Luận án Nghiệm dương của một số lớp phương trình toán tử

Chrtdng4 ,:J K" ? ,'? ,,'" " DIEM BATDONGCUATOANTV RONH(1PDONDI~U § 1.Toant11h6nh(1p,ddndi~uvadiim ba'tdQng Truochtt tadn mQtsQkhaini~mvaktt qualienquan.Xlakh6nggianBanach thvcdu'<Jcsa:pbdinonK . . Gia su'Dc K, roantU'.A: D x D -7.X duQcgQila h6nhQpdondi~untu A(x, y) la khonggiamtheebitn x vakhongtangtheebitn y. NghIala \iUl~U2V2~ Ul,V2ED ta. . , coA( u J, V-I) ~A( u 2;V2) . Bi~m(x",Y")E D2duQcgQila c~pdi~mtvabit dQngcila A ntu A( x",Y")= x" va""" . . A(y ,x ) =Y . Bi~mx" E D duQcgQila diembit dQngcilaA ntu A( x",x")=x" . ToantU' A: B(A) eX -7X duQcgQila16intu \i x,y E D(A) max ~y, \it E [0,1]co A(tx+(1-t)y)~tAx +(1-t)Ay (1) A gQila lornntu -A la 16i Dink if 4.1.1. . . Gia slYK la nonchua':n,A: KxK -7K la roantU'h6nhQpdondi~u,bonm1'a (i) voiyc6djnhA(. ,y):K-7K lalorn.Voi xc6djnhA(x,.):K-7 KIa 16i (ii) ::1v> 0,c >Yz saccho0 <A(v,O)<v (2) vaA(o,v)~c.A(v,o) . Khi doA coduynhitdiembit dQngX*Evatucacdayli[lp xn=A(Xn-l,Yn-l); Yn=A( Yo-!.Xn-l)u~1 (3) Voi cackhdid~u(Xo,Yo)E Xtuyy, taco II Xa:'" x* II -70, IJ Yn-x* II . -70 (n.-7oo)vOit6t dQhQitl;l : 29 II x,_x'II ,; N'( 1~c)' II v II } . }IYn-X*II~N~(l~C t II vii . (4) Chungminh: a.,T6n tc;lidilm beitdQng. B?t Uo=0,Vo =v tac6Uo<vo'Giasa Un= A(un-I, Vn~I), Vn=A( Vn-I,Un-I), n.=1,2,.., (5) VI A tangtheobie'nthlinha'tva.giamtheobie'nthli2 nen 0 =Vo <U I::;u2~'" ~Un~ Vn~... < V2~ U I ~ Vo=v (6) (6')TITgia thi~t(2)ta c6 Un~UI ~cv I~cVn B?t tn=Sup { t >0 : Un~ tvn} n=1,2,... Khi d6 Un~tn,Vn(7),Theo (6') va tinhcha't U~+J ~ Un~ tnVn.;:::tnVn+J ta c6 0 < <t <t < <t < <1 A t)l: t ' 1' t - t* ' 0<t* <1- c - I - 2- ,..- n - , nen on ql lID n - va - - n BaygiGtasechIfa t*=1 Th~tV~Y,tugiathie't(i) tac6caeh<$thlicsau rjx ]:,;X2,Y J :0;Y 2, t E [ 0,1]: A( tx I+(I-t) x 2,Y) ~t A (x bY) +(1-t)A( X2,y) A (x, t YI +(1 - t) Y2)~ t A (x, Yl) +(1 - t) A (x, Y2) (8) (9) A{x,y)=A(x,1-fly) ~t A( x, fly) +(1-t)A(x,o} (10) rjtE [0,1] Tu (10)talqi c6 A(x, fly) ~ fl[ A(x,y)-(1-t) A(x,o)] , rjt E[O,I] (11) 30 Tii'caehl$thuc(5)(6) (7)(11)(8) (9)(10),va giathie'tA la toantU'h6nhQpdon dil$utaco Un+1=A( Un,vn)~A(tnvn,vn)~tnA( Vn'Vn)+(1-tn)A(o, Vn) -1 . ~ tnA(vn, tn Un)+(1-tn)A(o, V) ~ tn[tn-lA(vn,un) - tn-l(l"-tn)A(vn,o)]+ (1-tn) A(o,v) . . . =A(vmUn)+(1-tn)[A(o,v)- A(vn,o)] 1 . ~Vn+1+(1-tn)[UI-V 1] ~Vn+l+(1-tn)(l--)u]c . 