Bất đẳng thức biến phân và ứng dụng

. Hadjisawas[18],S. Schaible[20],X.Q. Yangva C.J. Goh [8],B. Ricceri [21],Q.H. Anssari[28,19],P.Q. Khanh va L.M. Luu [3,4, 5],F. Giannessi[26,27],...Tu0ng l1ngv6i moi d1;tngcua bai toanbilt dilngthuc bi~nphan,ta c6 cacmohlnh can b~ngm1;tngiaothongtu0ngling,chilngh1;tn: canbAngm1;tngiaothongdam\lC tieu [8],din bAngm1;tngiaothongph\!thuQcthbi gian [2],can bAngm1;tngiao thongda tri [3,4, 5], ... Trongm\lCnaychungtoi trlnh baynguyen15'canbAngWardrop,cacmohlnh , ,? toanhQccuabai toancanbangm1;tnggiaothongvacacdieuki~ndemohlnh m1;tngd1;ttcanbAng. Nguyenly canbAngWardrop[13](1952)du<;Jcphatbi~unhusan: "MQtm1;tng giaothongdu<;JcgQila d1;ttcan bAngWardropn~umQingubid§u Iva chQnhanh trlnh c6chi phi thilp h0n". De apd\lngnguyenly naymohlnhtoanhQccuabai toancanbAngm1;tngiao thongdu<;Jcduara nhu san 3.1 Bai toan m~nggiao thong ? Giasvc6mQtm1;tnggiaothong,trongd6ngubitaphaiv~nchuyenhangh6atv diemA, gQila di~mb~td§.u(origin),d~ndi~mB, gQila di~mcu6i(destination), theomQtnhuc§.uhanghoanaod6.C~pdi~mA- B du<;JcgQila c~pd§.u-cu6i. MoimQtdubngdi tv A d~nB gQila mOthanhtrlnhn6ic~pA-B. Ta ki hi~uW 1at~ph<;Jptilt cacacc~pd§.ucu6itrongm<;LngvaP lat~ph<;Jptiltcacachanh trlnhtrongm(;Lng.Gia sv W vaP cohUllh(;Lnph&ntv. Vdi moic(Lpd&ucu6i wE W, ta ki hi~uPw la t~ph<;Jptilt cacachanhtrlnhn6ic~pd§.ucu6iw E W. " ??, /? / Ki hi~udwla nhuCalicanv~nchuyentv diemdatiAwdendiemcuoiBw.Ta ki hi~ud la vect0nhuctiutrongd6cacthanhph§.ncuad la cacdwtu0nglingva Trang40 s6thanhph§,ncuad b~ngs6ph§,nta cuaW. Trenmoihanhtrinhr E P, giiisa comOtluQnghanhhoa(flow)chuy@nqua,ki hi~ula Fr. Ta gQiFr la dongtren hanhtrinhr. VectClF macacthanhph§,nla cacdongFr trenhanhtrinh,gQila vectCldonghayphuCinganluuthong.86thanhph§,ncuaF b~ngs6ph§,nta cua P. Giii satrenmoihanhtrinhr E P corangbuOcv~tiii nang Ar <Fr <{tr, (3.1) vav6imaici;ipd§,ucu6iw E W cacdongtrongPw luauthoanhu c§,u,nghia180 L Fr =dw. rEPw (3.2) Xet ma tri;ins6 Kronecker =«I>w,r)xacdtnhnhu sail { I, r E Pw; w,r= 0, r E Pw. (3.3) Khi doh~thllc (3.2)duQcvi@tl[;1inhusau F=d. (3.4) NeumOtvectadongF thoamancach~thllc (3.1)va (3.4)thl F duQcgQila vectc5dongch~pnh~nduQc(feasibleflowvector).Di;it