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VOLTERRA STIELTJES-INTEGRAL EQUATIONS

This Page Intentionally Left Blank

NORTH-HOLLAND MATHEMATICS STUDIES

16

Notas de Matemgtica (56) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Volterra Stieltjes-Integral Equations Functional Analytic Methods; Linear Constraints

CHAIM SAMUEL HONIG lnstituto de Matema'tica e Estatistica, S o Paulo, Brazil

1975

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM OXFORD AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK

0 NORTH-HOLLAND

PUBLISHING COMPANY

- 1975

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

North-Holland ISBN .for this Series: 0 1204 2100 2 North-Holland ISBN for this Yolume: 0 7204 2117 7 American Elsevier ISBN: 0 444 10850 5

Publishers :

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NORTH-HOLLAND PUBLISHING COMPANY, LTD. - OXFORD Sole distributors for the U.S.A. and Canada:

AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017

PRINTED I N T H E NETHERLANDS

INTRODUCTION This work presents the results we obtained in the study of linear Volterra Stieltjes-integral equations with linear constraints i.e. in the study of systems of the form (K), (F) where

(F)

F[y]

= c.

These systems are studied in all their generality (see Remark 3 at the beginning of Chapter 111); we give a Banach space X, y,f E G(] a,b(,X) (the space of regulated functionssee the index for this and other definitions) and K:

] a,b(X)

a,b(

+ L(X) ,

that satisfies natural conditions defined in 81, Chapter 111.

FcL[G()

[ ,X>,Y] ,

a,b

where Y is a locally convex space, is called a linear constraint. In the item A of §1, Chapter 111, we show how linear differential equations, linear Volterra integral equations, linear delay differential equations, etc. are reduced to the type (K). As particular instances of linear constraints we have the initial conditions, boundary conditions, periodicity conditions, discontinuity conditions, multiple point conditions, integral conditions, interface conditions, conditions at infinite points etc. (see item B of 8 3 , Chapter 111). We give necessary and sufficient conditions for the existence of a Green function for the system (K), ( F ) i.e. a function G: )a,b(X)a,b( + L ( X ) such that for f continucus and c 0 the solution y of the system is given by

V

INTRODUCTION

vi

(see Theorem 3 . 2 8 of Chapter 111); we also characterize the Green function (see Theorem 3 . 2 9 of Chapter 111) and show that the solution y is a continuous function of f (and K). In order to obtain the representation (GI we have to solve preliminarily two important problems; I - Find a resolvent for the equation (K). I1 - Find integral representations associated to ( F ) .

I

-

The resolvent of (K) is a function

that satisfies

and such that the solution of ( K ) with tinuous is given by

y(to)

= x

and

f con-

(see Theorem 1.5 of Chapter 111). In the particular instance of an integro-differential equation r t

the existence of the resolvent (called harmonic operator in this case) was proved by Wall [W] and specially by Mac-Nerney [M] under the restriction that A is locally of bounded variation (and continuous). In this case Mac-Nerney also proved that there is a one-to-one correspondence between the space of coefficients A and the harmonic operators. We extend this correspondence to the general case (Theorems 2 . 1 and 2 . 3 of Chapter 111) and prove that it is bicontinuous in a natural way. The proof in the general case however is much more difficult than the proof given in [MI where one applies directly

v ii

INTRODUCTION

t h e Banach f i x e d p o i n t theorem. I n t h e g e n e r a l c a s e i n o r d e r t o prove t h e e x i s t e n c e of t h e r e s o l v e n t and s p e c i a l l y t h a t it s a t i s f i e s (R,)

we had t o r e p l a c e t h e n a t u r a l norm of

the

s p a c e of r e s o l v e n t s by an e q u i v a l e n t one (Theorem 1 . 1 2 o f Chapter 111) and g i v e a l a b o r i o u s p r o o f t h a t now w e can g e t a c o n t r a c t i o n . By t h e way, t h i s proof i s made d i r e c t l y f o r t h e r e s o l v e n t of (K) and t h i s g e n e r a l i z a t i o n e x p l a i n s why, f o r t h e r e s o l v e n t of (L), w e have t o prove f i r s t t h e s e m i v a r i a t i o n p r o p e r t i e s w i t h r e s p e c t t o t h e second v a r i a b l e and o n l y a f t e r wards w i t h r e s p e c t t o t h e f i r s t v a r i a b l e . T h i s g e n e r a l i z a t i o n a l s o keeps t h e symmetry between t h e e q u a t i o n s ( R " )

and (R,)

s a t i s f i e d by t h e r e s o l v e n t , a symmetry t h a t does n o t e x i s t f o r d i f f e r e n t i a l e q u a t i o n s or V o l t e r r a i n t e g r a l e q u a t i o n s when t h e y a r e w r i t t e n i n t h e i r u s u a l form. For e q u a t i o n s of t y p e

(K) w e a l s o prove t h a t t h e r e i s a n a t u r a l b i c o n t i n u o u s

respondence between t h e s p a c e o f k e r n e l s resolvents

R

K

cor-

and t h e s p a c e of

(Theorems 1 . 2 7 and 1 . 3 0 o f Chapter 111) and we

y of (p) i s a continuous f u n c t i o n of f , x and K . We mention t h a t p a r t of t h e r e s u l t s o f 8 2 o f C h a p t e r I11 on (L) have been extended by Maria I g n e z de Souza

show t h a t t h e s o l u t i o n

[S]

t o t h e c a s e where

A

a l l o w s d i s c o n t i n u i t i e s ; t h i s gener-

a l i z e s r e s u l t s of H i l d e b r a n d t [H-ie] I1

-

i n t h i s direction.

The i n t e g r a l r e p r e s e n t a t i o n a s s o c i a t e d t o

F

is

n e c e s s a r y e s s e n t i a l l y i n o r d e r t o p r o v e an i d e n t i t y of t h e form

t h i s i d e n t i t y ( t h e D i r i c h l e t formula) i s necessary i n o r d e r t o o b t a i n t h e Green f u n c t i o n f o r t h e system (K), (F)- i t e m D and E o f 63, C h a p t e r 111; i n t h e i t e m B w e do a f o r m a l ( a l g e b r a i c ) s t u d y of t h e system n o t u s i n g t h e i n t e g r a l r e p r e s e n t a tion for

F. Both f o r t h i s r e p r e s e n t a t i o n as for ( K ) w e need

t h e n o t i o n o f f u n c t i o n for bounded s e m i v a r i a t i o n .

The p r e s e n t work h a s i t s o r i g i n i n our a t t e m p t s t o e x t e n d

o u r r e s u l t s of [H-IME], where w e s t u d i e d systems o f t h e form

INTRODUCTION

Viii L[y](t) F[Y]

= y'(t>

f

A(t>y(t)

f(t>

t€

]a&(

= c

w i t h y ~ ~ ( l ) ( ) a , b ( , X ) ,f € G ( ) a , b ( , X ) , AEG()a,b[,L(X)) and F E L [G() a , b , X I ,Y] We wanted t o e x t e n d t h e r e s u l t s t o t h e

[

.

c a s e where t h e c o e f f i c i e n t A a n d t h e f u n c t i o n s f and y a l l o w d i s c o n t i n u i t i e s . I n i t i a l l y w e t r i e d t o work w i t h f u n c t i o n s t h a t were l o c a l l y o f

bounded v a r i a t i o n b u t i n t h i s case

w e had no i n t e g r a l r e p r e s e n t a t i o n f o r t h e l i n e a r c o n s t r a i n t F; t h e same w a s t r u e f o r o t h e r " n a t u r a l " c l a s s e s o f f u n c t i o n s . A f t e r w o r d s w e r e a l i z e d t h a t t h e r e g u l a t e d f u n c t i o n s , or t h e i r e q u i v a l e n c e classes ( s e e t h e end of 8 3 , C h a p t e r I ) , a r e t h e

a d e q u a t e o n e s s i n c e i n t h i s c a s e w e o b t a i n e d good r e p r e s e n t a t i o n theorems f o r t h e elements e t c . (Theorems 5 . 1 , merical case

5.6,

(X = Y = R)

FEL[E( ( a , b ) , X > ,Y] , F E L[G()a,b( , X I ,y]

6 . 6 and 6 . 8 of C h a p t e r I ) . I n tlie nu-

and f o r c l o s e d i n t e r v a l s t h i s

re-

p r e s e n t a t i o n i s due t o K a l t e n b o r n [K];

it requires t h e i n t e r i o r or Dushnik i n t e g r a l ( s e e §1, C h a p t e r I ) . We a l s o o b t a i n o t h e r r e p r e s e n t a t i o n theorems (see theorems 5.10, 5 . 1 1 , 6 . 1 0 , 6 . 1 2 and 6 . 1 6 of C h a p t e r I > ; Theorem 1 . 6 . 1 2 is a p a r t i c u l a r case o f more g e n e r a l t h e o r e m s on m e a s u r e s p a c e s ( s e e [D]). For t h e s e and o t h e r r e s u l t s w e need c o n v e r g e n c e t h e o r e m s of t h e H e l l y type (theorems 5.8, 5 . 9 ,

6 . 3 o f C h a p t e r I). O f f u n d a m e n t a l

i m p o r t a n c e t o o are t h e f o r m u l a s o f D i r i c h l e t a n d o f s u b s t i t u -

(101, (111, ( 1 2 ) and (13) o f C h a p t e r 11) which are deduced from Theorem 11.1.1; i n t h e n u m e r i c a l case ( i . e . X = Y = lR) and for c o n t i n u o u s f u n c t i o n s t h i s 'theorem i s e s s e n t i a l l y d u e t o Bray [B]. tion ((61,

All t h e s e r e s u l t s u s e a q u i t e c o m p l e t e s t u d y w e made i n C h a p t e r I of t h e f u n c t i o n s of bounded s e m i v a r i a t i o n ( f i r s t def i n e d by Gowurin, [GI), o f t h e i n t e r i o r i n t e g r a l ( d u e t o Dushn i k - see [ H - t i ] p . 9 6 ) and o f t h e r e g u l a t e d f u n c t i o n s . W e give much more r e s u l t s , s p e c i a l l y i n C h a p t e r I , t h a n w e n e e d i n t h e

r e s t o f t h e work. These a d d i t i o n a l r e s u l t s or o t h e r s t h a t may be r e a d o n l y i n t h e moment t h e y a r e a p p l i e d are g i v e n i n smaller p r i n t and/or i n appendices.

INTRODUCTION

ix

For o t h e r v e r s i o n s of t h e f o r m u l a s of D i r i c h l e t a n d of s u b s t i t u t i o n see [H-DS]. For a n u n i f i e d p r e s e n t a t i o n of t h e r e p r e s e n t a t i o n t h e o r e m s m e n t i o n e d a b o v e see [H-R] For a n abs t r a c t of t h e main r e s u l t s of t h e s e n o t e s see [H-BAMS2]. For r e l a t e d r e s u l t s see [ C a ] , [HI a n d [R].

.

The n o t a t i o n ( 1 1 1 . 2 . 5 )

r e f e r s t o 2.5 o f C h a p t e r 111; for

a r e s u l t i n t h e same c h a p t e r w e w r i t e o n l y

ized

2.5.

T h e s e n o t e s were w i t t e n f o r t h e A n a l y s i s M e e t i n g o r g a n by t h e S o c i e d a d e B r a s i l e i r a d e Matemztica a t t h e U n i v e r -

s i d a d e d e Campinas, f r o m 1 5 t o 25 J u l y , 1 9 7 4 . They r e p r o d u c e a n a d v a n c e d g r a d u a t e c o u r s e w e g a v e a t t h e I n s t i t u t o d e Matem z t i c a e E s t a t i s t i c a da U n i v e r s i d a d e d e Sao P a u l o d u r i n g t h e f i r s t semestrer of 1 9 7 4 . The a u d i e n c e of t h i s c o u r s e was v e r y stimulating. S p e c i a l t h a n k s are d u e t o my c o l l e a g u e P r o f . L.H.Jacy M o n t e i r o who t o o k i n c h a r g e t h e p u b l i c a t i o n of t h e s e n o t e s a n d w i t h o u t whose h e l p t h e y would n o t h a v e b e e n r e a d y for t h e Meeting.

C O N T E N T S INTRODUCTION

v

I

THE INTERIOR INTEGRAL

7

1 . The Riemann-StieL.

7 12 16 21 38 53

$0

NOTATIONS

-

-

§

52.

53. 94.

15.

§6.

/ e b i n t e g h a L and t h e i n t e h i o h inkeghue T h e Riemann i n t e g h a e and t h e Vahboux i n t e g h a e Regulated dunctionb FuMctiOnb 06 bounded B - v a h i a t i o n R e p h e b e n t a t i o n theonemb and t h e . t h e o h e m 06 Heely R e p h e b e n t a t i o n theohemb o n o p e n i n t e h v a l b

11- THE ANALYSIS OF REGULATED FUNCTIONS 5 1 . T h e theohem

06

Bhay and t h e 6ohmuLa 0 6 V i h i c h l e X

5 2 . E x t e n b i o n t o open i n t e h v a k h

111- VOLTERRA STIELTJES-INTEGRAL EQUATIONS WITH LINEAR CONSTRAINTS

06 a VoLtehha S t i e L t j e b - i n t e g h a e equation 8 2 . l n t e g ~ ~ u - d i 6 6 e h e n t i a eLq u a t i o n b and hahmonic opehatohb $ 1 . T h e htbO.tVent

5 3 . Equationb w i t h k i n e a h conhthaintb

69 69 79 82

85 116

124

REFERENCES

151

INDEX

157

SYMBOL INDEX

153

X

5 A

-

0

-

NOTATIONS

W e always c o n s i d e r v e c t o r spaces over t h e complex

f i e l d C, b u t a l l o u r r e s u l t s , w i t h obvious a d a p t a t i o n s , a r e v a l i d f o r real vector spaces. For i n t e r v a l s w e use t h e u s u a l n o t a t i o n , ]a,.[ , ( c , d ] e t c . l c , d l , where c < d , d e n o t e s any of t h e i n t e r v a l s ] c , d [ , ] c , d ) , [ c , d [ and [ c , d ) ; ( c , d ) denotes t h e i n t e r v a l ( c , d ) i f c s d and t h e i n t e r v a l [ d , c ) i f d s c. Given r e a l numbers s , t w e w r i t e s A t = i n d ( s , t ) and

--

s v t = 6LLP(S,t). h ( t , s ) E Y , f o r every Given a f u n c t i o n h: ( t , s ) E B x A t E B, ht d e n o t e s t h e f u n c t i o n S E A h ( t , s )E Y and f o r hs denotes t h e f u n c t i o n t E B - h ( t , s ) E Y. every S E A , Given a f u n c t i o n f : X v Y and A C X I f denotes t h e IA r e s t r i c t i o n of f t o A . Ix d e n o t e s t h e i d e n t i c a l automorphism of X . Given an A c X , xA denotes t h e c h a r a c t e r i s t i c f u n c t i o n of A: x ( x ) = 1 i f x E A and x A ( x ) = 0 i f x E X A and x @ A . Y : R -IR denotes t h e Heaviside f u n c t i o n Y =

x [ ~ , W~ e (d .e f i n e

sg: R

-

{-l,O,l}

by

sg t = 1 i f

= 0 f o r t = 0 and = -1 i f t < o . I f X and Y are t o p o l o g i c a l s p a c e s , E ( X , Y ) denotes t h e s e t of a l l c o n t i n u o u s f u n c t i o n s of X i n t o Y. I f a sequence xn converges t o x i n a t o p o l o g i c a l space x, w e t > O ,

write

x - xX n

x - x . n I f t h e sequence t n E R t e n d s t o t and i s d e c r e a s i n g w e w r i t e t G t ; i n an analogous way w e d e f i n e t n + t and 6 G O . For c~:~)a,b) X I a ( t - ) denotes t h e l i m i t a t t h e l e f t , when i t e x i s t s . I n an analogous way w e d e f i n e c r ( t + ) . where X i s a normed s p a c e , I l f ( 111 Given f : [ a , b ] - X I d e n o t e s t h e f u n c t i o n t E (alb] Ilf (t)II€ IR+ and u n l e s s o t h e r w i s e s p e c i f i e d If 11 denotes Aup (t)ll 1 a ,< t < b). The n o t i o n of summable series i s d e f i n e d i n t h e s e n s e of Bourbaki

-

.

or

-

{]If

NOTAT I O N S

2

Given a c l o s e d i n t e r v a l ( a , b ) c R

-

B

Id( = n

and

.. . < t n=b.

d: t 0=a < t l . D(a,bb'

D , denotes t h e s e t of a l l d i v i s i o n s of

o r simply

> 0 w e w r i t e DE = { d E D i s a f i l t e r b a s i s on D. E

Given two d i v i s i o n s d i v i s i o n of

[a,b)

I

Ad
0

t M

X

Jab

and

1

l]fn(tl-f(tll]dt

and

f E D((a,b),X)

6

llf(t)-f,(t)lldt

fnE D([a,b),X)

and

then

i f and

there i s a step function

such t h a t Let

--f

f:

-+

E.

+X

(a,b)

0,

then

be such

f E D((a,b),X)

5 3 - Regulated dunctionb

-

5

In t h i s

X denotes a Banach space. W e s a y t h a t a

t i o n f : (a,b) f E G ( [a,b) ,X) , i f

i s h e g u l a t e d , and w e w r i t e has only d i s c o n t i n u i t i e s of t h e

X

f

k i n d , i . e . , f o r every t E ( a l b ( t h e r e e x i s t s € o r every t E a,b) t h e r e e x i s t s f (t-)

1

T H E O R E M 3.1. G i v e n

f:

[a,b)

-

ahe e q u i v a l e n t a ) f i d t h e unidohm L i m i t

w

b ) f E G ( (a&) ,XI. c ) Fon evehy E > O p < E.

PROOF. a )

I b)

I1 f (tn) -f

. Given

(t,) II
0 , f o r t ~ ] a , b ( t h e e x i s t e n c e of

.

THE I N T E R I O R INTEGRAL f (t+) and

"1

that

6t > 0

f (t-) i m p l i e s t h a t t h e r e e x i s t s

[

(f)
0

and

E

t - 6 t ,t there exists 6 a > 0

exists

17

a

I

0

(f

1
0 suoh t h a t S B [ a h ] < M for all , E Lo;

g;m tE[a,b)

f o r aZZ

-

a: (a,b)

b ) there exists

a,(tl.y

E

su c h t h a t

= act)-y

~ E F .

and aZZ

Then we have S B ( [a,b] ,El

1) a 2)

eim

J

’ Y a

f E

G [ [a,b] ,F).

PROOF. F o r have

we

b

d

and

.da,(t).f[t)

D

and

yiE

=

F

SB[a]

b!

,< M ;

-da(t).f(tl

with

IIyill

6

f o r aZZ

.

1, i = 1,2,. , , I d ( ,

THE INTERIOR INTEGRAL By a )

the f i r s t

1,2,

i

...,I

summand i s

(M

A € Lo. By b ) f o r e v e r y

i f

there exists

dl

47

3

LiE

A € Li

such t h a t f o r

We

have

-

i

where

i.e.

-

we have

1).

By d l

2)

of

hence,

Theorem 4.12

II

6 SBbJ A for /IFa I I S M A

IFa IIFall,

Lim

[xJ,~,IY]

Fa

=

; c A =

... n L I d 1

X E L o n Lln

i,i - 1 . H e n c e f o r

Lim

3

aA(r).y

-

alrl-y

IIFall

A € Lo:

4 a I ’

b y b l we h a v e

b

Lim

.daA(tl.x],,,)(tly

=

a

jb

=da(tl.x

=

).,.I( t ) Y

Fa[x 1 a , T l ~ l .

=

a

e’m

F

and s i n c e

-

and

$

This implies that f E E([a,b],FI

we h a v e

= Fa[f]

If] ‘A

IIFaAII s M

f o r every f o r

A € Lo

we h a v e

l i m Fa [ f l Fa(f) por a l l f E G([a.b],F) i.e. ; ’ P A b y a1 o f Theorem 3 . 1 t h e r e e x i s t s fgE E([a,b],F)

11 f - f E11

6

3M , E

IIFa(f)-Fa

( f l

11

f E €

EL ( a , b ] . F l

LE€

3

such

such t h a t

(f-fE)l\,
0

and

q[Fa(f)] f o r every

‘‘llf

rG

there e x i s t

Il(c,d)

f € G()a,b(,F): w e take

= (aq,bq)

(c,d)

then w e

have

hence w e proved a l s o c ) .

