Luận văn Về sự tồn tại nghiệm bất đẳng thức biến phân và mở rộng

5Chu'dng1 , , K 2 , K A BAI TOAN BAT DANG THUC BIEN PHAN " , A ~ VA BAI TOAN CAN BANG 1.1MQts6mnhnghiavakhaini~m Trongvi~cnghienCUllslft6nt~inghi~mcuabaitmlnbfftd~ngthucbie'n phanvabailoandin b~ngthltinhlientl;lc,tinhdondi~uvatinh16i-16mthu'ong xuyendu'Qcsudl;lng.Trongph:1naycacdinhnghlachocaclo~ilienWc, don di~uva16i-16mkhacnhausedu'QcgiOithi~u. Giii suX va Y la haikhonggianvectdtopo.AcX la mQtt~p16i,dong, khactr6ng.L(X, Y) la khonggiatfftcii cacanhx~tuye'ntinhlienWctil X vao Y. DJNH NGHiA 1.1.1.ChoF:X ~ 2Y la anhx~datrio (i)A.nhx~F du'QcgQilanualienwc tren(vie'ta:tla usc)t~iXoEdomF := {xE X: F(x):;t:0}ne'uvoimQiHinc~nN cuaF(xo),t6nt~iIanc~nM cuaXosaD choF(M)eN. F du'Qcgoila nualientl;lctrentrenmQtt~pV ne'uF nualientl;lctrent~i mQidi~mthuQcV. Saildayla cacla cackhaini~mlien tl;lckhacdu'Qcdinh nghlatru'oclien t~imQtdi~m,dinhnghlatrenmQtt~pclinggi6ngnhu'd tren nentakhongnha:cl~imla. (ii) A.nhx~F du'QcgQila nualien tl;lcdu'oi(vie'tta:tla lsc) t~iXoE damF ne'uvoi mQiN mo, N nF(Xo) :;t:0 , t6nt~iIan c~nM cua XosaDcho Vx E M, F(x)nN :;t:0 , ho(Lcphatbi~utu'ongdu'ong:V y E F(xo),V xa~ Xo,::J Ya E F(xa)'Ya ~ y. Anh x~F du'QcgQila lien tl;lct~iXone'uF vila nua lien tl;lctrenvila nua lienWcdu'oit~iXo. 6(iii) Anh X(;lF du'QcgQila mYalien tl;lctrentheohu'ang(vie'ttat la uhc)t(;li Xone'uanhX(;ldatri tH F((1-t)xo+ty)usct(;li0+. Ne'uanhX(;lF Ia ddntri thl tanoi F Ia lien tlJCtheohu'ang. (iv)Anh X(;lF du'QcgQila dongne'ugraphFla dongtrongX x Y. DfNH NGHIA 1.1.2.Chof: A ~ Y la anhX(;lddntrioGia saY du'Qcsap thutlJ'bdimQtnon16i,dongvai ph~ntrangkhactr6ngC. (i) Anh X(;lf du'QcgQila naa lien tlJCdu'ai(trenK) (vie'ttat la Isc) ne'uvai mQiexE Y, t~pmuc {xE K :f(x) ~ex+intC}la dong. (ii) Anh X(;lf du'QcgQiIa naa lien tl;lctren(trenK) (vie'ttatla usc)ne'uvai mQi exE Y, t~pmuc {xE K :f(x) ~ex-intC}ladonghay co nghlala anh X(;l-f la lsc. DfNH NGHIA 1.1.3.Cho anhX(;ldatri T: A ~ 2L(X,Y),C: A ~ 2Y. Trang do,anhX(;ldatfi C co anhla non16i,dongvacoph~ntrangkhactr6ng.Ta co cacdinhnghlasau: (i) T du'QcgQila C_ddn di~u(trenA) ne'u'\I x,y E A, '\I s E Tx, '\I t E ty thltaco (s-t, x - y) E C(x). (ii) T du'QcgQila C_tlJ'addndi~une'u'\I x,y E A: ne'u[:3s E Tx, (s,Y - x) E -intC(x)]thl['\I tE Ty, (t,Y - x) E -intC(x)]. (iii)T du'QcgQila C_tlJ'addndi~uye'une'u"'\It" trong(ii) du'Qcthayb~ng ":3t". (iv)T du'QcgQila C_gia ddndi~une'u'\I x,y E A: ne'u[:3s E Tx, (s,Y - x) E -C(x)]thl['\I tE Ty, (t,Y - x) E -intC(x)]. (v) T du'QcgQila C_gia ddndi~uye'une'u"'\It" trong(iv) du'Qcthayb~ng ":3t". Ne'uT la anhX(;lddntri thlhaic~pkhaini~mC_tlJ'addndi~uva C_tlJ'a ddndi~uye'u,C_giaddndi~uvaC_giaddndi~uye'ula tIlingnhauvakhidota gQichunghai c~pkhaini~mdo la C_tlJ'addndi~uva C_giaddndi~utu'dng ung. 7Neu C(x)==R+,thl cac khai ni<%mC_ddn di<%u,C-W'a ddn di<%u,C_tlfa ddn di<%uyell, C_gia ddn di<%u,C_gia ddn di<%uyeu du'c;5cgQi l(;li tu'dngling la ddn di<%u,t1;1'addndi<%u,t1;1'addn di<%uyell, gia ddn di<%u,gia ddn di<%uyeu (vi) T du'c;5cgQila ml'alien tl;lCtrentheohu'angsuyrQng(viet Ult la guhc) t(;liXone'uanhX(;ldatri tH (T((1-t)xo+ty),y - xo)usct(;li0+. NeuT laanhX(;lddntrithltanoiT la lienWctheohu'angsuyrQng. DJNH NGHIA 1.1.4.Cho anhX(;lhaibienf: A x A ---+R. (i) f du'c;5cgQiIa ddndi<%uneu 'V x,y E A, taco f(x,y) +f(y,x)~0; (ii) f du'c;5cgQi la t1;1'addn