Luận án Sự tồn tại nghiệm của bao hàm thức vi phân dạng cực biên

~2 CACBAOHAMTHOCVI PHRNTHUONGc6 GIA TAlKHONGLal TRONGKHONGGIAN BANACH. Phan1.KHAo SAT SU TON TAl NGHIEM CUA BAO. . . HAM THUC VI PHAN THUONG DANG CUC BIEN. . . ChoE Ia mQtkhonggianBanachthl;tc,kha1y,phanx~vaF 1amQthamda . triduQcdinhnghlatrenmQtt~pconmd,khacr6ngcuaRxE vaoE co giatri 16i,compacvakhacr6ng. Phffnnaytrlnhbaysl;t6nt~inghi~md6ivoibaahamthucvi phanthuong d~ngqic bien. { X1EExt F(t,x) xeD)=Xo Y tUdngsad\lngph~mtrUBairechocacbaahamthucviphantrenR duQc Cellinaduaradffutienvaonam1980.Saudocorfftnhi€u k€t quaUrn duQC dl;t~.trendiM 1:9ph(;lmtrUnay.Nhungk€t quathuduQcsau1~inhovaomQtleY thu~tmoiIa sad\lngphanho(;lchtrenmi€n t~oanhcuaF, saudoapd\lngthich hQpdinh1:9ph~ngiaokhacr6ngcuamQtdaygiamcact~pcompackhacr6ng. Fhatbilu kit quachfnh: Niu F compacva lient~c(haytangquath{fn: niu F compacvathoadiiu ki~nCaratheodory)thit{jpnghi~mcuabailoan Ia khacrang.Kit quatlt{fngt1/niu F lalf1r;mga-Lipschitz,a Ia kj hi~udQdo khongcompacKuratowski. ~ ~ 14'cO' I ?;:, 1. MO'dau: I ! X6tmQthamdatri: ;p : I xX ~ WeE) -10- ddayI =[0,T] ;X =BE (xo.r) ;XoE ~(E);r>0 TanoiF thoa(H)neu: (Hi) F falient1J.ctrenI xX (H2)tQ,pA =F(l xX) facompactrongE (H3)0h (A,0) Voi F thoa(H),va XoE Xo,taxetcaebai toanCauchysail : { X'E F(t,x) xeD)=Xo (2.1) { X'E ext F(t,x) xeD)=Xo (2.2) Cho XoE Xo, kY hi~u: SF ={x:I ~E/x zanghi~mci'ta(2.1)} SextF={x:I ~E/x langhi~mci'ta(2.2)} SFIa compac,khacr6ngtrongCE(I)nenkhonggianSFtrangbi metrichQi tvd€u Ia duo f)~t9JF={f:IxX~ E /f latatcatlientlJCci'taF} Lty f E .9'lp,xetthembai toanCauchy: { x'=f(t,x) xeD)=Xo NSu d~tPf={x: I ~E/x la i nghi~mci'ta(2.3)} (2.3) thlPf Ia mQtt~pconcompac,khacr6ngcuaSF. Lty mQtday {In}C E* co Illnil= 1trUm~ttronghlnhc~udonvi cuaE. Ta dinhnghIatheoChoquetham<PF:I xX xE ~ [0,+oo[nhu'sail: <PF(t,x,v)= f (In(v)J n=l 2n V E F(t,x) +00 vEE\F(t,x) .G9iQ1/Ia t~pttt ca.caehamaphintuE vaoR. Tieptheoxayd1,1'ngham ip; : IxXxE ~ ] -00,+00[: IPF(t,x, v) = inf (a(v)/ a E Q1/vaa(z)> C{JF(t,x z) vdiz E F(t, x)} -------------------------- -=-~r~~~~;i;~i~~;----------- BaygiOtadinhnghla: dF: IxXxE ~ ]-00, +00[ dF(t,x, v) =iYF(t,x, v) -qJF(t,x, v) thltacocacke"tquadu<;cnha:cl<;litrongm<$nhdS sau: M~nhd~2.1 ChoF thoa(H),taco.. (i) Vaim6i(t,x) E I xX vav EF(t,x),taco0 :s;dF(t,x,v):s;M2.HCInniia d~t,x,v)=0khivachikhiv EExtF(t,x). . (ii) Vaim6i(t,x) E lxX,dF(t,x,0)ia lOmng(ittrenF(t,x) valOmtrenE. (iii) dFianllalientl;lctrentrenIxXxE. (iv) Vaim6ix E SF,hamt -f dF(t,x(t),x'(t))ia kh6ngam,giainQivakhd richtrenI. (v) Ntu day{xn}CSF hQitl;ld~utaix E SFthi taco.. fdF (t,x(t),x'(t))dt;:: Jim supfdF (t,x(t),x'(t))dt I n~oo I 2. Phanho~chtrenmi~nt~oanh Ta selamquellvOimQtphanho<;lchtrenmiSnt<;lOanhcuahamF vathie"t l~pffiQtsf)tinhcha'tcuacaephepphanho<;lchnayd€ sadl:mgtrongph~nsau. 'ChoF thoa(H).f)~t:C=Xo +Vet-a) coA tel thl c c X VaC E «?f(E) (2.4) Dinh nghia: QI\f={x..I -f C/ x iaLipschitzvaihling<M} Ql\flamQtt~pcon16i,compaeeuaCE(I)chuaSF. ~inhnghia2.1: :Cho F thoa (H) Va}i E E* ; IIIJ =1,i =1,...d ; a> 0 Cho{Ik}Z:\lamQtphanho<;lcheuaI cobuckp. ------------------- ___om----- ----- ---------- -12 --- Vdik=1,...kovah=(hI....,hd)E Zd,d~t: R: = {(t,x) E IxC It Elk, hia ~IiCx)-2Mt<(hi+ l)a, i =1, ...d}, (j dayM Iah~ngs6trong(H).HQm tfftcacact~pkhacr6ngR: duQcgQiIa IDQtphanho~~htrenmi€n t~oanhcuahamF tuonglingvdi {Ii}~=1va {Ik}~:1 cuabudckhonggiana vabuocthaigian~.DuongkinhIOnnhfftcuacact~p R: khiR: thayd6itrenm duQckyhi~uIa v(m)vaduQcdinhnghlaIa chu§:n cuapll.