bất đẳng thức biến phân tựa đơn điệu và thuật toán xấp xỉ giá trị 4

luận văn nàysẽ khảo sát sự hội tụ của thuật toán xấp xỉ giá trị để giải bài toán trong trường hợp toán tử F là ánh xạ từ không gian Hilbert X vào chính nó và F tựa đơn điệu. Ngoài ra luận văn còn làm sáng tỏ sự tồn tại nghiện , cũng như sự hội tụ của thuật toán được xây dựng dựa trên nguyên lý bài toán bổ trợ để giải bài toán

CHV0NG 2

NGUYENLYBAITOANBOTR0

2.1 Thu~it toan cd so

Trong [2, 3, 4], mQt nh6m cac thu~t toan dtS giai bai toan (1) dil dtiQc th6ng nha't trong cling mQt khuon khB va dtiQc gQi Ia nguyen 15'bai toan bB trQ Yttidng chinh cua thu~t toan nay d1.,1'aVaGnh~n xet san day

Xet ham bB trQ M : X ~ 9i Idi m;:tnhva kha vi Gateaux tren X , va E la mQt so' dtiong cho trudc Vdi x E X cho trtidc, xet bai toan bB trQ :

(11) mill (M(y) + ( EF(x) - M' (x) , y) ).

Y EOXad

Gia sa y (x) Ia nghi~m cua bai toan (11)

A.p dl.lllg bB d€ 1.5 vdi FI( ) = M( ) Idi, kha vi Gateaux tren X va F2(-) = <EF(x) - M' (x),.) la ham Idi ( vi tuye'n tinh ) Theo (9) , Y(x) thoa

-(M' (y(x», y - y(x» + (E F(x) - M' (x), y) - (E F(x) - M' (x), y(x» 2 0

v Y E Xad ,

hoi:ic d d;:tng ni t gQn

Ne'u y (x) =x thi (12) suy ra

(EF(x),y-X)20 VYEXad, tlic la

(FcY(X»,y-Y(X»20 VYEXad, lien y (x) Ia nghi~m cua bai toan (1)

D1.,1'atren nh~n xet nay, xay d1.,1'ngthu~t toan co sd san day dtS giai bai toan (1)

Thu(it tmln 1 ( Thu(it tmln co sO')

Cho tru'de day cae sO'du'ong {8k, k E ~} 0

(i)

(ii)

(13)

(iii)

Chc;m dii!m xuclt phat XO EX tily yo

abuac k, bier Xk, tinh Xk+1 := y(Xk) b!ing vi?c giai bai loan b5 trq ( 11) vai x thay bai Xk , 8 bai 8k

mil}cl(M(Y)+(8k P(xk)-M'(xk),y»).

YEX

Neu II Xk+l - Xk II nho hcJn melt giai hC;ln cho truac thi dang 0 Nguqc

B6 d~ 2.1

T~li m6i buac cua thu~t loan tren, Xk+lla nghi?m duy nhC[tcua bai loan bien ph{m:

(14)

vai:

(15)

(pk(Xk+l),X-Xk+l):2: 0 \1XEXad ,

pk(X)=8kp(Xk)+M'(x)-M'(Xk) 0

Chung minh

Go? ?

(14) / h

'

hOA k+l, k+l /

1,

la su co al ng H~mx va y , We a

(pk(xk+l),yk+l -xk+l):2: 0,

Su y ra

( P k (y k+1) , Xk +1- Yk + 1) :2: 0 ,

(pk(Xk+l) _pk(yk+l),xk+l - yk+l):::; 0, ho~c vie't d d,;lllg khac

(16) (M'(xk+l)-M'(yk+l),xk+l_yk+l):::;O,

M~t khac, do M Ia ham 16im~nh nen theo m~nh d6 1.3 se co h~ng sO'

a > 0 sao cho

(17) (M'(Xk+l)-M'(yk+l),Xk+l _yk+l):2: a II Xk+l _yk+1112 ,

Tli (16) va (17) suy ra Xk+l=l+l . 2.2 Djnh Iy hQi t\1d1!a tren Hnh t1!a don di~u m~nh