di<%uneu'V x,y E A, ne'uta co f(x,y);::::0 thl f(y,x)~0; (iii) f du'c;5cgQila gia ddndi<%une'u'Vx,y E A, ne'utaco f(x,y» 0 thl f(y,x) ~o; Con neuanhX(;lf: A x A ---+Y va Y du'c;5cs~pthli t1;1'boi mQtnon l6i, dong vaiph~ntrongkhactr6ngC thltacocackhaini<%msail: (iv) f du'c;5cgQila C_ddndi<%uneu 'V x,y E A, taco f(x,y)+f(y,x) E -C; (v)f du'c;5cgQila C_t1;1'addndi<%uneu 'V x,y E A, taco f(x,y)E intC =>f(y,x) E - intC; (vi) f du'c;5cgQila C_gia ddndi<%une'u'V x,y E A, taco f(x,y)E intC =>f(y,x) E - C. DJNH NGHIA 1.1.5.Cho anhX(;lg:A ---+Y. Gia suY du'c;5cs~pthli t1;1'boi mQtnonl6i, dongvai ph~ntrongkhactr6ngC. (i) Anh X(;lg du'c;5cgQila C_16mne'u 'V x,y E A, 'V tE[O, 1],g(tx+(1-t)y) E tg(x) +(1- t).g(y)- C. (ii) Anh X(;lg du'c;5cgQila C_16ineu-g la C_16m. (iii) Anh X(;lg du'c;5cgQila C_gia 16mneu 'V Xj,X2E A va y E Y, va neu {g(Xj),g(X2)}cy+C thl g(tXj+(1-t)X2) E y+C, 'V t E [0,1]. 8(iv)AnhX';lgduQcgQilaC_gial6i n€u -g la C_gia16m. (v)AnhX';lgduQcgQila C_gia16mchi;itn€u voimQixl 7:x2thuQcA vay E Y, van€u {g(XI),g(X2)}cy +C thl getXl+(1- t) X2) E Y +intC,V t E (0, I). (vi)Anh X';lg duQcgQiIa C_gia l6i chi;itn€u -g la C_gia 16mchi;it. Ta cob6d€ sail: A> ;:; BODE 1.1.1. (i) g:A ~ Y Ia C- gia16mkhivachIkhi, v Y E Y, t~pmilc{xE A: g(x) E y +C} la l6i. (ii) N€u g:A ~ Y la C- gia16mthl v Y E Y, t~pmilc {x E A: g(x) E y + intC}la l6i. 1.2.811t6n t~inghi~mbai toaDba'td~ngthuc bie'nphan 1.2.1.GiOithi~ucaebaitoaDba'tdiingthucbie'nphan Giasux, Y la cackhanggianvectotapa.AcX la mOt~pl6i,dongva khactr6ng.C :A ~ 2Y Ia mQtanhX';ldatrithai cactinhcha't: V x E A, C(x) lamOtnonl6i, dongvaintC(x) 7:0. Ky hi~ux* la khanggiand6ingftutapa cuaX, L(X,Y) lakhanggiancacphi€mhamtuy€ntinhlientvctuX vaGY. TrongtruonghQp T: A ~ x* la mOtanhX';ldontri, taco bai toanba't d£ngthilcbi€n phan(VI) phat bi€u nhusail : - (VI) TIm x E A saGchovoi mQiYEA, - - ( T( x ), y - x) ;::: O. * Conn€u T:A ~ 2x la anhX';lda tri thl taco bai toanba'td£ngthilcbi€n phandatri(GVI) nhusau: - - (GVI) TIm x E A saGcho V YEA, :3t Er( x), 9( t,Y - x) ~o. BaygiOtaxetanhx~T: A ~ L(X,Y). Khi do, ta co bai toanba'td~ng thucbie'nphanvecto(VVI): - (VVI) TIm x E A saGcho V YEA, - - - ( T( x ), Y - x) ~- intC(x ). Tie'pthea,ne'uT: A ~ 2L(X,y)la anhx~da tri thl ta co bai toanba'td~ng thucbie'nphanvectddatri (GVVI) sail: - - (GVVI) TIm x E A saGcho V YEA, :3t Er( x), - - ( t, Y - x) ~- intC(x). C ,.('., . ? ? K A 2x T A 2L(XY) 1 , h . ' h d . UOlcling,gla su : ~ , : ~ ' a al an x~ a tq trong doK(x) la mQtt~p16i,khactr6ngt~imQix E A. g:A~ A la mQtanhx~cho trudc.Khi do,tacobaitoangiiiba'td~ngthucbie'nphanvectddatri t6ngquat (GQVVI)sail: - - -- (GQVVI) TIm x E AnclK(x) saGcho V Y E K(x), :3t Er( x), - - (t, Y - g(x» ~ - intC(x). Tatha'y(GQVVI) labaitoant6ngquatcuab6nbaitoanconl~i. - Ne'uK(x)=:A va g(x)=:x vdi mQix E A thl bai toan (GQVVI) trd thanh baitoan(GVVI). - Ne'uK(x)=:A , g(x)=:xvdi mQix E A va T: A ~ L(X,Y) la anhx~don tri thlbai toan(GQVVI) trdthanhbai toan(VVI). * - Ne'uK(x)=:A,g(x)=:xvdimQix E A vaT: A ~ 2x la anhX~datfi thl baitoan(GQVVI) trdthanhbaitoan(GVI). - Ne'uK(x)=:A,g(x)=:xvdimQix E A vaT: A ~ x* la anhx~dontfi thl baitoan(GQVVI) trdthanhbaitoan(VI). 