1 . 1 ~Vn+l+(1-tn)(1--)Vn+ 1=[ 1+(1-tn)(1-- )] Un+1 C C Tii'dosuyra 1 1 tn+l~1+(1-tn)(1- -) =>I-tn-l$ (1-- ) C C (12) Do giathie't(ii),tatha'yco th€ coiY2<C$ 1va0< l' -1 <1, c do v~ytii'(12).suyra dU<;1C: . I-tn+I$ (1-tn)( ! -1)$ (! -I)\I-tI) $ (l-c )n+l C C C tii'd6suyratn-7I(n->oo) (13) Tii"(6)va(13),taco 0$ Un+p- Un $ Vn- Un$ (1-tn)Vn$ (1-tn) v. DoKIa nonchuffn,voiN lah~ngS9chuffn,taco: II Un+p-unll$N(1-tn) v II $N(l-c rllvllC .11vn:-unll $N( I-tn) II vii $N(l-c r Il,y II . . . C , Tii'(14)vaX la kh6nggianBanachnent6nt<;tiJimUn=u*n Tuongtl,l',tachi duQct6n.t<;tilim 'vn=V*.. n~~ (14) (15) 31 Cungtii'(14)khi chop-700tadu<;5C II un;:Jl * II ~N(.!.tc IIvii II vn-v*11~N(l-c t " v II c Vi Un ~v* ~v* ~vn,ta co 05;v* -u* ~Vn.:.Un~(1-tn)V Dov~y II v*-u*11 ~N(1-tn) Ilvll'-7o(n-7oo)vau*'=v* La'yx* =u* (=v*)thix* E 0 Tuun~,x*~Vo>Un+l=A( Un,vn)~A( x*, x* ) ~A(vn,Un)=Vn+l La'ygioih(,ln(khin -7 00 ) ta dU<;5c. X*~A ( x*, x* ) ~x*, Nghlalax* la di€m ba'tdQngcuaA. b.81/duynhfitcuadiim biltdQng. _Giasli'Xla mQtdi€m ba'tdQngnaodotren[o,v]cuaA, khido - -- Uo=0 ~ X ~ V =V0 A ( X' , X) =X Den, . U 1=A(uo, Yo)~A ( x/x)=X ~A( Yo,no)=VredoA 1ah6nh<;5pdondi~u).B~ngquy n(,lpta co Un~ X ~VnV n'~1 La'ygioi'h(,lnkhin-7oo,tadu<;5CX =x*, v~ydi€m ba'td~I).g1aduynha't. c. ToedQhQitl!. V Xo,YoE [o,v],tuongt1,1'( 6)ta.co Un~ Xn~ vn; Un~ Yn~ vn-Theo (15) ta dU<;5C II xn-x*II~NII Yo-Un II~N2(1-C)n II V II c II Yn- x*" ~ N "vn -Un II ~N2 (1- C)n II V II Binh 19dU<;5Cchung minh 0 C Dinh IV4.1.2.Giii sli'K 1anonchu5n;A.~1aroantli'h6nh<;5p dondi~u.,Giasli'A(o,v)>Y2Vva 32 (i) Voi y c6dinhE ,A( . , y) :~ 1a16i,voi x c6dinhE , A:(x,.) :~ 1a15m. (ii) c6h~ngs6c: y~ c:::;1 saccho A( v,o) :::;cACo,v)+(I-c)v (17) Khi d6A c6duynha't1di€m ba'tdOngx* E .HonmIa,voi xo,YoEtuyy, day l~pXn=v- A( v- Xn-I,V-Yo-I);Yn=v- A(v- Yo-},V-Xn-I)n=1,2...(18) C6 S1;1'hOit1;1 v-xn~ x*;V - Yn ~ x* (n~CIJ) (19) Chungminh: Ta d~tB (x,y)=v-A(v-x,v-y)'\Ix,Y E f5 rangB 1aroantli'h6nhQpdon di~u,NguQcl~ivoiA, voi y c6dinhY E ,B (. , y): ~ 1a15m;voi x c6 dinhx E,,B( x,.):-7 1a16i. - .. . Ne'uA( o,v)=v thlA( x,y)=v ('\Ix,Y E <o,v»vaA(v,v)=v(VIAla h6nhQpGon di~unenA tangrheabie'nthunha'tvagiamrheabie'nthu2 