THEOREM 6 . 6 .

Let

x

a € SVoo()a,b[,L(X,Y)

1

-

Fa

an i n j e c t i v e Linean a p p L i c a t i o n 0 6 t h e o n t o t h e decortd, whehe d o h f € G - ( ) a , b ( , X ) Fci(f) =

jb

a

a SSCLCS; t h e

L[G-()a,b(,X) ,Y]

i b

w e have

Y

be a Banach bpace and

mapping

dihAt

v e c t o h Apace

we d e d i n e

= d a ( t )- f ( t ) ;

ci(t)x = F , [ x ) ~ , ~ ) x ] .

PROOF. By Theorem 6.5

F,(f)

i s w e l l d e f i n e d and

Fa€ L[G-()a,b(,X) ,Y].

L e t u s denote by

0

t h e mapping

a

@

F a ; @ i s obviously

linear.

a ) @ is i n j e c t i v e , i . e .

a # 0 i m p l i e s Fa # 0 ; indeed: # 0 t h e r e e x i s t t E ) a , b ( and x E X such t h a t a ( t ) x # 0 ; hence i f w e t a k e f = x ) ~ , x~~)G - ( ) a , b ( , X ) w e have F,(f) # 0 s i n c e if

ci

(*I because f o r every

Fa(~)a,t]x) = a ( t ) x

qE

rY

w e have


since

q[a(an)x]

o

=

an < a

for

9

b) I n o r d e r t o prove t h a t

(1

59

a(t)x] =

o

. i s s u r j e c t i v e w e have t o

@

[

t h e r e i s an L [Ga , b , X ) ,Y] such t h a t F = Fa. By ( * ) w e know t h a t SVoo()a,b(,L(X,Y)) for i f t h e r e e x i s t s such an a w e have c c ( t ) x = F[X]a,tjx] a l l t ~ ) a , b ( and a l l X E X ; l e t u s t a k e t h i s a s t h e d e f i n i -

show t h a t f o r every

F

t i o n of a. i) a ( t ) E L ( X , Y ) ;

q[a(t)x-a(b

XEX

)XI = 9

nuous, f o r every

indeed, f o r every

w e have

9

0

if

qEry

t >b

€or

t < a

for

t > b q (iii) ~ E S V( ( a b

qi

indeed, s i n c e

F

is conti-

there e x i s t

f ~ ~ - ( ) a , b ( , X i) f; w e t a k e

f = xlart]x

w e have

and

9

xi€ X ,

qEry

q E r y t h e r e i s an [aq,bq) c ] a , b [ such w e have q [ a ( t ) x ] = 0 i f t < a and

ii) For every that for a l l

E

.

I(xi((\< 1, i =

) , L ( X , Y ) ) ; indeed: 1 , 2 , ...,I dl w e have

q t 9

For

d€m,

(aq,bq] and

THE I N T E R I O R INTEGRAL

60

Hence by ( i ) ,(ii)and (iii)w e have a~ Svo0()a,b(,L(X,Y) 1 .

( i v ) Fa = F from

because b o t h a r e l i n e a r continuous operators

into

G-()a,b(,X)

t h a t by ( * ) t a k e t h e same v a l u e

Y

on t h e e l e m e n t s of t h e form a t o t a l s u b s e t of Let

G-

X] a , t]

(1 a , b [ , X ) .

be a Banach space and

X

and t h e s e elements form

x E c ~ O C ( ) a , b ( , X ) i f f o r every

x: ] a , b [

(c,d)

C

)a,b(

-

X; we w r i t e

w e have

co([ctdltx)

(c,d]

REMARK 3 . The analogous of Theoram 3.12 i s t r u e i f w e r e p l a c e r e s p e c t i v e l y by G ( (a,b) ,X) , G- ( [ a , b ) , X I , co ( [ a h ] ,X) G(]a,b(,X) Let

u: ) a , b (

, G-(]a,b[,X) , cAoC(]a,b[,X).

X

be a Banach space and L(X,Y)

we w r i t e

UE s

following p r o p e r t i e s a r e s a t i s f i e d : 1) For every

f o r every

THEOREM 6 . 7 .

mapping

ry

w e have

XEX

2 ) sq[u]
a,b(,X) ,Y]

an i n j e c t i v e l i n e a h a p p l i c a t i o n o d t h e d i h 4 . t v e c t o h dpace o n t o t h e decond, whehe d o h x E c A o c ( ) a , b ( , X ) we dedine

i d

we have

u ( t ) x o = FuLltlxo]

d o h evehy

t E)a,b[

The proof follows t h e s t e p s of t h e Theorem 6 . 6 t h e Theorems 5.5 and 5 . 4 ) .

and

X ~ X E

.

(see a l s o

I n a n analogous way a s from t h e Theorems 5.1 and 5.5 follows t h e Theorem 5 . 6 , from t h e Theorems 6 . 6 and 6.7 lows t h e

fol-

61

THE I N T E R I O R I N T E G R A L

Let

THEOREM 6.8.

be a Banach bpace and

X

Y

a SSCLCS; t h e

mapping ( a , 4 E SVoo()a,b[,L(X,Y))

soo()a,b[,L(X,Y))

x

[

F = Fa+FUE L [G () a , b ,X) ,Y]

-

(1

(= L [G- (> a ,b [ ,x) ,YI x L [cAoc a , b [ , X ) ,Y] ) i b an i n j e c t i v e eineah a p p e i c a t i o n 0 6 t h e 6 i h A X v e c t o h s p a c e o n t o t h e s e c o n d , whehe d o h evehy f ~ G ( ) a , b [ , X ) w e d e d in e

b -

Fa(f)

=

* d a ( t ) * f(t)

and

t €)a,b(

and

and

Y = G(R,X),

F(f) (t) = f(t+p) w e have

=

1

u ( t ) [ f ( t ) - f (t-111;

u ( t ) x = Fhtt1x]

do& eVehy

XEX.

+

= f (a-)

2. Take

FU(f)

actcb

a we have a ( t ) x = F L

-

p > 0

f ( t ) for

a ( t ) x = F[x)a,t)x]

[f ( a ) - f (a-I]

.

and F E L [ G W , X ) ] fcG@t,X). For t e R

E GOR,X)

d e f i n e d by

and

XEX

and

u ( t ) x = FIXIt}x] E G C R , X )

hence f o r

UER


62 w e have

The Theorem of Helly extends too:

T H E O R E M 6.9. Let

(E,F,G)

be a LCBT

SBoo()a,b[,E)

and ao: ] a , b (

01)

q E

F O h eVLLhy

~

buch t h a t a l l bame

intehval 021

E

3

and

2

a d i l t e h on

buch t h a t

H E 9

t h e h e e.Xi6.t

3

and

M > 0 q

H have t h e i h q-buppoht contained i n t h e

[aq,zq]

lim a ( t ) y

rG

+E

c)a,b

(

with

= a o ( t ) y doh

SBs r [ a Ibq)

aLl

Mq

and

t~]a,bf

*

y€F.

T h e n we h a v e poht

f

E

1 ) ao~SBoo()a,b[,E) and doh e v e h y q E r G t h e q - b u p .a i b contained i n (aq,bq) with

06

G()a,b(,F). The proof follows t h e s t e p s of Theorem 5.9. The theorem

s u g e s t s t h e following d e f i n i t i o n

Let a filter

X

3

b e a Banach space and on

a

Y

SSCLCS; w e say t h a t

SVoo()a,b[,L(X,Y)) 0-convehgeb

a E SVoo()a,b(,L(X,Y)), and w e w r i t e

a

0

p e r t i e s u l ) and 0 2 ) are s a t i s f i e d .

U * ao, f

to i f t h e pro-

The following theorem has a proof analogous t o t h a t of Theorem 5.10

THEOREM 6.10. Let x

be a Banach b p a c e and

Y

a SSCLCS;

WQ

have

L [G-

(1a ,b (,XI ,G

moae p h e c i ~ e l y

()

c ,d [ ,Y 13

d~G(]c,d[,SV&(]a,b(, L ( X , Y ) ) ) i b

= G [] c ,d [ ,SVzo (1a ,b [ ,L ( X , Y ) ) ]

-

Fd

E

an i n j e c t i v e lineah a p p L i c a t i o n

onto t h e becond, w h e u doh

f E G-

L[G-(]a,b(,X) ,G(]c,d(,Y)] 04

t h e d i h b t vectoh bpace

(1a , b [,XI

we d e d i n e

63


.

(t)

XI

REMARK 4 . We define

A(t,a) = J,(t)(a); it is easy to see that +L(X,Y) is characterized by the fol-

then A: )c,d[x]a,b[ lowing properties: (SVu) - A is locally uniformly of bounded semivariation in the second variable (i.e. for any

[z ,a] x [z ,5]

c

] c ,d [x)

we have AtE SV ( [ i , g ) ,L(X,Y)) every q c ry we have -bup- SV csttd

a ,b

[

-

(G') A is weakly regulated as a function of the first variable (i.e. for every s~)a,b[ aLd X E X the function t )c,d( A(t,s)x€Y is regulated, that is, has only discontinuities of the first kind). I-+

REMARK 5 . Still apply the coments of Remark 2 .

REMARK 6. In Chapter I11 we will apply Theorem 6.10 with Y=X. APPENDIX

(i.e.

Let 1

P

X +

1

-

P'

and =

Y

1 < p

,XI

m

and

we d e f i n e

fEG-[(a.b).Xl t h e space

GLP endowed w i t h t h i s norm3 t h i s s p a c e i s n o t k o m p l e t e p l e t i o n is t h e space

o f functions

Lp([a,b),Xl

p'=

W

G - ( [a,b)

P-1

IIfllp .XI

( i t s com-

o f t h e equivalence classes

t h a t a r e p - i n t e g r a b l e i n t h e sense o f Bochner-

Lebesguel. I n t h i s a p p e n d i x we e x t e n d t h e m a i n r e s u l t s o f 1 5 4 , 5 6 t o t h e spaces For

a:

[a,b)

+L(X.YI

we d e f i n e

and


64

+ L(X,Y) I

S V ~ ’ ~ ( a , b ) . L [ X , Y I l = ~ a : (a,b)

THEOREM 6.11. For

a € S V E ’ I( a , b )

PROOF. a 1 F o r

F

dED

and

SV[a]6

x E X

i

with

I)xi\I

we h a v e

The c o n t i n u i t y o f

follows from

a

Ila[t)-a(sl\/ c b)

/t-sIP

F o l l o w s f r o m Theorems 4 . 1 2

cl For

Id,clE

D

we h a v e

hence t h e r e s u l t f o l l o w s

[a]
b

(00)

b

a9

9

9

9

rz )

and corresponds t o then

THE A N A L Y S I S OF REGULATED F U N C T I O N S

81

vhere

06 Dihichtet

B - T h e dohrnuca

T H E O R E M 2 . 6 . W i t h t h e h y p O t h e 8 i 8 of Theorem 2 . 1 we t a k e )c,d[ )a,b( and g c o n t i n u o u s ; t h e n we have

-

b

s

J [ J-da[tl.h[t.s)

[71

a

ho[t,sl

same p r o p e r t i e s a s

THEOREM 2 . 7 . G i v e n a

SVoo[)a,b(,

g e e (]a,b[,Xl

I[ s

a

X,

L(Y,Z)I,

[or

t

b h[t.s)dg[sl]

then

ho

has t h e

H e n c e we may r e p l a c e

a s i n Theorem 2 . 1 ,

Z

h E G U ( ] a , b ( X ]a,b(,

ge C ( ) a , b ( ,

L(XI1I

1

u

a

=

x ] a,

daltl[

h

REMARK 2 . The e x t e n s i o n

by

ho

1

h[t,u)dg(ol

t

h

Z

i n T h e o r e m 2.6.

i n [ 7 1 a n d we g e t

o f the other results o f

c a s e o f open i n t e r v a l s and

I.

(t,~lY(o-tlh[t,a)

s a t i s f i e s t h e same p r o p e r t i e s a s

H e n c e w e may r e p l a c e

and

S E )a,b(

S

a

s] x ] a, s)

L(X.YI1

for every

S

J - d a [ t ~ . h [ t , u l dg(u1 =

ho(t,ul hD

a Y[s-t)h[t,s)

=

and

Y

PROOF. If w e t a k e

then

J

i n Theorem 2 . q .

h

we have (81

j*da[tl[

i n C o r o l l a r y 2.5 and we g e t ( 7 1 .

ho

by

b

-

a

P R O O F . If we t a k e h

dg(s)

(81.

$1 t o the

a SSCLCS i s n o w o b v i o u s .

CHAPTER

Vol t e r r a

Stieltjes-Integral

with Let

X

I11

Linear

Equations

Constraints

b e a Banach s p a c e and

Y

a SSCLCS; i n t h i s

c h a p t e r w e c o n s i d e r systems of t h e form

[

y , f E G (>a ,b ,X) ,

where

F E L ( G o a ,b ,X) ,Y)

K: ) a , b ( x ) a , b (

[

and

L(X)

i s a c o n t i n u o u s f u n c t i o n which s a t i s f i e s t h e p r o p e r t y (SV uo d e f i n e d on 8 1 . I n 61 w e show t h a t t h e r e i s a r e s o l v e n t a s s o c i a t e d t o K ; f o r t h i s purpose

K

need n o t be c o n t i n u o u s b u t o n l y regulated

as a f u n c t i o n of t h e f i r s t v a r i a b l e ; by Remark 1 0 of 81

of

Chapter I1 t h e r e s u l t s of t h i s 8 may even be extended t o t h e c a s e where K i s o n l y weakly r e g u l a t e d as a f u n c t i o n of t h e f i r s t variable. I n 8 2 t h e r e s u l t s o f 6 1 are a p p l i e d t o t h e case of a S t i e l t j e s I n t e g r o - D i f f e r e n t i a l E q u a t i o n and new p r o p e r t i e s o f t h e r e s o l v e n t are found.

I n 6 3 w e s t u d y t h e system (K), (F)

and f i n d i t s Green

function.

REMARK 1 . We w i l l see a l o n g t h i s c h a p t e r t h a t t h e e x t e n s i o n of t h e r e s u l t s t o t h e case o f c l o s e d i n t e r v a l s t o m a t i c and i n 81 w e go t h e o t h e r way around.

[c,d)

i s au-

REMARK 2 . I n o r d e r t h a t (K) makes a s e n s e it i s o b v i o u s l y s l r f f i c i e n t f o r K t o be d e f i n e d o n l y on t h e s e t

r

= { ( t , u >~ ) a , b ( x ) a , b ( 1 t ( u & to or

to& u Gt)

a3

STIELTJES-INTEGRAL EQUATIONS and t h e same a p p l i e s t o t h e r e s o l v e n t

when w e c o n s i d e r

R

t h e e q u a t i o n ( p ) o f 81. I n o r d e r t o s i m p l i f y t h e n o t a t i o n s w e extend

r

from

K

to

)a,b(x)a,b(

or even t o

)a,b(xIR

by

defining K(t,a)=

lK(t,t)

u a t %to

if

1 K ( t ,-to) if If i n

r

K

u s tos

K(t,t)

if

u < t < to

K(t,u)= lK(t,to)

f

if

u>to3 t

i s r e g u l a t e d as a f u n c t i o n o f t h e f i r s t v a r as a f u n c t i o n o f t h e s e c o n d v a r i -

i a b l e and s a t i s f i e s (SVUo)

a b l e ( s e e d e f i n i t i o n b e l l o w or Theorem 1.13 of C h a p t e r 11:

(SVUo) = b ) ) it i s o b v i o u s t h a t t h e e x t e n d e d f u n c t i o n s t i l l s a t i s f i e s (SVuo) and it f o l l o w s i m m e d i a t e l y from 1 1 . 1 . 1 5 that it i s a l s o r e g u l a t e d as a f u n c t i o n o f t h e f i r s t v a r i a b l e ; t h i s would n o t b e t r u e i f w e had o n l y (SV'); more p r e c i s e l y , i f K A : t E ) a , b ( H K ( t , t ) E L(X) w e r e not regulated.

REMARK 3 . E q u a t i o n s o f t h e t y p e ( K ) o c c u r q u i t e n a t u r a l l y ; indeed :

a ) For d i f f e r e n t i a l and i n t e g r a l e q u a t i o n s it i s n a t u r a l

t o work n o t w i t h f u n c t i o n s b u t w i t h e q u i v a l e n c e classes o f f u n c t i o n s : two f u n c t i o n s

y1

and

y2

are e q u i v a l e n t i f w e

have

for a l l

s

and

t ; i . e . , i n s t e a d of w o r k i n g , f o r i n s t a n c e ,

w i t h G ( ) a , b [,XI w e c o n s i d e r t h e q u o t i e n t s p a c e i() a ,b [,XI ( s e e C h a p t e r I, 8 3 ) or t h e s p a c e G - ( ) a , b ( , X ) isometric t o

it (1.3.13).

T h e r e f o r e , as a g e n e r a l i z a t i o n of l i n e a r i n t e g r a l o p e r a t o r s it is n a t u r a l to c o n s i d e r o p e r a t o r s L E L[G-()a,b

t h e n , by 1 . 6 . 1 0

[,XI ,G()c,d[,Y)]

;

[, SVEo (1 a ,b [,L(X ,Y)) )

there e x i s t s a kernel

KEG such t h a t f o r every

(1c

,d

f € G-()a,b[,X)

w e have

84

S T I E L T J E S - I N T E G R A L EQUATIONS b g ( f ) ( t > = ja*dGK(t,~).f(U),

where

K(t,a)x

l k ) a , u ) x ] (t),

t ~ ) a , b [ ,U E ] c , d ( ,

X.

XE

b ) If we m1n.t f u r t h e r t h e o p e r a t o r k? t o have p r o p e r t i e s s i m i l a r t o t h o s e of V o l t e r r a i n t e g r a l o p e r a t o r s , i . e . , t h a t X and t h a t t h e r e e x i s t s a p o i n t t o E ) a , b ( )c,d[ = )a,b(, Y such t h a t f o r e v e r y t E ) a , b ( , L ( f ) depends o n l y on f

f

then t h e o p e r a t o r

l t a k e s t h e form

Now however i n g e n e r a l k ( f ) i s n o t anymore a r e g u l a t e d f u n c t i o n u n l e s s w e impose f u r t h e r r e s t r i c t i o n s on t h e k e r n e l K. Furthermore, w e a l s o want K t o have a r e s o l v e n t and f o r t h i s purpose i t i s n e c e s s a r y f o r K t o b e r e g u l a t e d as a f u n c t i o n of t h e f i r s t v a r i a b l e and f o r t h e f u n c t i o n KA: t E ) a , b (

C,

K(t,t)E

L(X)

t o be r e g u l a t e d ; indeed: i n o r d e r t o c o n s i d e r t h e r e s o l v e n t

I+

e q u a t i o n we have t o work w i t h i n t e g r a l s o f t h e form t d u K ( t , a ) o U(o) where U € G ( ) a , b [ , L ( X ) ) ; s i n c e t h i s i n t e g d - c

0

do n o t change i f w e r e p l a c e K ( t , u ) by K ( t , u ) - K ( t , t o ) suppose f o r a moment t h a t K ( t , t o ) = 0 . Then f o r - r a t0 t a k e U = X ) ~ , ~ ) I ~ € G ( ) ~ , ~ ( , L ( X )w)e ; g e t

I n an analogous way f o r get

T

s to w e t a k e

U

xkyb[Ix

we we

and w e

STIELTJES-INTEGRAL EQUATIONS T

85 k(U)

i s regul a t e d f o r e v e r y U t h e n K i s r e g u l a t e d as a f u n c t i o n o f t h e f i r s t v a r i a b l e and K A i s r e g u l a t e d t o o . Also i n o r d e r t h a t ( K ) h a s a r e s o l v e n t i t i s n e c e s s a r y

Hence, s i n c e

i s a r b i t r a r y we see t h a t i f

f o r t h e equation r t

t o have a unique s o l u t i o n . However i f K s a t i s f i e s t h e nec e s s a r y conditiom;we j u s t found and i s l o c a l l y u n i f o r m l y of bounded s e m i v a r i a t i o n and s a t i s f i e s even (SV3) ( i >does n o t have a unique s o l u t i o n ( s e e t h e Example a f t e r 1 . 3 ) . However t h e c o n d i t i o n

KEGUo

d e f i n e d i n 61 w i l l a s s u r e

t h a t a l l t h i s n e c e s s a r y c o n d i t i o n s are s a t i s f i e d and t h a t

K

has a resolvent.