10 1.2.2.811t6nt~inghi~mcuabili toaDba'tdiingthucbie'nphan D€ chungminhs1,ft6ntC;tinghi~mtac~nmQts6kStquac6 di€n sau. ",? ;:; BO DE 1.2.1.Gia saE la khanggiantapacompactvad~tF ={Fi: i E I} lamQthQcact~pcondongcuaE. NSumQihQconhuuhC;tncuaF d€u coph~n giaokhactr6ngthi nFi *-0 . iEI D:JNHNGHiA 1.2.1.Gia sa E la t~pconcuakhanggianvectdtapax . Anh XC;tdatri F : E ~ 2xdu<;1cgQila anhXC;tKKM lIen E nSuvoi mQit~p huu n hC;tn {Xl, . . .,xn}trongE, tacoco{XI.. . ., xn}c UF(xJ, d daycoAla kyhi~u i=l bao16icuat~pA. DJNH LY 1.2.1.(KKM - Fan).Gia saE la t~pconcuakhanggianvectd tapaHausforffX vaF :E ~ 2Xla anhXC;tKKM co anh dong.NSu co it nha"t mQtX E E d€ F(x)compacthi nF(x)*-0. XEE D€ chungminh s1,ft6ntC;tinghi~mchobai loan(VI) tac~ncacb6 d€ sau. '" ;:; BO DE 1.2.2.GiasaK lamQt16idongtrongkhanggianBanachth1,fcX, T la tmintat1,faddndi~utuA vaox* va lienWclIen mQikhanggianconhUll hC;tnchi€u. Khi do ~E A langhi~mcuabailoan(VI) khivachIkhi - ( T(y), Y - x) ~ 0 voi mQi YEA. (2.1) Hdnnuaduoicacdi€u ki~nlIent~pnghi~mcuabailoan(VI) la dongva 16i. Chungminh:Gia sa x EA la nghi~mcuabai loan(VI). Do T la t1,faddn di~unentaco(2.1). Ngu<;1cIC;ti,gill sa x E A la nghi~mcua(2.1).D~tYEA tuyy vavoi0<t- - ~1d~tXt =ty+(l-t)x. Nhuv~yXtE A vatu (2.1)taco t(T(Xt),Y - x)~ O. Dodo - (T(Xt), Y - x)~ O. (2.2) 11 Chot ~ O.Do tinhlien Wctrenkh6nggianconhUllh~nchi~ucuaT, lien T(xt)hQitl,lv~T( x) trongt6p6ye'u*.Do do tU(2.2)tadu'Qc(T(y), y - x)~ O. Cu6iclingdot~pnghit%mcua(2.1)la lai vadonglient~pnghit%mcuabai toan(VI) clinglai vadong.0 SaildaylamQtke'tquac6di€n. A' ~ BO DE 1.2.3.Gia sli' A la mQtt~pcon lai, compactcua kh6ng gian Banach Ullh~nchi~uX va T : A ~ x* la toantli'lienWc. Khi dobai toan (VI) conghit%m. Saildaylake'tquacua[Yao1994]v~sljtant~inghit%mchobaitoan(VI). DJNH LY 1.2.2.Gia sli'A Ia mNt~pconlai, compactye'ucuakh6nggian BanachthljcX, T la toantli'tljadondit%utii'A vaGx* va lien tl,lCtrenkh6ng gianconhUllh~nchi~uba'tkycuaX. Khi dobaitoan(VI) conghit%m. Chungminh: D~tM la mQtkh6nggianconhUllh~nchi~uba'tky cuaX saGchoAnM khactr6ng.D~tPMla mQtanhx~donca'utUM vaGX va P~ toantli'lienhQpcuano.Tii'Hnhlientl,lCcuaT taco P~TPMla mQtanhx~lien tl,lCtii'AnM vaGM* . Do K compactye'uliennobich~nvadodot~pAnM la dong,bich~nvalai tromgM. Ap dl,lngB6d~2.2.3,tant~iXME AnM saGcho * (PM TPM (XM), Y - XM) ~ 0 voi mQiy E AnM. (2.3) Tii' (2.3)ta d~ntoi (T(XM),y - XM)~ 0 voi mQiYEA nM. Theo B6 d~ 2.2.2taco (T(y), Y - XM) ~Ovoi mQiy E AnM. (2.4) Voi m6iu E A tad~t Sell)={x E K: (T(u),u - x) ~ O}. Ta se chungminhhQ {Sell):u E A} co Hnhcha'tgiaohUllh~n.Th~tv~y, voi ba'tky t~pconhUllh~n{Ul, . . ., un}cuaA, chQnM la kh6nggianconsinh bdi t~p{Ul, . . ., un}.Tii' (2.4),tant~i XME AnM saGcho (T(y), y - XM)~Ovoi mQiy E AnM. 12 Tli day ta co ( T(uj), Y - XM) ~Ovai mQi i = 1, . . ., n. Nhu v~yXM E n nS(Xj)' Hon nii'aSell)khac tr6ngvai mQiU E A. M~tkhac Sell)la t~pdong i=l yeuva K la t~pcompactyeti,nentli B6 d~2.2.1ta co nS(x) *- 0. D~t~ XEK E nS(x) thl ( T(x), Y - ~) ~0 vai mQiYEA. Theo B6 d~2.2.2thl ~clingla XEK nghi~mcuabaitoan(VI). D Saudayladinhly v~t6nt(;linghi~mrhobaitoan(GVVI) cua[Lin- Yang - Yao1997]. D'NH LY 1.2.3.Gia sii'X vaY la cackh6nggianBanachthlfc,A la mQt t~pconkhactr6ng,16i,dongcuaX. Gia sii'C :A -+ 2Y la anhX(;ldatri thoa tinhchit:C(x)la non16i,dong,intC(x)*-0, C(x) *-X vaimQix E A vad6thi cuaanhX(;ldatriW : A -+ 2Y duQcdinhnghla W(x)=Y\(- intC(x))ladong yeu trongX x Y. Neu T: A -+ 2L(X,Y)la C_tlfa don di~uyeti, nh~ngia tri compactkhactr6ngvanii'alien