Volterra Stieltjes-integral equations: functional analytic methods, linear constraints

VOLTERRA STIELTJES-INTEGRAL EQUATIONS

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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
i s a f i n i t e sequence

I

Ad = AUp{lti-ti-ll

a d i v i b i o n of

i = 1,2

We write

,..., Id[>. 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 <

€1;

dlld2E ID

write

d l < d 2 , i f every p o i n t of

w e say t h a t

U,={IDE

the class

o b t a i n e d by a l l p o i n t s of

Given

I

E

and d2.

<

i s an o r d e r r e l a t i o n on D t h a t makes it t e r e d on t h e r i g h t . For every d € D w e d e f i n e

relation

Dd = { d ' E D

vG

the class

= {Dd

I

d€D)

f i n e r than t h e f i l t e r b a s i s 0.1

-

I

> 0) the

d2.

dl, and

i s a p o i n t of

dl

or

d l v d2

dl

i s {Lneh t h a n

d2

(a,b]

w e denote by

dlld2EID

(a,b)

we The fil-

d,
i s a f i l t e r b a s i s on ID which i s V , and hence

L e t x b e a topoLogicaL bpace and f : D -X; the exibtence 0 6 L i m f ( d ) , t h a t i b t h e Limit o u c h t h e @Ad+O

t e h babia

VA, impLieb t h e exibtence 0 6

i b , t h e limit oveh t h e { i L t e h equal. If ACX

X

or

[ a , b) i = 1,2,.

U and

l a5 4=

..,Id1 . 0'

f:

f

WA(f) = b u p { l l f ( t ) - f ( S ) I I

V

VG , and b o t h

babin

i s a seminormed space and

we d e f i n e t h e o b c i l a t i o n of

Lim f ( d ) , t h a t dsID

on

I

(a,b]-

,...,5 I d / 1

with

X I for

A

s,tEA);

denotes t h e s e t of a l l p a i r s

(5,

ahe

( d , O where

CiE [ti-l,ti],

denotes t h z s e t of a l l p a i r s

(d,E')

NOTATIONS where

dED

5'

and

=

(5;

3

-

,...,5 'Id1 )

with

<~E)ti-llti(.

W e s a y t h a t a f u n c t i o n f: (a,b) X is a step f u n c t i o n , w e w r i t e f E E ( [ a , b ] , X ) , i f t h e r e exists a d e n s u c h t h a t f i s c o n s t a n t i n ) t t - l I t i [ , i = 1,2,. ,I d / .

..

A b i e i n e a h thipk?e (BT) i s a s e t of t h r e e v e c t o r spaces

-

C

F

mapping

+G; w e w r i t e

BT

by

ExF

B:

(E,F,G)B

BT (E,F,G) where

and

are Banach s p a c e s , w i t h a b i l i n e a r x - y = B ( x , y ) and d e n o t e t h e

E l F, G , where

G

(E,F,G). A X o p o e o g i c d BT i s

o r simply E

t o o i s a normed s p a c e and

nuous ; w e suppose t h a t

B

a

is conti-

11 B I I < 1.

E X A M P L E S - L e t W, X and Y b e Banach s p a c e s . 1. E = L ( X , Y ) , F = X I G = Y and B ( u , x ) = u ( x ) . 2. E = L ( X , Y ) , F = L(W,X), G = L ( W , Y ) and B ( v , u ) = v o u .

0.2

-

3.

E = Y, F = Y',

4.

E = G = Y,

G = C

F = C

and

and

B(y,yl) =(y,y*).

B(y,X) = Xy.

a ) E x . 1 i s a p a r t i c u l a r i n s t a n c e of E x . 2 :

take

b ) Ex.3 i s a p a r t i c u l a r i n s t a n c e of E x . 2 :

t a k e X = C and

w

= Y.

c ) Ex.4 i s a p a r t i c u l a r i n s t a n c e of Ex.2: Given a BT ( E , F, G ) B IIXllB =

f o r every

xEE

EB = E x E E

EB

w i t h t h e norm

l o g i c a l BT (EB,F,G) D

-

Let

d e f i n e d on

E E

I

we define

IIx\lB<m);

11 [ I B

and w e s a y t h a t t h e

topo-

i s a b b o c i a t e d t o t h e BT ( E , F , G ) .

b e a v e c t o r s p a c e and such t h a t

plI...,pm~

rE

b ~ p t P l , . . . r ~ m ] ErE.

rE

t a k e X = W = C.

IIYII .c< 1)

I

AuPE IIB(X,Y)ll

and

w e endow

W = C.

d e f i n e s a t o p o l o g y on

E: t h e sets

rE

a s e t of s e m i n o r m

implies

NOTATIONS

4

form a b a s i s of neighborhoods of 0 ; t h e s e t s xo+v form PIE a b a s i s of neighborhoods of xo€ E . Endowed w i t h t h i s topology E i s called a l o c a l l y convex space (LCS). A LCS E i s s e p a r a t e d i f and o n l y i f p ( x ) = 0 f o r implies x = 0. every P E r E A sequence xn of a LCS E i s c a l l e d a Cauchy beqUenCe i f f o r every P E TE and every E > 0 t h e r e e x i s t s an nE(p) such t h a t f o r n,m >,n,(p) w e have p(x,-x,) < E. A separated s e q u e n t i a l l y complete LCS (SSCLCS) i s a s e p a r a t e d LCS inwhich every Cauchy sequence i s convergent. A F r e c h e t space i s a SSCLCS whose topology can be d e f i n e d by a c o u n t a b l e s e t o f seminorms (and i s t h e r e f o r e m e t r i s a b l e )

.

EXAMPLES LCS 1 - Every normed o r seminormed space E i s a LCS. LCS 2 - I f X i s a LCS and K a compact space there i s a n a t u r a l s t r u c t u r e of LCS on E = & ( K , X ) : f o r every seminorm PE rx w e d e f i n e a seminorm p E r E by p ( f ) = b u p p [ f ( t ) ] , where

f E E = 6 ( K , X ) ; w e o b t a i n on

6(K,X)

tEK

t h e topology of

uniform convergence on K . I f X i s a Banach o r a F r e c h e t space, so i s 6 ( K , X ) . LCS 3 - L e t X be a normed space; E = 6 ( )a , b [ , X ) becomes a LCS when endowed w i t h t h e family of seminorms

where [c,d) runs over a l l c l o s e d i n t e r v a l s of ] a , b ( . I f X i s complete, i . e . a Banach space, E i s a F r e e h e t space; i t s topology may be d e f i n e d by t h e countable s e t of seminorms , n€m. I' "[a+:, b-

i)

LCS 4

-

Let

X

be a LCS; E = &(]a,b[,

X)

becomes

t u r a l l y a LCS when endowed with t h e family of seminorms P(c,d] where

PE

rx

and

[f]

= bup{p[f(t)]

[c,d]

C

] a , b [.

I

c < t ( dl

na-

NOTATIONS For LCS

and

X

Y,

5

d e n o t e s t h e v e c t o r s p a c e of

L(X,Y)

a l l c o n t i n u o u s l i n e a r mappings from

X i n t o Y ; i n o r d e r that a l i n e a r mapping f : X 4 Y be c o n t i n u o u s i t i s s u f f i c i e n t t h a t it i s c o n t i n u o u s a t t h e o r i g i n and hence f o r e v e r y q e r y

there is a every

PE

rX

X

i s a normed s p a c e and

a >0

and

such t h a t

q[f ( x ) ] 6 a p ( x )

for

xEX.

-

LCS 5

If

L(X,Y) we

a LCS, on

Y

c o n s i d e r t h e t o p o l o g y d e f i n e d by t h e seminorms p ( f ) = bUp{P(f

Y

If

I

(XI)

i s a SSCLCS s o i s

XEX,

L(X,Y).

-

I f X i s a LCS and f : [a,b) d e f i n e t h e o s c i l a t i o n s as i n B: f o r w

deD

and f o r w

qrA

q,i

IIxII G 1 1 ,

PE

ry. q € Tx w e

f o r every A C X we w r i t e

( f ) = bUp{q(f(t)-f(S))

X

I

tlsEAl

we w r i t e

(f) = w

(f)

(ti-1*ti)

W;lIi(f)

I

= w

q I ) ti-1 I t i

[ (f)

I

etc.

-

A t o c a e e y convex BT (LCBT) i s a s e t o f t h r e e v e c t o r E l F , G , where F i s a Banach s p a c e , G a SSCLCS,

spaces

w i t h a b i l i n e a r mapping E

-

B: ExF

G.

I n c h a p t e r I11 w e w i l l u s e the f o l l o w i n g

T H E O R E M 0.3

-

Let

X

i E I

te2

Ti:

X

that: 1)

-

b e a c o m p l e t e m e t h i c bpace and

p o t o g i e a l &pace. Foh ewehy

X

a&

I

be

huch

i6

LocaLty a unidahm c o n t h a c t i o n , i . e . , d o h evehy ioE I t h e h e k b a neighbohhood J and a c o n b t a n t c J < 1 buch t h a t (Ti)iEI

d[TiXr

doh

aeL

2 ) Foh e v e h y

x,yEX XE X

tinuoua. T h e n , id d o h evehy

TiyJ

S

cJd(xry)

and ewehy i E J . t h e d u n c t i o n i €I XEX,

xi

MT

ixEX

id

denotes t h e d i x e d p o i n t

con06

Ti

NOTATIONS

6

-

( w h i c h e x i b t d b y Banach c o n t h a c t i o n mapping t h e o h e m ] , t h e mapping i € I xi€ x i4 c o n t i n u o u b . P R O O F . Obviously i t i s enough t o prove t h a t t h e mapping i s continuous a t

i

0

. W e have

d(xi8xi

)

= d[Tixi

0

0

s

]

i

0

] +

d[Tixi

TiXio

s

cJd (xi

I

x

iO

d[Tixi

‘Ti 0

+ d[TiXi

0

‘Ti X i

hence d(xi8xi

) 0

s 1-c d[Tixi

t h a t by 2 ) goes t o zero when

i+io.

,T. 0

xi ]

lo

0

xi ]

0

0

0

]

0

THE

§I -

For

CHAPTER

INTERIOR

I

INTEGRAL

T h e R i e m a n n - S t i e t t j e A i n t e g h a l and t h e i n t e h i o h

in t eg h a t

Let (E,F,G) be a BT, a: (a,b) -+E (d,<)E 0 and (d,<’)E 0’ we write

f: [a,b]--,F.

and

and

We define

and

b

b

if these limits exist. The first one is called Riernann-Sfidt j e b i n t e g h a t and the second one, the P u b h n i k or i n a k h i o h i n teghat

.

1.1

-

Id

rda(t) .f (t) exibtA t h e n t h e h e exiAtA a b da(t) .f (t) *.da(t).f (t) = a a

\

P R O O F . Follows from

U'C

D

and 0.1.

Ib

.

a

THE I N T E R I O R INTEGRAL

THEOREM 1 . 2

-

-

(E,F,G)

Let

a : [a,b)

f: (arb]

E,

bounded duncXionh b u c h 2ha.t t h e h e t x i b t b a oh f i b c o n k i n u o u b t h e h e e x i b X b

Ib

da(t) *f(t) =

a

-

b e a t o p o l o g i c a l BT and

Ib -

F

J:

*da(t)-f(t); .id

.

.da(t) *f(t)

a

The proof is given in the appendix of this

-

THEOREM '1.3'

a : (a,b)

-

E

and

f: [a,b)

Ib

da(t) . f (t)

=

I

b

id and o n l y id t h e h e a

-

I n t e g r a t i o n by p a r t s

a

F

5.

G i v e n a BT (E,F,G), thehe e x i b t b

da(t) -f(t) a(t) .df (t) and t h e n we h a v e

exibfi

a(b) . f (b)

-

a(a) .f (a) -

PROOF. It follow5 immediately from

Ib a

a(t) -df(t).

F O R T H E I N T E R I O R I N T E G R A L THE I N T E G R A T I O N B Y P A R T S FORMULA I N G E N E R A L I S N O T

UNLESS

a

OR

f

IS

VALID,

CONTINUOUS ( C f .

T h e o r e m 1 .2). S e e a l s o T h e o r e m s 4.21 THEOREM 1 . 4

-

Given

and 4 . 2 2 .

c~)a,b( t h e n e

and o n l y i d t h e n e e x i b t \’-da(t).f(t) Ja

exibtb

and

Jb

.da(tl.f(t)

.id

a [b*da(t).f(t) JC

THE I N T E R I O R INTEGRAL

and t h e n we h a v e b (a) .da(t) -dt = -da(t).f (t) + a a

jC

Ib

C

9

*da(t).f (t).

P R O O F . The existence of the second member of ( a ) implies

the

existence of the first since the set of all divisions that contain c is cofinal with the ordered dED(a,b) of all divisions. The other implication filtered set D (. is trivial.

4

Even in the numerical case (i.e. E = F = G = IR) the analogous of Theorem 1.4 is not true for the Riemann-Stieltjes integral (unless a or € is continuous) as shows the examPIe and

ab= X(c.,b)

f

and

J da(t) .f(t) C

= 0

there exist

=

but

:1

da (t).f (t)=O

da(t) .f (t) does not exist.

APPENDIX THEOREM 1.2

-

(E,F,G)

Let

a: (a,b) - E ,

be a t o p o L o g i c a L B T , and

f: (a,b)

yF

bounded d u n c t i o n b buch t h a t thehe e x i b t b *da(t)-f(t); id la a oh f i b c o n t i n u o u b t h e h e e x i b t b b b da(t) .f ftf = =dcr(t)- f(t). J J a a PROOF.

Since there e x i s t s

there is a

for

d>dE

dE€

D,

w i t h Ad

6

d E : a = t: < t:

we h a v e

prove t h a t given

j I . d a ( t 1 - f [tl, f o r every

E

<

[ I U ~ , ~- U. d E , c E . / I < 20

there i s a

E > 0

... < t I dE

=

6 > 0

that for

E

E.

b. s u c h t h a t

Hence i t i s enough t o suc

dED

we h a v e

(a)

I

-

Let us f i r s t

c o n s i d e r t h e c a s e when

f

i s c o n t i n u o u s and

10

take Ad < 6

THE I N T E R I O R I N T E G R A L 6 > O

such t h a t

we d e f i n e

[ t j - 1 8"j) (61

E'=

[E;

,..., t i d [ )

llu;,;

- u;,;.

s i n c e we h a v e

6 > O

ij

d )d

=

J

Si

with

if

i h d

E'

II

4

E

i t follows that

a

i s continuous.

We

such t h a t

d E D

=

Si

,

we

i f

have

Since

6.

d E D

Iy).

Again f o r we t a k e

For

-

such t h a t

I1 - L e t us now c o n s i d e r t h e c a s e w h e n take

.

2 \ d E l * 2 1 1 ~ / l w ~ L5f l E .

d,S

[Yl

which implies

11 <

-

llu;,;

I IM

t h e n we h a v e

;

Let us define a

E

I

4 dE d v d E a n d we t a k e

d =

=

and t h e n ,

w6[fl 5

-

have

with

Ad <

(Ej-l,tj)

6 , we d e f i n e =

[ti-l,ti].

2

= d v d E and

Then b y

[a)

we

THE INTERIOR INTEGRAL

ti,:.,

Definition of

t.

[ i ) Let

i

:

d;

b e a p o i n t of

1,2,..., I d E 1 - l

=

-

and

11

i

t. J

'j

3

t.

if

# tf

J

-

f o r

t h e n we h a v e

'j+l

with

by

I f we e l i m i n a t e s u c h p o i n t s

[bl.

t.

land enumerate again

J

-

t

t h e r e m a i n i n g o n e s ) we w i l l h a v e t h a t i f 1,Z

i =

and

,...,I

'j+l

Liil

tj-l

t

tj-l,

=

"

t

t

tj-l < tj-l

-

E,; ~

ij

t

=

-

-

take

j-1-

j

t

( i i i ) If

and

We h a v e

j '

If

j-1

=

t:

It.-tj-ll 6 26. J = ij or t* J. =

Z tp

tj

slightly to t h e left".

=

E

-

t . = E, -1

j

';j € ) t j - l , t j ( . By

and

j

f o r every

such t h a t

and

# tp

for

-s j

we c a n n o t h a v e s i m u l t a n e o u s l y

dE]-l

=

j

tj '

i

and

t j - l > E,j-l

i.e.

ij

-

=

t

j

t h e n we t a k e

(a1 we h a v e

=

tj-1

we

[which i s p o s s i b l e

"we m o v e t h e p o i n t

i.

T h e n we h a v e

J-1

( i v l I n a n a n a l o g o u s w a y i f we h a v e t j z tp f o r e v e r y = t "we m o v e t h e p o i n t t slightly t o the j j r i g h t ''

i

and

.

By

cj

[ b l between any two p o i n t s of

o n e p o i n t of

cases i n w h i c h

-

d

dE

there is at

least

a n d t h e r e f o r e we h a v e c o n s i d e r e d a l l p o s s i b l e

Sj

is not an i n t e r i o r point o f

THE INTERIOR INTEGRAL

12 We h a v e

because i n t h i s

sum we h a v e

appeared i n ( i i ) i n ( i i i ) and

cldl

and

[ivl:

-
summands o f

the

t y p e that

summands o f t h e t y p e t h a t a p p e a r e d

t h i s completes t h e p r o o f of

(y'l.

