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View of On quadratic equations

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¡,IÂTIIEN4ATI./\

-

RE\¡UË D,ANALYSE NUtrIÉ:I].IQUI]

]]1' DI]'ft]ÉOIìIE DII L,r\IrI)RO\II\T¡\TION

I;Z¡INITI,}'SI' ,:. ., NU,}I ÉRTOUE E'1'

I,A

TI.ITIONIE

I)Iì.

L'APPROX I,!I;TTI0]I,

: j

. Tonrc

lll,

N"

l,

lÐ{lg, pp.

Xg:,26

j

ûN QU/\.DRATIC E(ùUÄ.IIONS

IOANNIS Ii. A1ìGYI'.OS (I-as Cluccs) ' ¡i I 'i

¡lbst is ¡rresentcd tor linding soluliòus oI lhc rluadralic ecluatìon a Br-

iti"t iJ"t',lJ'.',ïu:1ïij',î:lìî'jiï-îît"""îiii"'i;lï"iiiiiü"

r.hcories or rarria;

Intloducfion. fll

,¡ho 1ùcories

of

I,adia1,i

\

1,t'ansporl

l3l, l4l, [¡ l, il1]

an ilrrior.lanL

lolc tcgral equations of thò folrn

:l '..

vt¡ tlansfrrt, alrtL ncutlon is

pla¡'ecl

by lonlinear in-

0)

,r(s)

:9(s) fn'(e Í(s,

t) n(t)

dt,

rvhtl'e,g(s) rr,nrl

/þ, l)

ar._o givern

functions on

[0, 11.

Iìqrratiorr

(2) ca,'brì

considcl'cd as

* *pidirt'case of

thrr <rquation,

(2) n: lt

nl1(n)

1'-Ìrerc

is

a

linear

ope'ator

olra

lranach algcbra

-y;

and

ll € x¿is

fixecl"

obvioust¡',.equation (à) rectuces r,o

1ij it l*i;i* ; 1}fil

"r¿

1

irln¡¡s¡:

[.r,',

r)ø (r) dü.

The method of

s

rracrion lt,iãiiãi, rsl "ftÎ îåi

havo bcen uscd

to

oÈL¿r,i

rn

almosr

*ri tno ii""å

rhc

esl,imate (3)

lvhere

liø*

ll

<

t - [T- 4øm :d,

2b

-c

"

-'olltnt',

\

l/(s'

l)lttl

Proviclecl

that

+

tp,¿ < r.

uncler

the

above assumption rrowcve,r

it is

r<no¡¡,n thai;

the

corres- poncling

real

quadratic equation has

two solutio"r. w" ì"o"del if

bhis can

(2)

.) CIN QII'AD.H¡ìTrC oeu.{lrtol\g

ct,

-

a(H¡

r) :

lll{(ø)l¡

't- ll'( ll llK(elll

1

2fl I. K. ARGYÍIOS

:llt: tnrt:

irr a

lkrrra,crh s¡tzrCrcr,T.

Il

burrls oUl;1,hat

this is ttrtc uttdtlt

Cett'ain'

*rssunrnt:ions.

What ic -rleally

neccl

tc¡ clo

(rssttttring t'lrtl"

(4)

holtls), is

i,i"ö;i;,;il' Åü'*iî"t'*tion

1t,,1 convc¡gent [o,:r,

solir[ion r'* ol' (l]

(ot'

ìöllì-üi.fi

g'uarantets

iìtai

if 'lläi 11

>

(t

tñt'rt

licr,,

il > ,lancl

t'lr.relott'

(1,)

llcP*

ll

>' d'

\l¡e

$uggssl, 1,he iLera[iort

{6)

#ei+r '** (L(s';)Xf{(o*))

Tìor solvirrg

(1)

(or'

(2)),

n'hete

(î)

tr(atl

:'

xl

-

tl

"â,ffCt

J{(n+)

- t'

ffi(a,)

rrrovidecl tll.¿r,U,

f((ø) is well

defirrod anct

-t(r) + 0 on 'ti(zrr\ :

rrn e

Xal

'll\*

-

âll

<

?'l

fot

solrte

pe Xt and c'>

0'

ftl

thc

first

parb of bhis paper we E-ive conditions

tor

blre (,oilvefgonce

of

(6J

t; *îiri"d¡o" of (t) (or (2)) s'iitrout

rnaking ¡s'e of {,he stontl¿rrd.