tl,lCtrentheohuangsuyrQngtrenA thlbai toan (GVVI) conghi~m. Chungminh:Ta dinhnghlacacanhX(;lFl, F2 :A -+ 2Anhusau: Fl(y) ={xE A: t6nt(;lis E Tx saorho (s,y - x) \l - intC(x)}, F2(y)={x E A: t6nt(;lis E Ty saorho (s,y - x) \l - intC(x)}, vdi mQiYEA. Do Y E F1(y)nenF1(y)*- 0 vai mQiYEA. D~utien tase chungminhFI la anhX(;lKKM trenA. D~t{Yl, . . ., Yn}la mQt~pconhii'llh(;lnbit ky cuaA. Cho ~E CO{YI, . . ., Yn}thl se t6nt(;li Ai n - n - n ~O,i= 1,.. ., n; LAi =1saorho X=LAi Yi' Ta c~nchungminh x E UFI (yD. j=l i=l i=l - n - - Th~tv~y,gia sii'x \l UFI (Yi)thl x \l F1(Yi),'V i. Vai bit ky s E Tx, taco (s, i=l- Yi- x) E - intC(x), V i =1,. . .,n. Nhuv~ytaduQc n - - LAi (S'Yi-x) E - intC(x). i=l Di~unaydin Wi 13 n n- 0=(S,x - X) =(S, LAj Yi - LAj X) i=l i=l n - =(S,LAj (Yj - X)) i=l n - - =LAi (S,Yi - x) E - intC(X). i=l Nhu'v~y0 E - intC(x), di€u nay mall thu~nvdi gia thi€t C(x) *- Y. Do n d6CO{Yl,. . .,Yn}E UFI (yJ V~yFl Ia anhX(;lKKM. j=l Do T la anhX(;lC_giadondi~uy€u nenFl(Y )c F2(y),V YEA. M~t khac,F1(y)laanhx(;lKKM, nenF2(y)clingla anhx(;lKKM. Ti€p theotase chungminhF2(y)la compacty€u vdi mQiYEA. Cho Y E A tuyy. La'y{xa}Ia mQtlu'diba'tky trongF2(Y)vaxa~ x* E A. Vdi m6i a, do xaE F2(y) nen t6n t(;li taE T(y) thoa (ta, t - xa) ~ - intC(xa). Do {ta}c T(y), compactrongtapachuffnL(X,Y), {ta}co lu'diconhQitv d€n t* E T(y). Khangma'ttfnht6ngquat,taco th€ gia sa ta~ t*.Ta c6 t* E L(X,Y), t* lientlJCt11tapay€u lIen X d€n tapay€u lIen Y. Do do * * * ,.(' (t , Y - xa) ~ (t , Y -x ), yell trongY. M~tkhac,tac6 II(ta-t*,y-xa)t=llta -t*IIL(X,Y) Ily-xallx' Do Iita-t*IIL(X,Y)~ 0 va IIY-Xallx la t~pb~ch~n,nen (ta- t*,Y -Xa)~O, y€u trongY. V~ytaco * * * * ,.(' (ta, Y - Xa)=(ta-t , Y -Xa) + (t , Y - Xa) ~ (t , Y -x ), yell trongY. Do d6 (xa, (ta, Y - xa))~ (x*, (t*,Y -x*)) y€u trongX x Y. Vi Gr(W) d6ngy€u trongX x Y nen(x*,(t*,y -x *)) E Gr(W). Vi v~y,(t*,Y -x *)E Gr(W) nghlala (t*,y -x*) ~ - intC(x*).V~y F2(Y)dongy€u. Do A compacty€u va 14 F2(y)c A nenF2(y)compacty€u vai mQiYEA. Nhuv~y, theof)~nh192.2.1 (KKM-Fan),tac6 nF2(y) * 0. yeA f)~t~E nF2(y). Vai mQi YEA, ta xay dvng anh x<.lF: [0, 1]--) 2Y d~nh yeA bdi - - F(a) =(T(ay - (1-a)x), y - X). DoT ml'alient\lctrentheohuangsuyrQng,c6giatq compactkhactr6ng nenF nualient\lCtrenva clingc6giatq compactkhactr6ng.Do d6,F([O,l]) compacttrongtapachuffnY. f)~t{an}la daytrong[0,1]giamv€ O.Vai m6i an, t6nt<.litnE T( a nY+ (1 - a n)~) thoa - - (tn,Y- X) ~- intC(x). Do {(tn,Y - X)}c F([O,l])la t~pcompact,nenkhangma"ttinht6ngquat c6thegiasading(tn,Y- X)--) w,vaiw E Y. Vi F naalient\lCtren,nenWE F(O).Do d6t6nt<.lit E T(x) de w =(t,Y - x). Vi d6th~cuaW d6ngy€u, nen- - - - (t,Y - x) ~ - intC(x). V~y X E nF1(y) nghiala x la nghi~mcua bai toan yeA (GVVI). 0 Saudayla k€t quav€ sv t6nt<.linghi~mcho bai toan(GQVVI) cua [Khanh- Lu'u2000]. DJNH LY 1.2.3.Gia sa K compact,g: A --) A la anhX<.ldontq, lien wc, E:={xE K: x E clK(x)}*0 vacacgiathi€t sauduQcthoa: (i)AnK(x) la t~pl6i, khactr6ngvaimQix E A, K-1(y)la t~pmdtrongA voimQiYEA, vaclK(x)lausc; (ii) N€u xa --) x, Ya --) Y va n€u taE T( xa), thi t6n t<.lit E T(x) va cac luoicon x~,Y~vat~E T(x~) thoa(t~,Y~)--) (t,y); (iii)Y\ - intC(.)la d6ngva V x E A, ::3t E T(x) saocho (t,x - g(x)) E Y\ - intC(x). Khid6baitoan(GQVVI) c6nghi~m. 15 Chungminh:Tnfochtt tachungminhE la t~pdong.Th~tv~y,doeIR(.) la anhX(;luscva co gia tri