$ 2 - T h e Riemann integ ha d and t h e Vahboux in te g h a L I n what f o l l o w s w e r e c a l l t h e p r o p e r t i e s of t h e Riemann and Darboux i n t e g r a l s f o r Banach s p a c e v a l u e d f u n c t i o n s . more d e t a i l s see [H-IME]; A

-

Let

appendix of c h a p t e r I .

b e a Banach s p a c e and

X

For

f:

[aIb)

X;

we

say t h a t € i s Riemann i n t e g h a b d e , and w e w r i t e € € R( (a,b) ,X)? i f there exists

Si

(where

[ti-lIti)).

We w r i t e 2.1

-

2.2

- If

-

2.3

R([a,b)l

R((a,b)

(a,b) , X I C R ( (a,b)

we h a v e

(1

(bf(tldtll

la If

f,E

R ( [a,b),Xl

then

and ,XI

+

Jb

a R[[a,b),X)

f:

f E R [ [a,b)

fn(tldt

Hence

,El.

,XI.

f E R[(a,b),Xl

I ) f n - f l )+ O ,

that

=

[a,b]

<

+X

[b-a)

11fll.

are

such

and

f(t)dt.

a

i s a B a n a c h s p a c e when endowed w i t h t h e

sup norm. However f

and

Ilf(

fER[(a.b).X)

111

does n o t i m p l y

[If[ ][I€

R((a,b));

may e v e n b e n o t L e b e s g u e i n t e g r a b l e a n d e v e n

n o t measurable:

EXAMPLE.

We t a k e t h e H i l b e r t s p a c e

t h e space o f a l l f u n c t i o n s

x:

[a,b)

X

= L,[(a,b)l. ---*

C

t h a t is,

such t h a t

THE INTERIOR I N T E G R A L

1

( x ( t I ( * <

13

m;

a
e

t and

d e n o t e s t h e element of et[s)

for

= 0

I): t

E (a,b]

be any bounded f u n c t i o n ; f ( t 1

fERc(a,b).

a) a 4 s <

-

such t h a t

f:

we d e f i n e

lti-ti-l/z<

Id1

1 lti-ti-l[2

<

(a,b]

,,"

~ , ( ( a , b ] ) ~ and

Let us g i v e the proof f o r

-

$ [ t l E C

f(a)da

t 4 b.

[because

et[tl

1

=

s # t. Let

$[t)et.

w

PROOF.

L2((a,b)l

s E [a,b),

s

I t -ti-llAd i

a

=

Q

=

and

e,[ ( a , b ) l

by

f o r any

t

b;

we

I

and

have

and t h e r e f o r e

(h-aIAd1 w h i c h i m p l i e s t h e r e s u l t .

i = l

b)

/If[

)I]$ R[(a,b))

because

,llf[t)ll

=

I$[tl

this

f u n c t i o n i s o n l y supposed t o be bounded and n o t n e c e s s a r i l y Lebesgue i n t e g r a b l e n o r measurable. If

is a t o p o l o g i c a l B T ,

[E.F.GI

gERc ( a , b ) , F I

f E R((a,b).E)

and

f - g E R( [ a , b )

does n a t n e c e s s a r i l y i m p l y t h a t

,GI

n o r does i t i m p l y

EXAMPLE.

a

a

a

We t a k e

E

=

B(x,yI

F

-

=

L2~(a,b)l,

(x(y1

-

1

G = C

function

f E R((a,b),El since

-

x[t)y(t),

a
t h e i n n e r product o f t h e H i l b e r t space

f.f@RI[a,b)l

and

e2[[a,b]l.

For t h e

o f t h e p r e c e d i n g e x a m p l e we may h a v e [ f - f l ( t l

= l$[t1I2

be any bounded p o s i t i v e f u n c t i o n .

and

11'

may

THE INTERIOR INTEGRAL

14 by

Also

of

different

- Given

a

&

s 4

6

f

that

f E D l (a,b)

f E R((a,b),Xl

4 t 6 b

B - Let say

o f t h e p r e c e d i n g example t h e f i r s t

member be

f r o m t h e f i r s t one.

2.4

for

a)

( a ] i s z e r o b u t t h e s e c o n d m e m b e r may n o t e x i s t o r may

X

and

take g ( t )

dense i n

S

[s,t)

be a Banach s p a c e and

i s Darboux i n t e g r a b l e ,

,XI

if f

satisfies

2.5

- D((a,b),XlCR([a,b).XI.

2.6

- If

X

= g(al

+

we h a v e

c X;

[a,b)

f:

and

we

we

write

t h e Darboux c o n d i t i o n :

i s f i n i t e d i m e n s i o n a l we h a v e D ( (a,b),X)

R ( [a,b)

=

,XI.

F o r Banach spaces o f i n f i n i t e d i m e n s i o n a Riemann

inte-

g r a b l e f u n c t i o n does n o t n e c e s s a r i l y s a t i s f y t h e Darboux condition.

E X A M P L E S . 1. We t a k e

X = .12aN)

(a,b);

enumeration o f t h e r a t i o n a l s o f fE R([a.b),Xl

by

is i r r a t i o n a l ;

f o r

2.

We t a k e

f [ t l

en

5

i f

a c s < t 6 b

X = L_((a,b)l

rl,r2,

and l e t

t

be

any

we d e f i n e

rn

=

...

and

f(tl = 0

i f t

we h a v e

and

fctl

=Xla,~),

t E [a,b);

we h a v e a)

fER[(a.b),L_[(a,b))l; d Sd

that for

we h a v e

Ilui,;[f)

b) f $ D C [a,b],L-[(a,b)lI;

C)

f

indeed,

i t i s easy

t o verify

- u d,S ( f l l l c Ad. indeed,

i s d i s c o n t i n u o u s a t a l l p o i n t s of

(a,b);

indeed.

THE INTERIOR INTEGRAL for

a < s < t < b

d) f

we h a v e

i s n o t measurable

gue i n t e g r a b l e ) ;

indeed,

t h e r e does n o t e x i s t ure

<E

Ilf[t)-f[s)ll

IIx

=

(and t h e r e f o r e

from c l

i t follows

K E C (a.b)

a compact

and s u c h t h a t

15

n o t Bochner-Leksthat

for

0

E >

w i t h L e b e s g u e rneas-

i s continuous.

flCKE

T h i s example i s i m p o r t a n t t h e i d e n t i c a l automorphism o f

since

6 [ [a,b)

f

i s used t o represent

1 : f o r e v e r y $Eb[[a,b)l

we h a v e $ ( t l d f ( t l =

4.

T h i s e x a m p l e a l s o shows t h a t t h e f u n c t i o n 3,. where I t j,[.tl = J a f [ s ) d s , t E a,b , i s a b s o l u t e l l y c o n t i n u o u s b u t

0

i s

n o t d e r i v a b l e a t any p o i n t : L

and i t i s easy t o see t h a t

2.7

-

2.8

-

6 [[ a , b ] fE

b llJabf[tidtl/

every

D[ ( a , b ) , X )

If

n E N

formely

implies

[If[ 1

/I€

-

are such t h a t

f , f n E D[(a.b).Xl

and

f n ( t l 3 fctl

+0

Ilfn(tl-f[tllldt

2.10

,XI. D[(a,b)l

and

a llf(t)lldt.

-

2.9

, X ) C D[ ( a , b ]

and

I f t h e sequence

t o the function

Jabllfn(t)-f(tilldt w i t h t h e sup norm

4

o

f:

f o r every

fn[tldt

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

Ilfnll"

Ib

t $ (a,b]

+

a

for

then

f(t1dt.

converges

uni-

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

and

D[[a,b),X)

M

Endowed

is a Banach space.

THE INTERIOR INTEGRAL

16 2.11 and

- I f LE,F,G)

,F)

g E O ( (a,b)

imply

Ibd e n o t e s

If

i s a t o p o l o g i c a l BT, f * g E D ( (a,b)

fED((a,b),El

,GI.

t h e u s u a l u p p e r Riemann i n t e g r a l f o r po-

'a

sitive,

b o u n d e d n u m e r i c a l f u n c t i o n s we h a v e

2.12

- Given

f:

only i f f o r every fEE E ( [ a , b ] , X )

2.13

that

IlflI

-

E

[a,b)

> 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 <

.

X

06

I) f (tn) -fE

first and

t h e dollowing phopehrtieb b t e p dunctionb.

thehe existb tn.C t E (a,b

f(t+)

func-

(

(tn) II +

d€D

buch t h a t

w e have

II f ,

( t n -f, ) ( t +11) +

+ ( J f E ( t + ) - f E (+t m llf,(tm)-f(tm)l1 )l( < ,< 2 E +

Ilf,(tn)-f,(t+)Il + Ilf,(t,)-f,(t+)ll

and t h i s i m p l i e s t h e e x i s t e n c e of f ( t + ) . b) c ) Given E > 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) <

6b> 0

and

E

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

exists

17

a

I

0

(f

1<

E;

such

analogously

"1 a d + & , (( f ) <

such t h a t

such t h a t

(.

,t+6

w]

and t h e r e

E

[ ( f ) < E . The s e t s t+6 ] b-6b i b [

)b-6bIb

[, ] t - 6

form an open c o n v e r i n g of

I

[ I

[a,b]; i f

i s a f i n i t e subcovering w e t a k e d: t o = a < t < t < 1 2

...<

=

such t h a t

to = a ,

...,

si-SS

and w e h a v e

c)

s -6

i

t 2 = sl,

< t 1 < a+6,,

< t2i-l<

i-1

t2i

I

-

... Sit

. . . I

t

Id1

= b

wi(f)<E:.

+ a).

If

wi(f) <

and w e o b v i o u s l y have

COROLLARY 3.2. Given a ) Fok ewehy

E >

E

11 f - f d ,

we define

5.

E

E ( [ a h ) ,X)

by

11 s E .

f E G ( [a,b) , X I 0

fdls.

we have

-the b e t s

{ t E (alb(

1 [If

{t E ) a , b )

I 11 f ( t )-f

( t + ) -(t)II f a

E)

and

ahe 6iniZe.

(t-11) a 1

b ) T h e b e 2 06 dibcontinuitieb 0 6 f i b c o u n t a b l e . P R O O F . b ) f o l l o w s from a ) and a) f o l l o w s from c) o f Theorem 3.1.

THE INTERIOR INTEGRAL

18

- 6((a,b] , X ) C G ( [ a r b ) , X ) . 3 . 4 - BV( (a,b] , X I C G ( (a,b) ,X) . P R O O F . See [H-IME] , Theorem I . 2 . 7 . 3.3

a ) of Theorem 3.1 i m p l i e s

06

3 . 5 - T h e unido4m t i m i t gutated dunctian. T h e r e f o r e , i f w e endow

Ilfll

G ( (a,b) , X )

and

16 (E,F,G)

gE G ( [a,b) ,F)

T H E O R E M 3.8. G i v e n

agtsb

a Banach bpace; and

i b

t E [a,b)

f,g:

.

f . g € G( [a,b) , G )

impey

.

.

a t o p o e o g i c a t B T , t h e n f € G((a,b) ,E)

i b

a4e e q u i v a e e n t a ] f € G((a,b),X)

he-

a

w i t h t h e norm

G ( ( a h ) ,X) c D ( (a,b) , X I

3.7.

i b

nup Ilf(t)ll

=

w e have T H E O R E M 3 . 6 . G((a,b],X)

h e g u t a t e d dunctiond

+X

[a,b)

and

t h e 6oLtouring phopehtieb

g(t) = g(a)

t

+

f(s)ds

d o h eweny

Ja

b l F o h eueny

t E [a,b[

thehe

exibtb

g;(t)

= € ( t + )and

d o h euehy t E ] a , b ] thehe e x i a t b gL(t) = € ( t - 1 . and g i d a phimitiwe Od f (i.~., c ) € € G ( [a,b) , X ) g i b continuoub and o u t b i d e 0 6 a c o u n t a b t e b u b b e t 06 (a,b) thehe e x i 4 ; t o g ' ( t ) = f ( t ) I . PROOF. a) I b ) . We w i l l prove t h a t for to ( a , b ( w e have

.

g; (to)= f (to+)We have

-

l1i [ g ( t o + h ) - g ( t 0 ) ]

=

J

f ( t o )= 1

to+h

such that far to< s < to+& 3.2.a)

. By

f(s)ds

-

f ( t o +=)

[ € ( s ) - f (to+)]ds.

The r e s u l t follows becausk f o r every b ) +c)

I

to+h

b ) w e have

f

E

> 0

there is a

[I f (s)-f (to+)((<

w e have €

G ( (a,b] , X )

t h e subset of p o i n t s where w e have

E,.

S> O

and t h e r e f o r e by f ( t + )# f ( t ) o r

19

THE I N T E R I O R I N T E G R A L f ( t - ) # f ( t ) i s c o u n t a b l e ; a t t h e o t h e r p o i n t s w e have

then

.

g ' ( t ) = f ( t ) g i s obviously continuous.

c ) ==3a ) . By a ) of Theorem 3 . 1 t h e r e e x i s t s a sequence s t e p f u n c t i o n s f n E E ( (a,b) , X ) t h a t converges uniformly

of to

f. W e define

t

i,

+

gn(t) = g(a)

fn(s)ds;

i t i s immediate t h a t t h i s sequence

i where functions,

i(t)=

to

a)

S (a)

+

g(a)

= g(a),

['f

g,11

(s)ds. and g

'a

converges uniformly

-

g and g a r e continuous i s a p r i m i t i v e of f (by

9'

( t ) = g ' ( t ) outBy [C] 3.2.2 w e have

c ) )I t h e r e f o r e t h e r e e x i s t s

b)

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

(a,b)

.

= g.

then

Given

f:

(a,b) 4 X

every E > 0 t h e .set is obvious t h a t 3.9

-

THEOREM 3 . 1 0 . G i v e n ahe e q u i v a l e n t :

f:

i b

and

b ) fEG((a,b),X)

and and

c ) fEG((a,b) ,X) f €

IIf (t)ll 2

-

if

for

is finite.

E)

It

G ( (a,b) ,X).

a ck!obed b u b b p a c e a d

(a,b)

a ) fEG([a,bj,X)

d)

I

{ t € (a,b]

co( [a,b] ,X)

fe c o ( [a,b) , X )

we w r i t e

t h e d o l t o w i n g pkopehttieb

X

1;'

f(a)da = 0

f(t-) = 0 f ( t + )= 0

d o h dl

all doh a t e

s,tE

[a,.>.

t€)a,b).

dok

t E [a,b'[.

co( [a&] , X I .

PROOF. I f w e t a k e

t g ( t ) = Iaf ( s ) d s

t h e equivalence of a ) , b)

and c) follows f r o m t h e equivalence of a ) I b ) and c) of theo-

r e m 3.8; i t i s obvious t h a t d ) i m p l i e s b) and c) ; by c ) Theorem 3 . 1 b) and c) imply d )

of

.

DEFINITION.

G-((a,b),X)

= tfEG([a,bj,X)

1

f(a) = 0

f ( t ) = f (t-) f o r I f we r e c a l l t h a t the operators

@a: f E G ( ( a , b j , X )

d f (a) E X

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

f ( t ) - f ( t - ) €X ,

and

at:

-

and

t ~]a,b)}.

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

20

t ~ ) a , b, ) a r e continuous and G- ( ( a r b )

8x1

n

=

;

actgb

hence

-

3.11

G-([a,b) , X )

i b a cLabed b u b b p a c e

06

G((a,b) , X I .

(I-f ) ( t ) = f (t-) i f

D E F I N I T I O N . f E G ( (a,b] ,X): we d e f i n e and (I- f ) ( a ) = 0 .

t .]a,,)

T H E O R E M 3 . 1 2 . We have a ) I- i6 a continuoua p n a j e c t i a n G- ( [ a t b )

8x1

*

b ) The kehneL c ) G- ( [a,b) , X I

06

+

+

onto

c o ( [ a 8 b ) 8x1. = (01

Co( k t b ] r x )

w d

= G ( [arb]

8x1

;

m u g be w h i t t e n u f i i ~ u e L y W h e J ~ e f-E G-((a,b) ,X) a n d f o E cd(a,b) I X ) ;

muhe p h e c i d e l g evehy f = f-

id

nco([a,b) , X I

G- ( (atb) 8x1 ab

I-

G((a,b) ,XI

06

fo

€ € G ( (a,b) , X )

f- = I - f and fo = f - I - f . P R O O F . a ) L e t us prove t h a t I-f i s r e g u l a t e d : f o r every ( I - f ) ( t + )= f ( t + ) because i f t E (a,b( w e have

we have

I f (t+)-f

f o r every

s ~ ] t , t + 6[

(s)ll

t h e n also

<

E

IIf ( t + ) - f (s-111

<

E

i.e.

€1

t , t + 6 [. I n an analogous f o r every s (I-f) (t-) = f (t-) = ( I - f ) ( t ) f o r Hence I-f E G- ( ( a t b ) ,X) I t is immediate t h a t 1- is t a,b) a p r o j e c t i o n ( i . e . 1-(1-f) = 1 - f ) 8 i s continuous ( ) ) I - f ) ) < )If))) ( s i n c e I-f = f i f and only i f and i s o n t o G - ( (a,b> , X I f E G- ( (atb] 8 x ) ) b) I t follows from t h e equivalence of b) and d ) i n Theor e m '3.10. c) It follows immediately from a) and b ) . )If ( t +-)(1-f) ( s ) 1) 4

E

1

way w e prove t h a t

.

