On non-linear equations in a complex Banach space

O N NON-LINEAR EQUATIONS I N A COMPLEX BANACH SPACE'l)

By

L e o n B r o w n

in Detroit, Michigan, U.S.A.

In this paper we are concerned with the generalization of known

results in the theory of non-linear integral equations and in the theory of

several complex variables. Specifically, we are interested in generalizing the

Erhardt Schmidt Branching Theory [see 8, 9 and 11] (2~ and the Weierstrass

Preparation Theorem [see 2, p. 183].

We consider the following problem: let X be a complex Banach

space and given a function f , with domain and range in X, which is

analytic and bounded for ]]xl] <_-1, then what is the nature of the solutions

of the functional equation x - - f ( x ) = y , where y is a given element of J(?

This problem has been extensively studied when f is a completely continuous

(cQmpact) linear function. An excellent presentation of these results is in

Riesz and Sz-Nagy [10].

We wish to analyze the situation when f is a non-linear analytic

function with certain conditions. C3~ To this end we develop a specific tool,

namely, a generalization of the Weierstrass Preparation Theorem.

In paragraph 1 we present some pertinent lemmas in the theory of

complex variables.

In paragraph 2 we consider a function f whose domain is in J()< C

and range in C, where X is a complex Banach space and C is the space

of complex numbers, f is analytic and bounded for [Ix]] ~ 1 , and ]w] ~ 1 ,

x ~ X ' , w ~ C . Assuming f ( 0 , w ) has an s-fold zero at w = O , then in

a neighborhood N of the origin

I. Most of this paper is pat of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Minnesota. The dissertation was written under the direction of Professor Paul C. Rosenbloom whose encouragement and guidance were'most helpful.

A portion of this work was done while the writer was at Tulane University where he was partially supported by the National Science Foundation.

2. Numbers in brackets refer to the bibliography at the end of this paper. 3. Jane Cronin considered this problem. Her theory is based on the Leray-

Scbauder theory of the degree of mapping [see 4, 5].

2 LEON BROWN

8--1

V=0

where Hv are analytic functions on X to C, and ~ is a non-zero function

on N = X X C to C. The size of this neighborhood is estimated and P

and ~ are represented as integrals of the function f .

In paragraph 3, we summarize some results from Riesz's theory of

compact linear operators.

In paragraph 4 we consider the functional equation x - f ( x ) = y

where f ( o ) = 0. We assume that f is analytic and f ' ( 0 ) is a compact

transformation. If ( I - f ' ( 0 ) ) -1 exists then the above equation has a unique

solution if y is sufficiently small. I f I - f ' ( 0 ) has no inverse then there

exist complex valued functions f i , i = 1 . . . . . k and g on X X M where

M is a finite dimensional subspace of X such that x = g ( y . , u ) + u is a

solution if and only if f d ( y , u) = 0, i = 1 . . . . , k .

Assuming that for each i, f i ( 0 , u ) ~ 0, we can apply the Weierstrass

Preparation Theorem to these functions and then use the classical elimination

theory [see 16] in order to arrive at our "branching equations".

In paragraph 5 we prove that if R is the ring of functions, with

domain and range in X, which are analytic at the origin then R is integrally

closed. Applying a theorem of Butts, Hall and Mann [3] it is easily seen

that a monic polynomial in the polynomial ring of R can be factored

uniquely into irreducible monic polynomials.

The sizes of all above mentioned neighborhoods are estimated in this

paper.

w 1. L e m m a s on /=lnalytic Funct ions

By elementary means one can prove the following well known result:

L e m m a 1.1. I f F(z) is a n o n - c o n s t a n t a n a l y t i c f u n c t i o n

a n d IF (z ) l ~ l f o r I z l ~ t a n d IF (0 ) I = A > 0 , t h e n f o r ] z l ~ r < A ,

A - r A + r 1 - - A r ~- IF(z)[ ~ I + A ~

a n d

2 < 2 I l. IF(O-r(o)l lq_V. l~ ~z[ 2

ON NON-LINEAR EQUATIONS 1N A COMPLEX BANACH SPACE 3

We are interested in the nature of the zeros of an analytic function

if it is perturbated by a small constant. This is related to the classical

result of Hurwitz [15, p. 119].

L e m m a 1.2. I f F(z) is a n a l y t i c , a n d I E ( z ) ] ~ l f o r / z ] ~ l ,

a n d F h a s an s - f o l d z e r o a t t h e o r i g i n , a n d F(z )~O f o r

0 < ] z [ ~ 1, a n d

'l = rain IF(z) I < max IF(z) ] <~ 1, I~!=1 Izl=l

t h e n f o r 0 < ] ~ ] ~ , t h e f u n c t i o n F(z) - -~ h a s e x a c t l y s

s i m p l e r o o t s zi(Z), i = 1 ..... s in l Z l ~ l , a n d t h e s e r o o t s a r e

in t h e a n n u l u s

I~l~, < [z[ ~_

T h e zi(~.) m a y b e so c h o s e n s o t h a t t h e y a r e b r a n c h e s

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

c u t t h e n e g a t i v e r e a l a x i s .

Proof : I f [~.]<7, then I F ( z ) [ > ~/>[).] for ]z[-~ 1; and by Rouch~'s

theorem [15, p. 146], F(z) - -~ has exactly s roots in Izl < x.

For

and

Also

[z[< I~1'1~< 1 , lEO) l< [zi ~

[ F (z) - - Z l ~ I~1 - r F (z) l > I~l - - Izl ~ ~ 0 .

IF (z) l ~ ~ Izl s ,

so that for

Iz[ ~> (~-), 1f(z)-Zl~-lf(z)l-Tl>=~lz[s-T[>~

We proceed to show that the s roots of F (z ) - -~ . are distinct, and

also construct zi(~). Observe that F(z)z -s is analytic and not equal to zero in ]z]~ 1.

Let h(z) =z (F (z) z-s)'/s, where any particular determination of the s-th

root is chosen so that h(z) is analytic. For

4 LEON BROWN

I~[ -~ ~ , ih(z ) l = I~L I F ( O z - ' l * = I F ( z ) l * < ~. h ( o ) = o , h ( z ) r

for O < Izl ~ 1, and for Izl = 1 , Ih(z) l~ r l ' l s " Therefore, RouchCs

theorem implies that for Itl<~'/$, there is a unique z in [zl<l such that

h ( z )= t . Let g (t) be the unique z. The implicit function theorem implies

that g(t) is analytic for [tl<~'/s. The equation h ( z ) ~ t is equivalent to

F (z) = t~.

Placing t = ~ lls, a branch of the sth root of X and w - : e 2~'ils, we

have that zj(~) = g (w] - l ~l/s), j = 1 . . . . . s are thesroots of F ( z ) - - ~. q.e.d.

Note that if ~/----- max l F (z)l, then F (z) = Bz s . The zeros of F (z) -- ~. Lzl=l

are , and g(~) ---- is an analytic multivalued function.

We will have need for the following integral representation:

L e m m a 1.3. I f f ( z ) is a n a l y t i c f o r [ z l < t a n d f ( z ) r

f o r Izl---1, a n d f h a s e x a c t l y s z e r o s z t , . . . ,z$ f o r Izl<X, t h e n

f o r lz]> 1 ,

f (~') n = l ICt=t

w h e r e L o g z = log]z I + iargz , --~<argz_<__ n .

Proof: For Iz[>l , and I~l<=l, l o g ( 1 - ~ ) i s an analytic function

of ~'. One then simply evaluates the above integral with the aid of the

theory of residues, q.e.d.

The following lemma is a generalization of the Euclidean Algorithm.