dong,neneIK(.)dong.Ltiy {xa}c E, nghlala {xa}cA, xaE eIK( Xa) va Xa~ Xo.Do d6 thi cua eIK(.) dong nen Xu E clK(xo).V~yE la t~pdong.Voi m6iX E A, YEA tad~t reX) :={y E A: ( T(x), y - g(x))c - intC(x)}, { K(X) n rex) ntu xEE, (x):= / AnK(x) neuxEA \E, Q(y) :=K\ -\y). Tasechungtof5ngQ la anhX(;lKKM. Giii sadi€u nguQcl(;lila t6nt(;li~ n n - n - =I- ~ ~ i~ l(Yi),vaVIv~yYi E (~), voi mQii. Ntu ~E E, thl (~) =K(~)nP( ~).Tit day,tacoYi E P( x), i =1,. . . , n,nghlala - - - (T(x), Yi - g(x))c - intC(x). VI v~Y - - n - (T(x), x - g(x)) =(T(x),Iai(Yi -g(x))) i=l n - - - =I<Xi(T(X),Yi- g(x))c - intC(x), i=l di€u trenmallthu~nvoi (iii).Do do,chico th€ ~E A\ E. Tit dinhnghla- - cuaE, x ~K(x). M~tkhac,voi i=I,..., n, - - Yi E (x) =A II (x ). n Do do, x= IaiYi E K(x), la mQtmall thu~n.V~y, Q la mQtanh X(;l i=l KKM. Titp theotasechIfadng Q(y)dong,\j YEA. Voi mQiYEA, taco -l(y)={xEA:YE(X)} 16 ={xE E:y E K(x)nP(x)} u {xE A \ E:y E K(x)} ={xE E :x E K -1 (y)n P-1(y)} U {X E A \ E :x E K -I (y)} =[EnK-1(y) np-1(y)] U [(A\E) nK-1(y)] =[(Enp-1(y»u(A\E)] nK-1(y) =[(A\E)UP-l(y)] nK-1(y). Vlv~y Q(y) =A\([(A\E)uP-I(y)] nK-1(y» ={A\[(A\E)uP-I(y)]} u[A\K-1(y)] =[En(A\p-I(y»] u[A\K-1(y)]. (2.5) f)~utien,taseki€m tratinhdongcuaA\p-I(y),V YEA. Tli dinhnghla cuarex)taco A\P-I(y)={x E A: :3t E T(x), (t,Y - g(x» E Y\ - intC(x)}. Gia sa ding Xa E A\P-l(y), Xa --j>xu.Khi do, t6n t~i ta E T(xa) sao cho (ta,y-g(Xa»E Y\-intC(xa). sad1,lllgiathie't(ii),t6nt~itovacaclu'oiconxpva tpE T(xp)saocho (tp, y- g(x~»--j>(to,y- g(xo». Gia thie't(iii) d~nWi (to,y - g(XO»E Y\ - intC(xo), nghlala, XoE A\P-l(y). Nhu'v~y,A\P-I(y) la dong.Tli cacgia thie't rong(i) va tli (2.5)la mQtt~pcondongcuat~pcompactA. Nhu'v~y,Q(y) la compact. Ap dl,lllgdinhly KKM-Fan, t6nt~i ~E n(A \ -1(y»=A \ ( U-1(y» . YEA YEA 17 VI v~y,(~) =0. Co haitru'onghQpKayfa.D~utien,n6u ~E A\E, theo dinhnghlacua ,(~) =A nK( ~)-:F 0 , lamQtmatithu§:n.Nhu'v~y~E E,- - - - - nghlala x E clK(x). Hannlla,tadingco 0 =(x)=K(x)n(P(x). VI v~y, vaiffiQiy E K(~),Y ~P(~),nghlala, :3t E T(~),d~cho - - - (t,y- g(X))E Y\-intC(x). V~yx langhi~mcuabaitmln(GQVVI).0 1.3.811t6nt~inghi~mbaitoaDdin b~ngvabaitoaDbfft diingthucbie'nphiln~n 1.3.1.Gi6'ithi~ubaitoaDdin b~ngvabaitoaDba'tdiingthti'cbie'nphanffn Trangph~nnay,giasuX va Y la haikhanggianvecyatapa;Kc X la ffiQt~p16i,dong,khactr6ngvaf: AxA~ Y. N6uY=R,tacobaitoancanb~ngvahu'angla - - (EP): tlm YEA saGchof(x, y )S;O,v X E A. Baitoand6ing§:uvaibaitoantrenla - - (DEP): tlm x E A saGchof(x, y):2:0,V YEA. Trangtru'onghQpt6ngquathall,khi Y du'Qcs~p thli t1!bdimQtnon16i, dongvacoph~ntrongkhactr6ng,thlbaitoancanb~ngvecWla - - (VEP): tlm YEA saGchof(x, y)~ intC, V x E A. Lucnay,baitoand6ing§:utu'anglingvainola - - (DVEP): tlm x E A saGchof(x ,y)~ intC, V YEA. 18 Ne'unonC du'Qcthaybdinonchuy€ndQngC: A ~ 2Y thoaC(x)la mQt non16i,dongvoi ph~ntrongkhactr6ngt~imQix E A, thl c~pbai toantrentrd thanh - -- (GVEP): tim YEA sao cho f(x, y) \tointC( y), '\j X E A, va - - (DGVEP): tim x E A saochof( x ,y)\tointC, '\j YEA. Cu6icling,chungtaxetmQtbai toanco d~ngra"tgi6ngvoi bai toandin b~ng.Giasaf: L(X,Y)xAxA~ Y lahamdontItvaT :A ~ 2L(X,Y)lahamda tIt.Baitoanba"td~ngthlicbie'nphan~n(IVVI) du'QcdtnhnghIala: - - - (IVVI): tim x E A saocho '\j YEA, :3 t E T( x) thoaman - - - f(t, y, x) \tointC(x). Ta tha"ydn& bai toancanb~ng( haybai toancanb~ngd6ing~utu'ong