If

G ( (a,b)

denotes t h e Banach q u o t i e n t space

,XI

G( (ab)

/co ( (atb)

8x1

I

i t follows from Theorem 3.12 t h a t 3-13

-

G([a,b),X)

id

i4UmetJLiC

t U

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

21

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

06 bounded

54 - F u n c t i o n 6 A

-

Given a BT

B-vahiation a: [a,b] - + E ,

( E , F , G ) ~ and

f o r every

w e define

d€D

= S B (a,b)

SBd[a]

=

Id1

= 6 Up{ll

[a ( t i )-a (ti-l)] ’Yi i=l

1

1 1

Y i E F I / I Yill

and = SB

SB[a]

( a 4

La] = dup{SBd[a]

i s t h e ~-vaaiationof

sBl.1

I

deD);

a ( , o n ( a , b ) ) . W e say t h a t a i s and w r i t e a S B ( ( a , b j , E l I

a d u n c z i o n o d bounded B - v a h i a t i o n , if

SB[a] <

m.

REMARK 1 . If

c1

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

and w e c a l c u l a t e t h e B - v a r i a t i o n w i t h r e s p e c t instance, we w r i t e

S B ( ~ )C a ( s , t ,

to

s,t,.

..

t, for

...)I.

The f o l l o w i n g p r o p e r t i e s a r e immediate:

4.1

-

a€SB([a,b) ,El i b

a beminoam. 4.2

a) The dunction

t

t o n i c a t y i n chead i n g ;

DEFINITION.

La]

E

we h a v e (a,b) c3 SB a

( 1

SB a , c

SBo([a,b),E)

THEOREM 4 . 3 . Let

t o t h e BT

SB[c1]6 R+

a E SB((a,b) ,El

- Foh

b , SB(aIb)

-

i a a v e c t o h dpace and

SB( (a,b) , E l

[ a ] ~ lR+

[a] + SB(c,,] [a]

= {a€SB([a,b],E)

(EBIF,G)

(E,F,G)

I

i d

mono-

d o h e v e k g CE]a,b(.

I

a(a) = 01.

b e t h e t o p o t o g i c a t 81 a b d o c i a t e d

( b e e 5 O . C ) ; we h a v e

SBo ( (a&) ,EB) = SBo ( (a,b) , E )

and t h e d U n C t i 0 n b a SBo( (a,b) , E ~ ) ahe bounded. PROOF. Given a E SBo ( (a,b) , E ) w e have

since

a ( a ) = 0 . The rest i s obvious s i n c e t h e d e f i n i t i o n of

SBrci]

depends on no topology on

E.

T H E INTERIOR INTEGRAL

22

and

B

-

qe

If

rG

(E,F,G) is a LCTB and we define

and

SB [a] = Aup{SB 9

9 Id

a: [a,b) -E,

I

[a]

for d e n

deD}

is, by definition, the q-B-vahiation of a. We say that a is a ,junction 06 bounded B-vahiation, and we write if for every

REMARK

qE

rG

a

SB( [a&)

we have

I

SBq [a] <

m.

2 . Unless otherwise specified, all the results of this or, more generally, of this work are valid if we replace the BT (E,F,G) or its particular instances (see item C) by a LCBT (E,F,Z) or its particularisations; for this purpose we replace in the proofs the norm of G by the seminorms

5

qE

rz.

C

-

Examples

Ex. SV( (a,b) ,L(X,Y)) - Given a normed space X and a Banach space Y we consider the BT (L(X,Y),X,Y) (see ex. 1 of 50,C) ; then we write S V [ a ] instead of SBIa] and SV( [a,b) ,L(X,Y)) instead of SB((a,b] ,L(X,Y)). The elements of SV( [a,b) ,L(XIY) are called ,junctionA 06 bounded Aemiv ahiation

.

Ex. BV([a,b),Y’) - If we consider the BT (Y,Y’,C) (see ex.3 of 50,C) we write Via] instead of SB[a] and BV((a,b) ,Y’) instead of SB( b,b) ,Y I ) . We say that the elements of BV( [a,b) ,Y1) are 6unctionA 0 6 bounded vahiation. Obviously we have

THE I N T E R I O R INTEGRAL

23

and w e o b t a i n t h e r e f o r e t h e u s u a l n o t i o n of f u n c t i o n of bounded v a r i a t i o n , i . e . , a f u n c t i o n c1 such t h a t i t s v a h i a t i o n V[a] = b U p Vd[a] i s f i n i t e . Of c o u r s e t h i s d e f i n i t i o n d€D may be g i v e n f o r any normed s p a c e . For r e a l f u n c t i o n s w e obtain t h e usual notion, i . e . ,

functions t h a t a r e t h e d i f f e r -

ence of two monotonic o n e s .

REMARK 3 . I t i s immediate t h a t SV( ( a b ) , L ( X , C ) 1 = B V ( [a,b] ,XI)

X

REMARK 4 . I f

i s a Banach s p a c e w e have

have

g ( t ) = g(a)

V[g]

c B V ( [a,b] ,X)

( [a,b] , X f

R(”

i.e. i f

+

I,

(E,F,G)

BW[[a,b),YJ

S0,C)

-

and

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

we w r i t e ,Y 1

REMARK 6 .

W[a]

SB([a,b],Y).

i n s t e a d of BWI [a,b]

E

R([a,b) , X ) , w e

i s a t o p o l o g i c a l BT w e have

BV( [a,b] , E ) C SB( [a,b] , E )

of

f

f ( s ) d s , where

< (b-a)l(f [ l ; i n d e e d

REMARK 5 . I f

Ex.

.

i n s t e a d of We s a y

SB[o.) BT

IIBIIV[ol].

5

[Y,C;YI

SBca]

and

(see

ex.

BW[[a,b],YI

t h a t t h e e l e m e n t of

a r e f u n c t i o n s o f weak bounded v a r i a t i o n .

S V I (a,b]

,L[C,YI

1 = BW[ [ a , b ] , Y 1

obviously.

REMARK 7 . I t i s i m m e d i a t e t h a t SV[(a,b].L(X,YllC

THEOREM 4.4.

§O,C, whehe

G i u e ~.the BT W #

PROOF. It i s enough we h a v e

IL[x,YI,L(W,X),LIW,Y11

1 0 1 , we have

S B [ [a,b],L(X,Yll

dED;

BWI ( a , b ) . L [ X , Y 1 1 .

06 ex.2,

= SV([a,b],L[X,YII.

t o show t h a t

SBd[ct]

= SVd[ct]

for every

4

THE INTERIOR INTEGRAL

24

Reciprocally,

llxi\\ < 1

x i € X.

given

U i E LIW,Y)

there exist =

IIUill

i = 1.2

IIxill

and

[ b y t h e theorem o f Hahn-Banach)

WEW

Ui ( w )

and

,...,I =

dl,

with

such t h a t

xi

and t h e r e f o r e

SVd[a]6

SBd[a],

hence t h e e q u a l i t y .

4.5 - Let we have [a.y)'(tl

PROOF.

D

b e a BT; f o r a ~ S B ( ( a , b ] , E l and y E F and W [ a . y ] ~ s B [ o ~ ] I ( y I ( , where

(E,F,G)

a-yEBW[[a.b].Gl = act)-y.

For

-

dED

t E (a,b). we h a v e

(E,F,G) be a BT.

Let

Given

THEOREM 4.6.

a ) Foh e v e h y b

Fa(f) =

a~ S B ( ( a , b ) ,E) we h a v e

fE

6 ((a,b] ,F) t h e h e

da(t1.f (t)

a b ) F a E L [ 6 ( [ a h ) ,El ,G]

c ) Foh evehy b

1J

a

(d,C)

E

and

IIFa(f)\l

and IIF,II 0 we have

da(t)-f(t)-U,,E(f;a)

exibtb

(1

, I

s SB[a]\lflI.

, I SB[a].

SB[a]wAd(f)

By Theorem 1 . 2 , Theorem 4.6 f o l l o w s f r o m T h e o r e m 4 . 1 2 b e l l o w ; f o r a d i r e c t p r o o f o f Theorem 4 . 6 s e e [H-IME], Theorems 1.15 and 11.1.1.

-

THE I N T E R I O R INTEGRAL 4.7 a ( t ) = ci

a(b)

-

a: (a,b]

16

id

tc)ti-l,ti(,

Cldl+lt

jb

a 4.8

where

- If

E

i b

a b t e p ,junction w i t h

i = 1,2

f

d o h euehy

25

,..., ( d l

b([a,b),F)

and

thehe

a(a) = co,

exibtb

ldl+l

1

d a ( t ) .f ( t ) =

a

i=l

[ci-ci-lpf

i.e.,

D'll([a,b),E).

then

B E D [ [a,b] ,El,

a

S B ( [a,b)

(ti-1).

act1 = acal

+

l.i

B[slds

and f o r e v e r y

,El

we have

f E D([a,b),FI

b

b

(11

a

P R O O F . B y t h e r e m a r k s 3 ) a n d 4 ) a n d b y 2 . 5 we h a v e SB([a,b],El

and b y 2 . 1 1 t h e second i n t e g r a l e x i s t s .

have

SiE

where

[timl,ti).

f E O ( (a,b]

Since

follows the existence o f

,F1,

jabda(t)-f[t,

REMARK 8 . M o r e g e n e r a l l y [ l ) i s v a l i d i f and

f

E D [ [a,b] , F l ,

or, i f

a

D'l)[

[a,b)

from

as w e l l as

a ,El

(1).

R(l)[[a,b),El and

We

THE INTERIOR INTEGRAL

26

However,

f €R((a,b),F).

(1) i s not v a l i d i f

ceeds 2.4, f E R [ [a,b)

PROOF.

I

a€R'')[(a,b),E)

pre-

and

,F).

- Let

4.9

as i s s h o w n by t h e e x a m p l e t h a t

b e a Banach s p a c e :

X

BW((a,b),XJc

R((a,b),X).

b

a ( t 1 d t = ba(b1

-

acc(a)

-

a

By 4 . 5

4.10

tda(t1. Jab

and 4.9

- For

a

we h a v e

and

SB( (a,b) .El

YE F

we h a v e

a - y E R ( ( a , b ) ,GI. By t h e r e m a r k s 5 a n d 7

4.11

-

B V ( [a,b],L[X,Y) CBW( [ a , b ]

E

-

a n d b y 4 . 9 we h a v e

1

C S V ( ( a , b ) ,L[X,Y))c

,L(X,YI

1 C R ( [a,b] .L[X,YI I .

The f o l l o w i n g theosem i s fundamental:

THEOREM 4 . 1 2

-

Let

(E,F,G)

f E G ( ( a , b ] ,F) :

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

be a B T ,

a

S B ( ( a , b ) ,E)

and

rb

d€ID

we define

t h e s e s e t s form a f i l t e r b a s i s and i t i s enough t o show t h a t

w e have a Cauchy f i l t e r b a s i s because by d e f i n i t i o n t h e l i m i t

THE INTERIOR INTEGRAL c) of Theorem 3.1) t h e n

diam C

z>d

enough t o show t h a t f o r

27

c 2 ~ ;f o r t h i s purpose i t i s d w e have I l u z , ~ . - u d , < . I( < E :

SB [a] u i ( f 1 where

i (j) denotes t h a t

(Ej-l,tj)

i

E

{1,2,

c [ t i - l , t i ) ; from ( a ) and t h e hypothesis

E

u i ( f ) ,<

SB 1 c.

follows t h e conclusion.

c) From

/(Fa ( f ) l ( s S B [ a l Ilf 11.

and a) follows t h a t F,

is f i n i t e since hence i f w e t a k e

= { t E [a&]

For

E

> 0

t h e set

I 11 f (t)ll2 I I f - l l + E )

f-f-E c o ( ( a , b ) , F ) by c ) of Theorem 3.12; dElD such t h a t d 3 F E w e have ‘d

and t h i s i m p l i e s c).

,5 11 s SB [a] (Ilf-ll

b) follows from c ) because i f have F a ( f l - f 2 ) = 0 by c ) .

+E)

I

fl-f2 E c o ( [ a , b ] ,F)

we

d ) follows from c ) . e ) follows from ( a ) because

E X A M P L E . Given have

a =

u € E f o r T E [ a , b [ and a = - x [ ~ , ~ [ u w e b fa - d a ( t ) .f ( t ) = u - f (T+) and f o r T E ) a , b ) and

-X[,,,]U

w e have

j:.da(t)

- f ( t ) = u * f (T-)

.

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

28

PROPOSITION 4.13. L e t f E G ( (a,b) , F l ;

be a BT,

[E,F,GI

for e u e r g

we d e f i n e

t E (a,b]

1

and

aESB((a,b),El

t

gf(t)

=

*da(slf(sl.

a

We hatre

ff

PROOF. For

E

BW( [a,b]

d € D

By 4.9

, G I C R ( [a,b] , G I

and

w e have

we h a v e

&(If] < SB[ol] IlfII.

y f € R[(a,b].G].

PROPOSITION 4.14. W i t h t h e n o t a t i o n s o f t h e p r e c e d i n g P r o p o s i t i o n we h a v e : a ) Given

t h e r e e z i s t s j f [ t o + l for e v e r y f E G( [a,b),F) if and onZg if f o r e u e r g y E F t h e r e e z i s t s a[to+)y = Lim a C t 1 - y ; t h e n we have t o €( a , b [

t+to

Yf[tO+1 - ff[tO1

=~alto+)-altol]f~to+l.

there e z i s t s f f ( t o - l f o r every if and o n l y if for e v e r y y E F t h e r e e z i s t s f E G I [ a , b ) ,F1 u ( t o - l y = L i m c t ( t 1 - y ; t h e n we have b ) Given

toE)a,b)

t+to jf(tO1

c) I f i8

j f j f [ t o - =l [ d t o 1 -

(to-l]f[to-l.

is EB- c o n t i n u o u s on t h e r i g h t ( l e f t ) a t

to

so

jf. d)

if

a

-

a

PROOF.

ff

ie r e g u l a t e d f o r e v e r y is weakly r e g u l a t e d .

i f and on&

I n o r d e r t o show t h a t t h e

I t is enough t o p r o v e a l .

c o n d i t i o n i s n e c e s s a r y we t a k e

f E G( [a,b),Fl

f

:y .

The r e s t follows f r o m

29

THE I N T E R I O R INTEGRAL

i f we r e c a l l t h a t

W)to,to+E((fl

goes t o zero with

since

E

f

is regulated.

P R O P O S I T I O N 4 . 1 5 . L e t ( E , F , G I b e a B Y , a < S B o ( [ a , b ) , E l and f E G ( (a,b) ,F), t h e n a - f E R( [ a , b ] , G I . PROOF.

B y 4 . 3 we h a v e

a~ S B o ( [ a . b ) , E B I

Theorem 3 . 1 t h e r e e x i s t s

By a ) o f

( ( f n - f ( l3 0 .

Since

I I a m f n - a * f l+ l 0;

B:

J

that

i s c o n t i n u o u s we h a v e

it follows that

b e u BP, a €

a-fnE R((a,b),G)

SBol [a,b),E)

and

t

b

a(tl-f(tldt

a

PROOF.

such

a - f E R ( [a,b),G).

THEOREM 4.16. L e t (E,F,G) f E G ( [ a , b ) , F l ; we have (21

f n E E[(a,b),F)

E s x F c+ G

from 4.5

h e n c e b y 2 . 3 we h a v e

and

a(tI-dt[l

=

a

By P r o p o s i t i o n 4 . 1 5

flslds].

a

the first integral i n

and i s a continuous f u n c t i o n of

(21 exists

f:

a

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

g

i s continuous, where

t g[t) =

fcslds,

i t f o l l o w s from Theorem 4.6 t h a t t h e r e e x i s t s

~ a b d a ~ t ) - g ( t la n d u s i n g i n t e g r a t i o n b y p a r t s we h a v e

30

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

Hence t h e i n t e g r a l o f

t h e s e c o n d member o f

a continuous function

of

i t i s enough t o p r o v e i t f o r Y E F

and

s i n c e t h e s e t of by a1 o f

G([a.b),F)

( 2 1 e x i s t s and i e

[21

f . Therefore i n order t o prove f

=

where

X I d , G I Y

these functions

Theorem 3 . 1 .

But f o r

c E [a,b)

i s total f

=

x

i n

la,clY

(21

i s tri v i a l .

REMARK 9 . One c a n p r o v e t h a t 1 2 1 i s s t i l l v a l i d i f U E

R([a,b).EBI

and

f EDI(a,b),F),

see

APPENDIX I n this

[H-DS],

1

we w i l l p r o v e a t h e o r e m t h a t

appendix

the existence o f the i n t e r i o r i n t e g r a l f o r i s not obvious

at

Theorem 6 . 8 .

since i f

a l l

not n e c e s s a r i l y complete while t h e ordered set

ID

This result

i s a LCBT,

[E,F.G)

but only

LCBT.

implies

sequentially

h a s no c o u n t a b l e

G

i s

complete

cofinal

subset.

