View of On the monotone convergence of an Euler-Chebysheff-type method in partially ordered topological spaces

Tome

XXVtr,

No

l,

^199g,

^pp.

^23-31

ON TTTE MONOTONE CONVERGENCE

OF AN EULER-CTIEBYSIIEFF.TYPE METHOD

IN PARTIALLY ORDERED TOPOLOGICAL SPACES

IOANNIS K. ARGYROS

1,INTRODUCTION

- ^{In this study we}

^are

^{concemed with} the problem of approximating a solution x. ofthe nonlinear operator equation

(l) F(x) ₌

0

in

a

linear

space 81,

where F is defined on

a

convex

subset

D of D1 with values in

a

linear

space

82.

We

have

recently

shown

thât if .El

arid

Ezare

Bànach spaces, then

under

stan- dard

Newton-Kantorovich

hypotheses the Euler-Chebysheff-type method

of

the

fonn

(2) ln=xn-lxn,xn]-'¡'(r,)

(3) xn+1= yn -f)ín,xnf-'(fx,,!,f-l_x,,x,l)(yn-xn) ^xo eD (n20)

converges

with order almost three to a locally unique solution x*

_ç

D of equation (1). Here [x,y]

^{denotes a}

divided difference of

order one,

which

is a

linear

operator.

we introduce

and study

the monotone

convergence

of

the

iterations {a,}

^and

{x,l

@

> _{0) given} by

(4) F(a,)+fxn,xnl(wn -u,¡=g

(5) _F(x,) +fxn, xnl(yn - ^{x^)} =

⁰

(6) (fx,,!nl-[x,, x,f)(un-u,)+fx,,xnl(u,*,-w,)=0

and

(7) (lxn,l,f-lx,,x,l)(y, -xn)+lxn,x,f(x,*t-U)=0

to approximate

a

solution

x*

of equation (l).

(1991) AMS (MOS) Subject Classif¡cation Codes. 65H10, 6SJl5, 49D15, 47H17

rvhere

^Rro*r.',

=

9',0

*'h('

- ^x).

^Choos

^ing

^s

='2k

⁺

^2' in (6)' from the boundness of h, ^Lemma 2 md the Cauchy inequality it results that there exists a constant ^y which

^depends

^on

^fr and d; such

that

QlÏ)Rru.r.") ^(x)

^{< Ym-(k+r)}

'

^l

Theorem

3

follows from (3)

^and

^{Theorem 2'} REFERENCES

l.

B. Della vecchia,

on

the approxìmation ^of functions by ^means of the operators or D' D' stancu' Studia Univ. Babeç-Bolyai, Mathematica, ^37,

I

(1992)'3-36'

?..A.

Di

Lorenzo and

M.

^R.

-Occorsio,

Polinomi

tli

^Stancu, Rapp' Tecnico, no'

l2ll95' I'A'M''

Napoli, ^1995.

3.

ti. H.

Gonska and

J.

Meier, Quantitative theorems

oi

approximation

by

Bernstein-slancu operators,Calcolo ²¹ (1984), fasc' IV,

3l'l-j35'

4. G. G. Lorentz, Bernstein Polynomials,University of Toronto ^Press' Toronto' ^1953'

5

G.MastroianniandM.R.Occorsio,Sullederívatedeipolinomídistuncu,Rend'Accad'Sci'Fis' Mat. NaPoli, 45,4 (1978),n3-281'

6. G. MasUoianni ^and M. R. Occorsi o, I)na generalizutione (lell'operatore tli Stancu' Rend' Accad' Sci, Fis. Mat' Napoli, ^45, 4 ^(1978), 495-511'

'7.C.

p.

May, Saturation and inverse theorems

for

combinations

oJ'a

class

of

exponential-rype oPerators,Canad' ^J. Math., 2E (1976), 1224-1250:

8. G. Múhlbach, verallgemeínerungen der Bernstein- und lttg,range-Polynome. Bemerkungen zu einer Klasse linearer Polynomoperatoren von D. D. Stancu, Rev. Roumaine Math' Pures Appl. ^{15, S} (1970), 1235-1252.

