Luận văn Bất đẳng thức biến phân tựa đơn điệu và thuật toán xấp xỉ giá trị

CHV0NG 2 A '" " ,:? NGUYENLYBAITOANBOTR0 , ., 2.1 Thu~itoancdso Trong [2, 3, 4], mQtnh6mcac thu~ttoandtSgiai bai toan (1) dil dtiQc th6ngnha'ttrongclingmQtkhuonkhBva dtiQcgQiIa nguyen15'bai toanbB trQ.Y ttidngchinhcuathu~ttoannayd1.,1'aVaGnh~nxetsanday. Xet hambBtrQM :X ~ 9i Idi m;:tnhvakhavi GateauxtrenX , va Ela mQtso'dtiongchotrudc.Vdi x E X chotrtidc,xetbaitoanbBtrQ: (11) mill (M(y) +( EF(x) - M'(x),y) ). Y EOXad Gia sa y (x)Ianghi~mcuabaitoan(11). A.p dl.lllgbB d€ 1.5vdi FI( . ) =M( .) Idi,khavi GateauxtrenX va F2(-) =< EF(x) - M' (x),.) la hamIdi ( vi tuye'ntinh). Theo(9) , Y (x) thoa - - - (M' (y(x», y - y(x» +(E F(x) - M' (x),y) - (EF(x) - M' (x),y(x» 2 0 v Y E Xad, hoi:icd d;:tngnit gQn (12) (M'cY(X»+EF(x)-M'(x),y-Y(X»20 VYEXad. Ne'uy(x)=x thi (12)suyra (EF(x),y-X)20 VYEXad, tlicla (FcY(X»,y-Y(X»20 VYEXad, lien y (x) Ia nghi~mcuabai toan(1). D1.,1'atrennh~nxet nay,xay d1.,1'ngthu~toanco sd sanday dtSgiai bai toan (1). 14 Thu(it tmln 1 ( Thu(it tmln cosO') Cho tru'deday cae sO'du'ong{8k, k E ~} 0 (i) (ii) (13) (iii) Chc;mdii!mxucltphat XOEX tily yo abuack, bierXk, tinh Xk+1 :=y(Xk) b!ing vi?c giai bai loan b5 trq (11) vai x thaybai Xk, 8bai 8k mil}cl(M(Y)+(8k P(xk)-M'(xk),y»). YEX Neu II Xk+l - Xk II nho hcJn melt giai hC;lncho truac thi dang 0 Nguqc IC;li,tra v~buac(ii) vai k +- k +1 0 B6d~2.1 T~lim6ibuaccuathu~tloantren,Xk+la nghi?mduynhC[tcuabai loanbien 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, Suy ra ( P k(y k+1), Xk+1- Yk+1) :2: 0 , (pk(Xk+l)_pk(yk+l),xk+l - yk+l):::;0, ho~cvie'td d,;lllgkhac (16) (M'(xk+l)-M'(yk+l),xk+l_yk+l):::;O, M~tkhac,doM Iaham16im~nhnentheom~nhd61.3secoh~ngsO' a>0 saocho (17) (M'(Xk+l)-M'(yk+l),Xk+l _yk+l):2:a II Xk+l _yk+1112 , 15 Tli (16)va (17)suyraXk+l=l+l . . 2.2 DjnhIy hQit\1d1!atrenHnht1!adondi~um~nh Trangphftnnay,chungta se chungminhslj hQiW cuathu~tloan 1 tranghaitnionghQpvoi loantll'F trangbailoan(1)Ia dontr!vadatr!,voi giii thie'tF Iatl!adcJndi~um~nh. 2.2.1 TruonghQptm!ntll'dontrj Caegiathie't 1. F Ia tl!adondi~um~nhvoi'hangsO'e lIen xad, 2. F Ia lien WcLipschitzvoi hangsO'A lIen X , 3. M' Ia dondi~um~nhvoi hangsO'b lIen xad. B6 d~2.2 V6'igid thilt F la tZ!adondi?um(;mh,nlu bai loan(1)co nghi?m,thl nghi?m do la duynh{[t. Chungminh. * * d Giii sabailoan(1)cohainghi~mIa