THE METHOD OF SUCCESSIVE APPROXIMATIONS FOR FUNCTIONAL EOUATIONS.
11 y
L. KANTOROVITCH
()f LENINGRAD.
In the functional analysis abstract linear spaces are considered which may have for their elements mathematical objects of a various nature: numbers, sequences of numbers, functions etc. Therefore theorems established for such abstract spaces usually can be applied to very different branches of mathematical analysis. Thus the general theory of functional equations, i.e. of-such equa- tions where the unknown quantities are elements of a linear space, comprises the theories of differential, integral and some other equations considered in ana- lysis as well as the theory of finite and infinite systems of algebraic equations.
One of the most important methods for establishing the existence of solutions and for the investigations of these solutions is the method of successive approx- imations. We shall give here the geaera] theory of this method for linear and non-linear functional equations in a very wide class of spaces viz. the spaces normed with the elements of a semi-ordered space. This class comprises inter ,~lia Banach's spaces and semi-ordered spaces. ~ The theory of this method will be based on the principle of majorants. We shall give also some applications of the general theory to the systems of alo'ebraic equations and to the differential and integral equations.
1 F o r t h e case of semi-ordered spaces s o m e t h e o r e m s on t h e s e f u n c t i o n a l e q u a t i o n s h a v e b e e n p u b l i s h e d already b y t h e a u t h o r of t h e p r e s e n t pa.per. See L i s t of L i t e r a t u r e , vide ]~ANTOROVITC]-I, ~I.
Contents.
l~age.
w 1. T h e S p a c e s C o n s i d e r e d .
~. S p a c e s of t h e T y p e B . . . 64 2. S e m i - o r d e r e d Spaces . . . 65 3. S p a c e s N o r m e d w i t h t h e E l e m e n t s of a S e m i - o r d e r e d S p a c e . . . . 66 4- S o m e P a r t i c u l a r S p a c e s . . . (;7 w 2. F u n d a m e n t a l T h e o r e m s on F u n c t i o n a l E q u a t i o n s .
5. E x i s t e n c e T h e o r e m s . . . 68 5. U n i c i t y T h e o r e m s . . . 71 7. C o n t i n u i t y Q u e s t i o n s . . . 7 2 w 3. L i n e a r E q u a t i o n s .
8. O p e r a t i o n s A d m i t t i n g a M a j o r a n t . . . 74 9. F u n d a m e n t a l T h e o r e m s for L i n e a r E q u a t i o n s . . . 75 i o. A p p r o x i m a t i v e S o l u t i o n of E q u a t i o n s . . . 77 w 4. S y s t e m s of A l g e b r a i c E q u a t i o n s .
I~. A Class of I n f i n i t e S y s t e m s of L i n e a r E q u a t i o n s . . . 78 I2. A Class of n o n L i n e a r S y s t e m s . . . fq2 13. A T h e o r e m on F i n i t e S y s t e m s of n o n L i n e a r E q u a t i o n s . . . 85
~4. A u E x i s t e n c e T h e o r e m for I m p l i c i t F u n c t i o n s . . . 86 15. T h e C o n v e r g e n c e of N e w t o n ' s M e t h o d . . . ~7 w 5. I n t e g r a l E q u a t i o n s .
16. L i n e a r E q u a t i o n s of t h e S e c o n d K i n d . . . $8 17. S y s t e m s of F r e d h o l m ' s E q u a t i o n s . . . 90 i 8 . A n A p p r o x i m a t e S o l u t i o n of I n t e g r a l E q u a t i o n s . . . 90 19. A T h e o r e m C o n c e r n i n g F r e d h o l m ' s E q u a t i o n s in a n I n f i n i t e I n t e r v a l 92 2o. N o n - l i n e a r I n t e g r a l E q u a t i o n s . . . 92 w 5. D i f f e r e n t i a l E q u a t i o n s .
2 i . T h e C o n v e r g e n c e of P i c a r d ' s M e t h o d . . . 94 22. C a u c h y ' s M e t h o d of M a j o r a n t s . . . 96
w I. T h e S p a c e s C o n s i d e r e d . i. S p a c e s o f t h e T y p e B .
A l i n e a r (o1" v e c t o r ) s e t is b y d e f i n i t i o n a s e t X = {x} i n w h i c h t h e f o l l o w i n g o p e r a t i o n s a r e d e f i n e d a n d s a t i s f y t h e u s u a l r u l e s t: s u m x~ + .% o f t w o e l e m e n t s o f X , t h e i r d i f f e r e n c e x ~ - - x ~ , t h e p r o d u c t )~x o f x E X b y a r e a l n u m b e r ; a n d t h e z e r o e l e m e n t o o f X .
Cf. S. 7[L~.NACIr, l, p. 26.
T h e M e t h o d o f S u c c e s s i v e A p p r o x i m a t i o n s f o r F u n c t i o n a l E q u a t i o n s . 6 5
A l i n e a r set X is called a n o r m e d space if to every e l e m e n t x of X a (necessarily n o n - n e g a t i v e ) r e a l n m n b e r ~ ! is c o r r e l a t e d (which is called t h e n o r m of ;r) a n d t h e f o l l o w i n g 3 c o n d i t i o n s a r e satisfied:
'). i x i : = : o if a n d only if x = o, 2). !! X 1 -~ ;/' 2 / ~'~ ii '/'1 -}-! '9C2 i' 3 ) z;,. - - I z : , i l x l
W e shall say t h a t a sequence {x,,} of e l e m e n t s of a n o r m e d s p a c e X con- v e r g e s t o w a r d s . ~ ( x E X ) a n d w r i t e x , , - , x or x , - ~ x(b) if
lira i .r,~ . .r ii = o.
A n o r m e d space X is called a space of t h e t y p e B if i t is c o m p l e t e i . e . if w h a t e v e r be a sequence :r,, of e l e m e n t s of X s a t i s f y i n g t h e c o n d i t i o n
lira i I ,%, - x,,, : = o
t h e r e exists a n e l e m e n t :rE X such t h a t x,, ~ .r(b). ~
2, S e m i - o r d e r e d Spaces."
A senti-ordered space is a l i n e a r set Z in w h i c h t h e r e l a t i o n z~ > z,, is de- fined f o r c e r t a i n pairs of its elements. "~ T h i s r e l a t i o n m u s t s a t i s f y t h e u s u a l rules c o n c e r n i n g t h e inequalities. I n t h e s e spaces we c a n define (in t h e u s u a l m a n n e r ) t h e n o t i o n s of a n u p p e r a n d a l o w e r b o u n d of a set E ~ Z a n d also of its least upl)er a n d g r e a t e s t l o w e r b o u n d s . W e shall s u p p o s e m o r e o v e r t h a t e v e r y finite set is b o u n d e d a n d t h a t a n y b o u n d e d s u b s e t E of Z h a s a l e a s t u p p e r a n d a g r e a t e s t l o w e r bound. T h e s e b o u n d s we shall d e n o t e resp. sup E a n d i n f E . A space p o s s e s s i n g all these p r o p e r t i e s we shall call a s p a c e of t h e t y p e /s ~
1 I b i d . , p. 33.
" I(ANTOItOVITCII, l l, w 1 6 7 I - - - 4.
:~ F o r t w o g i v e n e l e m e n t s .z~ alld .z~ m a y s u b s i s t n e i t h e r of t h e r e l a t i o n s z t > z.2; ,'G < 7,.2;
Z I ~ Z2.
4 A l l t h e s e c o n d i t i o n s a r e f u l f i l l e d i f Z s a t i s f i e s t i l e f o l l o w i n g 5 " t x i o m s
I) O i s n o t > o.
2) zl > o a n d z 2 > o i m p l i e s z~ + z2 > o.
3) F o r a n y z E Z t h e r e e x i s t s a n .G ~ o s u c h t h a t z ~ . - - z ~ o.
