ll1ì\'f iì r)';rN.\t.tstì ¡-uNÌìnIQUIì Iì,I rltì
].iltioIìlIì
I)rt L',\t'ì,ltoxlìt.\1.ro\'l-cnre 23, No P, lf)94, ltp, l$7- 1221
ON A NE\ /TON TYPII. I{äTHOI)
IO]IN r..\zÃtì
(Cìu j-N:r¡toca)
I
.
IN'l.ltoì)u(;T l0Nrype :*:'åiä.il: 3d;,ä:îi*îi"ï,^"1î3il"iîå"å ",Ì"ìffiì
also
eiarnlrle.the
r,ootsof a
singlenonlincar
equation .f(n):
0,Moser [2]
has proposedthe
follorving- it-eratió;"sfin+r: r" - !/nÍ(rò, lJn+t:'!J" - l/n(f'(ròy* -7),
t¿:0,I,...
the first iteration is
sirnilar:to
Ne eclna,lLo 7lf'(n"). 'Ihe
seconcliter
that :I$2... rhe
orcrcror
con\¡er.sc,ìc"r""";lÏJiil:,:"1ä:i"t1-'?î Ì'¡;;T;":
.An
irnproved scrrc're \\'as proposecrby u|rn [4]
anclHafcl f1l.
tr'or' 1,he cquation
(1.1)
F(;n):
o,*.ltete
n' : D c -\'_- I, -L a'tl y
a,re tn,o B¿r,n:r,ch spaces ancl/) c -l a'
open set, the a'thors ha'e
consiclereclilre
iteratioî,s(1
.2)' r;n+t:
e:,,-
A,,F(ur),,(1.3)
An+t: an
- A^(n,(c,n*t)An- 1) : a,,(2I _
x,'(e:,,nr)A,),n:0, 1,..., n,ilìr
LIt.c'jnifial
qrr(rsses"roe
f)
arrr-l _zloe
/,(I,,
_\.), qlrer,c!-(y,.\') Y into
,ìí.rlcrrolcsllre
.Barraelr*iir.o ui 1l;"31*"ìì;ìi iiiLit;
o¡cr,aror,s ¡r,orr. ,Jterations (1'2).antl (1.8)
rreepthe
properticsof
n{oser's iterations anclthc
seqnencc (ø)_conr-e'gj"*q.ä,l"atiäall'i'to a
snr..tionof (1.1)
(scc Zehncler.[5]
anclrráia p
11.,
Oll a Ncrv[o¡r '-l')'pc l\.I0tììod168 I. Lazirr 169
Since,
by
(2.1)I]ncler l(a'tor.or.iCh t¡'pc
zìsisurnlltions)si.rilarl¡',.rvit'h
l,hose 01givc a
cónvelgence theoremfor
the bouncls are sharper t'han t'hosefrom
th:rrr in ì{en'tonts
llllocess.2. (lONYlìIìGlÌ\i;lì 1'lIIìolllìll
[)oncorningccluatiorr(1.1)ancltlreiter.ations(1.2)_arrcì11.3)l'ehar'e llrtnolturt
2.1.II It is þ'róclta 'tttfeì'ltti"iìL''"" 51'o' ì'; - !Y-t 'Y'
l,
r--a,orl < r'l c n',i,l,t'lni'"
r'o,1,,íi'"ì,'r"tí, ír o
ttttd r¡ >"tJthc 'fdlouing
c¡tn, tl'í,t i,c¡ns h,oltl
(2.
la) Ao is
inuet't'ible utttl-4t'
et
(-\'
Y) ;(2.
1b)
llr{oLi(øo))ll< I
;(2,
1c)
llÁ0(/¡'(r)-lr"1v¡¡ll < I
lln-all,
Vn'v eS(ro'
r') ;2.
1rt)
lir -
Àozr"(rco)ll <
q ;I
¿¿,(2.