Xlva X2E Xa .tucIa (18) (19) * * ad (F(XI)' X - Xl ) ~ 0 '1/X EX, * * ad (F(X2),X-X2)~O 'l/XEX . Thay X=X; trong(18),taduQc * * * (F(XI),X2-Xl) ~0 . DoF Ia tl!adondi~um~nhnentant~ihangsO'e >0saocho * * * * * 2 (20) (F(X2)'X2 -Xl) ~ ell X2 -Xl II . * M~tkhac,thayX= Xl trong(19)thi thuduQc (21) * * * (F(x2),X2 - Xl)::; 0 . * * Tli (20)va (21),suyra xl =X2 . . 16 DjnhIy 2.1 Gidsa riinghili loan(1)conghi~mx* . Ne'uM' la dondi~umqmhvai hiing sffh trenXld, thi t6ntqziduynhatnghi~mXk+lcila hili loanh8 trei(13).Ne'u F la tf:Cadondi~umqznhvai hiings6 e tren;:ad( thix*duynhd't) va lien tl;tc Lipschitzvaihiings6A tren;:adva: \j k E ~, ex0,~>0, A +~ thiday{Xk}hQi tl;tmqznhv~x* . Hon mla,ntu M' la lien tl;tCLipschitzvai hiings6B tren;:ad, thicouaclu(,fngsai s6: (22) (23) Ilxk+l-x*II~~(~ +A)lIxk+l-xkll.e c; Chung minh. a) Sf:Ct6ntqzivaduynhd'tnghi~m. Ap dl.mgb6d~1.5va (12),taco Xk+lla nghi<%mcuabai toan(13)ne'u vachine'u (24) <M'(Xk+l) - M'(Xk) +c;kF(Xk), x - Xk+l) ~0 \j X E Xad . Ap dl.mgb6d~1.1vdiA la M', f la M'(Xk),cpla c;kF(Xk)thiA va cpd~ tha'ythoa3giathie'tdftutiencuab6d~.Takit3mtfagiathie'tcu6i. R5rang0 E domcpvaM' dondi<%um(;lnhvdih~ngso'bnen < M' (x) - M' (x*), x - x* ) ~b II x - x* 112 , thayx*=0 thi tadu<;)c < M' (x),x ) ~b II X 112 +< M' (0), x ) . Dodotaduoc <M'(x),x)+cp(x)~bl!x 11+<c;kp(xk)+M'(O),x) II x II II x II ' suy fa 17 < M' ex),x) +cp(x)-} +ex) khi IIx II-} +ex). IIx II Do v~y,theob6 d~1.1thibai tmln(24)luautant<,tiit nha'tmQtnghi~m, nghi~mnayduQcgQiIii Xk+l. Tinh duynha'tcuaXk+lsuyra tub6 d~2.1. b) Day (XkjhQitl,im~mhv~x*. * x langhi~mct'iabaitoan(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' dondi~um<,tnhnentub6d~1.2,taduQc (27) ct>(Xk)~ b II xk - x* 112 ~0 . 2 X6t s1fbie'nd6i ct'iact>t<,tim6ibuckcuathu~toan1. 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' dondi~um<,tnhnentub6 d~1.2,taduQc M(Xk+l) - M(Xk) - <M' (Xk), Xk+l- Xk) ~ b II xk+l - xk 112. 2 Dodo sl:=M(Xk)-M(Xk+l)_<M'(Xk),Xk_Xk+l)::::; - b Ilxk+l-xk 112. 