vanha(17). Ne'uA(o,v) <v thl0<B(v,o)=~- A (o;v):::;Y2V, (dogiathie'tcuadinh1y:A(o,v» '12v vadinhnghlaB) Tli (17)tacoA(v,o)<cv+( l-c)v =v Vav- A(v,o)~v - [ c A(o,v)+(I-c) v ] =C [ v - A(o,v)] NghlalaB(o,v) ~CB(v,o).Do dataco(2)trongdint 1y4.1.1. Va nhaphepchungminhhoanroantuongt1;1'tasuyduQcB codungmOtdi€m ba'tdOng y*>O.Tuc la. * * * * * y =B(y , Y ) =v - A (v - Y ), (v- y ) * " * ". * hay A (v -;-Y , v - y ) =v- Y D~tx =v- y , taco x*=A (x*, x*).Honnua,'\IXo.YoE nha(18), (20)vadinhly4.1.1.taduQC " " Xn-+Y , Yn-+Y 33 " " . " V ,-xn~" x* ; v- Yn~ x* va""dinh19dU<;1Cchung:minh 0 Binh IV4.1.3 " Giii sacacdi~ukien(i) , (ii) ciladinh194.1.1du<;1cthoa.Khi d6t6nt(;lis6AO~1 saGcho.r\..A(v,o):::;v, va "i/A E [6, AO] phuongtrlnh u=AA (u, u)coduynha'tnghi~mU(A) Giii saUo(A)=0, Vo(A)=vva Un(A) =A A (Un-I(A), Vn-I(A)) Va vn(A)=A A (Un-I(A), Vn-I(A)) Khi do ta UOClu<;1ng . II Un(A)~U(A)11 :::;N (l~c)n Ij~ II ~o(n~:oo) (21) c "II Vn(A)-U(A)11 :::;N(l-Ct II v II ~o(n~oo) (22) C Chungminh Ne'uA=0 thlke't)u?nlahi~nnhien,va u())=(1f)~tAO=sup{t>0: tA(v,o):::;v} . giii sar~ngA E(O,AO) . Tit giii thie't(ii) A(v,o):::;v =>AO~1 " I Tli 0 <AA(v,o):::;AOA(v,o):::;v va " AA (o,v)~ coAA(v,o) (do(2))ta tha'yr~ngAA thoacac di~uki~ncila dinh 19 4.1.1.V?yAA coduynha'tdi~mba'tdQngU(A)Evau (A)>0. . Cac ke'tlu?nve uoclU<;1ngt6cdQhQit\1trong(21),(22)la r6 rangtheodinh19 4.1.1 f)~chungminhdinh19tie'ptheotac~nb6d~sau Bddl 4.1.1 Giii saX lakhonggianBanachdu<;1cs:1pbdith~nonduongKj, Y lakhonggian BanachdU<;1Cs:1pbdinonchu~nduongK2.Giii satoanta:A: DAcX ~ Y la loantalom Q" " ho~c16iXoED (A) khidoA lienWct(;lixone'uvachIneuA bi ch~ndiaphuongt(;liXo' .Nghiala t6nt(;li8 >0 saGchoA bi ch~ntrongHinc?nNo (xo)cilaXo 34 Chungminh a)Di€u ki~ncdn:A lientl.1ctC;liXonenVE>0,38>0: II x - XoII II A(x) - A(xo) II <E. , ,. V'~yne'ula'yNI) (Xo)={xED(A) : II X-XoII < 8} thlll A(x)11~II A(xo) II + E tren NI)(xo) b)Di€u ki~ndu:GiasaA bich~ntrenNs(Xo)=B(xo,8).B~ngcachgiam8,cothe coiB(xo,8)c D(A). X6t dayXnE D(A),JimXn=Xo'Ta cothe.vie'tXn=X6+tnYnvoi tn>0, n--+'" , Xn - Xo 8 limtn. =0/,11Ynll ~8/2. Th~tv~y,chidnla'y Y~rrx . [ " 2 ' 'I{~n- Xo! .t...