f'

(5.2.18)

is

finite.

also

finite.

• L 2 (0 , ~)

Hence

lim

f' (x)

=

0

for

f • D

.

(5.2.19)

X+~

Since

f , f'

are

locally

of

bounded

variation

on

[0 , ~)

251

Lemma 5.2.3 and

(5.2.19)

II fll

co

<

imply that

,

oo

II f ' l l

oo

<

oo

f

c D .

(5.2.20)

The above results essentially comprise Theorem 1 of

[18] but

in a more general setting.

THEOREM 5.2.3: The operator

TO ,

defined by

(5.2.7),

is semi-

bounded from below and

(T0f , f) > -C2(f , f)

where

Proof:

C

is defined by

Let

f e D(T 0)

f e D(T0)

(5.2.21)

(5.2.1).

Then

(5.2.16)

along with an applica-

tion of the Schwarz inequality shows that

II fll 200 <= 2 II fll 2 II f'll 2

(5.2.22)

where the subscripts are self-explanatory.

Moreover

co

Io If(x)19 Id(~(x) I < llfll2

Inserting

(5.2.22)

S

•C

(5.2.23)

into the latter we obtain

°If(x ) [2 id(~(x) l < 2Cllfll211f,ll2

f £ D(T0) .

0

(5.2.24)

252

Using

(5.2.15)

with

f e D ( T 0)

we

have,

on account

of L e m m a

5.2.3, oo

t { I f ' 1 2 d t + Ifl2da(t)} _-> IIf'l12 2 - [ I f 1 2 1 d ~ ( t ) l

(Tof, f) =

J

0

co

0

_-> IIf'l12 2 -

2cllfll 2 IIf'l12

f c ~(To) .

Since

we obtain,

from

the

(llf'l12

- cIIfll2)

former

equation,

(Tof, f) => -c21r fll22

This

proves

the

When lower

bound

a

=

_C 2

2 _-> 0 ,

that

(f ' f)

f ~ D ( T 0)

continuous

and

theorem.

is a b s o l u t e l y

a' = q

becomes

2

c = llqlli so t h a t

The

(T0f

f) > -llqLl2 (f

bound

was

obtained

Consider

now

the

latter

f)

f ~ 7)(T0)

by Everitt

recurrence

[16,

relation

p.

146].

,

the

253 2

-A

where

Yn-i

(b n)

we k n o w

is any

that

the

defined

by

D(T0)

= {Y

=

=

real

(5.2.25)

Consequently TO

+ bn'Yn

n= 0 , 1 ....

XYn

sequence.

From

is l i m i t - p o i n t

results

(Yn)C

£2

of C h a p t e r

_A 2

:

the

earlier

(in the 3 imply

Y n - i + b n Y n e"

£2

(5.2.25)

results

£2-sense). that

the

' Y-I

operator

= O}

2 (Tof) n

is s e l f - a d j o i n t .

If w e

=-A

fn-i

suppose

+ bnfn

further

that

oo

C =- [

Ibnl

(5.2.26)

< co ,

0

then

it w i l l

follow

that

co

I

Ibnl If n 12 < ~

f e D(T0)

(5.2.27)

0

(since any

the

f

sequence

÷ 0

n f

n

as

n + co

for

f e DO ) .

Moreover

,

n

Ifn 12 = I < 1 1 2 + ~ { f j - i A f-j -I -

3.af j-i }

0

where

Thus

we have

if

used

f e D ( T 0)

partial

,

the

summation.

Schwarz

inequality

gives

for

254

Ifn '2

{ ~ --
Ifj_II

{ i IAfj-i '2

2 }½

12

and so sup Ifn 12 __< 211fH211AfII2 n

f ~ ~(T o)

(5.2.28)

where IIfll2 =

Ifn 12 ½

and IIAfII2 =

IAfn_iI 2 ½

Inserting (5.2.28) into (5.2.27) we find oo

7.

[bnlIfn 12

<__ 211fII211AfII2" c

f ~ ~(To) .

0

Now, from (4.1.7) we see that, for

f ~ 0(T 0) ,

oo

(Tof, f) = ][ {IAfn_iI 2 + bnlfn 12 } 0

X IAfn_112 - X Ibnllfn 12 0

0

> I1~fll2 _ 211fl12 IIAfII2 c _->-c 2 IIfll~

(5.2.29)

255

Thus

(T0f , f) => - C 2 ( f , f)

consequently

we o b t a i n

in the case w h e n

o(t)

f e D(T 0)

the a n a l o g o u s is g i v e n

bound

of T h e o r e m

5.2.3

by

g(t n) - g(t n - 0) = -b n since

then oo

oo

ida(t)J o

The above [16] .

= [ JbnJ o

is the d i s c r e t e

analog

of E v e r i t t ' s

theorem

APPENDIX

§i.

FUNCTIONS We

cerning

be

DEFINITION

in

of

those

definitions

bounded

variation

Most

these

[i, p.

of

127ff.]

and

which

results

thus

we

results

were

con-

used

in t h e

are well-known

omit

the

and

proofs.

i.i:

A

O(x)

function

is of b o u n d e d

values

VARIATION:

here

chapters.

found

(a, b)

BOUNDED

summarize

functions

preceding can

OF

I

of

the

defined

variation

over

a finite (a , b)

over

real

if

interval

the

set

of

sum n-i (I.l.l)

Io(xr+ I) - O(x r) I r=0

is b o u n d e d (a, b)

,

for

all

of

all

such

over

(a , b)

and

Vf(a,

THEOREM

and

a = x0 < xI <

bound

or

n

sums is

all

... is

possible

subdivisions

< xn = b

called

sometimes

the

The total

denoted

by

least

{Xr}

common

variation Var{o(x)

upper

of ;

a , b}

b)

A: If

o(t)

is of b o u n d e d

variation

on

(a, b)

of

then

257

a)

a(x+

0)

, a(x-

0)

exist

b)

o(b-

0)

, o(a+

0)

both

c)

The

points

If

a

a

is

variation

over

requirement when

on

and

n

(a, ~)

that

DEFINITION

wQrk

form

say

a(~)

a finite

that

~

or

at m o s t

is of b o u n d e d

is d e f i n e d

boundedness

we

a(t)

shall

of

and

(I.l.l)

assume

= lim

cannot

that

be

the

satisfied

further

that

(I.i.2)

a(t)

have

a jump

at

infinity.

1.2:

A real-valued

function

absolutely continuous on is a

we

if

a(~)

so

(a , b)

=

In t h i s

exists

c

(a, b)

b = ~

of u n i f o r m

b = x

x

set.

is b o u n d e d

finite

each

exist.

of d i s c o n t i n u i t y

denumerable

d)

for

6 > 0

such

f

[a, b]

defined if

for

on

every

[a , b] e > 0

is there

that

n If(Xk+h

(I.i.3)

k) - f ( x k) I < e

k=l

for such

every that

n

disjoint

subintervals

(x k , x k + h k)

of

[a , b]

258

n

hk

(I.i.4)

<

k=l

THEOREM

B: Let

f

be an a b s o l u t e l y

continuous

function

on

[a , b]

Then

a)

f

is c o n t i n u o u s

b)

f

has

c)

f

is of b o u n d e d

variation

on

(a, b)

d)

f'

is L e b e s g u e

integrable

on

(a, b)

a derivative

almost

I~ f ' ( x ) d x

For variation

further

and

the

results

Lebesgue

= f(b)

concerning theory

of

everywhere.

and

- f(a)

functions integration

of b o u n d e d we

refer

to

[1].

§2.

THE

RIEMANN-STIELTJES

DEFINITION Let finite

For

any

INTEGRAL:

2.1: f, 0

interval

be r e a l

valued

form

partition

the

defined

on

some

[a, b]

{Xr}

of

a = x0 < xI < we

functions

(a, b)

..-

,

< xn

b

(I.2.1)

sum n-i S =

f(~r ) (O(Xr+ I) - O(Xr) ] r=O

(I.2.2)

259

where

~r

e

If, tends

[Xr' Xr+l] as

n + ~

to a u n i q u e

choices

of

~r

integral of

f

and

limit

c

m a X l X r + 1 - Xrl

for

all

[x r , X r + I]

partitions

the

limit

with respect to

i

+ 0

o

the

{Xr}

is c a l l e d

sum and

the

S for

all

Stieltjes

written

b f (x) do (x) .

(I.2.3)

a

If t h e

"distribution

integral

THEOREM

reduces

variation

f over

is

continuous

(a, b)

f over

is

improper

on

the

the

148].

[a , b]

integral

continuous

(a , b)

f (x) dO (x)

Integrals

[i, p.

is a s t e p - f u n c t i o n

and

(I.2.3)

o

is of b o u n d e d

exists

[i, p.