(Luuyr~ng,m Ia phanho~chuuh~ncuaIxC nghlaIa mIamQt hQ hii'llh~ncact~pdoimQtgiaonhaukhacr6ngvacohQpb~ngIxC ). 'BaygiataIffy9;={ldc E*, II Ii II =1Ia mQtdaytritm~tronghinhc~u donvi cuaE*.Voi m6in E N, IffyRn={R: (n)}Ia mQtphanho~chtrenmi€n t~oanhIxC cuaF, tuonglingvoi {Ii}:l va ~:tn=lcuabuockhonggianan,va budcthaigian~md dayan->0,~n->0khin ->00. F.S.DeBlasivaG.Pianigianidachirar~ngconoE N saGcho'v'n>nothi tacov(plln)<A.HonmIa,concob6d€ quailtrQngsau: B6d~2.1 Cho F thoa(H). Lily 8> 0 vaa > 0 thi c6 mQtphan hOc;lChmcho mi~n IxC,cuaF tu<JnglingveJi{IJ~=1va {Ik}::1 cuabu(Jckh6nggiana vabu(Jcthai gianj3, 0 <P<min{8II I, 813M};wYichudnv(m)<AsaGcho m(!!.;J<8 /1/ K ={l,...,ko} WJi mQix E QJ\f trongd6 va Kx={kEK:c6h EZsaochograph xlkcR:, R: Em}. Tub6d€ nay,dethffyKxIakhacr6ngntu0<E<1. Chopll={R:} Ia mQtphanho~chtrenmi€n IxC cuaF thoacactinhchfft da neu trongb6 d€ 2.1. Voi m6i R: E pll, hay xet mQtphanho~ch ~J~=lE @(Ik) (d dayp =p(R:), vaIkIakhoangthaigiancuaR:. h h" D~t Rk,j =Rk n(JjxE) j=I,...p --------------- -- -- ----------------------- -13- GQiJj 1akhoangthaigiancuaR:,j. La"yg(' 1ahQta"tcacact~pkhacr6ng R:,j nhu'trenkhi R: thayd6i tren g(.R6 rang g(' cling1amN phanhot;lch euaIxC va g(' 1amQtst;lammillcua g( . { C a E111 }Ti€p thea,d~tPo=mill -:;-, 2M' 2koPo voi Co=mill{I Jj 1/ R:. E g('},j Po=max{p(R:)/ R: E g(}.va 'La"y0<~<~o.Voi m6iR:,j E g(',dinhnghia: h h Rk .(fl) ={(t,X)ERk. /tE Ij(fl), hia+flM slJx)-2Mt 5{hi+l)a-flM,J J i =1,...d} (2.5) . trongdoJj(~)=[tj+~,tj+l- ~]; haidiSmtj <tj+l1acaediSmd~uva cu6icua khoangthaigianJj cuaR: ..,j Dinhnghiatreneonghiakhi 0 <~<mill{Co/ 2,a / (2M)}. Ky hi~ug('~1ahQta"tca cact~pkhacr6ng R:,j (~) khi R~,jthayd6i treng(', vad~t A~= U R:./,u) R;'j(ll)e!r{, (2.6) TITxaydt;lngtren,cothSchungminhk€t quasau: B6d~2.2 ChoF thoa(H).Diy 0 O.Chog( la m(Jtphtinho(fchtren mi~nt(fOanhIxC cuaF vaicaetfnhchtltdll(Jcneutrongb6d~2.1.Vdi g(', g('Jl vaAJlnhllJ trenthitaco.. m(I\Ix)<2 &/1/ ttxE cYV trongdoIx={tE1/ (t,x(t))EA~. H(fnriTlaAJlla tqpconcompackhacr6ngcuaIxc. .. 3. Caeke'tquaerunh ---------------------- --- -- ----- -----_n_n_-n_n_n--- -14- Baygiotasel~n1U<;ftxemxetcacb6d~chinhtruockhid~c~pWidinh1,9 t6nt(;lichobii loanCauchy(2.2)- mQtke'tquad\l'atrendinh1,9ph(;lmtrll Baire. ChoF thoa(H)vi 8>O. D~tSe={~ESF/ fdF (t,x(t),x'(t))dt<8} I K€t quad~uliendSsuydU<;fcngaytuvi~capdlfngm~nhd~2.1(v),doIi: :.B6d~2.3 Cho F thoa(H), V(yim6i ()>0,ttJpSBIa miJ trongSF' - B6d~2.4 ChoF thoa(H).LtJyf E.9Pva()>0,5>othicomQthamg EgpsaDcho VxE Q.I'\I;ta co .. fdF (t,x(t),g(t,x(t))< () I (2.7) va I supII f[g(s,x(s)) - f(s, x(s))]ds II<6 leI 0 (2.8) "Chungminh: La'y f E .9F,8 >0,8>0,. ChonEsaocho0 <E <mill { e 2 I I' 0 I I' I} , trongdoM Ii . 4(1+M ) I 2(1+4M) I h~ngs6trong(H). D~tZ=IxX Bl/fJc1 : XtJpxl dtaphuongf biJi cachamcogill trtgdnvdicacdiimq(Cbien cuaF. .ta'y (s,u) E IxC, (C du<;fcchotrong(2.4)).Do f(s, u) E F(s, u) nen theo dinh1,9Krein - Mil'man's,co mQts6Ps,uEN; co cac di~mv~u E extF(s, u),, , '- 1 " , "",\j . 