=1::I1X~-Xo~ ./ t' Khi nauIOndein~1,tavie'tXn=(1- tn)xo+t~(Xo+Yn)vaapdl.1ngtinhl6i cua A, taco Axn~(1- tn)Axo+tnA(xo+Yn). =>Axn- Axo~tn[A(Xo+Yn)- A(xo)] VI taclingcoXo=Xn+tn(-Yn),d6i vaitracuaxo,XntacoAxo- Axn~tn[A(xn- Yn)- A(xn)].V~y ta co . -tn[A(xn-Yn)-A(xn)]~Axn- Axo~tn[A(xo+Yn)- A(xo)](*) khin duIOntacoXn - Yn, xn,Xo+YnthuQcB(xo,8)nencacdaythll'nha'tvathll'3 trong(*) hQit1,1v€ O.Do K2 lanonchugnnendayAxn'- AxohQit1,1v€ 0.0 Dinh z.-v4.1.4 -' GiasaK lathenonchugntrongX; A:KxK --+K la toantahanhQpdondietl, A(v,0)~0vacacgiathie't(i)(ii)trongdinh194.1.1duQcthoa. Khi dophuongtrlnh AA (u,u)=u AE [0,do] (23) codungmQtnghi~mileA)thoa 1)u(.) : [0,Ao]--+lien t1,1C 2) VO<AI <A2E 'Ao,ta co 35 1..2 '\ u (A 2)~ - C .U (I'.I) A) (24) U (AI) ~ ~ c. u(A2) 1..2 (25) , . . Oday' Ao =sup[t>0: tA (v,o)::;v] Chungminh.. . 1.Ta d~tuo(A)=O;VO(A)~V un(A)=AA (Un-I+(A), v n-I (A); V'n~)=AA CYn-1 (A),' . . n-I (A); n cAI2/'" (26) Til dinh194.1.3,tacosvhQitvcuaUn(A)~ U(A) v (A)~ U(A) (t'I-~CtJ)la d~utheoA E [0,Ao].Til dosuyra . «(A) lien t1,ICtren [0, Ao]ne'uvoi m6i 'l\~ 1,uYi(A),VYI(A) lien t1,Ictren[0,AoO.] . , Bay giGtaseChIra un (A),vn (A) lien t1,Ic\in, ~ 1.. Th~tv~y,voi XO,yo E KG ,,;x,y E taco II A (x,y) - A (xn,Yn) II ::;II A {x,y) - A (xn,y) II + II A (xn,y) - A (xn,Yn)II. , f Theob6d~4.1.1, neBY codinh, A( .y)la bi ch~ntrennenA ( .y)lient1,IC tc,tixo,tu'dngtv A ( xo, . ) lient1,Ictc,tiyo.Nentheo(27)A(x,y) lienWctc,ti( xo,yo).vi ( xo,yo).latuy9nenA lient1,ICtren( KG(I ).R6rangtaco limileA)=limAA (U(A) . i~O i~O , ileA)) =0=u (0). Til (2)va (26)voi chu9dng A ( v,o)>0,\i A E [ 0, AO]coUI (A) =AA (O,u»O; VI (A) =AA (v,O»Ovaui (A), V.:t(A) la lientvc. , Til (6}( trongdiilh194.1.1)taco u~(A)~O;v~{A»;Qvab~ngquync,tptadu'QcUn (A), Vn(A) lient1,Ictren[0, \-J . NghIala tacoke'tlu~n1) 2) Vi UCA) E , nho(2)tacoU(AI) =AI A (u(AI),(u (AI) ~\I A (O,v)~AI CA (v,o) Al '\ . A) ~ - C 1'.2A(u (A2),U (A2)=-. Cu(A2) 1..2 1..2' 36 A . . tu'ongttftaco U(1.,2)~-2- e . U (AI) 0 , Ai. He !Iud4.1.1 Giasii'KlanonchugncuaX A: K x K --+KIa toantli'h6nh<;lpdondi~u,thoadi~uki~n(i) voi ~codinh. A (. , y ).:K --+K la lorn.Voi x codinh A (x, .) :K--+K 1<\16i. Va ::Je, ::Ju, ve.K saDchoY2<e ::;1 A«u-,v>. <u,v» c <u,v» (28) A (u,v) ~e A (v,u)+(1-c ) u. Khi doA codungmQtdi~mba't.dQngXE . Chungminh E>~tB (x,y)=A (x+ u,y+u ) -u "ifx,Y E K (29) Khi doB ( x ) c la toantli'h6nh<;Jpdondi~uthoa (i).Honm1'a B (v - u,0)=A (v;u)-u }B(o,v..