L e m m a 1.4. I f f ( z ) , g ( z ) a r e a n a l y t i c f o r [ z l ~ l a n d

f ( z ) ~ - O f o r Izl-~ 1, a n d f h a s e x a c t l y s z e r o s f o r Izl< x, t h e n

t h e r e e x i s t s a u n i q u e p o l y n o m i a l P ( z ) o f d e g r e e <s , a n d

u n i q u e q(z) a n a l y t i c in Izl~l s u c h t h a t

g ( z ) = q(z) f ( z ) + P ( z ) .

Furthermore for Izl< 1,

P (z) = e (z) w (z) f g ( 0 a~" 2 ~i zv (r (~" - z)

Ir

and

ON NON-LINEAR EQUATIONS IN A COMPLEX BANACH SPACE 5

1 W(Z) ; g(~) dr q ( z ) = 2ai f(z) , . , W(~)(~--Z)- '"

Ir

Proof: C a s e I: We assume f hasss imple roots zi, j = l . . . . , s .

The condition P(zi )=g(z i ) , j= 1 .... , s uniquely determines a polynomial

of degree <s . Let

w (z) = I I (z - zi). j = ,

Using Lagrange's interpolation formula we find that

g (zj) w (z) P (:)

w' (zj) (z -zs ) " j = l

g ( O If lz ]<l and z • z i , i = 1 .. . . . s then

~ ( 0 ( ~ - z ) z, . . . . , Zs and z. Thus, by the theory of residues

has simple poles at

i f g(~)d~ ~ g(zi) + g(z) 2 a i w (~_) (~ - z ) = - w" ( z j ) ( z - z j ) w ( z ~ "

Ir = t i --1

Thus for z r [z l< t ,

w (z) ( g (r) d; P (~.z. = g (~.z. 2,~r . ~ ~ (O (~" - z)

Ir

w (z) / ~ g (~) d~" Since P g and

' 2a i J w (0 (~-- z) ]r

formula is valid for all z such that [z]< 1.

Let

are analytic functions the above

q ( z ) = g ( z ) - P ( z ) _ 1 ~,(z) f g ( O dr . f (z) 2Jti f (z) w (~) (~ -- z)

IC]=t

w(z) is analytic for Izl <--1 q Cz) is analytic for I z ] < l . q(z) is Since f (z---~ -- '

uniquely determined since P(z) is uniquely determined. Note that although

the above representation of q is only valid for l zt < 1, q is analytic

for [zl-- 1.

6 LEON BROWN

C a s e I I : Let f have multiple roots. From Lemma 1.2 we see that

there exists a ~ such that if 0< l~ , l<8 , then f ( z ) + ~ has exactly s simpte

roots zsQ.) for Iz!< t . Thus for each ~ there exists a unique function

q(z, )~) and polynomial P(z , )~) in z of degree less than s, such that

I f

then

g(z) = q(z , ~) [ f ( z ) + X] + P ( z , ~).

w (z ,

and

s

x) = ~ I 0 - ~s (~)) j = l

1 w(z,~.) s g (Od~ q (z

' ~ ) = 2~i f ( z ) + Z J w ( z , Z ) ( E - z ) " Ir

One can easily show that

l imw(z,~.)-----w(z,O)----- w(z ) . ~ + 0

However we wish to show that w(z , ~) is an analytic function of ~..

For Iz i> l , o < l x l < ~

Ir

= z~exp~p(z, ~.) (Lemma 1.5).

if I.I > t and I~l < rain I f (01 = ~ then ~p (z, ~.) is analytic in ~.. Since I~l=l

w (z, ~.) is a polynomial in z with coefficients functions in ~., each coefficient

is analytic in ~. and thus for I~.t < r a i n ( g , ~), w ( z , ~.) is analytic in ~..

Thus for I~.l < m i n ( ~ , ~), P (z, ~.) is analytic in ~. and therefore

lim P ( z , ~.) = P ( z , O) = P ( z ) . k-N0

P(z ) is a polynomial of degree less than s since for each ~., P ( z , ~.) is a

polynomial of degree less than s.

(z, Z) w P (O d r g P ( , x) = g (z) . ]

' 2~i __ w (z, X) ( E - z ) ' I~[=i


w(z,X) = ~ /" w(~',X) a: f (z) + ~ 2~ti ~ ! ( f (~') + )~) (~" -- z)

17]=1

for Ikl < ~ implies that w (z , ~.)

f (z) + ~. is an analytic function of ~ and we have

We have for

l i m q ( z , ~ . ) = q(z,O)= q(z). )V+o

[ z ] < l , g(z) = q(z) f ( z ) + P ( z )

with the integral representations which are valid, q.e.d.

We will need one more lemma in the theory of analytic functions.

L e m m a 1.5. I f f o r Izl<--l, F(z) is a n a l y t i c a n d

oo

IF(z)l ~ t , r ( z ) = ~ a , z ~ , a s ~ o , s~_ ~, k z s

a n d f o r [ z ] ~ l , G(z) i s a n a l y t i c a n d lG(z) l < e w h e r e

r* (las{ -- r) 0 < e <= ~ ([asl) - - max

o<--,<_l~J l--]asl r

t h e n t h e r e e x i s t s rl(~)~ro<=r2(8)~]asl s u c h t h a t F + G h a s

e x a c t l y s z e r o s i n t h e c i r c l e Izl<r~(r;) a n d n o z e r o s in t h e

a n n u l u s r,(e)<=]zl<=r2(~). F u r t h e r m o r e , i f ]z l '~r2(O a n d

W (z) = z s exp 1 f ' (~) + G' (~') -- - T 2~i F ( z ) + G ( z ) Log 1 d~ r

[~'l=r2 (~)

t h e n W is a p o l y n o m i a l o f d e g r e e s w h o s e z e r o s a r e

e x a - c t l y t h o s e o f F + G in Iz[<=r2(Q. T h u s f o r

F (z) + 6 (z) fz I < r2 ( 0 , f~ (z) - W ~z)

i s a n o n - z e r o a n a l y t i c f u n c t i o n a n d F ( z ) + G(z)~- W(z)fI(z) . We define rl (~), r2 (~), ro ([as[)----r0 in the following manner.

1) ro is the solution of the equation r o 1 - lasIro

2) r l ( e ) i s equal to the smaller root of ~ = - r s { lasl--r }

r2(~) is the larger root.

8 LEON BROWN

Note that rl(e) is a monotonically increasing function and r2(e) is

monotonically decreasing.

F (z) H (z) is analytic for lzl ~ 1, and Izl = 1, Proof: Let H ( z ) = z ~

1H(z) I = IF (z) I ~- i . Thus for [z I s I , l H (z) l s I . H (0) ----- as :;z: O.

Therefore by Lemma 1.1, we have for Izl <=r <= ]a,I,

la, l - , i lasl , < I n (z) I < la.[ + , - = = 1 + l a . l '

which is equivalent to

J~]' 1 - la~l, -~ IF (z)] ~< jzl ~

Let r = rl (0 or r2 ( 0 , then for Izl = , ,

[a~ l + r I 1 + la, I r / "

]G(z) t < e = r~{ fa,l-, } l - [ a , 1 , _ ~ I F ( ~ ) I .

Applying Rouch4's theorem, we see that F + G has exactly s zeros in

[z I <. r~ (0 and [z I < r~ (e), respectively. Consequently F + G has exactly s

zeros in [z I < r, (0 , and no zeros in r, (0 ~-- Izl < r2 (0 .

It follows from Lemma 1.3 that

$

W O) = I [ ( z - zj) j = t

where Izi} are the s roots of F + a in the circle Iz[~r~(O. Hence for

F (z) + G (z) Lzl < rl ( 0 , f~ (z) = w (z)

is a non-zero analytic function and F (z) + G (z) = W (z) Q (z).

q.e.d.

w The Zeros of Pmalytic Functions of a Complex Banach Space Let X , Y and Z denote complex Banach spaces and C the field of

complex numbers.

