Nghiệm dương của một số lớp phương trình toán tử

x) r. R5 rangtu gia thie't3) va 2) mil Mo :f; ~_;vii A (Mo)c Mo. Taseapdlfngnguyen1;'Entropichot~pMovaphie'uham- Strongdo Sex)=sup~ II A (y) -A (x) II :y ~x~ur, ( uEMo) ) . xEMo YEMo 1.Truoche'tachungminh: M6idaytang~ Xnrn C Mo co c~ntren.Ta coday ~ A (xn)rn hQitlJ dogia thie'tA Iii toantli'compactddndi~u.GQix Iii gioi h<;tncuaday ~ A (xn)r,taco 10 Xn:$;A (xn):$; X.V~yx lamQtc~ntrencuai Xnr. Ta conphai chI ra XE Mo. TMt v~y,vi M dongne~XE M:Chon~ 00 trongba'tdangthac A (xn.) :$;A(x) ta., , , du'Qc X:$;A(x) v~yXE Mo. 2.Taco-S(x):$; 0 nenS bi ch?ntren.Ne'ux:$;x' thiiy : y E Mo"y ~x' rei y : y EMo,Y~xr,nen-s lahamgiam,dodoS lahamtang. ' Apdl,lngnguyen19Entropi,tatimduQcph~ntU'a E Mo saGrho : v x EMo,x ~a::::>S (x) =S(a). 3.TachungminhS (a) =O.Gia satnii l(,liSea)=2oc>0 khi do tatlmduQCXl E Mo,Xl ~asaGrho II A(x) -A( a) II >a. Vi Xl ~anenS (Xl)=Sea)=2a>O,dodota tlmduQcX2E Mo X2~Xl, II A(X2)- A(XI) II >a. L?p l(,li19lu~ntrentaxaydlJ'ngdlJ'QcdaytangiXnr c Mo saGrho IIA (XntJ- A (Xn) II > a. (\in=1,2,3,...)v~ydayi A (xn) rnkhonghQitl,lmallthu~nvoi giathie'tAla roan tll'compactdondi<%u. Ta chungminhdi~mb =A(a)'la di~mba'tdQngcuaA. Th~tv~y,do b ~a nen rheadinhnghlahaniS taco : II A(b) - A(a) II :$; Sea)=0~ A(b)=A(a) =b. Dinh19duQcchungminh0 HeQua2.'1.1.Giasa1.KIa nonchuin 2.A: -7 la roantadondi<%nva A( <uo,V0» la t~pcomp~ctuongd6i. Khi doA co di~mba'tdQngtren Chungminh: Vi K lanonchucfnenrheadinh191.2.1.la t~pdong,bich?n.Xetday tang {xn}n C ,ta c<inchungminh A( xn)nhQitl,l. Th~tv~y,VI A(<no,V0» 13.t~pcomp~ctu'ongd6i nenA(xn)nco dayconhQitV. SongVI {xn}n13.daytang,A dondi~unen{A(xn)n}dondi~ucochuadayconhQitvnen banthannoclinghQitV.Nghla13.A 1ato<lntli'dondi~ucomp~cdondi~u,theodinh1y 2.1.1.A codiSmba'tdQngtren0 . Hegila2.1.2:Giasli' 1. Kia nonchinhguy 2. A: -71atoantli'dondi~u Khi d6A co di~mba'tdQngtren' Chungminh: VI K 13.nonchinhguynenK cling13.nonchua"nva dodinh1y1.2.1.t~p 1ad6ng,bi ch?n. Ala ddndi~u,nenntu {xn}n1adaytangthl {A(xn)}cling13.daytang,vanonK chinhguynen{A(xn)n}hQiW.Nghla1aA 13.toantli'dondi~u,comp~cdondi~u.Ap dvng dinh1y2.1.1,A codi~mba'tdQngtren0 