-u)=A (u,v)-u (30) Tli'cach~thlic(28)(29)(30), taco B (v.-u,0)::;v -u;A (u,v)-u~e A (v,u)- eu,nenB (o,v-u)~e B (v- u,0) Ta coth~giathie'tr~ngB (v - u,0)>0(vIne'uB (v - u,0)=0 Thl A (v,u)=D, do (30),tlido "ify,x E X ,A (x,y)=uvauseladi~m ba'tdQngduynha'tcuapY.Nhu'v~ygia thie't(ii)trong dinhly 4.1.1dU<;lCthoa,nenB co duynha'tmQtdi~mba'tdQngx* E ,nghlala . . A (x* +u,x* +u)-u=x* hay A (x,x) =~, d dayi =x* +u0 lJe qua4.1.2 GiasaK lanonchugntrongX . A :x --+la toantli'h6nh<;lpdondi~u,giasli'di~uki~n(I) trong dinhly 4.2.1dU<;lCthoava::Je saDchoA (u,v) ~Y2(u+v) 37 Yz <C ::;1, A (v,u)::;C A (u ,v)-+(1- C) v Khi doA codungmQtdi~mbeftdQngx E <( Chl1J!gmillh D~tB ( x,y)=A (x+u,Y.+u ) -.u '\Ix,y TudngWnhuchungminhdh~qua4.1.1.T: B: x ~ thoaml B coduynheftdi~mbeftdQng. x* E <0,v - u;,.nghIa1ax*=B (x*,x*) =>A (x* +u , x* +u)=x* +ud~tx=x* cuaA 0- & -u> sa duQc ,ie'tcuadinh194.1.2.Nhuv?y ~ +u, x* +u)-u 5x ladi~mbeftdQngduynheft § 2. Di~m t1}.'aba'tdQngcua toaD ta hOnhQp ddnmen. 2.1.Caekhainiem x lakhonggianBanachdu<;Jcsa:pbdinonK D =vaD1=D2 =... =Dk =D eX Dinh nghfa4.2.1 . Tmintti'A: D1x Di x...Dk~ X du<;JcgQilabonhdpddndieune'uA tang d6ivoimoimQtrQngmbie'ndgutienvagiamd6ivoimoimQtrongcacbie'nconI~i. . GiasaA: D1x D2X ...Dk~ X la bonhi€m(x,y) E D x D du<;JcgQila capdi€m tu'aba'tdongcuaA ne'ux =A (S1,S2,'..., Sk)va Y =A (S'1 , S'2, ,..s\ ) ? d daySj=xvas\=yne'uA tangdbienthlii Sj=y vaS'j=x ne'uA giamd bie'nthlii Nhfinxet J) Ne'uA la roantti'hon.h<;Jpddndi~uvatang,d6ivoimbie'ndgu, giamd6ivoik -m bie'nconI~ithl taco th€ xetroantti'A' xacdinhtrenD' =D x D tuX' =X x X VaG X' nhusau: A' (x,y) =A (x , x, , x , y , ...,y) illbie'n m-kbie'n Ta coA' tangrheabie'nthlinha'tvagiamrheabie'nthli2 (1) 2)TrangkhonggianBanachX',taxetnonK' =K x(-K) . , Va ky.hi~u" ex"Ia quailh~thlitl;1'trongX' sinhbdinonK'. Ta tha'y (x,y)a (x',y') <=1x.::;x' , ly':6;yvoi"::;"Iaquailh~thlitvsinhbdinonK 39 D~dangki~mtraduQcdmgne'uK cotinhchit chu§'nhaychinhquyhay minihedta1thlK' clingcocactinhchit ~uongtv. , ' Xetanhx~B:D'-7X' xacdinhboiB(x,y)=(A'(x,y), A'(y,x)). (2) Bd dl 4.2.1 Giii sil'A:Dk-7 X 1atoantil'h6nhQpdondi~uvaB :D' -7 X' duQcxacdinhboi (2).Khido ' 1.(x,y)1ac~Pdi~mtvabit dQngcuaA ne'uvachine'uno1adi~mbit dQngcuaB. 