2 Thay x =x*vao(24)thi taduQc (28) S2 :=< M'(Xk) - M'(Xk+l), x* - Xk+l) ::::;Sk <F(Xk), x* - Xk+l ) ::::;Sk <F(Xk) - F(Xk+l) , x* - Xk+l) +Ek <F(Xk+l) , x* - Xk+l). Do F tl,l'adondi~um<,tnhva < F(x*) , Xk+l- X* ) ~0 , nen 18 (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 :::;_lll xk+l - xk 112- e Ek II xk+l - x * 112+Ek(F(xk) - F(xk+l), X* - xk+l )2 :::; _lll xk+l - xk 112- e'Ek II xk+l - x * 112+Ek II F(xk) - F(xk+l) IIII x * - xk+l II2 :::;-lll xk+l - xk 112- e Ek II xk+l - x * 112+EkA II xk+l - xk IIII xk+l - x * II2 - ( vib II k+l k II EkA II k+l * 11) 2 ( A2E2k k J II k+l * 11 2- - - x - x - - x - x + - eE x - x -Ii J2b 2b :::;E2k (A2 - +] II xk+l - x * 112.2b E 2 "1'. k 2eb ~ e A +/3 D d/Y1 a . 0 0 A +/3 E 2b E2k ( A 2 - ~ J <- /3a2. 2b Ek 2b 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)} giamva bi ch~ndtioibdi 0 lien hQiW. Do d6, L'.t+1-:; 0 va tli (30)suyra day {Xk}hQiW m(;lnhv6 x* . c) Changminh(23). * Thayx =x vao(24)vado(29),tadtiQc (M'(xk+l)-M'(xk),X* _Xk+l) +Ek(F(xk)-F(xk+l),x* -xk+l);::: ;:::-Ek(F(xk+l),x* -xk+l) 19 ;::::Ek e II Xk+l - X * 112. M~tkhac,doM' lien WcLipschitzvdi h~ngso'B va F lien WcLipschitzvdi h~ngso'A nen :::;B II Xk+l - x * IIII xk+l - xk II, :::; A II Xk+l - x * IIII Xk+l - x k II . Dodotaco 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 TruonghQ'ptmintii'datrj Trongph~nnay,chungtasexettru'onghQpF la loantii'datIt,giatricua F lucnayla illQtt~pconcuaX. Bai loan(1)lucnaytrdthanh: * dTIm x EX" SCWcho * * * * d ::3r E F(x ) : ;::::0'Vx E Xa . TrongtnronghQpF Ia loantii'da trt,cacdint nghlatti'1.5de'n1.13 tliongling choant X<;ldatri seco dliQcb~ngcachthayF(.) bdi r E F(.). Chingh<;ln,dint nghlatt,(adondi~uill<;lnhcuaant X<;ldontridliQcthaythe' bdidintnghlasanday: ::3e > 0, 'V Xl, X2 E xad , 'V rl E F(Xl), 'V r2 E F(X2), (31) (32) ;::::0=> ;::::e II Xl - X2 112. Thu~tloancosdvftnnhlicu,nhlingdoF(Xk)Ia illQtt~phQp,nenla'yba't ky rk E F(Xk)thayehoF(Xk)trongloan tii'dontIt, va day {Ek}trongtrliong hQpnaythoa (33) -tco . -tco Ek >0, LEk =+CXJ,L(Ek)2 <+CXJ k=O k=O 20 Thu~ttmin 2 Bdt dauta dilm xuc{tphatXOE X. Tc;zibuckk, bie'tXk,am Xk+lbangcach gidi bai loanbdIrq: (34) mill (M(X)+(ck rk -M'(Xk),X»), x E Xad /0 k F( k )VCllrEx. Caegiathie't Trangphftnnay,cacgiathie'tv~ngillnguyennhu'trong2.2.1.Riengtint lient\lCLipschitzcuaF du'Qcd6ithanh (35) ::3a> 0, ::3~>0 saochoV x E Xad,V r E F(x), II r II ~a II x II +~. Chti Y 2.1 Ke'tquacuab6 d~2.2v~ndungtrangtru'onghQpF la loantli'da trio Chungminhdi~unaytu'dngt11nhu'chungminhb6d~2.2,chuy dingluc * * ~ * * nayF(x})va F(X2)du'QcthayboiVrEF(x})va VsEF(X2)' Djnhly 2.2 Gid sa bai loan (31)co nghi?mx*. Ne'uM' la dondi?u mc;znhvdi hangso b trenxad,thi