4) I f z > o, z E Z a n d ~. i s a p o s i t i v e r e a l n u m b e r t h e n ).z > o.
5) A n y b o u n d e d s e t E p o s s e s s e s t h e l e a s t u p p e r b o u n d .
Cf. KANTOROVIT(?II, ~1.
9=-3932. A c t a mathematlca. 71. I m p r i m 6 le 2 m a r s 1(t3!).
T h e following definitions will be useful:
T h e positive a n d t h e n e g a t i v e p a r t s of z (z+ a n d z - ) are defined as follows:
z+ ~ sup (o, z) ; z - ~ sup ( - - z, o). I t m~y be easily seen t h a t z = - z + - - z - .
T h e ~absolute value>> izl of z is defined as i z i = s u p ( z , - - z ) - - z t _ + z - . This absolute value possesses t h e f o l l o w i n g p r o p e r t i e s : I). o i = o; !z > o if
A linfit. I f Z,~ is a b o u n d e d sequence, we shall d e n o t e
lim z,, ~ inf [sup (z,, z,,+~, . . .)l; lira z,, ~ sup [inf (.e',, z',,§ . . .)].
I f lira z'~ = : l i m z ~ - - z we shall say t h a t t h e s e q u e n c e .z,~ c o n v e r g e s (o) t o w a r d s z a n d write lira z n - - z or z,~-~ z ( o ) . This limit possesses all t h e
?l-~ cr
e l e m e n t a r y p r o p e r t i e s of t h e o r d i n a r y limits. I n p a r t i c u l a r it satisfies t h e cri- t e r i u m of Cauchy, i.e. if f o r a g i v e n sequence lim .z,~--Zm!----o t h e n t h e sequence .z'~ c o n v e r g e s (o) i . e . z E Z exists such t h a t zn-> z(o), or (which is t h e same) lim I z~ - - z ' "- o. 1
3. Spaces, l ~ o r m e d w i t h tile E l e m e n t s o f a S e m i - o r d e r e d S p a c e J L e t Y be ~ l i n e a r set a n d suppose t h a t to e v e r y e l e m e n t y E Y is cor- r e l a t e d an e l e m e n t ]!]1 of a c e r t a i n s e m i - o r d e r e d space Z (of t h e t y p e K~), so t h a t t h e following c o n d i t i o n s are satisfied:
I).
[y[ 0 if a n d only if y ~ - o ;2) I!/, + -< I + I:i_ l;
3). IZ? l=l llyl.
W e shall say t h e n t h a t Y is n o r m e d w i t h Z. F r o m ~)--3) follows t h a t f o r
~my y ~ o we have lY[ > o.
W e shall write !1,~-~ y ( b s ) i f [!/,,---y[--~ 0 ( 0 ) i n t h e space Z . I f t h e space Y is complete, i.e. if any sequence !t,~ such t h a t lira ]y,~ ~ y m [ = o c o n v e r g e s
751 m ~
(bs) t o w a r d s a n e l e m e n t y E Y, i.e. y , , - ~ y ( b s ) t h e n we shall call Y t h e space of t h e t y p e B s . This t y p e includes b o t h p r e c e d i n g t y p e s B a n d K~ or m o r e precisely a n y space of one of these types can be easily c o n v e r t e d i n t o a space of the t y p e B s . I n f a c t if Y is a space B we m a y t a k e f o r Z t h e space of real
1 ] ~ A N T O R O V I T C H , J [ . T h e o r e m X X b).
2 T h e s e s p a c e s were i n t r o d / l e e d i n ]{ANTOROVITCH, I l l , p. 272.
The Method of Successive Approximations for Functiomd Equations. 67 numbers and set I V l - ,!'Ji. In the ease when is a space n ; then we set Z - - Y a n d l Y I = l Y[. All the p r o p e r t i e s e n u m e r a t e d above of t h e space (Bs) are t h e n satisfied (this follows f r o m w h a t has been a l r e a d y said in 2. a n d 3.).
A c c o r d i n g l y in w h a t follows we shall consider chiefly t h e spaces B,~,.
4. S o m e P a r t i c u l a r Spaces. 1
I). T h e space of m e a s u r a b l e f u n c t i o n s . Consider t h e set S of all measur- able f u n c t i o n s defined in a given m e a s u r a b l e set E . T w o e l e m e n t s of t h e set S we shall c o n s i d e r as identical if t h e i r values coincide a h n o s t e v e r y w h e r e in E.
W e shall w r i t e 901 -> 90,2 if 901 (t)--> 90,2 (t) f o r a l m o s t all points of E .
This last c o n d i t i o n t r a n s f o r m s S into a semi-ordered space; 90,l -~ 90 (o) if 90,, (t) -~ 90 (t) a l m o s t e v e r y w h e r e .
suct~ m e a s u r a b l e f u n c t i o n s t h a t ; 1 9 o (t)i~ d t < + ~z (19 is h e r e
2). S e c o n d l y
a fixed n u m b e r > x) c o n s t i t u t e a n o t h e r semi-ordered space L~; in this case 90,~-7 90(0) m e a n s t h a t 90,,-~ 90 (o) in S a n d t h e r e exists a f u n c t i o n 900 e L~' such t h a t f o r a n y ~l: [q),,] <- 90o. T h e set L(P) m a y be c o n s i d e r e d also as a space of
/ p . 1
the t y p e B : if we set ]90
~-1 l:90(t):vdt)
v" T h e c o n v e r g e n c e ( b ) , , i l l be inx I I
1,1
this ease the c o n v e r g e n c e in m e a n of the o r d e r p, i.e. ~f,,,-~90(b) in L'P is equi- v a l e n t to
j
'190,,(t) -- 90 (t)i,'~tt ~ o.E
3)- Consider f u r t h e r the space s of all sequences :s] = (@), ~2(2), . . . ) of real nmnbers, w h e r e yx > y~ if f o r all i: V~) >-- ~2~ '). I n this space y,~ ~ y(o) if f o r e v e r y i we h a v e V~[)-> ~]()(when n--~ ~c). W e m a y c o n s i d e r also t h e space (m) I~(ol < + ~ This last of b o u n d e d s e q u e n e e s , i , e. of such sequences t h a t sup~
i
space anay be c o n s i d e r e d e i t h e r as selni-ordered or as n o r m e d , .with lY. =~
=
sup I~/')!.
T h e c o n v e r g e n c e will be different in b o t h eases.i
4). T h e m d i m e n s i o n a l v e c t o r space is t h e set of all systems y=(~](~), ~)(.2, . . . , ~i(,,)) of ~ real n u m b e r s . This space m a y be c o n s i d e r e d e i t h e r as s e m i - o r d e r e d (as above) or as a space of the t y p e B with e i t h e r
1 V i d e ]~ANTOROVITC/I ~], ~ I o - - I 2 a n d S. BANAC~I w 7,
9 rl , - s u p ',?'71 or i,~': i = ,~i,)),2
i
5). T h e s p a c e C of c o n t i n u o u s f u n c t i o n s x ( t ) defined in a s e g m e n t [a; b]
,vith t h e n o r m ! x l , - - s u p i x ( t ) i.
a : ~ t ~ b
w 2. F u n d a m e n t a l T h e o r e m s o n F u n c t i o n a l E q u a t i o n s . 5- E x i s t e n c e T h e o r e m s .
T h e o r e m I. L e t Z be a s p a c e of t h e t y p e K > C o n s i d e r t h e e q u a t i o n
=
W e shall s u p p o s e t h a t ( z ' > o b e i n g a c e r t a i n fixed e l e m e n t of Z ) :
I). V ( z ) is defined f o r every z E Z s u c h t h a t o - - - < z ~ < z ' a n d t h a t f o r a n y such z we h a v e V(z)E Z.