51 - T h e h e d o t v e n t eq u a t i a n of

06

a V a t t e h h a StieLtjed-integhat

A - L e t X be a Banach s p a c e , R , n o t n e c e s s a r i l y bounded, and

the Volterra Stieltjes-integral

where

y , f EG()a,b[,X)

and

) a , b [ and open i n t e r v a l to E a b We c o n s i d e r

. 1 " equation

K: ) a , b [ x ] a , b [

+L(X)

satis-

f i e s the properties (GI

(SVuo) exists a

-

K i s r e g u l a t e d as a f u n c t i o n of t h e f i r s t v a r i a b l e .

For e v e r y (c,d)c ) a , b ( and e v e r y 6 > 0 such t h a t for a l l s , t E (c,d]

(svUo)e x p r e s s e s t h e s e m i v a r i a t i o n of goes u n i f o r m l y t o 0

E

>0

there

w e have

t h a t i n every

K on an i n t e r v a l o f t h e second v a r i a b l e

with t h e length o f t h e i n t e r v a l , i . e .

STIELTJES-INTEGRAL EQUATIONS

86

It i s e a s y t o see t h a t 1.1. T h e p h o p e h t y

and

(SV')

(sv"): -

(SVu>

that

i m p t i e b t h e phOpe&,tieb

(SVUd)

UP

c 0

[c,d) C ] a , b (

<M. we h a v e

s,tE)a,b(

= 0.

[Kt]

6C0

We r e c a l l t h a t (SV'> s a y s t h a t l o c a l l y K i s uniforml y of bounded s e m i v a r i a t i o n as a f u n c t i o n o f t h e s e c o n d v a r i able.

1.2.

I{

[

Kt

t E) a , b

K has Xhe p h o p e h t y (SVo) LA a c o n t i n u o u b { u n c t i o n .

then

{oh

evehy

bo,t]

S i n c e Kte SV( ,L(X)) t h e i n t e g r a l i n ( K ) i s w e l l d e f i n e d and by II.1.14a) w e h a v e 1.3.

Let

K

A a t i b { Y ( G I and

(SVUO), t h e n

t

It

a ) The { u n c t i a n t € )a,b [ e d,K(t ,a> . y ( a > E X ~ e uL g at e d 0 b l Id K i b continuoub t h e { u n c t i o n

.

t )a,b(

It

ib

t

H

c o n t i n u o u b and a b o t u t i o n onty i d f i b continuoub.

ib

d,K(t,o).y(a)E 0

y

06

X

( K ) i b continuoub

h a s the p r o p e r t i e s (GI (SV’") t h e n f o r e v e r y X E X ( K ) h a s o n l y one s o l u t i o n t h a t y ( t o > = x (Theorem 1 . 6 ) . If however K s a t i s f i e s t h e p r o p e r t i e s (GI, (SV') a n d (SV d 1 t h i s i s no l o n g e r as i s shown by t h e We w i l l p r o v e t h a t when

K

and

and such only true,

STIELTJES-INTEGRAL EQUATIONS E X A M P L E . We t a k e ) a , b ( = 0 and f o r t >

K(0,s)

]-1,1[,

to

87 X = IR. W e take

0,

we define

0

and for t < O w e t a k e K ( - t , a ) K ( t , a ) . Then f o r y ( t o > = 0 t h e equation t y(t) + daK(t,u)y(a) o h a s two s o l u t i o n s ,

= 0

if

t = 0,

y 3 0

y ( t ) = Ag t

and

(= 1

f = 0 and

if

t > O ,

t (1).

= -1 i f

and w e want t o p r o v e t h a t for f G ( ) a , b [ , X ) XE X t h e r e i s one and o n l y one y E G ( ] a , b ( , X ) s o l u t i o n of ( K ) and s u c h t h a t y ( t o ) = x. O b v i o u s l y it i s s u f f i c i e n t t o p r o v e t h e same r e s u l t f o r any c l o s e d i n t e r v a l [c,d) C ) a , b (

In this

§

t h a t contains tosince i f the solution i n tion i n

[;,a)

3

w e have

[c,d)

s o l u t i o n s on t h e i n t e r v a l s I

(.

in )a,b ( R * ) , (R*)

is t h e s o l u t i o n i n

y

[c,d)

’7;

dl ,d)

(c,d)

and

by t h e u n i c i t y o f t h e s o l u y , h e n c e w e can u s e t h e C

1

a,b

[

t o get the solution

The same r e a s o n i n g a p p l i e s o b v i o u s l y t o e q u a t i o n s and o t h e r s considered bellow.

F R O M NOW O N W E W I L L C O N S I D E R A F I X E D I N T E R V A L (c,d) C ) a , b [ C O N T A I N I N G to We d e n o t e by G

( [c ,d] x [c ,d) , L ( X ) ) , or

s e t o f a l l bounded f u n c t i o n s

U:

[c,d]x[c,d)

s i m p l y by G

+L ( X )

, the that

s a t i s f y ( G I , i . e . , a r e r e g u l a t e d as f u n c t i o n s o f t h e f rst v a r i a b l e , t h a t i s , U s e G ( (c ,d] , L ( X ) 1 f o r e v e r y s G (c ,d] i s a Banach s p a c e when endowed w i t h t h e norm

G

UC6

6 IIUII

= bup{llutt,s)ll

1

.

s,t€(c,d]~.

W e d e n o t e by G' t h e vector subspace o f G formed by t h e f u n c t i o n s t h a t s a t i s f y (SVU). GU i s endowed w i t h t h e

norm

IIIUIII

IIUII

+ SVu[Ul

where w e r e c a l l t h a t


88

T H E O R E M 1.4. G U111 111 i b PROOF. L e t

a Banach Apace.

G u y n E N , b e a Cauchy s e q u e n c e . Then

UnE

uniformly convergent t o a function there exists

n

t t SV [Un-Um]

for e v e r y

<E

t t s sv[un-u

such t h a t f o r

for e v e r y

n,m> n

Guo

R E M A R K 4. I f

KE

w e have

and

E

[c,d)

n >n Un

f o r every

E

. Hence

GU

UE

Gu.

i n t h e norm o f

G' formed by t h e functions W e w i l l show ( P r o p o s i t i o n

i s a closed subspace of

Guo,

0

t

denotes t h e subspace of

Guo

E >

and t h i s i m p l i e s immediately

t h a t s a t i s f y t h e p r o p e r t y (SVuo). 1.14) that

E

is

Un

and f o r e v e r y

t E [c,d)

i s t h e l i m i t of t h e sequence

U

and

E

E

UEG

t E (c,d]

G'. w e have o b v i o u s l y

re-

hence t h e e q u a t i o n ( K ) d o e s n o t change i f e v e n t u a l l y w e

p l a c e K by K - K A y i . e . K ( t , o ) by K ( t , u ) - K ( t , t ) , that is, w e suppose t h a t K i s normalized: K ( t , t ) = 0 . Furthermore, s i n c e K E Guo, by 1 1 . 1 . 1 5 KA i s r e g u l a t e d and t h i s i m p l i e s i m m e d i a t e l y t h a t K-KAE Guo; n o t h i n g o f t h e k i n d would b e t r u e i f w e had o n l y K E G U . B

-

We g i v e now 4 i m p o r t a n t examples o f a n e q u a t i o n ( K ) .

EXAMPLE A

-

W e consider t h e Stieltjes integro-differential

equation

y'

(L')

where ( L ' )

t

A'sy

f'

is a n a b r i d g e d way o f w r i t i n g t h a t w e have

JS

for all

I (.

s , t E a,b

W e suppose t h a t

y,f€G()a,b[,X)

and

AEtSV1oC(]a,b[,L(X) 1

(i.e.

A

i s a c o n t i n u o u s f u n c t i o n a n d f o r e v e r y [c,d) c > a , b (

w e have A € SV( (c,d) ,L(X) 1 ) . I n t h i s case t h e c o n d i t i o n ( SVuo 1 becomes s i m p l y

STIELTJES-INTEGRAL EQUATIONS [A]

f o r every

0

S E

1a b4 .

The e q u a t i o n (L) i s e q u i v a l e n t t o y(t)-y(to) +

dA(u).y(u) = f ( t ) - f ( t o )

f o r every

tE)a,b(

I:t s i n c e it i s o b t a i n e d as t h e d i f f e r e n c e o f i t s v a l u e s f o r and s ; hence ( L ) i s a p a r t i c u l a r i n s t a n c e o f (K).

t

We r e c a l l t h a t (L) or ( L ’ ) c o n t a i n as a p a r t i c u l a r case

t h e o r d i n a r y l i n e a r d i f f e r e n t i a l e q u a t i o n s ; (L) w i l l a l l o w discontinuous s o l u t i o n s ( f o r f discontinuous).

EXAMPLE

B

-

We c o n s i d e r t h e V o l t e r r a i n t e g r a l e q u a t i o n r t

If w e d e f i n e U

K(t,o) = ItB(t,s)ds 0

t h e e q u a t i o n ( V ) t a k e s t h e form ( K ) ; h e r e w e suppose t h a t f o r every t E ) a , b [ t h e f u n c t i o n Bt i s Darboux i n t e g r a b l e (or Bochner-Lebesgue i n t e g r a b l e ) . I n o r d e r for

K

t o be regulated

as a f u n c t i o n o f t h e f i r s t v a r i a b l e it i s s u f f i c i e n t t h a t for every t E ) a , b ( t h e r e e x i s t f u n c t i o n s Btt and Btsuch w e have t h a t for e v e r y s E ) a , b [

and

I n o r d e r t h a t (K) s a t i s f i e s t h e p r o p e r t y (SVuo)

[

s u f f i c i e n t t h a t f o r e v e r y (c,d] c ) a , b t h e r e e x i s t s 6 > 0 such t h a t f o r every S+6

bup ccttd

1

IlB(t,o)l)da
0 s E [c,d) w e have E


90 s i n c e by 1 . 5 . 2

w e have

S+6

These c o n d i t i o n s a r e o b v i o u s l y s a t i s f i e d i f

i s a conti-

B

nuous f u n c t i o n or, more g e n e r a l l y , a l o c a l l y b o u n d e d h e a s u r a b l e ) f u n c t i o n which as a f u n c t i o n o f t h e f i r s t v a r i a b l e i s

regulated ( f o r almost a l l

E X A M P L E C.

s~ ) a , b ( ) .

W e consider t h e d i f f e r e n t i a l equation

A ~ ( A ~ Y )t ' BY = g ' loc

where gEBV ()a,b[,X), BEG()a,b[,L(X)) a n d Ao, A1 s u c h t h a t t h e r e e x i s t and a r e r e g u l a t e d t h e f u n c t i o n s

If w e m u l t i p l y t h e e q u a t i o n by tain A1 ( t) y (t1-A1 ( s y ( s 1 t

Ist.

L(X)

Ai(t)-'E

t E ]a,b(

Ail

are

1,2.

i

a n d i n t e g r a t e i t w e ob-

( a ) - l B ( a ) y (a1da

=

Is

t -Ao ( a 1 - l . dg ( a 1

t €]a&(

a n d t h i s i s t h e meaning t h a t must b e g i v e n t o t h e o r i g i n a l z(t> = A (t)y(t) 1 i n t h e form of t h e p r e c e d i n g examples:

e q u a t i o n . I f w e make

z(t> where

-

Z(S)

f

w e o b t a i n an equation

A ( a > z ( o > d o= f ( t )

-

f(s)

STIELTJES-INTEGRAL EQUATIONS E X A M P L E D. We w i l l

91

show t h a t a l i n e a r d e l a y d i f f e r e n t i a l

e q u a t i o n may b e reduced t o a p a r t i c u l a r case o f e q u a t i o n

T h i s example i s due t o Jos6 C a r l o s Fernandes de O l i v e i r a .

(K).

be a Banach s p a c e , t l > t o and 0 < r < t l - t o ( t h e d e l a y ) . Given y E G ( [to-r,tl) ,XI f o r e v e r y t E [toytl) a ) Let

w e define

X

,X)

y t € G( [-r,O)

yt(s> = y ( s + t > ,

by

SE

(-r,O).

A l i n e a r d e l a y d i f f e r e n t i a l e q u a t i o n i s an e q u a t i o n o f t h e form

(D)

+ g(t>

y ' ( t ) = cA(t,Y,)

where

YE G(

(to-rytl),XI ,

a:

gE G(

(toytl)xG(

(to,tl),X)

[-r,O)

,XI

4

and

x

has t h e following p r o p e r t i e s :

1) J t E L [ 6 ( [ - r , 0 ) , x ) ~ x J f o r e v e r y t E (to,tl), i . e . , as a f u n c t i o n of t h e second v a r i a b l e i s l i n e a r and continuous. 2)

SI,

f o r every

a

~ ( b

,tl] ,x) f o r e v e r y f E f E G ( T - r a O ) ,XI t h e f u n c t i o n

E

t E

(to+)

G ( [ - r , ~,XI ] ,

-+J(t,f)€

i.e.

,

x

i s r e g u l a t e d . Hence by 1 . 3 . 1 3 and 1 . 5 . 1 we have J t EL[G-((-r,O) L e t us d e n o t e by

responding t o

At

SVo([-ryO) , L ( X ) ) .

,X>,X]

At t h e element o f SVo((-ryO) ,L(X)) c o r , i . e . , for e v e r y f E G - ( [-r,O) ,XI w e have

Furthermore i f w e d e f i n e

5 ( f 1 ( t1 dt ( f 1 , by

the

hyp o t h e s i s above w e have Fd € L [ G ( [ - r , O ) ,X) , G ( (to,tl],XI] hence by Theorem 5 . 1 0 of C h a p t e r I and Remark 8 t h a t follows i t

-

t h e r e e x i s t s one and o n l y one f u n c t i o n A:

(toytl)x[-ry~)

L(X)

t h a t h a s t h e p r o p e r t i e s (SVu) and ( G a l o f t h e remark w e ment i o n e d and i s such t h a t f o r e v e r y f E G ( (-P,o) Y X ) we have


92

Hence (D) t a k e s t h e form

b ) But f o r Y E G ( (to-r,tl],X) t h e f u n c t i o n e r a l i s n o t anymore r e g u l a t e d , where

9

i n gen-

A ~ ~ U o ( ( - r , O ) x ( t o , t l ) ,L(X)) t h e n it f o l l o w s t h a t f o r e v e r y Y E G ( (to-r,tl],X) t h e f u n c t i o n 7 i s r e g u l a t e d ; indeed: w e have

However i f w e s u p p o s e t h a t

y(t)

=

-

rt d,A(t,s-t)’y(S) Jt-r

and by remark 6 of 51 o f C h a p t e r I1 t h e f u n c t i o n

B , where

B ( t , s ) = A ( t , s - t ) ( w i t h t h e e x t e n s i o n made a c c o r d i n g t o remark 2 ) s t i l l b e l o n g s t o G U o , i . e . s a t i s f i e s t h e h y p o t h e s i s

o f Theorem 1 . 1 3 o f C h a p t e r 11; h e n c e t h e r e s u l t f o l l o w s from a ) o f 11.1.14. T h e r e f o r e i f A € Guo, e v e r y r e g u l a t e d s o l u t i o n y o f (b) h a s a r e g u l a t e d d e r i v a t i v e y ’ , h e n c e y i s c o n t i n u o u s (for

t E [to,tl] 1

.

c ) For (D) or giving a function Y E G( (to-r,tl),X)

(6) t h e

i n i t i a l v a l u e problem c o n s i s t s i n 4~ G ( (-r,o) ,XI a n d l o o k for a f u n c t i o n t h a t i s a s o l u t i o n of (D) or (6) f o r = 4.

t E (to,tl) and s u c h t h a t

YtO

W e w i l l show t h a t t h i s problem may b e r e d u c e d t o a p a r t i c u l a r case of (K). ( 5 ) i s e q u i v a l e n t t o y’(t> =

I

t

d s A ( t , s - t ) ey(s.1 t g ( t ) t-r

We r e c a l l t h a t w e s u p p o s e t h a t x R taking A ( t , s ) = A(t,O)

bo,tl]

= A(t,-r)

if

A if

s < - r ; t h e n w e may w r i t e

t E (to,tl).

h a s been extended t o s> 0

and

A(t,s)

=


For

SE

(to-r,to) g(t)

w e have

g(t) +

we obtain

W e recall t h a t

y(to) = $(O>

ro

93

y ( s > = $ ( s - t o > ; i f we t a k e dsA( t

,S - t I*$( s -to

t 0-r

and w e t a k e

we get

A s w e s a w i n b)

= A(t,s-t)

B(t,s)

satisfies the

hy-

p o t h e s i s o f Theorem 1 . 1 3 o f Chapter I1 and a f o r t i o r i o f Theorem 1.1 o f t h a t c h a p t e r hence we may a p p l y (6’) o f 1 1 . 1 . 6 and w e o b t a i n

t

If w e make

A

K(t,s)

i n such a way t h a t finally obtain

where

= - J s A ( - r Y s - ~ ) d . c and i f w e n o r m a l i z e A(t,O)

0

i n s t e a d of

A(t,-r)

= 0 we


94

1 [\ t

d s A ( ~ , s - ~ > O ( s - t o ) + g dT (~) to to-r

f(t)

(and

y(t)

C

-

1

to

@(t-to) for

t E [to-r,to)),.

The main theorem of t h i s 8 i s t h e f o l l o w i n g

T H E O R E M 1.5.

we h a v e

K E Guo

Given

RE G , t h e h e -

I - Thehe e x i b t b o n e and ondg o n e e l e m e n t K, buch t h a t

b o l v e n t ad (R”)

f o r all

Ix- d a K ( t , a ) c R ( o , s ) Jst

R(t,s)

I 1 - R€GU0

and

I 1 1 - F o h euehg

= Ix

R(t,t)

fEG([c,d) ,XI

d o h aLl

and

s , t E (c,d).

t E (c,d].

t h e bybtem

x E X

y(to) = x hub o n e and ondg o n e b o t u t i o n

Y E G ( [c,d)

, X I ; t h i b bolution

by

i b given

t

(PI

R(t,to)x +

y(t)

I,

t E [c,d)

R(t,s)df(s)

0

and dependb c o n t i n u o u b l y on

IV

- 16

K

i 6

R(t,s)

(R,)

Ix

x

and

nohmalized ( i . e .

+

Jst

R

i b

KE Guo

od a L l

buch t h a t

buch t h a t R(t,t)

K).

K(t,t) = 0

K E Guo K(t,t)

d o h euehy

f o r all

s,tE(c,d).

adbociateb i t b

i n j e c t i v e and b i c o n t i n u o u b

t h e b e t od a d l REGUo

(and

R(t,o)odoK(a,s)

U - T h e mapping t h a t t o evekg heboluenf

f

( n o t l i n e a h ] dhom

E 0

onto t h e b e t

Ix.

REMARK 5 . The really d i f f i c u l t p a r t o f t h i s theorem i s t h e

proof of 11; t h e proof o f I i s q u i t e s i m p l e . However I1 i s


95

n e c e s s a r y i n o r d e r f o r t h e i n t e g r a l s i n ( p > a n d (R,) defined, t o prove t h a t

to

be

g i v e n by ( p ) s a t i s f i e s ( K ) a n d t o

y

prove V.

W e w i l l now p r o v e many p a r t i a l r e s u l t s till w e c o m p l e t e t h e proof o f Theorem 1 . 5 .