Definition 2.1. L e t f be d e f i n e d in an o p e n s e t D o f X

w i t h r a n g e in Y. W e s a y f is a n a l y t i c in D i f f is l o c a l l y


b o u n d e d and G - d i f f e r e n t i a b l e in D. T h i s is e q u i v a l e n t to

f p o s s e s s i n g a F r 6 c h e t d i f f e r e n t i a l at e a c h p o i n t in D

[see for example 7, pp. 109-112].

Using the concept of Baire continuity Zorn [18] has proven the

following generalization of Hartog's theorem:

Theorem 2.2. Le t DI and D2 be o p e n s e t s r e s p e c t i v e l y

c o n t a i n e d in X a n d Y. f is a f u n c t i o n on DIXD2 C X X Y to Z. N o t e t h a t X X Y is a c o m p l e x B a n a c h s p a c e w i t h

I!(x,y) l]= Ilxll+l!y]l. I f f o r e a c h xED1, f is a n a l y t i c in D2

a n d f o r e a c h y~D2, f is a n a l y t i c in DI, t h e n f is a n a l y t i c

in D1XD2.

We further generalize the Euclidean Algorithm.

Theorem 2.3. Le t f and g be t w o f u n c t i o n s , w i t h d o m a i n

in X X C and r a n g e in C, a n a l y t i c fo r [[xEl[~l, [w/_<_l. I f f o r

e a c h x s u c h t h a t I ! x ] l ~ l , f ( x , w ) has e x a c t l y s z e ro s ,

wj(x), j = l ..... s in I w [ < t , and f ( x ,w)@O f o r IWl=l , t h e n

t h e r e e x i s t s u n i q u e l y a p o l y n o m i a l P in w o f d e g r e e < s

w i t h f u n c t i o n s on X to C as c o e f f i c i e n t s w h i c h are

a n a l y t i c f o r I[xll<l, and a f u n c t i o n Q on X X C to C w h i c h

is a n a l y t i c fo r IIXH<l, [W]<l s u c h t h a t f o r Ilxll< 1, ]Wl<l,

g ( x , w ) = Q(x ,w) f ( x , w ) + P ( x , w ) .

In a d d i t i o n we h a v e f o r Iwl> 1,

= = e x p ~ J Log 1-- d~"

j=l [~-t= 1

and fo~ Ilxll<l, I~1<1,

Q(x w ) - 1 w(x ,w) ' 2~ti f (x, w)

and

wcx, ~) (~2--w) Ir

P ( x , w ) = g ( x , w ) W (x, w)

I;[=l

e (x, w) d~ W (x, 0 if--w)"

Proof: By Lemma 1.3 if Iwl > i for each x,

10 LEON BROWN

Ic[--~

Since W ( x , w) is a polynomial in w, in order to prove W is analytic, it

is sufficient to prove that for each w such that [w / > 1 W ( x , w) is analytic

in x. For each w, with Iw[ > 1, and for every hE X with % sufficiently

small,

W(x+~h,w) = wSexp ~ -~ - f(x+~h,~) /~[=1

and thus is an analytic function of ~, for ~.~--0. Consequently W(x, w)

is G-differentiable with respect to x. Since f ( x , ~ ' ) # 0 for I[I = 1, and

{~'EC I I~'1= i / is a compact set, W(x, w) is a continuous function in x,

and thus locally bounded. Therefore W (x , w) is an analytic function of x,

which implies that W is an analytic function of (x,w) for Ilx]l~l, wE C.

From Lemma 1.4, we see that there exist uniquely two functions,

Q and P, namely

and

1 w ( x , w ) s g(x,~)d~ Q (x , w) 2~i f ( x , w) ~I W(x,~)(~-w)

i~1--1

W (x, w) p g ( x , ~') d~" P ( x ~ b ) = g ( x w) |

' ' 2~ i J w ( x , ~') (~'--w) ' Ir

such that g = Qf + P . To complete the proof we need to prove that P

and Q are analytic for [Ixl[ < 1 and Iwl < 1 .

For IIxl] < I, [w I < 1, g and W are analytic. A proof similar to the

f g (x, O d~" one for W shows that W(x, ~)(~--w) is analytic in each variable

Ir and thus analytic (Theorem 2.2). Consequently P is analytic.

In order to prove that Q is analytic it is sufficient to prove that

W W ( x , w) - 7 - is analytic. For each x, ]]x][ < 1 , f ( x , w) is an analytic function

of w. Thus

ON NON-LINEAR EQUATIONS IN A COMPLEX BANACH SPACE 1l

w ( x , w ) _ _ ~ i " w ( x , O a 7 f ( x , w ) 2~i J f ( x , ( ) ( g - w )

I~]=t

Therefore W/f is analytic for ]]xl] < 1 and ]w] < 1. q.e.d.

The main theorem of this section is the generalization of the

Weierstrass Preparation Theorem.

Theorem 2.4. L e t f be a f u n c t i o n o n X X C to C a n a l y t i c

a n d b o u n d e d in D = { ( x , w ) ] [ [ x l [ ~ l , [ W i l l } , I f ( x ,w)]<=l t h e r e a n d

o0 / ( o , w ) = ~ a ~ w ~, a s : # o , s_~ I,"

k=8

t h e n

Q ( x , w ) a n a l y t i c in lw]<r<la,],

I]*l[ <= g~,) = ~ (

a n d H j ( 0 ) = O , ~] (O , O) -- as , Q(x

h o o d , a n d

t h e r e e x i s t u n i q u e l y s + l f u n c t i o n s Hi(x), j=o , . . . , s -1 ,

lasl - r .) 1 --ias] r '

,w)=260 in t h i s n e i g h b o r -

/ ( x , w ) = w ' - H i (x) wj ]=o

Q (x, w) = PO.

Furthermore if ro is the solution of

gO'o) = '~(l"~l) = max gO'), o<r<~ias l

and if for o = p ~ ct (la,]), G, (#) is equal to the smaller root of g(r)=p, and G2(R) is equal to the larger root (for R = ct (]asl), G1 = Gz = ro), then

for jlxll ~ p ~ ~(]asl), f (x , w) has exactly s zeros in I w] ~ G , (o) and none

in G,(o)~ [w] <=G2(,o). Thus for ['x]l ~ p~a(lasi), and ]w] ~G2(p), we have f : PQ and P and Q are analytic.

We have for [Ix]]~G(r) and Iwl<=r<=la,l,

and

( 1[ [ ( P x , w ) l s s 1 + - ~ -

I n ( . , w)l > 1 ( a s - r )

> 0 . (2,/, + 1)s 1 - l a , l r

12 LEON BROWN

Observe that

m a x

o_<r_< ]a s j

r ~(]as]-r ) < a( la~l)~ max - -

2 - - --o~r<[as I 2 la, r - , .). 1-1as7 '

therefore

s* s s (]as[)'+' 2 ( s+ l ) s+ t lasl s+' --<_ or(last) ~ 2 (s+l)S+ , (1--last 2)

Also if r <= lasI < 1, then

asl - - r 1 - F ~ , I ,

Consequently,

(lasl) ___ la,1 ~ax ,' - la'l'+' < • 2 ~ I 2 2

( x ) Proof: Let g ( ~ . ) = f ~ . ~ , w . We see that for I~l_<_l, gQ.)

is analytic and ]g(~)[~_ I . Applying Lemma 1.1, we have for ]).[ _~ 1,

2 I g ~ ) - g ( o ) l - - 1 -{- ~ / 1 --IJ(I 2 < 2 l~J.

Sett ing ~.:]]x[[, we have Ig(l]xll)--g(O)] : I f ( x , w ) - f ( o , w ) l < 2 ] ] x l ] .

w e let F ( w ) = f ( o , w ) and G ( w ) = f ( x , w ) - - f ( o , w ) and e = 2 Ilx[I

and apply Lemma 1.5. We have, if I l x l I < p < ~ ( l a , ] ) = 13([a,]) = : 2 '

f ( x , w) has exactly s zeros in ]w] < Gt (p), and none in

G,6o) ~_ I~1 ~_ c2(p).

