another cauchy peano theorem - cuhk mathematics
TRANSCRIPT
Another approach to Cauchy PeanoTheorem using
Ascoli's Theorem
Piecewise Linear Approximation
let R Eto a Tota XExo b Kotb slopeM W
M supplfit X as before xotb fg sMayassumeMZlaswealyneedanupperbd xo 1,4
I 1
Define xob
W Ct HER lx xok.ly t tol to a to tota
By symmetry projCW onto t axis is toastota'surn
for some a'Eco AI
Note that f CCCR f CCCW
f is uniformly continuous on WsinceWisclosede
bounded
HE 0,7 8 0 such that
F Cti XD Ctz Xz EW with
Iti tako and la Xiao
we have Ifctz XD fCtiXDIs E
On the half interval Eto tota's
choose to s t s ta s s the total
with Iti ti i k F f i i k se.ge aoxo
FftxaDefine a function ketti on Ito tota's
gfigh
l ketto Xo to
2 keIff gis linear with slope f Itai Xi i
l ti
where Xi can be determinedsuccesinely by
is x detenuiedbyKel to is linearpasewig
through tojo and with slope f Ito XD
Ciii Note that IfctgxoskMiXi Xo I E M It tot
c Hi xD EW CR andhence fit xD welldefined
is then Xz determined by Betty is linear
pasewig through taxi and with slopefitsxD
ius Swiilarly Hitoko l IfitishkM we have
1 2 Xol EM Itz tot
tz xD EW CR and f Ita Xa welldefined
Kotb
g 1443 graph of keltXo1
Xob i
to a to a to t t total tot aun d
E tzM
And so on the function kelt is defined on Ito total
Note that
4 he is piecewise linear
G1 theft ke I E M It St t t seEtoTota'T
By slopes HitixisKM on eachsubinterval
Ike is equicontinuous as subset of CEto tota's
ke's is also uniformly banded to totab
In fact it is convex and theends points Cti xD with
Xi Recti belongs to W we have ft kits C W
by piecewise linearity As WCR thefts x Kb
and hence Reits IEHd t b V tCEtoTota's andVero
Hence Ascoli'sTheorem wiplies that the is precompact
In particular the sequence I Kenshihas a
convergent subsequence that in Cftotota'se
with tenets felt C Eto Tota'S as l to
To show Bits satisfies the differentequation we
first show that he is an approximatedsolution
for any E 0
Consider t C to Tota'S and t f to E Qb k
Then 75 1,3 h suckthat For E the o e corresponding
tj stet8 o
Using It tjykltj tj.lk Ey we have
1kelt keltj.is EMIt tj iIsOHence
I f ftp.kectj iD fftkeltD sE
Since he is piecewise linear
hefts fctj ykectj.is by ourconstruction
Henie
feldt fct kectD ISE ft C Ito totabiltgtystab
As ketto Xo kett is an approximated
solution.to
Irpdd f tix on Eto tota's
Xlt D Xo
in the sense that ddkf fit he remainder
CtoF 0
with 11remainderHSE
Integrating the ODE we have
kelt ketto t fat Kelssds SgtkdDds
Xo t kids ds
kelt Xo fetoffs kecsddskf.tok'ecss fls.kecsDlds sea
In particular if we denote 9e kate lie E he o
thenI gfts xo fttfcsgelsddsfeaef.tt
ss
Hence
1k xo
ftfcskcsdds.IEklts xo SfffCshcsDds gftitXotSfffCsgcesDdsl
I geits xofeffcs.gecsbdslsllk
gjho ffflfcsgecsD ffs.HNdstKne
Since 11g hills o and f is uniform continuity
t
15945 fcskiss Idso as j to
Therefore by letting j to we have
kits XotfetoffskGDd5t te Etotota's
date fit kits htt CEto tota
facto Xo
Similarly argument I tem t EEto asto
satisfying date f Ct Ect htt CEto a to
htt Xo
Note that by construction
dd ftD fCto Xo dateCto
Hence
yhtt t E Eto Tota's
kits t E Eto a to
is C Eto a tota and solve the EVPxx
Remarks i This proof doesn't need the Picard Lindelof
Them
c'is The spirit of this proof is more in line
with solving the IUP numerically
Lili's The 1st proof solve approximated problems
the 2nd proof solve the original problem
approximately
4.2 Baire CategoryTheorem
Def let I d be a metric space A set E in E
is dense if t X C I and E 0
Bean E 0
Notes i d Easy to see that E is denseE E
Ci's E is dense in EE d
et If E discretemetric then for Os Est
X E E Beca 2 3 Therefore L is
dense in I uirplies E E i.e I
is the only dense set in CI discrete
GL In HR standardmetric Q IRI Q I
are dense
GI Weierstrass approximation theorem implies
the set of all polynomials P fams
a dense set in C CEO I do
Det het CI d be a metric space A subset
ECE is called nowhere dense if its closuredoes not contain any metric ball
ie E has empty interier
G i 21 1 z l g l znowhere dense
in IR However
Oh has empty interior but OTER has
nonempty interim so 0h isnot nowheredense