1e) d::lt'rt* q < t +Íl
ct'tttlt" >
'1t+ hvfFl
rtcr (r',,)
is
wrIL tLe,finett by(l
'2¡ nrrrl (-l '3)'.rettmin's irr'S(rn' r)
ttt tt'
stlttliott' t:* oJ
,1ql"ìtlo'i(l
' l)'
'l'l¿'is solwl'iot¡is
tr'n'iqu'er), iJ r <(1-q)/tu'
tire ioltotoitìg
u þriori enr¡r
esl'ì'¡t'¡tt'tes¡ a crÐr*!, , IA, - It''(r'r¡-tll <
cz(2d)'¿"'n:r'2'"'
tntd the
tr, Ttosteri'ot'i errot' estitt¿u'tes(2.3)
li e,,- T'i'li <
(2cl¡'"-t ll"to- r"-tll'
n':2t 3¡"'
r ¡3l I tr)
ll /.11.wherc
Ct = .'- '-,'
-'ntttl
C"-
7,,1
t -ar12) o
I -sdz
ProoJ'
Fot'rr
n'nrl "1,givcn b¡'('l'2) lnrl
(1'3) rvc shatltinrl sitnilat
lelal iottsto
a*stttt'ptions (2'1)' ,Irrorn (1.3)
ib
lollorvs iltat
Arrt)
: (2I - Ao\''(rr))Aow, lor all
weÍ' I -
tlrÞ''(rut): (I - Aul'(n'))z
I-A'A;1 -AoE'(n')-I'
(2.7) llI -
Aop,(ør)lJ< lll _ Aolt,(ur)ll + ll
Ao(I.,(ao)_ IÌ,(ar))ll
< tul *
e: it,
u,eobtain,
using (2.6), llI _
A Á_å i/< d < 1, vhich irnplies that il
îhis .,rere exists
(,4r,4;1) eL(X) ancl ll(:lr:l;,¡_r il <
f /(1 __ a¡.(2.8)
-4, is invertible, _4¡r eL (X, y)
ancl /1,4f,ll < li-A;, IyÍ _
d,).}-rom e.4)
ancl(2.7)
tve obtain(2.s)
llArî(Ør)ll <
(1+
r¿)ilAor- (ur)ll.
and, by
(1.2)Ao,(nr) : Aoln(rr) _F(ro) _F,(u)(ø1_no)l
+ Aol,(ro) | F,(øo)(rr_nòl
:
A olF ( ur) _ tr, ( uo) _ ]., (. r o)(
ut*
n òI _f ll
_ A oF, ( r o) JA op( n r).But, fi'orn the
assnmplions(2.1) it
follorvsthat lhe
is
dirrercntiabte onSü;;^;);;ir',i. j',,
Lipschirz,so operator' ÁoF(.)
ll
AolÛ(nr) _
F(æo)_ F,(a)(n, _ nòlll
_.1
ll
nr_rollr, thus, frorn the
aboveinequality
and (2.g)(2.10) lla,r'(æ,)it <
(1* ,[* n, + tj:t \u (d,_q,).
üre
alsoobtain from
(2.4),(2.7)
and (2.1c)(2'11)
ll Area'(æ)- x'(v))li <(1 +
d,)[email protected],@)- p,(E)), <Ãi(l *
d)lrn*y
ri,;iîlii?íue
8(ø0, r'),(2'72) lI - Ar?,(#l)il < lll _,qop,(nr)ll,
<
ar.n"o'r"
uJ$"inIT:?,åtqJ' r0
-_12)are simitar to
assumptioi*,,
r,r,i,årï;îîïi;T)113J ,'l"HT;î1i z¡ ana (r ãi";;;"#;u'- á|ittl"ä,",,i1î
** ,"rilil_r, let us
consider ttre"sequences (/c,),(n), (q^) and(d,)
definetlho:'k, ïo: I;
Qo- gt do:
d,(z rB) r::_':,i;Iî,:^_,,,
!" :
dÊ-r, n^:-rjþ kti-'
-
etr-t),n ='r,
2,. . .(2.4) (2.5) ancl (2.6)
77 (\
\yhcncc it, folloti's th¿r,t
(2.16) anrl
[, [,az-irl 4
Jr'or' (rln) \vc obatirì 1,he follorving
recrLlrcrt
t'elal,ion(9.11)
d17:1(.1 f
1 r/,,-r)2(r[u,,--
c1i,) !
tLi_r,ancl,
nsirg
(2.1e) wc caÌr pror.cby intlucliou
tha,t(2,15) I,4 il,u(
(1,-r' <
-7+V2 -l -= rtntl
cl,4 2tti ,, fol all n >
L,r\ow, b¡.
(2.18c)and
(2.19)
llr,,,*, - rnll " "'f'
.n,<Qd,)'_'
.r --
(2¿¡r,,,,,¡!,, 4.lt I _(2¿ty
!t,) i. a
Carrchv sr:J,cL x:,r,
--ìì;ì,"ä,.