K:= {FIAr F=d}. (3.5) T~pK duQcgQila t~pch~pnh~nduQccuabaitoanm[;1nggiaothong. Nh~nxet 3.1.1.K la tt)pl6i, dongvabi ch(invan€u A{tthi K =I- cpo Ch71ngmink.Gia sa FI, F2 E K thl, vuimQit E [0,1], A < tFI +(1- t)F2<{to va tFI + (1- t)F2=tcpFI+ (1- t)cpF2=d. V~yK la t~p16i.Tfnh dong,btchi;invakhacrangcuaK suyfa tu:tinhlien tl,lCelm va (3.1). 0 Trang41 B:= {s E Pw/Hs >As}. VI H thoaman (3.7) nen Cq > Os,\/q E A, '\IsE B. Do d6 t6n t<;1i"YwE R saocho inf Cq>"y >supOs. qEA sEE L§,ytuy ymOtvectadongF E K. Khi d6vdimQir E Pw, n@uCr <"Yw thi r tj. A, nghiala Hr =f-lr.guy ra Fr - Hr o.Tllangt\i,n@u Or>"Ywthi (Cr - "Yw)(Fr- Hr) >o.guyra L Cr(Fr - Hr) >"YwL (Fr - Hr) ="Yw(dw- dw) =o. rEPw rEPw V~y (C,F - H)= L L Cr(Fr - Hr) >o. wEWrEPw Cia S11ngll<;1cl<;1iH khangla vectadongcanb~ng.Khi d6t6nt<;1iWE W, q,s E Pw saochoCqAs.D$,t <5:=min{f-lq- Hq,Hs - As}, va Fq =Hq + <5,Fs =Hs - <5,Fr = Hr, \/r =1=q,s. Khi d6F E K va (C, FH)= <5(Cq- Os)<o.V~y(3.7)khangthoa. D 3.2 M~ng giao thong ph\1thuQCthai gian 3.2.1 M6 hlnh kh6ng co rang buQc Cia s11T c R va£,P(T),p> 11akhanggiancachamf : T --+ R dodll<;1Ctren T va IlfilP khii tfch Lesbesguetren T. Xet mahinh m<;1nggiaothongd tren vdi giathi@tt<;1imoi thai di~mt E T, dongtrenhanhtrinh r E P la Fr(t) ph1.lthuQc bi~nthai gian t, nhu cilu cuac~pdilu cu6iWE W la dw(t)va Ar(t),f-lr(t)la cac r~ngbuQctiii nangtren hanhtrinh rEP. Cia S11r~ngFr(.), Ar(.),f-lr(.)E fl(T), vdip > 1vathoaman,hAukh~pnai(ghit~tla h.k.n)trenT, Ar(t)<Fr(t)f-lr(t),\/r E P, Trang43 va cpF(t)=d(t). vdiCP,F(.), d(.) dli<;1CdtnhnghiaCJm1.lC(3.1). Khi d6t~pchfipnh~ndli<;1Ccuabai toancanb~ngm<,tnggiaothongdli<;1Cvi~t l~inhli sau K:= {F(.) E .cP(T)/[cpF(t)=d(t), Ar(t)<Fr(t)<f-lr(t),VrE P], h.k.nT}. (3.9) Nh~nxet 3.2.1.K Latt),pMi, d6ng,bi chiJ,nva compacty€u. N€u gilLsitthem A(t). Kf hi~u.c = .cP(T)va.c*= .cq(T)(~+ ~= 1)1akhonggiand6ingautapa p q cua.c.VdimoiG E .c*vaF E .c,ta dtnhnghia ((G,F))=!(G(t),F(t))dt, (3.10) Khi dochiphi trenm<,tnggiaothong1aanhx<,tC(.) tll K vao.c*vavectcldong canb~ngtren m<,tngdli<;1Cdtnhnghianhli sau Dinh nghia 3.2.1. (Xem[2j) M(Jt vectddongH(.) E K d'l1QcgfJi Lacan b1ingn€u H th6aman, h.k.n tren T, [VWE W,Vp,S E Pw,Cq(H)(t) <Cs(H)(t)] ::::;.