0 ,\j <1 '- I h? ,\1 ~Ps,u_ 1J - , ...,PsU vacacso '" . <'" -, J - , ...,Psu t oa '" + ... + /\,s U -, s,U s,U 's,U' saocho: -- --- - ----------- -15 --- P"u" 8III AJs,u V~,u - res,u)11<- j=l 3 (2.9) Theom~nhd~2.1(i va iii) thldFla u.s.eva dFtri~ttieut~i(s,u, v~,u), VI v~yco 0<p;,u <~ sac ehovoi m6i (t, x) E Bz((s,u),p~,u)va m6i v E BE(V~,u'P~,u),j =1,"'Ps,u,tacodF(t,x,v)<E. (2.10) VI F lientlfet~i(s,u) va vj E F(s,u)nenco 0 < pI < pO sacehovoiS,u S,u s,u m6i(t,x) E Bz'(s,u),pI ), taco:F(t,x)n BE(Vj , pO ):f:.0,j =1,...Psu., ~ s,u s,u s,u ' .Theodinhly Michael,secoPs,uhamlienlientlfe: zj :Bz,(s,u),pI )~ E sacehovoi m6i (t,x) E Bz,(s,u),pI )taco :s,u ~ s,u ~ s,u j - (j 0 )Zs,u(t,x) E F(t,x) n BE vs,u' pS,u . . Ke'th<;$pvoitinhlientlfeeuaf t~i(s,u)vatu(2.10),suyfaco 0 < pS,u < P~,u sac ehovoim6i(t,x) E Bz'(s,u),p ), taco:~ S,u II f(t,x)-res,u)II < ~3 (2.11) IIZ~,u(t,X)-v~,ull<~ dF(t,x, zj (t,x» <ES,u j=l,...,ps,u (2.12) j =1,...,Ps,u (2.13) Theoeachxaydvngtrenthlcaehamz~,unh~ncaegiatrig~nnhungdi&m evebieneuaF(s,u)vaxa"pXlg~ndunghamf (theo2.13,2.9,2.11,2.12). B1iuc2: Xayd1!ngmQt[atcdtr khonglient1jCcuaF trenIxc. HQ {Bz((s,u),ps,u)l(s,u)ElxC}la mQtphumdeuat~peompaeIxC, VIthe' nocophilconhUllh~n: {BJSn' Un),P'n'"n)::,} (2.14) .GQi A.>0IamQts6Lebesgueeuaphilconnay.Theob6d~2.1,ehoEvaA., secomQtphanho~eh:97l={R~}theemi~nt~oanhIxC euaF, tudnglingvoi {lJ ~=Iva {Ik}::0 ..eua buoe khong gian a va buoe thai gian p. 0<P<min{EI I I , a / (3M)},thGav(:~)<A.sacehoba"td~ngthlietrongb6 d~ -------------- - - - n_---- Un------- n -- --- -16- . 2.1dU<;fethoa.R6 rangm6i R~ E fill d€u dU<;feehliatrongit nha'tmQthinhe~u euahQ(2.14).DoeachxaydlfngnenR~coduangldnhnhohonhin 'A. BaygiGtal~y<D: fill ~ N la mQthameholingvoim6i R~E fillmQtva emffiQts6nguyene6dinhtheomQtquit5etuyy giuacaes6 tlf nhien n, 1~n ~nosaGeho Rkh C Bz(smun),Pn), (ky hi~uPn= P )Sn'Un R: E fillvagiasa <D (R:)=nthi(2.15)thoa. (2.15) La'y Theobuoe1,comQtPn= pEN; coPns6 'Aj='Aj voi 0 <'Aj ~ 1, Sn'Un n Sn'Un n 'AI+'A2+A +'APn= 1; va co Pn di~m vj = vj E ext F(sn, un) saGeho n n n n sn'Un Pn . . II 8" '\ J VJ - f(s U) <.- L.../"nn n' n 3j=1 D6ng thai co Pn lat e5t lien tve z~ = Z~n,Uneua F dinh nghIa tren Bz((smun),Pn)saGehovoi m6i (t,x) E Bz ((sn,un),Pn),taco: "f(t,x)- f(sn'un)"<~3 (2.17) Ilz~(t,x) - v~II <~, j =1,..Pn (2.18) dp(t,x, z~(t,x))<8 , j =1,..Pn (2.19) BaygiGtaxaydlfngmQtlate5ty coth~khonglientveeuaF trenI x C. La'y R~ E fill ba'tkY va (R~)=n. La'yPmJJn,v~va z~,j =1,..Pn,tuonglingnhutren.GQilkla khoangthai gianeuaR~.xaydlfngphanho<;leh{Jj }~~1E @{lK) co dQdai IJjl =JJn Ilkl j =1,...Pn. (2.20) vad~t R~,j= R~n (Jj x E),j =1,...Pn, tasecophanho<;lchfill' la hQ~~et~pRL khaer6ng. DiM nghIay : I x C ~ E bdi Y (t,x) = Z~(t,x)n€u (t,x) E R~,j' -- - ------ - - -- ---- -17- . thldinhnghlanayIa xacdinh.Honm1adocachamz~,j =1,..Pn, 1acac1at dt lientt;[cuaF trenBz((sn,un),Pn)::) R~,jneny 1amQt1atcatco thSkhong lientt;[cuaF trenI x C nhu'ngthuhypcuano trenm6it~pRL Ia lien t1;lC, Blluc3 : Caetinnchatcuahamr Tircachxc1ydltnghamy voi m6ix E Q/V,tacodu'<;5C: Jd F (t,x(t), y(t, x(t)) dt<() I 2 (2.21) 'va I sup II J[y(s,xes))- I(s, xes))]ds ll <5 leI 0 2 (2.22) Tasechungminhtinhcha'tnay. Do-mdu<;5cxftydltngthoab6d€ 2.1nentaco: m ( ulk ) <E I I I keK\K" (2.23) trongdo K ={I, ..,ko}' vaKx={kE K / coh E ZdsaochographXlkC R~,R~E p/(.} VI 0 < E < 1 nenKx Ia khacr6ng,suyra co mQtR~ E p/(saocho graphXlkC R~,vavdi 1::;;n ::;;no,(2.13)dU<;5cthoa.Honnua,1u'uycuaBu'oc2, taco: graph XI C R~J' C BzC(sn,un),Pn)j = 1,..Pnk ' (cac {Jj};~1va RL du<;5cxftydltngtrongBu'oc2) (2.24) vay(t,x(t))=Z~(t,x(t)), t E Jj, j = 1,..Pn. Truoche't,voik E Kx,taco: * Jdp(t,x(t),y(t,x(t)))dt < Ellkl Ik * IIJ[y(t,x(t))-f(t,x(t))]dtll<Ellkl Ik .. Th~tv~y,vdik E Kx thl: ---------- ---------------------------------------------------------------------------------- -18 - Pn . * Jdp(t,x(t),yet,x(t))dt=I Jdp (t,x(t), z~(t,x(t)))dt lk . j=lJj <8 IIJjl =81Ikl, j=l theo(2.19va2.20) * "/fr(t,X(t»- !