9. R. K. S. Rathore, Linear Combinøtions of Linear Posítive Operalors and Generating Relations in

S p e c iul l' ^u nctions, Thesis, I.l.T., Delhi, ^I 973'

10. D. D. Stancu, Approxirw.ttion ^of functions by a new class of linear polynomial operators,Rev' Roumaine Math, Pures Appl., 13,8 (1968), 1173-1194'

11. D. D. Stancu, IJse of prohabílisric ^methocls ìn the theory of

uniþrn

appruximation of continuotts .functions,Rev. Roumaine Math. ^Pures Appl', 14 (1969)'

673491'

12. D. D. Stancu, Recurrence relations

for

the central ^moments of some discrete prohability laws, Stuclia Univ. Babeç-Bolyai, Cluj, ^Ser. Math.-Mech', 15,

I

(1970),5542'

13.

D. D.

Stancu, Approximaiion properties

ttJ'a

class of linear positive oper\tors, Studia Univ' Babeç-Bolyai, Cluj, ^Ser. Math'-Me¿h., ^{15, 2 ( 1970)'} 33-38'

14.

D. D,

Stancu,

on

the remainder

of

approximation of functktns by ^{means oJ'}

a

pararneter- dependenr linear polynomial operator, Studia Univ. Babeç-Bolyai, Cluj, Ser' Math'-Mech',

16,2

(1971),5946.

Received November 15, 1997 ^{" Bab e} S - B o lYai

"

U nive rsitY, Faculty ^oJ Mathematics and Computer Science

l,

M.Kogölniceanu ^St., 3400 Cluj'NaPocø,

Romania

L

Ioannis K, ArgYros

The

Euler-ChebYsheff method

(or

otra [7]. Ul'm used divided divided diffe- orem

[7,

p'91]'The

^order

ofconvergence

of our iterations is ^almost

s

oforder

^one

^onlY'

2. MONOTONE CONVERGENCE

On the Monotone Convergence 25

Moreover, if

the

operators Ar=fx,,x,f are inverse nonnegative, then any solution

^u

^of

^the

^equøtion F(x) = 0 in (uo, xr) ^{belong to}

^(u*

, x')

_'

Proof. Let

us

define

the oPerator

.ff

:(0,.t0 -un)-+ 81, PtG)

=

x - ^{Bo(F(u,r) + Ao@)'}

This operator

is

isotone

^and

continuous. We

can

have in turn 4 ^(o)

⁼

^{-aoF(uo) 2}

^o,

4 ^(xn -t o) =Ío -uu- BuF(xo)+4(F(xo)- F(uo)-lo(xo -an)) 1

xtt

-

^{uo + Bo ([-xo}

,uo]-

[x6 , xe

]) (xo -

^uo

) þy ⁽⁹⁾⁾

1xo-uo,

since

_[-xn

,urlS

^{[xu ,}

t¡ ] ^{by (12).}

By Kantorovich's theorem [4], ^operator Pt has a fixed point z,

e

(0,

x o

-

^u o)

:

Pr(z

r)

^{= zr}

. Set

u _o = ^u ₀ + z

r, ^thenwe

have estimates

F(uu)+

An(w,',

-ro) ⁼

^0,

F(uo)

=

F(wo) - ^F(uo

⁾

- ^{Ao(wo- on)}

^{< o}

and

uo

<llo ^< xo, We define the

operator

Pr:(O,x0 -ut¡) + E', ^Pr(x) =

^tt

- ^{Bo(F(ro)+ ln(x)).}

This operator

^is

isotone

and

continuous. We

can have

in turn

Pz(0)=Bu¡(-ru)>0,

P,

(*

_u

-

^to

¡) ⁼

^x o

-

^u o

r

B oF ^(u.t

u)+

^Bo

(F(xu

₎

-

^{F (w} n)

- ^A,

^(x u

-

^u

o)) <

lxo _- ^wo+ Bo(fxr,wr,]-lxo,xo]) ^(¡o -Ø,,) ^{< (by (9))}

1 _Xç _ _lJ)ç¡

r

since

_[.ru

,ur]3

^{[xu , xs}

]

^bY

^(12)'

By Kantorovich's theorem there exists z, ^e(0,

^x,¡

- uo) ^{such that}

Pt(rr)

₌

zz.

