View of The Halley-Werner method in Banach spaces

RTì\¡[]E D'¿TT\ALYSE I{UNIÉRIQIIE

ET DD

THÉORIIì

DE

L,TTPPII0XIiIIATI0N

rutl.urì

Ð'rtNÀr,yslr Nu

trinrquri rì-f

ÐIr

a,lIiltxiln

D¡ì r,,rtppnf¡xrMA,fION Tome !13, ¡ìlo 1, lg9¿. pp. I _ tr/*

CO]TTITìi I)II RIìI),ACTION

THE HALLEY-\4/ERNER _I\{E1'HOD IN BANACH

_SPACES

Dircclcur

NICOLT\E Tli0llOlÌnSCU, lìlcrnl)re cìe l,Acarlénrie Roumaine

Io'\NNIs I(. ;\tì(ìY[ìoS' r{oFIÀ}IÀ'IAì)

r\.

'I'ArJA'r'Àr]r\I anrr DONG (rI{uN

(Larv

Lon)

(Fa¡,ctter.iltc)

Rórlactcur en chef ION P,,IVÃLCTU

1. tN:t'{t{)I)ti(;TI0¡ú

Rótlaclctrr en cltef arìJoinl 1ìLlìNr\ l,Ol'0\¡ICIU

The

stuclr-

of

cotlr/elgence

of

Chcb.l.shs1,

and I-Ia[cy netlìods ilt

Ranach spaces

iras

a

rrr*tu.!-üs r""g' ;r that of

Nenúon's _{rnethod. 11; has}

l-¡eeu conside'ecr sincc

ilrc

eàr'Iv

fiftìes.

The

fir,stìã*"ri ìu*

clue

to M.

A.

Mertvecor.a

iDohl.

^{.{kacl. _r\airh. SS}

a iol, of leseat'clì stuclies surfaco(l J.)\,

SSSIì. .Kazan Scr'. .triz.-Ì\{at.,

g

₍₁₉

A.

I\{ertr-ecor¡â

[Izr'. ^Akad.

,r\auì<.

he'ratics,

_21(1988),

Z5-g6l; \r.

Cantlela,

and A. l\{arqtii'a lOornputi'g,

4'1(1990)'

169-184; Co'rputirrg,

45(1990),

35ii-36?1. r' this

1,r,1,o,,,

.*

c'onsicler a farnjlY

of Ilalle¡'t¡'pcr

rnethods ri.hich cont¿r,ins Clheb],slicr. and tr{alleY metl'tocls

tts

spccizrl caj'i(ìs.

Undcl

stanclar,cl

I(antoloyicjr

assurn-

ptìons,

_rve c¡sta,blisir 1,he e,ristcncc-uniclueness

theolem

anrl gir.c

the

bcst upDer alrcl

lol'er,

ltorurcls for,

all

^ø.

in

_[0,2].

2' îIID tl{lìlllSlIlì\'-If'\LLrì\''¡'\'Pla lIlì.Ï1{{)l)S rlllt it'I¡Ittît I't'Iift,t,¡,lONFi

r¡ìrst

n'c rerlefine_tho er¡ui'arcnt itcr,ation

fr.m fo'

Halle¡, tr.pe mct_

hods. r3y

tbc

or'ìg'inar

rlarlc¡'

t¡'pr-r rncthotrs int'oduc"atuj.^iv"i:r1n'i,'Lsi,^iu.

can

u'lite,

for,

all

n,

> t,

(2.7)

t1(,L,,)

: t!'Yr)t':(-{l

/'( _'\

,,)¿/"(-L,,) illernbrcs

GHDORGI{N CO]IIAN

II¡\lìIUS IOSIITBSCU, tncnrltle cort'cspouclant clc l'z\carlélnie Rour¡ r ainc

mer¡ble cort.espondant cìe l,Académie Ronmaine

SOLO}IION II.É\RCUS, ÀLEX¡\NIIRU NEìIIIìTH 'I'ITIJS PNTRI[,,\

I}I}IITRIII D. STANCU

l\¡r\N SINGIìIÌ¡ nremltlc colrespondant de l,Ac¿rdéntie Rottrnaine

Secrétaire rle réilactlo¡¡

DOREI, I. DUCA

'{rr-t : X, -

^{7'( 'Y ,) I l!" ( X ,,)} EDrrurìA AC^DEÀ,rIEI noärÂNn

Calca 13 Scptcmblie rrr, 13, 1eì. 641.19.g0

?6117 llucur.c;ti, fìonrânia

lt - _î ^HeY,))

