Luận văn Bất đẳng thức biến phân và ứng dụng

Chu'dng1 Cae ki~nthue ehu~nbi. Chuang1 : Cacki@nthucchu/inbi Lu~n van thf;Lc81Toan hQc Trang chuangnay chungt6i nhl1c19>imQt86khai ni~m,tfnh ch§,tcua t~p 16i,ham16i,hamdondi~u,anhX9>da trt dU<;1c811d\mgtrang1u~nvan. 1.1 T~p 16i.Ham 16i 1.1.1 T~p16i Dtnh nghia 1.1.1.Cia SV:X la kh6nggiantuytnlink. M c X dur;cgQila tf)p affinentuthoaman,\la,bE M,\la E R, aa+(1- a)bE M. Tf)paffinechV:aOx dur;cgQila kh6nggian concuaX. Dtnh nghia 1.1.2. Cia SV:X la kh6nggiantuytn link vax, y EX. Tf)ph{fp [x,y]:={zEX/z=ax+(l-a)y,aE [0,1]} dur;cgQila do(lnthdng[x,y]. Dtnh nghia 1.1.3.Cia SV:X la kh6nggiantuytnlink. M c X dur;egQila tf)p l6i ntu thoaman,\la,bE M, Va E [0,1],aa+(1- a)bE M. Dtnh nghia 1.1.4. Cia SV:X la kh6nggiantuytnlink. Phitn hamf : X -7 R dur;cgQilaphitnhamtuytnlinh ntu thoaman,\Ix,y E X, a,/3E R, f(ax +/3y)=af(x) +[3f(y). Dtnh nghia 1.1.5.Cia SV:x, Y lacackh6ng iantuytnlink.AnhX(lT :X -7 Y dur;egQila anhX(ltuytnlink ntu thoaman,\Ix,y E X, \la,/3E R, T(ax +[3y)=aT(x)+[3T(y). Dtnh ly 1.1.1.Cia SV:x, Y la caekh6nggiantuytntinh,A, B la caetf)p16i tmngX va{Ao}la hQtf)pl6i biukztmngX, 1 la mOtsf;th'ljCbiukz.Kh-id6 (i) noAolatf)pl6i tmngX; (ii) A +B, 1A la cactf)pl6i tmngX. Dtnh 1y1.1.2.Cia SV:X, Y la caekh6nggiantuy€n linh,T : X -7 Y la anhX(l tuy€ntinh,A la tf)p16itrongX vaB latf)pl6i trongY. Khi d6T(A) la tf)pl6i tmngY, T-1(B) la cactf)pl6i trongX. Trang3 " ? Dinh nghla 1.1.6. Gid s'u:X la khonggian tuyentinh. x E X du(JcgQila to h(Jptuytn tinh l6i cuaXl, X2,...,XmE X ntu tOntr;liAI, A2,...,Am> o th6aman 2:7:1Ai = 1 vaX = 2:7:1AiXi. Dinh ly 1.1.3.Gidsv:X la khonggiantuytntinh.A c X latiJ-plOintuvachi ntuA ch{tamQitifh(Jpl6i cuacacdiemdla A. Dinh nghla 1.1.7. Gid sv:X la kh6nggiantuytntinhvaA eX. TiJ-plOinh6 nhdtch71aA du(JcgQila baal6i cuaA, ki hieulacoA. Nh{inxet 1.1.1. TiJ-pA lOikhivachikhicoA=A. Dinh nghla 1.1.8.Gid sv:X la kh6nggiantuytntinhvaA eX. TiJ-plOi,dong nh6nhdtch71aA du(JcgQila baalOidongcuaA, ki hieucoA. Nh{inxet 1.1.2.coA=coA. 1.1.2 Non 16i Gia 811X la khonggiantuy@ntfnh. Dtnh nghla 1.1.9. (i) TiJ-pK c X dur;cgQila non(cone)ntu,Vx E K, VA> 0, AXE K. (ii) TiJ-pK c X dur;cgQila non co dinh (pointedcone)ntu K la nonkhongch71a batki du()ngthiingnaG. (iii) TiJ-pK c X dur;cgQila non lOintu K dOngthai la non va t(lpl6i. Dtnhly 1.1.4. (i) TiJ-pK c X la