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

BẤT ĐẲNG THỨC BIẾN PHÂN TỰA ĐƠN ĐIỆU VÀ THUẬT TOÁN XẤP XỈ GIÁ TRỊ NGUYỄN VĂN THÙY Trang nhan đề Mục lục Phần mở đầu Chương1: Ký hiệu và định nghĩa. Chương2: Nguyên lý bài toán bổ trợ. Chương3: Thuật toán xấp xỉ giá trị . Kết luận Tài liệu tham khảo

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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 d@1.7du<;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

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