159].

D:

If variation

a sum

o

C:

If

THEOREM

to

function"

over sense

<

infinite so

and

o

If(x) I • V a r { o ( x )

:

a , b} .

understood

in t h e

on

[a , b]

is of b o u n d e d

then

sup [a,b]

intervals

are

that

f(x)do(x)

= lim b-~oo

f(x)do(x)

(I.2.4)

260

whenever

the

THEOREM

limit

exists.

E:

bounded [a, ~)

If

f

on

[a , ~)

then

is c o n t i n u o u s

the

and

o

for

all

is of

real

bounded

finite

x

variation

uniformly over

integral

f

f(x)do(x)

(I.2.5)

a

exists

[3,

THEOREM

F:

p.

If variation

422].

f

is c o n t i n u o u s

over

b I f(x)do(x) a

THEOREM

a)

then

= f(b)o(b)

[a , b]

[i, p.

and

o

is of b o u n d e d

144]

- f(a)o(a)

-

rb I

o(x)df(x)

G: If

bounded

[a , b]

on

f, g

are

continuous

variation

over

F(x)

f(s)da(s)

=

[a, b]

on

[a, b]

and

~

is of

then

is of b o u n d e d

variation

[a , b]

b) g(s)dF(s)

=

gfdo

.

over

261

c)

If

~

is c o n t i n u o u s

F(x)

d)

If

is c o n t i n u o u s

for

some

continuous

e)

the

(I.l.l) fine

d)

(a , b)

a),

follows

integral

as

b),

from

F(x0+

c)

the

0)

:

is

n ÷ ~

so t h a t

follows

THEOREM

from

the

at

f(x 0) = 0 ,

a , b}

x x

then

then .

F(x)

is

the

be

Ido(x) [

as

subdivisions

m a X l X r + I - Xrl

can

s2

=

interpreted

found

in

+ 0

being

as

[i, p.

the

limit

of

increasingly

r +

161-62].

relation

- F(x0-

0)

r = |

x0+0

&

=

e)

at

x0

Var{o(x)

where

Proof:

(right-continuous)

x0 c at

(right-continuous)

f(s)do(s) 0-0

f ( x O) [ O ( X o +

O) - O ( x 0 - 0)]

definition.

H: Let

f

be

a real-valued

right-continuous

function

on

[a , b]

Then lim h+0+ for

all

x e

[a , b)

l x+h f ( s ) d s

[

= f(x)

(I.2.6)

262

Proof: there

The

right-continuity

is a

6 > 0

such

of

f

implies

that

given

that

(I.2.7)

If(s) - f(x) I < e

whenever Let

x -< s -< x +

x E

[a, b)

,

6 ,

x e

[a, b)

E > 0

1 ~x+h f(s)ds-

We

restrict

as

that

find

h

in

> 0 ,

so

f(x)

that

(I.2.7).

=

1 ~x+h ~ (f(s) - f ( x ) ] d s

1 =< ~

~x+h

If(s) - f ( x ) I d s

0 < h < 6

Using

now

where

(I.2.7)

the

6

' I

.

is

the

0 < h < 6

where

same we

that

f(s)ds-

f(x)

i

=< ~

< 1 = ~.

whenever

THEOREM

0 < h < 6

This

If(s) - f ( x ) I d s

e'h

completes

= e

the

proof.

J:

Let

f

be

absolutely

continuous

on

[a , b]

and

I

suppose x

E

that

[a , b]

bounded

f Let

variation

has

a right-derivative

g over

be

f+(x)

a right-continuous [a, b]

then

at

each

function

of

point

263

lim h÷0+

for

each

Proof: and

x c

The

x £

prove

integral

above

exists

by v i r t u e

theorem

of b o u n d e d

= g(x) f+(x)

(I.2.8)

[a , b)

[a , b)

the

1 fx+h ~ ~ g(s)df(s)

when

variation

f

for

h

of T h e o r e m

F.

sufficiently It n o w

is i n c r e a s i n g

is the

difference

(since

of two

small

suffices every

to

function

increasing

functions). Let [i, p.

x e

160,

[a, b)

Theorem

and

7.30]

assume

implies

f

the

is i n c r e a s i n g . existence

Then

of a n u m b e r

c

where

inf se[x,x+h]

with

the

property

x+h

Dividing

(I.2.10)

continuity

of

g

g(s)

~ c ~

sup sc[x,x+h]

g(s)

(I.2.9)

that

g(s)df(s)

by that

h

,

= c[f(x+h)-

we

see

c ÷ g(x)

from as

f(x))

(I.2.9)

(I.2.10)

and

the

h + 0+

Hence x+h

lim h÷0+

[

g(s)df(s)

= g(x)

lim h÷0+

f ( x + h) - f(x) h

!

= g(x) f+(x)

for

x e

[a, b)

The

result

now

follows.

right-

264

THEOREM

K: Let

be R i e m a n n

f, g

satisfy

integrable.

the

g(s)df(s)

Let

A(t)

of T h e o r e m

J and

f'

Then

la Proof:

hypotheses

, B(t)

=

g(s)f+(s)ds

be d e f i n e d

(I.2.11)

.

by the

left

side

and

right

!

side

of

(I.2.11)

is c o n t i n u o u s continuous.

and

in

(I.2.11)

holds

(I.2.11)

then

that

of

A

at

t

and

so

§3.

GENERAL

g

THEORY

In t h i s required

for

equations theory

general

was

and

would

modifications. uniqueness

all

type

,

we

result.

continuity must

of the

shall

In t h e

[3, p.

only

state

to t h o s e begin

in

f

the of

case

from those

if

t

at the on

[a , b) .

EQUATIONS:

basic

tools

integral when

p(t)

= 1

case

of the

the

case

when

results the

is

t

The

[3] w i t h

by p r o v i n g

is

case

at

B(t)

INTEGRAL

339].

different

hand

everywhere

theory

A(t)

in w h i c h

of

equal

summarize

B(t)

the d e r i v a t i v e

derivative

= B(t)

shall

in

shall

similar We

A(t)

(i.0.0).

is n o t v e r y

be

,

On the o t h e r

the

A(t)

g

is i n t e g r a b l e continuous

VOLTERRA-STIELTJES

developed

so w e

gf+

of

t

since

the d e v e l o p m e n t

p(t)

= 1

OF

section

of the

jumps

is a t w o - s i d e d

for a l m o s t

of

Since

is a b s o l u t e l y

the

g

points

proofs

from

of

jump

p(t)

f

point

implies

the

since

Apart

appearing

a jump

respectively.

whose

appropriate

an e x i s t e n c e

and

265

THEOREM

1.3.1: Let

over

o(t)

[a, b]

function

(for a r e l a t e d

and

result

see

be right-continuous let

of b o u n d e d

p(t)

[79])

and

of b o u n d e d

be a p o s i t i v e

variation

on

variation

right-continuous

[a, b]

with

the p r o p e r t y

that 1 - p(t)

Then

the

integral

y(t)

has

a unique

Proof:

We

successive

= e +

ja t

B

i + ~

solution

essentially

y d o ds

p(s)

on

[a , b]

use

the

Picard

the

function

(I.3.2)

for g i v e n

method

~ , B

of

approximations.

(I.3.1)

implies

that

Yo(t)

is a b s o l u t e l y

continuous

= ~ +

on

[a , c]

and

is f i n i t e .

We

now define

Yn

sup [a,c]

Y0(t)

defined

by

(I.3.3)

B

I

M =

exists

(I.3.1)

equation

continuous

shall

E L(a,b).

where

c < b

I

so

(I.3.4)

lY0(t) l

by r e c u r r e n c e

and

on

n

266

Yn+l (t) =

p(s)

for

n = 0 , 1, 2 , ... , t c [a , c]

Let

V

be d e f i n e d

by

v =

ioeo

V

is the total

that

(I.3.7) for

variation

of

o

n!

n = 0 , 1 , .... holds

n = m

,

if

We p r o v e

n = 0

(I.3.6)

Ida(s) [

vn{la}n lYn(t) [ <

for

(I.3.5)

Yn d o d s

over

[a, c]

We claim

t e [a , c]

this by i n d u c t i o n

If we a s s u m e

that

(I.3.7)

on

(I.3.7)

n

.

is true

then

[Ym+l (t) I =<

Jap(s) fa]Ym(X) [ Ido(x)]ds lat

<

1 p(s)

(I.3.8)

f a ~ { M V m [ fax l ] m } [d(1 (x)]ds -~.