[Hq(t)= f-lq(t)hayHt(s)= As(t)]. (3.11) M6i lienh~giuanghi~mciiabaitoancanb~ngvabaitoanbfitd~ngthacbi~n ? phan the hif;jn(j dtnh 1ygall. Dinh ly 3.2.1. VectddongH E K canb1ingn€u va chi n€u ((C(H),F - H))> O,VF(.)E K. (3.12) Changminh.Gias11H(.) canb~ng.Khi d6H thoa(3.11).VdimoiWE W, d~t A ={qE Pw/Hq(t)<f-lq(t)h.k.nT}; B ={sE Pw/Hs(t)>f-ls(t)h.k.nT}. Trang44 Tli (3.11)vadtnhnghlacuaA, B suyfa,vdimoiqE A, s E B vah.k.ntrenT, Cq(H)(t)>Cs(H)(t). Suyra t6n t<;Li"Yw(t)E £(T) saocho,h.k.ntren T, inf Cq(H)(t)>"Yw(t) >supCs(H)(t). qEA sEB Vdi F E K va r E Pw tliy y, n~uCr(F)(t) < "Yw(t)h.k.ntrenT thl Hr(t) = J.lr(t)h.k.ntrenT. Suyra Fr(t) - Hr(t) < 0 h.k.ntrenT. Dod6(Cr(F)(t)- 1w(t),Fr(t) - Hr(t)» 0 h.k.ntrenT. N@uCr(F)(t) > "Yw(t)h.k.n tren T thlI I Hr(t) = Ar(t) h.k.ntren T. Suy ra Fr(t) - Hr(t)/ > 0 h.~.ntrenT. Vi v~y I (Cr(F)(t) -"Yw(t),Fr(t) - Hr(t)» 0h.k.ntrenT. Cu6icling,neuCr(F)(t)="Yw(t) : h.k.ntrenT thl (Cr(F)(t) - "Yw(t),Fr(t) - Hr(t))= 0h.k.n.trenT. Suyfa,h.k.n trenT, 'E (Cr(F)(t), Fr(t) - Hr(t)» "Yw(t)'E (Fr(t) - Hr(t)) =O. rEPw rEPw Dod6,h.k.ntrenT, 'E (Cr(F)(t), Fr(t) - Hr(t) » o. rEPw Nhli v~y l (C(F)(t), F - H)dt >0, nghlala ((C(F), F - H) » 0,VF E K. V~yH thoaman(3.12). GiaS11ngli<;5Cl<;LiH thoaman(3.12)vaH E K kh6ngthoa(3.11).Khi d6t6n t<;LiW E W,q,s E Pwvat~pE c T, c6dQdodlidngsaochoh.k.ntrenE Cq(H)(t)As(t). VdimoitEE, di;it 6(t)=min{J-lq(t)- Hq(t),Hs(t)- As(t)}. Trang45 SHYfa, h.k.ntren T, 0 va Hs(t)- As(t);Hq(t)+<5(t)< f-lq(t). DM Fq(t) = Hq(t)+<5(t),Vt E T; Fs(t)=Hs(t)- <5(t),Vt E T; Fr(t) = Hr(t), Vt E T, r =I-q,r =I-s va F(t) =H(t),VtE T\E. Khi d6ta c6F E K va ((C(H), F - H)) =IT(C(H)(t),F(t) - H(t))dt = r (C(H)(t),F(t) - H(t))dt JT\E ~ ~ v- 0 +IE(C(H)(t),F(t) - H(t))dt = IE <5(t)[Cq(H)(t)- Cs(H)(t)]dt<O. Di§u naymall thuanvdi vi~cH thoaman(3.12). D Nh~n xet 3.2.2. DiJu kien (3.11) fa tlCdngdudngvCfidiJu kien sau : Vw E W,3'YwE £ sao rhoVr E Pw, h.k.n trenT, Cr(H)(t) <'Yw(t) * Hr(t)=f-lr(t), Cr(H)(t) >'Yw(t) * Hr(t) =Ar(t). (3.13) 3.2.2 M6 hlnh co them ding buQc Giasatrongmohlnhm~ngiaothongdtren,cacvecteJdongF thoamanthem rangbuQcFED, VF E K vaD c £ la t~p16ithoa