2 I. I{. .Argylos, l,L A. Tabatabai, Ð. Cherr ⁿ 3 l-hc I'Iallcy-Wcl.Ícr. _Mcllìorl o

We

rervrite the

above

itelatjons in opelatol foun in a

^Banach space as

follows :

(2.2) Yo: Xn- P'(X,,¡-LP(X")

Proof. trlsjnq (2.2) l,ve have

in tuln,

P(X,+l) :

l?(_y"+l)

_ p(y',) _ p,(y,) _(x,+r _ ^y,) + ^P(y,,)

-l- _-P(Y,,)

(Xn*, _

_Yn)

H(X", Y,) : - P'(X,ìt-rP"(X,) (Y" - ^X")

xr+r : ^y, -

^L-

^p'(x,¡.f, - ^t, ü(x,,y,)l]-tp,'( x,)(y, -

^xn\2.

Florn

norvv on

the

aborre

rvill

be called

r.Iallev-welnel

methocls

in

B¿nach space. Note

that x''(rnl

is a

linear

operator' &nd lrr"(rir) is a

bilinear

opera-

tor

evaluatecl

at fi :

fin. n{oreovel

the

cleli.rrati-rres

ale

assurnecl

in

the tr'réchet-sense

[1],

[2].

Now

rve

try to find the

expression

of ^P(-Y,.,)

^relatecr

n'ith

^the

{(tr"*t) ^so

that P(X**r)

can

be estlnatecl by

g(1,,,¡1).'

Ln¡lrvr¡,

2.3.

Let

t: D e _Dr - ^Ilz.

^Assume:

@)

^I_lte nonlinear olterator

P

^,i,s ttoice Frécltet cl,iJJercntictble onDo;

(b) Ihe

itarates generatetl by (2.2) are well ileJineil Jor att

n _> 0.

"ú'h,en

the

follouing

appronimcúion

is

true

for

^{all r¿ >- O-:}

1

:\P"(Y' +

^I

^t(X,*, -

^{Yu)) (1}

-

f)cUX,,¡1,

-

^yn)z

0

+

^-P(Y,)

+ P'(Yn)(X,*, _ _Y,).

Obselve

that flom

(2.2), we have

x

¡t+.t

-

^{y n}

- ^{- l^} 2 ^o'tx')-tf t. 2 ^[ - : II(x,,)ì',,).l-t ''-"')

So,

we

can have

P(y,,) + ^{P,(Y,) (X,*,} _

_Y,,)

:

-P(y,,)

- ₎ o,ry,,)p,(-y,,)-tlt - ;i¿(.\-,,, y,,)l

P"(Xu)

(Y

"-X^)2

-1

f'"(X,) (Y"-Xn)t

: P(y,.)

- ! ^fn,ty,) - p,(xn)fp,(xn)

t

'P"(X,)(Y^ -

^Xo)z

CI -1

(2.4) P(X',+r)

: P"(Yu I ^t(Xu*t - ^Y,))(1 - ,)cU-Yn¡1-

Ya)2

I II(X,,,

Y,,)

2

- _+i P"(x, + t(Y,- x"))d( Y, - x,,)P'(x,)-' - * *'ru,,)p,(x,,)-r

lt ^{- +}

:P(Y,,) -+[r -to@,,

il(x,,)

-1

.P"(I,) (Y, - ^r,),

-1

P"(x")(Y,, -

^x,,)z

* ;[t ^- å ^{u(o,,, v,)f-'} r'{xn)-tp"(xo¡ (y, - ^x,) -+\ p"(xn+

_¿(y,,

- ^x,))d(y

^u

- xn)r',(-r,,)-,

1

lt _L2J -*H(x,,y,,)l p,,(x,)(yu - ^r,,),

P"(X, + ^t(y" -

^{X,,)) (L}

-

^r)d(

^{Y, _}

^x,,)2

Ðonote -by Å-n-

]U1X,,¡

*[t- å"r",,,u,,)-|

^-- _^-l

lu *

"r

p'qx,,¡-t p,,(x,,)(ï,,

-r,l] ^{-er].,) --}

I

P"(X, + ^l(Y, -

^-Y,,))

⁽¹ -

^{t) 1o}

P"(X")

d¿(Y"

-

^-Y')r.