nonl6i khivachikhi,VA>0, AK c K vaK +K c K; (ii) TiJ-pK c X lanoncodinhntudOngthaiK lanonvaK n (-K) ={Ox}. Dtnh ly 1.1.5 (Dtnh ly Carathedory). Gid sv:Y c X la tiJ-p affine co s6 chi€u la n vaE c Y. Khi do vdi mQiX E coE, tOntr;likh6ngquan + 1 diem Xl,X2,...,Xn E E va tOntr;licacs6 AI, A2,...,An > o th6aman 2:~=1Ai =1 va X =2:~=1Aixi' Dtnh nghla 1.1.10. Gid sv:A c X. Kh6nggianconnh6nhdtcuaX, ch71aA dur;cgQila baatuytn tinh cuaA, ki hieuspanA. Trang4 D!nh nghla 1.1.11. Cia 8V:A c X, Xo E A vaX* la khonggian a6i ng6:uar,Li86 cua X. Khi do t(ip NA(XO):= {yE X*/(y,x - xo)< 0,Vx E A} (1.1) du(JcgQ'ila nonphaptuytn (normalcone)euat(ipA tr,LiXo.Ki hi~u(y,x) la gia tri euaphitn ham y E X* tr,Lix EX. ~ MiJi Y E NA(XO) du(JegQi la phap tuytn eua t(ip A tr,LiXo. Nh~n xet 1.1.3. Ntu A la t(ip 18ith1,non phap tuytn NA(XO) la t(ip fiJi, dong. D!nh nghla 1.1.12. Cia 8V:A c X lat(ip 18i, khae cPo (i) T(ipA du(JegQi la liti xa theophudngliti xa d E X, d -#Ox ntu thoamiin, VA >0,A +Adc A. (ii) T(ipgiJmcaephudngliti xad vaOx du(JegQila non liti xa euaA, ki hi~u0+A: a+A:= {dE X/A +Adc A,VA>O}. (1.2) 1.1.3 Ham 16i Gia811X la kh6nggiantuyt!ntfnh,D c X vahamf : D ~ R U {::l:oo}. D!nh nghla 1.1.13. Ta ki hi~u damf := {xE D/ f(x) <+oo}; epif:= {(x,1) ED x R/f(x) < 1}. D!nh nghla 1.1.14. Hamf du<!egQila hamehinhthudng(proper)trenD ntu damf -#cPva f(x) >-00, Vx E D. D!nh nghla 1.1.15.Ham f du(JcgQila hamliJi (tudng71nglam) trenD ntu epif la t(ipliJi (tudng71ngt(iplam) trenX x R. Nh~n xet 1.1.4. Nt'uf la ham18itrenD th1,damf la tap liJi trenX. D!nh Iy 1.1.6. Cia 8V:f la hamchinhthudngtrenD. Khi dof la ham18itrenD ntuvachintu VXI,X2E D, VA E [0,1],f(AX +(1- A)Y)<Af(x)+(1- A)f(y). (1.3) Trang 5 Dinh ly 1.1.7(B§.t diing thuc Jensen). Cia sitf la hamchfnhthuiJngtrenD. Khi dof la ham l6i tren D n€u va chi n€u, VX1,X2, ..., Xm E D, VAl, A2, ..., Am > 0: 2::1 Ai = 1, m m f(L AiXi)<LAif(Xi). i=l i=l (1.4) ~ Dinh nghia 1.1.16.Cia sitD c X la tt)pl6i,f : D -+ R vaXoE D. (i) Hamf durjcg9i la l6i tq,iXon€u, Vx E D, VA E [0,1], f(AX+(1- A)XO) <Af(x)+(1- A)f(xo). (1.5) (ii) Hamf durjcg9i la l6i ch(j,t(strictly convex)tq,iXo n€u, Vx E D, x -=IXo,VA E [0,1], f(AX+(1- A)O) <Af(x)+(1- A)f(xo). (1.6) (iii) Ham f a'Ltrjcg9i la l6i mq,nh(stronglyconvex)twi Xo n€u, Vx E D, VA E [O,l],:3p>0 th6aman, f(AX+(1- A)XO) <Af(x)+(1- A)f(xo)- pA(l - A)llx- xo112. (1.7) Ham