^Set

lo

⁼

^{ro -} zr, thenwe have the

estimates

F(¡u)+Ao(yo-r,,)=0,

F(Yu) = F(lo) - ^F(xo) - ^AuQu-rr,) )

^o

2 ³

24

Weshallassumethatthereaderisfamiliæwiththemqaningofadivided difference of ^{order on" un¿} the notion of a partially ^ordered topological

^space

(POTL) ([1], [2]' tZl, iql)'Moreovet' ^{from now on we shall}

assume that

ù

^{and E2}

are POTL-sPaces.

We .átt no*

^{state the main}

^result'

Tlæongvt|,LetFbeanonlinearoperatordeJìnedonaconvexsubsetDofa regularPoTl-spacenlw¡thvaluesin-qPoTL.spaceE2'LetU0andxgbetwo

SupposethatFhasadivided'dffirenceoforderoneonDn=(uo'ro)=

= {x ^eE1

^{luo <}

¡ I ^xo} ç ^D

^satisJYing

(10)

^Ao

^{=fxo, xol}

^{has a}

continuous nonnegative left

subinverse Bs

' points of D such that

(8) (e)

(1 1)

(r2)

and

(1 3)

(4]'l.7), (14)

(1s)

uu3xo

F(u,,)<0<F(ro)

u0

<wo <ut ^< 3ll), 1un*t 3

^{xr+t 3}

ln ³ "' ( rl ⁽

Yo

3 xs' limr)-=t)' , ^lim

,-+* "

^n-+ú

and u'

,

xo

^e

D, ^{with u*} (x'

[xo,Y]20 ^for all ^tto3YSxs' [x,u]-[x,Y]<0 iÍ ^u<Y

lz,wl+lw,ql-lz,zl-lu,zl20 if ^uSw<z for some ^q e(u'z)

_'

Then there exist

^two

^sequences {un\'{x'\ ⁿ²⁰ satisfuing approximations

xn=x

and

Lo(yo -"u)<0.

From approximation (6)

we

now have

Ut

- ^{uo =}

^tÐ0

^{* Brlr(wu} -

^uo)

-

^wr,

^{= BnLu(wo} -ro)

^{2 0.}

Hence, we obtain

u)o

1uo.

Moreover, from approximation (5) we have

x't - lo= l, i ^Bulu(y, -rr) - lo= ^BoLo(yo -xo)10.

That is, we get

)ct

I _lo

Furthermore, we

can

obtain in turn

u

t - ^x'¡

=:',:7,'lï:r:,;,' _-l:...7 ^!;jí=

- "'

⁼

- uo BuLoF(uu) - ^(xo - BoF(x,,)) +

+

BuLo(u, - ^{BuF(un - BnF(uu))} -

^{BnZo (uu} )

+ +BrLoQo) - ^BrLo(xo - ^{BrF(xo)) =}

.

= u0

-

^x,0

- ^Bo(F(uo) - ^J¡(¡o

))

-

B0L0E0(F(¿o )

- F("r,, )) ⁼

=

(I - ^{Bslus, xof} - BoLrBofuo, xrl)

(ao

- ^ro

).

But using

hypotheses

(12)

and

(13) we

have

B0L,B,lt)r, xul+ Bofun, xo)S

B0L0B0A0

+

Bofuo,

ro ]l

<

BrLu +

Bn[uu, _'xo ]

I ^Bo(Lo

_{+ [uo,} ^xu

_])

^<

3 ^Bolyo, qlS BrAo 3,L

We

now obtain ut 1xt.