(t, x(t»]dtll=II~Jrz~(t,x(t» - f(t,x(t))]dtJ '~I J[llz~(t,x(t))-V~11+ Ilf(t,x(t))-f(sn'Un)IIJdt+ I J[v~-f(sn,un)]dt j=l J. j=l J.J J < t(~+~)IJ;I + t(v~-f(Sn,Un»)IJ;1 = = ~"II,I++ II~AJnv~- f(sn'Un)IIII,1 < 81Ikl theo(2.24,2.18,2.17,2.20,2.16) Lffyx E QJ\f.SadvngcaekStquatren,taco: 01<JdF(t,x(t),y(t,x(t))) dt I = I Jdp(t,x(t),y(t,x(t))}dt+ I Jdp(t,x(t),y(t,x(t)))dt keKxlk keK\Kxlk < I81Iki+ IM21Iki <8III+8M2III keKx keK\Kx . e e < - (do8< 2 II ), suyra(2.21) 4 4(1+M )I t * lIJ[y(S,X(S))-f(s,X(S))]ds ll ~ I II J[y(s,x(s))-f(s,x(s))]dsl!+ 0 keKx lk + I II J[y(s,x(s))- f(S,X(S))]dS II + ]1 y(s,x(s))- f(s,x(s))llds keK\Kx lk lie .. Nho l<;tiding IIiel=~<8 III, kSthQpvoi (2.23)taduQc -------------------------------------------------------------------------------------------------- -19 - tI[Y(S,X(S»-f(S,X(S»]dsll<k~x8lIkl+kEt~MIIkl +2M81I1 <8 III+28MI~+28M I~< ~ (dO8<2(1+~MJIIIJ Vl t E I tuyy nensuyra(2.22). Bu:fjc4 : Xfiy d1!ngmf)t[at cdt lien tIJCg cua F biing xdp xl 1- ThuhvphamylenmQtt~pcompacthichh<;1pAll c I x C SeHimchoytrd t4allhmQthitdt lientl,lccuaF trenAwTheodinhly MichaelthiycomQtmd rQnglient\lCgclingIa mQtlatcfitcuaF. Ta seth1ydinghamgnhu'v~ythoa cactinhch1tdilneutrongb6d~. L1yg{vag{'nhu'dtrongBu'dc2. D~t~o=mill {co/2, a / (2M),8111/ (2kopo)}, trongdo Co=mill{IJjl/ RL E 9['}, Po=max{Pn/ 1::::; n ::::;no}. co'dinh0 <~<~o. f!ll;la hQt1tca cac t~pR~,j(~)khacr6ngchobdi (2.5)khi R~,jthayd6i tre1).,g{' va All du'<;1cdinh nghlabdi (2.6).Theo b6 d~2.2 thi All Ia t~pcon compackhacr6ngcuaI x C.DoythuhvptrenAllIa mQtlatcfitlient\lCcuaF nentheodinhly MichaelcomQtlatcfitlientl,lcgcuahamF saocho: get,x)=yet,x) vdi m6i (t, x) E All Ta l1yx E <2/Vtuyvad~tIx={tE 1/ (t,x(t)E All}thitheob6d~2.2: m(I \ Ix)<28 I IJ . (2.25) fdp(t,x(t),g(t,x(t»)dt::::;fdp(t,x(t),y(t,x(t»)dt+ I I Taco + ~dp(t,x(t),g(t,x(t»)- df(t,X(t),y(t,x(t»~dt I\Ix 8 8 <- +M2m(I\ Ix)<- +28M21I!<8 2 2 .. theo(2.21va2.25) Nhu'v~y(2.7)dil du'<;1Cchungminhxong..---------------------------------------------------------------------------------------------------- -20- vOim6itEl, taco: t l[g(S,X(S»- f(s,x(s»]dSII~ flg(S,X(S»- y(S,X(S»11ds+ t + lIf[y(S,x(s»-f(s,x(s»]ds D < ~lg(s,x(S»- y(S,X(S»11ds+ () I\I 2x () () < 2Mm(I\ Ix)+- <4EMIII+- <() 2 2 theo(2.23va2.22). Suyra (2.8)clingduQcchungminhKong. B6 d~2.4duQcchungminhhoantoan. B6d~2.5 ChoF thoa(H),! E 9Fva B> 0 thzco 5 =5jB) > 0 saDcho v<Jix E SF btitkY: I sup II f[x'(S)- !(s,x(s»]dsll < 5suy TaX E BSF(Pfi B) leI C (2.26) Chungminh: Giii sa phat biSu tren khongdung thl co f E SF , E > 0 va mQtday {xn}C SF\ BSF(Pt, E) thoa sup 'I S[X1n(S)-f(S,Xn(S»]ds ll < l}n EN tel 0 n Do SFla compac,COmQtdayconcua {xn}hQitlJ d~uWi mQtdiSmx E Pi, VI v~yvdi n du lOn,ta co XnE BsF(Pt, E),mall thu~ngiii thie't.B6 d~duQc chungminh. .. .