Furthermore if for ] w ] > G2 (#), ltx II <= p,

w ( x ' ~ ) = ~ ' e x p 2~i 7 ( ; , O Log 1-- ac

I ~'I-----G2 (p)

then W is a polynomial i n w whose zeros are exactly those of f ( x , w).

With the aid of this representation of W (x , w) we have that W is analytic

for Ilxll < P ] w [ < G 2 ( p ) . Let Q(x w) -- f ( x , w ) ' ' W ( x , w ) " We have for

ITxll < p, ]w[ < G,(R,


O(x w ) - 1 y f(x,..') 1 d~ ' 2~i W ( x , 0 ~ - w "

Ir

Note that for [ ~'[ = G2(p), W ( x , r ~ 0. Consequently

analytic there. We define H(x) by the equation

(x, w) is

$ - 1

P ( x , w ) = W ( x , w ) : w ' - - Z H j ( x ) w]" j----0

Since W ( 0 , w ) : w*, H i ( 0 ) = 0, j = 0, 1 . . . . , s - - 1 . The analyticity of

W implies that H i is analytic for lixl! < p < ~ (] as I), [ w I ~ 62 (,o).

In order to complete our estimate we need the following result: for

Y II x II s g (r) , 2,/--7 < [ w [ <= r , f (~ , w) :/: o .

A - - u Let h ( A ) = 1 - - A t ' 0 < r < [ a s [ < l . For A < I , h(A) is

strictly monotonically increasing because h'(A) = ( l _ A r ) 2 > 0 for r ~ 1.

We know that for 0 < 1 ~ 1 < , I/(o w ) l < l w l s l a s l - , > o . . . . 1--1aslr

< x ) Applying Lemma 1.1 to f I [[xll , w and then setting ~.= I[xll, we

have for w ~ 0, and []xll --<__ rt < ] f (0 , w) } ,

lY(~, w)l ] f ( 0 , w ) ] - - r t >

1 - - [ f ( 0 , w ) [ r l

>_

- ~ZlZ, i ; / r~

If

rS ( ]a*[ - - r ~ ~ i f ( 0 w)[ and rl ~ - ~ - 1 - l a s l r ] ' '

we have

- - rt > - - - - - rt = 0 . 1 - - ] a , ] r 2 I - - ] a s l r

14 L E ON B R O W N

r So we have, if Ilxll ~ g ( r ) and 2,/~ ~ Iw[ <=r, f ( x ,w)r

Consequently, for []xll <=g(r), ]w] ~ r ,

f (x , w) ] IflCx'w)l = w(x,w) f

Applying the minimum modulus theorem we have

IQ(x , w)] >- min IQ(x , w) l Ew:=,

min I f (x, w) I rain I f ( x , w) l I,o:=, I~I=,

Iwl=' r s I + ~[;

We have shown earlier that for [w I = r, and for

I / (~, w)l >_- \-i ~ I a~l;! - "

tt~ri <__r, < I / (o , w)l,

2g (r) - - r, I~s l -r ) 1--2g(r)r,'

Let r , = g ( r ) , then for

If(x, w) l _~

I w J = r , l [ x i [~g ( r ) ,

g (r) > g(0 1 - - 2 ( g ( r ) ) 2

since

g(~) ~ ~(la~l) < ~ . 2

Therefore

IQ(x, w)l > ( 1 ), = h - l a s t r ( 2 ' I ~ + i ) , �9 r" 1 + ~-~ q.e.d.

Observe that the qualitative part of Theorem 2.4 can be proved by

applying Theorem 2.3. This would give us a different integral representation

of P.


We introduce the following notation: if A is any open sphere in C,

then H~ (A)----- {f I f analytic and bounded in A with ilfll = sup ]f(z)i }. ~EA

If A={z~CIIs we write Hoo instead of H| and let

S-- {f~Ho~l Ilflls

T h e o r e m 2.5. L e t T b e a t o p o l o g i c a l s p a c e a n d g a

c o n t i n u o u s f u n c t i o n f r o m T i n t o S. I f

00

f(to) fZ)= ~ak~k, a , ~ 0 , s>--i k ~ s

t h e n t h e r e e x i s t s a n e i g h b o r h o o d o f to. N, an o p e n s p h e r e

i n C, A0, w i t h c e n t e r a t 0, s c o n t i n u o u s f u n c t i o n s

Hi(t), j = 0 ..... s--1 f r o m N t o C a n d a c o n t i n u o u s f u n c t i o n Q

f r o m N t o Hoo(Ao) s u c h t h a t f o r a l l tEW, zEAo, Q(t)(z)~kO and

f (t) (z) = z*- ~ Rj (0 zi ~ (t) (.). j = 0

Proof : Using Lemma 1.5 the proof is essentially the same as t h e

t~rst part of the proof of Theorem 2.4.

w Summary of Known Results on Banach Spaces

We summarize some results in Riesz's theory of compact operators

(See Riesz and Sz-Nagy [10]).

Let X be a Banach space and T--= I - - K where K is a linear

compact transformation on X to X. If A ~ X then

T-~ (A) = {x~XlT(x)~A } and

Definition N . = T ( N . _ I ) .

T h e o r e m 3.2. M . a r e f i n i t e d i m e n s i o n a l s u b s p a c e s o f X,

a n d N~ a r e c l o s e d a n d t h u s s u b s p a c e s o f X.

T h e o r e m 3.3. T h e r e e x i s t s an i n t e g e r v s u c h t h a t

c C C C

Mo r M1 4: M2-'~ . . . r M~ = M~+, = . . . ,

T(A)= {y~X]~x~A~T(x)=y} .

3.1. M o = {o} and M. = T - ~ ( M . _ I ) . No = X" and

16 LEON B R O W N

a n d

No : # N , +e . . . +aNy = N~+, + . . . .

a n d T r e s t r i c t e d t o Nv is an i n v e r t i b l e o p e r a t o r .

T h e o r e m 3.4. X = M v ( ~ N v ; t h a t is, i f x E J ( t h e r e e x i s t s

u n i q u e l y uEMv a n d wEmv s u c h t h a t x = u + w .

T h e o r e m 3.5. T h e r e

t h a t

T (xq) = I xi'J+' t o

e x i s t s x q ~ X , a b a s i s f o r My, s u c h

j = . 1, ..., V i - - 1

y_-~,

k

w h e r e v=v,>=v2~. . .>-vk, a n d y v j = n = d i m e n s i o n o f My. j_--t

We now consider the adjoint transformation T * = ( I - - K i * = I - - K ~

Since K is a compact transformation K* is a compact transformation on X"

to X* and we define M* and N* as we defined M~ and m , . One easily

proves that :

L e m m a 3.6. M* = (Mn)*.

L e m m a 3.7. T h e r e e x i s t s xij a b a s i s o f M* v s u c h t h a t

x* (xkt) = ~ ,~ j t w h e r e xkz a r e t h e b a s i s e l e m e n t s o f M,, ij

A simple computation and we have

L e m m a 3.8.