hipothcsis

(4).

hr

t;hc seconrl ¡rat'1, rve

ptol'itlc

conclitions

fot

thtt solubion of th0 atrs-

trac't qrttctlttbic

equation

(9) ü-tt {

t'}(n,rn)

whcrc /j

is ¿r, llounclctl

llilinear

operator on a lJanzuch spelce

x

arrcl 'y e

'{

is

fixul,

ttsing t,ttc

itelation

(10) r'¡r+L:

lì(tt:,,\-L(t;,,

- a), 'tt: 0' [' 2' "'

for, Somc ¿rn e

X.

Morouver tvo ishot.t' l,hal, (10) Ìra's

the propcrty

(5)

(11) 4ll1;ll '

ll:Yil

<

1'

I.inall"v

lurl,c l;lrrr,t lìot, 1Ì(r,u, rt)

-

r.t¡'kt¡

and X - Xt

cqualion (1)) r'odr^ctls

to

(2).

I.

tr|asic llcsull,s.

wc

d.clroto

by tfû, [l

bbe iìiìDach space of atrl l¡oal

corrl,irllrotts llttuctions t'rn 10,

1|

-lvittr

tlte

rrr¡"xitnurtl nol'rn'

(12) lli'll": rn''r''r,lr(s)l'

-t\oto

tltat,

1,ltu

slrilcc X¡: C[0t1] with

r"t¡r'nr

given tly

(12)

is

a

,lJa'ach algol''a. ¡¡ bti. r'esl-ot thii part lløll de'otes

llrcll"'

W*

can Low trrïove

â

oolì.seqllerrcro

of

bho

contlatition

Irrap¡Jing prirt-

ciple theorerlr

tl11.

'llflrconrctr

l.

Ássu,nte :

(i)

lherc cs,isl, r; e

X

u,ttd an, ,htlct,actil:

I :

[r,.., ?,zlt ?,t

Þ

0 sr¿c/¡

tltaî

lhe ogtcral,or'

!l'

g,iaen, by

(13) ,t(c) -

(/i(a,))(.I((,r)),

'tol¿et'a

rt

a,nd,

K

are ¡iaen lt,u (T) unú

(gl

respeotiael,,g

it-weil

d.e,fùned, the oyte-

rator R(z) is

bou,nd'ed, ancl"tt'iun¿bei

,í >t

fu gì,ni,'rrì"tö""'"

{1,4)

Jor

(15) (16) and, (1? )

laltere (18)

I'

0< r' <

liif

lÍ. l1¡r(ø) lJ

(i,il for

e,,,!t c

I

tlrc J'oll,rwirt¡¡ inegwl.itías c,,e sa,litfi,ed,:

tti:fitr l-lle -

yl!)a, -F ¿¿ _--

r ( 0;

(rlll-¡..fr11 -r{o;

_ qiËeìL_

1-a lil

_- r-,iilj (8)

< ?'<

l1z

- yll

P(p): zit¡z¡

1-

y -

e.

'1'lten,

- - (")

rlclttøtiorr'

(z)

lt'us. u .'unique sorutiott, n* e (I(e,

rrl

wrticrd cøtt, I¡e

obluined.

øs

tl¿e l,í.ntit o.f

tlte

,itu'u[ion

#t+t : (L

(n"))

(I{

(,t:,,)¡

Jor

a'n,y c.:o e U(2, t'r).

(lt)

lWoreor¿r, I'a, e -ti(2, t'r¡.

Proo.l'. LeL

r

e

I antl

choose ?t), þ e

û(2, r¡.

Clulí,tn,

/.

!I' ,ís

a

cotttra,ct,io,u, on,

t@,

r).

\\;'o lrar.'c,

(,19)

'I'(w)

-

'1'(a)

- Ifltu)

(to

- y) -

K(u¡ (a

-

o¡)

: Itþ)Ê@ -

oa)It(rn) (ut

- y)

l-

K(a)

(w

-

,1)).