T H E O R E M 1 . 6 . Given K E G U o , euehg X E X t h e equation

dotr

euehy

duK(t,u>.y(a>

f(t)

and

f E G((c,d),X)

-

f(to)

tE(C,d)

0

hub at t?IOAk one A o e u t i o n YE G ( [ c , d ] ,XI p m v e d in CuhoLLahg I . 1 6 ) . P R O O F . I f y1 a solution of

and

(the exiAtence

a r e two s o l u t i o n s t h e n

y2

and w e w i l l p r o v e t h a t

0, hence

z

y1

E

i A

z = y2-y1

y 2 . For

t > t o we

have

t

and i f w e t a k e

Since

1

> t w e have

s a t i s f i e s (SVuo> t h e r e e x i s t s

K

t l >to

such

that

Aup

SV( to,t&Kt]

< 1

t a t l 0‘

hence

z(t>

for

0

t: and w e have z(t)

o

for

td, t E

t E

(to,tl).

W e define 03

= AUp { t > t d

and h e n c e

(to,tA)

z

zI

(toYd) satisfies

0s

= 0 ; indeed s i n c e


96

and t h e n , as above w e prove t h a t t h e r e e x i s t s t l > t0' such that z ( t ) 0 f o r a l l t E [to,tl) i n c o n t r a d i c t i o n t o t h e d e f i n i t i o n of t:. I n an analogous way one p r o v e s t h a t

for

z(t) = 0

c g t c t0

COROLLARY 1 . 7 . G i v e n

.

KEGUo, t h e hCbOt.Vent

t&d i e b (R")

R(t,s)

= Ix

-

t ~sduK(t,o)oR(a,s)

REG

for a l l

that

ba-

s , t E (c,d)

i n u n i q u e ( t h e e x i b t e n c e i a phoued i n Theohem I . 9 1 . P R O O F . For e v e r y

S E (c,d)

= Ix

Rs(t)

-

w e have R s E G ( (a,b) ,L(X)) t d,K(t,a)oR,(G) j

and

js

t h e r e s u l t f o l l o w s immediately from t h e Theorem 1 . 6 i f w e c o n s i d e r y ( t > = R s ( t ) x where X E X . THEOREM 1 . 8 . G i v e n

R(t,s)

(R")

w c haue a ) Fon cuehy

i n the botution

and

KEGUo

= Ix

-

batiddying

R€GU0

t ~sduK(tyo)oR(oys)

f E G ( [c,d] ,X)

and

XE

X

t h e dunction

06

y(to) = x.

may b e w h i t t e n a6

= f ( t 1+ R ( t ,to1 [x- f ( to13 -

and

y dependa continounLy on

x

and

PROOF. a) It i s enough t o prove t h a t i f t v ( t ) = \ t R ( t y . s ) d f ( s ) , t E [c,d), w e have 0

( t,s 1 f ( s )

t E (c ,d]

f. u(t)

R(t,to)x

and


97

r t

and r t

t h e f i r s t e q u a l i t y i s immediate i f w e a p p l y ( R " ) take

s

x

to

and

t o . I n o r d e r t o prove t h e second one w e have t o

show t h a t

t R(t,s).df(s) Jt

U

t

JtR(o,s)df(s)]

= f(t)

-

f(to).

0

If w e r e p l a c e t h e e x p r e s s i o n o f

R from ( R " )

i n the first i n -

t e g r a l w e have t o p r o v e t h a t

i.e.

and t h i s i s t h e formula of D i r i c h l e t (Theorem 1 . 1 3 o f Chapter 11).

b) ( P I ) follows from ( p > u s i n g i n t e g r a t i o n by p a r t s and t h e c o n t i n u o u s dependence i s a l s o immediate s i n c e ( 0 ' ) i m p l i e s

T H E O R E M 1.9. Foh e v e h y K E Guo R E G Auch t h a t (R")

R(t,s)

= Ix

PROOF. F o r e v e r y

-

UEG

theae e x i ~ t do n e a n d o n l y o n e

t

d,K(t,u)oR(u,s)

w e define

for a l l

3 ' U = gKU

by

s , t E [c,d].


98

By 1.4.12 and 1.4.4 t h e i n t e g r a l i s w e l l d e f i n e d s i n c e a n d s i n c e w e h a v e U s € G ( (c,d) , L ( X ) ) ; K t € G ( [c,d] ,L(X))

( r U I s € G((c,d] , L ( X ) )

II.1.14.a) w e have

II(rU)(t,s)IIs 1 +

sv

Is

4

II 7 UII i.e.

.

7UEG Hence a n e l e m e n t

[Kt]

6 1

REG

I(UII

by

a n d s i n c e w e have

it f o l l o w s t h a t

SVUCK3 IIUIIY

+

t h a t s a t i s f i e s (R")

point of t h e transformation

is a fixed

7 o f G . I n o r d e r t o prove t h e

e x i s t e n c e and u n i q u e n e s s of t h i s f i x e d p o i n t w e w i l l i n t r o d u

ce a norm i n G e q u i v a l e n t t o i t s n a t u r a l norm a n d show t h a t w i t h r e s p e c t t o t h i s new norm 7 i s a c o n t r a c t i o n . L e t us t a k e X > O ; f o r U E G we define I(U(IX

dup{((U(t,s)e-XIt-slI(

I

s , t € (c,d)l;

it i s immediate t h a t w e h a v e I I U I I X < IIU(1 s e X(d-c) II U l l h , h e n c e t h e norms 1) 1) and 1 ) I I X on G a r e e q u i v a l e n t . W e w i l l now prove t h a t t h e r e e x i s t s

X> 0

7

such t h a t

i s a contraction;

i t i s enough t o p r o v e it f o r t h e linear t r a n s f o r m a t i o n where

(~oU>(t,s>=

J:

duK(t,a)oU(a,s),

L e t u s f i n d a n u p p e r bound for t a k e 6 > 0:

s , t E (c,d).

Il(CI',U)(t,s)e

1) For

It-sl 4 6

w e have

For

lt-sl 2 6

l e t us suppose t h a t

2)

tt6 (s s d

5

-1 I t - s

c <s 6t-6;

t h e c a l c u l a t i o n s are a n a l o g o u s . W e have

II1 ; w e

if


99

and l e t us f i n d u p p e r bounds f o r t h e s e i n t e g r a l s : a)

s

b)

Hence

Hence w e have

Since SV6 [K]

then

K

< 1

s a t i s f i e s (SVuo) w e may t a k e 6 > 0 s u c h t h a t and a f t e r w a r d s w e t a k e X > 0 s u c h t h a t

i s a contraction of

6

I1 Ill

REMARK 6. I n t h e case o f t h e example (L) w e w i l l show i n t h a t f o r a l l s , t E (c,d) w e have R(t,s)E Isom X ( i . e .

52

R(t,s) i s a b i c o n t i n u o u s l i n e a r i n j e c t i o n from X o n t o X) and w e have even R ( t , s ) - ‘ R ( s , t ) . I n t h e g e n e r a l case t h i s

i s n o t t r u e ; w e h a v e a l w a y s R ( t , t ) = I X E Isom X a n d i f K : (c,d)x[c,d) + L(X) i s a c o n t i n u o u s f u n c t i o n s o i s R ( b y Theorem 1 . 2 5 ) s

hence w e have t h e n

R(t,s)E Isom X

buddicien.tk!g ctobe. I n g e n e r a l however

n o t be i n j e c t i v e .

doh t

R ( t , s ) E L(X)

and

may


100

E X A M P L E . I n o r d e r t o prove t h a t

R ( t 1 ,t0 1 i s n o t i n j e c t i v e it i s enough t o show t h a t t h e r e e x i s t s an x # 0 s u c h t h a t R(tl,to)x = 0. I f we define y ( t ) R ( t , t o ) x , t E c d , then y satisfies

($1

r t

and w e have t o prove t h a t y ( t l ) = 0 . We t a k e X = R , to 0 and c o n s i d e r t h e e q u a t i o n )a,b( = )-n,n(,

i t s solution is the function 71 a t tl 2 .

y(t)

x

1,

c o s t which h a s a z e r o

D - W e w i l l now b e g i n t h e proof t h a t t h e r e s o l v e n t uo in G

is

.

P R O P O S I T I O N 1.10. 1 6

TUE G'

'2Vehf.j UE Gu w e have denoted t h e than6 dohmation dedined b y

whehe 7

(JU)(t,s)

KEGUo,

= Ix

-

doh

j:duK(t

s,t

,c?)oU(~,s)

E

P R O O F . I I . 1 . 1 4 . a ) i m p l i e s t h a t f o r e v e r y S E [c,d) ( 7 U I s € G( [c,d) ,L(X) ; 7 U i s bounded s i n c e

hence every

K

(c,d].

w e have

IlTUll 6 1 + SV['K] . I I U ( \ . Fram 1 1 . 1 . 4 i t f o l l o w s t h a t for w e have (yUltC SV( (c,d) ,L(X) 1. We s t i l l

t E [c,d]

have t o prove t h a t ' f U is u n i f o r m l y of bounded s e m i v a r i a t i o n as a f u n c t i o n of t h e second v a r i a b l e : by ( 9 ' ) of §1 of Chapt e r I I w e have

hence

SV’

[TU]

O

and

6 > 0

for e v e r y

UEGU

we

de-


102

where, w e r e c a l l , SV(') [ Z ( t , s > ]

denotes t h e semivariation

calculated with respect t o the variable

It i s immediate t h a t

PROPOSITION 1 . 1 2 .

In

G"

111 111X,6

s.

i s a norm on G

t h e nohm

111 111

U

M

~

U

.

111 111X,6

Uhe

equiuaeent.

P R O O F . We w i l l show t h a t f o r e v e r y 1

7

IIIuII1~~4 IIIUIII

6 4e

UE

X(d-c)

a ) I t is immediate t h a t

GU

w e have

111 UII 1 1,6

l l U l I X y 6 < I1UII s e bl) Lemma 1.11 i m p l i e s t h a t

X(d-c)

llull

and a n a l o g o u s l y

hence

and t h e r e f o r e

11 Ulll g 6

4 4 lllUlll

.

b 2 ) Again by Lemma 1.11 we have

< svA Y 6 For a l s o have

[uIeX'd-c'

SV[

, )rut]

t+6 d

+

h(d-c) IIUIIX,6e

w e have an analogous m a j o r a t i o n ;

we

S T J E L T J E S - I N T E G R A L EQUATIONS

103

and t h i s completes the p r o o f .

THEOREM 1.13.

Tkehe

exist

a c o n t h a c t i o n o d G~

Ill

X

> 0

and

d > 0

buck t h a t

7

ib

111Xy6

P R O O F . Obviously it i s enough t o prove t h e same r e s u l t f o r the l i n e a r transformation wherc f o r U E G' w e define

Ist

(JoU)(t,s) =

daK(t,5)oU(a,s)

L e t us f i n d an upper bound for I

-

We b e g i n w i t h

a ) For

It-s

14

6

t , s E [c,d].

)11~oulllX,6-

1170ul156. w e have

11 ('TOU> ( t, s ) 11

(1

t

d a K ( t , a > o U ( a, s )

11
0 such t h a t 1 7SV6 [K] < 7 ; w e f i x ' such a 6 > 0 , t&w there e x i s t s 2, > 0 s u c h t h a t 5e-"SVU[K] < 1 , hence 7, i s a c o n t r a c t i o n i n

G"

111 IIIX ,6

The p r e c e d i n g theorem i m p l i e s t h a t t h e r e s o l v e n t R , sol u t i o n o f ( R " ) , i . e . t h e f i x e d p o i n t o f 7 , i s an element o f G'; however w e want t o prove t h a t REGUo; for t h i s purpose we w i l l show t h a t Guo i s a c l o s e d subspace o f G' and t h a t ?'GuoC

Guo

.

P R O P O S I T I O N 1.14. Guo PROOF. L e t

U

i b

u c t o b e d bubebpace

be i n t h e c l o s u r e o f

t h e r e e x i s t s U E E Guo s u c h t h a t 111 t t t E c d . w e have SV(c,d) [U 4

(4

-UEl

06

G".

Guo; t h e n f o r e v e r y E > 0 U-UEII( < E hence f o r e v e r y E.

Let

6 >0

b e such t h a t


106

sv6[uE3E

i.e.

t h e n we have for a l l

sv (s-6 ,s+6) Cut] hence

for a l l

y s - 6 , s + 6 ) [U;]&E

6

s , t E [c,d]

sv (s-6

s , t E (c'd];

that

,st6)

UEGUo.

P R O P O S I T I O N 1.15. T h e t ~ a n s , j o h m a t i o n 'Y o d GU t a k e b

Guo

P R O O F . O b v i o u s l y i t i s enough t o p r o v e t h e same r e s u l t

for

cue.

into

7,.

If

UE

Guo

w e h a v e by (8’) o f C h a p t e r I1 t h a t

+ sv

hence t h e r e s u l t s i n c e

K

and

U s a t i s f y (SVuo).

By Theorem 1 . 1 3 and P r o p o s i t i o n s 1 . 1 4 and 1 . 1 5 w e h a v e immediately

C O R O L L A R Y 1.16. T h e h i x e d p o i n t R 06 7 i d i n Guo, i . e . . d o h K E Guo t h e m i b o n e and onLy o n e R E G U o t h a t b a t i A & a (R*); we h a v e a ) a n d 6 ) 0 6 Theohem 1 . 8 .

rK

Given K E GUo l e t u s d e n o t e f o r a moment by the t r a n s f o r m a t i o n - d e f i n e d by K , a n d by RK i t s r e s o l v e n t i . e . t h e fixed p o i n t of

K. L e t u s p r o v e t h a t t h e mapping K E

Guo

c,R K E

Guo

i s continuous. b e t h e norm of Guo s u c h t h a t 7 I n d e e d : l e t 111 111 i s a c o n t r a c t i o n i n t h i s norm w i t h c o n t r a c t i o n c o n s t a n t cK g i v e n by ( y ) of Theorem 1 . 1 3 cK = 5e-"SVU[K]

t

7SV6[K]

< 1

.


107

I t i s immediate ( C f . t h e proof o f P r o p o s i t i o n 1 . 1 4 ) t h a t i f

KEGUo t h e n f o r a l l i? s u f f i c i e n t l y c l o s e t o K w e a l s o have c~ < 1 , hence t h e r e i s a neighborhood V o f K such

(yk)kEv

t h a t t h e f a m i l l y of c o n t r a c t i o n s f o r every

UE

GUo

KE

Guo

111 2'KUlll

-

Y(to)

t

satis-


108

f2 hence

and t h e same a p p l i e s t o

f l = f2

f . Hence by

subtraction w e obtain t h a t

I, t

du (K2(t ,a)-Kl(t ,u)) . y ( a )

for all

YE T E

XEX,

G( [c,d) , X )

t

and e v e r y

(to,t) and

y

-

for a l l b i t r a r y , or by Remark 2 , w e have Kl(t,-c)

XEX

i s a r b i t r a r y imply

K2

-

to i s ar-

i . e . w e proved t h a t

K1,

Guo R E Guo e l e m e n t s , i. e . duch t h a t

when hebXhicted to nonmatized K ( t , t ) Z O , i6 i n j e c t i v e .

-

= 0.

T E ( t o , t ) ;s i n c e

C O R O L L A R Y 1 . 1 9 . The continuoud mapping

E

If w e t a k e t h e n

[K2(t,r)-Kl(t,r)]x

The n o r m a l i z a t i o n and t h e f a c t t h a t K2(t,T)

[c,d).

= xfTYt)x w e o b t a i n

[K2(t,t)-Kl(t,t)JX

that

E

= 0

KE

W e w i l l now complete t h e p r o o f o f Theorem 1 . 5 .

T H E O R E M 1.20. Given

KE

duch Xhut

Guo

K(t,t)

5

0 and

R

iXb

h e d o e v e n t , we have

t

(R*) (R**)

R(t,s)

= Ix + ~ s R ( t , u ) o d u K ( o , s ) f o r a l l = R(t,s)

.K(t,S)

-

s,t€(c,d)

Ix + lstd0R(t , ~ ) o K ( u, s >

for a l l

s,tE

d . (c 4

P R O O F . L e t u s f i r s t remark t h a t t h e i n t e g r a l i n (R,) makes sense because R t ~SV( (c,d ,L(X) 1 ( s i n c e R E Guo by Coroll a r y 1.16) and Ks€ G ( c , d , L ( X ) ) . By ( R * > w e have I t R ( t ,o)odaK(u,s> =

’S

K(t,s)

(D)

-

= K(t,s)

Ist[ -

0 j:[Ix

-

I

lutdTK(tyT)QR(',o) oduK(u,s)

[dTK(t,r)oR(r,o)

I

od,,K(U,s)

(.D1

=

t Js d T K ( t , r ) o [ JsrR(r,a)oduK(o,s)

1

=

=

STIELTJES-INTEGRAL EQUATIONS (D)

where i n

w e applied 11.1.13.

I

109

Hence w e proved t h a t t h e

function T

S(T,S)

Ix + ~ R ( ~ , u ) o d ~ K ( u , s )

s a t i s f i e s the equation = Ix

S(t,s)

-

[dTK(t,r)oS(r,s),

i.e.,

it

R ; t h e r e f o r e w e have S = R , i . e . ( R i t ) . We g e t (R,tb) from ( R * ) u s i n g i n t e g r a t i o n by p a r t s . ( R 1, whose o n l y s o l u t i o n i s

T H E O R E M 1.21. G i v e n a n d o n L y one

K€GU0

(R")

= Ix

R(t,s)

-

RE Guo with

i:

w i t h R ( t , t ) z Ix t h e u K ( t , t ) 1 0 buch t h a t

duK(t,u)oR(u,s) R E Guo

a n d t h e ( n o n Lineah) m a p p i n g

for all

M KE

Guo

Lb

s,tE

one

[c,d)

i b cowXnuoub.

P R O O F . The u n i c i t y o f K s a t i s f y i n g ( ? ) f o l l o w s from Coroll a r y 1 . 1 9 . L e t us prove i n i t i a l l y t h a t t h e r e i s one and o n l y one

KE

Guo w i t h

K(t,t)

2 0

such t h a t ( % , I

o r , equivalent-

lY

(&*)

i . e . such t h a t

mation

= R(t,s)

K(t,s)

Q,

K

-

Jc

+ ~stduR(tyu)oK(u,s)

i s t h e f i x e d p o i n t of t h e a f f i n e t r a n s f o r UEGUo w e d e f i n e

where f o r

The l i n e a r p a r t o f R h anabgcms to To d e f i n e d i n Theorem 1 . 1 3 and P r o p o s i t i o n 1 . 1 5 ; hence a p p l y t h e same c o n c l u s i o n s o f these theozems tD t k t r a n s f o r m a t i o n A ( t h a t i s , t h e analogous of C o r o l l a r i e s 1 . 1 6 and 1 . 1 7 ) . b) We s t i l l have t o p r o v e t h a t K d e f i n e d i n t h i s way, i . e . , s a t i s f y i n g trt, a l s o s a t i s f i e s (R* : u s i n g i n t e g r a t i o n by p a r t s i n (R")

we obtain


110

By (€$&I w e have

and it i s s u f f i c i e n t t o p r o v e ( F?") . T K ( t ,u ) od,R(

JS

Q

t

t

s1

t

[.(

s

U)

- 3 + Ju dT R ( t ,T 1

OK

( 'I ,U

I

1

0 duR ( 0 , s

).