From all the

above

we

have

uoSuo1ut1x,1yo3xo.

By hypothesis (12),.it follows that the operator An has a continuous nonnegative left subinverse 4 ^{for all n> 0.} Proceeding by induction, we

can

show that there exist two sequences {u,), {xn} ⁽ⁿ

^>

0) satisffing (4) - ⁽⁷⁾

^and

(l _{) is}

a

regular

space E1 and as such

they converge to some u' ,x' ^{eDo. That is,}

we

have

)y_r, =u' ^3x.

⁼

]*x,.

l

and

urr

Swrr 3

_Yo

< xo.

We now define the

oPerator

P¡:(0, xo -uo)-+ Et, Pr(x) = ^x - Bo(LgBrE(uo)+'4n(x))' where I, _{= lxo}

, r,r _]

-

lxo,.Yn

]'

'Ihis operator is isotone

^and

continuous' We

^have

in turn Pr(0) = -BoLoBoF(un) ² 0 ^{bv (9)}

P,

(xo -

^It

o)

^{= xo}

-

^u o

- BrrLrBoF(x

_o)

+ +Br(LoBo(F(x,,

)

- ^F(uu)) -[¡n, ^x'

^l

^(ro -'o ))' But, try (11)

and

(12), we

can

liave

L.B,,F(x

_o)

- ^{F(ao)) - [xo, xo] (ro -}

^{/u )}

=

..

= (Lo/,o[xo,

uo]

- [x,,,

^{xo D}

^(xo -

^{uo )}

^<

<(Io _-lxo,xo]) (x, _-uo) I _-[xt,

J¡o]

(xo -uo) I ^0'

Therefore, we

^have

P,(to -uo)1xo-u,¡'

By Kantorovich's ^theorem there exists ^{z, e(0,}

^xo

- uo) ^{such that}

Pr(rr)

=

z¡. Set ut = uo

⁺

zr, thenwe have

estimates

- Lr(wr-ro)

⁺

^Ar(uo -Øo)

^{= 0}

and

Lo(wr-au)20.

Furthermore,

^{\rye can}

^define

^the

^operator

&:(0, ^xu -

^0o)

^-+

^Er

, ^{Pr(x) - x}

⁺

Br(LoB,rF(x

u)

- ^AuQÐ'

This operator is isotone

and

continuous. We

have

in turn P4(0) = BoIo-BoF(xo)> 0 ^bY(9),

Po(xo -uo)=

^xo

^-uo l BoLoBoF(ur)+

+

Bo(LoBo(F(xo

)

- ^F(uo)) - ^Ar(xo -

^uo

))

^{< xo}

-uo

(by using the same approach as for P¡). tsy Kantorovich's theorem there ^exists

z o ^e

(0,

x o

-

^U

o) ^{such that}

^Po

^(, ì

^{= z}

^{o. Set}

^X

^t

⁼

^!

^o

^-

^z

^{o, thenwe have}

^estimates

-Lu(yo - ^{ro) +} Ar(x, -./o)

^{= 0}

On the Monotone Convergence 29

Hence, we

get

0> _9-t F1u,))

un

-u),,

⁰

^{< 9-I F1x,)3}

^xn

- ^yn.

Since E¡ is notmal and 1im

^(un

-w,)

⁼

H ^{(x, - y)}

⁼

^{0, we have} ,lim Q-t Flun¡=

=,lg Q-t F(*,)

=

0.

ftrence,

by continuity, we

^get

F(u' )= F(r. ) =

^0.

(c) As

above,

we

get

02 F(a,)2 R(u, -w,),

^{0 <}

^F(x,)

^<

^R(x, -y,).

Using

the

norrnality

of Ez and the

continuity of F

and R, we

get F(u.

₎

₌ F(x* _{) =}

0.