_-] A-P"(N,)(yn-

]i,,)2

4 I. I(. Alg¡'ros, I\{. A. 'l'aìratabai, I). Chcn ^{4 l)} l-he l Jaìlcy-\Vcnrcr ilIcthorl

('\3) llP"(-Y')ll < l"(t,,)

sft):r!r, _29þ -Lt-rJ

- _{+ I} ^{P"(xn + t(Y,} - ^ã,

^))d(

^vu - x,lP'(x;)-'

(,{4)

L,-1P"(X")(Y,, -

^{X,,12 Then}

:

a-1

[",",,, - _* n"ro*) (Y, - ^{*,)'] +} !

^-,P'(x,)-, ^(c1)

ll

, *; P'(N,,)-1P"(x,,)( n., - ^..rll

.P"(,Y")(y"

- X")P(y") >

1

+ ^-f

n'(t,,)-tg(ú,) (s,,

-

^fn)

(c2)

_ll

xn+t - ^{y" ll < f,+r} -

^s,

- _å i P"(x,, + ^{t('Y1t- x,,))d( Y,} - x,,)P'(x,¡-r

Jjlrr""rr, ^{+ t(y,} - ^x,))(1 -

^r)

- p"(x,)l,r¿l _< + ^y, -. x, oi3

L-IP"(X,,)(Yu -

^{Xn\z, (c3)}

v'ith

P(y,,) : P"(X,, + ^t(Y,, -

^{X,,)) (1}

-

^¿)clr(Yu

-

^X")2.

(C4)

_Il

P(X"nr)ll <

_9(f,'-r)

(C5)

_ll

Y,*, -

^Xn+t

^ll ( s,*, -

^t,+t

tohere to

:0, t,

^cttt¡J, s)t Q,rc tl,eJined Jar

all

¡t,

>

^{O as} Jollotus:

(8.2)

^sn

:

tn

- _g'ft,j

^uQ")

hu(t,,, s,r)

: -U'G)-tg"(t,)

^(sn

-

^t,)

aurl S0, $'e have

1

- _^-'I

_JL

^I

?(y,,) + ?'(Y,,)(xu*r - ^y")

p"(xu + ^t(y,, - ^{x,)\ (1} -

^ú)

- î ,"r*,)] ^a{r,, -

^xo)z

1,,+t-rr_ L(sr-tn)'

{l'(tò-1!1" ^(t,,)

-t

,d L-IP',(X,,)-r.P"(X") ^(Y

" ^Ã,) P"(X,,+,(Y, _-,Y',)).

t -

!h,n(tn,s,) (1

-¿)d¿(Y"-X,)'

-

^-1

ⁱ p"(x,, +t(y,,- ^{x,))dr( y}

^,

- x,)F'(x,,)-ra- Lp"(.yur(y

_,

-

_^,,)2.

2J

Proof

.

(C1)

: lYe

have,

in turn

The

result

norv follorvs

lþ .; ^P'(xn¡ tP"(x,)(u,,

- ^-,,/l

)- t -

^tr'11

^{n' ru,-ll}

^{ltn,, <*^) il il}

^y, -

^{.y,, tl}

> r - | ^t - g'(t,))s"(t,)(s, -

^ú,)

:

1

+ |

g"Q,)t"(¿,,)(sn

-

^r,,).

3. SOrìIll U.S[ìttUl, INIìQU,.I L lT'llìS

L¡rm,ra 3.7. Stlyt1tose LlLal,:

(41) _llY,, -

^X,,ll

^{< s,} -

^t,,

(r\2)

ll P'(

X,)-t

_ll

< - l'(t")-r

6 L I(. r\r'gvr:os, N{.Â. Tabatabai, D. Chen ^{6 7 ?} he I-Iallcr'-\\¡cnie¡ l\,Iclhod 7

(C2)

: From

(2.2),

we

have

xr+, - y,: -T r'rt,,)-,Ir -+ ^H(x,,, "",] '*","n,¡(yn- x,lz.

['hen l:¡. estimating both

sicles, \\.e lìâ\re

rr'{'*, -

^{v,, ll}

^< _I

_ll

_"','t ",-

_ll

ll[t - _å

^{ur x n, Y}

^u\]-'

_ll

rr""t"")

^ll

llv'- ^x' ll'

< i tt'tr',,1' ll[, - ]tl ^{rttxu,, T}

^uttt]-'tt

e"tx,)ll llr'-x'll'

= - _; ^s'(',) _[r - t

^{uoû,^,}

")]

s"(t"')(s"

-

^t'"1'

:

fr,+1

_

_s.,

(C3)

:

r\'{oleover, rve get

,,l_

j/ i rr""t-.,, + (v" - x"))l(r -,) - p"(x,,lr,,,ll

+- .¡r

_(ì

^{llY,, - if,,llr}

1

q.

