View of On the midpoint iterative method for solving nonlinear operator equations in Banach space and its applications in integral equations

138 II. tJugg..isclì, P. ìlazilu trtt<l II. ^{\Vcl¡cr' 12} Anfahlzeiten

: ^?t :

1000 s

f', :

100 s

?'s:10

^s

Rclaxationszeii,

: l Stuntle;

jeder \rclsuch

\yurcle drcitnal l'iederholl, _;

Streunng

: ^2-5%.

Die

experirnentellcn

l)aten

sincl

in Abb. 1

clar'.qcstellt.

Aufgruncl

cles oben er.çr'áhnten AJgorithmns und mit, Flilfe eines I¡OIìTRAÌ{-Pt'ogramlnes ^-wtlr'- den clic folgenclen \4¡clto bestitnmt ^:

cq: +7.7

^Àr

:

0.00

u,r: IÐ.7

^Àz

:

1.05

r¿s

: ,11.i1

Àj

:

14.10

Dcr. \rcrgleicÌr

^zs'ischen

den expclirnentcllen uncl

bercchncl,olì Wel'ten

ist in ;lbl¡. 1

zrL sehcn.

ilr.'ulì

D',¡\N,\Ll'Srì xurrrirtlgun _{rì.r r)ìì}

rrrticlnr'

r)r,r r,,r\r,r,RoxtìIr\'r'ñ

'I'orno 23, No 9, lgg4, ¡¡r. lJ$_lJ2

ON THI' N,IIDPOINT ITERATIVE _{*{ETHOÐ F'OR}

SOLVING NONLINEAR OPERA'IOR E.SUATIONS IN BANACFI _SPACE

AND I'I'S APPLICATIONS TN INTEGRAL _ESUATIONS

DONG _CII.IJN

(1.a5'c l- l_cv ilÌ e)

and IO..\NNIS ì(.,,\IìGY}ìOS (I.al _ton)

1. INlIIOI)UCI'ION L

I1'tt Ìì.\

^{-r tJ} lì v Èl l'. z E Ic,

lt

N I S

21. Il. \\Icbcr., ðdn nicltllincrucs Sloff¡lcscLz lür ^ttic cbcrtc pttoLouísl;oelaslisclrc SpnruttLrtgstutuLgsa,

Iìhool. À,cla ¿2 (1983) 774-722

22. 1I. \Vcì;cr., .Einnchsiges niclttlittccu'cs ¡thotouisl:ocltLsliscltcs V¿rlruIIett uott PYC-weÍclt ^ttttd

U P-I.cgttuttl, Iìhcol. r\cta 20 (1981) 85-93.

2g. lL llrri¡giicìr, ir. llazilu ^unrl lI. \\'eltcr', Pc¿rrnteler idenlificctliott for uiscoelaslic tnaletials, ììhooÌogica Acla 27 (1988) :163-368

r' this

stucr¡' 1ve ¿ìr'e

conce.ned'.iilr ilrc

_¡rrobrer'

of

appro_rilììâi,ing _a ìocall.v

unique

zero

n4 of the .q.*tion

(1'1) _{P(u) :11,}

in

a Banach space -Y¡r where _-Z)

is

r,

nonlineal

operator

clefi'erl on

soûre coD\:cx sul¡set

of

,,l"o

wiilr

r.alues

in ^y¡.

incle.x

point of the vicrv

_{[19, 20],}

ch better than that of

r\ervtonis s.

fn

flre sccontl section of

this

study nvergence flreorem and give ur,

"*l

ryhich

is a function

oT

ilre initial h is

ca,lled

ilre rnidpoint

_method

of

orv

flrat

flre miclpoint mothocl is also rcler u,hich \yas definod

ì:y first

au_

error

_bouncl

is

thc satrtc

as that of

appri c a'on s o

r

rh e rnidpo

int,"ïï,ff

_:

ii'

"li,;:Tl

^J':

:îîXlå:,ï _å"#13*

fo the

sorution

of nonlinea' i'tegrar equations

a¡r¡roarÍng

i'

neutron transport.

