¡,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.ITIONIEI)Iì.
L'APPROX I,!I;TTI0]I,: j
. Tonrclll,
N"l,
lÐ{lg, pp.Xg:,26
jûN 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 i¡ a Br-
iti"t iJ"t',lJ'.',ïu:1ïij',î:lìî'jiï-îît"""îiii"'i;lï"iiiiiü"
r.hcories or rarria;Intloducfion. fll
,¡ho 1ùcoriesof
I,adia1,i\
1,t'ansporl
l3l, l4l, [¡ l, il1]
an ilrrior.lanLlolc tcgral equations of thò folrn
:l '..
vt¡ tlansfrrt, alrtL ncutlon is
pla¡'eclby 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 givernfunctions on
[0, 11.Iìqrratiorr
(2) ca,'brì
considcl'cd as* *pidirt'case of
thrr <rquation,(2) n: lt -¡
nl1(n)1'-Ìrerc -Ê
is
alinear
ope'atorolra
lranach algcbra-y;
andll € 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
srracrion lt,iãiiãi, rsl "ftÎ îåi
havo bcen uscd
to
oÈL¿r,irn
almosr*ri tno ii""å
rhcesl,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,rit is
r<no¡¡,n thai;the
corres- ponclingreal
quadratic equation hastwo solutio"r. w" ì"o"del if
bhis can.) 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,hatthis is ttrtc uttdtlt
Cett'ain'*rssunrnt:ions.
What ic -rleally
neccltc¡ clo
(rssttttring t'lrtl"(4)
holtls), isi,i"ö;i;,;il' Åü'*iî"t'*tion
1t,,1 convc¡gent [o,:r,solir[ion r'* ol' (l]
(ot'ìöllì-üi.fi
g'uarantetsiìtai
if 'lläi 11>
(ttñt'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 eXal
'll\*
-
âll<
?'lfot
solrtepe Xt and c'>
0'ftl
thcfirst
parb of bhis paper we E-ive conditionstor
blre (,oilvefgonceof
(6Jt; *îiri"d¡o" of (t) (or (2)) s'iitrout
rnaking ¡s'e of {,he stontl¿rrd.hipothcsis
(4).hr
t;hc seconrl ¡rat'1, rveptol'itlc
conclitionsfot
thtt solubion of th0 atrs-trac't qrttctlttbic
equation(9) ü-tt {
t'}(n,rn)whcrc /j
is ¿r, llounclctlllilinear
operator on a lJanzuch spelcex
arrcl 'y e'{
isfixul,
ttsing t,ttcitelation
(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)iÎ
(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^ctlsto
(2).I.
tr|asic llcsull,s.wc
d.clrotoby tfû, [l
bbe iìiìDach space of atrl l¡oalcorrl,irllrotts llttuctions t'rn 10,
1|
-lvittrtlte
rrr¡"xitnurtl nol'rn'(12) lli'll": rn''r''r,lr(s)l'
-t\oto
tltat,
1,ltuslrilcc X¡: C[0t1] with
r"t¡r'nrgiven tly
(12)is
a,lJa'ach algol''a. ¡¡ bti. r'esl-ot thii part lløll de'otes
llrcll"'W*
can Low trrïoveâ
oolì.seqllerrcroof
bhocontlatition
Irrap¡Jing prirt-ciple theorerlr
tl11.'llflrconrctr
l.
Ássu,nte :(i)
lherc cs,isl, r; eX
u,ttd an, ,htlct,actil:I :
[r,.., ?,zlt ?,tÞ
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,,git-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 {Í cI
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¡eobluined.
øs
tl¿e l,í.ntit o.ftlte
,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
eI antl
choose ?t), þ eû(2, r¡.
Clulí,tn,
/.
!I' ,ísa
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
abovcilequality
non, arrti (1b)justify tlro
clairn"Clai,m
2. I'
ntalts 't@,r¡
i,ttto'tþ,,
r).- 'l. K. ARGvIìos ', 4
22
Tho claillt
czr,sill'follol's ft'oltl the
itlcc¡ttalit¡''il trQo)
- all ç
rr'(llI-
zI( llrÌ llP(¡)ll) <
t''nsing;
(16)
nnct (17)'on
U(ø,r)'
'i,o inclirclô equat'ion
(9)
atr<Iitela-
(Notc
tlrtrt
¿lsivtur by
(S)for
á:, li/;ll is
suoh t;ha,t de 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,
,,.rll )
Tr Trlrnll
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¡lli \
,,ll i;f,;l
r,,ll
à t)oï
llt'.1!) = 11
ririaltr- it,
s,¡¡f¡1,,"*to srrou. --'ltättll
l'.,1 ll
1'> t - prltjli
or
ll trllp¿
-'!