2). I f o --< Zl ~< z,2 ~ "" <~ z; a n d lira z,, = z (in t h i s case e v i d e n t l y o --< z ~ z') t h e n l i r a V (z,) = V (z).
3). V ( z ) is m o n o t o n o u s f o r o--<z--< z', i . e . if o - - < z < z + H z - - < z ' t h e n 9 v (.-) _< v ( z +
4). (o)~> o.
5). v ( z ' ) _<
I f all t h e s e c o n d i t i o n s are satisfied t h e n t h e e q u a t i o n (1) h a s a s o l u t i o n z*
such t h a t o ~ z* ~ z' a n d this s o l u t i o n c a n be f o u n d by t h e m e t h o d of succes- sive a p p r o x i m a t i o n s .
P r o o f . L e t z' o - : o, z ' , , - - V ( z , ~ _ ~ ) f o r a n y n a t u r a l u; we shall p r o v e t h a t f o r a n y ,i,
O .... 2' o ~<~ Z 1 ~ . . . . ~ ~ ~ ~P.
F o r u - - o these i n e q u a l i t i e s subsist. S u p p o s e t h a t t h e y a r e p r o v e d f o r ~ = - " 0 ; we shall p r o v e t h e m f o r ,, ~ ' 0 + I . I n f a c t Zno+l = V(.Yno) ~- V ( z n o - 1 ) ~ ' n o (by 3). a n d 5).), if "o > o a n d Z , o + l - - V(,Z,,o)-~ V ( o ) >~ o = z~,o (by 4).), if , o = o ; t h u s in all cases Z,oH >-- Z,o; o n t h e o t h e r h a n d z,,o+x =- V(z',,o)<-- V ( z ' ) < - - z ' (by 3). a n d 5).).
W e h a v e p r o v e d t h u s t h a t t h e s e q u e n c e zl, z., . . . . is m o n o t o n o u s (increasing) a n d b o u n d e d . T h e r e f o r e it c o n v e r g e s t o w a r d s a n e l e m e n t z* of Z such t h a t o ~< z * - - < z'. H e n c e follows by c o n d i t i o n 2).
The Method of Successive Approximations for Fmletional Equations.
V(z*) = lira
V(~n)
69
or, V(z,~) being identictd with z.+~ a n d lim V(z,,) lira ,-'n+l = z*, w e have
~t~ ~_
being t h e r e f o r e e q u a l to
. 9 = ~" (~*).
As besides o--< z * - - < z' a n d z * = lim
z,,
T h e o r e m I is c o m p l e t e l y proved.R e m a r k . Suppose t h a t besides c o n d i t i o n s I).--S)., V satisfies t h e f o l l o w i n g c o n d i t i o n : 2'). if z' --> z 1 --z,, --> . . . >-- o a n d lira z,~ = z , t h e n V(z) = l i m
V(z~).
r ~ t t
T h e n s e t t i n g z o
z, z'~+l-- V(,z~)
we o b t a i n as b e f o r e a solution z'* of t h e e q u a t i o n (1). This solution will n o t necessarily coincide withz*.
I t is easy to see h o w e v e r t h a tT h e o r e m I I . L e t Y be a space of t h e t y p e
Bs
n o r m e d by t h e e l e m e n t s of t h e space Z. T h e space Z a n d t h e o p e r a t i o n V, us well as t h e e l e m e n t ~ ' E Z shall be the same as in T h e o r e m I. W e shall c o n s i d e r t h e e q u a t i o n:y = u (:q)
(~)
w h e r e t h e o p e r a t i o n U is such t h a t t h e e q u a t i o n (t) is a m a j o r a n t of (2).
By this we shall m e a n t h a t
I). U(.q) is defined f o r e v e r y !]E Y such t h a t
a n d f o r a n y such y we have U(y) E Y,
2). IU(o)l < V(o),
3). I t7(:,t + ~ 4 , j ) - u(~,~)l -< v(~ + 3 ~ . ) - r(~) if only I:'tl --< ~, iA.,Jl--< ~ ;
+ J ~ ~ Z'.
T h e n t h e e q u a t i o n (2) possesses a s o l u t i o n y* satisfying" t h e i n e q u a l i t y l y * l - - < z' (or m o r e precisely l y * l - - < z * ) which can be f o u n d by t h e m e t h o d of successive a p p r o x i m a t i o n s .
P r o o f . Set Yo .... o,
y,, U(y,,-~).
W e shall prove t h a t f o r any n and m such t h a t m > uI n f a c t ] f f l - - frO ] = I Yl [ = ] U ( o ) ] --< t z ( o ) - - ,~1 - - 2/1 - - ,~'o. Suppose that w e have p r o v e d the above i n e q u a l i t y f o r any m and ~ such t h a t ~ < m --< mo(m o >-- I).
W e shall prove it f o r /t < m <~ m o + I; t h e n (by 3).) if ~o -> ~ >- I
I ~ . ~ o + , - :,J,,I-- I u ( y , , - 1 + ( ~ j ~ o - : , j , , - 1 ) ) - v ( ~ n - 1 ) l _< v ( ~ , , _ , + (~.,; - ~,~_,)) - v ( ~ , , _ l ) = ~mo+, - - ~-,~
b e c a u s e I ~ n _ l I = ] y n - - 1 - - if01 ~'~ ~n--1 - - z 0 = zn--1 (,1 - - I ~ $~)'0)' II1 t h e e a s e w h e n n - - o we h a v e
I ~ o + 1 - y , , I
=I y,,,~+, I - I :/~ + I :'/, I-< (,~,,,o+,-~,) +
, ~ , - - z . , o + , - ~ , , . W e have p r o v e d t h u s o u r i n e q u a l i t y f o r any n,,m such t h u t ~t < n~. B u t n o w asz , , ~ - z,~-~ o (o) in Z
it f o l l o w s t h a t
]ym--:q,*[-+
o (o) (in Z ) w h e n c e (by C a u e h y ' s e r i t e r i u m f o r t h e space) t h e r e exists a( b , ) lira :,/,, - - , / ' 'Jo + (:Jl - :'/o) + (;I~ - 'J,) -I .. . . . As on t h e o t h e r h a n d [y,, [-< z,,, it follows t h a t [ y * [ - < z * - - ~ z'.
Besides
I u ( : ; ) - u (:,~,~) I -< v ( ~ " ) - v ( ~ . ) - - ~ * -
~,,~.,
( b e c a u s e I t * - r,, I
-< ~-+
- ~,, and I :'~,, I-< ~,,)-
H e n c e follows U ( y * ) = (b s) ]im U(y,,,) = l i m 11.,+~ - .q*.So t h a t y* is t h e r e q u i r e d solution.
R e m a r k . W e shall call t h e s e s o l u t i o n s
z*
a n d :q* of resp. e q u a t i o n s(I)and
(2) o b t a i n e d by t h e m e t h o d of successive a p p r o x i m a t i o n s a n d s t a r t i n g w i t h t h e v a l u e o, t h e p r i n c i p a l s o l u t i o n s of t h e s e equations.
C o r r o l a r y . ~ I f t h e i n e q u a l i t y
I
u(,j + a : y ) - u ( , j ) I - < ~ I ~,,al, (3)w h e r e o < a < i, subsists f o r all values of y a n d J y t h e n t h e e q u a t i o n
,J = U (y) + ,~o (4)
h a s a s o l u t i o n w h a t e v e r be Yo.
' In the case when Y = Z i s a space of lhe type B (and l y [ = i l y i l ) this corrolary coincides with the so-called principle of CACCIOPOLLI-BANACIt; vide, e.g., S. ]~ANACtI II, p. I6I and CAC- CIOPOLL~; I.
The Method of Successive Approximations for Functional Equations.
L e t in f a c t t h e n setting
v(~) = I u (o) + .Vo I + ,
,
Iv(o) +,~ol
I - - - C r
71
we have
V(z')= z'.