If i n t h e s e c o n d i n t e g r a l w e u s e i n t e g r a t i o n by p a r t s f o r t h e f i r s t two summands a n d a p p l y 11.1.13 t o t h e t h i r d w e o b t a i n rt

hence i f w e d e f i n e

-%

S('I,S)

+ R('I,s)

- ['IK( JS

w e have j u s t p r o v e d t h a t -S(t,s)

-

Ix + R ( t , s ) =

'I

u 1oduR (a, s )

-I:

dTR(t,T)oS(T,s)

i . e . S satisfies t h e equation (R-1 whose o n l y s o l u t i o n (by p a r t a > ) i s K; hence S = K i . e . w e have (R**). QED I f w e d e n o t e by that satisfy

K(t,t)

that satisfy

R(t,t) r e s u l t s w e have

GZo E 0

=

t h e subspace o f t h o s e a n d by

$ y

GYo

KEG''

t h e subspace o f t h o s e

t h e n i f w e group the preceding

T H E O R E M 1 .22 - The mapping t h a t t o euehg K E G': abbociateb i t a hebolvent REG? LA one t o one and b i c o n t i n u o u b dhom

onto GYo; i n evehy one 06 t h e equationb ( R 1 and ( R * I Y K detehmined u n i q u e l y R and R detehmined u n i q u e l y K. 2%

GE0

T H E O R E M 1 .23. I I Rs i b d i b c o n t i n u o u b t o t h e l e d t ( I r i g h t ) a t t h e p o i n t t o n l y i d d o h home U E { s , t ) Ku i b d i b c o n t i n u oub t o t h e l e d t ( h i g h t ) a t t . 2 ) y , b o l u t i o n 06 ( K ) i d dibcontinuoua t o t h e L e d t ( I r i g h t ) a t t h e p o i n t t onLy i d f i b dibcontinuoub t o t h e L e d t ( h i g h t t ) a t t , oh d o h borne S E (to,tl) K s ( o I r R s ) i b d i b c o n t i n u o u b t o t h e l e 6 t (bight) a t t . PROOF. 1) By ( R * ) and by (15’) o f C h a p t e r I1 w e h a v e Rs(t-) hence

Ix

-

duK(t-,a)oRs(U>


111

and t h i s i m p l i e s 1). 2)

By ( p ' ) o f Theorem 1 . 8 w e have

t h e n (15’) of C h a p t e r I1 i m p l i e s t h a t

hence

.f(s) R; f o r

K

it f o l l o w s

P - W e w i l l now prove t h a t i f t h e k e r n e l

K

satisfies

which i m p l i e s t h e a s s e r t i o n i n 2 ) f o r from 1).

certain additional properties the same i s true f o r

R

and

reciprocally.

DEFINITIONS. W e d e n o t e by & 6( (c,d)X(c,d) ,L(X 1 ) t h e c l o s e d subspace of G formed by t h e c o n t i n u o u s f u n c t i o n s ( i . e . 6 = &([cad)X[c,d) ,Lo( = G U ( (c,d)X(c,d) , L ( X ) ) deU formed by t h o s e e l e m e n t s of n o t e s t h e c l o s e d subspace of G

Gu t h a t ar e c o n t i n u o u s f u n c t i o n s . GUo = 6 u r \ G u o . bCo de n o t e s t h e subspace o f t h o s e e l e m e n t s UE Guo t h a t have t h e property


112

(SF).

hence t h e c o n t i r u i t y of K f o l l o w s from (SVc) and I n o r d e r t o prove t h a t ECo i s a c l o s e d subspace of buo

it i s enough t o prove t h a t e v e r y element of

GCo

is

KEG

go

&

belongs t o

&"

GC0; i f

such t h a t

KE

K

of t h e closure

for e v e r y E S i n c e w e have

111 K-KE 111 < E .

7

0

there

the r e s u l t follows.

if

REMARK 8. I n t h e Appendix o f t h i s U

s a t i s f i e s (SVc) and (SVo>

T H E O R E M 1 .25. G i v e n 1 . 1 0 , doh eVehy

§

w e w i l l prove t h a t (SVUo>.

it s a t i s f i e s

K E S U 0 , w i . t h .the n o t a t i o n d

UG&

We have

qU€&.

06

Ptropob&n

PROOF. We have

by I.5,9 t h e f i r s t summand goes t o

0

if

t2

-

tl

since w e

have SV[Kt] < SV['K] f o r a l l t and K ( t 2 , a ) ----* K ( t l , a ) t h e c o n t i n u i t y o f K ; t h e second summand i s bounded by

s v t s l+ 2 1

[Kt2]

IIUll

which goes t o

0

if

s2

+ s1

since

s a t i s f i e s (SVuo); f o r t h e f o u r t h summand w e have analogous SV'[K]lIUs -Us21 1 which goes t o 0 i f s 2 + s1 s i n c e U i s c o n t i n u o u s .

r e s u l t ; t h e t h i r d summand i s bounded by

Hence w e have ( Y U ) ( t 2 ' s 2 ) _$ ( 7 U ) ( t l , s , > if (t2,s2) ( t l , s l ) , i . e . , 7 U is c o n t i n u o u s .

by

K


113

By Theorem 1 . 2 5 and P r o p o s i t i o n 1 . 1 5 w e have

COROLLARY 1 . 2 6 . T h e t h a n d ~ o h m a t i o n 7

GUo

tahed

GUo.

into

It f o l l o w s t h a t i f K E GUo t h e n t h e f i x e d p o i n t o f , i . e . , t h e r e s o l v e n t R i s i n G U 0 t o o and Theorem 1 . 2 1 shows t h a t r e c i p r o c a l l y i f KE Guo and i t s r e s o l v e n t i s i n

EUo t h e n

KE

buo.

W e define

& ':

&n G:O

and &

yo

= 6n G

yo

t h e n by

Theorem 1 . 2 2 w e have

T H E O R E M 1 . 2 7 . T h e mapping d e d i n c d i n Theohem 1 . 2 2 when bthicted

to & ':

&yo.

onto

id

i n j e c t i v e and b i c o n t i n u o u d dhom & ':

t h i s i s o b v i o u s f o r t h e f i r s t summand s i n c e

TU

i s a contin-

uous f u n c t i o n . For t h e second summand w e have

By

he-

(9’) and ( 3 ) of Chapter I1 t h e f i r s t and second summands

a r e bounded r e s p e c t i v e l y by SV [Kt2-Kt1]

which go t o (SVc)

and

0

[llUll

+ 2SVu[U]]

if

t g + tl

(SVuo).

and since

QED

By C o r o l l a r y 1 . 2 6 w e t h e n have

svIt, K

[K' st23

1 ' sv'

[u]

has t h e p r o p e r t i e s

114

S T I E L T J E S - I N T E G R A L EQUATIONS

COROLLARY 1 . 2 9 .

Zd

We define

&zo

KE GCo t h e n

&ConGEo

and

GUo i n t o GCo.

taheb

& ;o

& ’ O n

then in the same way as Theorem 1.27 one

G;O

;

proves the

&zo

THEOREM 1.30. T h e mapping d e d i n e d i n Theohem 1 . 2 2 when d t h i c t e d t o 6zo i b i n j e c t i v e and b i c o n t i n u o u n dhom to & FIo e

he-

on-

REMARK 9 . One can still impose other restrictions on K and prove that R satisfies the same restrictions and reciprocal-

ly. For instance, we denote by

-

GBYUo GIBYUo( (c,d] X (c ,d) ,L(X))

the space of all functions U: (c,d)X(c,d) L ( X ) that are regulated as a function of the first variable and which as functions of the second variable satisfy

(BVuo) V(s-6

-

F o r every

,s+6) [Ut]

0

.

6 > 0

such that

if and only if R E G ~ ~ Y O ; Then we have that KEG~Y:' this correspondence is bicontinuous with respect to the obvious natural norms. The same is true if we consider the restriction to the subspace b6Yu0 of those functions of G63YUo that are continuous. In an analogous way we can define eevco,etc..

REMARK 1 0 . As we explained at the beginning of this item B, we did reduce the study of the solutions of the equation(K)

in ]a,b( to their study in closedintervals (c,d)cJa,b(. In this way all the results for the existence and unicity of the solutions of (K), of the resolvent etc. are true for ]a&[. The topological results, i.e., the results that use the topology defined on the spaces of functions over (c,d) or [c,d)X[c,d> are easily extended to ]a,b[ if we introduce in the corresponding spaces the topology defined by the corresponding seminorms on the intervals (an ,bn] where anfa we consiand bn+b. F o r instance in GUo(] a,b[X] a,b ,L(X)

[


115

d e r t h e l o c a l l y convex t o p o l o g y d e f i n e d by t h e sequence of seminorms

((1 111 (a

of t h e s p a c e s n 'bnl

Guo( [an,bn]X(anybn) ,L(X)). The l o c a l l y convex s p a c e s w e o b t a i n i n t h i s way a r e F r e c h e t s p a c e s and it i s immediate t h a t t h e c o n t i n u i t y and b i c o n t i imply t h e (c,d) c o n t i n u i t y and b i c o n t i n u i t y i n t h e c o r r e s p o n d i n g theorems on

n u i t y of t h e mappings i n t h e theorems on

APPEND1 X

THEOREM 1.31. 16

K:

(c,d]X(c,d)

pehtieb

t'lltSVIKt-K

tl

+ L(X)

]]

= 0

doh

batib6ieb

eueny

t h e pho-

t l € [c,d).

t+t

P R O O F . For e v e r y

a) V6

6 >O

we c o n s i d e r t h e f u n c t i o n

i s upper semicontinuous i . e . i f we have < c

V6(tl'S1)

( t l y s l ) t h e same i s t r u e f c r a l l p o i n t s (t,s> of a neighborhood of ( t l y s l ) . Indeed: i f f o r some p o i n t

t h e n by ( S V O ) t h e r e e x i s t s

E~

>0

such t h a t


116

> 0 such t h a t f o r It-tll < E~ €[K2 t ] < c ; i f ls-sll < E ~ lt-tll , < E*

and by (SVc) t h e r e e x i s t s we have sv[s1-6-E1 ,S1+6+E1]

w e have

(s-6 , s + 6 )

c

( ~ ~ - 6 ,s1+6+c1) - c ~

b ) By (SVo) we have

i . e . given w e have

E

> O

there exists

V6(t,s) c

for all

E

if

V6(t,s)+0

hence by t h e theorem of D i n i

V6


6+0

f o r every ( t , s ) ;

converges u n i f o r m l y t o 6€> 0

0

such t h a t f o r 0 < 6 < 6 €

s , t E (c,d>.

Q.E.D.

REMARK 1 1 . I n an analogous way one p r o v e s t h a t

Vg

i s lower

s e m i c o n t i n u o u s , hence c o n t i n u o u s .

0 2 - I n t e g h o - d i , j , j e & e n t i a l equationd and hahrnonic

0peha.tOhb I n t h i s 8 w e w i l l s t u d y t h e example A o f 8 1 i . e . t h e i n tegro-differential equation y(t>

(L)

-

Y(S)

+

I:

dA(U)*y(O)

y,f E G()a,b(,X),

where

i s c o n t i n u o u s and satisfies (SVO)

and

f(t>-f(S)

[A)

]a,b (

A E ~ S V ~ ' ~ ( ) ~ , ~ [ , L ( (Xi .)e) .

A € SV( (c,d) , L ( X ) )

Cim 6 + 0 sv(s-6

s,tE

0

f o r ' every

f o r every

A

[c,d)c)a,b[)

s€)a,b(.

,s+6)

W e d o n ' t know if e v e r y element has t h i s p r o p e r t y ; i f

X

AEQSVl°C()a,b(,L(X)) i s r e f l e x i v e t h i s is t r u e .

We d e n o t e by A = J \ ( ) a , b ( , L ( X ) ) AE6SVioc(]a,b(,L(X))

t h e s e t of a l l

t h a t s a t i s f y (SVo).

A - We r e c a l l t h a t ( L ) i s a p a r t i c u l a r i n s t a n c e of ( K ) from 8 1 , w i t h K ( t , s ) A ( s ) or K ( t , s ) = A ( s ) - A ( t ) , if K i s normalized ( i . e . K ( t , t ) Z 0 ) and t h e r e f o r e a l l t h e r e s u l t s of P 1 a p p l y t o (L). K d e f i n e d by A o b v i o u s l y h a s t h e prop e r t i e s (SVuo> and (SVC) on e v e r y i n t e r v a l ( c , d ) c ) a , b [ (see Theorem 1 . 3 1 ) and by Theorem 1 . 3 0 w e have


&yo

RE

= &Po()a,b[X)a,b(,L(X)),

is t h e resolvent associated t o

R

where

D E F I N I T I O N . For U:

117

)a,b(X)a,b(

4

L(X)

A. we c o n s i d e r t h e

following properties:

For e v e r y (c,d) c ) a , b e x i s t s 6 > 0 such t h a t

[

-

(SVuo)

SV (SVo) (SVc)

b-&,t+6)

s

[Us]

-

L i m SV(t-6,t+6)[Us]

-

For e v e r y

E

= 0

and e v e r y

for a l l

E

> 0

there

s , t E (c,d>

for all

.

s,tE)a,b[.

6+0

[c,d]c]a,b[

SE

for all

w e have

d . Ic 4

The p r o p e r t i e s (SVuo), (SVo) , (SVc)

are t h e analogous f o r t h e f i r s t v a r i a b l e of U of t h e p r o p e r t i e s ( S V U o ) , ( S V O ) , (SVC) which a r e f o r m u l a t e d w i t h respect t o the 2nd v a r i a b l e of U. I n a n a n a l o g o u s way w e d e f i n e ( S V u > , e t c . . The fundamental r e s u l t s o f t h i s 0 are c o n t a i n e d i n t h e Theorems 2 . 1 and 2 . 3 .

T H E O R E M 2 . 1 . G i v e n AEQI i . e . A€.&SV1oC()a,b(,L(X)) d u L i b d y i n g (SVo) we h a v e : I - T h e h e i b one and o n l y one R E 6, t h e h e b o h e f i t 0 6 A, buch t h a t

(R")

1 - R

(E*) I1

Ix

R(t,s)

-

R

Ix.

11'

-

R

111

-

Foh

equation

= R(-c,s) i.e.

RE&;' I

I:

bazibdieb

R(t,s)

R(t,t)

-

datibdieb

evehy

-

dA(u)oR(u,s)

i:

doh a l l

dA(u)oR(o,s)

bUti4dieb

(SVc)

doh

all

s,T,tE)a,b[.

a n d (SVuo), a n d ,

(SVuo) and (SV 1.

toE)a,b(,

s,tE)a,b[.

fEG(ga,b[,X)

and

XEX

the


118 y(t)

(L)

-

+

Y(S)

I:

dA(o).y(a) = f ( t )

- f(s)

hub one and o n l y one b o k u k i o n y € G ( ) a , b ( , X ) y ( t o ) = x; t h i b b o l u t i o n i b g i v e n b y R(t,to)x +

y(t) (P)

buch t h a t

i:,

R(t,s)df(s)

and dependb c o n t i n u o u b e y on f and x (and A ) ; y and oney i d f i b c o n t i n u o u b . I V - R(t,T)oR(-c,s) = R ( t , s ) and R ( T , t ) R(t,T)-’

tinrtoub i d s ,T ,t

V

1

Foh

-

a,b

VI - R

we have

u,v~]a,b[

:1

-

A(v)

Ist

R(t,-r)dA(r)

=

sE)a,b(.

doh

alL

dtR(t,s)oR(s,t)

bdiddieb

(Ra)

R(t,s)

= Ix

(R,)

R(t,s)

= R(t,a) +

-

PROOF. I and I11 follow.

doh a l l

R ( t , r ) d A(T)

s,tE)a,b(

doh ale

S , U , ~ E

immediately from t h e analogous

s u l t s o f 81 ( s e e I and I11 of Theorem 1 . 5 and 1.3.b);

lows from I ; S E

con-

(. A(u)

do& any

i b

?

refol-

I1 w a s proved a t t h e b e g i n n i n g of t h i s i t e m . 11 : L e t us t a k e k , d ) ~ ) a , b ( , c c t < t 2( d and 1 (c,d). By (R*) w e have

sv h By 1 . 5 . 2

t 2 1 [R,]

SVp:,t2)[

and 11.1.9 w e have

[dA(oloR(o,s)

I.

STIELTJES-INTEGRAL EQUATIONS If w e t a k e

[ t l y t 2 )= [ t - & , t + 6 )

by

and t a k e

RS

Rs+&-Rs

h e n c e (SVc) s i n c e

119

w e p r o v e ( S V o ) . If w e replace

(tlyt2)

we obtain

(c,d)

i s c o n t i n u o u s . By Theorem 1 . 3 1 w e

R

have

then (SVuo). IV:

(R")

t

point

(R")

of t h e s o l u t i o n of

T

s. A t the point

at the point

ue

R(t,s)

and ( L ) show t h a t

i s t h e value at

the

which t a k e s t h e v a l u e

Ix

t h i s solution takes t h e val-

R ( T , ~ ) . On t h e o t h e r hand i f w e a p p l y ( p ) t o f u n c t i o n s

w i t h v a l u e s in

f : 0 , to = T

L(X), with

x = R(T,s)

and

we

see t h a t R ( t , T ) o R ( r , s ) i s t h e value a t t h e point t of t h e s o l u t i o n o f (A") which t a k e s t h e v a l u e R ( - c , s ) a t the point T. Hence t h e f u n c t i o n s

t s a t i s f y t h e same e q u a t i o n R(.r,s),

at

and t c--j R ( t , - c ) a R(.r,s) and t a k e t h e same v a l u e ,

R(t,s)

I-+

(E")

T. By t h e u n i c i t y

o f t h e s o l u t i o n w e have

R ( t , T ) ~ R ( T, s >

R ( t ,s).

= t i n t h i s e q u a l i t y and i f w e recall t h a t R ( t , t ) = Ix w e g e t R ( t , ' c ) o R ( T , t ) = I x j a n a l o g o u s l y w e have R(T,t)oR(t,.r) = Ix hence R ( T , ~ ) R ( t , - r ) - ' . If w e t a k e

V:

s

If w e Apply s u c e s s i v e l y (R")

fV

rV

-Iu

11.1.9 and I V w e h a v e

r

1

V

=

,

dA(t)oR(t,s)oR(s,t)

=

-Iu"

d A ( t ) = A(u)

-

A(v).

V I : I t f o l l o w s from I V of Theorem 1 . 5 ( w e r e c a l l t h a t K i n I V o f Theorem 1 . 5 i s n o r m a l i z e d and t h e r e f o r e w e h a v e t o

take

K(u,s) = A ( s )

-

A(u)).

B - O b v i o u s l y ( L ) d o e s n o t change i f w e r e p l a c e A by A + c , where c E L ( X ) ; h e n c e w e may f i x a p o i n t o E ) a , b ( and s u p p o s e t h a t A(;) 0. W e write = { A € & A(;) = 0 ) .

A,

W e s a y t h a t a mapping

I

R: ]a,b(X)a,b[ 4 L(X) i s huhR s a t i s f i e s (svUo),tsvC),

monic o r an hamtonic opeautoh i f


120 (SVuo), (SVc) (0)

R(t,t)

and

= Ix, R ( t , T ) o R ( ’ I , s )

= R(t,s)

for all

(.

s , ~ ,E t) a , b Then w e h a v e o b v i o u s l y R ( . r , t ) R(t,‘I)-’. W e d e n o t e by 3-1 = J - t ( ] a , b [ X ) a , b ( , L ( X ) ) t h e s e t of harmonic o p e r a t o r s .

