problems - math.cuhk.edu.hk · Recall the notion of Initial Value Problem: Def An Initial Value Problem I IV P) consists of the following equations Htt = tax, X to ) = Xo where f:

Tutorial 10 : Selected

problems

of Assignment 9

Leon Li

13/11/2019

Recall the notion of Initial Value Problem :

Def An Initial Value Problem I IV P ) consists of the following equations

{Htt = tax ,

X to ) = Xo

where f : R := I to - a. Tota ] x Exo - b,

Xotb ] → IR is continuous .

- -

Ia Ito ) Ib No )

An IVP is uniquely solvable for a' ECO , a ) if there exists a unique function

XH ) : I alto ) → Iblxo ) such that Xlt ) is C'

and solves IVP :

{X' HI = TH.Hts )

,

t

te Ia . Hos

X to ) = Xo

Thin ( Picard - Lindelof ) Given an IVP as above.

① If f satisfies a Lipschitz condition l Uniform in t ) ,i.e . I L > O

such that V-H.xd.lt , xD ER,

ITH. xD - fit . xd ) s L Ix ,

- xd,

then

IVP is uniquely solvable for anya 's min { a

,

Ty,

'T },

M : -

- SIP HH.nl Cassuming M > o )

② IS in addition f E CYR ),

⇒ kzi,

then X HI E C' ' "

I Taito ) )

Q1 ) l HW 9,

Q

4)Using

the

perturbationof identity ,

prove

①

forany

a 's min { a.

÷b,

-4,

where Mo 's

t.EE?ul5H.xdl

Pf ) Recall that by Lecture note Prop .3.11

, it suffices to solve the integral equation

Xlt ) = Xo tf ! Tls,XIs ) ) Is

,

where

Xlt) : Ia .

Ho ) → Ib l Xo ) is continuous.

Equivalently : Xlt ) t ( - Xo - SIKHS. xcss ) - Fcs

, × odds ) = Stotts .

xoldsApplying the perturbation of identity with ( X. Hill ) = ( Citra : total

, Hello)

to Io :X → X, where Iocxlt D=Ht) t ( - Xo - SIKHS.

xcss ) - Fcs,

× odds )

= ( It E) CXHD,

where ZCXHD = - Xo - Sit ( Tls . xcss ) - As

. xo ) )ds

Let Xoth, Yott ) EX be defined as Xolttxo

,Ft E Iad to )

{yous -0

then 8 Hoth ) = You ).

Checking Z is a contraction : V-X.CH,

Xzlt ) E X,

V

TE Ia .

to)

II Kith ) - Exits ) I =/ 4THG. x. on - Hs. x. isD) dsl

S Sj L . I x. G) - xzlsslds E L . llxi-xdts.lt - Tol Ska' ) . Hx .

-

Hbs = 811k -

Hhs

, where 8 La 's 1.

c

'

.HEAD - Elwha E thx ,

- all.

: . By the perturbation of identity .

choose r --

b,

12=11 - La 't b,

then tf y

theBiH,

It !

XHIEBr

Hothsuch that

Bathe

yet,

.

CheckingY Hsia Sfo -56

. x olds E

Brlyoth) i V

t E Ia . to),

ly th - yo Hit = IS It -56 , xD dsl S

Mo. It - tot E Mia

'sCl - La

'

) b = R

I

( since a 's

M¥4⇐

Moa't Lba

'

Cb ⇐

Moa's H - La

'

)b)

Therefore,

I ! XTHEBrcxosuch that faith ) = Stotts ,kids

i. e . I !

XHI: Ia ,

to ) → Ibl Xo ) satisfying the integral equation .

QZ ) ( HW 9,

Q

5)Prove �2�

.

Sol ) Prove by

induction on 1<70 : FE CKCR) ⇒

xtseckttza,

#k=O : By 01

,zxlt ) E C [ to . a

'

, tota ' ] such that XHK x. + It 1 s,

xls ) ) dt

which is therefore C ? by the FundamentalTheorem os Calculus I FTC ) .

Suppose the statement holds for k= K,

then for k= Ktl,VTECK"

( R ),

thenFECYR),

hence by Inductive hypothesis XH ) e C" '

( Ia , Ho) ).

Therefore,

Flt.

#isCK"

,

then XH ) isCK"

by FTC.

I. By Induction

.fkzo

,

FE CKCR) ⇒ xtheckttza ,#