1-llyil<
O rytrichis
tlr-rc 1ior,2
eII¡t,I¡zl.
That
cornplctcsthe
'¡tloofof thc
¡loposil,ioir.-.
._
Using 1,he lJanach lernln¿l for-tìre
iln'ctLibili1,,r- oJli¡silr
opcratols-f1l ]
rvcr czr,n o¿lsil.r- shou'thc follorviirg lcsult.
llnolosJrlrox. Let
s ex
ba su,clt, l,lt,ui.trt,e. l,inear rtpercr,ktrß(z) i,t
.in_rertil¡le.
l!'hen, l)(tr;)is
r¡Lso inrerti,hle J'rn' aLL e,e
U(2,ll,,,j,
tnlutratto ..t
il
/jli'tJ
/;1r¡1¡l
'' \4¡c rvill
lrercrltho
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
byP
(*) :
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
al¡aritich *p^ã",ã".1
thai rin 1o
peratorl1l, [f1']. 'Ihe
opeator
,IJis
loss ofgcnclalitv sincc
1j canall'ays
tlefirredl'by
Eþ;,t¡)
= : (I)(n,Yl I
I3(1¡,u)), t:,Y
eX'
\\re
lravcB(.n,
n) :
B(n¡r) flor nll r
eX'
ì)cnoto lly il(ø),
n eX tho lineal opelatol on 'T
dtdinecl b'yB(u)
(y): ß(q
!J),t,
Y eX'
Wc
alc
ttorr'gt-ring 1o sltow t^hal^iternÛi"llÍ4
giver,r by (uf
cottvclgcncc 1,<¡ä
s.rñrf¡on r'ho[
(10)js
srtchl'hat llrr'll >'l
1,ain assumptions.
Pnorosrrrorq 2'
,Lssume t(1)
The itera,tiotaÌr+t: f3(t:)-t(no -
E)is
toelll rleJined'.foratt'n:0¡'I,2, "' lor
son\'enoeX
an'rJ con'aorgesto
a solutiott'u of
(IrJ) '(2) 'Ilie
fol,lowingis
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 equotiottll
Bllp' -' P -l
llY ll':
o'ri,
lluoll
> tt
1 0)
in
caseunclet cer-
an.il
â'nd
rçn¡
:(-Il
(ø))-1 ('a-
ll)are thcn 'rvtll
tle-finetlou U(z'4)'
Define
'1"
""åÏ*It.l"ti"l'ioìi'i;r'í autl -[" o'
lRhby
l¡1(1¿)
:
c¡11,2| erll -l
etON QUADRA'.I'ÍC EQUI'IXITONS
is
well ilej,ineil and, i,t conuctgcs -t.o n ,[oruny
r'oel:I(--, J3) Mot'eoaet,, ,iJ'7.-4ll/rii .llstl >0
antl
lJ a;o ll
Þ f-
2ll
rttlI'h,en"
"l tr-
I. K. .ARGYRÐS f¡
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 lJll '
ll l-Ì (ø)-1ll',o
-
lll)(ø\-r(r -
/J(z))ll- l'
and
eø':
ll ?l(ø¡*r "P (*)H'll¡orking
asin
Theorem1 ve
-can easily shorv t'hefolloving
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 te">.O,
øu{0t a\o-4e^er)0
tnd,
,r -rwr?w
(3)
tihere enistsE' ) 0
suclt' tlml' -É'r[E)>
0''n"(-R)
<
ou'nrl
R<ll ø-ull'
Then'
(a) tlw
oTterøtor'fi
gi'oen' bYi -: n6¡-'(o -
Y)is
weII d'ef.i,ned, ct,t¿d,,it ll,¡¡s tt,'ull,it¡ua ,fined, lloi,tt,Il; e't@,R).