ñt'qllcncc' so
it is
corrvelgort.rorn tJrc abovc
iuctlrratiiy"ü
Ona
1/t,
Ncrvt.ou TJ'pc ì{eilrocl
(2¿¡'"
7
-
(2cl)2" 1777
,
for all
tt,tn ÞIt
flrer.efotefollorvs ilrai, for,
all lr, I,
(2.20)
llr,-r* //< I
.1,
<
.1 (!r1)r" , l(:'l i)
(1,, tli,;t
< I ,1., < l
(2tr\.¿,t,' -t; ' I/
foL' a1t n
>7..So,
it
calì bo shou,n lrv ilclrLctiotr tlli¡l,fot all
n, ÞI
(2.
tLla) r,
e rS( rrru, t') :(2,,t3b) thclc
crisl.s,l;r
e,(-\) I) arxl ij,1;'i <
l,l;.1,lii(l --rl,-t)) (2.18c)
liÄ,,11(r',,) I{
.4, ,(2.1,!tl)
), ,4,,(11'(a;)- lt"(ll)) i(
/:,,' ,.,-'!l
1,fol :ll[ nt ll eB(ro,
r') ;(2.
tEt,)
ii/
':1,,11"(:r,,) 'a
rtu.Lr'<.r' r¿
.:
1, (2.18)holl.
\\¡c suplri)se ttrril,(2.t8)
holcllol
i,: I,
...,,u,SitL<rc,
fot
r¡11j, n >
1iiDt t .,,,4-7.''i'l ',:l¡ll:'( t 'i,'(Zt¿)-ittt¡-
,8, ' ' 4t; 1¡ .ri
,r1,(l'
Il))
ed)' 't
12¿ ¡''''
rl l;: (2rl)t'
zr,rcl f/ cr;
,-
noli<//
,i _
noll 1_ ll n,_n*
/l<
.n _L_d2
[Jr¿Í, is a,'n.e S(ro,
r,). ¡ \ '/ '
fu(.1 -_ J¿-i)*
r'', l.ru,thct,
¡is,,e r'riltcn ,
,, wprol¡e
1,Itats,r is
¿t,solution of (1.1).
Silcc (L2)
cattil,.is
sufficie't to sh
A;',(n,*t
-
nu):
-F(u,,),
Jlrrsi;, b1,
(2.1g)
owtJi t the
sequence (//A;til)
is bou¡clecl.(2.27)
!!t _ tl,,F,(#*)
ii
<
llI _
A,,L.,(c),,)ll
+
liA,(,.,(n,)
__ Lt,(rx))< q,
+_ l;,Jl a,,_
ø* ¡¡i/, for
al_ rt,¿
J, ancl,br. (Z.llj) anrl (2.1õ)
").tn+tl-4,t
-- ,:(, -l
el,,_,)(r:,,l_
rt)],_r¡<1.
tl"
-- 1'' '4"- -
,,.,r-< ¿/,, !t:L lt _ r!',4, _ hl;
l
l\-lìelì cr_-,
tl'h
ich
iu Lplics¡/Ð o.), ,, æ
\a'4¿)
Lr,,
,,* i]<
X
2 'r¡, 1
1-rt {
[r,'au, I'r,t' ,¡¡1 tt , ttt ) .l .
?u r-ra
(
4,, 1,[
-r) '4,,, 1ìrl a,ll n> l"
¡¡t:O2'r.