<

m ft 1 {~ [f~l] } f~

<

Mvm it p(S)i -~--! • V

a p(s)

--

p

"

(I.3.9a)

ld° (x) Ids

m ~

ds

a m+l

( m + i)'

(I.3.9b)

267

where

(I.3.9)

for all

is

n .

(I.3.7)

with

Consequently

n = m+ 1 .

Hence

(I.3.7)

holds

the series

oo

Yn(t)

(I.3.10)

= y(t)

n=O

is u n i f o r m l y

convergent

Y0 (t) +

p(s)

on

[a, c]

and

y (x) do (x) ds

1 = Y0 (t) + Sa p(s)

= Y0 (t) +

[ n=0

"la

~ Yn (x)d~(x)ds n=0

(s)

Yn (x)do(x)ds

P

oo

: Y0(t)

+

[

Yn+l(t)

n=O

= y (t)

where

the interchange

is justified

by the u n i f o r m

[i, p. 225]. (I.3.2)

Yn

and therefore because

So

and hence

that each

(I.3.10)

satisfies

the limit

the integral From

essentially

function

y(t)

and summation

of the series,

a solution.

is continuous,

the convergence

lemma

convergence

represents

To show that Gronwall

of the order of integration

equation

(I.3.5)

because

Y0

we see is,

must also be continuous

is uniform.

y(t)

[3, p. 455],

is unique we use the extended which

states

that if

p(x)

,

268

x

c

[a , b]

negative

is

and

continuous,

where

,

non-negative

continuous and

cO > 0

(I.3.2).

~

for

a ~ x

p(x)

=< c O + c i

p(x)

to

and

, cI > 0

Assume

and

if

~ b

continuous

is

and

~

non-decreasing

and

is

non-

right-

,

p(t)da(t)

(I.3.11)

then

< c O exp

possible

(I.3.12)

{c I [o(x) - o(a) ] }

that

y(t)

, z(t)

are

two

solutions

Then

Jy(t) - z(t) J _<

St i

=<

y

ly(x) - z(x) l do(x~ I

~

ly (x) - z (x) I do (x) 1

-

a

Jy(x) - z(x) J d a (x) Jd s

p(s)

<

(I.3.13)

a

The

extended

ly(t)-

Gronwall

z(t) J =

[ a , c]

for

0

any

for c

< b

lemma t ~

(I.3.12) [a , c]

.

Hence

to

show

now and

y(t)

implies

so =

y(t)

z(t)

that =

z(t)

everywhere

on on

[a, b]

REMARK

:

It

is

possible

Theorem

1.3.1

by

a transformation

269

of

the

independent

variable.

For

if

e L ( a , b)

then

the

P transformation

t~->

T(t)

:

f

t 1 p

a

will

transform

(1.0.0)

y' (T)

into

f

= cI +

an

(I.3.14)

equation

of

the

form

T T •

y(s)do(s)

[0 , T(b)]

(I.3.15)

0

where

(')

ness

results

THEOREM

_

d dT

[i,

for

pp.

(I.3.15)

The

45].

are

existence

discussed

in

and

[3, p.

unique-

341].

1.3.2:

The satisfying

solution the

(I.3.2)

where

B = p(a)y'(a) If b o t h y'(t)

is

Proof:

The

that

[3, p.

in

(I.3.15)

can

,

is

=

a right-derivative

are

theorems found

so

[a , b]

(I.3.16)

y(s)do(s)

[a , b] continuous

at

t

or

if

y(t)

= 0

derivative.

similar, and

in

equation

8 +

t c

an o r d i n a r y

346]

be

t)

o, p

proof

Other

has

integro-differential

p(t)y'

then

144-

with

the

appropriate

changes

is o m i t t e d .

dealing

with

in C h a p t e r

equations

ii of

[3].

of

the

form

to

270

THEOREM

1.3.3:

There z(t)

of

exists

(I.3.2)

two

such

linearly

independent

the

[3, p. t = b

"derivatives"

348].

In fact

are

generally

the W r o n s k i a n

= 1

(I.3.17)

right-derivatives (I.3.17)

is c o n t i n u o u s

at

.

We with

y(t) ,

that

p(t) (y(t)z' (t) - z(t)y' (t))

where

solutions

shall

(I.3.2)

y(t)

write

the

inhomogeneous

equation

associated

as

= ~ + B

st a

1 + p

s

p(s)

a

ydods

it,s

+

a

- -

ds

p(s)

(I.3.18)

where on

f [a,

is a r i g h t - c o n t i n u o u s

function

of b o u n d e d

variation

b]

THEOREM

1.3.4: Let

bounded

o , f , p

variation

and

be r i g h t - c o n t i n u o u s p(t)

> 0 ,

t e

functions

[a, b]

Then

of the

function ft ~(t)

= y(t)

la z(s)df(s)

- z(t)

y(s)df(s)

a

is a s o l u t i o n

of

(I.3.18)

for

~ = 0 ,

8 = -f(a)

(I.3.19)

271

Here

y , z

are two l i n e a r l y

independent

f

p(t)y' (t) = 1 +

solutions

of

t (I.3.20)

yd~

a

p(t) z' (t) = I t a

chosen

so that

(I.3.21)

zdo

p(t) ly'(t) z(t) - z'(t)y(t) I = 1

for all

t e [a , b]

Proof:

The p r o o f

the a p p r o p r i a t e , by

c o u l d be c a r r i e d

modifications

as g i v e n by [3, p.

(I.3.19),

out as in

or by u s i n g

[3, p.

351]

with

differentials.

For

has a r i g h t - d e r i v a t i v e

~'

given

352]

~'(t)

= y'(t)

and so m u l t i p l y i n g is of b o u n d e d

d(p(t)~'(t))

(I.3.22)

variation

=

zdf - z'(t)

by

over

zdf

p(t)

w e see t h a t

[a , b]

p(t)~'(t)

so t h a t

d(p(t)y'(t))

+ p(t)y'(t)d

(I.3.22)

ydf

[ftz d f ]-

-

a ydf

p(t)z'(t)d

a

d(p(t)z'(t))

Ift ] a ydf

(I.3.23)

=

zdfld~(t)

- z (t) [ f t ydf

(t)

a

+ p(t)y' (t) z(t)df(t) - p(t) z' (t)y(t)df(t) (I.3.24)

272

=

y(t)

zdf

+ p(t){y'

-

z(t)

(t)z(t)

-

= $(t)do(t)

on

account

That

of

ydf

y(t)

dc~(t)

z'(t)}df(t)

(I.3.25)

+ df(t)

(I.3.19)

and

the

constancy

is c o n t i n u o u s

can

almost

be

of

read

the Wronskian.

out

from

(I.3.19) .

For t+0 ~(t+

I

0

-

~(t-0)

=

y(t)

,t+0 zdf-

z(t)I

: y(t)z(t)

(f(t+

0) - f ( t -

- z(t)y(t) If(t+

=

so t h a t

Thus

we

~

is c o n t i n u o u s

can

integrate

p ( t ) ~ ' (t)

0)]

0) - f ( t -

0) I

0

at

t

(I.3.25)

= p ( a ) ~ ' (a)

ydf

t-0

;t-0

f

+

and

over

thus

everywhere.

[a , t]

to g e t

t ~(s)do(s)

+ f(t)

- f(a)

a

or d i v i d i n g [a , t]

we

~(t)

throughout

by

p(t)

integrating

again

over

obtain

= ~ +

(6-f(a)l

st

1 + ~

a

where

and

~ = ~(a)

,

6 = p ( a ) ~ ' (a)

st~ f s ~ d o d s a

+

a

But

~(a)

= p ( a ) ~ ' (a) = 0

273

thus w e (I.3.18)

find t h a t with

the l a t t e r

~ = 0 ,

equation

is e q u i v a l e n t

to

B = -f(a)

REMARK: If

h(t)

is the s o l u t i o n

f

p(t)h' (t) = c +

of the h o m o g e n e o u s

equation

t (I.3.26)

h(s)do(s)

a

satisfying

the i n i t i a l

conditions

h(a)

p(a)h'(a)

then

the s o l u t i o n

~(t)

~

of

= h(t)

= e

(I.3.27)

(I.3.28)

= 8 + f(a)

(I.3.18)

+ y(t)

is g i v e n by

ft

zdf - z ( t )

(I.3.29)

ydf

a

i.e.