THEOREM 4 . 1 7 . L e t (E,F,GI b e a L C B T ; f o r any c o u n t a b l e s u b s e t A of [a,b] we d e n o t e b y G A ( [ a , b ) , F l t h e s u b s p a c e of e l e m e n t s of G [ [a, b ] , F 1 t h a t h a v e no d i s c o n t i n u i t i e s o u t s i d e A. L e t dnE ID, n E m , b e s u c h t h a t

a)

U d n > A

n€N

bl

Adn

-

and a

T h e n for any

d n n A C dn+lnA;

0.

and any

SB((a,b),EI

f E GA( (a,b),FI

we

have :

PROOF.

A:

If

there exists have

F C d E

f E GA( ( a , b ) , F ) ,

dcE

D

where E

such t h a t

by c l

of

Theorem 3.1

ui ( f ) < E

E

f o r

and t h e r e f o r e

E

> 0 we

THE INTERIOR INTEGRAL

f

u{t

I

(a,b(

We d e f i n e

(*I

&E

d € D

dE

t:E

I

(lf[t+l-f(tl((>

€1

= ifld{It’-t’’l

I

with

Indeed: point

FE = { t € ) a , b )

d3FE

any i n t e r v a l

dE

IlfLtl-f(t-lIl>

I

u{tE)a,b(

l / f ( t + l - f ( t - l ( ( >€

FE, t ’ #

t’,t”E

and

EIU

t”};

of

]ti-l,ti[

d

I

w l t i - l a t i

)ti-l,ti(

of

d

( f l

that

c o n t a i n s no o t h e r p o i n t o f

< w

1.

we h a v e

w i ( f l < 4 ~ .

Ad<&€ i m p l i e s

that

contains

]t:-l,tE(

i s c o n t a i n e d i n some i n t e r v a l

and t h e r e f o r e

interval

31

no

of

t c ( f l < E . And any k( c o n t a i n s some p o i n t t‘c dE

lti-1.

A d < 6 E l and

dE [ s i n c e

j

t i L FE;

hence

8 : We w i l l now p r o v e t h a t t h e s e q u e n c e

i s a Cauchy sequence and i s t h e r e f o r e a SSCLCS.

I t i s enough t o

quence i s a Cauchy s e q u e n c e : take f o r

n

such t h a t

r,s a n

we h a v e

have a q-Cauchy

i(jl

given and

q[odr,S.-

sequenceli

s h o w t h a t if d = d r

[where

dn>FE

and

convergent since

show t h a t f o r E >

0

uds.ll

.]

qErG

by a)

the

i s

G

se-

a n d b l we c a n

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

Adn<

<

e E s ~[a] q

( h e n c e we

f o r t h i s p u r p o s e i t i s enough t o

;I = d r v d s we h a v e

denotes t h a t index

i

such t h a t

but t h i s follows from the definition of C.

any

SBq[a]

I n order t o prove t h a t t h e r e e x i s t s

and f r o m

[*I.

32

THE I N T E R I O R INTEGRAL

and t h a t we h a v e .da[tl-f[tl

= Limod

n [i.e.,

i t is

21)

e n o u g h t o show t h a t f o r a n y

q crG there exists

>

dE.

n.6'

dEE

ID

and

d = d

E.>

and a n y

0

such t h a t f o r every

n

we have

(**I For

t h i s p u r p o s e i t i s enough t o t a k e

and

Adn < tiFE; t h e n ,

a s i n B we

COROLLARY 4 . 1 8 . L e t

f o r any

there e x i s t s

(E,F,Gl

n

(**I

have

such t h a t i f

2 d

dn3FFE d

=

n

.

be a LCBP;

a E SB[ [ a , b ] , E l

f E G[ ( a , b ] ,F)

and any

J:.da(ti-f[ti.

PROOF. The r e s u l t f o l l o w s f r o m t h e p r e c e d i n g t h e o r e m i f w e A

take as [Cf.

t h e s e t of

a l l p o i n t s where

APPENDIX

6

2

THEOREM 4 . 1 9 . G i v e n a t o p o Z o g i c a 2 BT a:

is discontinuous

f

b l o f C o r o l l a r y 3.2).

[a.b]

exists PROOF.

E

such t h a t f o r e v e r y then

da[tl.f(tl [H-INE]

See

,

[E,F,GI

and

f €&[a,b),Fl

a ESB[(a,b],El.

T h e o r e m I. 1.11.

I n t h i s a p p e n d i x we e x t e n d T h e o r e m 4 . 1 9

t o the interior

integral.

TH-EOREM 4 . 2 0 . G i v e n a BT

[E,F,GI

@ ( a 1 = 0, such t h a t f o r any

PROOF. A : (Cf.

a : [a,b)

and

f E G[ [ a , b ] , F l

Let us f i r s t prove t h a t

50,C)J

there

i t i s obvious t h a t i f

-

E

with

t h e r e emists

a

takes i t s values i n

XE

E,

x@EB

then there

EB

THE INTERIOR INTEGRAL exists

y n E F.

hence

x-yn

that

nEN,

We may h a v e

I

i f

-,

tE]a,b]

such

c = b

1.

s E [a,t]

f o r a l l

[ I n t h i s case

cr(bl$!

EB

t < bl.

a ( c l € EB

CASE 1. I f

s Cc w i t h

t h e r e e x i s t s a sequence

and a sequence

0

Yn

IIa[sn)-Ynll hence

IIx-ynII +

but

0

If there i s a

a ( s 1 E EB

o r even

c = a

a c t 1 E EB

a [ s n l $ EB

--f

we d e f i n e

= b l l p { t E [a,b]

c

yn

does n o t converge.

a [ t l $ EB

but

such t h a t

33

such t h a t

-,

~ r ( s ~ 1 - y d o e s n o t c o n v e r g e . I n t h i s c a s e we d e f i n e n = yn, n E N , and f ( s l = 0 i f s % sn, n = l , Z , . . .

f[sn+ll

f E G [ [a,b] ,Fl

O b v i o u s l y we h a v e

.

but

r - d a i t l .f [ t l does n o t e x i s t s i n c e f o r any that

ti-l

-

s

n +1

= s i f we t a k e 6;' n' such t h a t f [ q i * l = 0

nk.

and

there i s a

t[

and

c

=

d E D

a(snlyn

does n o t converge,

d

such

for

i%k,

-

0

= 0;;'

then

t h a t d o e s n o t become a r b i t r a r i l y s m a l l s i n c e but

d' 2

a[clyn

hence

t l .f t l

[*da does n o t e x i s t .

-

CASE 2 .

F

yn

f

such t h a t

We d e f i n e

verge. s

0

Sn+l* nE

;I -

-

d

-

take

6;

that

f[h.p11

we t a k e

a(snl.yn

f[sn+ll

=

y,,

s

- 0

n€N,

such t h a t

ni'

if =

0

i<

t

f(sl = 0

124

= s

-

n

If

b u t then! dEm

161-1 and

s

a[clyn

and

f E G([a,b],Fl

e EB 1 a n d n does n o t con-

[hence

+c

n and

does n o t e x i s t s i n c e f o r any

and

ci;lr = s

tidl ~

there ( a 4 = c a n d i f we

-

a+n d ~

ni;ll

we h a v e

IIu;I,E. that,

EB

!t

N. We h a v e

Jac.da[t).f[tl i s a

a[cl

If

-

0d -n n

.I)

- 11

[aCcl-aCsn1]yll

a s i n C a s e 1, d o e s n o t b e c o m e a r b i t r a r i l y

small.

such

THE INTERIOR INTEGRAL

34

a : [a,b)

8: Next l e t us prove t h a t

i s bounded.

i s n o t bounded t h e r e e x i s t s a m o n o t o n i c sequence t

I f

a

for

instance

tn4c,

such t h a t

and t h e r e e x i s t s a sequence

is n o t c o n v e r g e n t . n€N,

+ EB

and

f ( t l

f E G ( [a,b],Y).

give

E

dn:

a

for

a l l

0

> 0

t

0

=

at

0

F

yn

We d e f i n e

f ( t )

1

<

...

n'

since

such t h a t

i f

= y,

t < t n'

tn-lC

obviously

Jac-da(t~.fit)

nEN

< t n< c

p 21, because

0

a l l other points;

L e t u s show t h a t there exists

< t

I I a ( t n l l l B > 2'",

such t h a t

does n o t

exist:

for the division

we h a v e

f(t1

-

i f

yi

t€)ti-l,ti(,

i€m.

Hence

w h i c h becomes a r b i t r a r i l y b i g w i t h

11 a [ c l - a ( ti becomes a r b i t r a r i l y s m a l l w i t h C:

bounded.

Theorem 1 . 2

(EB.F,G) f o r any

n +P

-

increasing since y

p

--i,

n+P

0.

a

i s by

a:

f E b ( [a,b)

daltl.f(t1

Ib a

hence b y Theorem 4.19

111 ' Y

because

i s a t o p o l o g i c a l BT a n d t h e r e f o r e

H e n c e we p r o v e d t h a t But

n

-

Ib

[a,b]

,F1

EB

and t h a t

there exists

. d a ( t l .f ( t

I,

a we h a v e

f o l l o w s s i n c e by Theorem 4.3

a

SBo( (a,b),EB)j

we h a v e

the result

aESBo~[a,b),E].

THE INTERIOR INTEGRAL APPENDIX

3

I n t h i s a p p e n d i x we g i v e a m o d i f i e d formula

[a,b]

a

G‘(

i n t e g r a t i o n by p a r t s

for the interior integral.

G i v e n a BT

a:

35

+E

[a,b],El,

(E,F,G) i s

iff o r

yE F

t E [a,b]

,El

THEOREM 4.21. G i v e n a ,Fl,

-

every

the function a(tI.yE G

Definition:

G*SB[ ( a , b )

f E G [ [a,b]

a function

w e a k l y r e g u l a t e d , a n d we w r i t e

a-y: i s regulated.

we s a y t h a t

=

G[

BT

[a,b) , E l

(E,F,Gl,

n s B ( (a,b) , E l .

a~ G ‘ S B ( [ a , b ) , E )

and

there exists

and we h a v e

+

c

[ f [ t + l - f ~ t l ] - [ a : u ( t l - a [ t - l ][ f [ t l - f [ t - l ] l

C[a[t+l-a(tl]

astcb

where

h(a-l

PROOF.

a[t+hly,

A:

Let

3.2

US

f i r s t prove that

for

h = a,f,

and

i.e.,

i s finite;

t h e s e c o n d member o f

t h a t t h e s e r i e s i s summable.

[al i s

By a1 o f Co-

the set

E t

FE =

€

[a,b)

1 \ l f ( t + l - f [ tI l\ > € 1

g i v e n any f i n i t e s u b s e t

0 < h
hence

h [ b + ) = h(b1,

etc..

hJ.0

w e l l defined, rollary

and

= h(a1

lim

a(t+)y =

I

t ’ , t ” E

F’,

F’C

t’ % t”}

[a,b[

r l cFB, for

we h a v e

THE INTERIOR INTEGRAL

36

Therefore by the Cauchy criterium there exists

1

ast6b

[a[t+l-a(t)]

Cf(t.1-f

(t1-J.

For the other series the proof i s analogous. 8 : In order to prove n o w that

exists and that we have (0.1 E >

0

d E ID

there exists

-f( ti - 1

11

-

-

such that f o r

d

E

<

Id1

I:

in1

we have

d

[a (t 1 -a[ ti- I] [f [ ti 1 -f (ti- I]

-

11 11

The sum inside

it is enough to show that f o r any

I I1 ,<

E

.

is e q u a l to

[acti-l-a[S;l]

Id1

1 [a(ti1-a[6;l]

i=l

[fLtil-f(ti-l] [f (ti-l-f

-

+

The norms of the first and third sums i n ( y 1 a r e cSB[a]w;(fl hence if

d’

they are

6

is such that

4E for a l l

d 2 d’.

THE INTERIOR INTEGRAL

37

L e t u s now c o n s i d e r t h e s e c o n d sum i n ( y l : 3.2

rollary

is f i n i t e ; d”€ID

the set

l e t

b e t h e number o f i t s e l e m e n t s

kE

u;,, [ a - y l 6

be such t h a t

where

t h e n we h a v e f o r

t ETE;

,<

k

E

2-

f o r svery

A

2 + SB[a] 8kE

we h a v e ( 8 1 .

and l e t = f(t1-f(t-1

2 d”:

d

BSB [a]

=

4 4 .

d”’EID

t h e n o r m o f t h e l a s t sum i n ( y ) i s ,<

d >, d ’ v d ” v d ” ’

y

BkE

I n a n a n a l o g o u s way t h e r e e x i s t s d >, d ” ’

b y a 1 o f Co-

such t h a t f o r

$

hence f o r

Q.E.O.

Reciprocally

-

THEOREM 4 . 2 2 . a:

[a, b ]

Given a BT ( E , F , G I and a bounded f u n c t i o n such t h a t t h e r e e z i s t s

EB

I,“.a[t

f o r every

f

E G ( (a,b),F1

P R O O F . By t h e s y m m e t r i c a l l o g i c a l BT [EB,F,GI for

1.df[ t )

then

a € G‘SB(

o f Theorem 1.2

(a,b] ,El.

applied t o t h e topo-

IY& ( [ a , b ] , F 1

every

there exists

and

a

a

Hence u s i n g i n t e g r a t i o n b y p a r t s we s e e t h a t e x i s t s f o r every u E SB( (a,b]

,El.

fE &([a,b],Fli

by Theorem 4.19

we h a v e t h e n

THE INTERIOR INTEGRAL

38

I n order t o p r o v e t h a t we t a k e

YE F

and any

t h e r e exists

f

a € G'([a,b),E)

= ~ ] , , ~ ) Ey G ( [ a , b ] , F l ;

Jab.,[t)-df(t)

[

.a(t)-df(t)

f o r any

T

E)a,b)

by h y p o t h e s i s

and

- Bim

cr(s')y

= a ( ~ - ) y .

E'+T

I f we t a k e T E [a,b[ t h a t t h e r e exists

and

=

we h a v e a n a l o g o u s l y

'(-r,b('

b *a(tl.df(tl.

CI(T+)Y =

a

EXAMPLE.

q u e n t i a l l y complete,

i s r e f l e x i v e o r w e a k l y se-

Y

One c a n show t h a t i f

a13 functions of

SV[ (a,b) ,L(X,YI

1 are

regulated.

-

55

RepheAenkatian theohemb and t h e theohem a6 tfctLy

In Theorem 4.12 we saw that the functions of bounded B-variation define naturally linear continuous operators. The next theorems shows us an important situation where all linear continuous operators are represented by functions of bounded semivariation. A

-

THEOREM 5 . 1 . Let

X

and

Y

SVo((a,b) ,L(x,y))

be Banach Apaceb; t h e mapping c--*

i d an i b o m e t h q ( i . e . [ ( F a [ (= SV[a] dpace o n t o t h e becond whehe d o h f Fc,w =

Jr.a.(t) -f(t);

w e have

Fa

L[G-((atb)j,X

1

E

06 t h e .

tY]

6.ihb.t Banach G- ( (a,b],X) we dedine

a(t)x =

-

~ c x i xa,t)xl* l

Q the mapping 01 Fa. a) By Theorem 4.12 the mapping is well defined and is

PROOF. Let us denote by

.

obviously linear and continuous (1) Fa )I 5 SV [a] ) b) Q is one to one: if Q # 0 there exist and x X such that Q(T)X f 0; we take f = x)~,,Ix G-([a,b] ,XI

T c

]a,b]

39

THE I N T E R I O R I N T E G R A L and w e have

# 0,

Fa(f)

i.e.

Fa

# 0

since

b Fa(f) = c) Q

(t)x

*da(t)x

l a 4

a i s onto: given

= a(T)x.

[a,b) , X ) ,Y]

F E L[G-(

by a ) w e know

t h a t i f t h e r e e x i s t s an

such t h a t F=Fa c1 E S V o ( [ a , b ] , L ( X , Y ) ) and XEX; a ( ~ )= x F ( X I ~ , , I X ) for a l l T c)a,b)

t h e n w e have

l e t u s t a k e t h i s a s a d e f i n i t i o n ; we m u s t prove ( i ) sv[aJ

6

(ii) F

IIFII

a

= F.

i)

ii) W e have F = Fa

F , F a E L[G-

( [ a , b ] , X ) ,Y] ; i n o r d e r t o show t h a t

i t is enough t o prove t h a t t h e y t a k e t h e same v a l u e since these elements

on t h e elements of t h e form form a t o t a l s e t i n

x)a,T)x

(by a ) of Theorem 3 . 1 ) ; w e

G- ( [ a , b ] , X )

have b

COROLLARY 5 . 2 .

Fan: evekg

a

S v ( [a,b) , L ( X , y ) )

we have

b SV[a]

= bup{Ilj - d u ( t ) . f ( t ) l l

I

fEG-((a,b],X),

Ilf11s1).

a

REMARK 1 . I f w e t a k e

Y = C , by Remark 3 of $ 4 w e have

G- ( [ a h ),XI

REMARK 2 . I f w e t a k e

'

=

BVo( [ a h ) ,X’)

.