K n intD=I- cjJ. (3.14) Khi d6,dinhnghlav§ dongcanb~ngcuabai toanm~nggiaothongdu<)cphat bi~unhu san. Trang46 D!nh nghia 3.2.2. (Xem[2j) VectadongH E K nD a71rjcg9i to,canbangn€u H th6aman ((C (H), F - H) )>0,\/F E K n D. (3.15) Quanh~gil1adongcanb~ngcuabai toanm:;tng iaothongtrongtntdnghQp c6themrangbuQcva bai toanb§,td~ngthucbi~nphanth~hi~nd dinh ly sail D!nh ly 3.2.2. (Xem [2j) Vectddong H E K nD can bangn€u va chi .n€u tan tf;li8(.) E £* saDcho ((8, F - H))<0,\/F E D, ((C(H) +8,F - H) » 0,\/F E K. (3.16: (3.17: Changminh.Gia S11cacdi~uki~n(3.16)va(3.17)thoaman.Khi d6 ((C (H), F - H) )>((C(H) +8,F - H) )>0,\/F E K nD. guyra H thoaman(3.15). GiaS11ngllQcl:;tiH thoaman(3.15).Ta dl;tt A = {(F,~)E D x R/~ <O}, B ={(F,~)E K x R/((C(H),F - H))<~}. Ta th§,yA i= cP vaB i= cP. TachungminhAn B =cP. Th~tv~y,n~u(F,~)E A thl~0,\/F E K nD.Suyra(F,~)~B.Vi A =DnR- vaD la t~p16ilienA la t~p16i.D6ivdit~pB, giasU:(Fl, ~d,(F2,~2)E B va u E [0,1]tuyy.Ta c6 ((C(H),Fl - H) )<~l, ( (C (H), F2 - H) )< ~2 va ((C(H),uF1+(1- u)F2- H)) = fr(C(H),uF1+(1- u)F2- H)dt =ufr(C(H),F1- H)dt +(1- u)fr( C(H),F2- H)dt =u((C(H),Fl - H)) +(1- u)((C(H),F2- H)) <U~l+ (1 - U)~2' Trang47 Suy ra B la t~p16i.Tit intD -I-cPsuyra intA -I-cPoTa thiiy caegia thi~tcua dinh IS'tacht~p16ithoamanvai hai t~pA, B. Do d6t6n ti;ti(8,k) E £* x R sao cho(8,k) -I-0 va t6n ti;ti0;E R saocho ((8, F) )+k1<0;,\f(F,1) E A, ((8,F,))+k1 >0;,\f(F,1)E B. (3.18) ~ (3.19) Tit (3.18)suy fa k > O.N~uk = 0 thl ta chQnFo E K n intD, khi d6 ((8, Fo))= 0;.VI 8 -I- 0 nent6n ti;tiFED saocho ((8,F))> 0; (di@u nay mati thuanvai vi~cH thoa man (3.18)).Do d6 khongmiit tfnh t6ngquat ta c6 th~ gia s11k =1.Khi d6 vai mQi1 >0dunhovavaimQiFED, ta c6 ((8,F))< 0;. (3.20) N@uchQn1 = ((C(H), F - H)) thayvao(3.19),thl vaimQiF E K ta c6 ((8,F) )+((C(H),F - H) » 0;. (3.21) NeuchQnF = H thayvao(3.20),(3.21)thl vdimQiFED, ta c6 ((8, H))= 0;,((8, F) )< 0;, vavaimQiF E K ((C(H)+8,F-H))> O. 0 3.2.3 Di~u ki~n t6n t~i nghi~m Dinh nghla 3.2.3. (Xem[2j) Gids'llX lakhanggianvectotapath'l,tC,K c X la Uj,pl6ivaC :K ---+X* finhxq,dontrio (i) Anh xq,C du(jc99i la rinhxq,t'l,tadondi~u(pseudomonotone)ntu C thoaman vcJim9ix,y