Trang phftn nay, chung ta se chung minh slj hQi W cua thu~t loan 1 trang hai tniong hQp voi loan tll'F trang bai loan (1) Ia don tr! va da tr!, voi giii thie't F Ia tl!a dcJndi~u m~nh

2.2.1 Truong hQp tm!n tll' don trj

Cae gia thie't

1 F Ia tl!a don di~u m~nh voi' hang sO'e lIen xad ,

2 F Ia lien Wc Lipschitz voi hang sO'A lIen X ,

3 M' Ia don di~u m~nh voi hang sO'b lIen xad

B6 d~ 2.2

V6'i gid thilt F la tZ!adon di?u m(;mh, nlu bai loan (1) co nghi?m, thl nghi?m

do la duy nh{[t.

Chung minh

Giii sa bai loan (1) co hai nghi~m Ia Xlva X2 E Xa tuc Ia

(18)

(19)

(F(XI)' X - Xl ) ~ 0 '1/ X EX,

(F(X2),X-X2)~O 'l/XEX .

Thay X= X; trong (18), ta duQc

(F(XI),X2 -Xl) ~ 0

Do F Ia tl!a don di~u m~nh nen tan t~i hang sO'e > 0 sao cho

(20) (F(X2)'X2 -Xl) ~ ell X2 -Xl II

*

M~t khac, thay X = Xl trong (19) thi thu duQc

(21)

(F(x2),X2 - Xl)::; 0

Djnh Iy 2.1

Gid sa riing hili loan (1) co nghi~m x* Ne'u M' la don di~u mqmh vai hiing

sff h tren Xld , thi t6n tqziduy nhat nghi~m Xk+lcila hili loan h8 trei (13) Ne'u

F la tf:Cadon di~u mqznh vai hiing s6 e tren ;:ad ( thi x * duy nhd't ) va lien tl;tc Lipschitz vai hiing s6 A tren ;:adva:

\j k E ~, ex< C;k< ~eb , vdi ex> 0, ~ > 0,

A +~

thi day {Xk} hQi tl;tmqznh v~ x* Hon mla, ntu M' la lien tl;tCLipschitz vai hiing s6 B tren ;:ad, thi co uac lu(,fng sai s6:

(22)

e c;

Chung minh

a) Sf:Ct6n tqziva duy nhd't nghi~m

Ap dl.mgb6 d~ 1.5 va (12), ta co Xk+lla nghi<%mcua bai toan (13) ne'u

va chi ne'u

(24) < M'(Xk+l) - M'(Xk) + c;kF(Xk), x - Xk+l ) ~ 0 \j X E Xad

Ap dl.mgb6 d~ 1.1 vdi A la M', f la M'(Xk), cpla c;kF(Xk)thi A va cpd~ tha'y thoa 3 gia thie't dftu tien cua b6 d~ Ta kit3mtfa gia thie't cu6i

R5 rang 0 E dom cpva M' don di<%um(;lnhvdi h~ng so'b nen

< M' (x) - M' (x*), x - x * ) ~ b II x - x* 112 , thay x* = 0 thi ta du<;)c

< M' (x), x ) ~ b II X 112 + < M' (0), x )

Do do ta duoc

<M'(x), x)+cp(x) ~ bl! x 11+<c;kp(xk)+M'(O),x)

suy fa

<M' ex), x) + cp(x) -} +ex) khi IIx II-} + ex).

IIx II

Do v~y, theo b6 d~ 1.1 thi bai tmln (24) luau tan t<,tiit nha't mQt nghi~m, nghi~m nay duQc gQi Iii Xk+l

Tinh duy nha't cua Xk+lsuy ra tu b6 d~ 2.1

b) Day (Xkj hQi tl,im~mh v~ x*.