T H E O R E M 2.2. 1 6

R: )a,b(X)a,b(

RE^ a n d

(SVo) t h e n

R

i b

-

-3

L(X)

batiddied

t h e heboLuent o d

0

A(u) = i d t R ( t , s ) o R ( s , t )

PROOF. R

(0)

and

A, whehe

,

0 6 Zhe paaticuLaa

t h e d e d i n i t i o n being i n d e p e n d e n t

all

SE

)a,b[.

i s c o n t i n u o u s as a f u n c t i o n o f t h e f i r s t v a r i a b l e

s i n c e i t s a t i s f i e s (SV 1. R i s a l s o c o n t i n u o u s as a f u n c t i o n 0

of t h e second v a r i a b l e because i f

( t , s n ) 4 ( t , s > t h e n by

( 0 ) w e have

,s)ll

IIR(t ,sn)-R(t

l1R(sn,t)-l-R(s

and t h i s e x p r e s s i o n g o e s t o z e r o when

-

n

-+

,t)-’Il OJ

because

R

is

c o n t i n u o u s i n t h e f i r s t v a r i a b l e and t h e mapping R(u,T)

R(u,T)-’

i s continuous. ilence w e have R s e & SVlo

() a ,b [, L ( X )

and

[

R‘E 6 (>a ,b ,L(X)


-

A ( u ) = j u0d t R ( t , s ) o R ( s , t ) i s w e l l d e f i n e d . By ( 0 ) w e h a v e 0

A (u

i d t [R ( t

,T

i . e . t h e d e f i n i t i o n of SE

]a,b[. From

O R ( ‘I,s >] O R ( s ,t 1 A

f d t R ( t ,T 1oR( T ,t 1

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


121

it f o l l o w s t h a t A s a t i s f i e s ( S V O ) . We w i l l now p r o v e t h a t R s a t i s f i e s (R") i . e . R i s t h e r e s o l v e n t of A and i s t h e r e f o r e harmonic. By 11.1.9 we have

-

QED

i s d e f i n e d by t h e

W e r e c a l l t h a t t h e t o p o l o g y on seminorms

A where

(c,d)

-

IIIA1ll [c ,d ] = IIAll

(cyd)

+

sv(cyd]

LA]

r u n s o v e r a l l c l o s e d s u b i n t e r v a l s of

i s a F r e c h e t s p a c e and

4,

>a,b(.,A

i s a c l o s e d subspace o f 4 .

d e n o t e s 3-1 w i t h t h e t o p o l o g y induced by bCo or i . e . w i t h t h e t o p o l o g y d e f i n e d by t h e seminorms

3jco

Guo

where, w e r e c a l l ,

We d e n o t e by

Gc0

t h e s e t of a l l

U : ]a,b[X)a,b(

4 L(X)

t h a t s a t i s f y (SVc) and (SVo) (and hence (SVu0) by a r e s u l t analogous t o Theorem 1 . 3 1 ) . d e f i n e d by t h e seminorms

On

bco w e c o n s i d e r t h e t o p o l o g y


122

= {UEGco

We d e f i n e &:o

the set

I

U(t,t)

J j w i t h t h e t o p o l o g y i n d u c e d by

Ix).

3Cc0 d e n o t e s

Gc0.

T H E O R E M 2 . 3 . On Jd t h e t o p o L o g i e b 06 3-tco and Jjc0 coinc i d e and t h e mapping AEJ; H R E 34 i b i n j e c t i v e bicon-tinuoud ( n o n Lineah) dhom t h e 6 i h d . t pace o n t o t h e hecond. P R O O F . We d e n o t e by

RA

t h e resolvent associated t o

A

and

R E 3.1 w e d e n o t e by AR t h e e l e m e n t o f 4 0 d e f i n e d i n The r e s u l t w i l l f o l l o w from t h e f o l l o w i n g f a c t s t h a t w e s h a l l prove s u c e s s i v e l y :

for

Theorem 2 . 2 .

A

1) For e v e r y A E J ; w e h a v e KAE Gco and t h e mapping KA i s o b v i o u s l y l i n e a r a n d c o n t i n u o u s .

&

GCo

2 ) KAE GCo I A € J,} i s a c l o s e d v e c t o r s u b s p a c e of and t h e mapping A c,KA is b i c o n t i n u o u s .

I n d e e d , w e have

{K€ Gc0

1

K(t,s)

= {KEGco

I

= K(o,s)-K(o,t)

@ t , s( K )

for a l l for all

Os(K)-Ot(K)

s,te)a,b(} = s,te)a,b(},

Q (K) = K ( t , s ) and Q o ( K ) = K ( 0 , a ) ; t h e and t,s t ,s a r e l i n e a r c o n t i n u o u s o p e r a t o r s and t h e r e f o r e t h e v e c t o r aO s u b s p a c e d e f i n e d above i s c l o s e d . The mapping A cj KA i s o b v i o u s l y one-to-one and c o n t i n u o u s ( b y 1)) hence b i c m t i n u o u s

where

by t h e i n t e r i o r mapping p r i n c i p l e . 3 ) The mapping

KE&Zo

i s i n j e c t i v e , bicon-

RE&;o

t i n u o u s and o n t o . I n d e e d , t h i s w a s p r o v e d i n Theorem 1 . 3 0 .

From 2 ) , 3) and Theorem 2 . 2 i t f o l l o w s t h a t 4 ) The mapping

t i n u o u s and o n t o . 5 ) The mapping

AEJ-

0

RENcO

&

RAEJ-\co

HAREA

0

i s i n j e c t i v e , biconi s i n j e c t i v e a n d con-

tinuous. I n d e e d , i n Theorem 2 . 2 w e saw t h a t t h e mapping i s i n j e c -

t i v e . L e t u s p r o v e t h a t i t i s c o n t i n u o u s . For w r i t e A1 A and A AR. We have R1

R1,RE

I-[

we


123

hence

which i m p l i e s

I n t h e same way one p r o v e s

hence

A

AAO

__j

6 ) The mapping

A1 AEA;

PROOF. a ) By 4 ) t h e mapping

if

R RA€

AEd;

"O

>Ico

i s continuous.

++ RAc 6 i s

b) We s t i l l have t o prove t h a t g i v e n ficiently close t o

> R1.

Al€J;

for

A1

continuous. AEA;

suf-

becomes a r b i t r a r i l y SVu ,[c ld) [R1-R] s m a l l . I n t h e proof o f 11’ of Theorem 2 . 1 w e s a w t h a t

"(c

,d]

ERsI ' ''(c ,d) CAIllKsII(c ,d)

and if we proceed as in 5) we have


124

-

by a > w e h a v e

A +A1

and t h e r e f o r e , s i n c e f o r

11 R1-RII (c ,d) it f o l l o w s t h a t

[Rl-R]

By 5 ) and 6 ) t h e mapping

+0

h e n c e b).

-

RAE>[co

AEd

0

i s injective,

b i c o n t i n u o u s a n d o n t o and w i t h 4 ) t h i s shows t h a t

31

7 ) On

t h e topologies of

>Ico

and >[co

coincide.

T h i s c o m p l e t e s t h e p r o o f o f t h e theorem.

-

13

Equationd w i t h tineah conbthaintd

I n t h i s 8 w e s t u d y t h e s o l u t i o n s of t h e s y s t e m ( K ) ,

(F)

(see t h e i n t r o d u c t i o n o f t h i s c h a p t e r ) when w e h a v e u n i c i t y o f t h e s o l u t i o n s and w e f i n d t h e Green f u n c t i o n . W e recall F i s c a l l e d a l i n e a r c o n s t r a i n t . I n A w e g i v e examples of t h e main l i n e a r c o n s t r a i n t s We s u p p o s e t h a t K i s conKE &uo) ; i n B w e make a p r e l i m i n a r a l g e b r a i c tinuous (i.e.

that

.

s t u d y where i t i s enough t o s u p p o s e t h a t

The a n a l y t i c

KE G u o .

r e s u l t s of C w i l l allow us t o transform t h e formulas of B

in

f o r m u l a s o f t h e Green f u n c t i o n t y p e (D and E l . A

-

I n what follows w e g i v e t h e main examples o f l i n e a r

c o n s t r a i n t s t h a t a p p e a r i n A n a l y s i s , i . e . , of o p e r t a t o r s

F E L[G()a,b( ,X) ,Y] 1

Fb]

-

F E L[G( (a,b) ,X) ,Y]

I n i t i a l conditions: we take

.

Y = X

and

to~)a,b(j

=

y(to). W e r e c a l l t h a t when w e have a l i n e a r d i f f e r e n t i a l equa-

t i o n of order

(N)

or

N[z]

n

:z ( n )

where, f o r i n s t a n c e ,

+

a l ( t ) z (n-l) ZE

t

O(")C)a,b(,Z)

... t

a,(t)z

and

= b(t)

6

(1

= cn

,

b ,aiE

with i n i t i a l conditions z(to> = then we take

C1’

X = Zn

z'(to) = and

C2'

yi(t)

..., z - ) ( t o )

a ,b ,Z I ,

= z (i-1)( t ), i = 1 , 2 , . .

[

. ,n,


125

and the n-order equation is transformed into the system

................. n

that is, of the form y’(t>

t

A(t)y(t)

= f(t),

y(to)

c.

Boundary conditions: we take Y X and ( a 4 ; 3 Ay(a) + By(b) where A , B E L(X). We recall that if we have the n-order equation (N) and boundary conditions

Fb]

2

-

where oij, B . . E L(Z), by the transformation given in 1 we =I get an example of the type 2. given

3

-

Periodicity conditions: we take )a,b( p > 0 (the p e h i o d ) we define Fb](t)

Fb]

4 5

5

-

We give

t E IR.

y(t+p) - y(t>,

Left discontinuity: We take Y (y(t,) ,y(t,-) 1

-

-

= IR, Y=GCR,X);

X 2 , to

€1 a,b [

and

Multiple point conditions (the Nicoletti problem): tl tmE)a,b( and A1 ,Am E L(X,Y); m FLY] E 1 Aiy(ti). i=l

,...

,...,

If for the n-order equation (N) we give Fi[z]

3

m

n

1 aijhz (h-l)(t.I ) 11 h=l j=

.

i=l,.. ,n, aijh L(Z)

the transformation of example 1 gives us an example of type 5.


126

-

6 t n E

(a,b)

Conditions at i n f i n i t e p o i n t s : W e g i v e a sequence

,n

= lYZy...

and u = ( u ~ ) s a~N , ~L ( XE, Y ) ) Fb] uny(tn). nc N

- I n t e g r a l c o n d i t i o n s : We g i v e

7

and

Fb]

(see B

1

o f 8 5 o f C h a p t e r I > and

a

SVoo()a,b(,L(X,Y))

:I f d a ( t ) . y ( t ) . a

-

8

I n t e r f a c e c o n d i t i o n s : We g i v e

A, A_, A+€ L(X,Y);

-

9

~ [ j r J:A - y ( t o - ) + equations: W e take

Integral

toE)a,b(

Ay(to) + A+y(to+). Y

= G()c,d[,Z)

A E G ( ) ~ , d ( , S V ~ o ( ) a , b ( , L ( X y z ) ) ) , F[y] ( t ) : (see (1.6.10)

B

and

and

duA(t,U).y(a)

>-

-

We w i l l now make a n a l g e b r a i c s t u d y of t h e r e s o l u h n o f t h e s y s t e m (K), ( F ) ; w e r e c a l l t h a t K E G U o and hence by I1 o f Theorem 1 . 5 w e h a v e R E Guo h e n c e f o r e v e r y R s € G ( ) a , b [ , L ( X ) ) . Given FE L[G()a,b(,X),Y] 1 . 6 . 8 w e have F = Fa + Fu. We r e c a l l t h a t

SE

w e have

and Fu continuoub mapping4 d h o m 3.1.

F , F,

PROOF. Given

1a 4 b

by

have naZuhaL e x t e n b i o n b ah Cineah G()a,b(,L(X)) i n L(X,Y).

U E G()a,b(,L(X))

f o r every

we define and h e n c e Flux] is XGX

F[U]x F[Ux]. We h a v e U x E G ( ) a , b ( , X ) w e l l d e f i n e d and depends o b v i o u s l y l i n e a r l y a n d c o n t i n u o u s l y on x . For q & ry t h e r e e x i s t (c,d)c)d,b( and c > O s u c h 9 and h e n c e t h a t q [F ( f )] G cq((f (1 'dl

I.

-

sCFCUIJ 4

i*e. t h e mapping proofs

.

U

Cq

which p r o v e s t h e c o n t i n u i t y of llUl!(c,d) 3 FLUJ. For Fa and F, w e have a n a l o g o u s

DEFINITION. F o r e v e r y = F t [ R ( t , s ) ] = F[Rs], By 3 . 1 w e h a v e

sc)a,b(

we define

J C r ( s ) Fa[Rs],

JU(s)

Js = J ( s ) =

Fu[Rs].

STIELTJES-INTEGRAL

and

3.3. Ja€ SVloC(]a,b(,L(X,Y))

60h

ewelry

SE ]a,b(

and

qE

127

EQUATIONS

rY.

P R O O F . By d e f i n i t i o n w e h a v e

[

q E ry, let ]a b c o n t a i n t h e q - s u p p o r t of f a ; by 9’ 9 ( 3 ) of 81 o f C h a p t e r I1 ( a n d Remark 8 of t h a t 5 ) w e h a v e

given

which i m p l i e s a l l t h e a s s e r t i o n s i f w e r e c a l l t h a t K i d c o n t i n u o u b we h a v e J = Ja and i s c o n t i n u o u s so is R s , h e n c e J u [ R s ]

3 . 4 . 16 PROOF. I f K

THEOREM 3 . 5 .

eq ua t i o n J(t)

P R O O F . By

R E Guo.

Foh t h e equation (L) J

- J(s)

ti?,>

-

I:

J(a)dA(a)

0

butibdieb

doh

0.

0.

the adjoint

s,tE)a,b(.

o f Theorem 2 . 1 w e have R(T,t)

-

R(T,s)

=

i.'

R(T,U)dA(a),

h e n c e , i f w e r e c a l l t h a t by 3 . 4 w e h a v e

J(t)

aLe

Ju

-

J(s)

j:da(T)o[

J = Ja, w e o b t a i n

/:R(r,u)-dA(o)

1

=

where w e d i d a p p l y ( 5 ) o f 51 o f Chap. I1 a n d Remark 1 of t h a t 8 .

We now d e f i n e


128

Kb]

and we w r i t e

f

K[y](t)

if

= f(t)

-

f(to).

We d e f i n e

THEOREM 3 . 6 . Given t h e b y b t e m ( K ) ,

(F) t h e 6 o t l o w i n g

PhOpCh-

t i e s ake e q u i v a l e n t : ( i 1 y : 0 i4 t h e o n l y b o l u t i o n 06 K[y] : 0 , Fly] = 0 . (iil F o h evehy C E Yo t h e b y b t e m K[yJ f 0 , F b ] = c had exactly one b oeution (iiil T h e m a p p i n g YE K - l t O ) Yo i d one-to-one

.

Fb]

onto. UOUb

(iul J ( t o ) :X -+Yo 1.

o n e - t o - o n e onto ( b u t n o t b i c o n t i n -

i b

+ ( i i ) .Given

Yo

w i t h K[yl] E 0 i f t h e r e were a y 2 # y1 w i t h K[y2] E 0 and F[y2] = c t h e n y = yl-y2 # 0 would be a s o l u t i o n of K[y] :0 , F [ ~ J = 0 i n PROOF.

(i)

c = F[YJE

contradiction t o (i). (ii) ( i )i s o b v i o u s . ( i i ) W (iii) i s immediate. (iii) ( i v ) . L e t yx be t h e s o l u t i o n of K[y] :0 , y ( t o > = x (by I11 of Theorem 1.5). Hence t h e mapping XE

X

yX€ K - l ( O )

i s a Banach s p a c e isomorphism. T h e r e f o r e t h e mapping Y E K"(0)

is one-to-one xE X

c-,Fry]€

Yo

o n t o i f and o n l y i f t h e composed mapping F[yx]

= F[R

x] t0

J ( t o ) x E Yo

i s one-to-one o n t o .

R E M A R K 1 . I n t h e case of t h e example (L) o f 1 2 w e may t a k e as t any p o i n t s E ) a , b [ and t h e n t h e p r o p e r t i e s above a r e a s t i l l e q u i v a l e n t t o t h e f o l l o w i n g ones: (iv') For every s E ) a , b ( , J ( s ) : X + Y o i s one-toone o n t o .

-


(v ) There exists is one-to-one onto.

-

s~)a,b(

such that

129

J(s):

X

+

yo

NOW ON W E SUPPOSE T H A T THE E Q U I V A L E N T P R O P E R T I E S O F THEOREM 3.6 ARE S A T I S F I E D FROM

F o r every

t E) a,b(

we define

j(t) = R(t,tQ)oJ(to)-l: 3.7. a ) j(t>

t i ve.

6 ) J(t)

i d

c ) z(t)

i d

Yo + X.

i f l o t COntiflUOUd i f l g e n e h a t ) . i n j e c t i v e i d a n d onLg . i d R(t,to) i d i n j e c i d Cineah

carntinuoub i d J(to)-l i 6 continuous. d ) 16 J(to)-l i d c o n t i n u o u d , Yo i d a Banach dpace and i d c e o d e d i n Y. el I n t h e example (L) 0 6 2 2 we have R(t,s) = J(t>-i J ( s ) a n d S(t) J(Z)- . 4 ) I n t h e exampee (L) 04 2 2 , j(t) i d b i j e c t i v e and id 3(t> id cona2nuoud d o h dome t )a,b( it i d c o n t i n u o u d doh evehg t E)a,b(. PROOF. a > , b) and c) are obvious by the definition of

z(t).

d) If J(to)-l is continuous then J(t ) is bicontinous, 0 hence Yo is isomorphic to the Banach space X and therefore complete, hence closed in every separated LCS. e) By Remark 1 for every tE)a,b( there exists J(t1-l and in order to prove that R(t,s) = J(t)-loJ(s) it is enoupj~ to show that J(t)oR(t,s) J(s): by 0 of Theorem 2.1 we have

J(s) = F ~ [ R ( T , s ) J = FTIR(T,t)oR(t,s)]

= F, [R(T ,t ) ] oR(t

,s

= J( t)oR(t , s 1.

The second assertion follows from

f) follows immediately from el.

=

130

STIELTJES-INTEGRAL 3.8.

Foh evehy

-

EQUATIONS

t h e dunction

csYo tE)a,b(

i b a e g u L a i e d ( c o n t i n u o u d id

z ( t > c EX

.LA c o n t i n u o u d ) ,

K

z(t>c = R(t,to)J(to)-lc; followsfrom t h e fact t h a t the f u n c t i o n (continuous i f K i s continuous). P R O O F . We have

T H E O R E M 3 . 9 . T h e dunc-tion

a) ?(t>c- j(to)c CE

Yo, t E a , b ( . b ) Ft[j(t)c]

PROOF. W e have 1 . 5 w e have

= c

t

:tI

5:

had t h e

hence t h e r e s u l t t0

doLLowing p h o p e h t i e d :

d,K(t,o).j(a)c

d o h euehy

-

= 0

d o h euehy

c€Yo. by ( R " )

;(t>c= R(t,to)J(to)-'c;

R(t,to)

i s regulated

R

t Ix t I t d u K ( t , u )

o f Theorem

R(u,to)

0

0

and i f w e a p p l y t h i s t o

we get a).

J(to)-'c

BY Ft[z(t.)c]

w e have b )

= Ft[R:R(t , t o ) * J ( t o ) - l c ] =

.

C O R O L L A R Y 3 . 1 0 . T h e d o L u t i o n yc 0 6 K[yl : 0 , F[y] c€Y0 i4 g i v e n b y y c ( t > j(t)c, t ~ ) a , b ( .

b ) T h e Lineah m a p p i n g

t inuoud id a n d onLy id i d bicontinuoud)

.

CE

Yo

G()a,b(,X)

r-,

J ( t o ) - lLA continuoud

= {(f,c)EG()a,b(,X)XY such that

'K,F

{ ( g , c ) E G()a,b(,X)XY such t h a t

13 y ~ G ( ) a , b ( , X )

K[y]

K[y]

f,

I

F[y]

c}

YE G(]a,b(,X)

g,

Fry]

i d con-

( a n d hence J ( t o )

DEFINITIONS 'K,F

c whehe

= c)


131

I n t h e case o f t h e example (L) of S 2 w e w r i t e

Given

(f,c)E S

K,F o f Theorem 1 . 5 w e h a v e

if

= f

K[y]

and

Fry]

= c

SL,F and by ( p )

r t

For

fEG()a,b(,X)

we define

'

t0

tE-)a,b[; F(f)

by b) o f 1 1 . 1 . 1 4 w e h a v e

i s w e l l d e f i n e d . I f we a p p l y c = Fry]

? ~ G ( ] a , b [ , X l , hence F t o (1) w e o b t a i n

= J(to)y(to)

t

F[f],

hence y ( t o ) = J(to)-l[c

-

F(?)]

and i f w e r e p l a c e t h i s v a l u e i n (1) w e o b t a i n y(t> = R(t T h i s p r o v e s t h e f i r s t p a r t of

T H E O R E M 3.11.