X = C , by Remark 6 of 5 4 w e have

L [G- ( (a,b]hYI

=

BWo ( [ a , b ) , Y )

.

EXAMPLES 1. Take

Y = X,

t o €] a , b )

t h e a r r e s p o n d i n g element

a

€

and

F ( f ) E f ( t o ) ; then f o r

SVo ( [a,b] ,L ( X ) )

w e have

T H E INTERIOR INTEGRAL

40

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

x ] ~ , ~( t )o ) x =

(to,b) ( t ) x

a = X[to,b)lX.

hence

2 . Take

t o €[ a , b ]

Y = X,

and

F(f)

f ( t o + ) ; then w e

have a ( t ) x = F[x l a , t l x l

= Xla,t) (to+)x = x]to,b] (t) x

= X]to,b)lX*

i.e.,

3. Take Y = G-([a,b),X) and F ( f ) p f ; then f o r t h e corresponding element a c SVo( [a,.) , L ( X , G - ( ( a , b ] , X ) ) ) w e have a ( t ) x = F [ x ] ~ , ~ ) x ]= x ) , , ~ ) X E Y

u

and f o r

Since UE

-E

[a,b)

w e have

= f

J:-da(t).i(t)

[a,b]

= G-((a,b] ,XI

f o r every

f ~ ~ - ( [ a , ,bX )I ,

for

w e have b

5. Take CIE SVo (

hence

[a,b] ,L ( X I

and

b F ( f ) = i f ( T ) d T ; w e have

and

b

a ( t ) = (t-a)IX. 6. Z

fox

Y = X

is a

Banach s p a c e ,

€€G-([a,b],G([a,b],Z))

X = G([a,b] , Z ) ,

w e define

Y = 2

and

F ( f ) (0) = f ( a ) ( u ) ,

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

41

u c [ a , b ] . Then a SVo([a,b),L[G([a,b],Z),Z]) g c G ( ( a , b ) ,Z) and u E ) a , b ) we have

where

[a(t)g]

[ a ( t ) g ] (a) = 0 .

and

7 . Take every Then

[x)~,~) (a)g]

( u ) = F [ x ] , , ~ ) ~ ]( 0 ) =

Y = c,((a,b),X)

U E (a,b[,

and

=

(0)

a SVO((a,b],L(X,co((a,bj,X)))

4

f , i.e.,

for

F(f)(b) = -f(b).

and

and

x [u ,b) ( t )

-

F ( f ) = f,

F ( f ) ( o ) = f(u+)-f(u)

and f o r

C (a,b(

for

Q

-

XIalt]

we

have

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

-

(0)

X]u,b] ( t ) x

-

= X)a,t)

(U+)x

(a)x =

X ( u , b ] ( t ) x = -XIa} ( t )x

and [ a ( t ) x ] b = F[x 8 . Take

for

1-1

( b ) x = -x{b} ( t ) X .

u c R ( ( a , b ] , L ( X , ~ ) ) and

f E G - ( ( a , b ) , X ) . Then

a

i.e.

X] ( b ) =

b F(f) = Jau(t).f(t)dt

aESVo([a,b],L(X,Y)) b

a

and

THE INTERIOR INTEGRAL

42

b

6 -

Let

X

S

be a normed space and

Given a f u n c t i o n

u:

+L(X,Y)

(a,b)

Y

a Banach space.

f o r every

dED

we

define

and SCU]

sru]

I

= 4Up{sd[u]

dED}.We

write

THEOREM 5.3. PROOF.

.LLX,Y 1 1

s ( (a,b)

w i t h t h e norm

i f

u

+ s [u]

.

is a Banach s p a c e when endowed

One c a n g i v e a d i r e c t p r o o f o f T h e o r e m 5 . 3

sult also f o l l o w s f r o m Theorem 5.5

REMARK 3 . of

u~s[(a,b),L(X,Yll

a.

<

[a,b];

I n the definitions hence

we r e p l a c e

EXAMPLE 1.

(a,b)

If

Y

a b o v e we d o n o t u s e a n y s t r u c t u r e

may b e r e p l a c e d b y a n y s e t

by t h e f i n i t e subsets

dED

C

=

but the re-

bellow.

F

of

I).

I

[and

we g e t

absol u t e l y summable s e r i e s w i t h i n d i c e s i n

E X A M P L E 2.

If

X

-

C

o n e c a n show t h a t

11 1

IIyIIc = bup yill) i.e., FCI i€F m a b l e s e r i e s w i t h i n d i c e s i n I. [where

T H E O R E M 5.4. L e t and

UE

t h e space o f Y

b e a normed s p a c e , S [ I , L ( X , Y ) ) ~ tJe have X

I.

Y

-valued

sum-

a Banaah s p a c e

THE INTERIOR INTEGRAL PROOF.

a)

mable,

b y t h e Cauchy

every

E

1 u(i)xi

I n o r d e r t o show t h a t t h e s e r i e s

> O

subset

i€F o b v i o u s if s[u]

U(i1Xill i f

0;

=

4

such t h a t

we h a v e

E .

s[u]

# 0,

since

xEco[I.X1.

the set

> 0

E/S[U]

’

F E C I

F’nFE = 0

F’C I with

11 1 This i s

i s sum-

iE I

c r i t e r i u m i t i s enough t o p r o v e t h a t f o r

t h e r e e x i s t s a f i n i t e subset

f o r any f i n i t e

given

43

FE = { i E

is f i n i t e ,

and f o r

F’nFE

0

b l I t i s immediate that

we h a v e

is l i n e a r and t h a t

Fu

rUiI I ~ I.I

llFu(x)Il hence c ) .

The n e x t t h e o r e m c o m p l e t e s t h e p r e c e d i n g o n e .

THEOREM 5 . 5 .

Let

X

b e a normed s p a c e and

t h e mapping UE

s(I,L(X,Y)l

i s an i s o m e t r y ( i . e . llFU/I o n t o t h e s e c o n d , zjhere f o r

FUE L[c,[I,Xl,Y]

o f t h e f i r s t Banach s p a c e we d e f i n e

= s[u], XE

=

F,[XJ

-

a Banach space;

Y

co(I,Xl

1 u(ilxi; i EI

and x o E X tje h a v e u ( i l x o = F u ( e x I t j h e re e i i o i s t h e eZem e nt of c o [ I ) t h a t t a k e s t h e v a t u e 1 a t i and is z e r o a t a22 o t h e r e Z e m e n t s of 1). for

i E I

L e t us d e n o t e by

PROOF.

u

t h e mapping

Fu

b ) The m a p p i n g

i s injective i.e.

F u # 0; that

0

a 1 By T h e o r e m 5 . 4

indeed,

u(ilxo

# 0;

0

u #

i f

0

there exist

0

Fu.

u # 0

i E I

and

I I F u I I 4 s[u]. implies

xoE X such

[e x 1 = u c i l x # 0. u i o i s surjective: given F E L[co[I,Xl,Y]

t h e n we h a v e

cl T h e m a p p i n g

C,

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

F

44 we w a n t t o s h o w t h a t

F = Fu fine

If

I N T E R I O R INTEGRAL

THE

and

there exists

IIFII = s [u],

UE

s(I,L(X,YII

i€ I and

For every

u[i)x0 = F ( ~ ~ X ~ u I ; i i ) L~[ X , Y I

J

6

s [u]

hence

IIFII.

F = Fu

7

because b o t h o p e r a t o r s a r e

t i n u o u s and t h e y a r e e q u a l on t h e elements which form a t o t a l subset o f

-

T H E O R E M 5.6. L e t [a,u)

we d e -

since

r

to

such t h a t

xoE X

F =

and

X

,L(X,Y)lxs[

S V o [ [a,b]

(I L[G-[

,Xl,Y]

L[G[ [a,b)

.

1 0’

~ i E I,x o E X,

be Banach s p a c e s ; t h e mapping

Y

from

F,+FU

e

co(I,XI.

con-

[a,b)

,Xl,Y]xL[co~

[a,b)

[a,b]

,L(X,YII

,XI,Y]

1

is

a b i c o n t i m o u s i s o m o r p h i s m of t h e f i r s t Banach s p a c e o n t o t h e

s e c o n d , where for e v e r y

f E G [ (a,b) b

we d e f i n e

,XI

and Fu[f)

1

=

u[t).[f[t)-f(t-l)

(f[a-)=Ol;

act
o e have

and

a ( t 1 x = F[xJ.,~)x]

t E [dab)

and

PROOF. By

Theorem 3.12

fc

X E

for every

u c t l x = F[x{,lx]

x.

G[ [ a , b ] , X l

t h e mapping C,

( 1 - f , f - I - f ) €G ~ ~ ( a , b ) , X I ~ c o ~ [ a , b ] , X l

i s a b i c o n t i n u o u s i s o m o r p h i s m o f t h e f i r s t Banach space o n t o t h e second. mapping

F

Y

Hence f o r any Banach s p a c e e *

(F1,F2)

from

L[G[(a.b),Xl,Y]

-

L I G ~ ~ ( a . b ) . X l . Y ~ ~ L ~ c o ~ ( ~ . b ] . Xwl h, e Y r~e

F2[fI = FIf-I-fl and,

F2(f-)

-

[hence 0

since

Fl[f-I-f) f--I-f

we h a v e t h e n a t u r a l

=

01

0

t o

Fl(f)

= F(f-1

and

since

(f-1-f)-

= 0,

which i s a l s o a bicon-

45

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

t i n u o u s i s o m o r p h i s m o f t h e f i r s t Banach s p a c e o n t o t h e second; so t h e r e s u l t f o l l o w s f r o m t h e Theorems 5.1

REMARK 4 . I n t h e n u m e r i c a l c a s e ( i . e .

-

X = Y

=

RI

t h i s theo-

[K].

rem i s due t o K a l t e n b o r n C

and 5.5.

The t h e o r e m o f H e l l y o f t h i s i t e m i s f u n d a m e n t a l i n

t h i s work.

THEOREM 5 . 7 . L e t a

x

and

b e Banach s p a c e s and

Y

E SVo~[a,b),L~X,Yll,~~N s u, o h t h a t f o r e v e r y

- &krn I

there e x i s t s F [ f l

n+w

then a ) There e x i s t s b ) There e x i s t s

t E (a,b]

and

PROOF.

We d e f i n e

.dan(t).f(tl,

a

s M f o r a l l ncN. a € SVo( (a.b) , L ( X , Y I I such t h a t

J

=

x E X.

SV[an]

b .da[tl.f(tl

a z d we h a v e

fEG([a,b],Xl

every

b

suah t h a t

M > 0

Fcfl

for every

f E G-[(a,b,),X)

-

an[tl.x

a[tl.x

for

1

rb

4.12

we h a v e

. d a n [ t l * f ( t ) ; by d l of Theorem a F n E L[G-((a,b),X],Y], hence by t h e theorem o f

Banach-Steinhauss

all

n€:N

F E L[G-[

Fn[fl

=

there exists

(and t h e r e f o r e

[a,b) , X I ,Y].

M z 0

SV[an]

such t h a t

By T h e o r e m 5 . 1

IIFn(I s M

b y T h e o r e m 5.11

s M

f o r

and

there exists

a~SVo[(a.b),L~X,Yll such t h a t F(f)

=

Jb

.daltl.fltl

a

f o r every

fEG-((a,b),Xl.

If we t a k e t h e n

f

=

x]a,Tlx

we

have an[T1x

b

-(

*dctn(t)'f[tl

--*

F [ f l

= \-da[tlX]a,T][t)~

a

= a ( ~ ) x

and t h i s c o m p l e t e s t h e p r o o f o f b l .

I n what f o l l o w s

WE

g i v e a r e c i p r o c a l o f Theorem 5.7.

THE I N T E R I O R INTEGRAL

46

THEOREM 5.8. T H E THEOREM O F HELLY - G i v e n a BT (E,F,GI and a sequence a n S B ( (a,b] ,El w i t h SB[an] 6 M f o r e v e r y n :N and such t h a t t h e r e e x i s t s a: [a,b] E with an[tly a[t)y f o r e v e r y t E [a,b] and aZZ Y E F , t h e n 2 ) aESB([a,b],El and SB[a] 6 M . 2) F o r e v e r y f E G ( [a,b] , F 1 we have

-

-

Jb a

a

-

.dan[t).f(tl

- -

In t h e numerical case ( i . e . , E

n

F

G = R

- hence

are functions of bounded variation) this theorem

the

was

proved by Helly f o r t h e usual Riemann-Stieltjes integral. We

will obtain Theorem 5 . 8 as a particular case of the

THEOREM 5 . 9 . L e t [ E , F , G I b e a B T , 9 a f i Z t e r on a s e t L and f o r e v e r y , E L Zet b e a, SB((a,b),El su c h t h a t a) *here e x i s t L o € 2 and M > 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\,<

,< ( I F a ( f - f E ) ~ ~ + I ~ F a ( f E l -IFf a E ] ) )+ ! \ F a

A

llf-fEll

+

IIFa(fg)-Fa

cM

t h e f i r s t summand i s t h i r d one I f

indeed:

hence

A

IIFall

2);

A E Lo;

E E = 3M

h

(f,]

II

+

A IIFa

A

II Ilf-fE.ll;

a n d t h e same i s t r u e f o r t h e

s i n c e we j u s t p r o v e d t h a t f o r

tim

we h a v e

that f o r

J

AELE

11 s E

F

(f

1

= Fa(fg),

t h e s e c o n d summand t o o

A E Lo

hence

11 F a ( f I - F a

REMARK

5. We w i l l u s e t h e e x t e n s i o n a b o v e w i t h

x

[ fI

for

there exists I s < 3

n LE. L

a topolo-

,

THE INTERIOR INTEGRAL

48

3

g i c a l space and

t h e system of

neighborhoods o f a p o i n t o f

L.

We g i v e now some e x a m p l e s t h a t s h o w t h a t theorem of

H e l l y cannot

--

- There e x i s t

Ex.1

a(t1

an(tl f:

[0,1]

'

V[a]

-b

-

Ex.2

sen

=

t

Jol

* f[ t 1 d a ( t

f:

with

4 0

[hence

f

We h a v e

where

A

but

such t h a t

for

haus,

1.2.13

v[an-j

an

u

[H-IPlE)l

and such

$

1 -

[$s

s]-

+

2"

is s u c h t h a t

an ~ b B V [ ( a , b ] l m

then there exists

fE

bC ( a , b ) 1

i s not convergent.

-

then,

M > 0

If

F,(~I

is c o n v e r g e n t

by t h e theorem o f Banach-Stein-

such t h a t

IIFnll d

M

i s a c o n t i n u o u s f u n c t i o n we h a v e

- -. of

=

n=l 2

b Jaf(t1dan(tI.

b ( (a.b) I

there exists

REMARK 6 . that

fE

~ , ( f )

2

bounded

-1.

-

V[aJ

f[t)da,[t)

Jab Define

every

Since

-

xA

- I f t h e sequence

Ex.3

PROOF.

=

\ol*f(tldan[tl

+0

llanll

f

174

i s not regulated): m

,

a n = x11,2nl

=x

an

V[aJ

-I3

[O,l]

I

take

.

an€ BV[[O,l])

and t h e r e e x i s t s

*f(tldan[t)

take

and

and t h e r e e x i s t s

r l

that

Q 1

V[an]

i s not regulated):

f

f[t1

There e x i s t

0

11

exists but

(hence

and

a = X)O,l]

5

with

bounded such t h a t

J:*i[t)dan[tl

does n o t e x i s t

an(t)

an€ BV[[O,l))

[hence

R

i n general the

be improved.

f o r a l l

))Frill

nEN.

= V[an]

i n contradiction t o t h e hypothesis

[Cf.

that

By a n a r g u m e n t o f c a t h e g o r y i t i s e a s y t o p r o v e

Jabf[t)dan[t)

does n o t converge f o r " a l m o s t "

continuous functions

f

(i.8.

f o r a dense

b u t i t i s v e r y d i f f i c u l t t o g i v e an e x p l i c i t continuous f u n c t i o n

f

such t h a t

G6

of

all

6

[(a,b)Il

example o f a

49

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

Jabf( t i cia,

d o e s not c o n v e r g e ,

1:

already

b y parts.

-

D

that

b e c a u s e if only

f(tldan[tl

-

Let

and

X

Y

SVo( [a,b] , L ( X , Y ) )

a

t h e norm

SVla].

0

simply

is a Banach space when endowed w i t h Theorem 5 . 7 and 5.9 s u g g e s t s t i l l an-

-

a1

for a l l 02

-

-

3

SVo ( [ a , b ] , L ( X , Y ) )

on

a O E SVo([a,b) , L ( X , Y ) )

a

have

we

b e Banach s p a c e s ; Theorem 5.1 shows

W e say t h a t a f i l t e r U

f 6BV((a,b)1

as one can s e e using integration

o t h e r n o t i o n of convergence on

uehged t o

tI

.

SVo((a,b] , L ( X , Y ) )

, and

we w r i t e

a

u-cona, or

"f-

a0’ i f t h e following c o n d i t i o n s a r e s a t i s f i e d

There e x i s t

HE

3

and

M >0

such t h a t

SV[a]

<M

a€H.

t i m a ( t ) x = ao(t)x for a l l

tcz (a,b)

3

SV: ( ( a ,b) ,L (X , Y 1

W e denote by

, the

SVo( [ a h ] , L ( X , Y )

and

XEX.

space

1

endowed w i t h t h i s n o t i o n of convergence.