He gila2.1.3.Gia sli' 1.X 1akhonggianphanX<;l,KIa nonchua"n. 2.A: -7 13.toantli'dondi~u.. Khi doA codi~mba'tdQngtren. Chungminh: Vi K 1-anonchua"nentUdinhly 1.2.1.suyra 13.t~pdong,bi ch?nva 16i.DoX 1akhonggianphanX1acomp~cytu. X6tdaytang {xn}nc,tachungminh{A ( xn)n}hQitV.Th~tv~y. {A(xn)}nc6daycon {Ykh= {A(Xnk)}khQitvytuv~y. V8i m6if EX* tacof (yk) ~feym)(m ~k) rho m -700tadu'QCf( yk) ~ fey)v~yYk:;Y (\ik) 12 Tachungmint JimYk =Yk-+", Dodint19Mazur,V 8>0,-3Z =tlYkl +"" + tmY kmE C(yk)k : II z- y II <8/ N.Voi N Iii h~ngs6trongdint nghIanonchua:n. Ta d~t: Kg=max{kI,k2 km},taco Vk ~Kg : z ::;;Yk~ 0 ::;;Yk - Z ::;;Y - z ~IIYk-23II::;;Nlly-~11<8. V~ydaytang{A(xn)}coday canhQit1,1nenclinghQit1,1.Nghla la:A la:toantli' compacd(jndi~u.Theodint192.1.1.,A codi~mba'tdQngtren0 s=> "§2.Diim bili d{)ngcuatoantitddndi?utfJi hqn D,inhllgh'ia2.2.1 * T6antU'A : Me X ~ X duQcgQila compactdondi~utoi h;:tllne'umaiday iA n ( Xn) rn thoadi~uki~n A (Xl) ::;A2 (X2)::; An (xn)::; XnE M d~uhQit\l (1) Dinh If 2.2.1 GiasU' 1.M Ii'!t~pd6ngvabi ch~ntrongX 2.ToantU'A :M ~ M dondi~uvacomqactdondi~utoih;:tn 3.,T6n t<:tiXo E M saGcho Xo::;A(Xo). Khi d6A codi@mba'tdQngtrenM Chungmink D~tMo ={x EM, x ::;A(x) }tacoMo :f-~vaA (Mo) c Mo. Tren Mo tadinh nghladayde phie'nhamSnnhusail: Sn(x):=Supi II An (u) -An (v) II : u,v E Mo, x::;An (u)::;An (v) r , , T~pMex)=i (u,v) : u,v E Mo, x::;An (u)::;An (v) r:f-~VI x::; An (x)::; Ant-(x); Do v~ySn(x)xacdinh.Ngoairane'ux ::;x' thlM(x') c M(x) ne'uSn(x') ::;Sn"9' Taconh~nxet ill An+l(u)-An+l(v) II :u,vEMo,x::;An+l(u)-An+l(v) r ci II An(u')-An(v') II :u',V' EMo,x::;An(u')- An(v') rnen Sn+1(x)::;Sn(X).Do v~yt6nt;:tigioi h;:tnSex)=lim Sn(x) .n->"" vahamSex)cling1ahamgiamtrenMo Taseapd\lngnguyenlyEntropichot~pMovaham( - S) 14 1.Xetdaytang~xn~nC Mo; ta1~pbangvo h~nhai phis.sail : A (XI)~A2 (XI) ~...... . A (X2)~A2 (X2)~'".... ~ An (XI) ::0;..... ::;An (X2) ~..... A (xn)~A2 (xn)~...... ~An ({(.n)~..... . . VI ~ Xn~ n1aday tangnencacph~ntUlIenmQtCQt1adaytang(do Ala loantU'cion cii~u).Dov~ytac6daycheo~ An(xn)~n1aday tang,VI A 1aloan tU'compactcion cii~ut6ih~n,daynayhQitl,lv~ gia tri gi6i h~nXva Xn~ An (Xn)~X. Nghla1aX1ac~ntrencua~xn~ n.Taphaiki~mtradingx E Mo Th~tv~y,tac6A n(xn)~An+I (xn)~A (X)V n C~on =- ex>, ta cil1QcX ~ A (X), nghIa 1aX E Mo, . .. . 