Iiir)g(ìS-arìg(ÌrÌ anr 1 r\pt'il ¹⁹⁹¹

Inslilu.l fiir Alccltanísclre VeLfnlu cttsleclttilt untl ÀIecluutili cler Ut'tiucrsildl l{arlsrulrc (7'II)

-1)- 7500 liru'lsrulrc, ^Gcrtttuttg

74J l). Chcn cucl J. J(. Ar9)'r'os ^,:)

Ìlclatilc trlcthotl ₇₄₁

2. ft,tStC tlIlRÀTION ITDLATIO¡;--S

Iìilst

rve define the niethod as lollor.vs :

: _ltn ,l+

!t,

! ^r:,)]

T "t+

^Qln

t

^,:,)

:

lrn

- _'l+

^(!tu

^* t:;l-'[r'l]ur"

- ^P'(r") lP'(u"¡1'tP(rn) :- Fo' ar arhitra'v

cJroice {roe

x¡¡, rel, us clefinc the .ridpoi't

pro-

ceclurc by _{| .n}

- ^{P'(r") (y"} _-

^un)

(2.r)

!/n

: rn

- P'(r,r)-lP(rn),

c)nrt

: ." - ^o,l+

^@,

I ,rf-'

'l-helefore

it follol's

that,

P(y,) -

?'(yn)(,t:,,

¡t -

^!/,)

:

P(a:")

\4ie.orv

1,rv to Tind

a'

cxpression for

?(rn*,)

*,hich can later bo ¡16¡1i- natecl

by a

real" function.

LDlrivlÄ 2.2. Assu,nt,e _tJtcLt

P

:

D

,Yp

-

^y-n

^{is ¡oice}

trrécttef,-d,iJfe_

renti.cr,ble,

u

_in, incttLctüd

in

ct

riìrt

nnna"n,'i,iLáe

.À'3, with aa 'ftlten

II

P(rn*t) :

\p,,(.!J,,

-l t(t,,,r -

^{.1,))(1 __}

^t)

tjt,(r,,.r,

_

_!/,)z

_

J

P(yI

-r

^I

t

I L

P' klò - ^P'

---Qt"i l

2

1

- (Y,*

2

l1 L2

*rr)f

",) ] r1 t;

(y"

-l ^r") j

^-t(t:r*t.

_{- !lu)*}

+P',

^(rn.rt

_-

_!/n)

:

- l'(y") I ^{P (ru-¡} _-

!io) l-'

- *\",,[

0

I _¡r''(Y') - ^P'

^{(y^}

l ^rr,)]]

^{,.,,,

_t -

^{,!tn) ==}

î ^r..

^1-

^{yò r} *,r, - ^{.,)f at|- (u.} _-

^cn)

_-

J

llr'{r,, I ^t(y* - ^rì,)) - p'(r^)tùt(y,

-

^an)

-

- l.l*,r,,+ ^{",)] -} P'(c,,)] (y, -

^rn)

-

-fr',r,,, - ,l+ ^{(v, j-} *òll"'i+

^{(E^}

_)- ,,,,)f

Ii

0

P"(r, lt(u,-

^n,))(7

-

^¿)

- +p,,i.,+*rr,-r,l]]artu,_n*)2.

Proof. We obtain in turn

P(nnr) : P(nn,t) -

^I?(y,)

-

p'(y,)(trn+t

-

^!ln)

-l I

^P(y")

I p'(yn)(r^+t-

!tn)

:

aI

:

t

\p,,(,,," *

t,(rn.rt"

_ _y*))(l _ t)rlt(rnr, _

_Un)L j_

J

+

I

Ir'' li,t,,, * ^{r^)]} - "',t,]

^@,

- ^*u) :

:5

i l

ï

-5

I?"(n,,

*

^t(8,,

- n.))(I-

^{f) ttú}

^(ll, -

^n,,)2

P"I *, + +

^(yn

- ^{n,)] .,} ₊ ^{(y, --}

^!),;)z

* ^P(llo) I

]?'(y,)(nn+t

-

!t¡,)

,

^Oìrserve

^{that from}

^{(2.1), r,r,e have}

?n+t

^tt,

- P'(fin)-t]''(a^) l-

P'(rn1,):1p(u:,)

- p,f

! {r.+*-)f-'o,*,, : ^{P"[-]- L2}

^l@,"

^{+ $,) * *rr,- r,)] dt!-@,, -}

^r*7

:

_Un

- l* ^{r" + '',,)] -} p'1*,,)-'1P(n,,) : ,l; ^{(u*i s,)l} '5o"[',,++(y,- n,)la,lø, -

^û,;)2.

That cornpletcs

the proof of the

lemna.

742 D. Cllen ancl I. I(. ^Algylos 4 ^Ð Itclatìve llcLhocl ¹⁴³

3. SOITT USDIìUI, INUQUAI,III]]S

By tahing

noÌnìs

in the

abol.e a,ppt'oximation,

\'e

^ltave

llr"*, - ^y,ll

^<

- +

_ll

^p' ]- _(r. t ,^f

^'ll

i

^ll

'"(",, ⁺ | ^{{u,,- ,,,,}

^ll

^{t!r, - n,t,}

^<

u - ^o'[å

^{(r, -]- s,)}

_{)- +}

^þn

- ^t*)'(

^ra11

-

^,e2.