On the other h a n d V(o)--> o and V evidently satisfies also all other conditions of Theorem I. Also the f u n c t i o nU(y) + Yo
satisfies all conditions of Theorem I I (con& 3) is satisfied becnuse of (3)). Therefore we m a y apply Theorem I I to this f u n c t i o n and conclude t h a t a solutiony*
of the equa tion (4) exists such t h a tI~,~* I ~ ! ~ (-~ .%1.
I - - c ~ "
6. U n i e i t y T h e o r e m s .
Generally speaking there can be more t h a n one solution of t h e equation (2), however if the m a j o r a n t - e q u a t i o n (I) has only one solution t h e n t h e same is t r u e of the e q u a t i o n (2), or, more exactly:
T h e o r e m I I I . I f the functions U a n d Y satisfy all conditions of Theorem I I and V satisfies besides the condition 2'). (see 5. Rein.) and if moreover z * - - z ' * t h e n there exists but one solution of t h e equation (2) s a t i s f y i n g the condition
I y I g z ' This solution can be obtained by t h e m e t h o d of successive approxima- tions s t a r t i n g w i t h a n y value Y'o of y s a t i s f y i n g the inequality [Y'o[ -< z'.
v t _ _ : v v t
Proof. L e t y o E Y a n d suppose t h a t
l y
o l < Z'o Set yn =U(yn-I).
Then we have (using the n o t a t i o n s of Theorems I and II)
r v : v
I , ' o - y o l = I?Jol-< ~o ~ o - ~ o .
W e shall prove now t h a t in general
l Y' I '
F o r ~ : o this inequality is true. Suppose t h a t it is true for ,~ : ~%; we shall prove it for ~ = n o + I. I n f a c t
I.r -- ?I,,o+~ I = l ~" (J,,o) -- U(.~,o) I = I U(?Ino + (~/,,o -- i~,,o)) -- U(y,o) I --<
_< r(~,,o + (~',,o - ~,,0)) - v(~,,o) = / , , o + 1 - Z , , o + l .
,P ,'#
The above inequality is t h u s p r o v e d inductively. B u t we have lira z , ~ - z - - z* -- lim z~ and consequently lira (z'~, --- z,) ~ o. H e n c e lim
y',,~--lim yn~-y*;
t h u s starting' with an a r b i t r a r y Y'o s a t i s f y i n g only t h e condition lY'ol -< z' we arrive at the principal solution
y* of
the equation (2).t t P
Suppose in p a r t i c u l a r t h a t Yo satisfies the e q u a t i o n (2); t h e n y , ~ Yo a n d y * ~ lim y',~ = Y'0, Thus we have proved t h a t there can be no other solutions y of (2) s a t i s f y i n g the inequality
l yl-< z.'
except ?f.0orrola, ry. The equation (4), where U satisfies the condition (3), a d m i t s of b u t one solution which c a n be obtained by the m e t h o d of successive approxima- tions s t a r t i n g w i t h a n a r b i t r a r y element of y.
I n f a c t the ma.~orant e q u a t i o n
Iu(o)+ :iol+
has evidently one solution only.
T h e o r e m I I I ' , I f U a n d V satisfy the conditions of Theorem I I t h e n there exists only one solution of the equation (2) s a t i s f y i n g the condition l y l ~ z*.
This solution can be obtained by the m e t h o d of successive approximations s t a r t -
r !
ing w i t h an), y o such t h a t [y 0 [ - - ~ z ~.
T h e o r e m I I I ' is proved in exactly the same m a n n e r as T h e o r e m I I I with this difference only t h a t everywhere in the proof we m u s t substitute z* for z',,.
7. (',ontinuity Questions.
W e shall prove now t h a t if an equation of t y p e (2)depends in a c o n t i n u o u s m a n n e r on a p a r a m e t e r t h e n (if certain conditions are satisfied) its solution is also a continuous f u n c t i o n of the parameter.
T h e o r e m IV. I f an equation is given
:,~ = l ' (.'I; Z) (5)
where U is a f u n c t i o n of y a n d ~ defined for a certain set
/1.
of values of c o n t a i n i n g )~0(40 E l / ) ; a n d if the following conditions are satisfied: I). the equa- tion (T) is m a j o r a n t to (5) for every value of 4; moreover V is defined a n d satisfy the above conditions for o--~ z ~ 3z'
and condition 3)- of Theorem I [ is satisfied for every y and z?!/ such that andThe Method of Successive Approximations for Functional Equations. 73
2). U(y, ~)
as function of ~ is continuous in the point ~o, i.e. liraU(y, ~ ) =
2 ~ Io
U(Y,~o); 3). V satisfies condition 2') of R e m a r k to T h e o r e m I; t h e n t h e principM solution y* (~) of the equation (5) converges towards
y*
(~o) w h e n ~-> ~oi.e. ?/*
(~) r e g a r d e d as a f u n c t i o n of )~ is continuous in the point ~o.Proof. W e have (for a n y .n)
. : (x) = :~, (z) + (:F (z) - y,~ (~)) and besides
I : (~) - - Y,, (X) I -< : - - Zn.
W e shall prove now inductively t h a t for any .~
lira yn ()~) = y,, (Zo).
2 ~ 2o
I n fact for ~ = o this equality is true (because t h e n both members are o) suppose t h a t it is true for ~ = n o. Then we have
I :/,,o+, (X) --
:+,o+,
(go) I = I U (,:/,,o (Z), Z) - - U (Y.o (Xo), Zo) l -<--<l l; (.~,o (Z), Z ) - U (Y-o (Zo), Z) I + I g (Y,o (Zo), Z ) - v (y~o (Zo), go) l-<
-< v(ly,,~ + ly,,o(Z)- y , , ( ~ o ) l ) - v(l.~no(Zo)l) + + I u(y~o (Zo), z ) - u ( y , o (Zo), Zo) l
because [
y,~()~o)[ ~ z'
a n d [ y,o (~) --y,o()~o)[ ~ [ Y,,o()~)[ + ] Y,o ()~o)[ ~ 2 z'.
The first s u m m a n d in the right h a n d m e m b e r of this inequality converges towards 0 be- cause y,o(~)->yno(]~o)(bs)
and V satisfies t h e condition 2'); the second s u m m a n d converges towards 0 because U is continuous for ) . = s This proves t h a t Yno+l(Z ) -=> ff,o+i(~o). Thus we have proved t h a t for any n: y,,(2)-+ y,,(Z0).2 ~ 2 0
Now f r o m the inequality
Iv* (z) - nn (z) l -< : - - ,~,~
follows
I ~,:" (Z) - - : (Zo) I -< I . : (Z) - - y,, (Z) I + I . : (Zo) - - Y,, (Zo)I + +
I:/,
(z) ---y.
(Zo) l -< 2 ( : - .~,) + I.v,, (z) - y . (Zo) I.I f 2-~ 2 0 t h e n (according to w h a t precedes)
lira l y + (z) -
:/*
(Zo) l -< ~ ( : - ~.),2 ~ 2 o
n being a r b i t r a r y there follows
lira y* (Z) = y* (Zo). q . e . d . (6)
2.~ 2o
] 0 - - 3 9 3 2 . Acta mathematica. 71. I m p r i m 6 le 3 m a r s 1939.
w 3. Linear Equations.
8. Operations A d m i t t i n g a Majorant.
In this w we shall apply the general theorems of the preceding w to the special case of linear equations.
Let Y be a space of the type B.s.. Consider an additive operation trans- forming the space Y into itself, i.e. such t h a t whatever be :]e Y there exist always
f(!]) e Y
and t h a t for any y, e Y, y.~ e Y we have f(Yl +!/~) =f(?]:l) +f(Y.2).