tbntc;ziduynht/tmiltnghi?m~+1chobai loanb6 trq (34).Ne'u F la ti!adondi?u mc;znhvdi hangso'e tren)(ld(x*la duynha't) va thoaman (35),vane'uday{I} thoaman(33),thiday{Xk}hili tl;tmc;znhv~x*. Chung minh. a) Si! tbntc;zivaduynha'tnghi?m. Chung minh hoan loan tu'dngt1fnhu'dinh 1:92.1, ChI thay F(Xk)bdi k k )V r E F(x . b) Day {Xk}hili tl;tmc;znhv~x*. x* la nghi~mcuabailoan(31)ne'uvaChIne'u * * * * d ::3r EF(x):(r,x-x ):::::0 VXEXa. 1.5va (12),taco Xk+lla nghi~mcuabai loanb6 trQ(34) (36) , ? ' Ap d\lngb6de ne'uvaChIne'u 21 (37) (M' (Xk+l)- M' (Xk), X - Xk+l) +8k (l, x - Xk+l ) ~0 ~. k F( k )VOl rEX. \j X E xad , Xet ham * * (x)=M(x ) - M(x)- ( M'(x),x - x) . VI M' la dondi~um<;lnhnentub6 d~1.2,tadtl-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' dondi~um<;lnhentub5d~1.2,taco M(Xk+l) - M(Xk) - ( M' (Xk), Xk+l- Xk) ~b II xk+l - xk 112. . 2 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),taduQc S2 := (M'(Xk) - M'(Xk+l), x* - Xk+l) < k ( .k X* - k+l)- 8 I, X < k ( k * k ) k k k k+1 -8 r,x-x +8(r,x-x). Thay x =Xkvao(36)thitaduQc' * k * (r ,x -x )~O. M<.Hkhac, F tl,(adon di~um<;lnhnen (rk , Xk- x* ) ~ ell Xk - x* 112. Dodo 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 II2 [ k ~ 2 k 2 :::;_b Ilxk+l-xkll-~llrkll +lillrkIl2-e8kllxk-x*1I22 b 2b 22 ( 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~::::; ~2(a211 Xk -x* 112+(a II x* II+~)2)2b 2b 2 a k * 2 l ( * ) 2 ::::;-11x -x II +- a IIx II+~ b b ::::;Yllxk_x*112+8, trongdo y =~ 8 =(a IIx*II+~) b' b Do do ,0.~+1::::;-eEk IIXk -x* 112+(Ek)2(y II xk -x* 112+8). Voi bat ky s6 tlf nhien N, ta co N-l N-l I,0.~+l::::;I(-eEk IIXk -x* 112+(Ek)2(YII Xk -x* 112+8)), k=O k=O ke'thQpvoi (27), ta duQc b IIxN - x * 112::::;<:D(xN) 2 N-l [ ](39) ::::;<:D(xo)+ I -eEk IIXk -x* 112+(Ek)2( Y IIXk -x* 112+8) k=O N-l ::::;<:D(xo)+I(Ek)2(Yllxk -x* 112+8), k=O SHY ra 23 N * 2 II x -X II ~ ° o 2 28N-1 N-1 2 * ~ 2cD(x ) + 2(£ ) Y IIxO-X* 112+- L(£k)2 + L ~(£k)211 Xk -X 112 b b b k=0 k=l b N-I ~l1N + L~Lk Ilxk-x*f, k=l ydi 28 N-1 N = 2cD(xO)+ 2y (£°)211XO-X* 112+- L(£k)2, 11 b b b k=O 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 kE~ b kE~ bdi vi L(8k)2 <+00. kE~ Vi I(8k)2 0 saGcho l1N ~11, \iN E ~. kE~ Luc nay,haiday{II Xk - x* 112}va {l1k}thoagiathi6tcuab6d61.6,suy ra day {II xk - x* 112}bi ch~nvadododay{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 =AX +(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 baa16iciiaday {Xk},va doK bi ch~nnensuyra (40) Han Hila,tu (39)taduQc f lien WcLipschitz. 