*

x la nghi~m ct'ia bai toan (1) nen

(25) < F(x ), x - x ) ~ 0

EHit

v X E Xad

(26)

ct>(x)= M(x ) - M(x) - < M' (x), x - x) .

Vi M' don di~u m<,tnhnen tu b6 d~ 1.2, ta duQc

(27) ct>(Xk)~ b II xk - x * 112 ~ 0

2 X6t s1fbie'n d6i ct'ia ct>t<,tim6i buck cua thu~t toan 1

6.t+1 :=ct>(Xk+l) - ct>(Xk)

=M(Xk) - M(Xk+l) - < M'(Xk) , Xk- Xk+l ) + < M'(Xk) - M'(Xk+l), x*- Xk+l ).

Vi M' don di~u m<,tnhnen tu b6 d~ 1.2, ta duQc

M(Xk+l) - M(Xk) - < M' (Xk), Xk+l - Xk ) ~ b II xk+l - xk 112.

2

Do do

sl:=M(Xk)-M(Xk+l)_<M'(Xk),Xk_Xk+l)::::; - b Ilxk+l-xk 112.

2 Thay x = x* vao (24) thi ta duQc

(28) S2 :=< M'(Xk) - M'(Xk+l), x* - Xk+l )

::::;Sk < F(Xk), x* - Xk+l )

Do F tl,l'adon di~u m<,tnhva

< F(x *) , Xk+l- X* ) ~ 0 , nen

(29) (F(Xk+l) ,Xk+l~X*);::: ell xk+l-x* 112.

Do d6

S2:::; - e skllxk+l - x*112 + sk( F(xk) - F(xk+l), x* - xk+l).

V~y

k+l

L'.k

:::;_lll2 xk+l - xk 112- e Ek II xk+l - x * 112+Ek (F(xk) - F(xk+l), X * - xk+l )

:::; _lll xk+l - xk 112 - e'Ek II xk+l - x * 112 +Ek II F(xk) - F(xk+l) IIII x * - xk+l II

2

:::; -lll xk+l - xk 112 - e Ek II xk+l - x * 112 +Ek A II xk+l - xk IIII xk+l - x * II

2

-(

vib

II

II EkA II

11)

2

(

J

II

11

2

:::;E2k

(A2 - + ] II xk+l - x * 112.

2

" 1' k 2e b ~ e A +/3

D d/

E2k (

A2 - ~

J

< - /3a2

V~y

(30) L'.~+1 :::;- a2/3II xk+l - x* 112

2b Tli d6 ta suy ra L'.t+1:::;0 tilc la <D(Xk+l):::;<D(xk) Do v~y, day {<D(Xk)} giam va bi ch~n dtioi bdi 0 lien hQi W Do d6, L'.t+1 -:; 0 va tli (30) suy ra day {Xk}hQi W m(;lnh v6 x*

c) Chang minh (23)

*

Thay x = x vao (24) va do (29), ta dtiQc

(M'(xk+l)-M'(xk),X* _Xk+l) +Ek(F(xk)-F(xk+l),x* -xk+l);:::

;:::-Ek(F(xk+l),x* -xk+l)

;::::Ek e II Xk+l - X * 112.

M~t khac, do M' lien Wc Lipschitz vdi h~ng so' B va F lien Wc Lipschitz vdi h~ng so' A nen

< M' (Xk+l) - M' (Xk), x* - Xk+l > :::; B II Xk+l - x * IIII xk+l - xk II,

< F(Xk) - F(Xk+l), x* - Xk+l > :::; AII Xk+l - x * IIII Xk+l - x k II

Do do ta co

Eke!! Xk+l -x* 112:::;II Xk+l -x* IIII Xk+l -xk II (B+EkA),

tlic la

II Xk+l - x' II ,; : (~ + A) II Xk+l - Xk II

.