I{encc,

Il'L'(w)

-'l'(a)ll < tll/rll(r'4-

llø

-

ytl)o,z

+ alllw --

all.

The

abovc

ilequality

non, arrti (1b)

justify tlro

clairn"

Clai,m

2. I'

ntalts 't@,

i,ttto

'tþ,,

r).

(3)

- 'l. K. ARGvIìos ', 4

22

Tho claillt

czr,sill'

follol's ft'oltl the

itlcc¡ttalit¡''

il trQo)

- all ç

rr'(llI

-

zI( llr

Ì llP(¡)ll) <

t''

nsing;

(16)

nnct (17)'

on

U(ø,

r)'

'i,o inclirclô equat'ion

(9)

atr<I

itela-

(Notc

tlrtrt

¿l

sivtur by

(S)

for

á

:, li/;ll is

suoh t;ha,t d

e lpr.ryt,ll.

Proof. Using (g) rvc

lrave

})(c.'r, ltr,*) : tt, -- ?!

orr

lle,,

- yll :ll I)(r,,,tr:,,*,,)ii< jj/,jll.lj,¡,,Ij.

jle;*,11, S0t

:

il

,'',, - ltli >

_tl

!,,1i_ llf li

.

rr;rr+ril

t

¡tl¡'¡,1q

l,

lilf

. lr x,,, ¡¡

Assr¡urc

llrail,

,,.r

ll )

Tr Trlr

nll

Å,

_.0, 1,2,

.

..,

)¿. Sirrcr, il

,li

lt>

p

>

> Iiltil to

slto\r,

ll,r;*.ll 2

1t,

it is.tnouglr io'slít,s.

_11,r.,,

t,_

li¡l

li \

,,

ll i;f,;l

r,,ll

à t)

llt'.1!) = 11

ririaltr- it,

s,¡¡f¡1,,"*

to srrou. --'ltättll

l'.,1 ll

1'> t - prltjli

or

ll trllp¿

-'!

1-

llyil<

O rytrich

is

tlr-rc 1ior,

2

e

II¡t,I¡zl.

That

cornplctcs

the

'¡tloof

of thc

¡loposil,ioir.

-.

.

_

Using 1,he lJanach lernln¿l for-

tìre

iln'ctLibili1,,r- oJ

li¡silr

opcratols-

f1l ]

rvcr czr,n o¿lsil.r- shou'

thc follorviirg lcsult.

llnolosJrlrox. Let

s e

x

ba su,clt, l,lt,ui.trt,e. l,inear rtpercr,ktr

ß(z) i,t

.in_

rertil¡le.

l!'hen, l)(tr;)

is

r¡Lso inrerti,hle J'rn' aLL e,

e

U(2,

ll,,,j,

tnlutra

tto ..t

il

/jli'tJ

/;1r¡

1¡l

'

' \4¡c rvill

lrercrl

tho

clcTinit.iou :

I)rll¡ri'lrllrclx. T.¡trt z e

X

l¡c srLch thaL_.thc lirLcat, opelri,ùoi /3(a)

is i¡-

ur:r'tiblc. I'et Il ) 0 lte fixecl and

_Zl

<

JÌ0.

The

oper,ators Þ,

i'givon

by

P

(*) :

lì(tr,, ,t) +- ,!J

-

,q)

ON QUADRA'I'TC ¡JQUAI'IONS 2i

l,ltcn,

ct'¡t tl lt,,r',ll>

?,, n -

t), 1,2,

Ilnl

> p.

tion

(10).

l , ,.i'

trf

.

Iìxtensiort

- llem¿rtlis. ["Lor 'Y is

a

l¡aritich *p^ã",ã".1

thai rin 1o

perator

l1l, [f1']. 'Ihe

ope

ator

,IJ

is

loss of

gcnclalitv sincc

1j can

all'ays

tlefirredl'

by

Eþ;,t¡)

= : (I)(n,Yl I

I3(1¡,

u)), t:,Y

e

X'

\\re

lravc

B(.n,

n) :

B(n¡

r) flor nll r

e

X'

ì)cnoto lly il(ø),

n e

X tho lineal opelatol on 'T

dtdinecl b'y

B(u)

(y)

: ß(q

!J),

t,

Y e

X'

Wc

alc

ttorr'gt-ring 1o sltow t^hal^iternÛi"l

lÍ4

giver,r by (

uf

cottvclgcncc 1,<¡

ä

s.rñrf¡on r'h

o[

(10)

js

srtch

l'hat llrr'll >'l

1,ain assumptions.