(d) From the equicontinuity of the operator

An

we have

,lim A,

^(u _n

-

^u

n) ⁼

=,lg ^An(x, - ^{yn\ =0.}

Hence, bV

(a)

and

(6)

F(u")=F(x')=0.

(e) Using

hypotheses

(10) - ^{(14), we}

^get

^{in turn}

0

( _{F(Y,) =} F(Y,)- _{F(x,) -} A,(Y, - x,)

⁼

=

(An -1.!n,

^xn))

(x, - y,)

^<

^([xo, xof-l**, x' _l) G, - y,).

Since Ez is normal and ^lim (x, -!,)=0, ^{we get lim F(x,)=Q. Moreover,}

from hypothesis (12)

[x*, ¡*] ^("n -

^{*n )}

I _[,r-, xn)(x, - ^x'

)

=

= F(xn) -_F(¡.) ( [ro,

^xn ]

(x, -x')

and by the normality of E2,F(x-)

=

j_T F(x,).Hence, we g:t F(x-) = 0. The result F(u- ₎

=

0

^{can be}

obtained similarly.

The proof of

the

theorem is now complete.

As in Theorems I

^and

2, we can prove the following result (see also _[7, Theorem

6.21:

Trmonuvt

3. Assume

that hypotheses of Theorem I are true.

^Then

the appro- ximations

lr=xn-BnF(xn),

xn+t

=yn+ BnLr(yn- xn), Ln=fx*xrl-fxn,!rl

,:

( _,i , ^{un =un - B,F(un)}

7

lf

^{u,, < u}

^{1 xtt aîd F(u) =0, then we}

^can

^obtain

An(Y o

- ^u)

⁼

^Ao(x

^o

-

^-Bo

^{F(xn )) -}

^Asu

⁼

=

^Ao()(o

-

^u)

- ^AnBr(F(xo

⁾

^{- F(z)) =}

=Ao(I_Bo[¡n,u])(xo_u)20,sinceBu[x9,uf<BoA,,<1.

Similarly, ^we show Au(w, -

^{a) <}

^0'

Iftheoperator,4gisinversenonnegative,thenitfollowsfromtheúï.:

*o 3,3 yo. i'ro...Airrg by induction, weãeduce that

^{w n}

3

u <

l" from which it follows ^that wnlunlun*t31t3!n+t3xn1!n' ^{fot all} n>0' ^{That is' we}

^have

u, ^1u3 x, for ^all n>0'

^Hence,

^{we get} u' ^{1tt< x'}

'

That

completes the

proof

^{of the}

^theorem'

ln what follows, we shall give some natural conditions under which

^the

points u*

and

x* aresolutionsof equation F(x)=Q'

TITEOREM

2. Under

hypotheses

of Theorem I

^suppose

^{F is} continuoîts at

t)*

and x'

^{. IJ one}

^of

^the

folowing conditions

^is

satisfied (a) N' =y" ' _terator

Q:E', -+ Er' (Q(0)=0) ^which

(b),El

¡s

normal and there

exists

an

op

has an isotone inverse continuous at the origin ^{and such} that An<T for sfficientlY ^large

^n,

(c)EzisnormalandthereexistsanoperatorR:8,-+82(R(0)=0)conti-

nuous

at

the

origin

^{and such}

that A, 3 R for stfficiently larg

^n,

(d) operators

^An

are

equicontinuous

Jbr all n20'

qnd

(e) Ez

ß normal and lu,u)<Íx, yl if

u <

x ønd

^{u <}

y'

thenwe

^have

F(u.)=F(x.)=0'

Proof. (a) Using the continutty of F and F(u')30<F(x')' ^we

^get

F

(u. )<

⁰

^{< F(u.} ). ^That

^is,

^{we obtain} F(x

^-

_{) =}

^F

^(u"

) --

0' (b) By (a)

^and

(6)

0

) ^F(u,) = An(u, - ^u,)>- Q@, - ^w,)

0

I

^F

^(r,

)

= An(x

n

- Y,)> QG, - ^Y,)'

Ioannis K. Argyros 6 28

6.