('u) "'I
hc'í'l'er u'iiot¿
fit¿+1
: B('l')
-t ('fr''--'!J)' \L:
0' 1¡ 21 '"
li emm,li:s 2.
tion (9)
has twoand
llnrlj
>
d." (Þ) can
easil.v bor.olified.
heoremZ'tnny
be
diffic.rn<l,r¡
bc ilìäi.î ?fíJ
solution.
r.forvcvor
thc othet ts'o popnlar
r¡rethodsfor 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:
1¡-l
|t(n*,nol,
n,: (\, lr.Z, ...
share
similar
difficulf,ies., tn
parl,icular Newton)s mothotlalso
requiros øto
bc .(close"to
ti¡e solrrtiorr ¿rncl bheinvertibitily_of
l,lro ope.ra,tor.I -28(n,,1
al, each stc¡r (ortlro 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 theinvertibilit.v
ot thelinear opelator
,li(a),but
s ínust,still
Lrt' closgtr¡
thosoltrtion
andlløll
{
rf,trnd.el h"vpothosis (1"1)
[1], [2],
f101.'Ihcre,fo¡eit
cannobbe
uscd t:ofind
;l solrrbiou e; such 1,hablÌmli
>
r/,,sinco
tlte solutiot
ol¡tainerfthen
satisfies llølf<
d.llc'll
> -1-.
2llrìli
(4,) l.'t.1;he hypotheses
of
Tl.eorern2
¿r¿,el,rue
{}¡¡r1¡ 1¡¡1¡¿-,solutions
ø, antl
e,, suchthat
ll et
ll<
dI. K. ARCY]IOS ß ati
20) nor' (21) tloes itr:r'tr'-
shal to filrtl
thtrtiolt
('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\ cì 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,,'"'. juii'eJiíliii- iltò
. parcr.'io, r0 (1e61),I0.
ll
n Il, l'
lì,
2".'l'ìl'""ii""""""ìiì'1'1"'f
i
tuun'.,tr sp(ct' lltrrcl oilc' \Ial31'1-: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'AÀIALYSENUIIÉ|ìIQUO
,:.liT
DIì 1'IlIìOììIliDll
L'i\Pl'}f ìC)XII{A'I'IONL',ÄN¿lL
YSIÌ
Nult,f EITIQUE IJ't'LA
:1'HÉOIìrE
tr)Iì
ti'AI't'tì
OX I ÈtATlON 'l'ornc18,
rT"l.
1989, pp.27-Í|{i
ON ']IIÌE SECÄNT }IEiIHOÐ AI(D IiON'ÐtrSOTìE'.I]]I
MATTIE]\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) -
rti\1tet'o.l
is
zl tìclnìircâr'opo'âtot
rnal;1tiuga
srll)scljfl
Olia
lìarr¡ic,tr sltÍtcrl-/r,',
into ànotlìcl
l-ìan¿lch sl)zìco 1t2.Ilelo l-e
2ìr'o concerìrcd u,ithfinding
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'ci,l'o
¡toitrlsín tht' rlonlùilr
of,/, arrtl
ò./is a
cr¡ìisis-tcnt altltloxinull,ion o[ ,/'.
:'llhjs l'orli is
llaseclupon
l,lìc elcg¿ìntl'olli of F. l'ott'a
irrchrrlcdin
l-4
I colctrlrring
thcr ctrol'zùn?rl)-sis oJithc
lJecant rnothocl. Onc oiì
L'trt.l'a'st¡asic ztssumptious
is
1,he fact,thal,
cssontiâllvthc liliear'
olx,t'¿ttot'/''
ìs.lripschitz
contjr-Luclus.I]ou'et'er jn ihe
plescnceof
sornc jrrttlr'csl,iug e-rampkrs (see pa,rt(IrI)),
l\rlìcref
isonl¡.
Hölclcr cotrtinuous \\'(ì ('\tr-ì-n([rnost
of tlte lt¡sulfs
contâincdin [a]
for, 1,herjtela,ticn (3). \\i'lt'lvc
thç,extension
of thc
resultsfor' (2) to tho
urotir-atetl l'eãdct'.\ì¡e
furnish
tr\¡o oxamlrlcsin
¡rarl,(lII) to shori'th¿t oli'r'cijulLs
can :be a,pplicrrl \\'ììolezìsthe
cc¡rir.zr,lcll, r'csultsin |4l c¿ìrruot. 1
ìSinct¡ orlr
losults ale
dr'âr\-ìr ¿r,hnoiitin
tlrtr sanroliBcs
r¡'i1-h 'LJrc orrcsìtr
l -1], r'crl'jll
nccclto
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,eLnof l; points frorrr -Li.
\'ì-c q'r-urlio
.¿ttach l,o t¡¿r,ch
pajr
(,f,a)e û a
soquon(1o{"v,,),n:0rI,2,... oI:
poilìtär.tl [r)¡ ('otÌl'el':g,i nS^