* " / x ' , J - 1 j = 2 . . . . ,~'~

T (xq) = i 0 j = 1.

w Non-Linear Equa t ions and the Schmid t Branching Equat ions

Let X be a complex Banach space and f a function on X to X

which is analytic at the origin. Let f (0)----0 and if(O), which is a linear

transformation from X to Jr, be a compact operator. We then consider

the functional equation ( f - - f ) ( x ) = y where x and y are in X'. If

F ( x ) = f ( x ) - - f ' ( o ) x , then F ( 0 ) = O and F'(o)(x)-=O. In this situation

( I - - f ) (x) = y is equivalent to (I -- f ' (0)) (x) = y + F (x). I f ( I - - f ' (0)) -1

exists, we have x = ( I - - f ' ( o ) ) -~ ( y + F ( x ) ) which has a unique solution

for all y in a neighborhood of the origin (see fixed point theorem;

ON NON-LINEAR EQUATIONS 1N A COMPLEX BANACH SPACE 17

Theorem 4.3 or [6]). The more interesting situation occurs if the inverse

o f I - - f ' ( o ) does not exist.

Consider first the linear case: ( I - - f ' (o))(x)= y, and let T = I--f'(o)

W e use the notation introduced in paragraph 3. We have

Mx = T - ~ ( o ) = l x ~ X l x s , ( x ) = o , j = 1 . . . . , kJ

since

( T ' ) - ' (0) = {X~l, j = 1, ..., k}

and the range of T is closed (see [1]). Let

y . = { x ~ x l x* ( x ) = o i = l, k}. iv i , . . .

T h e o r e m 4.1. T h e r e e x i s t s a WE J( x s u c h t h a t

k

TW = I - - E xi' @ x" and W ( X ) c i l

i=1

C o n s e q u e n t l y T W ( x ) = x f o r a l l x~N1.

P r o o f : Let P---- X x i s(~)x* and P ' - I - - P . We have P~ = P , �9 . i . / t , J

(t:") 2 = P', and for each x ~ X , x ---- p(x) + p ' ( r ) where P(x)~M~ and

p'(x) ~N~(X = M~ 0 Nv).

T(N~)= N~ and T-t[Nv exists. Let V = T-11N~: V ~ N ~ .

Define W = Q + V P ' where

For x E X , we have

and

k v i -I

Q = Z Z ~=1 ~=1

k v l

x = P (x) + P'(x) = E ~ bO xij + P'(x) i=1 ]=I

W (x) = (Q + VP') (x) = (Q + VP') (P (x) + P' (x))

= QP (x) + QP' (x) + VP" P (x) 4- VP'P (x)

= QP (x) + VP" (x)

k ~ i -1 k v i

= j = l "= j = l

18 LEON BROWN

T W (x) =

k v i - 1

~=1 j = l

bi,j+, x 0 + VP ' ( x ) ~ ~ .

k v i -1

Z Z b, j+, x,,+, + P'(x i~ l j = l

k v i

E l i=1 j----2

P (x) - - ~ b~l xit + P ' (x) i=1

--'~=- X - -

k

q . e d .

w h e r e Corol lary 4.2: T h e g e n e r a l s o l u t i o n o f T ( x ) = y ,

5 ' ~ N l i s x = W ( y ) + u w h e r e u ~ M 1 .

Proof : I f y is not in N1 = T ( X ) then there does not exist any

solutions of T ( x ) = y . The condition y ~ N1 is equivalent to the conditions

x ~ , ( y ) = 0 for i = l . . . . . k .

We return to our original problem: x - f ( x ) = y which is equivalent

to x - - f ' ( 0 ) x = y + F ( x ) . We have: there exist solutions if and only if

x* ( y + F ( x ) ) = 0 for i = 1, k and these solutions are il ""~

x = W ( y + F ( x ) ) + u

where u ~ M 1 . I f we let x = x l + u where x1~3~ and u ~ M l we have

1) x l = W ( y + F ( x , + u ) ) , and

* ( y + F ( x ~ + u ) ) - - - - - 0 i = 1, k 2) xil ..., .

From equation l it is seen that a fixed point theorem is indicated.

T h e o r e m 4.3: A F i x e d P o i n t T h e o r e m (Hildebrandt and

Graves [6]). L e t X b e a c o m p l e t e m e t r i c s p a c e a n d

s(x0,a)= {x~XIp(x,x0) ga}.

Let f be a f u n c t i o n o n S ( x o , a ) t o X s u c h t h a t

1) A L i p s h i t z c o n d i t i o n is s a t i s f i e d ; t h a t is

p ( f ( x l ) , f ( x 2 ) ) s a n d o < k < l .


Therefore

Therefore

8

p(x , y) = l - - k "

q.e.d.

For Lemmas 4.5 to 4.9 we assume that F is a function from one

complex Banach space X to another Y.

L e m m a 4.5: I f F is a n a l y t i c f o r Ilxll<~r a n d F ( O ) = O ,

]IF(x)iI<=M f o r I lx l l~ r , t h e n f o r

[Ixll I l x l [ ~ r , [IF(x) I I < = M - -

Proof: For a fixed x ~ X , there exists a y*EY* such that [ly*l[ = 1

and y* (F (x)) = liF (x) ll (Hahn-Banach Theorem [1]). Let

\ 1, 11 J /

G(0) = y* (F (o)) = y* (0) = 0 and for Ilk.l] ~ r ,

I(x) [G(X) I~[ly'll F ~ _< M.

[G(~,)l ~ M [~'[ - - r

for I~.] ~ r (Schwartz's lemma). Setting ~.= Ilx]], we have the desired result.

2) p(Xo, f (xo)) : b <= a ( l - - k ) ,

t h e n t h e r e e x i s t s a u n i q u e x E S xo, c S ( x o , a ) s u c h

t h a t f ( x ) = x ( f ( x ) r f o r a l l o t h e r x E S ( x o , a ) ) .

L e m m a 4.4: L e t y ~ S ( x o , a ) s u c h t h a t p ( f ( y ) , y ) ~ , t h e n ,

i f x is t h e s o l u t i o n o f f ( x ) = z in T h e o r e m 4.3,

8

p ( x , y ) = l - - k "

Proof :

p ( x , y) ~ p ( x , f (x)) -k p ( f ( x ) , f (y)) + p ( f ( y ) , y)) ~ 0 + ko(x ,y) - r e.

20 LEON BROWN

L e m m a 4.6: I f F i s a n a l y t i c f o r Ilxll ~ 1

IIF(x)lr ~ M f o r Ilxll ~ x , t h e n f o r

M Ilxll < 1, llF'(x)[I

1 --Ilxll

a n d F ( 0 ) = 0 ,

P roo f : Let x o ~ X and x E X such that [lxll = 1, then there exists

a y * ~ Y * such that IlY*II = i and y*(F ' (xo) x ) = IIF'(xo) x l l . Let

G ( Z ) = y ' ( F ( x o + Z X ) ) for [~l_<_l - - I[xoll.

[G(~)[ __<_ fly*l[ ItF (xo+ ZX)[[ <= U

which implies that

M IG'(o)l

1 - - 11 x0 II

(Cauchy's Inequality). Thus

M G' (0) = y* (F ' (Xo) x) - - li F ' (xo) x II

- 1 - - I l x o l l

Hence

Ii F ' (x0)II = sup II F ' (Xo) x II < It,11=1

L e m m a 4 .7 : I f i n a d d i t i o n t o t h e

L e m m a 4.6 w e a s s u m e F ' ( 0 ) ( x ) - ~ 0 , t h e n

Hxll 1 11F' (x) II <~ M - -

r 1 - - r

M

1 - Iixoll

q.e.d.

a s s u m p t i o n s o f

f o r I l x l l ~ r < 1.

P roof : For

M M Ilxl! ~ r , IIF'(x)ll ~ ~ - -

1 - !lxll 1 - - r

(Lemma 4.6). Since F'(x) is an analytic function from X" to yx, Lemma 4.5

implies that

M I!xll IIF'(x)ll < - -

l - - r r

q.e.d.