REMARK 6 . Obviously t h e a-convergence can be d e f i n e d on f o r any BT (E,F,G).

SB((a,b],E)

REMARK 7 . By Theorem 5.7 w e see t h a t t h e d e f i n i t i o n above is H E 3 that is

e q u i v a l e n t t o saying t h a t t h e r e e x i s t s an bounded i n

SVo ( (a,b] , L ( X , Y ) )

1

lirn ' f o r every

a

* d a ( t ). f ( t ) =

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

Ib

and t h a t

b

-dao(t).f (t)

a

.

W e w i l l now g i v e a r e p r e s e n t a t i o n theorem f o r t h e ele-

ments of

L[G- ( ( a , b ) ,XI , G ( (c,d)

THEOREM 5 . 1 0 .

Let

X

and

Y

8Y)].

be Banach & p a c e d ; we h a v e

L[G-([a,b\],X) , G ( ( c , d ) , Y ) ] = G ( [ c , d ) ,sVg((a,b) 8 L ( x # y ) ) ) .

Mohe p h e c i b eLy, Xhe. mapping

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

50

Banach Apace o n t o t h e Aecond, whe.he ioh evehy we d e d i n e G- ( [a,b) , X ) .the

06

fE

dihdt

b

a t E [c,d];

a

w e have

E (a,b),

FAkla,,) XI( t )

J ( t )( a ) x =

doh

t E (c,d),

X.

XE

5

PROOF. L e t us f i r s t o b s e r v e t h a t

( f ) ( t ) is w e l l defined

J ( t )E SVo( [ a , b ) , L ( X , Y ) and f E G ( [ a , b ) , X I . The funci s r e g u l a t e d s i n c e by t h e d e f i n i t i o n o f t h e aconvergence on SVo( [ a , b ] , L ( X , Y ) 1 and Remark 7 , t n + t E ( a , b ( since

tion

FA ( f )

implies t h a t

)

( a ) - f ( a ) is c o n v e r g e n t , i . e .

5

( f ) (t,)

i s convergent; t h e same a p p l i e s i f w e c o n s i d e r t n + t ] a , b ] . L e t us now d e n o t e by @ t h e mapping d C, FA ; @ i s

i s i n j e c t i v e since i f w e take A # 0 t o €( c , d ] such t h a t oh (to)# 0 , hence a E]a,b] and x € X such t h a t J ( t o ) (u)x # 0,

obviously l i n e a r . @

then there exists a there exist

I n o r d e r t o prove t h a t t h a t f o r every

A

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

@

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

F E L[G-(

E G ( [c,d] , S V o ( (a,b) , L ( X , Y ) 1 ) such t h a t

see t h a t i f t h e r e e x i s t s such an

u h ( t ) (a)x for

tE[c,d),

f i n i t i o n of

4

aE[a,b)

F = Fd

d w e have

and w e w i l l prove t h a t

llFj

( i l l CA ( t )( u ) E L ( X , Y )

(a) w e

x € X ; we take t h i s a s the

and

(ii) F = FA.

. By

= F[X]~,~]X ( t] )

( i )c h E G ( ( c , d ) , S V z ( ( a , b ) , L ( X , Y ) ) )

W e w i l l a l s o show t h a t

t h e r e e x i s t s an

(1

=

llJll.

since

and

de-

THE I N T E R I O R INTEGRAL

and s i n c e

( t )(u)

51

i s obviously l i n e a r d€ ID

and

is regulated, i.e.,

for

( i "A ) ( t )E SVo ( [ a , b ] ,L ( X , Y ) ) ; i n d e e d f o r X ~ E X , IIxill ~ 1 ,i = 182,.. , [ d l , w e have

.

A

( i " ' )L e t u s prove t h a t

stances, t h a t vergent i n

tn+t E(c,d(

implies t h a t

d

(t,)

is

in-

o-con-

S V o ( [ a , b ) , L ( X , Y ) ) : by ( 8 ) w e have

sv [A

(',)I

II F II

f o r e v e r y n , i . e . t h e c o n d i t i o n a 1 i s s a t i s f i e d ; w e have also a2 i . e . t h e sequence A ( t n )( S I X i s c o n v e r g e n t f o r e v e r y s E [a,b) and X E X s i n c e by d e f i n i t i o n w e have

A and

h)a,s] x]

G-((a,b)

8x1

s i n c e both are l i n e a r continuous o p e r a t o r s into

G ( ( c , d ) ,Y)

and t h e y t a k e t h e

v a l u e on t h e e l e m e n t s of t h e form X E X , i . e . on a t o t a l s u b s e t of G-(

BY ( B ) w e h a v e

I(F(f)ll =

a , b ,XI.

Ild 11 < IlFll; w e a l s o have

IIFII = h U P { [IF ( f ) and

(tn)

F ~ ) a 8 s ) x ] E G ( [ C , d ) ,X). (ii)F = FA

from

(tn) ( s )x = F

1 I

f EG-

( [ a t b ] 1x1 8

E ( a * b ]8

IIFI s lloh 11

I( f l l

,<

1)

6 u p IIF(f) (t)II and ccttd b

I ( F ( f ) (t)II = I I I - d u A ( t ) ( a ) * f ( a ) l (s

a

same

sv[A(t)]IIfll.

since

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

52

REMARK 8. From now on w e w r i t e of

-

A ( t , a ) = J ( t )( a ) ; i n t h e proof

(it")w e s a w t h a t t h e c o n d i t i o n t h a t

J : [c,d]

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

i s r e g u l a t e d is e q u i v a l e n t t o s a y t h a t f o r every S E [a,b] As i s weakly r e g u l a t e d (see t h e Appendix 3 of

the function

5 4 , Chapter I ) . O r s t i l l ,

(c,d)x(a,b) c h a r a c t e r i z e d by t h e f o l l o w i n g p r o p e r t i e s : A:

-

is

L(X,Y)

(SVu) A i s 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 ( i . e . , f o r e v e r y t E [c,d) w e have

A t ~ S V O((a , b ] , L ( X , Y ) ) (GO)

A

and

b u p SV[At] <

c-
a).

i s weakly r e g u l a t e d a s 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 ( i . e . , f o r every

SE

w e have

[a,b)

[ , ,

,

AS€ Ga ( c d ] L ( X Y 1 1 1

.

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

A

(Aa i s weakly r e g u l a t e d , hence f o r e v e r y 6 u p IIAa(t)xll < m and t h e p r i n c i p l e of uniform wtsd boundedness i m p l i e s t h a t IIAall = 6uP I I A a ( t ) l l < -1. cstsd IIAaII

XE

X

being f i n i t e

w e have

From now on w e w r i t e a l s o FA i n s t e a d o f ’ ? I i s t h e element t h a t c o r r e s p o n d s t o F.

and

AF

E X A M P L E S . Using Theorem 5.10 it i s e a s y t o p r o v e 1

-

FEL[G-([a,b]),G((c,d])]

F ( f ) b 0 i f and o n l y if f o r e v e r y i s monotonic i n c r e a s i n g .

i s such t h a t t E [c,d),

f > O

implies

At~BVo((a,b])

2 - F ~ L ( G - ( ( a , b ) ) , G ( ( c . d ) ) l i s such t h a t f is c r e a s i n g i m p l i e s F ( f ) >, 0 i f and o n l y i f % :0 and f o r every t E [c,d)

.

inAt

<0

3 - F E L [ G - ( ( a , b ) ) , G ( ( c , d ) ) ] i s such t h a t f i s i n is c r e a s i n g i m p l i e s F ( f ) i s i n c r e a s i n g i f and o n l y i f %

a c o n s t a n t f u n c t i o n and

As

i s decreasing for every

%[a,b).

53

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

-

4

that

F€L[G-([a,b)) , G ( [ c , d ) ) ]

F(€)

i s such t h a t

€>,

i s i n c r e a s i n g i f and o n l y i f f o r any

c \ <1t< t 2 s d

and

o

implies

a < s l <s2< b

w e have

-

A(t2,S2)

A(tl,s2)

A(t2,S1)

-

A(tlts1)

*

I n an analogous way as Theorem 5.10 on p r o v e s t h e

THEOREM 5 . 1 1 .

x and

Let

be Banach bpuceb and

Y

K

a com-

p a c t b p a c e ; we huve L[G-((a,b),X),

e(K,Y)]

=&[K,SVz((a,b] , L ( X , Y ) ) ] .

-

Mohe p h e c i b e e y , t h e mapping &E b [K,SVz( [ a , b ) ,L ( X , Y ) I ]

SE

L [G- ( [ a l b ) ,XI

i b o m e t h y ( i . e . 11% 11 = IIJll w i t h IldII 6 i h b . t Banach bpace o n t o t h e becond w we dedine f E G- ( [ a , b ] ,XI

i d

UM

0 6 the

t (K,Y)]

= sup s v l d ( t ) ] J

h

t€K

doh

b

( t ) = j.dscA(t) ( s ) .f ( s ) I a t E K ; We have A ( t ) ( S ) x = [ x ) ~ , ~ ) x(]t ) , t E K , X E

SE(a,b),

x.

REMARK 9 . I n t h i s case, i n an o b v i o u s way, a p p l y analogous c o n s i d e r a t i o n s as t h o s e made i n t h e p r e c e d i n g remark.

REMARK 1 0 . I n an analogous way o n e c a n g i v e r e p r e s e n t a t i o n theorems € o r t h e s p a c e s of l i n e a r c o n t i n u o u s mappings of c o ( ( a , b ) ,XI

G ( (c,d] ,Y) o r 6 (K,Y) working w i t h endowed w i t h an o b v i o u s a-convergence.

into

s ( [a,b] ,L ( X , Y ) 1

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

54

96

-

Rephcdentation theohemd

A - I n this

doh

open i n t e h u a d d

I t e m we e x t e n d t h e r e s u l t s o f

554 a n d 5

to

SSCLCS.

T H E O R E M 6.1. f E G I [a,b)

Let

.

,F1

la

Fa(fl

=

d e p e n d s o n l y on

b) Fa(f]

a

*da(tl-f[tlEG.

o r on t h e cZass of

f-

.F1.

c ) For e v e r y

and

S B ( [a,b),E)

b

a) There e s i s t s

G [ (a.b)

b e a LBCT',

[E,F,G)

q E

r G we have

q [Fa[f

11

5

SBq[a]

in

f

I(fII -

and q[Fa] < SBq[a]. and e v e r y q E r G we have

d ) F a € L[G([a,b).F),G] e ) For e v e r y [ d , S ' l E l ) ' b

- u d,C

*da(tl.f[t) q[

.(f;al]

6 SBq[a]ui(fl.

a

t h e s t e p s o f t h e p r o o f o f T h e o r e m 4.12.

The p r o o f f o l l o w s We r e c a l l t h a t

G

for

was d e f i n e d

SB[(a,b],E)

SSCLCS,

I n

B o f 54 a n d t h a t t h e e x i s t e n c e o f

the item

Jab d a ( t

-

I f

[t

I

was p r o v e d i n Corollary 4 . 1 6 .

THEOREM 6.2. Let

X

b e a Banaoh s p a c e and

-

mapping aESVO([a,b),L(X,Yl)

Y

a SSCLCS,

the

Fa€ L[G-[[a,b),Xl,Y]

is a l i n e a r

b i c o n t i n u o u s i s o m o r p h i s m o f t h e f i r s t SSCLCS o n t o t h e seoond where f o r f E G - ( [ a . h ) , X ) we d e f i n e

I, b

Fa[fl

we have

a(tlx =

Fa

[XI

a, t

11'

. d a ( t l . f (t.1

J

The p r o o f f o l l o w s t h e s t e p s o f t h e p r o o f o f Theorem 5.1. We r e c a l l t h a t seminorms

SVo~(a,b),L~X,YI)

a 4

svq[a].

is endowed w i t h t h e s e t o f where

q(F)

0. LCSS 5 1 .

-

dup{q[F[f)]

I s endowed w i t h t h e s e t o f

-

q c r y , and t h a t seminorms

I

F

fEG-[[a,b).X).

L[G-((~.~).xI.Y] q[F],

IIfll 6

q E r y

1)

,

[see 50,

55

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

REMARK 1 . W i t h t h e o b v i o u s a d a p t a t i o n s we h a v e t h e a n a l o g o u s of

and 5 . 6 .

t h e Theorems 5 . 5

The Theorem o f H e l l y e x t e n d s t o o :

THEOREM 6 . 3 . L e t (E,F,G) be a LCBT and a f i Z t e r on a s e t L and f o r e v e r y XE L l e t a X E S B ( ( a , b ) , E ) be such

that a ) F o r e v e r y q E r G t h e r e e x i s t L q ~3 and s u c h t h a t SBq [aX] 6 M q for e v e r y X E L 4 b ) There e x i s t s a : [a,b] E such t h a t

-

Lim

3

aX(tl.y

f o r a22 t E [ a , b ) and a22 T h e n we have 2 ) a E S B ( [a,b] , E l and

I

b

2) Lim

J

f E G( [a,b)

-daX[t)*f(tl

a.

.

Mq> 0

= a (t1.y

~ E F .

SBq [a)

-[

< M

for e v e r y

4

qE TG.

for e v e r y

.da(t).f[t)

,FI.

The p r o o f f o l l o w s t h e s t e p s o f t h e p r o o f o f Theorem 5.9.

X

If

is a B a n a c h s p a c e a n d

t h e o-convergence a s was d o n e i n

qE

ry

0

on of

there exist

for a l l

aEH

THEOREM have

6.4.

. 4

Y

a

SVo( [a,b) .L[X,Y

1)

SSCLCS

55; t h e c o n d i t i o n 01

Hq€

3

and

M

4

> 0

we d e f i n e

i n a n a n a l o g o u s way becomes:

such

for

t h a t SV

every

11x3s 4

M

4

T h e n we h a v e

Let

x

be a Banach s p a c e and

Y

a S S C L C S ; we

More p r e c i s e l g ; t h e mapping a , b ] , X I ,G[ [ c , d ) , Y )

is

Q

l i n e a r b i c o n t i n u o u s isomorphism ( i . e .

o n t o t h e s e c o n d , where for Fa(f)[t) =

J a

f E G( [a,b)

,XI

b

*dud(tl(o).f[ol,

4ChI

1

= 4 p ] for o f t h e f i r s t SSCLCS

we d e f i n e

T H E INTERIOR INTEGRAL

56 t E [c,d]; X E

we

have

tt)(o)x

REMARK 2. 10 o f

for

tE(c,d],

l a 4

x.

5.11

x](tl

=

I n a s i m i l a r way we h a v e t h e a n a l o g o u s o f T h e o r e m

and o f t h e r e p r e s e n t a t i o n t h e o r e m m e n t i o n e d in r e m a r k 15.

-

B In this item we extend the previous results t o open intervals. Let (E,F,G)B be a BT; we say that a: >a,b( + E is a (normalized) dunctiun ad buundLd B-uahiation and campact duppoht, and we write a SBo0()a,b[,E) if a has the f o l lowing properties 1) There exist [.,6)c )a,b( and c E E such that a(t) = 0 for t < a and a(t) = c for t>G. 2) a SB( [z,';] /E)

.

The smallest closed interval called the nuppoht of a.

[ g J 5 ] with property 1) is

Let (E,F,GlB be a LCBT; we say that a: )a,b( d E is a (normalized) function of bounded B-variation and allmost compact support, and we write. a € SBoo()a,b[,E), if a satisfies the following properties 1) For every q E rG there exist kaq,bq] c)a,b( and c E E such that €or every Y E F we have q [ a ( t ) y ] = 0 if 9 t b q q q 2 ) a SB ((aq,bq) 3 ) .

.

9

The smallest interval (aq,bq) is called the q-buppuht 06 a.

with the property of

In the case of the LCBT (L(X,Y),X,Y) particularize in an obvious way.

1)

these definitions

Let F be a Banach space; we say that a function f: )a,b[ + F is JreguLated, and we write f G ( a b ,F), if f has only discontinuities of the first kind; this amounts to say that for every [c,d) C )a,b( we have

1"

57

THE I N T E R I O R INTEGRAL

-

f-(t) = f(t-)

W e define

for

tE)a,b(.