'-So, 1,hc estin¿l,tes (2.2.L) becorne
it
[o]lo u's t'[ttr,tli ,r',.,-,
-rii,li< r,-t 2, \i {.'r, i ,,, rl:1 ,., <r,
Ircnce
íril1.1c:s(oo,r'), -.,r,rt ,- ã.'s*l r,,,'
o''.' ,,,,ootlr'ono ,rrì
rr'o\.o (2..LBlr--e) fot',i,'=
tL -l-1, as
rveltavc plovetl
(2.8) and (2.'1.0 -2.12).(2.23) l!r -.1,,11,(or,)i/<
q,,i_2/i:,,.r,,,_-2d,,,_{lu{Bct)l_,=lo{ruir,,
13v
(2.18') a'ci (2,'5)
rvc'¡ìvc ltA;t]l<i/;t¡r llle__(ty,,a,nr[
rising(2,28)
llA;t -
I1t,(nt,¡i/<//;l;ril,ll ¿
_ tL,,n,,(nr,)J/< tlì ir,.r,,
(2t1¡2,,tvlreltcc lt
,l
.r-
rttt/ ^,;i.\u ,\ , 4
'"o l'
(7-¡.¡''
t*,:::,:i".'ii,";-í',Ííl,l[
rì.9',,"i'i ¡,un,,,,,¡.il,;, i
Í1_f,,',ur,,- .r,c,.,rrrtrrJ,lil.rr -
*'o == rrt. -_ tl ,rl!r(n,r) --_a,N ===. l,I -_.A,,'ltr,(n,) l(an
_
n4) _1. tI,l P(e*) ._I(n,,)
__ [tj,(n,,)(6,t,_
n,,)_],
en+t{ (+ +
e,,)r,,,ancl, b¡'
(2.22)(2-24) dn+r
(
(lt^n^*
q.)en:
rl,en,for all n > 1, thus, using
(2.16),L-1
., * (tt, ,,) ,, * + e,t)':"' ,, :rff,,,
-r, flol. Q.20) there follows
1,hefilst part of
(2.2). tror b5,Q2B) forn :1
thele lollorvs II _A,tì,,(u*) ll ì
SAh .<t
isl,s
.lAr7'("iJl_-l
eL(X) antr
ll ArV,1s*¡-rll i l¡1r _B(t\',
(2.4) and
(2.7)ll.Ér'(¿x¡-ril
<1/(1 -ldr)ll1,ll < I+d
ll/^¡.
r_3d2
"Finally,
because A,,- l'1¡*)-t : - lI - A,fÌ,(a*)lP,(n*)-r, using
(2.28)rve ge!
llA,,-fi'1ø'*)-t ll 4
Or(2d,)2''.ifhe a
posterioriörror
bound (2.S) follorvsflorn
au4
dn_t €n_L¡ €,,_14 2.qn¡ ancl
(2.16)..
- ithe-uniq,*enessof ilre solution u*, if r < (r -rr)lt¡,
follo.rvsfrom tlrc fact
thal, 1,he operatorp(r) : r -A9!(ø¡ i!_coìtrac"ii""-óu" Sir;;
;,i:rnclecd,-
+ ll/o(¡"( p no)- is
c'rifferenriabre, F'(n)) ll<
/LrI
ìl¡,'(r)
q< r,
ll: if
¡it ø e S1ä0,-!rn'irjl
'r¡.ãìl-¡-
,+rn;i*"ry,í
772 1. I.az.irr
by (2.18) rve
get3. NUì,IIìRIC^1, lìXr\tr[I,LE
Jret
us
considelthe
l_lammersteinintegral
ectrual,ionr'et ñ : (ø');rro'
D=:
á(s,)ilo.so thc
s¡'ste'r (3.3) can be *,rittclr (3.4) [email protected]): e - lf )e _ - _ 0,
rvhere?: Àr+, +
lJ.v1r.tr-or ø- e ll.\,
lr ll,e
set ,7:
orr l?À' r
j t,,(ã) :
./_
f7,1åjr"à,,'ìâx :2 ,ra* È",
eniç.r.
,so 'sircll Y¡
-
â.,/<
(2,v(3.4) rve shall use
lro :1
(rhe,.,""*,,Tli,ii"1i îî,i;,?
(3'5)
,r-(,{rr)
_-
ùth)- Arpl¡ut¡,
(3'6)
t7¿,¡¡-
A,_t(21_
I1t,(ù(i))Ai_1)u,
._
112, ...,
lt.5,?j1_:t_
eonrputing ilre rnatr,jx,4,
ivvely. So rve musùps.
sat'clhe
Fréchttof application of (8.5)
anates for
qo, c/.i
csrirnatäs (å.'Bi1:,. "ilrl,T"l
6 7
On a
^\c\\,ton
t of arìilunetic
operations\{/i'iting
s--
,s,jn
(8.2)for the
rnethod G,2,8) applierito
solrreI'e
obl,ainthe
nonlineâ,r s¡rsN6¡nø¿
- ErI(si,,e¡, ñ¡)ru¡:
ó(s,),l:
o¡1r..'l'l'pc lIcilìod atìl0un
(3.2)
(3.3)
,/Y
(3.1) ø(s)
-
s?(u(t))z U,::os,
s e[0,
1],in C
[0,1], anrllet H
(s,f,
ø):
stttz, ö(s):
9120 s.^ Using
t_he repea-ted trapezoidalrule rvlth
fr,¡:i-lM,
s,: iffl, [:
I 0',1',.-.., l, w .