2.B 1a.anhx~dondi~u(tiing)d6ivoiquailh~,!CX" 3. Ne'uuo~A(uo,"'uo,vo,...vo);A( vo,'..vo,uo,..uo)~Vothlv'0exB( u'0); B ( v'o)a v'0;trongd6 u'0=(uova);v'0=(vo,uo)'" , Chungmink: Cackh~ngdinhtrend~dangduQcsuyfa, ch~ngh~ntaki~mtrakh~ngdinh2). Voi z' I =(XI, y I) Z2=(X2,Y2)thuQcD' ,ta c6 : Zl a Z2=> [I ~X2 { A'(XJ,y I) ~A'(X2,Y 2) - bl~Y2.. => 'A'(YI,xl)~A'(Y2IX2)=>B(zl)aB(Z2} 0 2.2.Trliilngh(lptOlllltitlientac: DinhIV4.2.1Giiisil'Di'=DVi =1,k,A =DI X D2X...xDk-7 X 1atoantil'h6nhQp dondi~u,cotinhchit: Uo::;;A (x I".xm,Xm+J,...Xk)va A ( ' " , )Vo ~ X I;...,Xm,X m+I,"'Xk (3) ? d dayxi=uo,x'i =VovaXj=vo,X'j=Uovoi 1~i~ m,m+1~j~k. Giii sil',them nfi'a,mQttrongcacdi~ukl~nsailduQcthoa. (HI) K 1anonchidnvaA hoantoanlientl.JC 40 (Hz)K 1:1lionchinhquyv:1A 1:1tl.!a.IientlJCye'u,tue1:1ne'uXn-7x; Yn -7Ythl . . .A(xn, ...xn,Yn...Yn)ye'~A (x,...x,y...y). Khi d6,A coc~pdi€m tl.!aba'tdOng(u*,v*) nghiaIaA( u*,...u*,v*,...v*)=u*va A(v*,...v*.u*,...u*)=v*.ddayu* xua'thit$nambie'ndftulien,v* xua'thit$nak-m bie'n.conI<;ii,tronght$thuedftu.V:1aht$"thuc2,v* xu~thit$nambie'ndftu,u* xua'thit$na k-IILbi!nconI<;ii.Hon nuau*~v* vavoi.e~pdi€m t1,1'ab 'tdOngba'tky( x,Y ) eilaA taco: u* ~x ~v* ; u* ~Y~ v* . Voi un=A( un- J,... Un-I,vn-l,...Vn-I) Vn = A(Vn-I...Vn-1 ...Un-I Un-I).' , , Vn~ 1 Vauo~ u~...~un.~...~Vn~VI~Vo (~ Tacou*=Jimu v*=limv0' n' . . n-+oo n-+co Chungminh:X6t anhX?A': A:D,xD-7DxaedinhbaiA'(:t,'i!) '=A(')(...'J(.)1"'~-)- : ~'~~ D~tu=A'( uo;vo), VI =A' (vo,uo)'Til gia thie'tA h6nhQpdon dit$u,Uo~Vonen: Uo~UI~VI~Vo Ta xaedinh ~, Un+1 =A'( Un,vn) . Vn+1~.A'( vn,un)- , (5) Til gia thie'tUn-I~Un~Vn~Vn-Iva A h6nhQpdondit$unenUn~Un+1~ Vndo do ta co Cdt) TasechungminhUn-7u* EX. 1.Khico(HI)VI K Ianonehugnen {Un} lientlJCnen~p {u I ,...Un }Ia compiletu'ongd6i. 1:1t~pbi chiln.Do A ho~mloan Dodo3{und kc {un}n:Unk-7U*EX. 41 Hi<innhienUn~u* ~Vn \in ~ 1 Khil>nkchungtacoo~u*-li2~u*-unk.Tudo II U*-UI II ~N 1IIIu*-Unk..J1 C5dayN la h~ngsf)chugncuanon K. V~yill-> u*(l-7oo).Tu'ongtl;l',co Vn-7V*(n-7oo). 