(I.3.29)

initial

§4.

is the s o l u t i o n

conditions

CONSTRUCTION We b e g i n

UiY

=

~(a)

of

= ~ ,

OF THE G R E E N ' S by d e f i n i n g

(I.3.18) p(a)~'(a)

corresponding

to the

= B

FUNCTION

two l i n e a r

forms

UI , U2

2 { (j-l) (j-l) } [ Mij y (a) + N i j p ( b ) y (b) 5=I

by

(I.4.1)

274

for

i = 1, 2 ,

integral ing

in

where

equation (I.4.1)

derivative

is

(I.3.0)

will,

and

y

the

a solution

or

(I.3.2).

in

general,

M, N

are

be

of

The

taken

some

Stieltjes

derivative to m e a n

appear-

a right-

constants.

By

we

shall

mean

U(y)

= 0

Ul(Y)

= 0

(I.4.2)

both

(I.4.3-4) U2(Y)

We are

assume

at t h e

linearly

outset

that

independent.

y(t)

= ~ +

= 0

the

If

boundary

the

conditions

homogeneous

(I.4.3-4)

problem

Itl ftl fs

B

a

--+ p

a

p(s)

a

y d a ds (I.4.5)

U (y)

has

only

the

equation

with

incompatible solution

(I.3.2)

= 0

zero

solution

homogeneous [5,

satisfies

p.

It

is

say

that

the

conditions

compatible

homogeneous

(I.4.3-4)

is

if a n o n t r i v i a l

(I.4.5).

y , z

be

such

that

y(a)

= 0

we

boundary

73].

Let

p(a) z'(a)

then

Then

two

linearly

= 0

,

independent

p(a)y'(a)

= 1

solutions ;

z(a)

of

= 1 ,

275

p ( t ) { y ' (t)z(t) - z ' ( t ) y ( t ) } = 1

for all

t We

now define

a new

function

K(x,

0 K (x , t)

a continuous t

A

and

K

we of

t

0 , t)

and

shows

represents

for

x > t

fixed

similarly that

- Kx(t-

if

.

see that,

computation

( t + 0 , t)

z(x)y(t)

a < t < b

function

Kx(t+

where

x < t (I.4.7)

(I.4.7)

simple

by

= [

a ~ x ~ b From

t)

if

y(x)z(t)-

where

(I.4.6)

for

0 , t)

the

x of

,

K(x,

x

for

a < t < b

t)

is

fixed

,

1 p(t)

(I.4.8)

right-derivative

of

K

X

with

respect

to

x

Moreover defined

evaluated

at

from Theorem

t

1.3.4

the

function

~(x)

by

~(x)

satisfies boundary

the

the

function

should

satisfy

problem

(I.4.3-4) . K

(I.4.9)

K ( x , t)df(t)

inhomogeneous

conditions

modify

=

so that,

the b o u n d a r y

(I.3.18)

but

It is t h e r e f o r e as a f u n c t i o n

conditions.

not

the

necessary of

x

,

it

to

276

Therefore

let

G ( x , t)

where as

we

choose

a function

= elY(X)

the

of

~. i

x

so

+ ~2z(x)

that,

satisfies

for

UG

(I.4.10)

+ K(x,

t)

fixed

t £

(a , b)

,

G

= 0

i.e. Ui(G)

= Ui(K)

=

Since the

U. (K) 1

can

following

+ ~2Ui(z) i=l,

0

be m a d e

system

U

+ ~lUi(Y)

of

continuous

on

2

we

[a , b]

obtain

equations

(Y)

(I.4.11)

U2(Y)

(I.4.11)

can

determinant In

such

be of

a case

continuous We

U 2 (z)

solved

t

now

show

determinant

For

the

for

of

equation

t c

that

-U 2 (K)

for

t c

from

solutions

[a, b]

zero

~. (t) 1

if t h e

for

will

such

t.

be

[a, b]

(I.4.5)

the m a t r i x

is c o m p a t i b l e

appearing

is c o m p a t i b l e

if a n d

a non-trivial

solution

~

which

is

there

ci , c2

not

both

zero

exists

each

is d i f f e r e n t

resulting

in

the

uniquely

the matrix the

e2

in only

the

and

(I.4.11) if

case

such

if

is

(I.4.5) if a n d

that

only

if

zero. admits

only

if

277

~(t) =

cly(t)

+ c2z(t)

and

u l(~) The

latter

(u 1 ( ~ )

is t r u e

u 2(~)

=

if a n d o n l y

=

0

if

)

c I U l(y)

+ c 2 U l(z)

= 0

(u2 (~) --)

c I U 2(y)

+ c 2 U 2(z)

= 0

admits

solutions

=

nontrivial

determinant since we

assumed

determinant unique

of t h e m a t r i x

must

G ( x , t)

as a f u n c t i o n the problem

the

of

x

and

(I.3.18),

was

(I.4.11)

being

incompatible

from each

in

is e q u i v a l e n t

zero

and

t

This

(I.4.10)

is c a l l e d

the

will

the

so

to the zero. latter

(I.4.11)

said,

Thus

admits

the

satisfy

UG = 0

Green's function for

(I.4.3-4).

1.4.1:

function

following

a)

for

defined

Whenever unique

(I.4.5)

~. (t) 1

this

appearing

be d i f f e r e n t

solutions

function

THEOREM

that

and

G(x,

(I.4.5) G ( x , t)

is i n c o m p a t i b l e defined

for

there

a s x

exists

, t s b

a having

properties:

t)

is c o n t i n u o u s

absolutely

continuous

x , t

in in

x

for

jointly fixed

and t ~

[a, b]

278

b)

As

a function

when

x = t

for

c)

The

of

x

and

,

the

G

satisfies

boundary

(I.4.5)

conditions

except

UG

= 0

a s t s b

solution

(I.4.3-4)

of

the

is g i v e n

inhomogeneous

problem

(I.3.18),

by tb

y(x)

G(x, t ) d f ( t )

= I

(I.4.12)

Ja

d)

When

x=

t

,

G

(x , t)

has

a jump

of m a g n i t u d e

X

G

(t+

0 , t)

- G

X

(t-

0 , t)

-

X

- 1 ÷ ~i (t) {y' (t) - y' ( t - O) } p(t)

+ ~9 (t) {z' (t) - z' ( t - 0) } (I.4.13)

where

~i'

~2' y,z

were

defined

(I.3.2),

we

had

in

section

earlier.

REMARKS:

i. on

If,

[a, b]

solutions "extra"

in

then, would

terms

have

in

Gx(t+

(I.4.14) C' (a, b)

is,

as w e

saw

continuous

(I.4.13)

0 , t)

would

- Gx(t-

in p a r t i c u l a r ,

in w h i c h

case

the

that

p , ~ 1 of

derivatives disappear

0 , t)

satisfied Green's

were

this and

continuous

Appendix, therefore

the the

leaving

1 p(t)

(I.4.14)

when

~, p

function

is

are that

both of

an

279 ordinary

differential

2.

If

discontinuous then

also

Gx(X+

x the

have

equation

is a p o i n t derivative

a jump

0 , t) - G x ( X -

there

[9, p.

192].

at w h i c h

either

of the G r e e n ' s

~

or

p

function

is

will

of m a g n i t u d e

0 , t) = {el(t) - z(t)}{y' (x) - y' ( x - 0) }

+ {O~s(t) + z ( t ) } { z '

(x) - z ' ( x -

O) } (I.4.15)

This or

is to be e x p e c t e d p

derivative

since

then

at t h a t

at a p o i n t

the

point

solutions and

this

of d i s c o n t i n u i t y may would

have

of e i t h e r

a discontinuous

affect

the G r e e n ' s

function. Again

we n o t e

addition derivative

that

when

to the u s u a l (see

both

o, p

hypotheses,

(I.4.15)I •

are

continuous,

the Green's

in

function

has

a

APPENDIX

§l.

COMPACTNESS

IN

In t h i s pertain

We

shall

AND

Appendix

to Chapter

fundamental

Lp

we

2 and

II

OTHER prove

state,

SPACES: certain

without

theorems

proofs,

which

certain

theorems.

use

the

following

version

of

Schauder's

subset

of

a Banach

Fixed

Point

Theorem.

THEOREM

II.l.l: Let

be

X

be

a continuous

and [57,

such p.

THEOREM

map

that

AX

leaving

X

II.l.2: family

is c o m p a c t

is c o m p a c t .

if a n d

There

(Riesz F

of

only

is a n

[54,

Then

M

p.

A

i.e.

has

AXc

a fixed

and X

A ,

point

any

e > 0

for

all

f • F

L p ( - ~ , ~)

in

,

p

> 1

,

if

> 0

such

P

For

137]

functions

lif[l

b)

invariant,

space

25].

A

a)

a convex

there ,

that,

for

all

f • F

(!I.l.l)

< M

is a

,

6(e)

> 0

such

that,

281

II f (x + h) - f ( x ) I I

whenever

c)

If

lhl

EA =

all

< 6

{x e I R :

f • F

I x - x01

> A

The Ascoli

for

is

the

theorem

is

0

(II.l.3)

induced

the

LP-norm

" L P - a n a l o g '' o f

on

EA

the Arzel~-

II.1.2: Let

> 1

norm

above

then,

theorem.