(C2) : Moreo\.et'r \\'e har.c

.,I

_

llil"",", 1-t(y,-n,))(r _-¿) -+o"l*,+ltu, ""t]],ull -

l,nlruA

3.7. Assu,ttte tltnt ^,ítt, ctdditi,on, to ttte lrygtoth,eses o.f Letnmu Z.Z Ih,e Jollowing ^{estimates cn'e}

true;

(À1) _lly" - ^{#,ll <}

^sn

-

^{t,,t ll}

^P(r,)

^ll

( g(t), (,\2)

_lr

p,(r,,)-,ls-y,(r")-',

li",l* @--tyòl'll u - o,l+(¿,*,,) j

^,

(l\3) (44)

cnt,iL

(Aõ) Ihett,

(cl)

(c2)

uir+fr-] ⁱ

¹

_<r

llP"(n) ll < /1, llP"(y) - P"(u) ll

S-¿\rlly

- ^rll g(t):Lr,- 2(rp ^l_ t+!

ll!

1

rl -

lt\

lP"(nu I ^t(y" -

ø"))(1

- ^t) -

^(L

- t)P"(t:,,))c\t

ll"

il

r,*t _--

y,

ll <

^l,+1

-

^s,,

I ilr,"," ^{i- t(y} -

,,,,,,))(L

- ^r) -

0

+ lI o",*,,) - + P"l*i, ₊ ! ^{tu,- ',,]] *ll}

⁼

7,,,,

2

f ".* îr,. ^- n,,)]o']ll _u # ^{ty,- n,t1,}

liP(a,nr)ll

S t(¿,*r)

SrY t(\ -

t) rltäy,,

-

^,r,il

⁺ +

t,itri1y,,

-,

lcu

l: _#',1y,, - ^n,lj

(c3) ctnd

(03)

: Irrom

Lernrna 2.2, s,c hâr,c

llP(ø,*,)il < #ll¡r,n, - ^u,li' I ll' ^r, -.r'"lia *

(C4) _lltl,*t - ^{r,rr il}

^{S s,+r}

- ^fu+'

mhere

to:0t

t,, tnt[l S]i,are tle,Jined, cts Jolloms:

sn:tt- _-91''J-.

tl'(ün) ^'

+ + ll",[+

^{@, -t-}

^{r,,r] 'l]iry,} -

^r,,rB

(

(3.1) _oqt")

< f, ^i,,*, -

^sn)2

* _{:i-(s, -} ^tn)' +

tn+t

:

1,,

tl+(r. ^{+ *J] -r ,} _-{ ^ì1' _I

^(sr,

^-

^fn)B

-K (s,fl*) þ2

Proof

.

^(C1) :

Irrom

(2.1),

s'c

gct ffir+t

- ^A*

t# ^{*T1," -tu)ù}

: - r v[+@,*,^f-'io"(,," t-] {u,,--,")) dt(yu-

^æ,,)¿

0

< jf

{,,n,

-

^su)s

*

+ -+(r' r

^s")

^:

9(l"nt)

144 D. ^(lhen ancl I. Ii. Arg¡'r.os

(C4) :

Finally, frorn

(2.1)

we

^gel,

llU"*,

-

^{n,+r ll}

:

_ll

-

P'(ntn*r)-1P(c:,*r) ll S

llP'(r,,*r)-t]]llp(ø,*r)

_{ll S}

( - 9'(fn*r)-tl(f,*r) :

s¿+1

-

^tn+7.

That

completes

thc

^pr.oof

of the

I_,enlma.

\\ie

har-e norv built,

up the

necessâr,y estimatcs

to

pLor-e

¡rc

rn¿iirr

result x'hich is ilre

subject

ãt ilre next

sectiotr.

4. îIIIì l(r\NloIìOYICII COÀ-\'liIìGìì\Clì TIÌlìORfitI

^\I) ^{IÌRItOtt IìOUìiÐS}

lrrr¡onnu

4.1

. Let PtDo -,,i; *

^}-ar

Xu, To

1t.e Rrutaclt, spctcæt, rea,l or contpleu etnil Do i.s cuu o1te,n, cotlj:efr dornrtin,. ¡Lssu,me lhat

p

_{ltus Z,n,cl}

ortler cot¿tittttous Ilrécltct rIe¡'it:uti.t:cs cttt .I)o etttl, llLat tlte .fo['(,ott,iu.g coucldriqtts

are

søli.sfiecl:

(4.2)

i,,P"(u)

lj<

^,tfii

^p,,(t:) -

^p,,(tD

^ilS

^{,v il}

^{r _ y}

^jl,

J'or a,ll

u,

y e 1)o

(4.3) llp'(rn\-ril < ^p,

l)yo

- ^roll (

rr,

,llIoreo't,er¡

uc

ltu'¡*e tlrc Jollouin{l

eïtot

esli.tnettes tutd oQttitttu,l

en'or

^cons- Itntts :

(4.10)

_li,,;,,

-

^l;,,*

^]i<

^I'r

- t,, ^Jor

^{aLl n,}

(4.11)

_ll,t/,,

-

^rj'F

^ll<

^?'r

- ^{s,, .for}

^{ul'L n.}

tuutl

( L

t:r)

t',

-

^I,,

- (l _{- -ot)l}

6,,"-,.