Mlly,, - ^x,,ll

' ₊ -ffllx,,-xoll

t.+".äl _{llY', -} x^li3

\-- -M

2 Xu+r

- Y,ll'+

L,

- M,,llx"-xoll 1-

^o'

2

n:|lÍ,' -

^X,,ll

L-all,r,-Noll

that

is,

(s, -

^ún)r

llP(Xunr)ll< f

^(ru*r

^{- s,,)z*} ry l-

lJ

^m. :

g{úo*r).

-d.

t -.-

₂ ^If(s,,

-

^ú,,)

t -

^rtto

þ

|.

I

LZP"(X,,

*

^¿(Y*

-

^X,,\\

⁽¹ - _') -

^2(L

- ')-r')"(x*)ld,

(Cl-r)

:

Furtherrnore, 'lve get

ll

Y,*r -

^Xn+tll

:

_ll

- P'(X,*r)-tP(X,*t)ll

(

_ll

P'(x,*r)-tll llP(x"*r)ll ( -

g'(in*r)-tl(tn*t)

-

^sr+t

-

^fr+t.

2 _ll

p,,(xu + ^t(y,, - ^x,,)) - p"(x*)ll

₍₁

-

^ú)rtf

{

^{2r1r ll Y,,}

-

^{X,, I}

1

t ^{,(1 -¿)d':{tt r,-x,ll.}

4. TIID S'r',tNDAf u r X{Ât['l'OtìOt¡ | {rII î'IIfiORn.tf

(C4)

: From

I;Gmìnâ, 2.3, s,e have _rea,l

_or Tntonnm

_{compilen u,ncl} 4.1.

_Dois

Let

P: _un Do

_opem ^c: _úonvefr

'{t,-Yo,

_d.oma'ín.

^{Xo, Yn}

_Assunte ^{&re ßa,nach,} _thøt

_P

_has ^sl)a,cesl_Znd,

ord,er conti,n,uo,u,s L'réchet deri,aa,ti,aes on,

Ð,

u,ctd tltut, the Joll,otoinr¡ cottd,ilions

are

su,tisfi,ed :

(4.2)

_ll

P"(Xlll < ff, llP"(Xl- P"(y)ll < À'llX - ^yll

^,

^for

^a,l,l

^{X, Y}

^{e Do.}

lor

^a, giuen,ínitittl ^,ûtt,lue Xoe Do, assllme thu,t

P'(Xol-r

enists r¡,nd,

(4.3)

llP'1Xo¡-tll

<

P, ll Yo

- ^Xoll (

^.r,

M2

1

y"-x,,|!E

- ^nflll. - ^xnll

ll

P(Ã,*r)lls

#ll,r,*, - ^Y,ll'* _[+ *:;)

^t)U

1.

a.

MllY,, - ^x,,ll

a¡

!-m[Ã,-xoll

8 I. l(. r\r'g5.r'os, l\{. A. ^'fabat¿rba_{i, l). Chcn} I _Thc llallev-Wcrner. l{ethod

I

I (4.4)

(4.8) (4.e)

(4.l2)

1

for all ^a

^{i,n, 10,}

^7l

2

< }i,0(ct.í

²

nt

-

^P1oof.

ft

suffices

to

shorv

that the

follorving itenrs

aïe-tr'e for all

ø

bv

rnathelnatjcal incluction.

(r") x,,,

e ,S( Y o,

\ -

^{"q) i}

(II") _llY,,

- X,,ll <

^sn

- t,i (rlr,) _Y"

_{e S(} yo,

\ -

^{n) i}

(rv,,)

llP'(,Y,,)-1ll

{ -g,(r")

(v")

_ll

_{| l + *} 2

P'çx

,¡-t

^P"(.Y

,)(y,, -

^.y,)

^'ll

^lf

à ¹ + f ^($,)-'

!t"(t,,)(s,,

-

^t,,)

ancl,

(vr") _llxnr, -

^]oull

( Í,nr -

^s,.

!!oo.f.ft ^is

^ea,sy

^to

^chech

^the

case fot.

n:0

b¡r

fhs

initi¿-¡l coneli_

tions. Norv assurne th_at (I::L

-

^(Vr,)_are

^{tlue for}

^all

^intägel

valnes s¡raller

ol

eqrral

than

a,

fixed ø.