24 N-l N-l N-l eL Ek II Xk - X* 112:::;<D(XO)+ L Y(Ek)2 II Xk - X* 112+8L (Ek)2 . k=O k=O k=O Vi day {II Xk - x* 112}bi ch<;inen k * 2 3p >0: II x -x II :::;p \t k, suyra N-l N-l el>k Ilxk -x* 112:::;<D(xO)+(yp+8)L(Ek)2 . k=O k=O Vi 2.:(8k)2 <+00nen tli tren suy ra kE~ (41) 2.:Ek II xk - x* 112 <+00 . kE~ Thay x =Xkvao (37) ta du<;jc +Ek 20, Wcla -bllxk+l-xk 112+Ek Ilrk 1IIIxk+l-xk 1120. Ke'th<;jpvoi (35) thi tadu<;jc II Xk+l - Xk II :::; ~II rk II :::;Ek (exII Xk II +0) (ne'u II Xk+l - xk 11:;t0) . b b Vi day {Xk}bi ch<;innend<;it~=exII x:11+0 thi tli trensuyra (42) II Xk+l -xk II:::; ~Ek . (42)vftndungtrangtru'ongh<;jpII xk+l - xk II =O. Tli (40),(41),(42)tasuyra cacgia thie'tcuab5 [email protected]<;jcthoaman.Do d6 tadUdc hm Ilxk_X* 11=0. k-Hoo V~y,day {Xk}hQitumanhv@x* . . 25 2.3 DinhIy hQitv dtfatrentlnheMitttfaDunn Trangph~nnay,chungtasechungmint slfhQit~lcuathu~toan1trang tnionghQptoan111F trangbai toan(1) Ia dontri va F Ia tlfaDunn. Caegiathie't 1. F co tint cha't11!aDunnvdi hangso'E trenXad, 2. F lien WcHoldertrenXadngmaIa :3c>0vaD> 0saGchoV x,y EXad,II F(x) - F(y) II::;D IIx - y Ilc , 3. M' Ia dondi~umanhvdihangso'b va lient~lCLipschitzvdihangso'B A X adtren . DinhIy 2.3 Oidsabaitoan(1)conghi?mx*.NtuM' la dondi?um{;mhwJi hangsob tren;:ad, thit5ntqziduynh(Jtnghi?mXk+lchobaitoanb8tre!(13).Hannlla, ntuF la tf!aDunnvaihangs(/E tren;:advane/u: (43) \-I k \0..> k+I , I--'> , E+~ thi dc7y{ F(Xk) } h()i tl,{v§ F(x*), II Xk+l -Xk II h()itl,{v§ a.va dc7y{ Xk}bi ch4n.Ntu themgid thilt la M' lien tl,{cLipschitzvaF la lien tl,{CHolder tren ;:ad,thim6iddm tl,{ylu cuadc7y{Xk} la m()tnghi?mcua(1). Chung minh. a) Sf!t5ntqzivaduynhatnghi?m. Slf tan t~iva duynha'tnghi~mciia bai toanb6 trQ(13) da:duQcchung mint d dint 192.1. b) Sf! h()i tl,{. Bat (44) \P(x, E)=cD(x) +D(x, E), vdi (45) * * cD(x)=M(x ) - M(x) - , * * D(x, E)=E. 26 Theo (27),taco cD(xk) ::::b II x k - X * 112. 2 * Do x Ia nghi~mcuabai roan(1)lien (\ k k - k < * k- * »~l(X , S ) - S F(x), x x - 0 . Dodo (46) '¥(Xk,Sk);::::bllxk-x*112::::0. 