f durjcg9i la l6i ch4t (tuangitng l6i mq,nh)tren D n€u h~thitc (1.6) (tuang itng (1.7))th6a man vdi m9i Xo ED. Dinh nghia1.1.17.Ciasitf: x -+ RU {:!:oo}va a E [-00;+00]. Cac tt)p So::= {x E X/f(x) <a}; S~:={xE X/f(x) <a}; durjcg9i la cactt)pmitccuahamf. Dinh ly 1.1.8.Ciasitf :x -+ R U {:!:oo}va a E [-00;+00].N€u f la haml6i thi m9i tt)pmitccuaf codq,ngSo::={xE X/f(x) <a}vaS~{xE X/f(x) <a} la cactt)pl6i. Nh~nxet 1.1.5.N€u m(Jthamf :X -+ R U{:l:oo}co caett)pmiteSa vaS~la caett)pl6i thi ehuaehilef la haml6i. Trang6 Vi d\l 1.1.1. Cia sV:X = R, f : x -+ R U {::1::00}xae dink nhu sau f(x) = Ilxll,\Ix E R \ {O}va f(O) = +00. Khi d6 hamf khangl6i nhungcaet(ipmitela caet(ip l6i. Dtnh nghia 1.1.18.Cia SV:f :x -+ R U {::1::oo}. Hamf du<JegQila t7/al6i tren X neumQit(ipmiteSa la t(ipl6i. Dtnh Iy 1.1.9.Hamf : X -+ R U {::1::00}lahamt7/al6i khivachikhi, \lXI,X2E X, V)..E [0;1],f()..XI+ (1- )..)X2)<max{f(xI),f(X2)}' 1.1.4 Ham lien hQp Cia s11X 1akhanggianvectcJtapa 16idtaphl1cJng,X* 1akhanggiand6i ngau tapacuaX vahamf :X -+ R. Dtnh nghia 1.1.19. Ham f* : X* -+ R xaedinkbdi f*(x*) :=sup{(x*,x)- f(x)}, xEX (1.8) du<JegQi la ham lien h<Jpeua hamf. Tli dtnhnghla(1.1.19)suyra f**(x*):= (f*(x*))*= sup{(x*,x)-f*(x*)}. x' EX' (1.9) M~nhd~1.1.1. (i) f* la hamd6ngyeu*val6i trenX* . (ii) f(x) + f*(x*) > (x*,x)\lx E X, \Ix* E X* (BatddngthiteYoung-Fenehel). (iii) f**(x)<f(x),VxE X. Dtnh Iy 1.1.10 (Dtnh Iy Fenchel-Moreau). Cia SV:X la khanggianvecto tapal6idiaphuongHausdorffvaf :X -+[-00,+00].Khi d6f** =f khivachi khif la haml6i d6ngtrenX. Trang7 1.1.5 Topoy€u. Topoy€u* D!nh nghia 1.1.20. (a) Cia sV:X lakhanggiantuy€nfinkvap :X ~ R. Hamp du(Jeg9ilanv:aehudn trenX n€u p thoaman caedi€u ki~n (i) p(x)>0,'\IxE X; (ii) p(ax)= Iiallp(x),'\IxE X, Va E R; (iii)p(x+y)<p(x)+p(y),'\Ix,y E X. (b) Cia sv:X la khanggiantuy€n finkva(foJ la h9caenv:aehudntrenX. Cdsd tane{)neuaOx, ki hi~u;3(Ox),la h9caet{)pcod(lng n f;;l( -E; E) := {x E X/llfa(x)11<E,'\IaE I}, aEI v(jiE la m(5ts6 dudngvaI la t{)philu h(ln. ? TapatrenX sinh bCfih9;3(Ox)du(Jeg9i la tapasinh bCfih9 nv:aehuanf(a). Gia sti (X, 11.11)la khanggian dinh chu§,nva X* la khanggiand6i ngautapa cuaX. Vdi moi f E X* ta dinh nghia P!(x) := IIf(x)ll, '\IxE X. ? Khi d6hQ(P!)!EX*la mothQcaentiachuantrenX. TucJngtlj vdimoix E X, hQ(qX)XEXxacdinhbdicangthuc qx(f) := IIf(x) II,'\If E X*, lamOthQmlachu§,ntrenX*. D!nh nghia 1.1.21. a) Cia sv:(P!)