E K, (C(x),y - x» 0 ---+(C(y),x - y)< 0; (ii) Anh xq,C du(jc99i la rinhxq,lien t'{LChemintu C thoamanvcJim9iY E K, finhxq,x ~ (C(x),y - x) la rinhxq,n'llalient'{LctrenK; (iii) Anh xq,C du(jc99i la rinhxq,lien t'{LChemid9Ctheodoq,nntu C thoamanvcJi m9ix,y E K, finhxq,~~ (C(~),Y -~) lan'llalient'{LCtrentheodoq,n[x,y]. Trang48 D@ tlm di~uki~nt6nt<;ticanbAngcuabai toanm<;tnggiaothong,chungta xet , ~? ~ haidinh ly vebat dangthlic bienphanBali. Dinh ly 3.2.3. (Xem !2j) Cirl S71X la khanggian vectdtapava K c X la U)p lai,khacq; vaC :K -+X* la anhxr,£thoaman (i) Tantr,£iU)pA c K thoamanA latq.pcompact,khacq; vatantr,£itq.p.Bc K thoamanB la tq.pMi, compactsaocho\/x E K\A, 3y E B, (C(x), y - x)< 0; (ii) C la anhXr,£lien t7j,Chemi. Khi d6 tan tr,£ix E A saochovdi m9iY E K (C(x),y - x» o. Dinh ly 3.2.4. (Xem!2j) Cirl s71X la khanggianvectdtapa,K c X latq.plai, khacq; vaCK -+X* la anhXr,£thoaman (i) Tan tr,£itq.pA c K thoamanA la tq.pcompact,khacq; vaB c K thoamanB la tq.plai, compactsao choVx E K\A,::Iy E B, (C(x), y - x)< 0; (ii) C la anhXr,£t'{taddndi~'Uva lien t7j,Chemid9ctheodor,£n. Khi d6 tan tr,£ix E A saochovdi m9iY E K (C(x),y - x» o. Ap dl,mgDinhly (3.2.3)vaDinhly (3.2.4)chomohlnhm<;tnggiaothongvdi X = .cvaK la t~pch§,pnh~ndU<;1c,xacdinhbdicongthlic (3.9).Do nh~nxet (3.2.1)lienK la t~p16i,dongvabi chi;tn.Suyra K la t~pcompacty@utrong.c. Dododi~uki~n(i) cuacacDinhly (3.2.3)vaDinhly (3.2.4)du<;1cthoamanvdi A = K vaB = q;.Bai toancanbiingseconghi~mn@unothoamandi~uki~n Bali. Dinh ly 3.2.5. Cirl s71K c .c la tq.pchapnhQ,nd'l1(JCcuabailoanmr,£nggiao thongvaC : K -+ .c* la anhXr,£chiphi. Khi d6 bai loan can banggiao thongc6 nghi~mn€'Ucacdi€'Uki~nsauthoaman (i) C la anh Xr,£lien t7j,Chemid6i vdi tapamr,£nhtren K va tan tr,£itq.pA c K latq.pcompact,khacq; va tan tr,£itq.pB c K la tq.plai, compactsaochoVH E K\A,::IF E B, ((C(H),F - H))< 0; Trang49 (ii) C la anhX(llien t'l,lChemid6ivdi tapaytu trenK; (iii) C la anhX(lt'ljaddndi~utrenK va lien t'l,lChemindQctheodo(ln. 3.3 M~ng giao thong da fiVC tieu phV thuQc thai gian > ? ThljC te, nguoitham gia giaothongthuongIlja chQnhanhtrlnh v~nchuyen , ?? ,,? " hangh6adljatrennhieutieuchuan,changh(;LnchiphItoi