COROLLARY

p

the

fixed}

, x0

,

l i m IIfllE A A+oo where

(II.l.2)

< E P

,

F

a > -~

There

be

a family

,

satisfying

is a n

M

of

> 0

the

such

IIfllp If

EA =

and

for

{x : all

A < x f c F

functions

in

following

that,

for

L P [ a , ~) conditions:

all

f • F

then,

for

given

E>0,

,

IIf II~A --< ~ if

A

For f e F

is

£ > 0 ,

sufficiently

there

,

(II.l.4)

=< M

< ~}

,

is

(II .1.5)

large.

a

6(e)

> 0

such

that,

for

all

282

] f (x)

dx

< £

(II.l.6)

~a

whenever

d)

For

lhl

e > 0

f c F

< 6

there

whenever

Proof: a.e.

Set on

LP(~)

i)

ii)

Ihl

F

F =

to w h i c h

0

such

<

n

on

shall

is

F = 0

all

( - ~ , a) F

that,

for

all

clearly on

f e F

(II.l.7)

LP[a,

i.e.

Theorem

verified

~)-sense).

If

is a f a m i l y

apply

f e F of

functions

f = 0 in

II.l.2.

on

account

of

, ~ > 0

,

there

is a n

(II.l.4)

A 0 ~)

that

< e

A => A 0

clearly

implies

then

(-~ , a)

I-A If(x)[P dx

This

such

< e

P

(in t h e

IA[f(x) IPdx

and

> 0

•

Then

we

(II.l.l)

For

n(e)

h) - f ( x ) [ ]

is c o m p a c t

( - ~ , a)

and

an

,

[If(x+

Then

is

that

= 0 < e

(II.l.3)

A => A 0

is

satisfied.

> 0

283

Let

iii)

e > 0

If

0

<

h

Suppose <

If(x+h)

=

fa

6

,

f

[h I < 8" = m i n { ~ ( e / 2 )

e

F

,

{re

- f ( x ) lP dx =

If(x+h)

IPdx

, q(e/2)}

+

If(x+h)-

lf(x+h)

+

f(x)lPdx

a

- f(x)IPdx

a

=

2

If (x + h) Ip d x

S

+

If (x + h) - f (x)

-h

=

fa+

If(t) I P d t

+

If(x+h)

,a

<

on

If

-6

Ip

dx

a

- f(x)IPdx

a

e/2

+

e/2

account

< h < 0 ,

of

=

s

(II.i.6-7).

f(x+h)-

f(x)

= 0

a.e.

on

( - ~ , a)

that

I If(x+h)- f ( x ) [ P d x

= 0 +

If(x+h)

- f(x)IPdx

a

<

¢/2

<

for Thus

such if

h

lhl < 6"

I

~If(x+

for

all

f e F

which

h)-

verifies

f(x) I p d x

<

(II.l.2).

Consequently

so

.

284

Theorem

II.l.l

is,

That

implies

given

which

we

f e F

.

any

rewrite

that

F

sequence

as

f

n

is

(fn)

,

compact

e F

which

in

there

converges

the

is

LP(~)-sense.

a subsequence,

to an e l e m e n t

i.e. oo

I

Ifn-flPdx

÷

n

0

-~

o0

--oo

But

since

implies

f

= 0

n

that

a.e.

f = 0

on

a.e.

(-co , a) on

the

( - ~ , a)

latter

and

equation

so

co

la I f n - f l P d x

which

says

compact

that

in

this

÷ f

n

sense.

in

the

This

n

LP[a

-~

oo

, ~)-sense

completes

the

and

so

F

is

proof.

II.l.l:

LEMMA

In

Proof: Let

f

÷ 0

the

Let (t i)

proof

K be

be

of

Theorem

a compact

set

an e n u m e r a t i o n

B

2.3.1,

in

of

the

in

Bn

is

n

compact.

[T , T + n ) rationals

for in

fixed

n .

K

and

let

zk

for

xk

I

xk

be

(the that

an

arbitrary

sequence xk

of

has Since

hence

there

is

tI

Similarly

sequence

right-derivatives

a subsequence (Xk)

c Bn

which the

a subsequence there

is

zk

We w r i t e of

Xk)

.

converges are

Zl,n(t)

a subsequence

We will uniformly

uniformly which z2, n

on

K .

bounded

converges of

show

Zl, n

and

at which

285

converges

at

way

there

is a s u b s e q u e n c e

at

t I , t 2 , . . . , tj_ 1

subsequence Zm, m

t2

of

then

and

at

if

converges

and

at

of

Continuing

Zj_l, n

t = tj

k z j

at e a c h

of

so on.

zj, n

and

z. 3,n

is a s u b s e q u e n c e

tI

diagonal

rational

{Zk, m : m ~ i}

which

Moreover

The

Isince

and

in this converges Zk, n

is a

sequence { Z m , m : m ~ k}

so c o n v e r g e s

at

t:~). Rewriting

z

for

m

z

m,m

we

define

a function

z(t)

by

z(t i)

This

limit

= lim

exists

of d e f i n i t i o n

from

of

K

by

sn

be

a decreasing

, 2 .....

considerations.

c a n be e x t e n d e d For

sequence

is u n i f o r m l y

sn + t

i =i

above

right-continuity.

z(s n)

quence

the

z(t)

in

Then

Z m ( t i)

t

the

and

limit

domain

irrationals let

converging

so t h e r e

of

The

is i r r a t i o n a l

of r a t i o n a l s

bounded

for w h i c h

if

to the

(II.l.8)

z(s n)

to

t .

is a s u b s e exists

as

n ÷ For

such

t

we

let

z(t)

This

is w e l l - d e f i n e d

decreasing that

lim

to z(r n)

t

for

= lira z(s n) n-~oo

if

for w h i c h exists

as

r

n

(II.l.9)

is a n o t h e r

rational

sequence

there

is a s u b s e q u e n c e

r

n ÷ ~

,

the

then,

denoting

n

such

286

latter

limit

by

z* (t)

Iz(t) - z*(t)

,

I = lim

IZ(Sn)

- Z(rn) I r

-< l i m

a Ifs n do

+ Ixn-snl}

n-~oo

n

= 0

because

of

the

right-continuity

We

now

show

of

o

Thus

z

is w e l l -

defined. that

the

convergence

is

uniform

on

the

rationals. If p o s s i b l e e0 > 0 is

some

a rational m

~ N

assume

the

number

t

contrary. so

that

for

Then

there

all

N

> 0

a rational

r > t

such

I°(t) - ° ( r )

where further

t < r < t + 6(g0 ' t) so

an

there

for which

(II .i .i0)

IZm(t) - z(t) I -> gO

Choose

is

that

g0 I < 12a

If n e c e s s a r y

(II.l.ll)

restrict

r

that

go t < r < t + l ~ .

(II.l.12)

287 Since

z

m

(r) ÷

z(r)

aS

m

,

÷

there

is

N ( S , r)

an

for

which

IZm(r) for

all

m

gO

I

- z(r)

<

(II.l.13)

6

z NO

By

our

supposition

t

in

K

then,

for

some

m

a NO

such

,

a rational

s o -< Izm(t)-z(t)I _< IZm(r)-z(r) I + IZm(t)-Zm(r) [ + Iz(r)-z(t)[ -<

Izm(r)

gO

-z(r)

I +

2a

(gO]

+ 2ix

do

tl

Eo _ EO

< --~ + 2a ~

+ 2 • 12

2

a contradiction. In

the

above

we

have

Iz(r)-

the of

proof

of which

functions

Hence

the

,

(s n)

this as

the

inequality

z(t) I < a

is

almost

it

is

do

the

uniform

is n o w

earlier,

+

immediate

satisfying

convergence

From g > 0

each

used

same on

simple t

r-t

since

z

is

a limit

inequality.

the

rationals

to c o n c l u d e

fixed,

I

in

that

K for

. given

288

m,n

IZm(S n) - z(t) I < e

For given

s > 0 ,

IZm(S n) - z(t) I <

IZm(S n) - Z ( S n ) I

+

and

the r i g h t - c o n t i n u i t y

of

Iz(s n)

z

the u n i f o r m

convergence

- z(t)

implies

iZ(Sn) _z(t) I < s_ 2

while

(II .1.14)

> N e)

that

n -> NI(s)

z

of

m

on the r a t i o n a l s

implies

that

IZm(S n) - z(s n) I < s_ 2

where

N2

is i n d e p e n d e n t

inequalities

there

Finally,

t

if

follows

before

n

.