G()a,b[,F)

is a

F r e c h e t space when endowed with t h e f a m i l l y of seminorms ~ ~ f ~ ~ [[c 8 d~ ) 8 d ] a] , b~( *

THEOREM 6.5. f

E

Let

-

b e a L C B T , ~ E S B o o ( ) a , b ( , E ) and

(E,F,G)

G()a,b(,F). a ) Thehe e x i ~ t d

I

b

* d a ( t ). f (t) a whe4.e anSar bn4b. Fa(f) =

d8f

bn

- i i m \n

* d a ( t )* f ( t )

an

6 ) F a ( f ) depeMdA ofiLg O n f-. C ) Foh evehy q E rG we h a v e

d ) Fa€L[G()a8b(,F) t G ] P R O O F . a ’ ) For every

n

-

ibn

there exists

*da( t ). f ( t ); indeed:

an

f o r any

qE

rG

w e have

a

if

SBq ( (an,,bn) r E )

(anlbnl

I.,

3

tbql

and a f o r t i o r i f o r a s m a l l e r i n t e r v a l ; hence by Theorem 6 . 1 the integral exists. a ” ) There e x i s t s

le

r n * d a ( t ). f ( t ); indeed: s i n c e

G

an i s a SSCLCS i t i s enough t o prove t h a t t h e sequence j b : d a ( t ) - f( t )

an i s a Cauchy sequence and 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 every q E S G w e have q[J

bn * d a ( t ) - f ( t )-

-.

an

if

(,an,bn]

3 (aq8bq)

bq *da(tl.f(t)] = a

4

( s i n c e then

SB

b) F o l l o w s from t h e d e f i n i t i o n o Thereom 6 . 1 .

o

9r

(an aq I

Fa(f)

and from b) of

THE I N T E R I O R INTEGRAL qE such t h a t

d) W e have t o prove t h a t f o r every [cld)

r G ( (a,b) , F )

c >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

THE INTERIOR INTEGRAL

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] <

qE

a SSCLCS; given

Y 00

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

there e x i s t s q(.(t)x]

= 0

a.

Let

X

b ] such t h a t La9, 9 if t $ (aq ,bq) *

be a Banach dpace and

-

i f the

Y

a SSCLCS; t h e

u ~ s ~ ~ ( ) a , b ( , L ( X , Y ) ) FUE L[col o c (>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

THE INTERIOR INTEGRAL

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

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

.

(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 <

be B a n a c h s p a c e s ,

11. For

f E G((a.b),Xl

a n d i t is i m m e d i a t e t h a t t h e m a p p i n g

i s a n o r m . We d e n o t e b y

-

( (a,b>

,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

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

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]<

m}

f E G ( [a,b] , X I

Ib-aIPSV

P’

[a].

b ( =~ J Ia d a ( t l . f [ t I .

~

and

and

,L(X,YlI

P’

-1

ue have a) a ~ b S v ( a~ , b( ) , L ( X , Y I )

b) There e x i s t s

a ( a l = O and S V

from

d l I t f o l l o w s f r o m c).

s v P l [a]. and 1 . 2 .

c 1, i = l , Z ,

...,I

dl

THE I N T E R I O R I N T E G R A L THEOREM 6 . 1 2 .

65

The mapping

a E SV~'((a,b],L(X,YIl

Fa€L[G-

([a,b),XI,Y]. LP

is an i s o m e t r y ( i . e . 11 Fa 11 = S V p space o n t o t h e second, where f o r Fa[fl

we h a v e

actlx

=

F,(fl

t h e mapping

@

F E L[G-

hence

((a,b)

Q

i s w e l l defined.

14111.

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

i s s u r j e c t i v e we p r o v e t h a t S V E ' ( (a,b)

,L(X,YIl

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

since

I I a [ t ) x l l = IIFCX a t

( i i l a E S V E ' ( [a,b]

CJ

I n t h e u s u a l way

( s e e Theorem 5 . 1 1 .

t h e r e i s an a

, X I ,Y]

( i l a ( t 1 E LLX,YI

[a]

by Theo-

((a,b],XI,Y]

is i n j e c t i v e

0

LP s u c h t h a t F = F a ; we d e f i n e

P'

Fa;

a

P

b l I n o r d e r t o show t h a t

SV

t)e d e f i n e

Fa[x)a,t)~].

\ / F a l l <SV[a]1

hence

,XI

i s w e l l d e f i n e d a n d we h a v e

one p r o v e s t h a t

every

of t h e first Banach

f E G( [a,b)

da(tl-f(t1;

F a € L[G, with

[a]

1."

=

PROOF. a1 L e t u s d e n o t e b y rem 6 . 1 1

I

-1

l l ~ l l l l x ) a , ~ ~ l =l pI I F l l ( t - a l P

,L(X,YII:

for

d E D

and

xi€

IIXII

X,

we h a v e

<

IIFII.

( i i i l Fa = F

since both are l i n e a r continuous

t h e same v a l u e o n t h e e l e m e n t s

t E [a,b),

and t a k e x E X.

which form a t o t a l s e t i n L e t u s show t h a t

t h e theorem above extends a s i m i l a r

THE I N T E R I O R INTEGRAL

66

theorem o f Bochner and T a y l o r rem 6.14 If

given for

[B-T]

i s a B a n a c h s p a c e i n [B-T]

Z

-

V p ' ( (a,b),Zl

{a:

is given the follow-

+Z I

(a,b)

V

P ' [a]

<

m)

1

Id1 l I a ( t , l - a [ t

=

w &D [ i c= l

V:'((a.b),Zl

Iti-ti-l =

{a

THEOREM 6 . 1 3 . V E ' ( [ a , b ) , X ' l = SVPl

i-ll;pjF'

a(a1 =

01.

and

SV~'([a,b),L(X,ClI

C1.'

by t h e H d l d e r i n e q u a l i t y ;

hence

V

[a]

P'

SV P'

Reciprocally,

hence

I

Vp'[(a,b),Z) =

=

IP'-

PROOF. I t i s e n o u g h t o show t h a t

have

C [Theo-

=

bellow:

ing definition

v P ’ [.I

Y

we r e c a l l t h a t f o r

= SV

P'

[a].

We h a v e

[a] < V P l [ a ] .

"i, . . . , x i

ldlE

X'

we

67

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

V

hence

P’

[a] c S V p

Theorems

THEOREM 6 . 1 4 .

6.12 G-

[a]. and 6 . 1 3

[[a,b).Xl’

LP

F o r t h e spaces

imply 1 V:’([a,b),X*l.

SV~((a,b),L(X,Yll

t h e analogous o f

the

Theorem o f H e l l y i s t r u e :

L e t 2 be a f i Z t e r on S V ~ ~ ( ( a , b ) , L ~ X , Y l l and such t h a t -+ L ( X , Y l 0 1 ) There e x i s t s H€$ and M > 0 w i t h S V [a] 6 M f o r

THEOREM 6 . 1 5 . a o : (a,b)

P’

a € H. 02) eim a ( t ) x = a o [ t ) x

for a l l

J

t E (a,b)

and a22

xEX.

Then we have

f E G ( (a,b)

.XI.

The p r o o f

follows the steps o f the proof of

We s a y t h a t a f i l t e r

verges t o

3

on

SV~((a,b),L[X,YlI

aoE SV~’((a,b],L[X,Y11,

a n d we w r i t e

t h e p r o p e r t i e s 011 a n d a21 a r e s a t i s f i e d . denotes t h e space vergence.

SVE’C ( a , b ) , L ( X , Y I l

S V E ”‘(

a

O-E-

+

(a,b)

ao, i f

.L[X,YII

endowed w i t h t h i s c o n -

We h a v e

THEOREM 6 . 1 6 . L[Gi

Theorem 5 . 9 .

Let

((a,b).XI,G(

X

and

[c,d],YI]

P

more p r e c i s e l y , t h e mapping

Y

be Banach s p a c e s ; we have

’ G~[~.d).SV~”‘~(a.b],L~X,Yl~l

68

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

of t h e f i r s t Banaoh s p a c e o n t o t h e s e c o n d , w h e r e f o r e v e r y

Q

we d e f i n e

f E G-([a.b],Xl

JA(tl(ulx

( f l c t l

=

5

0

for

t E (c,d),

T h e p r o o f f o l l o w s t h e s t e p s o f t h e p r o o f o f T h e o r e m 5.10.

REMARK 7 . I n a n a n a l o g o u s way a s was d o n e i n R e m a r k 8 we fine

A:

(c,d)x(a,b)

+L(X)

by

A(t,ol

-J[tl[ol;

A

dei s

characterized by the following properties:

-

[SVu 1 P' a function

we h a v e [G'l

A

i s u n i f o r m l y o f bounded p ' - s e m i v a r i a t i o n

o f t h e second v a r i a b l e ,

At€

-

i s weakly

f o r every

as

t E (c,d)

bup

SV [At] < m. c x t ~ d 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

SV~'((a,b),L(X,Yl)

A

i.e.,

with

variable.

REMARK 8. I n a n a n a l o g o u s way o n e g i v e s r e p r e s e n t a t i o n t h e o r e m s for

L[GL

L [GiP( stc.

(.,El

((a.b),XI,G(K,Y)]

, X I .G( ( c , d ) . Y

where

11,

for

K i s a compact space, f o r -1oc L [GL ( ) a , b , X I . G o c . d (.Y I] P

[

CHAPTER

I1

The Analysis o f Regulated Functions 5 1 - T h e theohem 0 6 %hay and t h e 60xtnuLa A

-

X

Let

be a normed space and

06

UihiCkdCt

a Banach space;

Y

given a f u n c t i o n h:

we w r i t e

[c,d]

x

(a,b]

+L(X,Y)

he GU((c,d)x(a,b), L ( X , Y ) )

if

i s regulated

h

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 ( i . e . , hsE G ( ( c , d ) , L ( X , Y ) ) S E ( a , b ) ) and

f o r every

as

i s uniformly of bounded semivarwe i a t i o n i n t h e second v a r i a b l e (i.e., f o r every t E [c,d) have ht€ S V ( ( a , b ] , L ( X , Y ) ) and 6 u p SV[ht] < m ) ; hence h is h

cGtbd bounded (see Remark 8 of 55 of Chapter I )

.

T H E O R E M 1 .l. T h e t h e o r e m o f B r a y Let x be a nohmed 6 p a c e and Y , Z Banuch 6 p U C e 6 . Given a t S V ( [ c , d ) , L ( Y , Z ) ) , h GU((c,d)x(a,b), L ( X , Y ) ) and g € G ( [ a , b ) , X ) 10h w e have gE G ( [a,b) , L ( X ) I]

-

(1)

J

b

a

d

*..[I

1

d

*da(t)-h(t,s)).g(s) =

c

[I

b

*da(t)

C

*dsh(t,s).g(s)]

a

I n o r d e r t o prove t h e theorem of Bray w e need two lemmas.

L E M M A 1.2. W i t h t h e S E (a,b)

we dedine

FIE SV( (a,b)

,L(X,Z)

1

hypothe6i6

E(s)

06

Theoarem 1 . 1 ,

d

= Jc *dcr(t). h ( t , s ) ;

d o h euehy

t h e n we huue

and

SV[iiI

4 SV[a]

b u p SV[ht]. c
PROOF. By Theorem 1.4.12

K ( s ) i s w e l l defined s i n c e

h E S V ( (c,d) , L ( Y , Z ) )

and

hsE G (

,Y)

.

70

T H E A N A L Y S I S OF R E G U L A T E D F U N C T I O N S

[ I

For d € D a , b have

xi€ X

and

06

LEMMA 1 . 3 . U i t h t h e h y p o t h e b i b t E [c,d]

g(t)

we d e d i n e

iEG((c,d),Y)

[oh

t l , i = 1,2,.

Theohem 1 . 1 ,

.. ,I d ( w e

doh e v e h y

= / I * d s h ( t , s ) - g ( s ) ; t h e n we h a v e

g€G((c,d),L(X,Y))]

IlSll

(4)

IIxill

with

and

b U P SVtht1 11411. cstsd

PROOF. By Theorem 1 . 4 . 1 2 g ( t ) i s w e l l d e f i n e d s i n c e h t E SV ( [a,b) ,L ( X , Y ) 1 and g € G ( (a,b] ,X) [or g E G ( (a,b) ,L(X)); c f . Theorem 1 . 4 . 4 1 . I n o r d e r t o prove t h a t g i s r e g u l a t e d

we w i l l

show t h a t i f

t,+tE

g(t+)

(c,d(

= Lim

n

there exists

g(tn)i

for i n c r e a s i n g sequences w e have an analogous r e s u l t .

By t h e

hypothesis made on h i n Theorem 1.1, f o r every S E (c,d> t h e r e e x i s t s h ( t + , s ) = L i m h ( t n , s ) and t h e r e e x i s t s M such n t h a t SV[htn) ( M f o r every n; hence by t h e Theorem of

Helly (1.5.8) w e have SVbt+] ( M 1 g € G ( (a&) J ) [g G ((a,b) , L ( X ) 1

-g ( t n )

= (*dshtn(s) . g ( s )

a

+

and f o r every

Ib

-dsh t+ ( s ) . g ( s ) =

g(t+).

a

( 4 ) follows from c) of Theorem 1 . 4 . 1 2 .

of

PROOF OF THEOREM 1 . 1 . We w i l l prove t h a t both members (1) are w e l l d e f i n e d and a r e l i n e a r continuous €unctions

of g [a) and b) bellow]. Afterwards [c) bellow] i t w i l l be enough t o prove (1) i f g = Lor xla,aluJ xla,alx where a € (a,b) and xEX CUE L(X)] s i n c e the set of t h e s e

THE ANALYSIS functions is total i n

OF REGULATED FUNCTIONS

G ( [a,b) ,X)

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

71

.

a ) From Lemma 1 . 2 i t f o l l o w s t h a t t h e f i r s t m e m b e r of (1) i s w e l l d e f i n e d anddepends c o n t i n u o u s l y on g s i n c e b

d

b

a

a

C

b) From Lemma 1 . 3 i t f o l l o w s t h a t t h e second m e m b e r o f (1) i s w e l l d e f i n e d and depends c o n t i n u o u s l y on g b e c a u s e w e have d

b

I l J * d a ( t )[ J - d s h ( t , s ) . g ( s ) ] C a

(2)

1)

SV[a]llgll

4

c ) L e t us c a l c u l a t e b o t h s i d e s o f (1) for

a

C

C

g = ~l,,~lx:

C

a

C

b o t h second members a re o b v i o u s l y e q u a l .

REMARK 1 . If i n Theorem 1.1 w e r e p l a c e t h e h y p o t h e s i s "regul a t e d " for h and g by f t c o n t i n u o u s ' f w e may r e p l a c e J* by

J

and i n t h e n u m e r i c a l c a s e ( i . e . X

Y

Z

IR) t h e

cor-

r e s p o n d i n g theorem w a s proved by Bray [B].

COROLLARY 1 . 4 . W i t h t h e h y p o t h e b i b 04 Theohem 1 . 1 we t a k e [c,d] [a,b) and d o h S E (a,b) we d e d i n e

I, S

fi(s) = We h a v e

fiE

SV( [a,b) , L ( X , Z ) )

*da(t).h(t,s).

and

72

whehe

THE ANALYSTS (Ih((h=

OF REGULATED FUNCTIONS

bup llh(t,t)\l.

a
ho(t,s)

p r o p e r t i e s as

i n Theorem 1.1 and

h

svrh:i

Y(s-t)h(t,s);

Ilh(t¶t)ll

+

"(t,b)

The r e s u l t f o l l o w s from Lemma 1 . 2 s i n c e REMARK 2 .

ho

Thtl h

may b e n o t r e g u l a t e d even when

continuous ( s e e however Theorem 1.13).

same

has t h e

*

KO. a

or

P R O O F . I t f o l l o w s immediately from t h e d e f i n i t i o n of

h

is

I"*

Ja THEOREM 1 . 6 . W i X h Xhe h y p o t h e d i d 06 T.heohem 1 . I we t a k e [c,d) (6)

(6’)

(7)

= (a,b].

W e have

THE ANALYSIS

OF REGULATED FUNCTIONS

PROOF. W e define

ho(tys)

p r o p e r t i e s of

i n Theorem 1.1. Hence w e r e p l a c e

h

Y(s-t)h(t,s).

ho

73

h a s t h e same h

by

i n (1) and a p p l y 9 ) o f Lemma 1 . 5 i n o r d e r t o o b t a i n ( 6 ) .

get (6')

ho We

i n an a n a l a g o u s way. (7) f o l l o w s immediately from

(6) i f w e a p p l y ( 2 ) .

If we w r i t e

and a p p l y r e s p e c t i v e l y (7) and ( 2 ) t o t h e s e two i n t e g r a l s w e obtain ( 8 ) .

(9)

COROLLARY 1 . 7 .

G E SV( [a,b)

ltj

so

i s a m a j o r a t i o n o f (8).

h(s)

16

,L(X,Z))

a

WQ

IS

*da(t).h(t,s)

=0

and d o h e v e h y

have

[.,6)c

then

(a,b)

we h a v e

74

THE ANALYSIS

OF

REGULATED F U N C T I O N S

P R O O F . The proofs follow immediately from the Corollary 1.4 and from (8!, ( 9 ) and ( 7 1 , respectively, if we recall that by 1.5.2 we have

I, t

COROLLARY 1 . 8 . a] 16 gl(t) = haw e

i1e

G ( (a,b] ,Y>

b) Id

[or

g2(t)

g2= G ( [a,b)

,Y)

[or

-dsh(t,s)-g(s)-h(t,t)g(tt)

i1e G ( [a,b) ,L(X,Y))].

i,"

dsh(t,s).g(s)

g2E

+ h(t,t)*g(t-) we have

G( (a,b) ,L(X,Y))I.

P R O O F . We take T C E (a,b( and Then by (6’) there exists

B =

xla,.r(

IyE SV( [a,b] ,L(Y) 1.

but

In an analogous way one proves the existence of g-2(T+) and E 2 ( ~ - ) .

E’(T-),

-1 -2 REMARK 3 . The summands of g and g are not necessarily regulated functions since ht is not regulated in general.