' I _

03'¿

Proo,[.

tlsirte

m¿r'thenraticnl

jnrluction, it

strffices

to

^sliorv

ihnt

the

follol'ing

itctns

arc tlue for all

r¿.

(T")

r,, e

ßl(,rt¡,

).t

-

^{.r) ,}

(II,,)

_{li !/} ,,

-

^{1¡r,, j 5! .s,,}

-

^{1,, ;}

(Tir,,)

_!1, e I)(t10, t't

-

^'q);

(I1'")

_li

P'(r,,)-lLl < -

^{t'ft

^*)-1;

(\'") ll Lr

tl

u'l ^{I (,.,} ₊ " Jlr "13' ^{,/,,)l 'll}

<

-

^,,'l

^{L(¿,, +}

^s,,)'ì

^{'l '}

atr cl

(\'T',)

_il ,r,,*r

-

^j/,,

^{il S}

^l,+1

-

^s,,.

ProoJ.

rt

is r¿asv

to

^cJreck

in thc

case of

n:0 by

inil,ial conclitions.

Nol'assulne

integ'cl vahtcs. th¿r,t 'Jlhen, n-e har.c

(r") - ^(vr,,)

^are

^truc

^{Jlor a fixccl r¿ ancl ¿l,II smallel positive}

(r"nr)

: _lla,,*, -

^yoll

^S

^{li ni,*r}

- !J,,)i+lly,, -

^yoil

^S

^(1,,n,

-s,,)f

*

^(s,,

_-so) : f,+r -

^so

: _t,+t -

^'a

^11\ -

^"a.

(fI"nr) :

Ir-toLl't (C4),

l'e

^ìrar.e

lltl"*, - ^{r,,-rll S}

^úr+1

-

^s,¿-rl.

(fI-[,,+1)

:

[[oreover', rrre ^1'ÌA]re

lly,nr --

^yo ll

( _llt/,*r -

^nuorll

l

ll,nu.r,r

-

^!/,,il

I lly" -

^{yo ll S}

S ^(sr*, -

^{lnnr) -l- (ú,,-rL}

-,s,,) +

^su

-

^so

^:sn+r - ^so:

.:8¿r-1- 4(l'L-'1t.

(l\',,nr) :

.Fulthermore,

l'c

^hâr.e

6

(4.e)

tohere

r,

ís the smallest _root (2.L)

is

crntaer,ç1ent. Also un

is

^a, salution of tlte equation

^ I -l|-Ztt,

1

¡ 1fu= m'

ol

tgototio* (4.7).

Then

th,e rnirl.,poi.,nt gtrocedwre.

,3,

^e ,Sn7-70,7,

-

^{r¡ ,}

^for

^{all ø e}

Xl. nti Umit

u'r.

P(n) :0.

ltcl'ativc Àlctlìo(l

P"(fru*r.

* ^í(n,*t -

^{tro)) dt(n,,*,}

-

^no)

14t

uj+#l

h=:I{p,.rìU+,

ñ(.rlo,

t',

-

^{-r¡) c: I)o}

1 -]/T -

^ztt

1

I

(4.1)

( 4.5) altcl (4.ô)

S

1l'

uhere rS(r, r)

-- {r'. Xo; lla,-- rll(

r},

(1.?) _sØ - ^]- _2pp ^xr" _- ^-L

^¿

^{+ -a}

(4.8) attcl

rì: _\

P'(fro+r)

-

^?'(øo)

:

746 l). Chen alrrl I. ii Arg¡'r.os

I

Itel'ative tr,Ieilrorl ₁₄₇

so,

l[¡e obtain

p

llP'(n**r)-

^P'(ao)/l

_<

,Uil nn.t7

_

_#0ll

_<

_1f(4,+r

_{_}

_úo)

: , ^_ p,ll

¹

a'(æ"+r +

^!/1,+I

1

: I(t,¡1 { ^I(r. : lç t _-ltt

_h

-m

^.A

_K !_v!_?t

r(þ"n

\

t -(1--

ⁿP ^2y,

I

^{P'(uo1 1}

^I

^t)iJ

^lt[

² _ll

_n,*r-

fio

I

Au+t

- ^toll

and

by

l3anach Theorern [21,

pp.