2. Ap dl,lngnguyen1yEntropitatlmcil1Qca E Mo c6 tinhchatx E Mo, X ;;:::a => Sex)=Sea) TachungrninhS (a)=0 . Gia sU'-S(a)=2a> 0 I VI S2(a) ;;:::S (a)>a nen1uont6nt~iu~' v1~E Mo saDcho a ~A2(UI)~ A2 (VI), II A2(UI) ~ A2 (VI) II >a VI A2 (VI) ;;:::a nen S (A2 (VI) ) =2ex:dov~yS4 ( A2 (VI) ) >a va ta Hm cil1QCU2"V2,E Mo sao cho A2 (VI) ~ A 4(U2)~ A 4 (V2) IloA4(U2)- A4 (V2) II >a Tie'ptl,lC1y1u~nhl1lIentasexaydl,l'ngcil1Qccacday~l\.. ~, ~vl<\0 ~cMosaDcho A2 (UI) ~ A2 (VI) ~A 4(U2)~A 4 (V2)~ ~A2n(Un)~ A2n (Vn) ~... (J.) 0) II A2n(un)- . A2n(vn)ll>a R6rangday(2)c6d~ng(1)trongciinhnghla,songdo(3)n6khonghQitl,l.Mau thu~nv6itinhcompactGoncii~ut6ih~ncuaA. v~yS (a)=O. 15 . , 3)Cu6icungtachungminhA c6di~mb:1tdQngtIeDMo. Ta c6 a:::;A(a) :::;A 2(a) :::;...... daynayc6d~ng(1)Denno hQit1,1 Ta d~tb = limA n (a) ta c6 An-(l~)~~o/-An (a):::;A (b) V (b) n ~ 1 n-->a:J n~nb:::;A(b). Ta l~ic6 a :::;A n (a) :::;A n(b)V nDen II A n (a)- A Ii (b) II :::; S (a). Do limS (a). =S (a).=on-->a:J ne~1im[A n (b) - A n (a)] =o n~CX) Cu6icungtub:1tdgngthuc 0:::;A (b) -b:::;A n (b)- A n (a)tac6A(b) =b. Chuy: Ne'utrongdinh1y2.2.1tagiunguyencacgiathie't1),2) con3)duQcthay b~ng3) ::3XoE M saochoA (xo):::;Xothl v~nc6 ~e't1u~n: "Khi doA c6di~mba'tdQngtrongM. " Trencosachliytrentachungrninhke'tquac6lienquailde'nroantU'16md~u Dinkngkla2.2.2 ne'u Cho lio'~(t,roantU'A duQcgQi1aUo 16md~utIeD I) A dondi~utIeD II) V X E ,::3ex>0, ::3~>osaoexuo:::;Ax:::;~uo. III) V [a,b] C (0,1),::35 =5(a,b)>osaochoV x E V.t E (a,b)thl A (tx) ~( 1+5 ) t A (x). R6rangcach~ngso'ex,pph1,1thuQcvaox,Ne'uA 1auo- 16md~uthlA ( tx)~t A tx) V t E ( 0,1),V X E 16 DinhIf.2.2.2 , Gia sti' 1.K 1anonchuin 2.Ala toantti'uo-16md€u tfen 3.u:::;Au;Av:::;v Khi doA codi~mba'tdQngtren Chung minh Do 3)maA « u,V» c ;KIa nonchuinnenla dong,bi ch~n.Ta sechi fa A la toantU'compactdondit$ut6i h?n * Giasu:3a,~>0 : a uo:::;u,v :::;~Uova~<1~ Th?tV?y,ne'uu,v khongco tinhcha'trenthltUdi€u kit$n(ii) trongdinhnghlaA la Uo-16md€u