Suppose t h a t there exists an operation g transforming the space Z into itself, additive, positive (i. e. such t h a t z > o implies
g(z)
> o) and besides such t h a t for any y e Y the inequalityI f(?/) I -< :: (I Y I) (7)
subsists. We shall say then that the operation f admits a m a j o r a n t and t h a t g is a majorant for f.
In this case (i. e. if f admits a m a j o r a n t ) f is necessarily homogenous, i.e.
whatever be y e Y and a real number E we have always
f(~y) =).f(y).
In fact in the case of rational E this follows from the definition of an additive opera- tion. Suppose L irrational and let L~(~ ~ I, 2, . . . ) be rational numbers such t h a t I ) . , ~ - - ) ' I < ! . W e have thenJ/
I Zf(y) -:'(z?/)l
<-l(z - z.)f(y) I +
If(~..,/-;.,,~/) I --:
I
-<-;If(y)l + ~(Iz- z,,l. I.'/I) -< ~If(:'/) I ~ :,:/(l?/I)
whence ~ being arbitrary and in a semi-ordered space .z >--o always implying
(')
inf ;;z = o we obtain f ( ) , y ) - ~ . f ( y ) .
In the particular case when ] : = Z and I:,Jl--
:,/I
(i. e. when Z is a senti- ordered space and for the norm of an clement z of Z we take its >>absolute ,value>; J zl) the set of all operations f admittin~q" a majorant g, (i. e. such t~hatfor any z E Z
I f(~):--< :/(Iz I)
can be considered itself as a semi-ordered space (if we write f~ > f ~ when the operation f ~ - - f ~ is positive) which we shall denote (Z-~ Z). ~ Accordingly for
1 KANTOROVITCH, I I I , T h e o r . 3.
Tile Method of Successive Approximations for Functional Equations. 75 such operations can be defined f + , fi-, [fi (which are again operations belonging to Z - ~ Z and moreover positive) and hold the formulae f = = f + - - f ~ , ] f l =
:=-f+ + f - & c . We see thus that any operation belonging to Z - r Z is the difference of two positive operations. On the other hand every positive opera- tion is its own majorant so thus every positive operation and consequently every difference of ~wo positive operations belongs to Z - > Z.
I f Z satisfies some additional conditions then the class Z -+ Z coincides with the class of (0)-continuous additive operations, i . e . of such additive opera- tions that z~ -+ z (o) implies f{z,) - > f ( z ) ( o ) . ~
W e shall remark lastly that if Y i s a space of the t y p e B , i.e. i f [ y ] = y l then an additive operation admits a majorant if and only if a constant C exists such that for any y e 15
if(Y) ii < C Y .
The smallest possible constant here w e shall define as ] f ] .
This class of operations coincides with the class of {b)-eontinuous additive operations. 2
9. F u n d a m e n t a l T h e o r e m s for Linear E q u a t i o n s .
In the case when operations mentioned in theorems of w z are linear these theorems may be somewhat simplified.
Theorem V. Let f be a linear operation admitting a continuous majorant g;
then if for a certain z ' , z ' > - - o we have
s0 - - ~ ' - v (~') -> o ( s )
then the equation
Y = f ( Y ) + Yo (9)
has a solution for any ?to such that l Yol -< zo.
Proof. This theorem follows immediately from Theorem II, if we set U (y) =
= f ( Y ) + Yo; V ( z ) := g (z) + zo.
In fact all conditions of Theorem I [ are satisfied:
V(o) = ,% -> o ; v ( z ' ) = ~ ( z ' ) + zo - z';
V ( z + d z ) - - V (z) .= g (d z) --> o if J z ~ o ;
1 A s to w h i c h v i d e Ibid., T h e o r . IV.
2 Cf. S. B ~ A C H , ], p. 54-
V(z) is a continuous function,
I u (o) 1 = I y01 -< ~o = V(o);
[ U(y + A y ) - - U ( y ) [ = i f ( J y ) [ <-- g ( [ J y ] ) _ < g(Az) = V(z + J z ) - - V(z), if only [ J y ] - - < J z .
Hence follows by Theorem I I that the equation y - ~ U ( y ) - - f ( y ) + Yo pos- sesses a solution which can be found by the method of successive approximations.
Theorem VI. Let z* be the principal solution of the equation (8). Then there exists only one solution of the equation (9) satisfying the condition [y] --< kz* (k is a real number --> I), viz. the principal solution of (9). Besides if the equation (8) possesses only one solution satisfying the condition [z[--< z' then (9) has only one solution satisfying the inequality [y[--< k z'.
The theorem follows immediately from Theorem I I I if we take V ( z ) =
= g ( z ) + k z o and substitute k z' for z'.
0 o r r o l a r y . Let Y be a space of the type 13. Then if i l f [ i - - a < I then the equation (9) has exactly one solution whatever be go ~ Y. This solution satisfies the following inequality
!l i! ~ yo
I - - ~
and can be found by the method of successive approximations starting with an arbitrary y E Y.
This corrolary can be proved by applying to the equation (9)the corrolaries of Theorems I I and I [ I ; it may be found in Banach. 1
Theorem VII. I f
y = f ( y ; z) + .~fo (z) (~o)
is an equation depending of a p a r a m e t e r Z and the equation (8) is majorant to (IO) for any Z and if (whatever be y e Y)
f ( y ; z) ~ f ( v ) and yo(Z) ~ Yo then
lim y* (Z) ~ y*
2 0 ),o
where y*(Z) and y* are principal solutions "of (IO) and (9) respectively.
1 S. BANACFr, I, p. I58.
The Method of Successive Approximations for Functional Equations. 77 This theorem follows from Theorem IV where we set U(y; 2 ) = f ( y ; ,a,)+ Yo 0,), U ( y ; ~ , o ) = f ( y ) + y o and V ( z ) = g ( z ) + z o. I t is easy to verify that all the conditions of Theorem IV are thus fulfilled.
io. Approximate Solution of Equations.
In order to obtain an approximate solution of an equation of the type (9) we must substitute in (9) for f another simpler operation f ' such that the norm of difference f ' ( y ) - - f ( y ) be smM1 and that the solution of the equation
v' --f' (v') + v0 (11)
be known to us. T h e n this solution of (I l) will be the approximate value of the solution of (9). W e shall give now a theorem which will enable us (under certain conditions)to estimate the error of this approximate solution and in the same time will prove (in the case the above mentioned conditions are satisfied) the existence of a solution of the equation (9).
Theorem VIII. Let the equation (I1) have a solution for every Yoe Y and suppose that this solution is an additive function of Y0
,y = r 0Io)
and that the following inequality subsists for every y e Y
I r(f(v) -f'(v))l _< ~1; I. ( o < ~ < i) (i2)
Then the equation (9) has a solution y and the following inequality subsists
I;-v'l-< 1~,~1SI. (IS)
Proof. W e can write the equation (9) as follows
Y -= ( f - - j " ) ( y ) + f ' ( y ) + ~to. (I4)
This equation (I4) is evidently equivalent (because F(yo) is the solution of (1I)) with the equation
y = r ( ( f - - f ' ) ( y ) + Yo) = r ( y ) + y' (I5)
where T = 1 - ' ( f - - f ' ) . To this equation we can apply the corrolary to Theorem I I
78 L. Kantorovitch.
(condition (3) of this corrolary takes in this case the form of the inequality (T2) which we have supposed to be true). Therefore the equation ( I 5 ) h a s a solution y such that
I:,II-< - - I ~ I J I.
I - - ~
This solution is in the same time a solution of the equation (9). Deducing the equation (II) from (I4) we obtain the equality
: , / - v' = f ' (.',t - .~') + ( f - f ' ) (y), whence
:i -
.,/-
r ( ( f - f ) (y)) andI.,f-y'l ~ - I y l -< " - I y ' l
I - - C~which prove (I3).