2.2.2 Truong hQ'ptmin tii' da trj

Trong ph~n nay, chung ta se xet tru'ong hQpF la loan tii' da tIt, gia tri cua

F luc nay la illQtt~p con cua X Bai loan (1) luc nay trd thanh:

::3r E F(x ) : <r , x - x >;::::0 'V x E Xa Trong tnrong hQp F Ia loan tii' da trt, cac dint nghla tti' 1.5 de'n 1.13 tliong ling cho ant X<;lda tri se co dliQc b~ng cach thay F(.) bdi r E F(.) Ching h<;ln,dint nghla tt,(adon di~u ill<;lnhcua ant X<;ldon tri dliQcthay the' bdi dint nghla san day:

::3e > 0, 'V Xl, X2 E xad , 'V rl E F(Xl), 'V r2 E F(X2), (31)

(32) < rl, X2 - xl > ;::::0 => < r2, X2 - Xl > ;::::e II Xl - X2 112.

Thu~t loan co sd vftn nhli cu, nhling do F(Xk) Ia illQt t~p hQp, nen la'y ba't

ky rk E F(Xk) thay eho F(Xk) trong loan tii' don tIt, va day {Ek}trong trliong hQp nay thoa

(33)

Ek > 0, LEk = +CXJ, L(Ek)2 < +CXJ

Thu~t tmin 2

Bdt dau ta dilm xuc{t phat XOE X Tc;zibuck k, bie't Xk, am Xk+lbang cach gidi bai loan bd Irq:

(34) mill (M(X)+(ck rk -M'(Xk),X»),

x E Xad

/0 k

F( k)

VCll rEx.

Cae gia thie't

Trang phftn nay, cac gia thie't v~n gill nguyen nhu'trong 2.2.1 Rieng tint lien t\lCLipschitz cua F du'Qcd6i thanh

(35) ::3a> 0, ::3~ > 0 sao cho V x E Xad, V r E F(x), II r II~ a II xII +~.

Chti Y 2.1

Ke't qua cua b6 d~ 2.2 v~n dung trang tru'ong hQp F la loan tli' da trio Chung minh di~u nay tu'dng t11nhu' chung minh b6 d~ 2.2, chu y ding luc

nay F(x})va F(X2) du'Qcthayboi VrEF(x})va VsEF(X2)'

Djnh ly 2.2

Gid sa bai loan (31) co nghi?m x* Ne'u M' la don di?u mc;znhvdi hang so

b tren xad, thi tbn tc;ziduy nht/t milt nghi?m ~+1 cho bai loan b6 trq (34) Ne'u

F la ti!a don di?u mc;znhvdi hang so' e tren )(ld (x* la duy nha't ) va thoa man

(35), va ne'u day {I} thoa man (33), thi day {Xk}hili tl;tmc;znhv~ x*.

Chung minh

a) Si! tbn tc;ziva duy nha't nghi?m

Chung minh hoan loan tu'dng t1f nhu' dinh 1:9 2.1, ChI thay F(Xk) bdi

)

V r E F(x

b) Day {Xk}hili tl;tmc;znhv~ x*

x* la nghi~m cua bai loan (31) ne'u va ChIne'u

::3r EF(x):(r,x-x ):::::0 VXEXa.

1.5 va (12), ta co Xk+lla nghi~m cua bai loan b6 trQ (34)

(36)

Ap d\lng b6 de

ne'u va ChIne'u

(37) (M' (Xk+l) - M' (Xk), X - Xk+l ) + 8k (l, x- Xk+l ) ~ 0

F( k)

VOl rEX.

\j X E xad ,

Xet ham

<!>(x)= M(x )- M(x) - ( M' (x), x - x)

VI M' la don di~u m<;lnhnen tu b6 d~ 1.2, ta dtl-Qc

<!>(Xk)~ b II xk -x* 112~ O.

2 llt+l : = <!>(Xk+l)- <!>(Xk)

=M(Xk)- M(Xk+l) - (M'(Xk) , Xk- Xk+l) + (M'(Xk) - M'(Xk+l), x* - Xk+l).