------------------------------------------------------------------------------------------------------- - 21- B6d~2.6 ChoF thoa(R). Chof E SFva Ii> 0, e> 0 thicomQthamg E SFsaGcho Pg c::San BSF (Pfi Ii). Chungminh. Lffy f E SF,E >0,8>O.Tli b6d~2.5suyraco8>0 d~(2.26)xayravoi x E SF.Khi do,theob6d~2.4thlcog E .9F(tudngungvOif, 8,8)saDchovoi m6ix E <2/Vtaco: fdF(t,X(t),g(t,x(t)))dt<8 I (2.27) va t sup II J[g(S,X(S))- f(s,x(s))]dsll<8 tel 41 (2.28) Voi m6ix E Pgtuyy,dox'(t)=get,x(t)),tEl nenke'th<;1p(2.28)va(2.26) tasuydu<;1cx E BSF(Pf,E).Con (2.27)suyra x E So.V~yx E Sen BSF(Pf, E). . Doxtuyy nenPgc Sen BSF(Pf,E).B6d~duQcchungminh. Bay gia chungta chuy~nsangchungminhdinhly 2.1,clingla ke'tqua chinhcuaph~n ay. .DinhIf 2.1 ChoF thoa(H),f E.9},Ii> 0thiSex!F n BsF(PftIi)~0, vadij.chi?t,hai loanCauchy(2.2)conghi?m. Chungminh: ~ 1Bat 8 =- n E N. n , n Tli b6 d~2.6,voi 81> 0 da choco g1 E .9}: Pg!C BSF(Pf, E).Do Pg! Ia compacnenco0 <111<81saDchoBs (Pg ,111)c Bs (Pf,E) (2.29)F! F . Tudngtl,l'co g2E .9}: Pg2c So!n BsF(Pg!' 11)'So!md theob6 d~2.3. B6hg thai Pg2Ia compacnen l~ico 0 < 112< 82saDcho Bs (Pg2,112)cF So! n BSF(Pg!,111)'Tie'pt\lCthvchi~nta dU<;1cmQtday thong tangcac t~p --------------------------------------------------------------------------------------------------------- -22- compackhac r6ng BSF(Pgn' Tln)cua SF voi gn E 9F va 0 <Tln <en, thoa Bs (Pg , Tln+l)CSa nBs (Pg , Tln),n ENF n+! n F n La"yx E SFIa mQtdi~mthuQCvaom6i t~pBs (Pg , Tln)thlx E BsF (Pf, 8). F n theo(2.27).Hqnnlla,dox E Sanvoim6in ENnen fdF(t,x(t),x'(t))dt=0 I Yl v~y,nhomt$nhde2.1(i),suyra x'(t) E ExtF(t,x(t))h.k.n,tucIa bai loanCauchy(2.3)conghit$m. 4. Md rQng Tuynhien,k€t quavesvt6nt<:iivatrum~tchobailoanCauchyd<:ingqtc bien(2.2)moiduQcchungminhduoigiathi€t hamF la lient1,lc.Cac lacgia damdrQngk€t quachinhvuatimduQcchotrttonghQpt6ngquathan: F thoa . mangiathi€t Caratheodory. TanoihamdatriF (dachotrongph~nmdd~u)thoa(H')n€u (H'1)vfJimtiitEl, hamdatrix --+F(t,x) ia lientlJctrenX, vavfJimtii x EX, hamdatri t --+F(t,x) ia dodllf/Ctren1. (H'2)tfjpA =F(l xX) zacompactrongE (H'3)0h(A,0) ChoF thoa(H') va SF,SextF,Sava GVVduQcdinhnghIanhud ph~ntrUoc. T~pSFIacompac,khacr6ngtrongkhonggianCE(I),suyraSFvOimetrichQi1\1 deula khonggiandu. Lat c~tf cuaF duQcgQiIa latc~tCaratheodorycuaF n€u vOim6itEl, hamx -+f(t,x)Ia lien1\1ctrenX, vavoim6ix E X, hamt -+f(t,x)Ia doduQc Bochnertren1.Voi F thoa(H'),d~t 9'F ={f:I xX -+E / f Ialatc~tCaratheodorycuaF} Theocacdinhly cuaScorzaDragonivaMichael,t~p9'F la khacr6ng. .Lu'uY dingF thoa(H')thlcactinhcha"t(i)(ii)(iv)(v)trongmt$nhde2.1 deuduQcthoa.Hannlla,khithayhQSFthanhS'Ftuangungthltaco duQCta"t cacack€t quatrongb6de2.3-+2.6voi ly lu~nchungminhtuangtV.Rieng ph~nchungminhb6"de2.4sad1,lngdinhly ScorzaDragonid~thuhypcua hamF vaf trent~pJ x X Ia lient1,lc,voi t~pJ c I thoamQts6dieukit$n;sau dosad1,lngmQtdinhly mdrQnglient1,lcuahamdatrid~comQthamdatq ----------------------------------------------------------------------------------- -23- . compac,lien t1;IcF : I x X ~ '$tE)va leitGifttu'dngling f. K€t hc;fPvdi b6 d€ 2.4va m~nhd€ 2.1,ta thudu'c;fCke'tquac~ntim.Tli'cac ke'tquanay,19lu~n tie'ptheotudngtl!taclingthudu<Jcke'tquasau: DinhIy2.2 ChoF thoG;(R') thibili toanCauchy(2.2)conghifm. HayxetthemmQtmdfQngd6ivoihamF.Tan6ihamF thoa(K) ne'u: .(Kj) F Ia lientlJCtrenI xX (K2) TijpA =F(I xX) Ia gifJinl)i,tacIa h(A,0)<M, vacoml)thlingso" L >0saochoa(F(I x Y)) 5{La(Y)vfJim{JiY eX (Kj) 0<T <min{riM,l/L} Cacb6d€ 2.3~ 2.6v~ncondungne'uthaygiathie't(H)bdi(K).Tli'd6su d1;Ingdinh192.1c6th~chungminhdinh19sau: Dinh Iy 2.3: ChoF thoa(K) thibili toanCauchy(2.2)conghifm. Cu6iclingtaxetthemmQtke'tquaungd1;Ing .Voi XoE Xova tEl, d~t : (filp(xo,t)=(x(t)/ x..I ~ E litnghifmcua(2.1)} c£f1extF(Xo>t)=(x(t)/ x..I ~ E litnghifmcua(2.2)} DinhIy 2.4 ChoF thoa(R'). Ntu coa thoa0 a) cXo WYim6i XoEX thibititoangiatrtbien:' { X(t)EExtF(t,x) x(a)=x(O) (2.30) coitnhfftmQtnghifm. .Chungminh:Lty f E !