Remark if Y is a space of the type B then the condition (12) may be written as follows
cli.i!.f-f':i<_~ ( o < ~ < ~). (~2a)
w 4. S y s t e m s o f A l g e b r a i c E q u a t i o n s . I n this paragraph
to systems of algebraic of equations.
we shall apply the results of the preceding paragraphs equations chiefly to certain classes of infinite systems
T i. A Class o f Infinite S y s t e m s of L i n e a r Equations.
Consider first the systems of equations of the form
a v
~2i -= ~_j ci, k ~k + bz k=l
where (for any i)
( i - - 1, 2 , . . . ) (16)
c o
Y, I ~ i , ~ l - < ~ < i. (I7)
k = l
Such systems were considered, e.g. by Helge v. Koch, I.
Let I z be the space m of bounded sequences y = ( v l , V2,...) where i:Yi] is defined as sup I~/~[ and suppose thut the sequence b = (bl, b2, . . . ) belongs to Y,
l
i.e. that
:lb : s u p r])i[ ~2. 2c o~. (IN)
i
The Method of Successive Approximations for Functional Equations. 79
The operation f ( y ) = c;,~: w transforms the space Y into itself and we
k = l
have besides
Z vk Z
If(Y) [ = sup i
ei, k--< sup i(',~ J" t[:~iI--< ": V!
i i k ~ l i k = l
or ]W]I--<
~The equation (I6) is thus an equation of the type (9) satisfying the conditions of Corollary to Theorem u Hence follows
Theorem I X ? I f conditions (J7) and (~8) are fulfilled then the system 06) possesses one and only one bounded solution (i. e. one and only one such solu- tion t h a t sup t w l < + ~)- This
only
solution can be found by the method ofi
successive approximations starting with any bounded system a f values of {7;} and it satisfies the following condition
sup i T,[ -< I _ ~ _ sup
I bi'.
i I - - ( ~ t
Remark 1. If
(weaker) conditions'
instead of condition (I7) our system satisfies the following
0r
/; = N + I
a ~
2) y ,
k = N + l
ci, k i < +
( i = N + I , N + 2 , . . .)
( i = I, 2, . . . , N)
(~7')
and there exists an upper bound of all numbers
!ci, k],
then we can apply the preceding theorem to the systemv , = Z ( ~ , , k ~ +
b,+Z<~v~
(i = N + ~, N + 2, ) 0 6 ' )k = N + l 1.'~1
and express the unknown quantities V~'+I, . . . as functions of ~ 1 , . . . , ~]~-. Then after substituting these functions for W'+I, . . - into the first N equations of the system (I6) we shall obtain a finite system of equations. Hence, e.g., such a system does not possess more t h a n N linearly independent solutions etc.
V. KOCH, [, KANTOROVITCH & KRYLOFF, p. 2 8 - - 3 I .
80
any
Mark that conditions (~7') will be always satisfied when the system satisfies of the following two conditions (which were considered by Helge v. Koch)
~ v
a) Y~ I<~i < + ~
i, k~l
b)
where ~_~ z i < + m.2
R e m a r k 2. I f instead of the space of bounded sequences we shall consider the space lv (where p > I) of such sequences y = (Vl, W, . . . ) t h a t
is finite then we can arrive to the analogous conclusions.
In this case the expression of I]f'~ is rather complicated but it is easy to see t h a t in any case
Consequently if H < I then the system (I6) possesses a single solution satis- fying the condition
E I':,!" < + ~
i = 1
i f only
1 v. KOCH, I I . V. K o c r r , I I I . I n f a c t
I!f(x) [] = ci, k Xk
i=1~ 2 , . . .
V i d e K,t~'TOI~OVITCH, IV, T h e o r e m 6.
~]k=l ~' ~xk L i p -<
IF ~ i a,,~ I~ ~ ~ =
= H . L ! x I : .
The Method of Successive Approximations for Functional Equation s . 81 I f instead of the inequality H < I the (weaker)condition H < + ~ is satis- fied then for the system (I6') the corresponding number H ' will be < I provided t h a t N is sufficiently great. Consequently we may argue us in Remark I and reduce the system (I6) to a finite system. I n particular when p = 2 this condi- tion takes the form ~ e } , k < + ~ . This case (p = 2) has been considered by Helge yon Koch. ~
Consider again systems of the type (I6) but instead of (I7) we shall suppose the following weaker condition to be fulfilled
The space.
I say t h a t the system
set of all bounded sequences we shall consider now as a semi-ordered
is a majorant for the system (I6) provided only t h a t for any i we have
b~l --< ~e~. 01)
In fact if f * (Y) ---- {~k=X
[Ci'k{•k I"
thenf l ( y ) l : ci, kw. ~ , lei, kilVkl ~- f * ( l Y l ) .
1 i i
I t is also evident t h a t the operation f * is continuous, i.e. t h a t y - + o implies f*(y)-+ o. The sxstem (20) possesses a positive solution, viz. the solution
*A = V ~ . . . . K. Hence Theorems V and VI of w 3 may be applied and we obtain the following theorems.
Theorem X. The system (I6) satisfying the conditions (19) ~nd (2I)posses- ses a solution satisfying the inequalities / V~ I -< K (i = I, 2 , . . . ) ; the solution can be found by the method of successive approximations. ~
1 See H. v. KOCH, IV'.
" D I x o n , ~, I; FELLET, I I I ; WI~T~'ER, I I ; KOYALOVITCH, I ; KUZMIN, r ; KANTOROYITCH ~5 ]~_RYLOFF.
11--3932. A c t a m a t h e m a t i c a . 71. Imprhn6 le 3 mars 1939.
R e m a r k . Note t h a t condition (21) is always satisfying in the case when all
b~,
with the exception of a finite number of them, are o.Theorem XI. t I f {~*} is the principal solution of the system ( 2 0 ) t h e n there exists one only solution of (~6) satisfying the inequality iWl~< )~7". I n particular if the lower bound h of the sequence ~2~" is not 0 (i. e. if for any i:
V* > h > o) then each of the systems (I6) and (20) possesses a single bounded solution.
So, e.g., we find t h a t the system
( ~ - ~ - I , 2 , . .)
possesses exactly one solution if ]bi[~< =-. K On the other hand the system
I I
(i--- ,, z , . .)
has two bounded solutions, viz. ~2~ = I and ~h = ~ - ~ }" I
W e shall remark in conclusion that. from the next No. will follow t h a t the principal solution of the system (I6) may be obtained not only by the method of successive approximations but also as the limit of the solutions of (finite)
>)reduced)> systems.
i2. A Class o f Non-linear Systems.
Consider two systems of equations
oo r
v, = b,
+ Y, 4:i ~ vk, + 2~ 4:, ~, ~k, ~., + . . . . .f:
(v,, ~ , . . . ) and~, = B, + Z c(') ~~ + Z c~',!,,,
~, , " ~, ~, + . . . . S(~.,, ~ , , . . . )k t ~ l kl, k.~:l
( i =
, ,2, ...) (22)
( ? ' = I , 2 , . . .) ( 2 3 )
where we suppose t h a t the second system is a majorant for the first, i.e. t h a t for any i and kl, . . . ,
kj
.r
I K A N T O R O V I T C H t~ ~RYLOFF.
The Method of Successive Approximations for Functional Equations. 83 i b~l < B~ and c (~)
C (i)
i -- "~ ... ~:/ -< ~ ... ~:~" (24)
S u p p o s e moreover t h a t the second system possesses a positive solution, i. e. there exists such sequence of positive n u m b e r s z ' = - ( ~ " , ~'2, - . . ) t h a t if we s u b s t i t u t e in (23) ~'1, ~'2, . . . for ~ , ~s, . . . then (for every i) the series in the right h a n d m e m b e r of (23) will converge t o w a r d s the left h a n d m e m b e r of (23). 1 T h e n the equations (2z) and (23) w e m a y consider as the equations of the f o r m ( 2 ) a n d (I) respectively (see above, Th. II) in the space Y = - Z = s (see above, p.