!EX* la m(5th9 caenv:aehudntrenX. nv:aehudn(P!)!EX* du(Jeg9i la tapay€u trenX. b) Cia sv:(qx)xEX la m(5th9 caenv:aehudntrenX* . mJ:aehudn(qX)XEXdu(Jeg9i la tapay€u* trenX*. TapatrenX xaedinkbdih9 TapatrenX* xaedink bCfih9 D!nh ly 1.1.11.Cia sv:(X, 11.11)la khanggiandinkehudn. Trang 8 Chl1dng1 : Cacki~ntlH1cchuiinbl Lu~nvanth~c81ToanhQc (i) Tapaytu trenX la tapaxac dink bdicdsd fan cQun{3cuaOx, vdi13la h9 cac tQuP co dr,mg U:= {x E X/llfi(X)11a}. (1.10) (ii) Tapaytu*trenX* la tapaxac dink bdicdsd fan cQun{3*cua0x*, vdi13*la h9 cac tQuPco dr,mg V := {f E X* /IIf(Xi)11O}. (1.11) (iii) Tapaytu trenX la tapaytu nhatmam9ianhX(Lf E X* vanlient'(tc. (iv) Cia S71vdimoix EX, phitnhamx(.) xacdinktrenX* bdicangth11c x(f) :=f(x),"If E X*. Khi dox(.) la m(Jtphitn hamtuytntinh,lient'(tctrenX theotapachutln.Ntu tad6ngnhatmoix E X vdimoiphitnhamx(.) tudng11ngthiXc X**. Vdicac girlthitt(jtrentaco,tapaytu*trenX* latapaytunhatmam9ianhX(Lx(.)van lient'(tc. Dinh ly 1.1.12. (i) DayBUYr(Jng(fa) C X* h(Jit'(tytu*vi f E X* khivachikhi,"IxEX, (fa,x)-+(f,x), vdi(fa,x) va(f, x) tudng11nglagiatri cuafa vaf t(Lix. (ii) DayBUYr(Jng(Xa)C X h(Jit'l)ytu vi x E X khivachikhi,Vf E X*, (f,xa)-+(f,x). 1.1.6 Dinh ly Hahn- Banach v~tach t~p16i Dinhly Hahn-Banachla illQttrongbadinhly cdbancuagiaitichham.C6 nhi§ud:;tngphatbi~ukhacnhauelmdinh ly nay.ddaychungt6i trlnhbaydinh lyHahn- Banach0 d1;1ngtachti;tp16i. Dinh nghia 1.1.22. Cia S71X la khanggian tuytn tinh. TQupH ~X du(jc99i la m(JtsieuphdngtrongX ntu H la tQuPaffineldnnhattrongX, nghiala khangco tQuPaffinenaGkhacX mach11ahdnH. Trang9 / ? Dinh ly 1.1.13.Cia sitX la m(jtkhanggiantuyentinhvaH la m(jtsituphang tmngX. Khi d6t6ntr;Lim(jtphitnhamtuytntinhf trenX vam(jtsf;th7/ea sao rho H ={xE X/(f,x)= a}, vdi(f, x) la giatri euaf tr;Lix. Daolr;Lintuf la m(jtphitnhamtuytntinhkhae0 trenX vaa E It th'it(ip K:= {xE X/(f,x)= a}, la m(jtsituphdngtrongX. Dinh ly 1.1.14.T6ntr;Lituangilng1-1 giftat(ipcaesituphdngkhangehila0 tren X vacaephitnhamtuytntinhf E X*\{Ox}saGehoH ={xE X/(f,x)= I}. Gia sa X la khanggianvectatapa.Khi d6dinh15'saula di;ictn1ngchocae sieuphiingvakhanggiand6ingautapaX* cuaX. Dinh ly 1.1.15.CiaS'U:X lakhang'ianvectotapal6idiaphuong.Khi d6 (i) Situ phdngH = {x E X /(f,x)=a}lat(ipd6ngkhi va chikhif lien t7,lC, nghialaf E