ling)chliabaitoanba"td~ngthlicbie'nphandu'QcdtnhnghIaphlihQpnhu'mQt tru'onghQprieng.Th~tv~y: (a)GiasaT:X ~ X* la anhx~dontrt,ne'ud~tf(x, y):=( T(y), y - x), thl baitoan(EP)setrdthanhbaitoan(VI) . (b)Conne'uT:X ~ L(X,Y) vane'ud~tf(x,y):=( T(y),y - x), thl bai toan (VEP)' setrdthanhbaitoan(VVI). Ngoaifa, taclingtha"ygiii'abai toancanb~ngva bai toancanb~ngd6i ng~utu'onglingco m6iquailh~ra"tg~nnhau.Nghi~mcuabai toancanb~ng lingvoihamf(x,y)clingchinhla nghi~mcuabaitoancanb~ngd6ing~utu'ong linglingvoihamg(x,y):=-f(y,x).Conne'uxettrenclingmQthamthlgiii'ahai baitoannaycom6iquailh~sauday. MtNH DE 1.3.1.GidsitC fanoncodlnhvaanhX(lI AxA ~ Y thod j(x,x)E C, '\j X E A. Khi do: (i) nfuf( . , y) fa uhcvaimQiYEA vaf(x, . ) fa C_f6i vaimQix E A, thi nghi~mcuabailoan(VEP)fanghi~mcuabailoan(DVEP); 19 (ii) ne'uf za C_t1;tadan di~u,thi nghi~mcua bai roan (DVEP) ciing la nghifmcuabai roan(VEP). Chungminh:Kh£ngdinh(ii) la r5rangtudinhnghiaC_tlfaddndi~ucua anhX<:lf. Bay giOta chungminhkh£ngdinh (i). Gia sa yla nghi~mcuabai toan(VEP), tucla - f(x, y) ~intC, V x E A. CO'dinhx E A ba'tky. Voi m6i t E (0,1),d~tXt=tx + (l-t)y. The'XtvaGbitiu thuctn3ntaco:f(Xt,y) ~intc.Dotinh16icuaf(x, . ),taco - t.f(Xt,x) +(l - t).f(xt, y) E f(xt, Xt) +C =C. Dodo I-t - f(xt, x) E - ( - f(xt, y) +C ) t E 1- t ((Y\ - intC) +C) =1- t (Y\ -intC)=(Y\ -intC). t t Do f( . , y) la uhcva do(Y\ -intC) la dong,nend bitiuthuctrenchot ~ 0+ta- - du'Qc: f(y ,x) E Y\ -intC. V~y Y la nghi~mcua(DVEP). D 1.3.2.SItt6nt~inghi~mcuabai tmindin b~ngva bai tminba'tdiingthuc bie'nphanftn Sandayla ke'tquav~slft6nt;;tinghi~mchobai toan(EP) cua[Chadli- Chbani- Riahi 2000]. DfNH LY 1.3.1.Chox la khonggianvect{ftapaHausdorff,A la tqpl6i, dong,khacding. Gid sU:f, g:AxA -f R thoa (i) V6im9ix,y E A, g(x,y):5:0,thif(x,y):5:0; (ii)Co'dtnh XE X, hamf(x,.) nu:alien tl;lcdu6i tren m9i ti)p compactcon cua A; 20 (iii)VaimQit4pconL hiluhc;mcuaA, taco sup ming(x,y) 5{0; YEco(L) xEL (iv)Gid thiit vi tinhcompact.Gid sit t6n tr;tit4pcompact,I6i C c A sao choitnhfitm(Jttranghaidiiu ki~n(a),(b) thoa (a) tiYEA\C, 3xECsaochof(x,y) >0; (b)3XOEC saocho tiYE K'\C: g(xo,y)>0. Khido,bili loancanbling(EP) conghi~m.Hannila, t4pnghi~mIa t4pcompact. Chung minh: E>~tM ={Xl, ..., Xn} la t~pconhUllh<;lncuaA, va B = co{MuC}.Khi dodoC la t~p16ivacompactnenB cling16iva compact.X6t anhX<;lS:B ~ 2Bxacdinhboi SeX)={yE B: g(x,y) <O}. La'y{Zl,...,Zm}la t~pconhUllh<;lntrongB; tac~nchungminh m CO{ZI,...,zm}cU S(zJ. i=l Gia saphanchung,nghlala t6nt<;liYE CO{ZI,...,zm}saocho g(Zj,Y)>0, voimQi =1,...,m. Bi~unay suyJ;a .mill g(Zi'y) >0:mall thu~n(iii). M~t khac, tU (iii) va tinh l=l,...,m compactcuaB, voi mQiX E B tacoXE Sex);dodo Sex)khactr6ngva compact. ApdlJngBinhly KKM-Fan,taco n deSex))7'0. XEB Tli (ii)va(iii),taco n deSex))c n d({YEB:f(x,y)~O}) XEB XEB = n {YEB:f(x,y)~O}. XEB (3.2.1) 21 £)~t YEn deSex)). XEB Dung(iv),tachungminh Y E C. Ne'u(iv), (a) thoa,gia sa Y ~ C suyfa t6n t?i XOEC saDchof(xo,y) >0:matithu~n(3.2.1).Do do Y E C. Ne'u(iv), (b) thoa,tacoS(xo)c C. M~tkhacdoC dongnen Y E C. Dodo,voimQit~pM hii'llh?ncuaA, taco n {YEC:f(x,y)~O}*-0. XEM VOimQix E A, d~tT(x) ={YE C: f(x,y)~O}.Til (ii), T(x) compactvoi mQiXE A, doC compact.Khi do ( theotlnhcha'tcompact),taco n T(x) *-0. XEA V~yb~litoan(EP)co nghi~m.T~pnghi~mcuabai toan(EP) la n {yE C : XEA f(x,y)~O}compact.D £)~chungminhs1;(t6nt?i nghi~mchobai toan(DVEP), tac~nb6 d€ sau. ~ ;:. " BO DE 1.3.1.Cho Y la kh6nggianW!ctathf:CC,l6i djaphlfrJng,C la nonl6i dong,codlnhva intC,r:0. Laya,bEY, vaiaE - intC va b Ii!C. Khi do, t(ip ch(intrencuaa vab khacrangvacophdngiaovai Y\ C. Chungminh: Ta phai chi fa dng t6nt?i c ~C saDchoc -a E C va b - c E C.Do intC*-0, t6nt?i d E intC saDchod - b E C. Voi t E [0,1],d~t dt=td+( 1- t)b, dt E C, voi mQit E [to,1], dt~C, voi mQit E [0,to). £)~cbi~t,taco dtoE C, a E -intC;dodo dto- a E intc. 22 Dodo,voimQitl <to,dug§n to,tav~nc6 dtl - a E intc. D~tc=dtl .Khi d6c ~C, hannuatac6 c- a E C vac- b=tied- b)E C. 0 DJNH LY 1.3.2.Chox Ia khanggianvectatapaHausdorffth1!c,Y Ia khangianvectath1!C,Iaidtaphuong.CIa nonIai,dong,codlnhvaintC:;c0, A {;;X IatqpIai,dong,khactr6ng.Xerhamhaibie'nI AxA ~ Y saDchof(X,x) E C,tfxEAvathoacacgia thie'tsau (i)V6'imQicli!C va \7XEA,tqp(YEA:c- f(x,y)E intC}Ia tqpIai; (ii) Ne'uf(x,z)- f(x,y) E intC vaf(x,z) Ii!intC, thif(x,z) - f(x,zt) E intC, v6'i mQiZt=ty+(l-t)zvatE (0,1); (iii)f(',y) Ia lien t1;tCrheahuangvai mQiYEA vaf(x,.) Ia mla lien t1;tCdu6'i v6'imQix E A; (iv)f(x,y)Ia C- t1!adondi~u; (v)vdu ki~nhac.Tan tqi tqpcompactB c A va vectaY*E B saDcho f(x,y*)E - intCv6'imQixEA\B. Khidohai roan(DVEP) conghi~m. Chungminh:f)~t S(y)={XE A: f(x,y)~ - intC }, 'If YE A. Tli giiithi6tf(X,X)EC, tac6 y E S(y),voi mQiy.Bay giOtad~t S*(y) ={XE A: f(y,x)~ intC },'If YE A. Tli (iv),tac6 S(y)c S*(y), 'If YE A. DungtinhmYa lien tl,lCdu'oi(iii), S*(y) la t~pd6ng.Do d6 S(y)c S*(y). Ti6p theo,tachungminh 23 nS*(y)c n S(y). YEA YEA Th~tv~y,la'yXE n S*(y). Do d6 f(y,x) t1.intC, \j YE A. La'yy c6 dinh,YE A, YEA d~t Yt=ty+(1-t)y,tE (0,1). Khid6 f(YbX)t1.intC,vai mQitE (0.1). (3.2.2) Ta dn ki€m traf(YbY)t1.- intC, vai mQitE (0,1).Gia su phanchungnghlala t6nt~it*E (0,1)saorho f(Yt*,Y) E - inte. Ta xethaitruonghQP. TruonghQp(1).NSuf(Yt*,x)EC, thlf(Yt*,x)- f(Yt*,Y)Einte.Tli'giathiSt (ii),tac6f(Yt*,x)- f(Yt*,Yt*)E inte. Khi d6,dof(Yt*,Yt*)E C suyra f(Yt*,X)EintC :mallthu~n(3.2.2). . TruonghQp(2).NSu f(Yt*,x)t1.C, theob6 d~tIeDthl t6nt<;tic t1.C saorho c - f(Yt*,X)E C va c - f(Yt*,Y)Ee. Theogia thiSt(i), tac6 c - f(Yt*,Yt*)E C, di~u naYc6nghlala c E C : mau thu§.n. Dod6,f(Yt,Y)t1.-intC va dotinhlien tlJCtheohuangDenf(x,y)t1.- intC, vai mQi Y E K. V~YX E nS(y ). Do d6, YEA nS(y)c n S*(y)c n S(y). YEA YEA YEA TiSp theo, tli'di~uki~n(i) t~p{YEA: f(x,y) E - intC} la t~p16i,dungphan chung,tachungminhduQcS Ia anhX<;tKKM. Han nlla, tli'di~uki~nbuc(v) ap dlJngdinhly v~KKM, tac6 n S(Y)"*0. YEA Dod6 nS(y) "* 0, YEA nghlala,bai loan(DVEP) c6nghi~m.0 24 Tadn dinhnghlasauday. DJNH NGHIA 1.3.1.Anh xC;lg:A ~ Y duQcgQila C_16irnarQngnC'uvoi IDQix,YEA, tE [0,1]ta co g(tx+(1- Oy- [tg(x)+(1- t)g(y)]E intC(tx+(1- t)y)u {O}. [Chadli- Riahi2000]dilduadi€u ki~nduchosl;(t6ntC;linghi~rncuabai tmln(DGVEP). DJNH LY 1.3.3.Chox la khonggianvectCltapaHausdorff,Y la khonggian vectCltapa,K la t(lpkhactrang,dong,lbi concuaX. AnhX(II AxA -;. 2Y, co f(x,x)=0, \iXE A va C: A -;. 