1641 lr,(

o,+t)-r

exisl,s

alcl

¹

li P'(ø,,*r)-t _ilS ^li

l'(ao)-t

_il

TM

ll

^nn*,

-

^{0o ll}

- {llr,,*, -

^roll

s

1

- ^li?'(

^{øo)-t jl ll}

^P'(*,*r) _-

^{p,(no) ll 9 2}

s{ ^p 1 1

7 -

^{pKll nu+t}

-

^{roll 1}

^K

2 1

p

- ^K,l

^fr,,+t

-

^{0oll p}

(tn*r-tr)-+(s,*r-úo)

\< 1 1

1

a P

: -

!J'(tn+t)-r. ^{1 1}

2

_L

-

^I((tu+I

-

^{to) 1 g' (4,nr}

*

^s,*r)

t) I{tu+1

1_r

g ₂₍

^¿r+r

^t

^Sr+r)

(Y,'*r)

: _{Flom the}

_estirnal;c

'i+ ^{'"'+t *} a"*)f- ^P'(ro) : (Vf"*r) : Ifsing (2,1), rve

obtain

lluu*, - ^nu+tll:ll - _",LT ^r*,,-, r yn*)'l-'o,*,_r,

l]

u u ll ",li ^{,*,,., + u,,*,]'}

^{ll tt}

^or*,-,rrr

^<

s - s'lî ^{r".r+ ,,*r)]}

^'s

^(t,+) :

tn+z

-

^tn+t.

ll¡e

norv

prove

(4J.2).

Notice that P' Ir, * ]{r,,*rl

^ltu+t

-z*o)l** ^(nn*,í

ltn+t

-

^2no)¡

lYe get

ll"t+ (ru*ttv,,*,)f -

p'(,¡:o)i]

= +

^ltuu*,

^{-,0ir r} _!ür,,*,--

^#olrs

s f ^{ir,*, -}

^1o)

⁺ '' ^(r,,*,.-

^le)

:

{{,,,*,J_

^s,.,-,)

^s

u l!,,', *r',) - /(ri s r----r,7 ^_ l[1'-

^2t-t

,:

_ t ^{_Vl, _}

^ztt. _{o.( -1-}

.

P -

P *iJP'(n:o)-t,¡'

Thc'cfo'c, b¡,ilrc Baraclr

ilrco'crrL

,,1+(r,,*, * a,,rr)l ¹

exjsLs

artl

ll"'[+ ^(rr,*r t r,,.,,]-'ll u

g(t,,¡

:f;f,., - ^t,)(r, -

^t,,),

J'(tu)

: _- I ^,U.r-

^,,,)

^{+ (r, -- t,)),}

g'(s, ₎

K

l(rt-s,)f(r'r-s")l

o

and

\ P'(un)-t

I

¹

2

ú,

f

^s,)

I _-') t,,(ror '

^f _ii

n,f!

^(nn.,.,

_i

_!Jn+t)

_-

",(rr,]

^l]

K

1

(¿,

*

^s")

i

^t'z

-f

^tr,,

^{+ r,l]}

|',t -- ₂ 2

148 7t

Iterative _Method D. (lhcn atìd I. I{. Ar_ijy f os

10 149

Also,

_{'n.c get}

D'nrx*ron

_2.

-(B-order) Let g(r) be

a

scarar testing

function of

or.der

2 given by

g(r)

: rt ttJ"- _{tr_.} a p

¿

r*î *

^a

for some nonnegative

_constants

I(,

_9,

\

satisfying the

cc _A

_sequence

_of

iterations defined

in a

Banach spa

Ë"Ti"T"í-:#ä,à',j3"#: ilil-1d'ffå"i,å"1 _';n3

E(g(t,*r),

_{to, t,+r)}

:

g(tu+t)

-

^c(tn. tu*r)(tnr,

_

_tn)Þ

:

^g,

E(g(t,*r), tu.

sn)

:

g(tu*t)

-

^{c(fu¡ snXs"}

^--

^tu)e

:

e

for

some c

) 0,

rvhere

E(P(n**r),

_{tn,t Unt frn+t)}

: p(r,*ì _

_R(no,

_!/r¡

_fr*ç11.

;tJ"ir"-, fnîä:-i-t"med ^to

be functions

or

these variables

in rhe

corres- tr'inally _we ^.rvill _need

the definition

rvhich lvas also

givon in [lL,

^12].

DE.'V*row 3. The asymptotic error constant c(l-r¡ is

definod

by ,i-,9ß:'ì't-tr-

K

2 (\ -

t,)(r'"

- ^t,)

2

[(ti - ^t,) i

^(¡.,

-

^t,,)]

- ^t")'

r'1-tn!t,r -tu

1'ìren br.