Foh euehy ( f , c ) E S K,F K r y ] = f , Fry] = c i b g i v e n b y u

khe b o e u t i o n y

06

(2)

b ) ReciphocaLLy, i d

(f,c)E G()a,b(,X)XY

c - F ( f ) € Yo t h e n t h e b y s t e m K[y] = f , F[y] y g i v e n by ( 2 ) ( h e n c e ( f , c ) E SKyF1. Proof o f b : immediate s i n c e w e may

LA s u c h t h a t c hub a b o t u t i o n

"go back" t h r o u g h

t h e t r a n s f o r m a t i o n s w e made i n t h e proof of a ) .

R E M A R K 2. I n ( 2 ) w e c a n n o t w r i t e j(t)[c-F(f)]

J ( t > c-

because, i n g e n e r a l , w e d o n ' t have and

j{t)FC?) If

K

J(t)F(g)

c , F ( ? ) E Yo

hence

z(t)c

are n o t d e f i n e d .

i s c o n t i n u o u s so i s

R

h e n c e by b ) of 11.1.14


132

i s c o n t i n u o u s if g by 3 . 8 w e t h e n have

i s c o n t i n u o u s and t h e r e f o r e

F(g)=Fa(g);

THEOREM 3 . 1 2 . Let K b e c o n t i n u o u b . a) F o h ( g , c ) E S' t h e b o t u t i o n Y E G()a,b(,X) 06 K,F K [ ~ ] = g , Fly] = c i d a conttinuoud d u n c t i o n and i d g i v e n b y (3)

y(t>

It

t

=

R ( t , a ) d g ( a ) + j(t)[c-l,bda(T)[

jTR(.r,o)dg(u)]]. t0

0

b ) R e c i p h o c a L L y id

-

c

-the d y h t e m

( g , c ) E ~ ( ) a , b ( , X ) X Y i d duch t h a t

0

= g , F[y]

K[y]

I

] ~ R ( ~ , ~ ) d g (EoYo )

/:da(T)[

= c

had a d o e u t i o n g i v e n b y ( 3 ) .

U s i n g i n t e g r a t i o n by p a r t s i n (1) w e o b t a i n

I:

d u R ( t ,a) * f ( a ) .

f ( t ) + R ( t , t o ) ~ y ( t o ) - f(to)]-

y(t>

(4)

0

t For

w e define

fEG()a,b[,X)

r(t)

ItdUR(t,o).f(u), 0

and a b , and by a ) o f 11.1.14 w e have ?EG()a,b(,X) f i s c o n t i n u o u s if K ( a n d h e n c e R ) i s c o n t i n u o u s ; t h e r e f o r e F(g) i s w e l l d e f i n e d . If w e a p p l y F t o ( 4 ) w e o b t a i n c F[y] F [ f l t J ( t o ) b ( t o ) - f ( t o ) ] - F ( Z ) , hence

t E

1"

-

y(to)

f (to) = J ( t o ) - ' [ c - F ( f ) + F ( f ) ]

and i f w e r e p l a c e t h i s e x p r e s s i o n i n ( 4 ) we g e t

i.e.

THEOREM 3 . 1 3 . a ) Foh e v e h y K[y]

= f , F[y]

(5)

y(t)

c

i 4

f(t)

-

(f ,c)E S

given b y

6,

KYF

daR(t,a)*f(o) +

the batution

z ( t )[c-F(f)+F(r)].

b ) R c c i p h o c a L t y , i $ (f , c ) € G()a,b(,X)XY

c LZ

06

y

i d

duch t h a t

- F ( f ) + F ( ? ) € Yo t h e n t h e byd.tem K b ] = f, F[y] b o L u t i o n givefi b y ( 5 ) (hence (f ,C)E sK,F),

c

had

STIELTJES-INTEGRAL EQUATIONS .-

f

133

i s c o n t i n u o u s , s o i s R and t h e n by a ) o f 11.1.14 i s c o n t i n u o u s t o o , hence F ( f ) F , ( f ) ; t h e r e f o r e w e have If

K

Let

3.14.

y

bokution

be continuoub. Foh evehy

K

= f , F[y]

K[y]

06

= c

i b

( f , c ) E S K Y Ft h e

given by

r t

(6)

(6) i s a hegul?ahizing 60hmuta s i n c e a l l f u n c t i o n s of t h e second member, b u t f

, are

c o n t i n u o u s (by 3 . 8 and b e c a u s e

f"

is

y and f have t h e same kinds of d i s c o n t i n u i t i e s and a t t h e same p o i n t s .

c o n t i n u o u s ) ; hence we see t h a t

C

-

The theorem t h a t f o l l o w s w i l l a l l o w us t o make

fur-

t h e r t r a n s f o r m a t i o n s of t h e f o r m u l a s (21, (31, ( 5 ) and ( 6 ) .

W e recall t h a t i f and

X

uEL(X,Y), f o r every

i s a Banach s p a c e , qE

ry

Y

a

SSCLCS

we d e f i n e

L E M M A 3.15. G i v e n B E SVoo()a,b[,L(X,Y)), qE ry and a ~ ] a , b ( buch t h a t doh evehy X E X we h a v e q[B(t>x] 9 doh

pS)

0

t < a then q’ q[

doh euehy

.dg(s)

g~&()a,b(,X).

PROOF. It i s immediate s i n c e for

because

q[B(ci)l

= 0

if

Ei-s a

T H E O R E M 3 . 1 6 . G i v e n K€GUo,

R

J ( s ) = F[Rs] we d e d i n e

si-16

ci(

si’< a

9' i t b

h e b u l u e n t and

= F,[Rs]

9

w e have


134

la

S

H ( t ,s>

da(r)oR(r ,s>

- Y(s-t)J(s);

w e have

a ) Ht E SVoo()a,b(,L(X,Y))

t E ) a ,b

[.

b ) J ~ d a ( r ) l r t R ( r y s ) . d g ( s )=

and

H(t,b-)

0

J:

d o h evehy

d o h evehy

H(t,s).dg(s)

gE t ( ) a , b [ , X ) .

t

doh a l l PROOF. b

+ Y(s-t)J(s) +

H(t,s)

C)

+ Y ( u - t ) J ( u ) ] d U K ( u y s ~= a ( s )

\:[H(t,a)

s,t E)a,b(. a ) We want t o show t h a t for e v e r y such t h a t :

1c)a,b[ 9

( i ) For e v e r y

w e have

x€X

or i n c r e a s e (aq

b

9

= 0

qlHt(s)x]

of Ht i n s u c h a way t h a t t E [a

( i )We t a k e t h e q - s u p p o r t

q c Ty

(aqybq]

there is a

if

sg a

a

and decrease b

9’ 4

1.

4

Then f o r

we have 'a

and

for e v e r y s >, b 9 Ht(s)

XE

X

since

[a]

SV

= 0 . Analogously

for

w e have

= l asd a ( ~ ) o R ( r , s ) -

da(r)oR(r,s) =

and

f o r every

x

since

SV

9 Y (bq ’b)

[a]

0.

-Isb

da(r>oR(r,s)

9


135

( t i ) By 1 . 6 . 2 we have

ra Hence

I f we a p p l y ( 7 ) o f 11.1.6 and r e c a l l t h a t obtain

b) From t h e d e f i n i t i o n of

H

R(t,t)

z Ix

,

we

it f o l l o w s t h a t

b

H ( t ,s)

= J * d a ( r ) o s g ( t - r ) xf T , t ) ( s ! R ( ~ , s ) a

hence

W e define

a a h(.r,s) = sg(t--c)x

h

s a t i s f y t h e h y p o t h e s i s of Theorem 2 . 6 of Chapter

a

and

g

f. 4 ( s ) R ( ~ , s ) ; t h e

functions

a,

I1 h e n c e , by (7) of t h i s theorem, t h e l a s t i n t e g r a l i s e q u a l to

jbli -da(-c)

a

c ) By t h e d e f i n i t i o n of

R(T , s ) d g ( s ) .

'I

H

w e have


136

H(t

y

+) Y ( a - t ) J ( a ) =

~

f

a

da(T)oR(T

,a>j

by Theorem 2 . 7 o f C h a p t e r I1 w e have

and by (R,)

IS

of Theorem 1 . 5 we h a v e

T

R ( T , o ) o d U K ( o , s > = Ix

-

R(T,s);

if w e r e p l a c e t h i s i n t h e second member of D

-

(ak)

we obtain c).

We g i v e now a first form of t h e i n t e g r a l f o r m u l a s of

Green f u n c t i o n t y p e .

THEOREM 3.17.

Foh K E & " we h a v e t h e h e i b o n e and o n l y o n e d o a ) Foh e v e h y ( g , c ) E SE l u t i o n y 0 6 K[y] = g , F[yj = c; t h e b o l u t i o n y i b c o n t i n u o u b and i b g i v e n b y . (7)

c

then the by (7).

byb.tem

KEY]

t

b i H ( t o , u ) d g ( u ) E Yo

= g , Fry] = c

had a d a l u t i o n

y

given

P R O O F . It f o l l o w s i m m e d i a t e l y from (3) by b) of Theorem 3 . 1 2 .

I n t h e case of t h e example (L) of 8 2 f o r e a c h w e t a k e to = t i n (7):

T H E O R E M 3.18.

L[y]

(7')

g , F[y]

Foh

eveny c

( g , c ) E F,:S

%he b o t u t i a n i b c o n t i n u o u b and i b g i v e n b y

tE)a,b( y

06


f , Fry] = c

t h e n t h e AyAtem b y (8).

K[y]

P R O O F . K[y]

may b e w r i t t e n as

= f

hub a d o L u t i o n

137

y

given

where r t

a n d by a ) o f 11.1.14 w e have

gE b ( ) a , b ( , X ) ; h e n c e by Theorem

3 . 1 7 w e have

We have

F[y-f] /ttR(t,o)do[ 0

c

-

F[f]

and w e s t i l l h a v e t o p r o v e t h a t

]

ltudTK(u,T)*f ( T ) 0

[:TR(tyT)'f(T).

I n d e e d , u s i n g f i r s t i n t e g r a t i o n by p a r t s i n t h e f i r s t i n t e g r a l a b o v e , t h e n a p p l y i n g ( 6 ' ) of C h a p t e r I ' I ( a n d r e c a l l of Theorem i n g t h a t K ( t , t > :0 ) a n d u s i n g a f t e r w a r d s ( R * , ) 1.20 we obtain


138

REMARK 3. ( 8 ) i s a r e g u l a r i z i n g formula (see t h e comment a f t e r 3.14). I n t h e c a s e of t h e example ( L ) of 8 2 for each

t ~ ) a , b (

t ; w e r e c a l l t h a t i n Theorem 3 . 1 9 K i s normalized hence w e have K ( u , s ) = A(s)-A(u) and w e o b t a i n

we may t a k e

Theorem 3.20 - Foh e v e h y F[y] c hub a b o l u t i o n

E

( f , c ) E S L Y F t h e AyAtem y given b y

Ley] =

f,

- I n t h i s item w e give conditions i n order t h a t t h e

s o l u t i o n s of t h e problem K[y] g , Fry] = c may b e w r i t t e n i n t h e form g~b()a,b(,X)

with

(9)

We w i l l show t h a t t h e n y depends c o n t i n u o u s l y on g. W e e x t e n d t h e s e r e s u l t s t o t h e g e n e r a l problem K[y] f, F[y]=c.

THEOREM 3.21. Let K and F be Auch t h a t I ) T h e b o . t u t i o n y 0 6 K[yI = g , F[y] c whehe may b e w h i t t e n i n t h e 40nm ( 9 ) . g~t()a,b(,X) 2 ) Foh any c € Y 0 and g ~ Q ( ) a , b ( , X ) , y g i v e n b y ( 9 1 i d t h e d o e u t i o n 0 4 K[y] = g, FLY] = C. T h e n : a ) SFYF 6()a,b(,X)XYo

b ) FIG()a,b(,X)]

= Yo


139

P R O O F . a ) If ( g , c ) E S6 i s such t h a t t h e s o l u t i o n y of K,F g , F b l = c may b e w r i t t e n i n t h e form ( 9 ) t h e n c h a s K[y] t o be i n t h e domain of d e f i n i t i o n o f J ( t ) , i n p a r t i c u l a r o f j ( t o ) = J(to)-’, i . e . , C E Y o , h e n c e by 2 ) w e have t h e o t h e r i n c l u s i o n . b) W e h a v e

K-l(O)C

S ~ y F c e ( ) a , b ( , X ) x Y o and

O()a,b[,X),

Yo = F[K-’(O)]

C

hence

F[b(]a,b[,X)]

a n d by 2 ) w e h a v e t h e o t h e r i n c l u s i o n .

D E F I N I T I O N : Ya

Fa[G()a,b(,X)].

I n what f o l l o w s w e w i l l p r o v e t h a t i f Y a = Yo t h e n w e h a v e 1) and 2 ) o f Theorem 3.21. The example 3 a t t h e end o f t h i s 5 shows t h a t b ) o f Theorem 3 . 2 1 d o e s n o t n e c e s s a r i l y imply t h a t

=

Y,

THEOREM 3 . 2 2 .

*

16

Yo t h e n

Y,

PROOF. I t i s i m m e d i a t e t h a t

l y , given

SEYF = ~()a,b[,X)XYo.

$ , F C b ( ) a , b ( ,X)XYo; r e c i p r o c a l ( h , d ) E ~ ( ] a , b [ , X ) X Y o w e want t o p r o v e t h a t t h e

h , F[y] = d h a s a s o l u t i o n . By Theorem 1 . 5 there exists a G()a,b(,X) s u c h t h a t K[y] h and by a > system

K[y]

GE 7 -

i s c o n t i n u o u s , h e n c e F[y] = For[?]€ Yo and t h e r e f o r e d Fa[?]€ Yo; t h e n by b) of Theorem 3 . 1 6 , app l i e d t o g : 0 and c d - Fa[?] , t h e r e e x i s t s

of Theorem 3 . 1 2

z ~ G ( ] a , b ( , X ) , s o l u t i o n o f K[z] take y z t w e h a v e K[y] Fb]

= F[z]

= 0 , F[z] K[y] = h

+ F[-] = d

-

FaL]

d and

-

+ Fa[?]

Fa[-];

if we

= d. QED

3.23. eXidtA

16

Ya = Yo

t h e n doh euehy

gt6()a,b(,X)

thetre

I + b H ( t o y s ) . d g ( s ) E Yo.

PROOF. By Theorem 3.22 w e h a v e r e m 3.17 t h e s o l u t i o n of K[y]

( g , O ) € S E Y F h e n c e , by Theog , F[y] = 0 i s g i v e n by

140


t h i s i m p l i e s t h e r e s u l t if w e t a k e

t = t

Yo

J(to) =

i s t h e domain of d e f i n i t i o n of

0

and r e c a l l t h a t

J(to)-l.

J ( t o ) i s a ( c o n t i n u o u s ) l i n e a r b i j e c t i o n from X w e m a y u s e it t o t r a n s f e r t o Yo t h e Banach s p a c e norm of X . We d e n o t e by Yx t h e v e c t o r s p a c e of Y 0 endowed w i t h t h i s new norm, i . e . , for c e Y X w e d e f i n e -1 IICIIX IIJ(to) cu Since

onto

Yo

-

Obviously

z(t> =

f o r every

t ~)a,b[.

R(t,to)

o J(to)-': Y x

3 . 2 4 . 16 Yk = Yo t h e n d o h a l l ai H ( t , s > E L ( X , Y o ) .

6) H ( t , s )

+X

i s continuous

s,t E)a,b(

w e have

E L(X,Yx).

P R O O F . a) W e have

For e v e r y

x€X

w e have

H ( t , s ) x = F a b S y t x ] E Fa[G()a,b(,X)] hence

Yay

H ( t , s ) E L(X,YJ.

b ) By a) t h e graph of H ( t , s ) i n XXYo is closed, hence a f o r t i o r i t h e graph i s c l o s e d i n X X Y x j t h e n w e have b ) by t h e c l o s e d g r a p h theorem.


have

Ht E SVoo()a,b(,L(X,YX)).

have

Ht,

ib

= 0

H(t,s)

that

PRO0 F. a ) By

Y,

d o h aLL

= Yo

s < at

141

on

SV

()a,b[,L(X,Yo)).

( a t , b ~ ~ c ) a , b [d u c h

s >bt.

Fa L[G()a,b(,X),Yc] ; hence

w e have

i s c l o s e d ana a f o r t i o r i t h e graph of F a i n Gf)a,b[,X)XYo it i.s c l o s ed i n G()a,b(,X)XYX; t h e r e s u l t f o l l o w s from t h e c l o s e d graph theorem. b ) f o l l o w s from a ) . c ) f o l l o w s from

a

that

a~ SVoo()a,b(,L(X,Y))

takos i t s values i n

d ) By b ) of 3 . 2 4 w e have

and t h e f a c t

Yo. H ( t , s ) € L(X,Yx)

f o r every

s ~ ] a , b [ hence t h e r e s u l t f o l l o w s from a ) o f Theorem 3 . 1 6 a p p l i e d t o t h e Banach s p a c e Yx ( i n s t e a d of Y) s i n c e by b )

w e have O E SVoo()a,b(,L(X,YX)). e ) f o l l o w s d i r e c t l y from a ) of Theorem 3 . 1 6 . f ) f o l l o w s from a ) of Theorem 3 . 1 6 and from d ) . 3.26.

16

Y,

= Yo

t h e n d o h evehy

g ~ G ( ) a , b ( , X ) thehe

eXibtb

a n d .the 6.ihb.t

i n t e g h a t exid.tt6

boxh i n

Yx

and

Yo.

P R O O F . By d ) of Theorem 3 . 2 5 t h e f i r s t i n t e g r a l e x i s t s i n Yx ( h e n c e i n Yo s i n c e Yx C, Yo i s continuous) t h e r e f o r e t h e f i r s t member i s w e l l d e f i n e d . Again by d ) o f Theorem 3 . 2 5 and since

J ( t ) E L(Yx,X)

w e have

hence t h e second i n t e g r a l e x i s t s . The e q u a l i t y f o l l o w s from t h e c o n t i n u i t y of

J(t)

in

Yx.


142

F R O M N O W O N W E SUPPOSE T H A T DEFINITION. For every G(t,s)

we w r i t e

s,tE)a,b(

= J(t)oH(to,s)

t

Y a = Yo

[Y(s-to)-Y(s-t)]R(t,s)

i s c a l l e d t h e G h e e n dunction o f t h e s y s t e m ( K ) , ( F ) . By b) o f 3.24 a n d s i n c e z ( t ) E L ( Y x , X ) (and R ( t , s ) E L ( X ) ) and

G

w e have 3.27.

G ( t , s ) E L(X)

doh a l l

s , t E)a,b(.

R E M A R K 4. The t h e o r e m s t h a t f o l l o w a r e t h e f u n d a m e n t a l t h e o rems o f t h i s 5 . We d o n ' t know i f t h e y a r e t r u e w i t h o u t t h e h y p o t h e s i s Ya Yo, i . e . w e ignore i f t h e necessary condit i o n s 1) and 2 ) o f Theorem 3 . 2 1 imply t h a t Ya a € BV1Oc()a,b{,L(X,Y)) t h e necessary condition

T H E O R E M 3.28.

K

a) The dybtem

and K[y]

i b

given b y

Yo.

F a h e b u c h t h a t Ya = Yo t h e n = g , Fry] c hub a b o e u t i o n

y ~ b ( ) a , b ( , X ) id a n d onLy id

tthib b a L u t i o n

Ya

implies t h a t

( g , c ) ~ t i ( ) a , b [ , X ) X Y ~ t; h e n

eveay ( f , c ) E sK,F t h e b y b t e m hub o n e and onLy o n e b o l u t i o n g i v e n b y b)

If

= Yo

F[G()a,b(,X)] ( s e e Theorem 3.21)

Yo.

FOJL

P R O O F . a ) By ( 7 ) and 3.23 z(t)c rb

and

are w e l l d e f i n e d and if w e r e c a l l t h a t

K[y]

= f , F[y]=

c

STIELTJES-INTEGRAL EQUATIONS the result follows from the definition of b) We may write y(t)

-

143

G.

duK(t,o)[y(a)-f(~)]

f(t) +

g(t)

where rt

is continuous and if we apply a) we have

By a) of 11.1.14 g

hence the result. THEOREM 3.29. T h e Gheen 6 u n c t i o n

G: ]a,b[X)a,b(

hub t h e 6oLLowing p h o p e h t i e d (Go)

F[Gs]

(G1)

Gs(t)-Gs(to)

(G2)

= 0

I:

doh e v e h y t

4

L(X)

:

sE)a,b(.

doK(t,u)oGs(u)

0

[-Y(s-t)+Y(s-to)]IX s ,t

G(t,s) + Y(s-t)R(t,S)

t

€1a,b(.