X*,. (ii) SituphdngH (xacdinhnhua (i)) khangd6ngkhivachikhiH trum(ittmng X. Dinh ly 1.1.16(Dinh ly Mazur). Cia sitX la khanggianvectatapal6i dfa phuong,A c lat(ipl6i,ma,khaecPvav lat(ipaffinethoamanV n A =cPo Khi d6t6ntr;Lisituphdngd6ngH ehilaV saorhoH n A =cP,nghiala 3f E X*, 3/,E R, (f, v)=/,'\IvE V,(f, x)< /','\IxE A. Dinh nghia 1.1.23.Cia sitX la khanggianvectatapal6i dfaphuongvaA, B la cact(ipconcuaX. (i) T(ipA vaB dUr;fe99ila tachdUr;fCntu t6ntr;Liphitnhamf E X* saGrho sup(f,x)< inf(f,x). xEA xEB Trang 10 (ii) T~pA va B d'u(Jc99i la tachch(jtn€u t6n tr;Liphi€n hamf E X* saocho sup(f,x)< inf(f,x). xEA xEB B6 d~1.1.1.Cia SV:X la khanggianvectdtapal6i diaphudng,M c X lat~p mdvaf E X*\{Ox}.Khi d6f(M) lat~pmd. ~ D!nh ly 1.1.17(D!nh ly tach, d!nh ly Eidelheit). Cia SV:X la khanggian vectdtapal6i diaphudngX vaA c X, B c X la cact~pl6i,khaccj; thoaman di€u ki~nintA -=I- cj; va(intA)nB =cj;. Khi d6A vaB tachdu(Jc. Chitngminh. Vi A la t~p16ilien intA la t~p16i.Vi (intA) n B = cj; lien t~p U :=intA - B la ti;i,pm0,kh6ngclItiaOx.Theodinhly Mazur,t6nt1;tisieuphllng d6ngH clItiaOx saoclIo (intA - B) nH = cj;. Gia S11phi@nham f E X* va f-l(OX) =H. Vi intA - B la t~p16ilien f(intA - B) la khoangmdkh6ngclItia O.Do d6 f(intA - B) <0; sup f(x) < inf f(x). xEintA xEB Vi intA tru m~ttrongA va f lient\lClienSUPxEintAf(x) =SUPxEAf(x). V~y sup <inf f(x). D xEA xEB ? D!nh nghia 1.1.24. Cia SV:X la khanggianvectdtapa,A c X vaXoE A. Diem Xo du(Jc99i la ditfmt7/a(supportpoint) cua t~pA n€u t6n tr;Liphi€n hamf E X* vas6th7/c'"'Ithoaman (i) f(xo) ='"'I; (ii) f(x) <'"'I,\:IxE A va :3xE A, f(x) < T Khi d6siev,phiingH := {x E XI (f, x)= '"'I}du(Jc99i la sieuphiingt7/a(support hyperplane)cua t~pA tr;LiXovaf au(Jc99ilaphi€n hamt7/a(supportfunctional) cuat~pA tr;LiXo. D!nh nghia 1.1.25. Cia SV:X la khanggianvectdtapavaX* la khanggiand6i ngaucuaX. Phi€n hamS :X* ~ R du(Jc99ilaphi€nhamt7/a(supprotfunction) cuat~pA n€u rOH.IJ.H.TlJN~~IEN S(f) = Sup(f, x),\:IfEX*. i IH!! \!!E~:xEA ":'-. L ~l400Trang11 D!nhly 1.1.18.Cia S'l1X la khanggianvectatOpal6i diaphuong,UjpA c X l6i,d6ngc6intA=IcPo Khi d6m9idiembiencuaA d€u ladiemt7/a. D!nh ly 1.1.19. Cia S'l1X la khanggian vectdtOpal6i dia phuongva A eX, B c X la cact(ip l6i, d6ngthoamanB la t(ipcompactva An B = cPoKhi d6A vaB tachchijI ~ Ch71ngminh. GQi(3laeoSdIane~nelmOx,(3g6meaet~p16ituy~td6ivama.Gia