thieuvathaigiantoi thi§u,...Do d6vi~cxetbaitoancanb~ngm(;Lngv6ichiphIdaml).Ctieula c:1n thi@t. Cia Sltcacki hi~uva dOitu</ngcuamohlnhgiOngnhutrlnhbaycuaml).C (3.1)va(3.2).Cia Sltbaygiatrenmoihanhtrlnhr E P, chiphI la motanhX(;L Cr : K ---+£* phl).thuOcthai gian.Ta d~nhnghia C(H)(F - H) =(((C1(H),F - H)), ...,((Cr(H),F - H)), ...). D~nhnghiavectadongcanb~ngtrenm(;Lngiaothongdu</cvi@tl(;Linhu sau. D!nh nghia 3.3.1. M(Jt vectddongH E K du(jcgQilacanbilngntuH thoaman, h.k.ntrenT, \/wE W, \/q,s E Pw, [Cq(H)(t)- CS(H)(t)E -R~\{O}]=?[Hq(t)= f-lq(t)hayHs(t)- As(t)].(3.22) Quanh~cuavectadongcanbiingtrongbaitoanm<;Lnggiaothongvanghi~m cuabaitoanbittdiingthu:cbi@nphanvectadu</cth§hi~ntrongd~nhly sau. , ,? , D!nh ly 3.3.1.Dieuki~ndin devectddongH E K din bangla vdimQiF E K C(H)(F - H) ~ - R~\{O}. (3.23) Ch71ngmink.Cia SltH thoaman(3.23)vaH khongthoa(3.22).Khi d6t6nt(;Li WE W, q,s E Pwvat~pE c £ c6dOdoduangsaocho,h.k.ntrenE, Cq(H)(t)- CS(H)(t)E -R~\{O}, Hq(t)<f-lq(t), Hs(t) >As(t). Trang50 V6i moi tEE, d~t o(t) = min{jl;q(t)- Hq(t),Hs(t) - As(t)}. Khi do,h.k.n.trenE o(t)>0 va o(t)+Hq(t)< jl;q(t), Hs(t)- o(t)>s(t). D~t Fq(t)= Hq(t)+o(t),VtE E, Fs(t)= Hs(t) - o(t),Vt E E, Fr(t) = Hr(t),Vt E E, Vr i=q,Vr i=s va F(t) = H(t), Vt E T\E. Khi do, v6i moi i = 1,...,m ((Ci(H), F - H)) = JT(Ci(H)(t), F(t) - H(t))dt = JE( Ci(H)(t),F(t) - H(t))dt = JEo(t)(Cl(H)(t) - Ct(H)(t))dt. Vi Cq(H)(t) - CS(H)(t) E -R~\{O}, h.k.n tren E lien C(H)(F - H) E -R~\ {a},matithuanv6i (3.23). D 3.4 M~ng giao thong da tr! ph\! thuQCthai gian Xetmohint m(;Lnggiaothongv6icackf hi$uvadinh nghiadll<;1Ctrlnh baynhll trongcacm\lC(3.1)va (3.2).VectadongF(.) E £,dll<;1CgQiIa vectadongch§.p nh~ndllQCn@uF thoaman,h.k.n.trenT, F(t) E £',0<Fr(t) < J-lr(t). (3.24) Cia 811t~pch§.pnh~ndll<;1CK Ia motanhXI;1xacdinh bai K(H) ={FE R~\{O}/FE B(d(H),E(H)),0<F <jl;, h.k.nT}. (3.25) Trang 51 vdiB(d(H),E(H))1aquac§,utamd(H) bankinhE(H). GiasitchiphiC 1aanhxl;tdatri tli K(H) vao£*, vdi C(F)(t) = (C1(F)(t),...,Cr(F)(t),...,Cm(F)(t)) (3.26) vavdiC(H) E £*, F E £ ((C(H),F))= !)C(H)(t), F(t))dt. (3.27) D~t e(H)(.)E £,e(H)(t)E V(t) C mill [0,ILs(t)],"It E T.s=l,...,m Dinh