L e r n m a 4.8: W i t h t h e s a m e a s s u m p t i o n s as i n Lemma 4.7,

w e f i n d t h a t

4 [[XHMM f o r [lxl[ ~ 1/2 [IF' (x)[i <=

- 1-Z ]]xll f o r Ilxll >-- 1 /2 .

1 1 P r o o f : Note that (1 - - r ) r has a minimum at r = u W e obtain

I the desired results by Lemma 4.8, setting r = ~ - when llx[[ =< T and

1 r = l l x [ I when [[xll=>_y.

L e m m a 4.9: W i t h t h e s a m e a s s u m p t i o n s as i n Lemma 4.7, I

w e f i n d t h a t i f I]x,l], I]X2]] ~_~r~- , t h e n

[IF (x,) -- F (Xz)11

1 P r o o f : I f [Ix H ~ r <= -2-, then [l

Let x, and x2 be given, [Ixll[, H x~]l

y" (F (x~) -- F (x2)) ---- LI F (xt) - - V (x2)I{ �9

Let G ( } O - - y * ( F ( x l + } ~ ( x 2 - - x O ) ) for [~.[_<__ 1.

G (1) -- G (o) = y* (F (x2) - - F (xx)) = II F (x2) -- f (x,)I[.

IIx, - -X(x~-x, )r l = Ir(t - -X)x , + Xx~ll <= ] ~ - X l llx,!l + IX] l]x~ll

< = I I - X I , + [ X [ , : r for 0<_X<_*.

W e have

4rM I[ xl -- x2 [[ .

F ' (x)! l < 4M * " = [Ix] ~ 4rM (Lemma 4.8).

_< r . There exists a y*E Y* such that

<= 4rM II x2 -- Xx I] �9

L e m m a 4.1o :

t h e n [ [ I - - K H ~_ 1.

II ( I - - K)-~ [[ ~_ 1.

q , e . d .

I f K E X x i s a c o m p a c t t r a n s f o r m a t i o n

F u r t h e r m o r e , i f ( I - - K ) - t e x i s t s , t h e n

!

G ( 1 ) - - G ( 0 ) ~--- f G ' ( ~ . ) d ~ . ~. Max ]G'(~.)[ o~__~<t

0

= Max I Y" [F (xl + X (x2-- x~)) ( ~ -- xl)] I

< Max [[y*[I I IF ' ( x~+t ( x z - - x , ) ) [ [ [Ix2-xl l[

22 LEON BROWN

Proo f : There exist x, such that I Ix~l l= 1 and IIK(x,,)ll<=l/n.

If not, let a = rain I l K ( x ) l l > 0 . Thus for x E K ( X ) , K -1 exists, and I1~1!=1

[!K-l( x [I <= • Ilxil . For any bounded set B c K ( X ) , K - t ( B ) is bounded a

and hence K ( K -j (B)) is compact. K ( X ) is locally compact and therefore

is of finite dimension. This implies K is of finite rank which is a

contradiction.

We have

II I - K !l = sup It x - - K (x)[I >_ II x~ -- K (x.)11 HzH=t

1 II x., li - - 1[ K (x,,) ',l ~ 1 - - - -

n

Therefore I [ I - - K I [ ~ 1.

I f ( I - - K ) - ' exists, then let ( I - - K ) - ' = I - - / ~ .

( I - - K ) = 1 or I ~ = ( I - - K ) - ' ( - - K ) . Consequently

transformation and l i ( I - - K)-*[I = I ! I - - -KI I >= 1.

L e m m a 4.11 : I f

W e have ( I - - K )

is a compact

q.e.d

k v i - I

w = 0 + vp' : Z Z | + vP" i : 1 j : l

(see Theorem 4.1). T h e n IIWII ~ 1 .

P r o o f : I l W i l > l I W l l N v = [ l V P ' l I N v = [ [ V l l A , v > = l

P ' - - I on N~).

(Lemma 4.10 and

q.e.d.

Let us return to our original problem. We have

x, -- w (y + F (x, + u ) ) ,

where u E M1. F (0) = 0 and F' (0) = 0. W e make the additional assumption

that for lixll _ 1 , IJf(x)ll <= I . This implies that for Ilxll ~ 1,

II F (x) II = II f (x) - - f ' (0) x II <_-- II f (x) II + l[ f ' (0)I[ II x II <= 1 + II x II <= 2.

Therefore, IIF(x)ll <= 2( l lx l l )h W e now apply the fixed point theorem.

T h e o r e m 4.12: F o r f i x e d y a n d u, l e t G ( x ) = W ( y + F ( x + u ) ) .

1 L e t g ( r ) = r ( 1 - - 8 r l I W I I - - 8 , o l I W I J ) , w h e r e p <

1611WII + 3211WI[ 2


a n d l e t r0 be t h e s o l u t i o n o f g(ro)----- max g ( r ) . T h e n f o r ][y[[, 0 ~ r < l

I [u l l<p , t h e r e e x i s t s a u n i q u e xl i n t h e s p h e r e [ Ix l l l<r , s u c h

t h a t G(xl) = x~, a n d n o f i x e d p o i n t i n t h e a n n u l l u s

r, <= llxll <= ro, w h e r e r~ is t h e s o l u t i o n o f g ( r ) = ( p + 2 , o 2 ) l l W [ I

a n d o < & < r o .

P roof : By

We have

r 0 - -

the use of elementary calculus we find r0 and g(ro).

l - -811Wli and g ( r 0 ) = _ ( t - - 8 , o l l W I I ) ~ 1611Wil 32 ilWII

Thus

ro + p = 1 -1- 8,0 II W !l 3 q- 4 II W II 1 < <

16 IIWII 32 IIW[I ( 1 + 2 I]WII) 8 liWll

Since 3 + 4 [IWII

32 IIWII ( t + 2 IIWII) is a monotonically decreasing function of the

7 1 Thus if I[ull Ilyll < p then IIWII and [ ' .WII~ 1, r 0 + p < ~ - j < ~ - , , :

for I[xl[[, I I x 2 l I ~ r where r,<_r<_ro,

II G (x2) -- G (xl)[l = II W (F (x2 + u) -- F (x~ + u)II ~ II W ',1 8 (r + p) II x2 - - x, II

= k (r) [! xz - - x111

where k ( r ) < 1. Furthermore, we have that

[l G (0)I] --- W (y + F 0 0 If <---- II w 1[ (]]yl] + 2 I1,1] 2) <= IJ w If (p + 2p 2)

< r (1 - k ( , ) ) = g ( r ) .

Consequently, G (x) satisfies the hypothesis of Theorem 4.3 and the conclusion

of the theorem follows.

1 We see that for I]YI!, IIu[I < p < t 6 [ [ w l l _ I_ 32[iWi]~ we have a

unique function x ( y , u ) such that W ( y - l - F ( x ( y , u ) + u ) ) - - = x ( y , u ) .

L e r n m a 4 .13: x ( y , u ) is a c o n t i n u o u s f u n c t i o n f o r

I l y l [ , l ] u H < , o .

p r o o f : Let ]IYIII, Hu*ll , Ily211, Ilu2l] be less than p. We have

24 LEON BROWN

II w (yt + F (x (y2, u2) + u,,)) - x (y2, u,)II

= I[ w (y, + F (x (y , , .~) + . , ) ) - - W (y~ + F ( . (y~, ~ ) + ~))rl

-= ]] W(.yt--y2) + W ( F ( x ( y 2 , uz) + u 0 -- F ( x ( y 2 , u2) + u2))!l

<= ]I W [i ( l [ y ~ - y , [[ + 8 (ro +,o)][ u~--u2 [[)= ~.