S11,\IV E (3,(A+V)nB =IcPo Khi d6hQ{(A+V)nB IV E (3}lahQcact~pcompact c6tfnh ch~tla mQigiaohuu h1;1nd~ukhaccPoSuy fa nVE,e(A + V) n B =IcPo Do d6t6n t1;1iXoE B va XoE A + V c A + 2V,\IV E (3.Suy ra Xola di~mt1,lcuaA va XoE A (do A la t~pd6ng).Tli eaedi~utren suy ra XoE A n B (mauthuan voigiiithi@tAn B =cP). Vi;iyt6n tC;LiV E (3SeWcho (A +V)nB = cPo ChQn U =~Vthl(A+U)n(B+U)=cPoTh~tv~y,n~uc6x E (A+U)n(B+U) thl t6n t1;1ia E A, b E B va VI,V2E V saocho x = a + ~VI= b+ ~V2'Suyfa b=x - ~V2=a+~VI- ~V2E (A+V)nB (mauthuanv6i(A+V)nB =cP).2 2 2 Vi;iyhaiti;ipA +U vaB + U thoamancacgiii thi@tcuadjnhIf tachEidelheit, dod6A +U vaB +U tachdl1<;c,nghlala t6nt1;1iphi~nhamJ E X* saocho Sup(J, x)< inf (J, x). xEA+U xEB+U Do d6 sup(J,x)<inf(J,x). D xEA xEB Nh~n xet 1.1.6. Cia thiefv€ tinh compactcuaB la canthiefngaycakhi X la khanggianhiJ:uh(ln chi€u. Vi dl,l 1.1.2. Cia S'l1X = R2;A := {(x,y) E R2/y > .!.;x > O} va B :=x {(x,0)Ix E R}. Khi d6 m9i gia thietcua dinh ly (1.1.19)d€u thoa tril gia thief compactcuaB. Hai t(ipA va B trongtrudnghrjpnaykhangtachdurjc. H~qua 1.1.1.Cia S'l1X la khanggianvectdtapa16idiaphuong.Khi d6 ta c6 cackhdngdinhsa'u. (i) M9i t(ip l6i d6ngtrongX lagiaocua,tatcdcacn'l1akhanggiand6ngch71an6. (ii) M9i t(ip16id6ngtrongX d€u la t(ipd6ngyeu. Trang12 1.2 D~oham cua anh x~ 1.2.1 D~oham theo huang Djnh nghia 1.2.1. (Xem [23))Gid S71X la khonggianvectdva (Y, 11.11)la khong ? / giandink chudn,A c X la tt;ipkhacc/J,XoE A, hEX vaanhxq,f :X ~ Y. Neu gidihq,n f' (xo)(h) := Hm ~ (f(xo+Ah)- f(xo)),>'---+0+/\ (1.12) t6ntq,ithi f'(xo)(h) du(Jc99i la dq,ohamtheohuangcuaanhxq,f tq,iXotheo huangh. Ntu vdim9ihEX gidihq,nf'(xo)(h) luont6ntq,ithif du(Jc99ila khd vi theohuangtq,iXo. Djnh ly 1.2.1. Gid s71X la khonggianvectd,A c X la tt;ipl6i, khacc/Jva f:X~R. (i) Gids71XoE A la diim c7/ctiiu cuaf trenA. Ntu anhxq,f codq,ohamtheo huangtq,iXotheom9ihuangx - Xo,vaix E A, th1, f'(xo)(x- xo)>0,\/x E A. (1.13) (ii) Ntu anhxq,f la haml6i vacodq,ohamtheohuangtq,iXoE A theom9ihuang x - Xo,vdix E A va f'(xo)(x - xo)>0,\/x E A, thiXola diim c7/ctiiu cuaanhxq,f trenA. Ch(cngmink. (i) Tv gia thi§t ant X<;1f co d<;1ohamtheohudngt<;1iXotheomQihudngx - Xo BUY ra vdi moi x E A, ta eo f'(xo)(x - xo)= Hm~ (f(xo+A(X- xo))- f(xo)).