PROPOSITION 1.9. T h e formula o f s u b s t i t u i t i o n a and

(10)

SV([a,b)

g~ G ( [a,b) ,XI

!,"*ds

,L(Y,Z)),

[ oh

[ j:*da(t)h(t)

g

1

hEG((a,b)

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

.g(s)

c

-

16

,L(X,Y))

3

we have

=da(t).h(t

)g(t).

P R O O F . The result follows from (6) if we take there

h(t,s) E h(t).

T H E A N A L Y S I S OF REGULATED FUNCTIONS

The f o r m u l a o f D i r i c h l e t

THEOREM 1 . 1 0 .

06 Theohem 1 . I we t a k e t h e n we h a v e

[[

(11)

J:mda(t)

PROOF. Since b lads[

g

I,

(c,d]

= (a,b)

-

With t h e hypothedin and g COntiflUOUb;

1

.h(t,s) dg(s) = [-do(t)

1

[[h(t,s)dg(s)

i s c o n t i n u o u s , b y (1) a n d by 1 . 1 . 2 w e h a v e

b-da(t)h(t,s)]g(s)

\:*da(t)

obtain

j:-da(t).h(t,s)

I

[ [dsh(t,s)*g(s) 1 i n both i n t e g r a l s w e

and u s i n g i n t e g r a t i o n by p a r t s i n

I[

75

dg(s) = l I * d a ( t )

[ j:h(t,s).dg(s)

1.

If w e t a k e h o ( t , s ) = Y ( s - t ) h ( t , s ) t h e n ho h a s t h e same p r o p e r t i e s as h i n Theorem 1.1; h e n c e , w e may r e p l a c e h by ho

i n t h e l a s t e q u a l i t y and w e g e t

(11).

REMARK 4 . Theorem 1 . 1 0 may a l s o b e p r o v e d by u s i n g (6) and i n t e g r a t i o n by p a r t s . C O R O L L A R Y 1 . 1 1 . W i t h t h e h y p o t h e n i d 06 Theohem I . 1 0 we d e d i n e

then

id

hegulated.

P R O O F . I t f o l l o w s from b ) o f C o r o l l a r y 1 . 8 u s i n g i n t e g r a t i o n by -2

p a r t s i n t h e i n t e g r a l t h a t appears i n t h e d e f i n i t i o n of

g -

rb REMARK 5 . If t h e f u n c t i o n

g is only regulated, may n o t e x i s t (see Theorem 4 . 2 2 o f C h a p t e r I ) . COROLLARY 1 . 1 2 . W i t h t h e h y p o t h e b i d [c,d] = [a,b]; t h e n we have

P R O O F . We d e f i n e

06

t

1,- h ( t , s ) d g ( s )

Theohem 1 . 1 we t a k e

k(t) = j a g ( s ) d s . By C o r o l l a r y 1 . 4 and 1.4.16

t h e f i r s t i n t e g r a l i n (12) i s e q u a l t o

76

THE A N A L Y S I S OF R E G U L A T E D F U N C T I O N S

Again by 1 . 4 . 1 6 t h e second i n t e g r a l i n (12) i s e q u a l t o jab*da(t) [ / z h ( t , s ) d k ( s )

1.

Hence t h e r e s u l t f o l l o w s from Theorem 1 . 1 0 s i n c e t inuous

.

k

i s con-

THEOREM 1 . 1 3 . W i t h t h e h y p o t h e b i b 06 T h e o h e m I . I w e (c,d) [a,b); we d u h t h e h b u p p o b e ,that

By (8) ( w i t h

-

a = s1

and

s,)

take

and from a) and b ) follows

t h a t 6 i s c o n t i n u o u s . L e t us now prove (13) : S i n c e ht , t E [a,b) , are c o n t i n u o u s f u n c t i o n s (6) becomes JIds

[\I

da(t)-h(t,s)]g(s)

h,

\Ida(t) [h(t,t)g(t-)

+ /Idsh(t,s)*g(s)

a and

+

1.

Using i n t e g r a t i o n by p a r t s i n b o t h i n t e g r a l s (13).

Ids

we obtain

THE A N A L Y S I S OF

77

REGULATED FUNCTIONS

C O R O L L A R Y 1 . 1 4 . W i t h t h e Aame h y p o t h e d i b a6 i n Theohem I . 1 3 we d e d i n e

We have a)

i

i b kegutated;

ContinuouA id

i b

rt

(14)

h

ib

continuoub.

(14’) b) L A hegueated; c o n t i n u o UA ;

t

continuoud i d

i b

and

g

h

ahe

~~dSh(t+."*g(s).

The frrst and the second integrals go to zero when e+O, respectively by the Theorem of Helly (1.5.8) and by the hypothesis b) of Theorem 1.13; hence (14). In an analogous way one proves (14’). (14) and (14’) show that is continuous if h is continuous. b) Using integration by parts we have hence (a)

z(t)

h(t,t)g(t)

g(t+> = h(t+,t).g(t+)

-

h(t,a)g(a)

-

-

h(tt,a).g(a)

i(t),

-

i(tt).

Using again integration by parts, the expression for g( tt 1 becomes

78

THE A N A L Y S I S

O F REGULATED F U N C T I O N S

and i f w e compare ( a ) and ( B ) w e o b t a i n (15). I n an analogous

way we o b t a i n (15’). (15) and (15’) show t h a t i f g and h are c o n t i n u o u s .

i s continuous

-

COROLLARY 1 . 1 5 . W i t h t h e aume h y p o t h e a i d ab in Theohem 1 . 1 3 t h e dunction h A : t E [a,b) h ( t , t ) E L(X,Y) LA h e g u e a t e d . P R O O F . W e t a k e g = I X E G( [a,b] , L ( X ) ) ; h e n c e , w i t h t h e n o t a -1 t i o n s o f C o r o l l a r y 1.8.a) and 1 . 1 4 w e have hb = g - g hence t h e r e s u l t s i n c e g and g1 are r e g u l a t e d .

-

REMARK 6 . One c a n , o b v i o u s l y , g i v e an e l e m e n t a r y d i r e c t proof o f C o r o l l a r y 1 . 1 5 or, more g e n e r a l l y , one can prove d i r e c t l y (a,b)x(a,b) t h a t t h e r e s t r i c t i o n of h t o any segment of ( i . e . , t h e f u n c t i o n t w h ( t o + t c o s 0 , so+t s i n e ) ) i s r e g u lated.

REMARK 7 . I t i s e a s y t o see t h a t a l l t h e r e s u l t s o f t h i s § are s t i l l v a l i d i f w e r e p l a c e t h e h y p o t h e s i s g c G( (a,b) , L ( X ) ) by PEG([ a b , L ( W , X ) ) where W i s a normed s p a c e . ¶

]

REMARK 8. I t is n o t d i f f i c u l t t o see t h a t a l l t h e r e s u l t s o f t ' h i s § a r e s t i l l v a l i d i f 2 i s a SSCLCS (working w i t h t h e seminorms q~ rz and t h e c o r r e s p o n d i n g s e m i v a r i a t i o n s ) . REMARK 9 . More g e n e r a l l y t h e r e s u l t s o f t h l s I a r e v a l i d i f instead o f L(W,XI ( o r L ( X 1 o r X I and L ( X , Y I and L I Y , Z l we c o n s i d e r s y s t e m s o f s p a c e s w i t h b i l i n e a r mappings r e l a t e d i n an a s s o c i a t i v e way ( s e e [ H - D S ] ] .

REMARK 1 0 . F o r g : [a,b) + X t h e n , w i t h obvious adapt a t i o n s , a l l r e s u l t s of t h i s § b u t c o r o l l a r i e s 1 . 8 , 1.11 and 1 . 1 5 a r e s t i l l v a l i d if we suppose o n l y t h a t h i s 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 v a r i a b l e (and s a t i s f i e s (SVU 1); i n t h i s case, f o r i n s t a n c e , i n Lemma 1 . 2 (and i n Theorem 1.1 e t c . ) t h e i n t e g r a l

jd C

- d a ( t ) . h ( t ,s)

T H E A N A L Y S I S OF R E G U L A T E D F U N C T I O N S is defined in the weak sense, i.e., f o r every exists

Id

E(t)x

C

-

etc.

XE

79 X

there

*da(t).h(t,s),

g: [a,b] L(X) [or g: (a,b) * L ( W , X ) ] all the results of this 8 are still valid if we replace everywhere the property "regulated" by "weakly regulated". For

- E x t e n b i o n t o open

52

-

A

Let

intekvatb

be a normed s p a c e and

X

given a function h: we w r i t e

(.,a]

]c,d[

X]a,b(

hEGU()c,d[X)a,b(,

x (i,b]C]c,d(

L(X,YII

a Banach

space^

L[X,YI

i f f o r every

we h a v e

x)a,b[

hE

-

Y

Gu[(c,G)

X

(:,El,

The p r o o f s o f Theorem 2 . 1

L(X,Yll.

t o Corollary 2.5 that follow

are a n a l o g o u s t o t h e p r o o f s o f t h e c o r r e s p o n d i n g r e s u l t s o f

$1.

T H E O R E M 2.1. 2

Let

b e a normed s p a c e , Y

X

a SSCLCS. G i u e n

g E G[)a,b(.

a

SVoo[)c,d(,

and

L[XlI]

L[Y,ZlI,

hE Gu()c.d(

a Banach s p a c e and

[or

g E G[)a,b[,XI

X)a,b[,

L[X,YII

that

Satisfie8

(;,:I

for every (c,d) C ]c,d( there exists (i,b)C such t h a t f o r t E we h a v e h [ t , s l = 0 s and h ( t , s ~ = ~ [ t , b i~ f s > b

(00)


]a,b[

if

t h e n we h a v e

a

gioen

[i.b]

C

rz

[;,dl

and correspond t o q E

C

(;,dl

a

o o n t a i n i n g t h e q - s u p p o r t of b y (00) t h e n

a, i f

THE A N A L Y S I S OF REGULATED FUNCTIONS

80

LEMMA 2 . 2 . f o r every

W i t h t h e h y p o t h e s i s and n o t a t i o n s o f Theorem 2 . 1 , s E ] a , b we d e f i n e

(

COROLLARY 2 . 4 . W i t h t h e h y p o t h e s i s o f Theorem 2 . 1 we t a k e c o n t < n u o u s ; t h e n we h a v e

a

--c

PROOF. S i n c e rb

by

Ja ds;

C

g

g

a

is c o n t i n u o u s w e may r e p l a c e

rb

Ji

ds

in (11

t h e r e s u l t f o l l o w s using i n t e g r a t i o n by parts.

REMARK 1 . T h e preceding r e s u l t s a r e s t i l l valid if w e r e p l a c e ]a,b[ b y a closed i n t e r v a l [ a , b ) . In t h i s c a s e i t i s e n o u g h L(X,YII, i.e., t h a t f o r t o s u p p o s e t h a t h e GU((c,i)X (a,b), every ( C , i J C )c,d( w e have h E G U t ( c , i ) X ( a , b ) , L[X,YII. COROLLARY 2.5. )c,d( = ]a.b[

W i t h t h e h y p o t h e 8 i s of Theorem 2 . 1 we t a k e and f o r e v e r y s ~ ] a , b [ we d e f i n e

G(SI

-

S

J-du[tl.h[t,s). a

We haue SVoo()a,b(, L(X.211; g i v e n qE c o n t a i n i n g t h e q - s u p p o r t of a, i f (aq,bq) (aq,bq) by ( 0 0 ) ( w i t h c a and b > 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
Foh evehy

SV[cyd] [Kt] - Foh

(SVd)

evehy

buch

thehe CXibtb M > 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

STIELTJES-INTEGRAL EQUATIONS

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 <

S-6

it i s and e v e r y E > 0 s E [c,d) w e have E

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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)

=

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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].

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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]

<

SVu [K]

*

[I1

U1(+2SVu[U]]

.

R E M A R K 7. The proof above a l s o shows t h a t the a f f i n e

formation

7

i s continuous i n

@.

trans-

STIELTJES-INTEGRAL EQUATIONS 1.11

LEMMA

IS

Let

x

and

Y

sV( ( a y b ) , E ( X , Y ) )

101

b e Banach 6paceh;

and

A E BV( (a,b] 1.

W e have SVCAa] c SV[AJ

llall

+

IIAIIsv [a]

*

PROOF. W e r e c a l l t h a t B V ( [a,b] 1 = SV( (a;b) , L ( C ) ) w i t h SV[A] V[A]. For d E D and xiE X , i = l y 2 , . . . y l d l w i t h I(xill < 1

we have

hence t h e r e s u l t .

We r e c a l l (Theorem 1.9) t h a t t h e r e s o l v e n t i s a f i x e d

p o i n t o f t h e t r a n s f o r m a t i o n 3’. If

'Y w e r e a c o n t r a c t i o n of GU w e would have proved 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 i n G'. I n g e n e r a l however 3’ i s n o t a c o n t r a c t i o n w i t h r e s p e c t t o t h e norm of G U . But w e w i l l show t h a t w e can i n t r o d u c e i n Gu a norm e q u i v a l e n t t o 111 111 and s u c h t h a t J i s a cont r a c t i o n i n t h i s new norm.

DEFINITION. Given fine

where

and

with

k > O

and

6 > 0

for e v e r y

UEGU

we

de-

STIELTJES-INTEGRAL EQUATIONS

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 <

S

b) F o r have

It-sl 9 6 , l e t u s suppose t h a t

c4s

st-6;

we

STIELTJES-INTEGRAL EQUATIONS

104

t+6 t s t d

and when

we have

analogous bounds. a ) and b )

imply t h a t

I1

-

I n o r d e r t o f i n d an upper bound f o r

w e w i l l look s e p a r a t e l y t h e 4 t e r m s of t h e d e f i n i t i o n of

svA ,6 ;t

[r

0UJ:

a ) I f we a p p l y (8’)

s

0

o f Chapter I1 ( w i t h

-

a = t-6

and

= t ) and a f t e r w a r d s u s e Lemma 1.11 w e g e t

b ) For u p p e r bound.

SV[T:6

,dl [(?'oU)t(s)e

w e have t h e same

STIELTJES-INTEGRAL EQUATIONS c > By ( 7 ' )

105

o f Chapter I1 w e have

hence

By ( a ) and ( B ) we have

S i n c e K s a t i s f i e s (SVuo) t h e r e e x i s t s 6 > 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

STIELTJES-INTEGRAL EQUATIONS

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

.

STIELTJES-INTEGRAL EQUATIONS

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 <

i s continuous (

i s uniform; s i n c e

t h e mapping cj)

TKU€ Guo

111 U 111

SVu[K]

: w e have t r i v i a l l y

svu[Kl gull

I17KuII

and by ( 3 ) o f Chapter I1 w e have

<

SVu[TKU]

SVu[K]

f o l l o w s from Theorem 0 . 3 t h a t t h e f i x e d p o i n t t i n u o u s f u n c t i o n o f K , i . e . , we have t h e

C O R O L L A R Y 1.17. The mapping t h a t t o each i t d

( R'4

hedoevent

1

RE

Guo,

bOlUtiOn

= Ix

-

R(t,s)

i b a c o n t i n u o u b duncttion.

KE

RK

Guo

it

SV'[U]

)

is a

con-

abbociaten

06

I:

s , t E (0)

dUK(t,a)oR(U,s)

b ) of Theorem 1 . 8 and C o r o l l a r y 1 . 1 7 imply t h e

C O R O L L A R Y 1.18.

bolution

Foh

KEGUo,

YE G ( (c,d)

y(t)

-

x

,X)

I:,

elements

X

f € G ( ( c , d ) ,XI

and

duK(t,o).y(o)

.t

dependb continuoudLy on Let

XE

06

K,

and

x

= f(t)

-

the

f(to)

f.

now p r o v e t h a t i f w e c o n s i d e r o n l y n o r m a l i z e d

US

( i . e . with

KEGUo

K(t,t)

0 ) t h e n t h e mapping o f

Corollary 1 . 1 7 i s i n j e c t i v e . Indeed: L e t resolvent

K1 , K 2 E G uo

R . Given

By Theorem 1 . 8

b e n o r m a l i z e d and have t h e same

Y E G ( (c,d]

,XI w e d e f i n e

y ( t ) = R(t,to)y(to) +

R(t,s)dfl(s)

It-

f ies y(t>

-

Y(to)

t

satis-

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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>

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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]

<

E

there exists f o r all s ,t E (c,d) E

> 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)

[

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

and t h e r e f o r e

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

STIELTJES-INTEGRAL EQUATIONS

&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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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)

and t h e r e f o r e

-

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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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,

STIELTJES-INTEGRAL EQUATIONS

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.

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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 .

-

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

(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)

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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

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

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.

STIELTJES-INTEGRAL EQUATIONS

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.

STIELTJES-INTEGRAL EQUATIONS

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),

STIELTJES-INTEGRAL EQUATIONS

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
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

STIELTJES-INTEGRAL EQUATIONS

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.

STIELTJES-INTEGRAL EQUATIONS

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 [ ) '

STIELTJES-INTEGRAL EQUATIONS

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

STIELTJES-INTEGRAL EQUATIONS

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)

STIELTJES-INTEGRAL EQUATIONS

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

,<

(T+E>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

\<

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

150

u ( t > 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

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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]

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[MI [R]

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[S]

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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 .

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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

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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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