Then, _n'e hùve

(f,*r)

:

llÏ".', - ^{Yoll( ll-{,,*r} -

^Y,,ll

^{+ lly,} -

^yoll

(

(ún*r

- ^s,) * ^(s, -

^so)

-t- vttll oo

:tn+r - n(\-'A.

(II,*r) : Frorn

(C5), ^.rve have

ll

Y"*, -

^{,Y,,+1 ¡¡ gú,,+r}

-

^s¡,r1.

(fII"+1) :

Moreover', lve have

llY,*, - YollçllY,,+r -

^Xn+tll

*

ll-y"*,

- ^{y,,ll + lly,} _

^yoll

(

(s,r-r

- ^t,*r) * ^(Í,*, -

^s,)

*

^(s"

-

^so)

: Sr+l - ^SO

:

S¿+1

-

⁴

< ft-'Q.

Fultherrnore, florn the

iclentity (4.õ)

(4.6)

_S(Yo,

_r, -

^T)

-

^Do,

uhere rS(ø,

r) : _{u'

e

Xllc'- øl( ^r},

(4.7)

b

- /{pa( 0.485 i/ 0< a (

1

O.ii i/ 1<

u

{2¡

s(t)- !tçt"- t,¡J,

t). a o

I

1

T'T :

0:

rt-tu3

.l

2h

Tt

(1 - ^0')l

1

(1 -

^0')n

v

¹

t'Ìt

oot.lT eeyeation (4.7)- ?hen, t'lttt llu,lleu-llrerlLer proce-

is . Also

X,,,

y,,

_{e,S(yo, ì't}

-

^.rt),

,l'ot,'ítil

^{ru e No.}

fne

co

the ery.r,tttiott,

P(X) :

0. l[or.er¡uer, we tmtse th,à folto- toinç1

error

estimu,tes :

(4.10)

_ll,Yn

- ^X*

^ll

^{< rt} -

^tu,

^{.for all}

^n,

(4.11)

_llY,,

- ^{-{*ll< rr} -

^s,,

for ^nll

^n,

(1

-

^{0') ,l}

1

V2-a

1+ 7-2h

lliz -

^0,f,"1

lV2

-

qJ)13"-1

3,ì', - ^t,,

03t

"11

øll 02 0

l

Jor

all

v.

in lI,

^{2), eur,cl}

(4.18)

⁽¹

- ^0')n

^ga,,-r <

1-03n 2+g

1+20

⁰

'-lt#,um,l

3rr-tr,!

ll1f 201a"-t

1

l]/zny"

Vz

(IV,+r) : P'(X,,*r) -

^P'(,Yo)

: P"(Xo +

^t(,Yn+r

-

-{o))clt(Xn*,

- ^Io)

10 I. 1(. r\r'gylos, il'{. A. ì'abatnlrai, ì). Chen ¹⁰

tf

^{The t} {allct'-Wctnc¡ ì\Icthod i1 11'e get

>

n

+ ]O'(r,.)-'tr(s, -

^fu)

ll

P'(x,+r\ - ^P'(xo)

^{ll <}

^¡{

^ll

x"", -

^{xo ll}

<Ií(¿u+l

-

^¿o)

-

^Ktu+t

I I{r'l :

_{1 1-}

] ø'Q,\-tt"(t,,¡

^1s,o

-

^t,,1.

(YI"*r) : Note that

.tir+z

- ^-y.¡+r: -], n'(x,*,1-tIt _- i ^{*rx,,*r, y,,*,)]}

t.

.P"(;fu+l) (T" - ^x")r.

Thelefore, r'r'e can get

llx,uz -

^Yu+tll

₌ ;

^ll

F'(x"*,)

^,ll

ll

[r - ] ^nçx,*,, ",,.,)1

lj

rto"c",.,)¡ t{r"n, - ^ru*,ll

^*

= -; {t'(t*,-t\lt - ;

tto(tn*,,r,.')J

9"(lr*t\

^('eu*r

-

^fu*r)'

:

lu+z

-

^8¡+l'

Noi+. s.c arc going

to

^prove

(4.I2).

Notice

that

!j(t,) - _T ^rrr-

^{t.,,\ (r',}

-

^!,,,1

and,

{J'(t,,)

: -l ^tnr-

^¿,,)

⁺

^(r',

-

^tulf.