From

these

last

two

(II.l.14).

is rational,

z(t)

by d e f i n i t i o n

of

m -> N 2 (£)

while

if

t

= lim Zm(t) m+~ is i r r a t i o n a l ,

so that

z(t)

= lim n÷~

z(s n)

= lim Ilim

Zm(Sn) }

we

let

s

n

be as

289

= lim ( l i m m ÷ ~ -n÷~

on a c c o u n t

of

Zm(Sn) }

(II.l.14),

= l i m Zm(t) m+~

N o w an a r g u m e n t convergence on

similar

is u n i f o r m

to a p r e v i o u s

one

shows

on the i r r a t i o n a l s

t h a t the

and so e v e r y w h e r e

K . !

Thus

for

xk c B n

of a s u b s e q u e n c e of

,

x k H zk

which

we have

converges

shown

uniformly

the e x i s t e n c e on c o m p a c t

subsets

[T , T + n) !

Moreover that of the

the u n i f o r m

(xk)

and

Ilxk -xll~ =

thus

sup t~ [T, ~)

convergence letting

x

of the

(x k)

be t h e i r

implies

limit,

Ixk(t) - x(t) 1

max I sup
Ixk(t) - x ( t ) I ,

sup Ixk(t) - x ( t ) I } tc[T+n, ~ )

There

is no loss of g e n e r a l i t y

rational

so t h a t

be d e f i n e d

at

T +n

t = T +n

l i m i t w h i c h w e use x(t)

for

t a T +n

in a s s u m i n g

is r a t i o n a l and w i l l

to d e f i n e

and so the

converge

x(T +n)

by letting

that

x(t)

We

T

is

xk(t)

uniformly

can to some

then d e f i n e

- x(T +n)

= c

290

If w e

let

xk(t)

= ck

]Ck-C

I ÷ 0

t_>

,

T+n

,

we

must

then

have k

÷

Hence

sup te[T+n,~)

IXk(t ) - x(t) I =

Ic k

-el

÷0

as

k

÷~

sup te[T,T+n)

as

k ÷ ~

Thus

from

previous

llxk -xiI ~ ÷ 0

Similarly

IXk(t) - x ( t )

since

as

I ÷ 0

considerations.

k

÷

O(t)

is

bounded

[T + n

, ~)

away

from

zero

on

[T , T + n)

!

and

xk

~ 0

on

,

!

!

xk(t) sup te [T,~)

oo

- z(t)

Q(t) !

xk(t) sup tc IT, T + n )

- z(t) I

(II.l.15)

Q(t)

!

since Then

both

xk

uniform

(II.l.15)

z

are

convergence

of

tends

and

to

zero

as

identically the k ÷ ~

zk

to

zero z

(Also

on

[T + n

, ~)

implies

that

z = x'

follows

291

from

this.)

Hence

~~ !

llxk

and

so

B

n

is

xlJB = IExk

xll~ ÷

÷0

k÷~

compact.

,

!

APPENDIX

§l.

EIGENVALUES

related

OF

GENERALIZED

We

shall

mainly

to

eigenvalue

be

£[y]

THEOREM

was

defined

initial

basic

results

= ly

(III.l.l)

in C h a p t e r

value

Y(a)

where

e,

unique

solution

B ~ •

functions

,

3.

problem

fixed

follows

= ~

and y(x,

y ( x , I)

For

solutions I)

some

problems

~[y]

y(x,

with

EQUATIONS

III.1.0: The

Proof:

DIFFERENTIAL

concerned

i[y]

where

III

1 I)

,

from

is a n e n t i r e

the

~y

(III.l.2)

y' (a)

=

and moreover are

function

for

and

3.2.0 of

parameter

entire

existence

Theorem

(III.l.3)

8

is a c o m p l e x

, y ' ( x , I)

1

=

1

has

fixed

x

functions

uniqueness

and can

[3, p. be

,

the

of

1

of

341].

found

the

in

That [3,

293

p.

355].

p.

216,

[35]

The

complete

Theorem

also

2].

applies

result

is f o u n d

We o n l y

when

~

need

is,

in

[35,

p.

to n o t e

that

the p r o o f

more

generally,

250

and in

of b o u n d e d

variation.

THEOREM

III.l.l: Let

such

that

o

for

be some

a non-decreasing set

E c

[a, b]

~d0(t)

Let

V

be a r i g h t - c o n t i n u o u s

such

that

for

some

set

Then

the

function

,

(III.l.4)

e

function

F c

~

> 0

right-continuous

of b o u n d e d

variation

[a, b]

Id~(t) l > 0

(III.l.5)

problem

;x du(x)

y'

(x) -

}

y(s)de(s)

=

Xy(x)

(III.l.6)

a

with

Proof:

e,

6 c

y(a)cos

~ - y' (a)sin

~ : 0

y(b)cos

8 + y' (b)sin

8 = 0

[0, ~/2)

If p o s s i b l e

the p r o b l e m

and

has

let

y(x,

I)

only

I , the

real

(III.i.7"8)

eigenvalues.

Im I ~ 0 , corresponding

be an e i g e n v a l u e eigenfunction.

of

294 We m u l t i p l y

(III.l.6)

and i n t e g r a t e

by

with

respect

to

to o b t a i n

la

{

y(x,

l)d y' (x, X) -

Integrating

yd~ a

the left h a n d

}

Using

-tan

the b o u n d a r y

81Y' (b

Combining parts

'

, x) 12 a x +

fa~ ly(x ,

conditions

I) 12 - tan elY' (a

the latter

ly(x

l)

d~(x)

side by parts,

[..y ( x , t ) y ' (x , t ) ] b = I b { l y ' ( x a a -I

= -I

into the

I)

ly(x,X)

I~ dr(x)

(III.i.7-8)

'

I) 12 =

former

J2,~o(x)}

(III.l.9)

we

[y(x

find,

'

l)y' (x

and t a k i n g

'

I)] ba

imaginary

we o b t a i n

(Im I)

and thus,

since

= 0

(III.l.lO)

Im I ~ 0 ,

i Inserting

I~ly(x 'I) I~ d~(x)

the latter

b lY(x , 11 12 d~(x) a

into

(III.l.9)

(III.l.ll)

= 0 .

we m u s t

therefore

have

295

I~{ly

'(x,

= -tan

But

since

negative Hence

we must

i From

the

[a, b] zero.

to

the

,

{lY' (x

latter

11

it

12

dx

side

is

hand

side

is

necessarily

strictly

non-negative.

+ ly(x , I)I 2 do(x) } = 0

follows

(III.l.4)

Hence

that

finally

y ( x , I)

therefore

implies

.

H 0 y

y(x,

implies on

is a c o n s t a n t

that

[a , b]

is an Im

I)

this

constant

which

,

hence

must

is c o n t r a r y

eigenfunction.

I = 0

on

This

all

contra-

eigenvalues

real.

III.l.2: Let

variation

~

and

be

some only

Proof:

set

We

assuming

and, F c

real

a right-continuous

suppose

non-decreasing

by

right

have

that

THEOREM

has

hand

requirement

diction

on

left

the

(III.i.12)

a

and

be

are

b

[0 , ~/2)

the

~)12d~(x)}

l) J2 - tan(~ ly'(a, 1) J2

Bly'(b,

e , 8 e while

1) 12 dx + l y ( x ,

that

eigenvalues

the

as

Then if

in

the

existence

of

of b o u n d e d

is r i g h t - c o n t i n u o u s

additionally,

[a, b]

proceed

~(x)

function

that the

satisfy

problem

e , B E

proof

it

and

(III.l.5)

(III.i.6-7-8)

[0 , ~)

of

a non-real

the

previous

eigenvalue

theorem 1

and

296

arguing,

as

, B ~ 7/2

in t h a t ,

(III.l.10).

we

proof,

take

Since

until

imaginary

Im

I z 0

we

reach

parts

in

(III.l.9). (III.l.9)

If

obtaining

,

lab]y(x , I) 12 dr(x) = 0

But

since

y ~ 0

impossible

and

v

satisfies

y

F

is

has

positive

a solution

of

solution.

This

values

exist.

is

this

is

{ 0

on

F

be

the

unless

IF l y ( x ,

Since

(III.l.5)

can

treated

l)12d~(x)

~-measure

(III.l.6)

contradiction The

separately

case with

:

0

y ( x , l)

then

y

shows

that

when

.

must

either

a similar

trivial

no n o n - r e a l or b o t h

argument

eigen-

of

to

Since

e,

the

B = 7/2

one

above.