2YsaDchoWYimQiXEA, C(x) la non lbi, dongco dlnh.Gidsa themcacddu sauthoa (i)f la anhX(I t1jadelndi?u ma rQng,nghzala WYimQix, YEA, f(x,y)fi! intC(x)thlf(y,x)E C(x); (ii) Vai mQix E A, (y E A: f(x,y) E Cry)}la t(lpdong vaf(x,.) la anhX(I C_l6imarQng; (iii) C la anhX(IuhctrenA; (iv)VaimQiYEA, f(',y) lientl;lctrentheohuang; (v)rbnt(Iit(lpcompactBe X vavectClYoEAn B saDchof(x,yo)E intC(x), wYimQix E K\ B. Khido,baitoan(DGVEP) co nghi?m. Chungminh:Voi rn6iYEA, ~d~thaianhxC;ldatri nhusau FJ(y) :={XEK: f(x,y)~intC(x) }, F2(Y):={XE K: f(y,x) E C(x)}. Nh~ntha'yr~ngFJ(y), F2(y)khactr6ngvoi rnQiYE A, hannuaYE FJ(y)n F2(y), VYEA. D€ chungrninhbailoan(DGVEP)conghi~rn,tac~nn FJ(Y)* 0 .Tase YEA chungrninh 25 0 * n elF1(y)c n F2(y)c n FI(y). yeA yeA yeA 1.B~ulien taCOFI(y)c F2(y)(tinh tlfadondit%u(i)). Tli (ii) suyfa Fiy) la t~pdong,do do elFI(y) C F2(y).B€ chungminh n F2(y) c n FI(y), d~t yeA yeA X*E n F2(Y), nghTala yai mQiYEA, taco fey,X*)E C(x*). C6 dinhmQty trong yeA A vad~tYt=ty+(1-t)x* yai o x*) E C(x*). Do fey!>. ) la anhX~ C_16imdfQngtaco fey!>Yt)- [tf(Yt,y) + (1 - t)f(y!>x*)] E intC(Yt) U {O}. DointC(Yt)yaintC(x*)la nonkhacf6ng,taco fey!>x*) - fey!>y) E! (f(y!>x*) +intC(Yt)U {O}) t c C(x*) +C(Yt). DoC lauhcnenyaiV la Hinc~ncuacuao trongY t6nt~ikv E (0,1)saocho C(Yt)c C(x*)+V yaimQitE (0,kv).M~tkhac,dof lientlJCtheohuangnen - f(x*, Y)E C(x*) +V. Di€u nayv~ndungyaimQiHinc~nV cuaA nen- f(x*,Y)E C(x*).Dononco dInhnenf(x*,y) ~intC(x*), nghTala X*E FI(Y)' Do do n F2(y)c n F](y). yeA yeA 2.Tiep theo,tachungminh0 * n elFI(y).B~ulientachIfadng elFI(.) yeA la anhX~KKM. La'y {Xl,...,Xn}"C A, gia sa phanchung,nghTala',t6nt~ix E CO{XI'...,xn}saochoX ~elFI(xJ, yai mQii =1,...,n.Khi do,t6nt~iAi,...)on~o n n vaLAi =1yaX=LAi Xisaocho i=1 i=1 f(x,Xj)E intC(x), yai mQi iE {l, ...,n}. Honfilla,intC(x) Ia 16inen n LAj f(x,Xj) E intC(x). j=1 26 M~tkhacf(x,.)la anhX<;lC_16imarQngva f(x,x)=0 nentaco n n - LAj f(x,xj)=f(x,x) - LAj f(x,xj) E intC(x)u {O}, j=l j=l mallthll~n,do C(x) la non co dinh.V~y elFl la anh X<;lKKM. Con l<;li,ta chI dn sad\lngdi€u ki~nbuc(v) la t6nt<;lit~pcompactBe X va vectaYoEAn B saochof(X,YO)EintC(x), voi mQi XE K\ B. Suy ra Fr(Yo)e B va elFr(yo)la compact.Ap d\lng Dinh ly KKM-Fan, ta co 0 =t-nelFr(y).Til day suyra YEA n Fr(Y)=t-0, haybailoan(DGVEP)conghi~m.0 YEA Trang[ Kum - Lee 2002],caclacgiadu'arak€t quasauchobailoanba't d£ngthucbi€n phancin(IVVI). DfNH LV 1.3.4.Xb bili roan(IWl).Gid sa X la khanggian tapa vecta Hausdoiff,Y ding la khanggian tapavectaHausdorffsao chokhanggiand6i ngauy*cuanola tachdiim.GidsaC : A ~ 2Y la anh Xf;lda trt thod man: WYi mQix E A, C(x) la nonlbi thod - intC(x)=t-0 va C(x)=t-Y. Gid sa p:= nX(x), XEX K la m(Jtt(ipcompactytu va anhXf;lda trt W : A ~ 2Y, W(x)= A\( - intC(x)), thodtinnhscha'tla dbthtcuaWdongytu trongXxY. Gidsacacddu ki~nsau daythodman: (i) T la anhXf;lC_t{tadandi~usur r(JngtuangangWYig, nghfala, v x, y E A, :J s E T(x)saocho g(s,y,x) ~ - intC(x)=>V E T(y), - get,x,y) ~ - intC(x); (ii)T lanaalient~ctrentheohuangsurr(Jngtuangungvaig,nghfala, v x,YEA vaa E [0,1},anhxf;ldatrt a Hg(T(x + a(y-x)), y,x) lanrlalient~ctrentf;li0+; (iii)vaimQix E L(X,Y)vax E A,g(s,.,x) la P_Ibi,nghfala, v y,Z E A va exE [0,1], g(s, ay +(1- a)z, x) E ages,y,x) +(1- a)g(s, z,x) -P; 27 (iv)vaim9i t E L(X,Y) va x E A, g(t, ., x) la lien t1;lCkhi cd A va Y du:(!c tranghi tapaylu; (v)vaim9ix E A vaS E T(x), g(s,x,x) E P; (vi)vaim9is E L(X,Y), x,YEA va a E fD,l], g(s, y,x + a (y - x) ) =(1- a)g(s,y,x); (vii)vaim9it(ipcanhiluhqnN cuaA, t6ntqimQt(ipcanl6i,compactylu LNcuaA chu:aN saochoV x E LN\A,:3Y E LNthod-g(t,x,y)yhu- intC(x),V tET(y). Khido,bailoan([WI) conghifm.