Therefore

E H ~ ( y ~ , - ~ ) - ~ ( y , . . , ) 1 1 <=

1 - - k (ro)

(Lemma 4.4) which implies that x (y , u) is continuous.

Theorem 4.14: (see fbr example [10]) L e t L E X x a n d IJLJI<I ,

t h e n ( I - - L ) - ' ~ I + v~~L" e x i s t s .

Theorem 4 1 5 : x ( y , u ) is an a n a l y t i c f u n c t i o n f o r

xJl, l lul l < p.

Proof: Let G(x , y , u) = W ( y + F ( x + u ) ) . Then for I]xjJ--<_r0,

1 and I[u[[--<--P< 1611WI]+32(IIWII) 2 we have

{i G, (~, y , u)!! = i! w (F" (~ + ~,))II

liWll li F" (~ + ,,)II ~ 8 IIwll li ~ + ,~ ii -<- 8 l lwll (]I~II + !lull) I[wI[ s (,,o + p) < ~.

Thus we have IIG~(x,y,u)ll< t ana (Z--G~)- ' exists. Recall that

x - ~ G ( x , y , u ) defines x ( y , u ) . If A x - - - - - x ( y + X h , u ) - - x ( y , u ) , then

- - t - - hx = G,(x , y , u) hx + o( h hxil ) + Gy(x , y , u , ) (~h)

+ o (/[ xh I1) = G. (~ , y , ~) Ax + G~ (~, y , u) (~h)

+ o (II Zh II)

since x ( y , u) is continuous. Therefore we have

Ax-- (X -- G,)-' [(G~) (~h) + oCH ~h!l)] and

Consequently

and similarly we have

Ax lira ---- (I -- G,) Gy.

xy (y , u) = (I - G,)-' (Gy)


x . ( y , u) = (r - G+)-' G. .

Thus x has Gateau partial derivatives and since x is continuous it has

Fr&het partial derivatives which implies that x is an analytic function for

[lYlI, Ilul] < p .

q.e.d.

We originally had

1) x , ~ W ( y q - F ( x l q - u ) ) = G(xl , y , u)----- G(xl) and

1) x ~ j ( y + F ( , , + u ) ) - - - - 0 for i = l . . . . . k .

1 qt rl This is equivalent to for iiYJJ, :l~iu]] < P < i6 ffwft + 32 rlwif ~

1) x, ----- x ( y , u) and

2) x* ~--- it 0 ' -t- F (x, q- u)) ---- 0 for i 1 . . . . . k .

Thus we have for I ly l<p, if there exists a u ~ U , , I < i < P such

that x* , , it(Y -1- F ( x ( y , u ) - } - u ) ) = f i (Y u) = 0 for i ~ - 0 . . . , k then

x = x ( y , u) q - u is a solution o f the functional equation x - - f ( x ) ~ y , and

if x l q - u , x l E ~ and u E M x is a solution, then x* i; (Y -[- F (x, --b u)) = 0

and if IlYH, Ilull < P , x t = x ( y , u) .

So we see that f i , i = 1 ..... k are the branching equations that

y and u need to satisfy in order for a solution of x - - f ( x ) = y to exist.

It would be desirable if our branching equations were independent o f u.

Therefore further analysis is indicated.

We have for Llyli, [lull < p , ]+,(.", u) = o i = 0 . . . . . k where k

u = E Xixi~,+ and the f i ' s are analytic functions. Thus

oo

r i (y , u) = ~ , P o ( y , u), j = o

where Pij is homogeneous in u o f jth degree and analytic. Let vi be the

smallest j such that Pij(O, u ) ~ o . Note that vi exists if and only if

f i ( o , u ) ~ O . We consider Pi~i(O , u) as a function on Ck to C. For

i = I ..... k, {u EM1 I Pivi(O , u) = O} is a k - - 1 dimensional algebraic

manifold in Mr . Therefore there exists a u0 E M1 such that P~i(O, Uo)ck 0

for i = 1 . . . . . k. (This result can be obtained using the concept o f category.)

W e choose a basis o f M, of the form u0, x2, . . . , x, , and for

26 LEON BROWN

u ~ M l , u = ~uo + ~2 x2 + ... + ~, xk ,

where ~., ),2 . . . . . 3., e C. We have

p,o, (y , ~) = ~ . ~ , . . . . . ~ (y) ~1 )~2 ... ~;~ ,

~c~ i ~ vi

and

f i (Y , Uo) = avi,o . . . . . o (y))`vi + ... ;

Thus Theorem 2.4 implies that for i---- 1 . . . . . k

v i - I

av, . . . . . o # 0 .

f i ( Y ' U) ----- ( ~vi - E H i i ( Y , )~2 . . . . . )~k))`i) Q i ( y , u ) i_--0

where H i i (0 . . . . . O) ---- 0 and H i i , Qi are analytic in a neighborhood of the

origin. Consequently f i ( Y , u ) = 0 if and only if

vi-1

~.~i - - E He (y ' )̀ 2 . . . . . )`k) ~.i = gi (~.) = 0 . j-----o

Defini t ion 4 .16: L e t

hi ()`) = ao ~.' + ... + a ,

a n d

h2 (~) - - b0 ~s + ... + b,

be t w o p o l y n o m i a l s , t h e n w e d e f i n e

a o , a t , a 2 , . . . , a v , O , . . . . . , 0

O , a o , a t , . . . , a r - t , a r , O , �9 �9 , 0

O , - - - - O , a o , a l , . , a t R ( h i , h2) =

b o , b t , . . . , b s , O , �9 . . . . . . , 0

O , bo, . . . , bs-t , bs . . . . . . . , 0

O, . . . . bo , b l , ... ,b~

( s e e V a n d e r W a e r d e n [16]) .

S r o w s

r r o w s

T h e o r e m 4 .17 : G i v e n k


p o l y n o m i a l s , gi(X), i = 1 . . . . . k, l e t

k k

,9, = g, a n d = g,

' l : t ~ :L

w h e r e wi,v~, i = l ..... k a r e i n d e t e r m i n a t e s . T h e n R(ccl,q~2)

is a p o l y n o m i a l in t h e i n d e t e r m i n a t e s wi,v i , i - - - - i . . . . . k.

T h e c o e f f i c i e n t o f W~lW~2...W~kV~I...V~k is a p o l y n o m i a l

say ds, in t h e c o e f f i c i e n t s o f t h e gi's. T h e n g i ( ) . ) , i = 1 ..... k

h a v e a c o m m o n r o o t i f a n d o n l y i f ds-----O, s = l ..... m.

Applying this result to our problem we obtain 3, is a common root of

ui-1

X"~ - - E H e (y ' ~2 . . . . . Ik) ~J, i = 1 . . . . . k j = 0

if and only if ds(Hij(y2,~2 . . . . . J~,))=0 for s = 1 ..... m. Since d~ is a

polynomial in the Hi/s, we have ds(Hit) are analytic functions from

X X C ~ - t to C.

Continuing in this manner, we arrive at a finite number of branching

equations qj (y), j -- 1 ..... t such that if qj (y) = 0, j---- 1, ..., t then there

exists a finite number of solutions to the equation x - - f ( x ) = y .

w Unique Factorization of Monic Polynomials

In this section we consider another approach to the analysis of the

equations g~(~)= O. We consider the coefficients of these polynomials in

as elements of some ring R and ask whether there exists unique facto-

rization in R[~]. Observe that the Hij's are functions from a complex

Banach space to the complex numbers C, and these functions are analytic

in a neighborhood of the origin.