We bounded

note

variation

(III.i.6-7-8) The

latter

singular

§2.

need

case

LINEAR

refer

that

on

see

the

if

[a , b]

not

situation

For may

here

be

was

all

~

is a n

then

the

real

even

illustrated

papers

[i0],

OPERATORS

IN A H I L B E R T

the

notions

to a n y

basic book

on

function

eigenvalues if

~ , 6 E

in C h a p t e r

4.

[0 , ~/2] For

the

SPACE: Hilbert

analysis.

spaces The

of

of

[ii].

regarding

functional

arbitrary

one

space

.

297 L 2 (V ; I) such

defined

by

those

(equivalence

classes

of)

functions

that

llfll---{ I

If(x)]2 d V ( x ) } ½

< oo

(III.2.0)

I

is

a Hilbert

space

whenever

V(x)

is

a

non-decreasing

function. If

HI

, H2

consists

of A

to

x

Hilbert

pairs

if

given x

spaces

{x I , x 2 }

operator

xn ÷

that

A

any

, Ax n +

operator

y

xI

e HI

a Hilbert

sequence

Xn

then

admits

A

where

in

direct

their

x

sum, ,

space

c DA

c DA

x2 c H2 H

is

( domain and

a closure

HI ~

A

y

if

said

of

A)

= Ax

and

only

if

relations

6 DA

n

imply

, x ' e DA n

that

y

The of

two

linear

An the

all

closed

be

such

are

those

x A

fact

a

of

vectors

for

x

subspace

S

the for

÷ x

x

, x ' ÷ n

x

closure which

{AXn}

~ D(A)

set

÷

n

, Ax

÷

n

y

, Ax . ÷ y ' n

y'

domain

xn

Then

=

, x

,

Ax

a H S

is is

D(A)

there

is

converges

H

lim

dense

dense

in

Ax if

precisely

xn

satisfying

as

c DA

n

÷

n its

H

consists

if

closure and

only

S = H if

there

In is

H2

298

no

nonzero

vector

Let of

A

definition The

in

be in

set

H

any H

of

all

which

D*

for

of

all

the

defined

x

c D

operator

Denote

the

vectors

and

is

simply An

x,

y

z e H

e D

called

E D*

the

operator

A

A*y

adjoint is

said

A

to

said

be

by

defines

denoted

D

the

by

A*

hermitian

if,

.

domain ,

and

hermitian

operator

z

of to

A be

. for

all

be

An

operator

A

=

(x , Ay)

with

a dense

domain

of

definition

symmetric.

to

with

self-adjoint complex if

A

a bounded

operator points

if

A

a dense

the

1

1

is

inverse defined are

domain

of

definition

is

said

= A*

number

operator

regular

=

,

(Ax , y)

is

,

,

A

domain

by

y

A*

that

some

S

a dense

such

(x , z)

A

with of

y

to

to

domain

=

adjoint

operator

orthogonal

linear

(Ax , y)

holds

is

a

regular point

( A - II) on

called

the

-i

whole

of

exists

and

space

H

the represents

.

points of the spectrum

All of

nonA

299

The

set

of all

discrete spectrum of All

other

eigenvalues A

of

A

constitutes

.

points

of the

spectrum

(if any)

points of the continuous spectrum

(essential

The

constitutes

collection

of all

such

.

spectrum of

spectrum of

A

discrete

continuous

and The

Hilbert

spectrum

space For

operators

§3.

we

general

refer

to

general

product

spaces

[7] and

[40].

and,

A linear [ , ]

spectra

of a n y

OPERATORS

For

points

A

of

are

spectrum) the

called of

A.

continuous

is the u n i o n

of the

A

self-adjoint

operator

A

in a

is real.

the

LINEAR

product

The

the

theory [46,

of e x t e n s i o n s

information

SPACE: concerning

in p a r t i c u l a r ,

K

is c a l l e d

symmetric

~14].

IN A K R E I N

space

of

with

Krein

indefinite spaces

a generally

we

inner

refer

indefinite

to

inner

a Krein space if

K = K+[+]K_

where

K+(K_)

product

[

denotes

a direct

inner

,

is a H i l b e r t

product

] (-[

,

])

,

] ,

with

respectively.

sum which [

space

is o r t h o g o n a l

i.e.

respect The with

to the

symbol respect

inner [~] to the

300

~+ ~ K_ = {o}

whenever

f+

e K+

, f_

A positive defined

on

K

definite

is

orthogonal

inner

here

then

,

g = g+

of

K

that

different are

all

space

- P_

different

inner

to be

induced

Hilbert

Example:

Let

of b o u n d e d

[ H V]

by

the

function

V

is c a l l e d

(

,

)

can

be

=

g_]

and

K±

e K±

if

P±

denotes

The

important

of

K

generate

the

corresponding

the

then

(Jf , g)

+ P_

= I

,

)

but

Topological

interpreted

in

the

concepts norm

fact

norms

in a K r e i n

topology

of

the

space.

be

variation

(Im

is e q u i p p e d

(

f± , g±

decompositions

[7].

~(x)

m

,

space

on

P+

products

equivalent

are

and

- If,

+ g_

a Hilbert

projector

J = P+ is

= o

product

If+, g+]

[f , g]

where

f]

by

f = f+ + f_

(K , ( , )]

[%,

K_

e

(f, g) =

where

,

on

denote

the with

~

a right-continuous [a , ~)

the

signed

respectively

total such

variation a measure

,

function

a > -~

measure

measure and

the

Let (measure)

(Chapter

locally

3, of

norm

induced

section ~

and

i) . when

is d e f i n e d

L2 by

301

Ilfll

then

L 2Iv ; (a, =))

=-

f2

= K

f (x) 12 dV(x)

is a H i l b e r t

<

space w i t h

inner

product

oo

for

f, g e K .

(f, g)

-: la f(x)g(x)dV(x)

If one

introduces

[f , g]

-:

the

indefinite

inner

product oo

f

f(x)g(x)dv(x)

a

for

f, g e K ,

the m e a s u r e Im I almost

then

m

K

is a b s o l u t e l y

the R a d o n - N i k o d y m everywhere

and

dp(x) dV(x)

ImvI-almost

is equal

in this

continuous

to

with

dm / d l m (Chapter

I

For

since

respect

to

exists

Im I-

3.1)

and m o r e o v e r

=

[24, p. 134,

case,

space.

~_v(x+ h) - v ( x - h)} V(x h) - lim [V(x + h) h+O

everywhere

(See for e x a m p l e

a Krein

derivative

f(x)g(x)dv(x)

Thus,

becomes

for

f(x)g(x)

Theorem

dg(x)

dV (x)

B and

dV(x)

(i), p.

f ~ K

(Jf) (x) = dD(x) dV(x----~. f(x)

[V]

,

135].]

302

and

1 {i-+ ~d~(x) )

(P+_f) (x)

where It

is

with is

[V]

means

then

J

an o p e r a t o r

space

K

Imwl-almost

readily

defined

,

: ~-

seen

with

that

=

(Jf , g)

and

K+

H P+K

dense

by

definition.

domain

of d e f i n i t i o n

Ax

is c h a r a c t e r i z e d

J-adjoint

the

everywhere.

[f , g]

above

[v]

f(x)

in

If

the by

A

Krein

the

relation

[Af , g]

In

this

space such

sense

setting,

=

it

[f , AXg]

f c D(A)

is d e f i n e d

for

D ( A x)

analogously still

for

all

The symmetric

f c D(A)

closed

with

D(A)

respect

is d e n s e

and

operator

[Af , g]

and

as

in t h e of

Hilbert

those

g e K

that

[Af , g]

holds

consists

, g c D ( A x)

=

in

to

=

some A

[

[f , Ag]

K

[f , h]

h E K

is c a l l e d , ] ,

Then

A+g

J-symmetric

i.e.

f , g E D(A)

--- h. if

it

is

303

adjo{nt

The

operator

if

A = Ax

A

In c o n t r a s t

with

spaces,

J-self-adjoint

(see

example,

for

If and

Ax

is

Hilbert [ The

,

A its

by

operators

[7, p.

Krein

133,

space

adjoint (

,

adjoints

)

Ax

D(A)

in

K

operators

may

have

Example

adjoint

(J-adjoint)

the

defining

are

related

in H i l b e r t spectrum

6.4]).

operator

is d e f i n e d

J-self-

is

non-real

defined

A* in

, A*

dense

self-adjoint

is a d e n s e l y

space

]

with

as

usual

relation by

in a K r e i n

the

then

space

its

replacing

for

the

adjoint.

formula

A x = JA*J

where

J = J*

Moreover

if

adjoints

Sx

j2

and S, T

are

, Tx

then

= I

For

operators [45,

p.

this in

result

the

Krein

see space

[45,

p.