}J'F Trudehe'ttachungminhbai toansauc6 nghi~m: { x'=j(t,x)" x(O)=x(a) (2.31) ---------------------------- --------------- -24- Tit dinhIy ScorzaDragoni,c6 mQtday {In}cact~pcompackhacr6ng Inc I, InC In+bnE N, vam(I \ In)--+0 khin --+00,saorhothuhypcuaf Ien InX X lientl,lC.Vdi m6in E N, Iffy<Pn: I x X --+E Ia mQthamLipschitzdia phuongc6giatritrongcoA saorho . 1 sup l<Pn(t,x)- f(t,x)1< - (t,x)elnxX n TheoCellina,vdi m6i8>0 c6 mQtnoE N saoeho~n (Xo,a) c BE(Xo,8), \in .;?: no' Tti d6 xay dvng day con {<Pnk}eua {<Pn}sao rho nxI, (Xo,a)c B(Xo,lIk), kEN. TheodinhIy Kakutani& Fan,vdi m6ikEN,1nk hamdatri u --+P4 (~nk (u,a), Ilk) n Xo e6mQtdi~me6dinhUk.Suyra vdi m6i kEN, e6mQtnghi~mXkcuabai loan x' = rpm(t,x); xiO) =Uk sao eho IIXk(O)-Xk(a)II< ~.Doday{xdc SFIa compae,e6mQtdayconhQit\1d@uWi x naod6 E SF. R6 rangx(O)=x(a) E Xo.Honnuax Ia mQtnghi~mcua(2.31), . VIvdi m6it E I, tac6 : t X(t)-X(O)-!f(s,X(S))dsll ::;;CK+!llf(s,X(S))-<Pnk(S,Xk(S)~1ds (2.32) trongd6 Ck=sup Ilx(t)-Xk(t)II+ Ilx(O)-Xk(O)II, tel VavfSphaicua(2.32)--+0 khi k --+00. TifSpthea,sird\1ngcacb6 d@(2.6),(2.3)va Iy Iu~ncuadinhIy 2.1,taxay dvng mQt day giam cac t~p con compackhae r6ng Bsp (Pgn+l' TJn+1) c Selln Bsp(Pgn' TJn),n E N. Vdi m6i n E N, Iffy XnIa mQtnghi~meuabai loan x' = git, x); x(O)=x(a). Do {xu}compacnenmQtdayconhQit\1d@u tdi x nao d6 E SF'R6 rangx(O)=x(a),va x E SextF do x n~mtrongm6i t~p Sa 'n n E N. V~yx Ia mQtIoi giaieuabailoangiatrievebien(2.30).DiM Iy du<;Jc chungminh. Chziy: 'DinhIy khongcondungnuanfSuthaygiathifSt(H'l) bdi"F Ia niralient\1C trenlIenI xX". .. ----------------------- --------- -25 - ... -- ? Phan II. MQT TINH CHAT D.L).CTRUNG CUA NGHI~M BM ToAN CVC BIEN M\lc dichchinhcuaph~nnayla tlmmQtd~ctru'ng2,(x)chonghi~mcua baitmlnqic bien: x' E ExtcoF(t,x); x(O)=xo, d dayF lahamdatriU.s.C.,cogiatqcompac. (2.33) Chox lamQtloi gi~Hcuabaitoan: x' E coF(t,x) " x(O)=Xo. (2.34) Ta da:bitt dingnSuF la Lipschitzthl t~pnghi~mcua(2.23)la trUm~t trongt~pnghi~mcua(2.34).TasedinhnghIa" tu'ongthichmetric"cuax qua mQtsO'2,(x) kh6ngam,va se chi ra r~ngcacnghi~mcua (2.33)chinhla nghi~mcua(2.34)matu'ongthichcuanob~ngzero.Tir do chuy€nbai toan timnghi~mqic bienvemQtbaitoankhactu'ongdu'ong. v ChoI =[0, T] ,D =I xB(xo,r) ,T,r>0vac6dinh1~P <00. ..Choala mQtdQdokh6ngcompactrenLP(I,RN). Ta li~tkecacgiatruStsail: (Hi) F.. D ~RN za9:""Bdodllf/CvUicacgiatridong. (H2)VUi tEl h.k.n,vUim(Jix, //F(t,x)/ :5:let),11day1(.)E LP va T f let)dt<r. 0 v (H3) VO'itEl h.k.n,anhx{lx ~ F(t,x)la U.s.C. ChoF: D ~ RNla mQthamdatrioVoi m6iE >0,dinhnghIaanhX(;lFe: Fe (t,x): =F (t,x(t)+EB)+EB, B lahinhc~udonvimdtrongRN. KStquaxa'pxl saillac6ngC\lIcythu~tcmnhcuaph~nay. .. ----- -26- M~nhd~2.2: ChoF: D ~ R!'ia m()tanhXfldatrj thoa(Hi) va(H2),thiwJi m6iE> 0, hailoan: { XI(tJ.EFc(t,X(t)),tEl x(o)=Xo conghifm. (2.35) HCInnila, (i) V6'im6inghifmxcuahailoan: { XI(tJ E coF(t, X(t)), tEl x(0)=xa (2.36) vam6i E > 0, co m()tnghifmXecua(2.35)saochoxJT) =x(T)va //Xe-X//< E. (ii) Ntux thoa(2.36)nhLtngx'(t)~ExtcoF(t,x(t))v6'imQit trongm()thIP d()dodllClng1hzcothi xdydl/ngm()tday{xn},n~1, XnE SFlIncotfnh chd'tia khongco day con nao cua no h()i tlj mflnht6'ix trong W1.p(I, R!'). Chungminh: Chia I thanhM khoangconIj =[jT/M , (j+I)T/M], j =O,...,M-I , saocho fl(t)d(t)<E.L~y~(.)Ia hitc~tdodu'<;1ctuanhX(;lt ~ F(t,x(jT/M))trenIj. D~t:J . t x(O)=xo,x(t)=xo+ f£(s)ds, tElj' 0 thlx lien t\lctuy~td6i vax'(t) E F(t,x(jT/M)), t E Ij h.k.n. TacoI x(t)-x(jT/M) I < fl(t)d(t)<E , nen x'(t) E FE(t,x(t))h.k.n.J Bai loan(2.35)conghi~m. i) GiastYxIa m<)tnghi~mcua(2.36). Anhx(;l: / N+l N+l t ~ {(Ai'U) E [O,It+lxF(t,X(t))N+l :~:)'i=1,x'(t)=LAiUi} i=l i=l co giatri dongva dodu'<;1Ctheo(HI). Vi v~yco d.c hamdodu'<;1cA/.), u/.) saocho: .. N+l x'(t)= "A.