Definition 5.1: L e t Y be a c o m p l e x B a n a c h s p a c e a n d R

be t h e s e t o f f u n c t i o n s f r o m Y to C, w h i c h a r e a n a l y t i c

at t h e o r i g i n , w i t h t h e f o l l o w i n g i d e n t i f i c a t i o n : f = g i f

a n d o n l y i f t h e r e e x i s t s a s e t A o f 2nd c a t e g o r y w i t h

r e s p e c t t o t h e o r i g i n s u c h t h a t f = g o n A. T h e i d e n t i t y

t h e o r e m s t a t e s t h a t f ~ g o n i t s d o m a i n o f a n a l y t i c i t y .

f , g ~ n , t h e n ( f -i- g) (x) = f ( x ) + g ( x ) a n d ( f . g ) ( x ) = f ( x ) . g ( x ) .

One easily proves that

28 LEON B R O W N

1) R is a commutative ring with unit

2) R is an integral domain

3) if f ~ R , f is a unit if and only if f (0):~0

4) The non-units form an ideal.

Let Q be the quotient field of R, and R[y] , Q[y] the polynomial

rings of R and Q respectively. A polynomial in R [y] whose coefficient of

the highest power of y is a unit in R is called a tonic polynomial.

Definition 5.2: Le t .jr be an i n t e g r a l d o m a i n w i t h un i t ,

F i t s q u o t i e n t f i e l d and J[y] t h e p o l y n o m i a l r i n g o f J.

We say t h a t J is i n t e g r a l l y c l o s e d i f f o r e v e r y t o n i c

p o l y n o m i a l f(y) in J(y), any r o o t y0 o f f in F is in J.

Theorem 5 . 3 : - ( B u t t s , H a l l a n d M a n n [4]). Le t J be an

i n t e g r a l l y c l o s e d i n t e g r a l d o m a i n w i t h u n i t e l e m e n t and

F i t s q u o t i e n t f i e l d . Le t f(y)~ J[y] and f(y)=g(y)h(y) w h e r e g(y),h(y)~F(y). Let f ,g a n d h h a v e f i r s t c o e f f i c i e n t s

a, b,c, r e s p e c t i v e l y . T h e n ~g(y) , ~-h(y) h a v e i n t e g r a l

c o e f f i c i e n t s a n d t h u s af(y)= ( ~ g ( y ) ) ( ~ h(y)) is a

d e c o m p o s i t i o n o f f(y) in J[y]. Corollary 5.4: I f o r is i n t e g r a l l y c l o s e d and t h e t o n i c

p o l y n o m i a l f(y)~J[y] f a c t o r s in FLY], t h e n i t f a c t o r s in

J[y]. T h u s , t o n i c p o l y n o m i a l s h a v e u n i q u e f a c t o r i z a t i o n .

Theorem 5.5: R is i n t e g r a l l y c l o s e d

Proof: Let f ~ Q , where f is a root of a tonic polynomial: that

g is f = ~ - , where g , h E R and

fn + E a J f J = 0 where a j a R . j=o

Since h(x)~ O, there exists a Xo such that h(xo)=;s We have

h ()~x0)~ 0, and h (0)---= 0. Thus there exists an r such that for I Z l - - , , h(zx0) 0. Since Ix c I I Zl = '} is a compact set, there is a neighborhood

of the origin N(0) such that if y E N ( 0 ) and I Z l - - , , h(y+zx0) 0.

ON NON-LINEAR EQUATIONS IN A COMPLEX EANACH SPACE 29

I f P ( z ) = z " + E a J z i ' then all the zeros o f P are in the circle j=o

I z i ~-- F(ELa0 ! . . . . . I a " - t I) , where F is some func t ion : for example if

l a j [ ~ M , ] = 1 . . . . . n - - l , P ( z ) = o , then i z i ~ = m a x ( n M , ( n M ) l / " ) .

For a fixed y ~ N ( o ) , / ( y ~ZXo) is m e r o m o r p h i c funct ion o f ~.

(]~(y + Z x 0 ) ~ 0 ) . F rom the r emark in the preceding paragraph we see it

is bounded and thus f ( y + 3,x0) is analytic in 3, for ]~.] ~ r . Therefore

for y ~ N ( 0 )

1 (" f (y q- ~.Xo) f ( Y ) - 2~i , - Z dZ.

[~[--r

Thus f is analytic in N(o), and f ~ R.

q.e.d.

W e no te that monic po lynomia l s factor in to p roduc t s o f monic

po lynomia l s . The genera l iza t ion o f the Weie r s t r a s s Prepara t ion T h e o r e m

gives r ise to special types o f mon ic po lynomia l s , namely, the leading

coefficient is a uni t but all o ther coefficients are non-uni ts . Since the non-

uni ts form an ideal one can easily p rove by induct ion that these po lynomia l s

factor in to po lynomia l s o f s imilar type.

B I B L I O G R A P H Y

[1] B a n a c h, S. Th~orie des op6rations lin~aires. Warszawa, 1932. [2] B o c h n e r , S. and M a r t i n , R. S. Several complex varial:les. Princeton

University Press, 1948. [3] B u t t s H., H a l l M. Jr , and M a n n H. B. "On integral closure". Canadian

J. Math., 6 (1654), 471--473. [4] C r o n i n, J. "Branch points of solutions of equations in Banach space". Trans.

Amer. Math. Soc. 69 (1950), 208--231. [5] C r o n i n, J. "Analytic functional mappings". Ann. of Math. (2) 58 (1953),

175-- 181. [6] H i l d e b r a n d t , T. H. and G r a v e s , L. M. "Implicit functions and their

differentials in general analysis". Trans. Amer. Math. Soc. 29 (1927), ~27--L53. [7] rt i I I e, E. and P h i l l i p s, R. S. Functional Analysis and Semigtoups, Amer.

Math. Soc. Colloq. Pub. vol. 31, revjsed edition, New York (1957). [8] L i c h t e n s t e i n, L. Vorlesungen fiber einige Klassen nichtlinearer Integral-

gleichungen und Integro-differentialgleichungen. J. Springer, Berlin, 1931. [9] L y a p u n o v, A., Sur les figures d'~quilibre peu diff&entes des ellipsoides d'une

masse liquide homog~ne dou& d'un mouvement de rotation I, &ude g~n~rale du probl~me. Acad. Imp. Sci. St. Petersbourg, 1906.

30 LEON BROWN

[10]

[~11

[12]

[131

[14]

[15]

1161 [~7]

[18]

Wayne State University Detroit, Michigan, U.S.A.

R ie sz, F. and S z - N agy , B., Legons d'analyse fonctionnelle, Acad6miai Kiado, Budapest, 1952. S c h m i d t, E , "Zur Theorie der linearen und nichtlinearen Integrargleichungen I11. Ueber die Aufl6sung der nichtlinearen lntegralgleichung und die Verzweigung iher L6sungen". Math. Ann. 65 (1808), 370--399. T a y 1 o r, A.E., Analytic functions and general analysis. Unpublished Dissertation, Cal. Inst. of Tech., 1936. T a y 1 o r, A. E , "Analytic functions in general analysis". Annali Della R. Scuola Normale Superiore di Pisa, 6 (1937), 1--16. T a y I o r, A.E. , "On the properties of analytic functions in abstract spaces". Math. Annalen, 115 (1938), 574-593 . T i t c h m a r s h. E.C., The theory of functions (2nd ed). Oxford Univ. Press,

1939. V a n d e r W a e r d e n . B. L., Moderne algebra II. J. Springer, Berlin, 1931. Z o r n, M. A., "Characterization of analytic functions in Banach spaces". Annals of Math. (2) 46 (1945), 585-593 . Z o r n, M . A . , "Derivatives and Fr~chet differentials". Bull. Amer. Math. Soc. 52 (1946), 133--137.

(Received June 23, ~960)

On non-linear equations in a complex Banach space

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