VI M' don di~u m<;lnhnen tu b5 d~ 1.2, ta co

M(Xk+l) - M(Xk) - ( M' (Xk), Xk+l - Xk) ~ b II xk+l - xk 112.

Do v~y

Sj := M(Xk) - M(Xk+l) - (M' (Xk) , Xk- Xk+l):::; - b II xk+l - xk 112.

2 Thay x = x* vao (37), ta duQc

S2 := (M'(Xk) - M'(Xk+l), x* - Xk+l )

< k ( .k X* - k+l )

Thay x =Xkvao (36) thi ta duQc'

(r ,x -x )~O.

M<.Hkhac, F tl,(a don di~u m<;lnh nen

(rk , Xk- x* ) ~ ell Xk - x* 112.

Do do

S2:::;-e8k Ilxk-x*112+ 8k(rk,xk-xk+l)

V~y

llt+1 :::;- b II Xk+l - xk 112 -e 8k II xk - x * 112+8k II rk IIII Xk+l - xk II

2

[

k

~

:::;_b Ilxk+l-xkll-~llrkll +lillrkIl2-e8kllxk-x*1I2

( k)2 *

::::;~ IIrk 112-eEk II Xk - X 112.

2b

Tli (35) suy ra

(38) Ilrkll::::;allxkll+~::::;allxk-x*ll+allx*II+~

M~ t khac ta co

( u + v )2::::;2( U2+ V2) '\I u, V E iR.

Do v~y, tli (38) suy ra

IIr~::::;2b ~2(a2112b Xk -x* 112+(a II x* II +~)2)

2

::::;-11 x -x II +- a IIx II+~

::::;Yllxk_x*112+8,

trong do

Do do

Voi bat ky s6 tlf nhien N, ta co

I,0.~+l::::; I(-eEk IIXk -x* 112+(Ek)2(YII Xk -x* 112+8)),

ke't hQp voi (27), ta duQc

b IIxN - x * 112::::;<:D(xN)

2

N-l

(39) ::::;<:D(xo)+ I -eEk IIXk -x* 112+(Ek)2( YIIXk -x* 112+8)

k=O N-l

::::;<:D(xo)+ I(Ek)2(Yllxk -x* 112+8),

k=O SHY ra

N * 2

II x -X II ~

~ 2cD(x ) + 2(£ ) Y IIxO -X* 112+- L(£k)2 + L ~(£k)211 Xk -X 112

N-I

~l1N + L~Lk Ilxk-x*f,

k=l

ydi

28 N-1

N = 2cD(xO) + 2y (£°)211 XO-X* 112+- L(£k)2,

k - 2y

( k)2

~L - - £ b

Ta CO

\i k, II Xk - X * 112~ SUp II Xl - X * 112,

l::;k+1

"k 2y" k 2 LJL =- L./£ ) <+00

bdi vi L(8k)2 < +00

kE~

Vi I(8k)2 < +00 nen co 11> 0 saG cho l1N ~ 11 , \iN E ~.

kE~

Luc nay, hai day {II Xk - x* 112}va {l1k} thoa gia thi6t cua b6 d6 1.6, suy

ra day {II xk - x* 112}bi ch~n va do do day {Xk } bi ch~n

D~t f(x) =II X - X* 112,theo dinh 1y gia tri tIling binh

f(x) - fey)=(f(z), x- y)

*

=2(z-x ,x-y), vdi z = A X + (1 - A )y, A E (0, 1)

Ta co

If(x)-f(y)I~21Iz-x 1IIIx-yll~2supllz-x 1IIIx-yll,

ZEK vdi K la baa 16i ciia day { Xk}, va do K bi ch~n nen suy ra

(40)