Ilenote ([¡ -

1',

-

^{tu, bu}

-

^Tz

-

f,,. ll'ìten Ltc lì¿ìr'e, g(

f,) - Í( ^-;

^tr,,l;,,,

!j'(t,,)

- ^I!

^1,r,,

¡

^It,¡

ll,tul,

bn:

^øw

₊ (1 -

^0,)ti/0.

1- L-21¡

:I{ ï

n- ^1-2lt

p

515

¹

f3

-

_ll

P'(xo)-'ll

and

b¡'

Ba,nacl¡ Tl¡eorem [4,

pp.

164], PI,{n*1)-1 exists ^anci

llp,(X,,.,)-l¡< _{" -}

^{ll P(Xn)-tll}

1 -

^{ll P'(Xo)-t ll ll}

P'(X"*.) -

^P'(uYn)ll

Du

L-þKllx^*r-xull

1

å-r.,'xn*,-rnll

1"

l- x{t,,*,-Ío)

1

g'(to*tl-r

! _ _I{t,*r

p ( 1",,+r) : \1re can also have,

in turn

llt * t, ^P'(x,,)-' P"(xu)(",, - ^t,)li

=t- ;

llP'(,[,,)-1ll ^ll

P"(n,,)

_{ll ll} Y,,

- ^Í,,

^ll

12 I. Ii. Argyros, lrl. z\. labatab:ri, Ð. Ohen 12 ₁₃ 'f'ltc _I f zillc¡'-\\'olncl llc Urorl 13

Now by

(3.2), rve have ^Ca,se

(ii) : 1 (

ø

{ ^2. Ijy

^¿r

sirnìlal rrelhod,

\\'o can

ol¡tain the

lollowing

Jrou.rrcls :

el'ïol,

{ln: gt-,-

(L -

^0')',,

2-s.1s6-t3a,,,:Tt-tn

1 ¹

¡l/z -

"1a6"

al,-rl(2 -

^a)u|-rb

^2-o.

(ør-, -f

^ltn-r)s

-

øcÍ,,,-rb,,

t(úr-ti

^br-r)

. (l _ _0r)T

¡,a, ^r

= t- 03" v

'

That

corrrpletes

the ploof of llte

theorem.

Ilemaili:s

_

^__

^(a)

^'l'he tlteoretn estriblishes

thc

enistance oJ a, za.o

X* of

^equ,ation

P('Y) : 0. If

^use _fmrther a'sstune thtû

(4.14)

tÌ

< ^{r{(I( +}

^{211{) .}

2(M + ^r{¡z

^'

t.lten ,Y'N

is

l,he rntirlur: zero

of

erltrulion

P(-I) : 0,ín, U(yo,\- _\).

^tr,ndeed,,

let us assutne that

I*

^{zs anot.her zerc, o.f'} tlte etluaî,ir,t,tr, P(,Y)

:

_{O 'ín} the su,me

l¡all.

'f'hcn Jnr

all I

e _10,

I),

_front, the /stintote

By

a

sirnilar

r\¡a\r rve har/e a,n explession for' ^{ðr,, :}

,

^{Ll,-r -l-}

⁽² -

^a)bl,-p,*1

(n,-' *

^{l¡ u-r)"}

:

øau-1bn-r(a1,-1 _| _{Òu-, )} So, 'vve obtain

i:l

^Anb,-t,l

G,t -l

"

nuï_t;

* ⁽² -

^o')

b n-t

1+(2

^{u) U,il-lr}

br-t ),*1 /(x*_l/*)

Case

(i): 0 t

ø

<

^{1. Notå}

that 0

= ?11 ^011,

^so

_ hn_t

ll?'(y* l- ^r1x* - ^Y*)) - ^P,(y*)ll : ill

^{l,t P"(z)}

^{(X* -}

^Y*)da

-31.t+

t) n,t (2

-

^o)

<

^,41¿llX*

- ^Y*ll

^,

1<- -1+20- 2+0

^<

1+(2- r)?- ^{<2 we}

^oblut'tt,,

t) t,t

P'(

Y*¡

^t _lP'(Y'4, -ì-

t1I'. - ^1*)) -

P',(I{,)lctú

that

is,

li;:]'s i+ ^qlffi]'

= i,i,=, [iï]'

Then, \¡,e solve this

ecluation

for

ül¡t)

usitrg

tJre fac1,

of

^Zr¿

:

4,, -þ

+ ⁽¹ - ^0t)l/0. It ^is

^eas¡'

to

^{sec tha1,}

(r

^.,z)'4

n'"-' <

⁽¹

^{- L')--}

^r - t/i# ^u]"-'

r -*;-v ='-i/HLÍr-r-'l

'

* j

,, P¡(

v*)-r

_ll

/{

^' ll

,{* - ^Y*

^ll

* -;

1 g'(rr)-11[( _lin',,

- ^Y"

^ll

^{+ lly,} - ^x*

^{ll )}

3

^c1,,,

:

l',

-

^t,,

³

(1 -

^0')"n lJf2 g1s"-'

'

( -

9'(rì)-rll'Í(r't

-

^{¿,) <}

^-- g'(rr)-llfr, { ^1, by

^(4,14)

lll'h er e,l'or e t h e lin, eu,r op eralnr

î

1

\ ^l,'11

^r'

^{-¡ t1.¡*} -

^¡'r))rt/

t-È[T/lo1'.

J

it ""'-""ilrt

I'otlotos itnnt'eilícr,td1¡

tÌtal X*:

^V*.

aiuii,l

Ñnraton-I{cotttot'ouiclt' ctssunltttons,

Gy¡lg c,rxl T'apiu l4l

as weli ás otlrcts

t51-

t10l ^{proaiúed, th,e} J'o\totiitr,g ^Íto:rmcl Jo:r Neuloru's ^ntel'ho¡J

,t -

^t.,

uì- -

⁽¹

-

^(]2ln

^gz'-r _Jor all

^{n, >- o,}

1-ozo

wlticlt, cant¿ot be ,ím4troued,. ^Thc¡,t

is

tl¿e o'¡'cler o,f con"uergen'crt oJ Neutott')s ^'¡n'el'- äotl

is

^l,wo tulrcre (¿ri (4.13)) tlte Ítcttleu-trVernef nt'eth.ocl lms ot'iler th¡'ee.

t"i

Note utioi tiu.ct rue'lrc,ue shorun (by (a.13)) Lhe enisl'anca oJ infinitclu^

,nrrrry'irrllrocl,s

for''øelil,2)

^tuh,ere

^t¡'c

^up'per bou'tt'cls tt're Less

t¡un tlt'at

of Halley's 'ntetiJt'ocl

(u : L

tlrcn).

}IEIìI]RJìNCI'S

1, r\rg¡rfos, I. I\., ena<bQlfu crpuiioils antt tL¡:;pIítations lo Chrndtascithtu"s ^tut<i tdûlcdcqualions,

Íjtrll. r\rrstr:aì, lf aLlt. Soc., 3g ^(.t 9S5), 2-t5-292'

2, -, ^Ou tt cllss of ¡tottline¡rr it'tlcgral equirtiotts ctrisittcl irt ¡tt:nlrott !rttttsporl. Accluationcs l,IaLhcrn¿rtictrc, Jli ^{(1988), 99-1 11'}

3. Chcn, l)., Itlnloioniclt'-Osltóín:;1,i coilûaìgcn('c lhcorcnts and o¡'tlitnal u¡or ltotn<Is for,Iurntl's

itc't.atitie ntelhatls, lrtcru. ^.J. Coilrputcr J\talh., 3l ^{(3 -'r} 'l) ^{(1!90)' 221} --2:li:.

4. GIagg, \\¡. ll. ^{atrrl l'apia,} lì. t\.,0'ptintcl error boun<ls fun'lhe Netulott-l{ct¡tlorc¡oiclt l.luorcn, SJÀ;U.f. Nunrcr'.,\nal., lt I ^(1974), 10-líl'

5. I{atrttuovich, L' V' antl Ákilor', G' })" lìrulc/inrt¡tl

"lrlnlu'sis i¡t \tort¡t¿tl Spalcs' Pcrgarrotr Prcss, Ncrv YoìÌi, 196!ì.

(ì. OsL'orvsiii, A. ìI., Sglrrliott o[ ]Ìqttcrlions nrrd ,S1,'s/rnts of lÌc1tt<tliotts, Âcarlcnric Ptcss, ^New

Yolk, ^19(i(ì.