Pnorosrrrorq 2'

,Lssume t

(1)

The itera,tiota

Ìr+t: f3(t:)-t(no -

E)

is

toelll rleJined'.for

att'n:0¡'I,2, "' lor

son\'e

noeX

an'rJ con'aorges

to

a solutiott'

u of

(IrJ) '

(2) 'Ilie

fol,lowing

is

h'ue:

1- 4llllll.ll3/ll >

0,

(3)

tet P e lltttltr'.1, toh,are þt,t?z ut'e t'l¿e soh¡tiotts oJ the equotiott

ll

Bllp' -' P -l

llY ll

':

o'

ri,

lluoll

> tt

1 0)

in

case

unclet cer-

an.il

(4)

â'nd

rçn¡

:(-Il

(ø))-1 ('a

-

ll)

are thcn 'rvtll

tle-finetl

ou U(z'4)'

Define

'1"

""åÏ*It.l"ti"l'ioìi'i

;r'í autl -[" o'

lRh

by

l¡1(1¿)

:

c¡11,2

| erll -l

et

ON QUADRA'.I'ÍC EQUI'IXITONS

is

well ilej,ineil and, i,t conuctgcs -t.o n ,[or

uny

r'oel:I(--, J3) Mot'eoaet,, ,iJ'

7.-4ll/rii .llstl >0

antl

lJ a;o ll

Þ f-

2ll

rttl

I'h,en"

"l tr-

I. K. .ARGYRÐS

24

â,nd Pr(Æ)

:

e.flz

! e"ll I

eu'

dr (ïi

Blì'

ll?l (a)-1ll)'z' a,

:= -

2ll B1i

'

ll fJ(e)-r ll'

cs:L-ll B(z)*'ll -illJll ' ll/ì(¿){ll'lìø -vll'

wJre;te

ea

:

ll lJ

ll '

ll l-Ì (ø)-1ll'

,o

-

ll

l)(ø\-r(r -

/J(z))ll

- l'

and

eø':

ll ?l(ø¡*r "P (*)H'

ll¡orking

as

in

Theorem

1 ve

-can easily shorv t'he

folloving

oorrso-

ou.r'rå' :ïä'*" ä't,iJrTil mapping principlo l11l'

eXsu'ch,thatI'trc\;írlen¡,oltaratot.B(ø)i'si,naor"ì,'ihlb.

g are

trwc t

e">.O,

øu{0t a\o-4e^er)0

tnd,

,r -r

wr?w

(3)

tihere enists

E' ) 0

suclt' tlml' -É'r[E)

>

0'

'n"(-R)

<

o

u'nrl

R<ll ø-ull'

Then'

(a) tlw

oTterøtor'

fi

gi'oen' bY

i -: n6¡-'(o -

Y)

is

weII d'ef.i,ned, ct,t¿d,,it ll,¡¡s tt,'ull,it¡ua ,fined, lloi,tt,I

l; e't@,R).

('u) "'I

hc'í'l'er u'iiot¿

fit¿+1

: B('l')

-t ('fr''

--'!J)' \L:

0' 21 '

"

li emm,li:s 2.

tion (9)

has two

and

llnrlj

>

d.

" (Þ) can

easil.v bo

r.olified.

heorem

Z'tnny

be

diffic.

rn<l,r¡

bc ilìäi.î

?fíJ

solution.

r.forvcvor

thc othet ts'o popnlar

r¡rethods

for solvÌng (g),

rrarucly Nen'tonts rncthocl

(20)

rì¡+r

-

at,,

- (28(a¡,) -

-f)-r

(P(n¿,)), il :

O,

lr2, -.

.

and

thc

rnel,horl oF succc.siye strl¡pti.tutiorrs

(2t)

Õn+t

:

-l

|t(n*,

nol,

n,

: (\, lr.Z, ...

share

similar

difficulf,ies.