Y(s-to> [j(t)J(s)-R(t,s)]

(Gg) Foh e v e h y s~)a,b( Gs i b c o n t i n u o u b a t Le6t-continuoub a t t s.

t # s

t

and

(G4) Foh evehy tE)a,b( Gtc SVoo()a,b(,L(X)) and Gt(s)= 0 id 5 < i n 6 [t ,to,aJ oh s > d u [t,toybJ whehe (ao,bo) den o t e b t h e b u p p o h t 06 J(to)- X G i d eocaLLy u n i d o h m l y 0 6 bounded b e m i v a h i a t i o n i n t h e b e c o n d v a h i a b l e .

PROOF. (Go). By the definitions of G G(t,s)

-

Y(s-to)J(t)oJ(s)

z(t>

and

da(T)oR(T,s)

+ Y(s-to)R(t,s)

-

H we have

Y(s-t)R(t,s),


144

and by Theorem 3 . 9 w e have

ho-

and by a ) o f Theorem 3 . 9 t h e f i r s t summand s a t i s f i e s t h e

mogeneous e q u a t i o n . For t h e s e c o n d summand w e have t o p r o v e that

= [Y

(S-t

1-Y (s-to)]Ix.

fto,t)

since then everything i s which i s o b v i o u s f o r s $! zero; f o r t o ( s < t t h e equation reduces t o

i . e . , t o t h e e q u a t i o n of t h e r e s o l v e n t ( ( R " ) and a n a l o g o u s l y f o r t <s < t 0

.

o f Theorem 1 . 5 )

( G 2 ) f o l l o w s i m m e d i a t e l y from c ) of Theorem 3 . 1 6 and f r o m t h e d e f i n i t i o n of G . (G3)

(G,+).

f o l l o w s from t h e d e f i n i t i o n o f

G.

By d ) of Theorem 3 . 2 5 w e h a v e lit,€

SVoo(] a , b ( , L ( X , Y x ) )

and t h e r e e x i s t s (aoybo) C ) a , b ( s u c h t h a t H ( t ,s> = 0 for s < a o or s > b o ; t h e same i s t r u e for J ( t ) o Ht o O, h e n c e it f o l l o w s from t h e d e f i n i t i o n of G t h a t G ( t , s ) 0 for or s bup[t,to,b

J .The

REMARK 5 . I n t h e case of example (L) of 8 2 form

rest i s obvious. (G2)

takes

the


145

Indeed i n t h e case of example (L) w e have

z(t)J(a> -

R(t,b)

(by c ) o f Theorem 3 . 9 ) ; f o r e v e r y

-

K(u,s) = A ( s )

w e have

s

0 we may t a k e

to

s and

A(u).

REMARK 6 . (G1) s a y s t h a t Gs s a t i s f i e s t h e e q u a t i o n EdK ( G ~ ) = ~ ( ~ ] of D i s t r i b u t i o n s i . e .

i n t h e s e n s e o f Theory

d G (t)t dt s dt

t

ft

dUK(t,u)oGs(u>

~5(~)(t).

0

T H E O R E M 3 . 3 0 . a ] T h e mapping gcC i d

9gE 6 0 a,b [ , X )

() a ,b [,X 1

continuoub, whehe

b (gg)(t) = jaG(t,s)dg(s)

b l T h e mapping whehe

CE

Yx

d

,

t€)a,b(.

y , ~O ( ) a , b [ , X )

i b

continuoub,

yc(t> = i(t)c.

cl T h e d o t l o w i n g phopehtieb ahe e q u i v a t e n t : (il T h e mapping C E Yo HycczG(]a,b(,X) tinuoud. liil J(to)-’: Yo 4 X i b continuoub. (iiil Yo i b a Banach b p a c e . P R O O F . By ( G 4 )

Gt

id

con-

h a s compact s u p p o r t , hence rb

The c o n t i n u i t y of

where

a

9

= inb[cyto,aJ

f o l l o w s from

and

6 =

bup[d,to,bJ

( c f . (G4)).

b ) Follows from b) of C o r o l l a r y 3 . 1 0 a p p l i e d t o

Yx. c) Follows from b ) o f C o r o l l a r y 3 . 1 0 and d ) o f 3.7.


146

T H E O R E M 3.31. T h e mapping f E G() a,b[

,XI

i h conkinuouh whehe

P R O O F . The mapping

f

-

f

Q,fE@

c,gK€

() a , b [ , X >

i s t h e c o m p o s i t i o n of t h e con-

t i n u o u s mappings

fEG()a,b(,X) and g€Q()a,b(,X)

-

t-,kf€e()a,b[,X) SgEe()a,b[,X),

where rS

9

and

EXAMPLES

i s d e f i n e d i n Theorem 3.30.

1. W e take

X = Y = R

y'

t

and c o n s i d e r t h e e q u a t i o n A'y = f ' ,

more p r e c i s e l y t h e i n t e g r o - d i f f e r e n t i a l e q u a t i o n (L)

and A€GBV1""(]a,b(). y,f E G()a,b[) The r e s o l v e n t of (L) is R ( t , s ) = exp[-A(t)+A(s)] t h e g e n e r a l s o l u t i o n of (L) i s

where

y ( t ) = y(s)exp[-A(t)+A(s)]

t

JI

exp[-A(t)+A(a)] * d f ( U ) .

If w e t a k e now a l i n e a r c o n s t r a i n t (F)

and

FCYI = c

FE G ( ) a , b [ ) '


147

a BVoo()a,b() and U E s o o ( ) a , b [ ) i s z e r o o u t s i d e a n i n t e r v a l [:,6) c ) a , b ( and 1) such t h a t

by 1 . 6 . 8 t h e r e e x i s t

(i.e. UE

,

u

l,( [.,5]

We h a v e

hence

The c o n d i t i o n f o r t h e e x i s t e n c e o f t h e Green

is

J(s) # 0 , i.e. / abe - A ( t ) d a ( t )

If

function

J(s) # 0

# 0.

t h e Green f u n c t i o n i s g i v e n by

hence t h e s o l u t i o n

y

of t h e p r o b l e m a ( L ) ,

rb 2 . Y = X2 (K) w e h a v e

and

Ya

(F)

i s g i v e n by

rS

Fry)

= (y(to),y(to-));

Yo = Ax = { ( x , x ) E X

2

I

f o r any e q u a t i o n X E XI


148

hence if J(to) is injective (i.e., if y 0 is the only solution of the system K[y] :0 , Fry] = 0 - see Theorem 3.6) the problema (K), (F) has a Green function (and J(to) is bicontinuous). 3.

Y = X2

and

F[y]

(y(to),y(to+>); then we have

=I

a(t>x = F[XJ~,~]X]= (XIa,t] (to)x~X]ayt](to+)x =

u(t)x

= F[x{~~x]

(x,x> (x,O) (0,O)

if if if

t > to t = to t < t0

(x~~)(~~)x,x{~)(~~+x)) =

(o,o)

hence

if

t

+

to

and

for any

K.

If we take now

R(t,s) = Ix and J(s)x We

L[y]

= F[Rsx]

5

y

we have the resolvent

= (Rs(to)~,Rs(tOt)~) = (x,x>.

have da(T)oR(r,s)

since

= F a [ ~ ~ a y S ] R s ]6 L(X,Yo)


H ( t , s ) g L(X,Yo)

hence

and t h e r e cannot e x i s t t h e Green

(L), (F).

f u n c t i o n of t h e problem 4. We t a k e

149

pi,

)a,b( =

Y = G(R,X)

= y(o+l)

F[y](u)

-

and a€

y(a>,

n.

W e have Fa[f] = F [ I - f ) , hence Ya = G - (R,X) which i s a Frec h e t s p a c e and thus (by t h e c l o s e d graph theorem) h a s no finer Banach s p a c e t o p o l o g y hence f o r no K can we have Y a = Yo, so t h e r e n e v e r e x i s t s a Green f u n c t i o n w i t h t h e p r o p e r t i e s of Theorem 3 . 2 9 .

Hence f o r any

K E GU0(RXR,X)

t h e r e never

e x i s t s a f u n c t i o n G : RXlR + L(X) with t h e p r o p e r t i e s ( G o ) t o (G4) s u c h t h a t t h e c o n t i n u o u s p e r i o d i c s o l u t i o n s of period 1 of y(t)

-

t Y(to> +

daK(t,a)*y(a) Ito

g(t)

-

g(to>

are e x a c t l y t h e f u n c t i o n s o f t h e form

L W

y(t) X = Y

5. We t a k e

G(t ,s> * d g ( s ) .

= G ( ( a , b ) ,Z) where

Z

i s a Banach

s p a c e and d e f i n e 0

F[f](o) f o r every

f E G ( [a,b] ,X>

= laf(T)(T)dT,

[a&)

Y

G( [ a , b ) , G ( ( a , b ) , Z ) > .

a) I n o r d e r t o show t h a t

prove t h a t t h e f u n c t i o n

0 E

T

F(f)

i s w e l l defined w e w i l l i s regulated; 0 < E < 6 and T E [a,,-,]

E ( a , b ] w f ( . r ) ( . c )E Z

t h i s f o l l o w s from t h e f a c t t h a t f o r

w e have

d

11 f ( T + 6 )

11 f ( T + 6 ) ( T + 6 ) - f ( T i - € ) ( T + 6 ) - ( T + € ) (T+6 11 + 11 f( T + E )

b llf(T+6)-f(T+E)l\

since

f

and

,
II

(T+6

+ [lf(T+E>(T+6> Ia rb) f(‘C+E) are r e g u l a t e d .

1-f

-

(T+E) (T+E)

f(TSE)(T+E)ll

11

\
x = F(x { ) x )

Fu = 0 :

b ) We h a v e F = FCi

where

x E X ;hence

[ ( x i t i ( ~ ) x j ( ~ ) d r=

F(xItIx](a)

= j aUq t ) ( T ) X ( T ) d T =

0;

F = Fa.

t h e r e f o r e w e have

c ) We h a v e

Yo

I

G L l ) ( ( a , b ) , Z > = { g E G ( l ) ( [ a , b ) ,Z>

g ( a > = 0)

Y = G ( [a,b] , Z >

t h a t i s n o t a c l o s e d subspace of

but t h a t

a l l o w s a f i n e r Banach s p a c e norm g i v e n by

d ) If w e t a k e

Ilgll(l)

=

L[y]

yt

IIgll

+

f

E

= X , t h e space

L-’(O)

= x)

yo = F [ L - ’ - ( O ) ] indeed: given

*

w e have

(T

of c o n s t a n t f u n c t i o n s

Ilg’II

and

= ya; we take

G ( [ a , b ] ,G( ( a , b ) , Z ) > XfE X = G ( [ a , b ] ,Z>

such t h a t function

-

= f(a)(a) for all ~ , u ~ ( a , b h) e,n c e t h e i s constant ( 2 f ( ~ l > = S f ( ~ 2 ) f o r any

;,(T)(U>

xf

T ~ , T ~( aE, b

L[y]

and w e h a v e

Yo

and

F[cf]

= F[f].

= 0 and F[y] = 0 imply y = 0 ; i n d e e d : i f w e have y = 2 a c o n s t a n t f u n c t i o n , hence

e > L[y]

= 0

0

-

f o r every J(s):

F[Gf]

X

F[y](a)

=

I,

a ~ [ a , b ] implies

Yo

Yo

U

y ( T ) ( T ) d T = fa:(T)dT

S(T)

0

i.e.

2

0 . Therefore

i s i n j e c t i v e c o n t i n u o u s and o n t o b u t n o t

b i c o n t i n u o u s ; if however w e c o n s i d e r on Yo t h e Banach s p a c e s t r u c t u r e d e f i n e d i n c ) J(s) becomes b i c o n t i n u o u s . f ) We h a v e

R(t,s)

Ix and

J(s)(u)

(u-a)IX ;

REFERENCES

[ B]

H. E. B R A Y , Elementahy phopehtieb

-

06

Stieetjeb integhal,

Ann. of Math., 2 0 ( 1 9 1 8 - 1 9 1 , 1 7 7 - 1 8 6 .

S . B O C H N E R a n d A.E.TAYLOR,

-

[B-T]

c e h t a i n dpaced

06

L i n e a h dunctionald on a b d t h a c t l y - v a l u e d dunctiond , Ann.

of Math. 3 9 ( 1 9 3 8 ) , 9 1 3 - 9 4 4 . [C]

- H.CARTAN,

C.S. C A R D A S S I , PependEncia d i d e h e n c i a v e l das d o l u ~ 6 e n

-

[Ca]

C a l c u l D i f f g r e n t i e l , Hermann, P a r i s ( 1 9 6 7 ) .

de e q u a ~ o e d iflt~~g hO-d id CKe flC iUeiA m e d p a ~ o d de-Banach, Master T h e s i s , I n s t i t u t o de Matematica e E s t a t i s t i c a da U n i v e r s i d a d e d e S . P a u l o , 1975.

[D] - N . D I N C U L E A N U , Vector M e a s u r e s , P e r g a m o n P r e s s , Oxford ( 1 9 6 7 ) . [GI

-

[H]

-

M . G O W U R I N , fibetr d i e S t i e l t j e b I n t e g h a t i o n a b d th a h te h Funktionen, F u n d . M a t h . , 2 7 ( 1 9 3 6 ) , 2 5 4 - 2 6 8 . S.Z. H E R S C O W I T Z , C e a d d e d d e d u f l ~ o e d adbociadab p e l a i n t e g h a l d e Riemann-Stie&jeb, Master T h e s i s , I n s t i t u t o d e Matemztica e E s t a t T s t i c a d a U n i v e r s i d a d e de S . P a u l o ,

1975.

[H-ie]

T.H.HILDEBRANDT, O n dybtemd 06 l i n e a h d i 6 6 e h e n t i o S t i e l X j e d i n t e g h a l e q u a t i o n & , I l l i n o i s J .Math. ,

-

3(1959),352-373.

p-ti] -

T.H.HILDEBRANDT,

p-BAMS]

-

I n t r o d u c t i o n t o t h e t h e o r y of i n t e g r a t i o n , Academic P r e s s , 1 9 6 3 .

C.S.HbNIG, T h e Gheen d unc tio n 0 4 a l i n e a h didbehent i a l equation w i t h a t a t e h a l condition, B u l l . h e r .

Math. SOC., [H-IME]

-

79(1973),587-593.

C.S.HoNIG, T h e a b d t a a c t R i e m a n n - S t i e l t j e d i n t e g h a l

and i t b a p p l i c a t i o n 4 t o Lineah V i6 6 e h e n tia L Equations w i t h g e n e a a l i z 5 d boundahy c o n d i t i o n d , Notas do I n s t i t u t o i de Mctematica e E s t a t z s t i c a da U n i v e r s i d a d e de S . P a u l o , S e r i e Matemgtica n 0 1 , 1 9 7 3 .

[H-BAMS2]

-

c.s . H O N I G , V o l t e h h a - S t i e t t j e d intc2gha.t e q u a t i o n 6 w i t h l i n e a h condthaintd and d i n continuoud b o l u t i o n d , B u l l . Am. Math. SOC. 8 1 ( 1 9 7 5 ) .

152

[H-DS]

REFERENCES

-

C.S.HbNIG,

tion

The dohmuhA

06

V i h i c h L e t and 0 6 S u b b t i t u i n Banach ~ p a c e A ,

oh R i e m a n n - S t i e t t j e d integha&A

T o appear.

[H-OP]

-

Open phobLemA i n t h e t h e o h y o h diddehentiae e ~ u a t i o n d w i t h t i n e a h COnAthaintA, C o l e c a o A t a s ,

C.S.HdNIG,

vol. n Q 5 ( 1 9 7 4 ) , 1 6 1 - 1 9 9 , Sociedade B r a s i l e i r a de Matemstica, Ri-o de J a n e i r o . [H-RJ-

[K]

-

[MI [R]

-

[S]

-

H U N I G , An u n i d i e d h ep h e d e nt a t i o n t h e o h y d o h L i n e a h continuoub opetratohb between dunction bpaceb , To appear.

C.S.

H . S . K A L T E N B O R N , Linealr duncti0na.t o p e h d t i o n ~ on dunctiond having d i d c o n t i n u i t i e d 0 4 t h e 6 i h A . t k i n d , Bull. h e r . Math. S O C . , 4 0 ( 1 9 3 4 ) , . 7 0 2 - 7 0 8 .

J . S . M A C - N E R N E Y , S t L e t t ' e d i n t e g h a l b i n l i n e a h Apace4 A n n . of Math., 6 1 ( 1 9 5 5 3 , 3 5 4 - 3 6 7 .

,

G . C . da ROCHA F I L H O , l n c o m p a t i b i l i t i e d i n getzezat RiemUnfl-Sti&tjeA i f l i e g h a t i o n th e o hi e b T o a p p e a r .

.

M. I . de S O U Z A , EquaC6eA d i d c h e n c i o - i n t e g a a i ~ d o s i p 0

Riemann-StieLtjeb em e d p a ~ o d de Banach com A $ . ~ U C . O ~ AdeAconztl'nuad, Master t h e s i s , I n s t i t u t o de M a t e m a t i c a e Es-

t a t i s t i c a da U n i v e r s i d a d e d e Sao Paulo, 1 9 7 4 .

[W]

- H .S .WALL,

Concehning kahmonic m a t hi c e & , A r c h . Math.

5(1954),160-167.

,

SYMBOL

A

116

Jb

119

~~"()a,b[x]a,b(,L(X))

6Z0 GU

3

BT

B V ( [a,b) ,X)

BW( (a,b) ,Y)

G(X,Y) c o ( [a,b)

c

'

0

(1a ,

111 111

&F

23

113

&yo

1

19

113 20

f-

60

[

b ,X)

61

F a y FU

85

d

2

(GI

Id1

2

(G")

Ad

2

G o a ,b ,F)

52

56

[

G ( [a,b)

16

,XI

dl 6 d 2

2

D y ’[a,b)

2

G - ( b , b ) 9x1

14

E(

D( (a,b) , X I

', "'

EB (E,F,G)

E( [a,b) , X >

3

G(t,s)

3

6,

GU

2

( E B ,F , G )

&, &( (c,d)

(a,b> ,X)

3 X (c,d)

20

35

G"SB( (a,b) ,E)

2

"(a,b]

19

Ga( [a,b) ,E)

2

'(a,b)

,L(X))

bCO

111

L g 0

114, 117

111

117

114

hUO

22

,X)

INDEX

G((c,d]

35 142

x

[c,d) ,L(X))

69, 87

GO;

110

G :o

110

rE H(t,s)

3

134

a7

154

SYMBOL INDEX

I-[, H () a

b (x) a b

JICO

121

li C O

122

1

IX

1-

20

J

127 127

Jcl

127

JU

J(to)

1 22

7

129 84

KA (K)

85

K LYI (L'),

127

116

(L)

LCBT

5

LC s

4

Lim

Ad-tO

Lim dell

[,L ( X ) )

2

(SVJ

2

( SVU)

117 52,

86

155

SYMBOL I N D E X

3

II I I B

Ilfll 'I

1

[c ,d)

1 4


157

INDEX 3

associated topological BT bilinear triple B-variation

Riemann-Stieltjes integral

3

21

Darboux integrable

support 75

8

linear constraint

12 4

q-support

56

variation

23

Volterra Stieltjesintegral equation

4

22

regularizing formula

3

uniformly of bounded semivariation 52,69

7

q-B-variation

46, 55, 6 7

topological BT

interior integral locally convex space

69

7.4 Theorem of Helly

142

integration by parts

56

Theorem of Bray

formula of substitution

7

22

semivariation

L4

formula of Dirichlet Green function

1 6 , 56

regulated function

weak variation 133

weakly regulated

85

23 35, 63


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