Han Hila, tu (39) ta duQc

f lien Wc Lipschitz

N-l N-l N-l

e LEk II Xk - X* 112 :::; <D(XO) + LY(Ek)2 II Xk - X* 112+ 8L(Ek)2

Vi day {II Xk - x* 112}bi ch<;innen

3p > 0: II x -x II :::;p \t k, suy ra

el>k Ilxk -x* 112:::;<D(xO)+(yp+8)L(Ek)2

Vi 2.:(8k)2 < +00 nen tli tren suy ra

kE~

(41) 2.:Ek II xk - x* 112 <+00

kE~

Thay x=Xkvao (37) ta du<;jc

< M' (Xk+l ) - M' (Xk), Xk - Xk+l >+ Ek < rk , Xk - Xk+l > 20,

Wc la

-bllxk+l-xk 112+Ek Ilrk 1IIIxk+l-xk 1120.

Ke't h<;jp voi (35) thi ta du<;jc

II Xk+l - Xk II :::; ~ II rk II :::; Ek (ex II Xk II +0) (ne'u II Xk+l - xk 11:;t0)

Vi day { Xk } bi ch<;innen d<;it~ =exII x:11+0 thi tli tren suy ra

(42) vftn dung trang tru'ong h<;jpII xk+l - xk II=O.

Tli (40), (41), (42) ta suy ra cac gia thie't cua b5 d@1.7 du<;jcthoa man Do d6 ta dUdc

hm Ilxk_X* 11=0

k -Hoo

2.3 Dinh Iy hQi tv dtfa tren tlnh eMit ttfa Dunn

Trang ph~n nay, chung ta se chung mint slf hQi t~l cua thu~t toan 1 trang tniong hQp toan 111F trang bai toan (1) Ia don tri va F Ia tlfa Dunn

Cae gia thie't

1 F co tint cha't 11!aDunn vdi hang so' E tren Xad ,

2 F lien Wc Holder tren Xad ngma Ia

:3c > 0 va D> 0 saGcho V x, y E Xad,IIF(x) - F(y) II::;D IIx - y Ilc ,

3 M' Ia don di~u manh vdi hang so' b va lien t~lCLipschitz vdi hang so' B

A

Xad

tren

Dinh Iy 2.3

Oid sa bai toan (1) co nghi?m x* Ntu M' la don di?u m{;mhwJi hang so b tren ;:ad, thi t5n tqziduy nh(Jtnghi?m Xk+lcho bai toan b8 tre!(13) Han nlla, ntu F la tf!a Dunn vai hang s(/ E tren;:ad va ne/u:

(43)- \-I k \0 > k + I < k, k 2b / O n 0

v E "", E - E , va a < E < -, VOl a > , I '> ,

E+~

ch4n Ntu them gid thilt la M' lien tl,{cLipschitz va F la lien tl,{CHolder tren

;:ad, thi m6i ddm tl,{ylu cua dc7y{ Xk} la m()t nghi?m cua (1).

Chung minh

a) Sf! t5n tqziva duy nhat nghi?m

Slf tan t~i va duy nha't nghi~m ciia bai toan b6 trQ (13) da: duQc chung mint d dint 19 2.1

b) Sf! h()i tl,{

Bat

(44) \P(x, E)=cD(x) + D(x, E) ,

vdi

(45)

cD(x)= M(x )- M(x) - < M' (x), x - x> ,

D(x, E) = E <F(x ), x - x >

Theo (27), ta co

cD(xk) ::::b II x k - X * 112

2

*

Do x Ia nghi~m cua bai roan (1) lien

»

Do do

(46) '¥(Xk,Sk);:::: bllxk-x*112::::0

2

Ta xet s11'bi€n d6i cua ham '¥ doi vdi m6i budc cua thu~t loan 1

B~ng cach dung cac ky hi~u va tinh loan tltong t11'nhu trong chung minh cua dinh Iy 2.1, ta duc;fC

1:+1 := '¥(xk+1, sk+1)- '¥(Xk ,sk)

=cD(Xk+1) - cD(xk) +Q(Xk+1 , Sk+1) - Q(Xk , Sk)

= s] + S2+ S < F(x ), x - x )- S <F(x ), x - x )

=s] + S2+ S3 ,

vdi

S] ~ - b II xk+1 - xk 112 ,

2

< k<F( k) * k+1)

S3= S < F(x ), x - x ) - S < F(x ), x - x ).