67) i.e.
the semi-ordered space of all sequences of real numbers. W e shall prove t h a t all conditions of Theorems I and I I are satisfied.
I. W e k n o w t h a t
V(z')
exists. I t follows at once t h a tV(z)exists
for any z = (~'1, ffs, 9 9 .) such t h a t o ~ z--< z'.2. I t is evident t h a t in the region o < z ~ z' each of the series ]} (~'~, ~'~, . . . ) converges uniformly and c o n s e q u e n t l y if for every i we have lim L-(n)= ~ and
~t~ao ~ i
o --< ~ ) < ~'~ then limfi(~(f), ~(n), . . . ) = ] ~ ( ~ , ~.~, . . . ) . H e n c e follows the con- tinuity of the operation V (in the considered region o ~ z --< z') and in particular the conditions 2) and 2').
3.--5. Are evident
[V(o)-{~,}; v(~')=~'].
As to operation U
I). U(y)
evidently exists for any y such t h a t !y! --<z', i.e. t h a t for everyi:
l w [ < ( i .~). I U(o) i = {Ib,:l <-- lB,}.
3). W e m u s t p r o v e t h a t if l Yl --< ~, ~ ~'~'i -< ~ ' ~ a n d ~ + ~ ' Z --< ~' t h e n
i U ( y + ~ y) - U(y) I -< r(,~ + ~ ~) - v ( ~ ) or in other words, if
~]k ~ ' k a n d I -/J~]k!---~/~" ( k - - ,, 2 , . . . )
t h e n
Y~ P.~,, ..,
~ [(~.,
+ ~ ~ ) . . . ( ~ + ~ ~ ) -~k~... ~ ?
-<< Z Z C/'l
-- I;1,
j = l k . . . 9 = l
., ~j [(;~.i + J ~;~,) 9 9 (~;~; + ~' ~ j ) - ;~:, . . . ;kj]
B u t this last inequality is almost evident.
F i n i t e systems of this form were Considered by PELLET (ef. I and /I).
84 L. Kantorovitch.
Thus all conditions of Theorem I I (see w 2) are satisfied; its conclusion is therefore also true and we have thus
Theorem XII. Under the above conditions:
I). the system
(22)
possesses a solution satisfying the condition-<
(2s)
which can be found by the method of successive approximations;
2). if {.~'~} is the principal solution of the system (23) then there exists only o n e solution of the system (22) satisfying the condition (25) viz. its principal solution ;
3). if coefficients of the system (22) depend in a continuous manner on a certain parameter ~ (so that
bi
--bi(~), ((~)
"k ... kj-~ c~. ... b' (~)) and satisfy the in-(~) equality (24)
for all values of Z, and if {Vi().)} is the principal solution of(22)
then the functions ~ are continuous.
This theorem follows immediately from Theorems I I - - I V . But from Theo- rem IV follows besides another interesting property of the systems of the t y p e (22) (satisfying (24)).
Theorem XIII. Consider together with the system (22)the following ~)reduced>>
system (finite)
N N
= y ,
. . .+ be
2 , . . . ,(26)
, i = 1 ,~ .. . . 1,~ :.: 1
where N is a natural system (26). Then
number, and let t~(~v)~ be the principal solution of the i'J~ / lira ~(N) - ~ : ~o~o) "]i ( 2 7 ) where r is tile principal solution of the system (25). In fact if we add to the system (26) the equations ',2i--0 (i = N + I . . . . ) we obtain an infinite system of equations with coefficients depending on 5 r. W h e n N - + ~ these coefficients converge towards coefficients of
(22).
Besides the system (23) is a common majorant of all these systems. Consequently, by Theorem IV the principal solution {~I iv)} of the system (26) converges towards the principal solution {V~ ~ of (22) which proves (27).The following theorem is a corollary of Theorem X I I .
The Method of Successive Approximations for Functional Equations. 85 Theorem XIV. 1 I f there exists such positive n u m b e r s a and b t h a t (y~ and
~- denoting the same operations as above, see (22) and 123))
(~, b, b, . . . ) --< M (28)
for every i, then the system of equations
~ ] i : 2 f i ( Z , W, V.2, . . . ) ( i : I , 2 , . . . ) (29) possesses a solution for a l l ~ ' s such t h a t [;~[--).o where ;to denotes M i n ( a , b ) ; this solution {V~(~)} depends on ~ in a c o n t i n u o u s m a n n e r and vanishes for ;~-~o.
Proof. The system (z9) is evidently a system of the t y p e (2z) with coeffi- cients d e p e n d i n g
continuously
of 2. The following system will be its m a j o r a n t~ ' ~ = ~ o f i ( 4 , ~ , ;4, . . . ) + ( b - - 4 ~ ( 4 , b, b, . . . ) )
(here the differences b - ;~oy~(Xo, b, b , . . . ) are non-negative because of (28) and of the definition of 2o).
This m a j o r a n t system possesses a positive solution ~'1 = ~.~ . . . b.
A p p l y i n g now T h e o r e m X I I we arrive at once at the conclusion of our theorem.
I3. i Theorem on Finite Non-linear Systems.
Theorem XV. -~ L e t i). y~(~, ~,, . . . . , ~',) (i--: i, 9_, . . . , ,) be n increasing continuous functions of their a r g u m e n t s ; 2). f,.(o, o, . .., o) ---- o (i---- I, . .., ~) and 3). lim I .j~(x, x, . . . , x ) - - o for i = I, . . . , ~ (30)
x ~ X
then the system of equations
~ , - J ; ( ~ l , . . . , ~,,) + ai (i = ~ , . . . , ~,) (3I) has a solution w h a t e v e r be ai ~ o.
Proof. I f we denote by Z the ~udimensional Vector space considered as a semi-order space, z ' - ~ (H, H , . . . , H ) where H is a positive n u m b e r such t h a t for ( i = I . . . . , ~)
2 ']]his t h e o r e m follows also f r o m t h e B R O W E R ' S t h e o r e m o n , F i x p u n k t e m , .
86
(according sufficiently
H - - f ~ ( H , H , . . . , H ) -- ai > o
to (30) this inequality will be always satisfied provided t h a t H is great) a n d V(z)--= V ( ~ , . . . , , n ) - - { j ~ ( ~ ' , 9 9 ~ ) + hi} t h e n all con- ditions of Theorem I are satisfied a n d t h e r e f o r e its conclusion is true q. e. d.
I4. T h e E x i s t e n c e T h e o r e m f o r h n p l i e i t F u n c t i o n s .
W e are going to show t h a t the classical existence t h e o r e m for implicit f u n c t i o n s 1 can be obtained as a p a r t i c u l a r case of the general t h e o r e m s of w 2.
Theorem XVI. I f a system of equations is given
F~.(~],, . . . , ~2,~; ~1, . . . , ~ m ) - - o ( i = I, 2 , . . . , n) (3 2 ) where I). F i as well as ~ OF, are c o n t i n u o u s f u n c t i o n s of W, - . . , V,, ~i, . . . , ~m in a n e i g h b o u r h o o d of a c e r t a i n point (Yo, Xo)= (V~, . . . , 7 ~ s~, . . . , ~ ) ; 2). in ~o the point (Yo, Xo) the equations' (3 2) are satisfied; 3). in the same point t h e
D (/71, . . . , F~)
J a c o b i a n D(V~, . . . , W) does n o t vanish, t h e n this system possesses a c o n t i n u o u s solution in a n e i g h b o u r h o o d of the point (Yo, Xo)
Vi~---~(~I, ' ' ", ~m) (,:--- I, 2 , . . . , n) (33) where
.f/(~i 0), . . . , ~(m 0)) = /]i 0). (i - - I, 2 , . . . , , ) (34) Proof. Consider the vector space of the points y =- (~]~, . . . , Yn) w i t h the n o r m
~ F I - - sup (iv11 . . . . , Iv~l).