2 Ta xet s11'bi€n d6i cuaham'¥ doi vdi m6ibudccuathu~tloan 1. B~ngcach dungcac ky hi~uva tinh loan tltongt11'nhu trongchungminh cuadinhIy 2.1,taduc;fC 1:+1:='¥(xk+1,sk+1)- '¥(Xk,sk) =cD(Xk+1) - cD(xk) +Q(Xk+1, Sk+1) - Q(Xk , Sk) k+! * k+1 * k * k * = 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 )S2- S x,x - x , k+] * k+l * k * k * 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 ) . DoF cotinhcha'tuaDunnva * k * <F(x ), x - x );::::0, lien <F(Xk),Xk-x*)::::! II F(xk)-F(x*) 112. E Dodo k S2~-~ II F(Xk) -F(x*) 112+sk<F(Xk),Xk _Xk+1) . E 27 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 S2+S3 :;;-~IIF(Xk)-F(x*)112 +Ek(F(Xk)-F(x*),Xk_Xk+l). E Suy 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 *) 11IIXk - Xk+l II :;; (Ek)2 II F(x k) - F(x *) 112+2~ +~ II Xk - Xk+l 112, nentadudc rk+l < ~-b Ilxk -xk+1112-Ek ( ~-~ ) IIF(xk)-F(X*)112. k - 2 E 2~ V /' '\ b ' k 2~ h' /01I\,< va ex<E <- t 1taco E+~ rk+l :;;-b-~llxk+l-xk 112- ex~ IIF(xk)-F(x*)112. k 2 E(E +~) . Ne'u Xk+l=Xkva F(Xk ) =F(x* ) thi tU(25),suyra Xkla nghi~mcuabai 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;indudi 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~ytaduQc II xk+1 - X k II ~ 0 khi k ~ +00, 28 IIF(xk)-F(x*)II--+O khik--++oo. Hon nlia, day {'¥(Xk, 8k)} hQi tl.llien bi ch~nva do do, tU(46) suy fa '¥(Xk,8k) ~~II Xk -x* 112. 2 V~yday{II xk -x* II} bi ch~nvadododay{Xk}bi ch~n. Bay giG,gia sii'z la mQtdit5mtl.lyeticila day {Xk},vagia sii'day con {Xkj} hQi tl.lv~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 WcLipschitzvdi h~ngsO'B lien II M'(Xk+l)-M'(Xk) II ::=;Bllxk+l-xkII. M~tkhac,VI 8k>a lien (F(xk), X - Xk+l) ~- B II xk+l - xk III1 x - xk+l IIa vX E Xad . Do do taduoe (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 --++00lien lieU F(z) =0 thlfa rangz Ia nghi~mcua(1). NeuF(z)"*0 , d~t yki = Xki - (F(z), Xki -z) II F(z) 112 ' thl (48) ( F(z) , ykj - z) =(F(z), Xki - Z ) - 29 - 12(F(z),Xki-z).(F(z),F(Z»=0 II F(z) II D@thfty Ilykj-xkill::;;llxki-zll~O, nen II yki - Xki II ~ 0, ki ~ +00 . Do F lien tueHoldernen II F(yki ) - F(xki ) II ::;;D II yki - Xki Ilc (e>O,D>O). Do d6 tadude (49) F(yki ) ~ F(x *) . Mat khae Ilyki -zll ::;;llyki-Xki 11+llxki-zll, nen II yki - z II~ 0 khi ki ~ +00. Do F tlfaDunnnentil (48)SHYra (F(yki), yki -z):2:~IIF(yki)-F(z)11 . E Cho ki ~ +00thi SHYra (50) Til (49) va(50) SHYra F( yki ) ~ F(z) * F(z ) =F(x ) . Do d6 (47) trdthanh ( F(z), x - z ) :2:0 V~yz langhi~meuabailoan(1). v X E xad. . 30