, tn

parl,icular Newton)s mothotl

also

requiros ø

to

bc .(close"

to

ti¡e solrrtiorr ¿rncl bhe

invertibitily_of

l,lro ope.ra,tor.

I -28(n,,1

al, each stc¡r (or

tlro invcrtibilil,.v of (r - 2B(e:)) if we arc rcferriirg'to flre

ltroctlitìcrl Newbon's mcthotl).

lïÍorcove,r

thc

methocl of succossiye sulrstil,ul,ion nnhes no lr¡l€ of the

invertibilit.v

ot the

linear opelator

,li(a),

but

s ínust,

still

Lrt' closg

tr¡

tho

soltrtion

and

lløll

{

rf,

trnd.el h"vpothosis (1"1)

[1], [2],

f101.'Ihcre,fo¡e

it

cannob

be

uscd t:o

find

;l solrrbiou e; such 1,hab

lÌmli

>

r/,

,sinco

tlte solutiot

ol¡tainerf

then

satisfies llølf

<

d.

llc'll

> -1-.

2ll

rìli

(4,) l.'t.1;he hypotheses

of

Tl.eorern

2

¿r¿,e

l,rue

{}¡¡r1¡ 1¡¡1¡¿-,

solutions

ø, antl

e,, such

that

ll et

ll<

d

(5)

I. K. ARCY]IOS ß ati

20) nor' (21) tloes itr:r'tr'-

shal to filrtl

thtr

tiolt

('lrrt'

tìtiIrti11l¡¡(ìES

tttl tr¡tplÌcaliot¡s /o C/lcl¡ldr'¡ts chlmr's antl rclttlctl 5)' 2' 275-292'

- '"'"" ir¡ tlett[]ott lrrurspotl'

i ttlnr<tl ctlttctl iotts (tI tst tt(/

49.

¿i,ion. r'n¿¿.,. rt ittlegrrtl erltalions' '] ()onrprtL'

I t t r r t s 1t o rl1 hcorrl' r\ rlisorl \\r'slc)' Ì' tl llì' lìcldi ll g'

s/rr' l)()\cl'' l)rtl;1" '\crr '\ orìi' 1llCr0' t'' t|"i'l'í ì -ììì""' t ¡ iì''- t''¡¡ s,

',,

t

" "' s nt rt t ntt

"'

J' rl rtt l r'

:1;ì:.iiïìl';,)llll' i IiÍlli;1'1ì'',11' ['i'11,

"

,' o" ''

462 -46,s.

T. l-1.. -{il ittrtúit¡c solu'lktil of the qnaclralíc uqtalirnt ín l}attacltsPr¡c¿s'

'

,]];,1,1,ì1,:]ì,1]i',.'r,,'"'. j

uii'eJiíliii- iltò

. parcr.'io, r0 (1e61),

I0.

ll

n I

l, l'

,

2".'l'ìl'""ii""

""""ìiì'1'1"'f

i

tuun'.,tr sp(ct' lltrrcl oilc' \Ial

31'1-:132.

11.1ì'lll,'I,.1Ì',CoiltpuLtttitltlals¡¡ltttiono|'lttlltlitlc:at()])el(lloI'ctlttulirlns'Joìllr\\ilc),1,rrll]',

Nt:rv Yol'ìi, 1!168'

lìct.trilorì 20 \¡II 1988 .V¿rri -'U¿:riro Skltc L) rtíD( t'silll J'ns Orttccs, '\lll 3\i00l

tIA't-IIIiN'iu\frcA

-

ììIìYI.Xì D'AÀIALYSE

NUIIÉ|ìIQUO

,:.

liT

DIì 1'IlIìOììIli

Dll

L'i\Pl'}f ìC)XII{A'I'ION

L',ÄN¿lL

YSIÌ

Nult,f EITIQUE IJ't'

LA

:1'HÉ

OIìrE

tr)

ti'A

I't'tì

OX I ÈtATlON 'l'ornc

18,

rT"

l.

1989, pp.

27-Í|{i

ON ']IIÌE SECÄNT }IEiIHOÐ AI(D IiON'ÐtrSOTìE'.I]]I

MA

TTIE]\IIATTCÁI INDUCTION

IOÄr.\NIS

I(.