7. I)otr.a, ¡i.4. ¿¡rl l,ialt, \.., ^shru,p trror bctuntls for N¿¡D¿on,.r ¡nelht¡tl, Nurrct'. \lath., ^lÌ4,

(.1 9{i()), 63-72.

g. Wcr.ncr.,'\V,, Snrne intprcnentcnls ol'clttssìtul ìleì'(liúc ntdltotls fctr ilic søIulions of nottlitttar eclttrrliotts. l,cctÙ-r'e llOtes in I,Iat.li., Nrrrnci i¿:rù solttlíott of rtortlitteat cquali<ttrs. Ìtroccttlittgs, Illetttcn, ^{878, (19tì0), 'l'2'1} _-'1'lO'

g. y¿¡lliarnoto, 'l'., À cnrttiir.eetu:c lhect¡e¡tt for Netuk;rt-lilr'c ntclJtotls in Bantu'lt .spcccs, Nunrct' l\'Iath., 5l ^{(19t17), 545-:iir7.}

10. Zabr.ejl<o, lr. lr. rn<l Ngìlell, ,t. 1;., ^',l'Ìtt nrtjorturl ¡tttlltod in lhe lhtotu of -'\rurlon-linttlotopítlt trppt'ot:intulÌ<tns r¿ntl lltc Ptnlr r,r;ril eslitt¡¡tles. Nrtrutlr. lìtrtlct. ,\rirl . atrd Optinriz , ^{f) (lf)ij7i,}

l!71 -(:ill4'

tRÌìvulì Ð'AN/\t,ysE lrlllrfÉIltQuE

n'r

ÐIì T-IrÉ()Itììì IJH r,'ÀppÌt0xlur\T.lolrj

X'omc 9Í1, Iiìo

I,

^IÐÐ4, ¡rp. t5-p3

ON A CLASS OF FITNESS FUNCTIONS FOR

GENETIC AL GORITHMS USING

PROPORTIONAL SELECTION

RIIÁìì,ION-IjRNö BAI,ÁZS

(Cìu j-Napoca)

1. tN'rnonllf.'iloN

74 l. l(. Àrsvros, ìf ^. À. 'fabat¿rbai, l). ^(ìltctt

is

inaertittle ct"ìLd .frctnt, the estriltùGte

P(X*) - ^P(r'r¡ -

^P',(Y'r

^{+ (X'r} -

Y'F))d(x',¡

- ^Y*),

lìeceilcd B X ^19U2

^ ^{-[n [2]}

Grefenstette

and Bakel

cliscussed

the impact of the

fitrress

ftlirctions

on

the

behavioul of genel,ic algorithms.

lhe¡i

shorved situations rn'hele Hollancl's _schema

theoremlsae

_¡ã1¡ cloes

not

have a cleal iritelpr,e-

tation

ancl su_ggested,

for

a

lather

large class of genetic algor,itìrrns a

iirn-

ple

but

useful chalacterization of

the implicit

pãrallelìsrnl

- ^rn

^{the present pâpel we_ s1,udy a clais} of fitness functions

for

genetic algorithms using

proportionâl

seie¿tion.

'Ihe first

sectjon contains

îhe

re- hrns using a monotonic fjtness func-

2. GNI,N[I'¡'IC AI,GOIII';'IINS USINç A NTONÛ'¡'(}NIc FI'|¡{IìSS IìUNC'I'TON

^ß*

r) A tf ON OTONIC S ll ¡,tr(ìTION

^LG(XrI'tìÉIlU

,

Selgcti_on is probably

the

rnost

irnpoltant

step

in

a ^gen beca,nse

it

clel,err'ines _r.vhich

inrlividrials rvill cdltribuie

amonnt)

to

l,he creatjon

of a

^lrer.v

population. ¡ls jn [2]

^rve

selection

to

be par,titioned

into twô

steps :

eàch cÌ'ea-

rentlale on the selectjon algolithrn,

algolithrn

is

siven.

it'lius ive sìra,li

rdividual as .i,yell &s the effect of such er,planes.

ilhis

latter,

rvjll lie

cìr¿lr,ac_

morrent f by the targel, sam,pliu,g rale t,sr'(H,r)

: I _,.y ^Yy+,

,n(H, l)

14

i ⁾ t ^p il ! ù1 r:!i I ^ct [ ^],[ ut hetnt.tI itrLI Scicltcrs, {:Qtlt¿l ott U ttioersilg

Lontlott, Oli 73505, Ii.S.A.

))r.ptttlmettÍ ol Ìlalh,5ci¿rlc¿s

l-lrt i¡r¿'¡.silil ^o I ^{Ã t )i<i ti} -<tt s

I¡ttUtllct¡iIl<, ^A Il 7 2'10 1, ^rJ'S.A