(s. 2),

_{we har.o}

1'7

-

^t,,+t

: I\ - ^t, I (rr-t,)(t.r-

t,,\

(ri-

^r,,)B

It,r_tnlrr_tr1lr,

ancl sirnilar.l¡r, rve get

-t, I

^?.1

_-,sa { rz- lrf ^r.r_su]

?'2

- tr*, -

(r, -

^{to)ts a(ú*)}

: lim

^9(tn*rl

Lt',

-

^tu

i

^t.z

-- t,llt,t -

^t,

So rve

olttain

f- _?'1

"--

^(tn*r-to)D

-

^sn

l,¡'z -

^tu

I

^t.,

_ _{s,l for the}

single step, _rvhereas

for the

murtistep c'se

it is

definetr

by tt-tu f ,',-t.u--, lt ..:[?',:ål,:6.,,.

L r;- ^t\-, _J : "':L;;

-r.

^J

tlris

eqnal,ion

for

r,,

-lo,by

^{using ilre}

^{fact ilrat} 1.r_t,:

'q10.

k is

cas.y

to'sec

_ilrab

2't-

- ^t, -

⁽¹

- ^o')? _e.'-,.

1 ^__orJ"

c(ú*)

: li1¡

^{s( t1} t."

-

^t,,

llten

_{x,e solve}

-?'1 -1,,+(1 ^_0r)

n eoo (so

- ^úole

x""riårt:"rfirflrhe;s-order

^and

asvmptotic e''or

bounds

for

rhe mrdpoinr

9(fn*r)

:f _{$^nr- s,)s*} ^åa*r" -

^r,)q

1

p

_K ,

^{(úo * an)}

5' SOITE CIIÀIII\L.'TIìISTICS UNI)IìIì 'r.IID DtrI¡INIlION _{OI,' S-ONDììI.}

To find thc sufficient conilitions of

orcler

of

conr¡erg.erce, ohen

J*,1¡,iiî,?i;nttv

srrsgcstcd

a nà*l-ïcri'ir,ion-ãi

"'ìr"""åt

convcrscnce,

\Ye rvill

neerl

the

clefinil,ions

:

111, ¹²¹

Dur¡rnrrrox 1. A

se*quence

of

iterates

1i,¡, n,) ^{0 in} a

l3anach space .Y¡ is saict

to

converge

rvitï óìáã;p-;i _to

_a

_point

_{æ* e}

_{X3 ir}

Il û,,+t

-

^ø*

^il (

cl!

r, _

n",,!lp

for

snralter solne

or c>

equal

0.

rvrre'c.c

ro l. wc is usuaily i,,iti;üd"Ë a funcbiou _ìç**¡

_t

y,. of ør'.iilr the no'm of

c

K

2

-g

_T ^{1.) üu}

^{

^su)

^-tK

^{(s" t")2}

t' ^T

u,ru. -

^r.)e

o

+ :

ú.

*

^a")

Ks

8 (s*

-

^tn) _K2

n'[+

_(f', ²

+ (¡. -

^úo)?

:

*

^,sn)

B-z

^(ú"

⁺

^so)

-

^{Cu (ún, sn) (s"}

-

^¿")9

3-c. ¹¹⁴⁰

150 l-). ()ìlcn nnd l. Ii. ."\r'gllos

rirþ

L-2lt

l'-l

so

b¡r

clefinition

I, p :3

¿rucl

Onr(l'F)

: lirn

_C¡¿(f,, su)

:

_

^{Ire1, u,s noì\¡ cl}

trie

olre.atä;,

'Ë';n"'lTTiJ' :

_{'2õ, v(s)}

: _! for all s e _f0,

1l

; ^anrl

^clefite

n( ^o-rb

--- ^;(t)rìt -

^e(s)

^{+ l.}

_ ^ìiote

^{thaü eve}

(6.r).

^{vvcl'J¡ Zel'o of}

^the

^equàtion

P(r) :0

satisfies

're

^eqnation

_

^{Set ,rô(r) __}

l.

trc.ir ¿tìi

eî'Ëi

, ri.',11,f1,11,.,,1t'Ë

fftliìiiilî .?5lå",l]ä-t f;:,]",ìiiT:,

^{rr,éer cr-}

ilr :

,1¿

:

2l),1

,r^^.

_|

(__-

_u, ^,

0<s<r lJ.ç _-l-.

¿ l:'l)'lln2 :

^.3465T8õ9,

P: ill"(1)-,li :

1.5B0SÐ42].,

\ ^> ilP'(7)-'"(1) _li >

^pÀ ln

2 :

.2651s7r07, À

:

.61993304õ,

I :.25.160J18 çl _,),

l3

and

Itcratirc lJcUlorì

151

f-

I(2

_- xr*

D

:

C,,(F),

4 1

r

fr(

I

_s

\\'lrere

Cr(t*) is

definecl

in

_[11].