suyfa,:3a,~>0 : , a uo:::;Au; Av:::;~Uo. Ta d~tuj=A\.I;Vj= AV- Ta co ocuo:::;uj,VI:::;~uo,a <1 ~ Uj:::;A Ul AVi :::;Vj DoK lanonchuinnendongbi ch~n4>\;jx E :3M >0 : Ihll:::;M " * A la toantti'comp"actdondit$uWih?n.Th?tV?y,Gia s11( xn)n C thoadi€u kit$n:A (Xj)~A 2(X2)~ A n(xn) ~ (*) tasechifa ( * ) la dayCauchy( khidosehQitvVIX la khonggianBanach) La'yE>0 dubed~ a <1-~ (N - h~ngs6chuincuanonK) ~ M.N DoA la uo16md€u nen:38>0 saccho\;jx E . 'v't8- [a ,1- 8 ] tacoA (tx) ~( 1+8) tAx ~ M-N, , 17 GQinola s6tlfnhienrheadieuki~n 0:(1+8)no-1:::;1- ~ ~ . MN 0:(1+8)no>I..!... ~. .MN' Bangeachgiams6(), tacothecoi 0: ~ (1+()to <1 Ta.chungminh\:fn:2:no,.\:fkEN thl II A n+k(xn+d II - An (Xn )11<8 Do A k(Xn+k)E va XE nen a.~ ~A k(x n+k)va Xn~p .uo k A (Xn+k) . Xn xn k 0: 2:uo2:-:::::>-:::::> A (Xn+k)2:-xn, 0: ~ ~ ~ ;ac6A ,.. (x..,):>A(;x. )"(1+0); x, A k+2(Xn+k)2:A[(1+8) 0:A(xn)] 2:(1+8)20:A 2(XJ - . - P. P Dodo Ak+no(Xn+k)2:A[(1+8)nO-I.CXAn.1 (xn)]2:(1+8)noexAnO(xn) . P P A n+k(Xn+k) = An-no [A k-no(Xn+k) 2:An-no [(1 + 8)no CXA n(Xn) J . P 2:(1+8)no0:.An(xn)2:(1-~)An(xn) P MN Ke'th<;lpvoidieuki~n A'n+k ( . )<A n ( )t ' . Xt>+k - xi, aco E 0:::;A n(Xn)-An'-k (Xn+k):::;-A n(Xn).Dod6 MN lI~n(Xn)-An+k(Xn+k)I/s~.8 IIAll(Xn) 11~~.M=8M M.N V~y(*) ladaycauchy.Ap dl,mgdinhly 2.2.1,A codiemba'tdQngtrenO 18 t(f §3:D~~mba'tdQngcua to~intdddn di~utren khong gianvoi nonMinihedral- m~nh x IiikhongianBanachthvc,s~pboinon.MinihedraIm<;lnhK. Tacoke'tquanhu sau: DinhIf 2.3.1 Gia s11' 1.A :--7 Iii toant11'dondi~u 2.IeIiinonrninihedralmc . Kh,idoA codi~mbatdQngtren. Chungmink D?tB=i x E :x ~Ax .~. Khi doB:;t~vi ue B. Anh XB,vi V x E B :x ~Ax, IAx E B V X.E B. TasechIram6it~pcons~ptuye'ntinhtrongB d€u coc~ntrenthuQcB. Th~tv~y, Giii s11'C Iii mQtt~pcons~ptuye'ntinhtrongB. Ta thayC bi ch~ntrenboiV. Vi K Iiinonniinihe9raImu ~Co ~v. V X E C tacox ~Co, A dondi~u,nenA (x)~A (Co)=>x ~A (x) ~A (Co)dodo A (Co)Iii ffiQtc~ntrencuaC nenCo ~A (Co)(do dinhnghlasupremum)=>Co E B. Theob6d€ ZorntrongB coph~ntli'toi d<;lix*. Ta se chungminhx* Iii di~mbat dQngcua'A. . Tacox* E B nenx* ~A (x*). LA (x*) E B =>A (x*) ( vi x* -I?h~ntli'toidA (x*)=x*.O f? 19 ._.