122].

with

122],

(ST) x z T X s x

§4.

FORMALLY SELF-ADJOINT EVEN ORDER DIFFERENTIAL WITH AN INDEFINITE WEIGHT-FUNCTION: In

and

P0 (x) The

the

following

Pk

E C

> 0

on

(n-k)

boundary

we

shall

(a, b)

[a , b] problem

assume

k=

that

0 , ... , n

EQUATIONS

304

(n-k)

n

(-i) n-k (pkf (n-k) ]

=

If

(III.4.1)

k=0

f(J) (a) = f(J) (b) : 0

is s e l f - a d j o i n t values

[9, p. 201, Ex.

are real,

accumulation. eigenvalues

Also

3].

in

(III.4.1-2),

L 2 ( a , b)

point

f

of

to d i s t i n c t

Moreover,

then

series

(III.4.2)

the e i g e n -

no f i n i t e

corresponding

(III.4.2)

convergent

Consequently

and h a v e

eigenfunctions

satisfies

into a u n i f o r m l y of

below,

are o r t h o g o n a l

f 6 c n ( a , b)

Yk(X)

bounded

j = 0 , .... n-i

if

can be e x p a n d e d

of the e i g e n f u n c t i o n s

i.e.,

co

f(x)

:

[

(III.4.2)

(f , yk)Yk(X)

k=O

where we assume result,

see

that

[9, p.

llyklI : 1

197,

Theorem

As u s u a l w e d e f i n e

D(A)

for all

k

latter

4.1].)

an o p e r a t o r

= {f ~

(For the

C n

A

with

domain

(a, b) }

such t h a t

Af :

If we

D(A)

n [ k=0

(n-k) (-i) n-k (pk f (n-k) ]

(III.4.3)

let

=

{f ~ D(A) : f(i) (a) = f(i) (b) = 0

for

i= 0 , 1 .....

n-l}

305

and

Af = Af

then

integrating

f E D(A)

(Af , f)

by p a r t s

we

find

fb n (Af, f) = I~ [ P n _ j l f ( J ) 12 dx a j=0

f • D(A) (III.4.4)

Now

from

Yk(X)

(III°4.2)

in

and

L 2 ( a , b)

the

we

completeness

also

have,

of the

for

eigenfunctions

f • D(A)

co

(Af , f) :

[

(III.4.5)

lkl (f , Y k ) I2

k=O

Thus

if w e

negative have

let

I0' 11'

eigenvalues

of

(f, yk ) = 0 ,

" ' " ' IN-I (III.4.1-2)

k= 0, 1,

rest

For,

of

the

argument

an a d a p t a t i o n

non-real

now

of L e m m a

eigenvalues

and

N

distinct

if for ,

some

f

we

4,

§2.

then

> 0

follows 4.2.2

that

shows

in C h a p t e r

that,

if

I , ~

are

of

f(i) (a) = f(i)(b)

Af = Irf

the

. . . , N-I

(Af , f)

The

be

= 0

i = 0 , ... , n-i

,

(III.4.6)

where

r(x)

is,

say,

continuous

on

[a , b]

and

changes

sign

306

at l e a s t

once

there,

and

I ~ [ ,

then

~b J|a f ( x ) g ( x ) r ( x ) d x

i

b

n

where

f , g

are

(j)~(j) P

a

j=O

the

(III.4.7)

= 0

.f n-]

(III.4.8)

dx = 0

eigenfunctions

corresponding

to

I ,

respectively.

Thus

we

let

(III.4.6)

H 0 , ..., HM_ 1

such

that

eigenfunctions

be the

H i ~ [j

%0 ' ~1 . . . . .

,

non-real

eigenvalues

0 =< i , j =< M -

%M-I

Since

1 ,

%i(x)

of

with ~ D(A)

we

have

for

i = 0,

fb

n

a

j=0

P n-j I ~i(j) 12 dx

..., M-

1

Thus

as in C h a p t e r

possible

to c h o o s e

(f , yk ) = 0 ,

4.2, the

we

ej%j

we

see

that

substituting

(x)

coefficients N- 1

(Af , f)

But by

let

M-I [ j=0

k= 0 .....

(III.4.9)

(~i' Y k )12

0~ Ik

=

f(x)

and,

=

(III.4

if e

This

M > N ]

such would

then

it is

that then

imply

> 0

in the

latter

relation

and

that

9)

307

expanding

the

form,

we

shall

find

(Af, f

on

account

the

of

(III.4.7-8).

that

= 0

This

contradiction

then

proves

result.

Note: idea

The

is

to

problem

here

approximate

is

the

the

following:

eigenvalues

of

Richardson's the

continuous

problem, !

(py')

+

y(0)

by

the

eigenvalues

of

the

(q+ Ik)y

= y(1)

= 0

= 0

discrete

problem,

2 m

A ( P i A Y i _ I)

i = 0 , 1 , 2 .... points The

i/m

claim

eigenvalues

,m

are

where

by

to be

that

the

discrete

Y0

are

approximations

problem.

However

the

denoted

appears of

+ qiy i + IkiY i = 0

to it

the

values Yi

' Pi

for

of

problem

y , p , q , k

' qi

large

,

' ki

values

at

the

respectively. of

m

the

with

= Ym = 0

eigenvalues

is n o t

at

all

of

clear

the that

above if

continuous

the

discrete

308

problem

has

non-real

eigenvalues

then

approximate

non-real

eigenvalues

in t h e

it is c o n c e i v a b l e

that

seem

information

the

as

if e n o u g h

latter

eigenvalues criteria

possibility. may

on the

existence.

these

exist

and

limits

so o n e

may

be real.

in s o m e needs

which

will

necessarily

continuous

is p r o v i d e d

In f a c t

coefficients

these must

in

[53]

cases

case.

For

It d o e s

not

to e x c l u d e

no n o n - r e a l

to e s t a b l i s h guarantee

some

their

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Some

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extensions

Rep. Acad.

223-226.

of

Sci.

the

Sturm-Picone

Canada.•

i,

317

[75]

• A

limit-point

criterion

C.R. Math.

term

recurrence

relation,

Sci.

Canada•

(1981),

[76]

~,

Scotland• Verlag,

Equations

three-

Rep. Acad.

Sturm-Liouville

in the Proceedings

Differential

a

171-175.

, Indefinite

To appear

for

of the C o n f e r e n c e

1982,

Lecture Notes

problems,

University

on

of Dundee•

in Mathematics, Springer-

N.Y.

[77]

, function

Some

remarks

associated

with

a

on

the

second

order order

of

an

entire

differential

%

equation,

Proceedings Operators• Notes

appear

TO

I

in Tribute

of the S y m p o s i u m on Differential University

of Dundee,

in Mathematics•

[78] of

Scotland,

Springer-Verlag,

, Asymptotic values

to F.V. Atkinson,

non-definite

To appear

Sturm-Liouville

Scotland•

Springer-Verlag W.T.

Reid•

Journal

of Math.

and

related

J. Math.•

H. Weyl,

Uber

Notes

of

University

in Mathematics,

8,

die

Functionen,

(1959)

matrix

i_00, (1966),

und

differential

linear

Riccati

gewohnlicher

singularitaten

willkurlioher

Proceedings

Operators,

linear

and Mech.,

Illinois

mit

eigen-

N.Y.

• Generalized systems

[81]

Lecture

Generalized

[80]

the

problems,

to F.V. Atkinson,

the S y m p o s i u m on Differential

[79]

of

I

in Tribute

of Dundee,

N.Y.

distribution

&

Lecture

systems,

705-726.

differential integral

equations,

701-722

Differentialgleichungen zugehorigen

Mat.

Ann.,

Entwicklungen

68,

(1910)•

220-269.

Subject

Conditional

Dirichlet,

Difference Equation,

Index

180 15

Dirichl et property,

180

Generalized derivatives,

120 ff.

Generalized ordinary differential

expressions,

Generalized ordinary differential

operators,

Green's

function,

25-27,

273 ff.

Indefinite weight-function, J-sel f-adjointness,

197 ff.

156 ff°

Limit-circle,

132, 147 ff.

Limit-point,

132, 147 ff.

Non-oscillatory

equation,

30

Non-oscillatory

solution,

30

Oscillatory

equation,

30

Oscillatory

solution,

30

Picone 's identity,

3

Strong Limit-point, Sturm comparison

180-181

theorem,

i0 ff.

Sturm separation theorem,

4, 22 ff.

Three-term recurrence relation, Volterra-Stieltjes

integral

Weyl classification,

16

equation,

129 ff.

29

123 ff.

156 ff., 225