6. oN 'r'rìrì sor.u.J.roN oF

,\

(ì..ù-s$.lì[rÌ,?,y,1ì.]{ï¿ol,ìår,"r,n" LQU^r.roNs .{nrsrìG

rn this

section \ye rÌse Theorern

4.1 to

suggest

ne\y

approaches to

fhc sol,tio' of

cluadratic

integral

equations

of ùrä fo''

(6.1)

¿(s):

u(s)

f

^{Àr(s) q(s, ú) æ(r) r1¿}

,_l

tt]g

spac_e

xo : cl0, 1l of aII functions

continuous

o' ilre inte'r.al

i0, ^1], l'itìr nolm

ii ø ll

:

oma,x lr(s) l.

Frcre

we

assurne

tirat

7,

is a real

nurnber _callecl

the

^(¡albedo, fol, scatteling

alcl the

ke,rnel q(s,

/) is a

continuous

function,of t*o .,Àrin¡t.s s,

ú

l'ith 0 (

sr

I < 1

ancl satisfying

(i) 0(q(s,¿)<l,o(s,Í<1;

(ii)

q(s,¿)

|_q!,s) :1,0{s,¿<1.

'I'he

function

y(s)

is a

gir.en ancl

finall¡.

a;(s)

is

t]re

unhnol,n f

ììcluations of this

type

are clos drasekhar

l7l,

^(Nor.el

^prize of

^ph¡

of radial,ive transfer, neutron trairsli

[1], l2l,

^t7.ì.

lhere

exists

an

extensive

literature or

equatious

lihe

(6.1) unrlel various assumptions

o!

the rrerrrel q(s,

ú) and

À

ii

a real or

coùplex i*i*

be,r.

one

can

lefer to ilre

recent ruo-r,r.'ii

l1l, [2]

^{anct itre}

ret""ef"us-tlió"ã.

l{ele

rye dcmonstrate

that

Thcorern

¿.f ì.iä

t-tré

ite pror.ides existence results for (1.1).

lVloreo¡,er

ilr

(2.1) convcr,ges faster

to the

solution

than all

the

Ilurthelmol'e

a better infotrnation _on the

locatiolof

thcsolu.l,ions is

gi'e'.

Note that the

computationar cos1,

is not higher

trraìi iire corresponding one

of _ltrevious

rnelhocls.

rìor sir'ìrlicity (rvithout loss of generality) rve l,ill

assunre

flrat

q(s,ú)

: s+¿ i . ^{for all o} (

s, ú

<

^1.

Note

that q

_so clefinecl satisfies

(i)

ancl

(ii)

above.

ri :

.311111702 0

:

.173133865.

11l, i2l).

TìIJI.'ERENCES

I ica t iotts lt¡ C ha n tlr ast L, lto r

, 275-292. ^{'s uttd tclaletl tqrr<t-}

,al _n cqualiotts arisíng in neulron Iransporl.

!";

l,tn' ;,

",,

",

^{ol., *,} ",,,

i *l

rn e t h o d s,Á,ìi\, G : 1

uillt nondif ferenliablc antl ^ptek crror cslitnales, Ie.þr nonlinear operalor eono^t,Íott-s on<! Iheir rtis_

,,,'i"',,

*,ll)'Ìll':q :

^{nF ( l-e s 0)' 2 6 5}

-

^{2i; ¡}

"'

"' "

r Durre(n _spaces, Applíctl _ÃIalh. út ^()ontp¡1!, . Nerv yorl<, _1g60.

csli.ntolas oI Iiittg,s ilcrttlio¡t ptot,cdurcs

r trIaut., åc: ₁₃

i

^{4) (1ossr),}

zZS:ùî.

í52 D. Chcn anrl I. Ii. ^{r\r. gyros f4} 9. Clren, Dong l(attforouich-Ostrouski coiuetgcnce llrcore¡tts atrd oplinlal et¡ot l¡ounds _f or Jarcall's

iteralioe metltods, Intern. J. Computel Mârh., 31 : (3 + 4\ (7990),221-235.

10. Chcn, Dong, On lhe conuergence of ct class of gcneralizcd Steffettsen's iteraliue prccedures dtrd eÍror analgsis, Inter'. J. Conputer Math., 31 (3 -l- 4) (1990), 195-203.

11. Ctren, Dong, On n nen clef ittiliott of order of conuergetrce in general íteratiue metltocls I : One-

poittt ileraliotrs, suìrmitted,

12, Chen, Dong, On ^(t neù def initi.ott of order of conuergence itt getrcrttl iluatiue ntethods II ^: Mulíi- poittt iteralions, submittecl,

13, Dcnnis, J. E., On the conuergettcc of Netulon-likc melhods, In Nr¿nr¿ric¿rl Methocls for ^Non-

linear Algebraic Drlualiotts, edited by P. lìabinorvitz, Goldon and Blcach, Nerv Yolk,

1 970.