nghia3.4.1. (Xem[3,4, 5]) VectadongH E K(H) d'l1{JcgQilil vectadong dinbangyeuneuH thoaman,h.k.n.trenT, VwE W,Vq,S E Pw,3c(H)(t)E (C(H)(t) saDcho [cq(H)(t)(Hq(t) E [lLq(t)-e(H)(t), ILq(t)]hay Hs(t) E [0,e(H) (t)]) Dinh ly 3.4.1. (Xem[3, 4, 5])Neuvectadong H E H(K) lil vectadong din bangyeuth'iH lil nghiifmcuabili toangid batdangthitcbienphan(QVI) vdi f(x, y) =2M.e*(H), trongd6 M =h(lE(H)(t) +m.e(H)(t))dt. (3.28) Chitngmink.Gia sit H 1adongcanb~ngy@u.Khi do vdi moi w E W, d~t A ={qE Pw/Hq(t)<ILq(t)- e(H)(t), h.k.n onT}; B ={sE Pw/Hs(t)>e(H)(t),h.k.nonT}. Khi dotontl;ti1'(t)thoaman,h.k.ntrenT, infcq(H)(t)>1'(t)>supcs(H)(t), qEA sEB vdi c(H)(t) E C(H)(t). L§,ytuy yF E K(H), w E W, r E Pw,n@uer(F)(t) < 1'(t),h.k.ntrenT thl r ~A. guyra Hr(t) E [lLr(t)- e(H)(tt), J-lr(t)],h.k.ntrenT. Do do,h.k.ntrenT, (er(H)(t)-1'(t),Fr(t)- Hr(t)- e(H)(t)» 0. Trang 52 Vi;iy,h.k.ntrenI, (cr(H)(t),Fr(t) - Hr(t))>"((t)(Fr(t)- Hr(t))+e(H)(t)(cr(H)(t)- "((t)). N@uCr(H)(t) >"((t)h.k.ntrenI thl r t/:-B. Suyra Hr(t) E [0,e(H)(t)]h.k.n trenI. Tit dota co,h.k.ntrenI, (c(H)(t),Fr(t) - Hr(t)» "((t)(Fr(t) - Hr(t)) - e(H)(t)(cr(H)(t) - "((t). N@ucr(H)(t) ="((t)h.k.n tren I thl (Cr(H)(t),Fr(t) - Hr(t))="((t)(Fr(t) - Hr(t)). Do do,v6iillQiw E W, r E Pwvah.k.ntrenI, (c(H)(t),F(t)- H(t)) = ~wEW~rEPw(cr(H)(t),Fr(t)- Hr(t)) > ~WEW"((t)~rEPw(Fr(t) - Hr(t) + e(H)(t) ~WEW~Cr(H)(t)<'Y(t)(Cr(H)(t)- "((t)) - ~cr(H)(t»'Y(t)(cr(H)(t)-'Y(t)) > ~WEW"((t)~rEPJFr(t) - Hr(t)) - 2e(H)(t) ~WEW~cr(H)(t)~'Y(t)Icr(H)(t) - "((t)I > ~wEW"((t)(d(H)(t)- E(H)(t)- (d(H)(t)+E(H)(t))) - 2e(H)(t)~WEW~cr(H)(t)~'Y(t)Icr(H)(t) - "((t) I = -2e(H)(t) ~WEW"((t) - 2e(H)(t) ~WEW~Cr(H)(t)~'Y(t)ICr(H)(t)- "((t)/ > -2M(lE(H)(t)+m.e(H)(t)). Vi;iy((c(H), F - H) )+2M.e*(H)>O. D Dinh nghla3.4.2.(Xem[3,4,5})MOtvectadongH E K(H) du(Jc99ilavecta dongdin bangn€u H thoaman,h.k.ntrenI, vdim9iWE W, q,s E Pw,c(H)(t) E C(H)(t) [cq(H)(t)< cs(H)(t)] =}[Hq(t)E {lq(t)- e(H)(t), {lq(t)hay Hs(t) E [0,e(H)(t)]]. Dinh ly 3.4.2. (Xem[3, 4, 5})MOtvectadong H E K(H) la vectadongcan bangn€u H langhi~mcuabailoangirlbatddngth71cbienphan(QVI) tuang71ng. Trang53 Chitngmink.Cia 811H kh6ng1avectC1dongcanbf1ng.Khi do t6nt<;1iW E W, q,s