Onecangeneralize eg3.6 to

PropI5 let thetas U R ked

m some open set VCRcontaining 0 suchthat

if Coto and Tejo cxS o Vij

Then I r o and R o such that Hye Brb

oIcxj y has a unique solution in Brio

Pt Fu Xi xzEBrC0 r 0 to bedetermined

Using remarked

consider get I Kitt Hz Xp fate 0,13i

Then 410 Elick 4ft Iida

Htt Eicxittaa xe IIE

TEikittah XD Hz Xi

IEilxd Eilxnl 19CI 9104

15 Ht dtl

E 11047cxittlxz xiD.kz XD dt

StolthEicxitthztaDldt HzXel

E ITEilxittixz HDL HzXel

forsome E'CQ1 by MeanValueThen since is d

Note that Xi cBHO xit Gz xDC Big

attire FI aoijsoFaiiaEi u

Eas Exist t.EE Iix

E EI ITEilxittacz Dl HzH1E Nr Hz Kl

By ffY Cx o ti j n and Iisc

I r o sit MrEtz NICKI EADIEElk XII

Take 13 4 E h E By Thur3.4Remark

F YEBELO IX CBHOJ set OI L

eg3 let glxjcC o.is and Kuyt E BELOIT

let 14 11 KH5 cffffco.rs oplKHtY

Then t gECCO B with 119116 sty

7 unique solution yEC6,13 with

113115444

Set

ycxt gcxs J.lk y7tsdtIntegralEquation

Pf Note that QB Il Ks is a Banachspacer o to be

consider OI BETO CEO is defined by determined

4

Icy sit V XE

QD.ICyscxiycx f.lk

yZt2dthetuIcyjBrcoT sCE9B be definedby

Icy folk yttsdt

Note also o a zerofunction

G 45670

b y Yz C BIT

HEIN EH HE EE I sokcx.tsyfctsdt fokcxpyiittdtf

folfff.ae lklxitsDlyices yicesl da

E M HYEyikes

E M HyztyHoHye Yello

E zrM Ily z y Ho

choose F then

HILYD Elba Host HyeYello f y YztBIHence Them3.4

FgeBETO with F t E r t Tm sty 0

7 YEBEICO sit Icy 9

ie ycxs fjkcxtsyktdt gcxs.tt XELED

which is the required solutionto the integral

equation

3.3 The Inverse FunctionTheorem

Recall ChainRule

Let Gi VCR IRM

F f CRY agedifferentiable

V openin Rh Rm respectively and

GCU CV

Then H Fog e U Rd differentiable and

DH X DF Gaa Da x

mere oa e iand similarlyfor DF DH

We alsoneed

PropI6 Let F B Rn be C where B ball in R

Then f X i Xz EB matrixacts on

FAD FAD DF tix g dtjohann

Mota

In componentfam F f n this is

Feck FIND JI IE CxzttC xzDdt Xi xD

Pf Fa each i L n

E Xi Fi Jo Fickett ki xD dt

So II EffyHatta xD Hi xD dt

g pffxzttlxi XD.HNDndtypdrotductof

TFoCxzttlxixzddtj.CHXD oectas

F Xi F DF text xD dt XiXa

Recall If F U CIR IRM bedifferentiable

at a paint p in an openset U ofR's

Then

Fcp tx Fcp Dflp x ofxD

t x f sufficientlysmall

xn ie 1 1 small

where o is a remaining term such that

04 11 1

o as 1 1 o

Thmet InverseFunctionTheorem

let F i U IR be a d map from an openset VCR

suppose Xo EV and DFCx o 5 invertible as a matrix

or linear transformation

a Then I open sets V W containing xo and

f No respectively such that the restriction

of F on V is a bijection onto W with a

C inverse

b The inverse is Ck when Fe's Ck Kkk 8

in V

W

IxHI

is invertible

Note We only have local invertibility by the IFT