14. I)ennis, J.8., Touartl a wtified colùergence lheorg for Neulott-likc ^methods, In No¡rldnear Funclional rlnalgsls and Applicaliolts, edited by L. B. Rall Acadcmic Press, Nerv York, 1971.

15. Glagg, W. ll. and Tapia, R. A.., Oplimci Error ßouncls for IIrc NewLott-I{ctntorouiclt Tluorent, SIAM J. Numcr. Anal., 11 (1974), 10-13.

16. I(antorovicb, L, V. and Akilo, ^G. P., Futtctiotml Analgsis in Nor'¡n¿d Spaccs, Pcrgernon Pt'ess, Nel' YorÌ<' 1964.

17. Ortega, J. l\,I. and Rheinboldt, \\¡. C., ffernfiuc Soluliott of Nottlincur Equaliotts itt Scuetal Yariables, A,cademic Press, Nerv Yorl<, 1970.

18. Rlreinbolt, \\¡. C., A Unified Conuergence 'l'hcory for ^{¿ Clr¿ss of lteratiue} Processes, SIAÀ.I

J. Numer. Anal., 5 (1968), 42-63.

19. Ostrorvsl<i, Ä. X,I., Solttliottof Equatiotts itt Euclídean and Banach Spaces, Academic P.r'ess,

Nerv Yolk, 3r'd ed., 1973.

20. Traub, J. F. Ile¡aliue lt[elhods for ^{tlrc SoluLiott} of Equalions, Plenticc llall, Englonood

Cliffs, 1964.

21, Taylor. A. E., /llrodltclion to ltunctional4.tralysts, I4tileg, Neu Yorkr 7957,

AEVUN D'ANAI,YSE NUìII1IIIQUII NT DE TIIúONID DR L'APPNOXIMATION Tornc 2J, No 2, 1C$2r pp. 153-fG5

SUR LA FORMULE D'HERMITE

ION ICIIII{ ct. GÌìIGORE AL}ìEANU (Bucharcst)

Received B X 1992 Departmenl of Mathemalicql Scí.ences

Uniuersilg of ^Arkauses

tr'agelteuille, ,4.r,lcansos 72 701, USA Deparlmenl of Mathematics

Cameton Uniuersitg Laulott, Oklahonta 73505, USA

Dans cet ouvl'age, en

utilisânt

cles résultats

de [4], [5] et

_{16l, nous}

présentons une

formule

de

quadlature

trigonometrique de type

Hèrmite et une

nou\¡elle mêthocle

pour

déterminer dans

un point la valeur

du

polynôme trigonométriquc

cf interpolation.

On sait

^que

ftr,,[x]':

{årtucos(ix) *år,sin(ix)lø¡, ^b¿elR pou' .r, "}

(,.*n. ñ,,-,[r],:

{å(,, *r(+") *

f

^b¿sin

(r+ ^{x)l o,,}

^{t¡, €}

^{R pour} i.*Ì)

est

un

espace de

llaar

sur

tout intervalle _la, bl

avec b

- ^{a 1}

^2n.

_

L,'opérateur clifferentiel associé

à

I'espace

Rr,[X]

(resp.

à

I'espace IRr,,_r¡x1¡ est

(1) I'rnr: D(D'+L2I)(D2 +2,1)...(D, {tr,2l)

(rcsp.

(2) Lzu-tt:

(D2

+

(712)zI)(Dz

+

^(312),1)

^.. .(D, | ((2n,-I)12)rI)).

= ^Lenoyau

^{9ru (resp.}

^0r,-r)

^{associé à} I'espac"

ftru [,T]

(resp. à I'espace

Rrn*r[x]) \'érifie

l'égalité.

(3)

^0¡(r,

y)- 2'(.r"

(+))i, ^{¡ ena-t3n}

^¡+1.

l

TrrÉonrìu

L

^Soi,ent

E

u,n esltace de

Ilanctcll,

fiot

frt¡

.

.t

tm¡ m

+ I

poitxts tlistitrcts

cle

linteraalle la, bl,

^t(no,

nr,..., n,,) un

^{élëtnent cle}

(hf*)'l'*t /,,\

et (f6o),..,,1Ú'o),...,,fr,.)

,...,|ft' ,...,Íf1, ,...,1Í1,,,t,

^{ulL 1)ecterLr tl,e} -D"+'(rr

* I ^: :

n.t).

^{Alors iL}

^y

^{u un, seul} polynôtne trigotrotnëLriqtrc Tv

à, coefficients dans

Il

^atsec

O(fy) < n

tel ^(fxLe

(4) (Tr)u)(n):lr?, j eO,

m

ct i

^e 0,

v.¡Qt¡::n¡ -

^I).