AlìGYrìOS (Lxs Cr.¡sçs)

-4.1)strlìcl. 'l'lto nrt'iltocl oI nortlisclclc uralbcrn¿ttical iu<ìucLiorr is uscd t.o liud cl't't¡l']tr¡uncls

Iol lhc Sccattl" rtciltotl. \Vc assruuc onlv lltaL thc opcrator'has I[öldcr cr¡rlirnlons (l(t.i\,rìtivcs.

In casc lltc lirécìrc[-tìcrivaiive of the o¡rcratol salisfics a Lipschitz colrlition oul r.csulLs ,rt:rluce to thc oncs oblainccl b¡' F. Potll (Nì.uÌì. llal"ìr. 1982).

(1)

Iulrotlrrelion.

Consir.lcr'

1hc

crluatiol't

,[(r) -

rt

i\1tet'o.l

is

zl tìclnìircâr'

opo'âtot

rnal;1tiug

a

srll)sclj

fl

Oli

a

lìarr¡ic,tr sltÍtcrl

-/r,',

into ànotlìcl

l-ìan¿lch sl)zìco 1t2.

Ilelo l-e

2ìr'o concerìrcd u,ith

finding

soltrtìons of (1

)using'ihc

lstrca,ni;

itclati

ons

(2)

c'¡,¡,

-- r; -

8,f(¿,-

r,

x'n) 1l

(!r;

:

(3); ,'

r,i,+r

:

a;,

-

ò/(a|-r,

eo)-t./(er,,) ,

i:

t\'lìcro íl'-t

?-rìlcl i?"0 ?ìt'c

i,l'o

¡toitrls

ín tht' rlonlùilr

of

,/, arrtl

ò./

is a

cr¡ìisis-

tcnt altltloxinull,ion o[ ,/'.

:

'llhjs l'orli is

llasecl

upon

l,lìc elcg¿ìnt

l'olli of F. l'ott'a

irrchrrlcd

in

l-4

I colctrlrring

thcr ctrol'zùn?rl)-sis oJi

thc

lJecant rnothocl

. Onc oiì

L'trt.l'a's

t¡asic ztssumptious

is

1,he fact,

thal,

cssontiâllv

thc liliear'

olx,t'¿ttot'

/''

ìs

.lripschitz

contjr-Luclus.

I]ou'et'er jn ihe

plescnce

of

sornc jrrttlr'csl,iug e-rampkrs (see pa,rt

(IrI)),

l\rlìcre

f

is

onl¡.

Hölclcr cotrtinuous \\'(ì ('\tr-ì-n([

rnost

of tlte lt¡sulfs

contâincd

in [a]

for, 1,her

jtela,ticn (3). \\i'lt'lvc

thç,

extension

of thc

results

for' (2) to tho

urotir-atetl l'eãdct'.

\ì¡e

furnish

tr\¡o oxamlrlcs

in

¡rarl,

(lII) to shori'th¿t oli'r'cijulLs

can :be a,pplicrrl \\'ììolezìs

the

cc¡rir.zr,lcll, r'csults

in |4l c¿ìrruot. 1

ì

Sinct¡ orlr

losults ale

dr'âr\-ìr ¿r,hnoiit

in

tlrtr sanro

liBcs

r¡'i1-h 'LJrc orrcs

ìtr

l -1], r'cr

l'jll

ncccl

to

l'ostâto sornc hr¡r'c.

I.

l'rolimin¿rrics. Cjonsirlcl a class C oI-

pails

(,/, t:o)

llict'c./

is as âbor'û tìncl 1r0

: (!i)-t,+tt...,

oo)

is

zr, s¡.st,eLn

of l; points frorrr -Li.

\'ì-c q'r-url

io

.¿ttach l,o t¡¿r,ch

pajr

(,f,

a)e û a

soquon(1o

{"v,,),n:0rI,2,... oI:

poilìtä

r.tl [r)¡ ('otÌl'el':g,i nS^

to

a

loclt

¡r,*

of (l).

'1lo a,chieyr-. lhisi

l-tr

ussocintq;u'ith

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