(t)U.(t),~ 1 1 i=l voi tEl h.k.n -- ------ ---- --- -27 - Chia I thanhhuuh:;mcackhoangIj=[tj-l,tj],(to==0) sao cho voi m6i j: 2 Jl(t)d(t)<E. ) Voi m6ij, gQi (E~).- IamQtphanho~chdodu</ccuaIj saocho:1-1,...N+1 fu.(t)dt=fA.(t)u.dtr 1 1. Ei I.j J (2.37) ~act~pnayt6nt~itheoh~quacuadinhly 16iLjapunov. .'Binhnghla: N+l ug(t)=LLXEi. (t)uJt), j i=l ) x (t)=x + ~u (s)dsE o.b E Bftu tientahIDyr.~ng,voimQij :xE(tj)=x(tj). . Th~tv~y:XE(0) =xeD) , vagiasan€u XE(tj-1)=X(tj-l) thitu (2.37): N+l { \ Xg (tj)- x(tj)=f(Ug(t)-x'(t)dt)=L f\XEi (t) - Ai(t)Pi (t)dt=0 I. i=l I. ) ) ) Yi v~y, n€u tE Ij : N+ltj I ~ Ixg(t)-x(t~~~f XE\(t)-Ai(t~IUi(t)ldt< 2fl(t)dt<81=1t. 1 I.~ ) Hon nua,dox'E(t)E F(t,x(t))nensuyra : x'E(t)E F(t,x(t)- xE(t)+xE(t))c FE(t,xE(t)) V~ytadachangminhiI. ii) La'yx langhi~mcua(!;.,'56)saochox'it) fiE xtcoF(t,xlt)), VtEE, m(E)>O. Bi~udi~n N+1 N+I x'(t)=LA/t)Vi(t), AiE[o,11LA/f)=1 vaViEF(.,x(.)). i=l i=l Khongma'tinht6ngquat,giasar~ngco 11>0saocho: IVi(t)- x'(t)J~17,VtEE,Vi =1, ,N +1 (2.38) ----------- -28 - Xay dV'ngday {xn}tuongtV'nhutren : voi m6i n21,chia I thanhhUllh(;ln ImmingmakhonggiaonhauI~ saocho2I~l(t)dt<Yn.J Voi m6in,j thl (E~(n)L IamQtphanho(;lchdo du<;5Ccua I~saochol-l,...N+l (2.37)khong~6i(voiE~(n)thayvl Ej ) .Tadinhnghla: N+l Vn(t)=I IXEi(n)(t)Vi(t) j i=l } t Xn(t)=xo+ fUn (s)ds 0 NhaKaydV'ngIIXn-xii <Yn, bonnuanha(2.38), N+l ~p Ilx'n-x'll:2 fl~~XEj(n)(tXUi(t)- x'(~)1dt = N+l I I fluj (t)- x'(t~Pdt 2 m(E)t,P>0 j i=l EnEJ(n) DoclU<;5ngtrenrorangkhongphl;!thuQcvaon,nenmQidayconcua{xn} d~u,khongth6hQitl;!m(;lnhv~x. ,Tachungrninhdu<;5cii/. Bay gio ta dinhnghIamQts6 kh,Hni~mse du<;5cstYdl;!ngtrongtoanbQ ph~ntie'ptheo. X6tcacbaitoansauvoiF Ia hamdatritu D vaoRN: { X' (t)eF(t,x(t)) x(O)=Xo { X'(t)eco F(t, x(t)) x(O)=Xo { xl(t)e Ext co F(t,x(t)) x(O)=Xo tel tel (2.39) tel tel (2.34) tel tel (2.33) Ta leYhi~uSF,ScoP,SextcoPl~nlu<;5tla cact~ph<;5pnghi~m,Rp(T),RcoF(T) va Rextcop(T)Ia cac t~pd(;ltdU<;5ct(;lithai di6mT cuacacbai toantuongling (2.39,2.34, 2.33).(Nh~cl(;li : Rp(T) ={x(T) / x Ia nghi~mcua(2.39)}. Ta dinhnghIa" tuongthiGh" cuax E SpIa s6 : ----------------- -- - ------- -29- cf; (x) := UmaLP({u'/ UESF n WI,p,Ilu- xii0 Tudngtv, taco cacdinhnghIasauvdi x E ScoF: .2:oF(X):= UmaLP({u'/UEScoFnWI'P,/lu-xll0 21;,(x) := !~a£p({u'/UESF, nWI'P,llu-xll<E}} trongdoa Ia mQtdQdokhongcompac. 'CacdinhnghIanayconghIakhicacbailoantudngling(2.39,2.34,2.33) conghi~m. M~nhd~2.3 ChoF..D ~ ~ lam(Jtanhxt;lthoa( Hi ), (H2),(H3)vax E ScoFthi.. J1(x) ~ ~F(X) ~ Jt(x) (2.40) Han nT1antu F( t, ~ ) laLipschitzthi.. J1(x) = ~F(X) = Jt(x) (2.41) Chungminh: Bit d&ngthucd~uIa hiennhien. D€' chungminhbit d&ngthucthu2 , liy x E ScoFva £ >0 chotrudc,xet hai t~psau: K& ={u' / uEScOF,/iu- x II <E } H&={u'/uEsF"llu-xll<E} Theom~nhd~ 2.2,KE duqcchuatrongbaadongyeutheoLPcuaHE, ta co: ~~F(X)=lim a(K&) ~ lim a(clwHJ = lim a(HJ = 21;,(x) &~O + &~O + &~O + ChoF( t,.) laLipschitz,liy k(.)E LPsaDcho: H( F(t,u), F(t,v)) ~ k( t) I u - v I, tEl h.k.n., \iu,v. 