llyti - !,...,T11
and,_wo:
r,oN:
n,*¡2,*iaþpr*
ximate the
exact solution ø(s):
sl2of
(S.1)byihe sölution'oî iheËci;_
tion
(3.2)
ø(s)- r&t(r,
s¡,ã(s¡))w.:
ó(s), s e [0, 1].v[e are not
concernecll'ith the
existenceof the
solutionsof (3.1)
and(3.2) or
wil,hthe approximatiou error but we
stratt,rai¡ãr
estirnate the114 I, I.ttzät' tJ NIì\'UIì ì},.,\N'\I,YSII NUI.IIìIQUIì ]'.f ])I] TIII¡ORII, ])]I I,!\I'PROXIIìI,{TION
^V a plior i
cstinute.s
a p os tclior i
cs tiìnâ Lcs
'lorle 2J, No 9, lgg¿, pp. f Zã.-f7fì
(lo do
4 16 0.1
1.32 .10 1
1 .25 .10-1 7.24.70-1
2 .96 .10-1 2 .81 .10-1 2.78 .1C-1
2 . 1r) '10-s
1 .32 .10-3 1.29.10-3
1.03 .10-ã 3.31 .10-a 3.12 .10-0
6 .62 .10-3
4 .03 .10-'r 2.51 .70-É
ON (u,
p, m.)-- CON\/trX trUNCTIONS
lìItIrItP. Ilr\CllS \/,\SILI' J\,III I-ESAN
(Cìu j-Na¡roca)
'1' I'l'trr, S, Ott tttc Ilcratiue llcll¡octs utillt SinttútantotLs ,\¡t¡tro:uitnaliot't oI tlte Inuersc rsf ltc
O1tcrrLlor. Iz.r'...\cad. Na[ì< ]jsLoDsltoi S.S.lì., 1,{:,4,,IOi_ ,111 (1g67).
õ. Zclrtrtìcr', .lj. .L ¡! llcn¡.¿r'li ttbotLL \4olt¡tt's lleiluttt'. (ìonrl¡. l.)r¡.c ¡\ppl. ttaf tt., 32, 3G1_366
(1 e74).
1, t\:l.Itotllr(;1.t{)ì\
lìccr.ivc(l lã XII 1903
Ilr
13l *'c trcfillcrl a
cr:ìs:i ol,gerrr*itrizc(-[c.,r\.c\
l,lru(.ti..,rr)t \vrlichå,;ll1:ì rl,i,,Tr1":,:]l,o
,ro,,urnl,iä'i,,n,,"r"*i,,s., _i^,ì_l,i;;n,'r.àn,.,."*
arrrt rlrrir,si_an aflit,tÌìai,ìr,c ¿ìns\\.el
to fltis
orras_s alrcl 1,o
sive
i:iomoclìa,],a.t."ir;iiì;i.
Inslilule oI Oulctilus llcpttbIicii,:)'l
PO l.lo¡: 63
l,l0t) Clu.j-Nullc:rt llontrinitt
2, lz, þ, nt) - [:ON\ ItX Iì{;\(:.t,tOÀiS
I)r';rrr''rl'r'ro'r
:.1.
,1,.'J
./:
10,irl
1,j,,:..,,,,1t l.ì ,t);
1,,?/¿c [0, 1]=.,{
T";, ;:'ì:î','oì,)r,lr'i ,'ï1"'1,,,(,t il' ,,í¡l..""'
''' 'rr
fr), ú| 'ir
¿,,r1c,ri'
ir :(2.I) tft*
1- nt(.I -_r)y) <
1¡:r,1.,,(t)
-j r¡(1 _
te)J,1t¡¡Str,tu, ,1,'f'-3'å, otå"11,,,,1ì!t?r,ji:l;t
'11
l,lrc (o., p,
,,)-conr.c,r.ru'cbio's
on ,,,,, ,¡rÏ*Il,;IosÌ'f'to'N2'r.
f,etae[-0,
Lt]. lt'tretr.Jer(l¡;t))(tt) i7 untt orlq
,i,fl2'2)
Í{,,,!1,)(,,1..:Ii_ll) _
"_!:l'!!:)_(t.__ 1¡¡1¡?,
,is ittcreusittq ou, (,nLu, l).1.