E Pw,c(H) E C(H) va t~pE c T codOdo dliC1ng8aochoh.k.ntrenE, cq(H)(t)e(H)(t). Di;tt J(t) :=min{J-lq(t)- Hq(t)- e(H)(t),Hs(t)- e(H)(t)}. Ta coJ(t) >O. Xet vectC1dongF xacdinhnhusail F(t) =H(t),Vt rJ:E vavOimQitEE, Fq(t)=Hq(t)+J(t), Fs(t)=Hs(t)- J(t), Fr(t) = Hr(t),Vr -#q,r -#s. Suyra F E K(H) va (c(H)(t),F(t) - H(t))= J(t)(cq(H)(t)- J(t)(cs(H)(t))<0,"ItE T. Dodo ((c(H)(t),F - H))= £b(t)(cq(H)(t)- b(t)(c,(H)(t))dt<O. Di§u naymall thuanvdi vi~cH 1anghi~mcuabai toangiab§,td~ngthacbi@n phan(QVI). D Dinh Iy 3.4.3.Ciasucacdieuki~nsauthoaman (i) C(H)la anhX(Llient7,lCtrenhemimrJrf)ng(g.u.h.c)trongK(H) vacogiatri compact,khaccjJ; (ii) (C(H), f) la anhXr;Lt1,tadondi~u(pseudomonotone)trongK(H) vavdim9i x E K(H), f(., x) la anhXr;LC(x)-l6i trongK(H); (iii) Vx E K(H), Y E K(H) va m9iludi (xa)hf)it7,lytu denx, t6ntr;Liludicon (x/3)cua(xa)vat6ntr;LiU E -C(x) +f(y, x) saGchof(y, x/3)hf)it'l,tytutdiu; (iv) Vuim9ix E K(H), K-l(X) la mdytu tmngK(H), K(.) dongytu vaVx E K(H), VyE K(x), "1"( E [0,1],"(y+(1- "()xE K(x) Trang54 (v) T6n t(li t(ipD c K(H) 10,t(ipcompactytu, khaccjJva t(ipXo c K(H) 10,t(ip 16i,compactytu saDchoVx E K(H)\D, 3z E Xo nK(x), (T(z), z - x)+f(z,x) C -intC(x). Khi d6 bailoangirl batdangth7J:cbitn phan (QVI) c6nghi~m. D!nh ly 3.4.4. Girl sV:cacdi€u ki~n(iii) va (iv) trongDjnh ly (3.4.3)th6aman va cacdi€'uki~n(i), (ii) va (v) cuaD'tnhly (3.4.3)d'l1{fcthayt'l1ong'u:ngbiJi (i') C{.) 10,anhx(llien t1,tCtrenhemimiJ r6ngtrenK{H); (ii') (C,f) 10,anhX(ltv dondi~u(pseudomonotone)trenK{H) vaVx E A,f(.,x) 10,anhX(lC(x)-16i trenA; (v') T6n t(li t(ipD c K 10,t(ipcompactytu, khaccjJva t6n t(li t(ipXo c K 10,t(ip 16i,compactytu saDchoVx E A\D, 3x E Xo n K(x), (\C(z),z - x)+f(z,x))n(-intC(x))# cjJ. Khi d6bai loangirl batdangth7J:cbitn phan (QVI) c6nghi~m. Nh~n xet 3.4.1. Ntu thaycacgirl thitt (ii') trongDjnh ly (3.4.4)biJigirl thitt, {ii'~(C, f) 10,anhX(l tv don di~uytu trenK va C(.) 10,anhX(lnv:alien t1,tCtren (u.s.c)d6i vdi tOp6ytu trenX thi djnhly (3.4.4)vandung. Nh~nxet 3.4.2.Ntu thaygirlthitt (i') trongDjnhly (3.4.4)b(Jigirlthitt, (i'~C{.)10,anhx(llien t1,tCd'l1dihemimiJr6ng(g.l.h.c)trenK thiDjnhly (3.4.4) vandung. Trang 55 ._.