'B~t G& = { u'/ uESF , II u - v II < E } Va p ~ R k(t) 1el.k(""(1+k(S))ds+1+k(t)}t .. = {u'/u'(t)EF(t,u(t)+6B)+iB,u(to)=xo,llu-xll<E} C {u'/u'(t)EF(t,u(t)+E(l+k(t)))B,u(to)=xo,lIu-xll<E} Thl Hs -- ------------------------------ -30- C Ge +&pBp (theob6deGronwall) VI V?y, theodinhnghiacuaa, tasur du'<;5C: .2;(x)= lima (HJ ~ lim [a(GJ+a(&pBp)] = l':(x)e~o+ e~o+ M~nhdedffdu'<;5Cchungminh. SaudaytaxetthemmQtvi d\l dgminhhQachoke'tquacuam~nhde:ne'uF khongphaiLipschitzthltakhongco(2.41). Vi d\l3.1 ChoF Ia viphandu'OicuaI x I , { -I F(x).= [~l,l] ne'ux <0 ne'ux=0 ne'ux >0 F Ia U.s.C.T?p cac nghi~mcua x' E F(x) ,.x(O) =0 la compactrong W1,p( 0,1;R ) , VI V?y hamtu'dngtmch ~~F la d6ngnha'"tb~ng0 tren ScoF. Nhu'ngm~tkhac,vi phancuanghi~mx ==0khongla Cvcbien,nentheodinhly (2.5) C1ph~nsauthl cft(x) >0, tucIa ~F(X) < Jt(x). Khaosatthemcacham"tu'dngthich",taconcoke'tquasail: M~nhd~2.4 ChoF thoa(Hi) - (H3) thicacanhX(l ~" 2:, va~ fa mla lien t{tC A ' s ' R+trenta coFvaG . Be}d~2.7 ChoF..D -)oJ?Vfamtjthamdatrjthoa(Hi) -( H3),vacho~E RcoF(T). Cacphat bilu (1a) va (1b) , (2a)va(2b)saudayfanhtr;ltU:ClngdU:Clngnhau.. (1a) V(ji m6i x E WI,P( I, J?V ), m6i day ( Xn}n2:1trang ScoFsaD cho xiT) -)0 ~vaXn-)0 wXGangfa htji t{tmr;mhtrong WI,P(I,J?V) tai x. (lb) ~~~(~) = 0 --------------- --- - ------------ - 31- (2a) Cho 8n J 0 . Vai m6iXE W1,P(/, JtV ), vai m6iday ( Xn}n~lsaDcho XnESPe ' xnrT)~;; va xn~wx ciing fa hQi t1;lmc;mhtrong W1,P(/, JtV )n , taix. (2b) .£1;,(q) = 0 Ntu F(t,.) fa,Lipschitzthi ta con co (1) va (2) cung tuangduangvai ~P(~)=O Chungminh: Truoc tien, hill y ding ~~F(;;)=0 tu'ongdu'ongvoi tinhchit : m6i day {xn}I2IC ScoFhQit1,1y€u Wi x trongWI,P(I, RN ) cling hQit1,1mg.nh.Th~tv~y : Cho~~F(x)=0 va xn~ X, {Xn}C ScoF' Tu a({X'nIn~l})=a({X'n In~k })~a({u'IUEScoF,llu-xll<&}) , voi k du lOn,day {Xn}n~11acompactu'ongd6i mg.nhtrongWI,P( I, RN ). L~p lu~nnaydungvoi mQidayconcua {xn}nenroanbQdayhQitv toi x. Ngu'Qc19.i,n€u m6iday {xn}EScoFhQitv y€u Wi x d6uco mQtdayconhQitl;! mg.nhtrongWI,P(I, RN ), du'CJngkinh ( rheadQdo kh6ngcompact) cua t~p {u;;UE ScoF'Ilu- xii<s}dffnWi0khic~ O.Suyfa ~~F(x)=O. Tachungminh(la) (lb) : la::::>lbl L{yx E ScoFsaochox(T)=~.Theonh~nxettrenthl .2P (x)=0,nen.2P (;;)=0coP coP lb::::>lal Tu Xn~wx,theomQtdpmlyhQitvcuaCellinavaAubin,x E ScoF vax(T)=~. Chungminh(2a)(2b)b~ngeachly lu~ntu'ongtl;(. N€u F(t, .)1aLipschitzthltUmt$nhd6 (2.3)taco du'Qc (1)~(2)~~(;;)=o. Tachuy~nsangchungminhk€t quachinhcuaphffnnay: DinhIy2.5 Cho F :D ~ gvJa meltham da trj thoa (Hi) - (H3). Ltfy x E ScoFthi .2JF(x)=0 khi va chl khi x E SextcoF' Chungminh: ------------------------------------------------------------------- - 32- ~/ La'yx E ScoF,ghl sa x ~ SextcoF'Theom~nhde (2.2), co mQtday {xn}n ~ 1,XnESPlin'XnhQit1,1deuWi x, nhu'ngkhongco dayconnaGcuano hQit1,1m<;inhtoi x trongW1,p(I,RN).Theochungminhcuab6 de (2.7)thl 2:F ph:Hdu'dng, matithu~ngia thie't.Suyra XE SextcoF. La'yXE SextcoF,va {Xn}n ~ 1Ia mQtdaytrongSFc ' £n~0,hQit1,1, n ye'utrongW1,P(I,RN)toix.Tathuanh~ndingXnclinghQit1,1m<;inhtoi x. (Theo mQtk€t quacuaRzezuchowski).Do {xn}du'Qcla'ytuyy, tuchungminhcuab6 de(2.7)suyra 2~(x)=o. Nhijnxet: <;:::::./ Dinhly naychIradingvi~ctImnghi~mcuabaitoanqtcbien(2.33)cling gi6ngnhu'vi~ctImcacgiatqminimize(t<;iib~c0)cuahamu.S.C.2:F trent~p compactScoF. Ta connoi dinghamtu'dngthich2:F la d:ftctru'ngchocac nghi~mqtc bien.TrencdsadatrInhbay,ngu'ditaconKaydl;tngdinhnghIa tu'dngthichcuacacph~nta thuQcVaGt~pd'<ltdu'QC,d@tirdoxetcactinhcha't . t~pnghi~mcuabaitoanqtcbien.