The system (32) m a y be t h e n written as F(y, x) = o
where x is a system (~1, . . . , ~,~) of m real nmnbers. Besides F ( y ~ x ~ = o.
Set
A (~, x) = a F ( y , x).
d y 1 Cf. VALL~:E POUSSIN, Ch. IV, w I.
The Method of Successive Approximations for Functional Equations. 87 T h e n A(y, x) is a m a t r i x d e p e n d i n g in a c o n t i n u o u s m a n n e r on y a n d x;
if D (A) is the d e t e r m i n a n t of t h e m a t r i x A t h e n we have D (Ao) = D (A (y0, x0)) ~ o.
L e t g ~ ?/0 + .4o~ z, t h e n we shall have the following equation for z
z = [Ao(AT ~ z ) - F ( y ~ + A o l g , x)] ~ - - - U(z, x) .... (35) d z ~ 0 o a u d consequently in a n e i g h b o u r h o o d of the point z = o the Lipschitz's condition (with a n arbitrarily small coefficient) is fulfilled. I n p a r t i c u l a r there exists such d t h a t IIzi! + ii.dz~!< d implies (for x sufficiently n e a r t o x0)
( 0 < : {~ < I)
T h u s the condition (3) of t h e Corollary to T h e o r e m I I is fulfilled, whence we conclude t h a t the considered equation in a n e i g h b o u r h o o d of the point z = o possesses a continuous solution
z = O ( x ) where O(x ~ B u t t h e n
y ~_~/o + A o l O ( x )
a n d
where )~i(i = - I , . . . , n) are c o n t i n u o u s f u n c t i o n s a n d
( i - - I, 2 , . . . , 3)
( i = I , 2, 9 3 )
The Convergence of Newton's Method.
~5'
An approximate solution of a system of e q u a t i o n s
/~;'(~21, " ' ", V n) = O (i ~--- I . . . . , , ) (36) which can be written shorter as
F(y) = o (37)
o f t e n m a y be f o u n d by t h e N e w t o n i a n m e t h o d of successive approximations. These successive approximations are expressed ~ by t h e following f o r m u l a e ....
y,,~ ~ .~n--1 -- A - 1 (~]n--1) ~l(yn--1) (38) where
1 ScA~no~owGn, No. 63; STONers, I; OSTROWSKI.
A (~j) - d F ( y ) _ 0~)i"
d ~j 6 9/]k !!i =1 . . . .
W e shall establish now (subject to certain conditions) the convergence of this process t o w a r d s the solution of (36). W i t h o u t d e t r a c t i n g of t h e g e n a r a l i t y of our r e a s o n i n g we can suppose t h a t Yo = o = (o, o, . . . , o).
T h e o r e m X V I I . I f all f u n c t i o n s /+'i as well as all t h e i r first a n d second derivatives are c o n t i n u o u s t h e n the N e w t o n i a n process converges t o w a r d s a solu- tion provided t h a t !F(o)II be sufficiently small.
Proof. Consider t h e e q u a t i o n
~l -~- A --1 (11) [A (y) ?! -- 1~'(?/)] = U (:/r (39)
This equation is equivalent to t h e e q u a t i o n (37) a n d the expression (38) gives the u s u a l process of successive a p p r o x i m a t i o n s for it. Consequently the process will surely converge towards a solution if t h e condition (3) is fulfilled. B u t this condition will evidently be fulfilled with c~ = I if ~F(y), is so small t h a t
2
; - d i ; ! = F(?~) __<-'
for such values of y t h a t ~y!i < 21 F(:~r ) .
(40)
w 5. I n t e g r a l Equations.
i6. Linear Integral Equations of the Second Kind.
Consider a n equation of Volterra's t y p e t
(t) - [ K (8, t) 7/(4 ~t,~ =.f(~) (4i)
Y
. J
a
a n d suppose t h a t in a certain interval (a; b) we have: I K ( s , t)l<~ 21I, !f(t)!<_ N.
T h e n t h e equation d gf.v)
i t t e r e
-dy
is a m a t r i x ( d e p e n d i n g on y) i . e . a n o p e r a t i o n t r a n s f o r m i n g t h e space y i n t o itself. I t s n o r m is t h e n o r m of an o p e r a t i o n .The Method of Successive Approximations for Functional Eqtmtions. 89
t
N (42)
will be a majorant equation f o r (4I).
But the equation (42) evidently possesses a positive solution which can be found e.g. by the method of successive approximations
M ~ N ( t - - a)"
~.(t) = N + M N ( t - - ~ ) + + . . . . x ~ " ( ~ - ~ .
(43)
I 2
Hence follows (if we apply Theorems V and VI) t h a t Volterra's equation has in an interval (a, b) one and only one bounded solution and t h a t this solu- tion can be found by the method of successive approximations; this single bounded solution will be also the only integrable solution because it is evident
1 '
t h a t any integr~ble solution of Vo terra s equation is in the same time bounded
c '
in (a, b).
In ~ similar way we can make analogical conclusions concerning' Fred- hohn's equation
b
(t) - z ~ K (~, t).,~ (~) d ~ = f(t). (44)
Y! a
I n this Case we shall obtain different boundaries for ,~ by using different spaces of functions (i. e. by giving different definitions to : ]Y:!). E . g . if we set y ==sup
y(s)
then we can prove t h a t the solution of Fredholm's equation8
exists when
b
sup f ,
K(,~,t) l a 8 < ~. (45)
Theorem V I I will allow us to make several conclusions as to the con- tinuity of the solution regarded as a function of parameter and as to lawfulness of .passing to a limit in the solution when the nucleis converge towards a >>limit nucleus>>. I n particular the solution of an integral equation can be obtained as a limit of solutions of a system of algebraic linear equations.
] 2 - - 3 9 3 2 . Actamathematica. 71. I m p d m 6 le 3 m a r s 1939.
I7. Systems of Fredholm's Equations.
Consider u system
.,<_t<_b,; , : , ( t ) - ~ fK,..~(.,t),a(,,)d.=9,(t)
(i-~-- 1 , 2 , . . . , n ) . (46) j = l a?'Define in a n y way t h e n o r m of a f u n c t i o n vz(t) e . g . l e t : W l ] = s u p ] w ( t ) ] ,
a i ~ t ~ b i
a n d take for y t h e set of systems of f u n c t i o n s y = ( V l , . . . , V,) a n d for Z the semi-ordered space of systems of real n m n b e r s z = (~1, . . ,, ~.,) a n d set ] ! y - -
= (]!V,I!, 9 9 '*],, ). Denote, besides, t,j
a i < t ~ b i
.j
Then if we consider the system (46) as a single equation of the t y p e (9) t h e following equation of the t y p e (8)
will be its m a j o r a n t . H e n c e follows :
r
- ~ < ~ C; =i= ~,
i' (i = i,..., ,,,)
(47) j = lTheorem
X V l I I . I f the algebraic system of equations (47) has a positive solution z = (~1, . . . , ~n) t h e n the system (46) also has a solution.This t h e o r e m follows i m m e d i a t e l y f r o m T h e o r e m V.
~8. A p p r o x i m a t e S o l u t i o n o f I n t e g r a l E q u a t i o n s .
W e shall n o w apply T h e o r e m V I I I which will allow us to estimate the error of a n a p p r o x i m a t e solution of an i n t e g r a l equation.
Theorem
XIX. 1 L e t k(s, t) a n d K(s, t) be two nuclei such t h a t t h e resolvent 7(s, t, Z) of t h e nucleus k(s, t) be k n o w n a n d letb
J K ( ~ , t) - - k (8, t) I a . = h s u p
a ~ t ~ b 3
a t KANTOROVITCIt & KRYLOFF, p. I59.