M.

T. Necepurenko,

on

chebysheffs ^method for functional equatiors (in Russian), usephi Mat' Nauk., ⁹ (1954). 163-170.

7. F.

A.

Pota, on an ileralive algorithm of order L839 ..' Jor solving nonlinear operqlor equations' Numer. Funct. Anal' Optimiz'' ⁷

(/),

(1934-1985)' 75-106'

g. s. ul,m, Iteration ^methods with divided diferences of ^{the second} order

(n

Russian), Dokl' Akad' Nauk SSSR, 158 (1964),55-58' ^Soviet Math' Dokl' 5' ll87-1190'

g. J.

A.

Vandergraft, Newton's method

for

convett operators in partially ordered spaces,

sIAM

^J'

Numer. Anal.

4

(19ó7), 406

-

^432'

Received August 10, 199ó Deparlmen! of Mathem atics' Cameron (lniversity, Lawton, OK 7i505'

U,S,A.

30 Ioærnis K. ^ArgYros

and

un*t

=ll)n

⁺

BnL'(wn -l)')

\)here ^the operators

^Bn

^are nonnegative

'ubinverses of

Ar, generate two

^sequences

{u,) ^and {x,\(n'0¡' satisfying"approximations (4)-(7) and(14)' Moreover'for iny solution ^u e(un, xr) ^of

^the

^equation F(x)

=

0 we

høve

ue(ur,x,) ^(,¿>0).

Furthermore,ctssuruethatthefollov)ingaretrue:

(a)

E2

is

a

POTL-space

^and

E¡ is

a

normøl POTL-space;

(t ) _,l1x ^{x, = x* and} lim Ü' =Ü*;

(c) f'rs continuous at u* and x'

;

and

(d)thereexistsacontinuousnonsingularnonnegdtiveoperatorTsuchthat ß,,2T _for sfficientlY ^{large n'}

^Then

F(u.)=F(x*)=0'

Remarlc.(a) our conditions coinci

^te

with (44) and(5Ð in _[7, ^p. 98].In

case

El=åz=[R,ourconditions(12)and(13)aresatisfiedifandonlyifFis

ditferentiable

^{on Ds}

, ^{aîd F,F'are convex}

^{on Do '}

(b) It followu ao- ui tlr" ^above that otlr ^method

^uses

the ^same

⁰¹

^simpler

conditionsthanthoseusedinallpreviousresults(t4]_t9])buttheorderof

convergence is faster [3]' (c) Similar ,.roi,.

"* immediately follow if the divided difference

^[x0 _'

xs'l is replaced by lxo ,xttfuttlzo3x" ^{in (10)'} lx"xn] ^{is replaced by [x" ln-tf}

(n>l) in(a)-(7).

REFERENCES

1.I.K.Argy'os,andF.Szidarovszþ'OnthemonotoneconvergenceofgeneralNewton-like

m e t h od s,B ull. Austral' Math' Soc'' ^45' (1992)' 489-5 02'

2.LK.AtglTos,andF.Szida¡ovszky,TheTheoryandApplicationsoflterationMethods'C'R'C'

Press, [nc. Boca Raton, Florida, U'S'A' ^(1993)'

3.

I.

K. Argyro s, On lhe

,oní"'g'n'"

o¡ ChebysheffHatley-type nethod ^under Newton-Kantorovich

h yp o t h e s e s,Appl. Math' Letter, ^6, 5' (1993)' ⁷ l-7 4'

4.

L .v.

Kantorovich, The method

of

successive approximatiott for functional equotions' Acta Math''

T

(1939),63-97.

5.

M. A.

Merwecova,

An

analog

of the

^process

of

tangent hyperbolas

for

general functional equations(in ^Russian), Dokl' Akad' Nauk ^{SSSR' 88} (1953)'

6ll-614'