Ta co

S2 ~ Sk < F(Xk ), X* - Xk+1 )=Sk < F(Xk), X* - Xk ) + Sk < F(Xk ), Xk - Xk+l )

Do F co tinh cha't tua Dunn va

< F(x ), x - x );::::0, lien

<F(Xk),Xk -x*)::::! II F(xk)-F(x*) 112

E

Do do

k S2 ~ -~ II F(Xk) -F(x*) 112+sk<F(Xk),Xk _Xk+1)

E

M kh d k+l < k

';it ac, 0 E - E nen

< k * k+l k) S3 - E (F(x ), X - X V~y

k

E

Su y ra

r:+1 :;;-~llxk+l_Xk 112-~ IIF(Xk)-F(x*) 112 +

+ Ek II F(Xk) - F(x*) 1111Xk - Xk+l II Vi

Ek II F(Xk) - F(x *) 11II Xk - Xk+l II :;; (Ek )2 II F(x k) - F(x *) 112+

2~

+ ~ II Xk - Xk+l 112 , nen ta dudc

rk+l < ~-b Ilxk -xk+1112 -Ek

(

~-~

) IIF(xk)-F(X*)112.

V/' '\

01 I\,< va ex< E < - t 1ta co

E+~

rk+l :;;-b-~llxk+l-xk 112- ex~ IIF(xk)-F(x*)112.

. Ne'u Xk+l=Xk va F(Xk ) = F(x* ) thi tU (25), suy ra Xkla nghi~m cua bai roan (1)

. NguQc l';ii, ta duQc

r:+1 = \Jf(xk+l,Ek+l) - \Jf(Xk,Ek) < 0 ,

do do day {\Jf(Xk , Ek)} giam, bi chi;in dudi bdi 0 nen hQi tl,l Do do, ta co

r:+1 = \Jf(xk+l, Ek+l) - \Jf(Xk, Ek) ~ 0 khi k ~ +00.

Do v~y ta duQc

II x k +1 - X k II ~ 0 khi k ~ +00,

IIF(xk)-F(x*)II +O khik ++oo.

Hon nlia, day {'¥(Xk, 8k)} hQi tl.l lien bi ch~n va do do, tU (46) suy fa

'¥(Xk,8k) ~ ~II Xk -x* 112

2 V~y day {II xk -x* II} bi ch~n va do do day {Xk}bi ch~n

Bay giG, gia sii' z la mQt dit5m tl.lyeti cila day {Xk}, va gia sii' day con {Xkj} hQi tl.l v~ z.

Ta viet l~i (24) la

(M' (Xk+l ) - M' (Xk ), X - Xk+l ) + 8k ( F(xk ), X - Xk+l) ~ 0 v X E xad M' lien Wc Lipschitz vdi h~ng sO'B lien

II M'(Xk+l)-M'(Xk) II ::=;Bllxk+l-xkII.

M~t khac, VI 8k> a lien

(F(xk), X - Xk+l) ~ - B II xk+l - xk III1 x- xk+l II

Do do ta duoe

(F(xki),x-xki+l)~_Bllxki+l-xki II Ilx-xki+l II VXEXad.

a

VI

F(xki) +F(x*), xkj +z, Ilxki+l-xki 11 +0 khi ki ++00,

lien

(47) (F(x*), x - z) ~ 0 V X E Xad

Ngoai fa, VI (F(z), x kj - z) + 0, ki + +00 lien lieU F(z) =0 thl fa rang z Ia nghi~m cua (1)

Neu F(z)"*0 , d~t

yki = Xki - (F(z), Xki -z)

II F(z) 112 '

thl

(48) ( F(z) , ykj - z) =(F(z), Xki Z )