2. Khi co (H2).Tu (4)vatinhchinhquycuanonK, suyfa f~ngUn-7 U*, Vn-7v* (n-700). Vi A' la t1;lalient\lCnen Un+l=A; (un,Yn)ye'u~A'(u*,v*)(n~>oo) ...J Vn+1=A' (Vo.un)~0.A'(v*, u*) (n-7oo). !. Cho n-7 00,apd\lng(3) ta duQc.u*=A'(u*,v*) . . .. v* =A'(v*,u*). Nhuv~y(u*,v*) la c~pdi<imt1;laba'tdQngcuaA'. Va f6 rangu* ~v*. BaygiotagiastY(~,yo)la mQtc~pdi<imt1;laba'tdQngnaodocuaA'. Khi do x = A' (x, y) ; y=A' (x,y).Vi uo~x~YO;uo ~y~vo . NencoUI< x<v; ; UI~Y ~VI; t6ngquatco: - . - . un ~ x'~vn; un ~ y~vn . Cho n ~ cOco : u*~x~v* vau* ~y ~v* 0 lJinh 1£4.2.2. Gia stYcacdi~uki~ncuadinhly 4.2.1duQcthoa,bonnua:3oc:0 < oc<1saDcho IIA(x,...x,y,...y)-A(y,...y,x,...x) II ~ocll x-yll (6) \i x, Y ED. Khi do,A codungmQtdi<imba'tdQngx trongD Chungminh : Ta stYd\lngdinhnghlaA' quaA vacacky hi~unhutrongdinhly 4.2.1,tu(6)ta co: 1/ v,n.+1-1l!:.n+1 II ~ II A' (v~,'u~)- A' (u.wv~)II ~:ocII v~ -LJ J II (11=1,2... ) 42 L~pl<:liI~plu~ntrentac6: II v n+I - Un + I II ~ ex:\<, II VI - uI11~ 0 (n~ oo)(vl c{E (0,1)) Tli'ke'tlu~ncuadinhIy4.2.1.suyrac6X=v* :=u*, Xladi~mba'tdQngduynha't cuaA. 0 R5rangkhik =1thl (6)18.di€u ki~nLipschitstruy€n thongdii bie'tcho anhX<:lco. 2.3.Trztifnghdptoantitkhonglientuc FJinhif 4.2.3. Gia sii'Uo, VoEX, K la n6n.Minihedralm<:lnh, . . A: ~~ X la tOaDtii'h6nhQpdondi~usaDcho(1)trongdinhIy 4.2.1 duQcthaa Kpi d6,A c6c~pdi~mtl,l'aba'tdQ.ng(u* , v*) v:diu* ~V*.Honnuavdic~pdi~m tt,I'aba'tdQngba'tky (~,y)cua A taluanc6u* ~~~vk , U* ~y~Vk Chung minh : Ta chidn apdl;}ngb6d€ 4.2.1vake'tquatuonglingtrongdinhIy.2.4.1chotOaD tii'B xaydl,l'ngtrongb6d€ 4.2.1 0 Chli v :C~pdi~m.Waba'td6ng(u* , v*) trongdinhIy 4.2.3c6th~duQcxacdinh rabon.Ch~nghk: A (XI"" X , y, ...y) ~X va A (y, ...y ,X ; ... ,x) ~y ~ . Tli giathie't(1)tasuyduQcD :f. ~ G,iasii'DI =~ X : (x,... X., Y , ...y) E D voiyna?d6~ D2 =~y: (x, ...,x , y, ...y)E D voix naGd6 ~ V~yv* =supDI, u*=infD2 FJinhif 4.2.4 Gia sii'Uo, VoEX, Uo<Vova A :k~ X la tOaDtii'h6nhQpdondi~uthaa(1)trongdinhIy 4.2.1va A « Uo, Vo>k)la t~pcomp~cttuongd6itrongX. 43 Khi doA coc~p di~mtl,tab1td6ng Chungminh 'fa chI.dn sll'd1,1ngb6d€ 4.2.1hc$qua2.1.1cho.anhx~B xacdint trongb6d€ 4.2.10 EJinhIf 4.2.5 Gia Sll'Uo, VoEX, Uo<'Va A: k~ X 1aroantll'h6nh<;fpdon dic$uthoa(1) trongdint 194.2.1; va K 1anon chinhquy. Khi do,A coc~pdiemt1;lab1tdQng(u* , v*) voiu* ~V*.Honnu~ne'u(~,y)1a diemtl,tabit dQngnaGdocuaA thlu*~x ,y~v* Chungminh Ke'tquasuydU<;fCkhitasll'd1,1ngb6d€ 4.2.1,hc$qua2.2.1varoantll'B duQcxac dint trongb6d€ 4.2.1.0 r5 44