>'---+0+/\ Do Xola di@meveti@ueuaf trenA lienvdiA >0dulito, taco f(xo+A(X- xo))>f(xo). V~y f'(xo)(x- xo)>0,\/x E A. Trang13 (ii) Tli f la ham16isuyra v6i mQix E A,v6imQiA E [0,1],ta co f(xo +A(X- xo»= f(AX +(1- A)XO)<Af(x)+(1- A)f(xo). Do do 1 f(x) > f(xo)+A(f(xo+A(X- xo» - f(xo»). Vi f co d<;1ohamtheohuangt<;1iXotheomQihuangx - Xonen f(x) >f(xo)+f'(xo)(x- xo). K@thQpvai gia thi@tf'(xo)(x - xo)>0,\/xE A, taduQc f(xo)<f(x),\/xE A. V~yXola di@mQic ti@ucuaanhX<;1f trenA. D 1.2.2 D~oham Gateaux va d~oham Frechet Dinh nghia 1.2.2. (Xem[23j)Girl8U'(X, 11.llx)va (Y, 11.lly)la cackh6nggian dinkchuan,A c X lat4pmd,khac4;,XoE A vaanhX(,Lf : A -t Y. N€u, \/h E X, gidi h(,Ln . 1 f'(xo)(h) := hm \ (f(xo +Ah)- f(xo»),A-->O/\ (1.14) t6n t(,Liva f' (xo)(h) la anhX(,Ltuy€n link lien t'I,LCtitX VaGY thzf' (xo)(h) dur;c gQila d(,LohamGateauxcuaf t(,LiXova anhX(,Lf dur;cgQila khrlvi Gateauxt(,LiXo. Dinh nghia 1.2.3. (Xem[23j)Girl8U'(X, 11.llx)va (Y, 11.lly)la cackh6nggian dinkchuan,A c X la t4pmd,khac4;,XoE A va anhX(,Lf : A -t Y. N€u t6ntq,i rinhxq,tuy€n link lien t'I,LCf'(XO) : X -t Y thoaman 1. Ilf(xo+h)- f(xo)- f'(xo)(h)11 01m -t , Ilhll-->O Ilhll (1.15) thzf'(XO) dur;cgQila dq,ohamFrechetcuaf tq,iXova anhxq,f dur;cgQila khrlvi Frechettq,iXo. M5i lienh~giUad<;1ohamFrechetvad<;1ohamGateauxth@hi~ntrongk@tqua sail. Trang14 ?D!nh 1:5'1.2.2. Gia sV:(X, 11.llx)va (Y, 11.lly)la cac kh6nggian dink chuan, A c X la t(ipl6i, md,khacrjJ,XoE A vaanhX(lf : A ~ Y. Khi d6n€u f khavi Fnichet t(li Xo thzf ding kha vi Gateauxt(li Xo va hai d(lo ham nay trung nhau. M6i lien h~giila d1;:Loham Frechetva tint lien t1).C,tint 16icua ant X1;:Lth@hi~n trongcacdint ly sau. ? D!nh 1:5'1.2.3. Gia sV: (X, 11.llx)va (Y, 11.lly)la cac kh6ng gian dink chuan, A c X la t(ipmd, khacrjJ,XoE A va anhX(lf : X ~ Y. Khi d6 n€u f khavi Fnichet t(li Xo thzf lien t7;,ct(li Xo. D!nh 1:5'1.2.4. Gia sV:(X, 11.11)la kh6nggian dink chuan,A c X la t(ipMi, md, khacrjJva anhX(lf : A ~ R kha vi Fnichet t(li m9i diim Xo E A. Khi d6f laham l6i n€u va ch:tn€u, Vx,YEA, f(y) >f(x) +f'(x)(y - x). (1.16) Changmink. (i) Gia S11f la ham16i.Khi d6 Vx,YEA, V)'"E [0,1] f(x +)...(y- x) <)...f(y)+(1- )...)f(x). Tli d6 ta c6 1 f(y) > f(x) +)...(f(x) +)...(y- x)) - f(x)). VI f khavi Frecheti;1imQix E A nen f'(x)(y - x) = lim ~ (f(x + )...(y- x)) - f(x)).A~O+/\ Do d6Vx,yEA f(y) >f(x) +f'(x)(y- x). (ii) Dao11;:Li,gia S11(1.16)thoaman.Do A la t~p16inen f(x) >f()...x+ (1- )...)y)+ f'()...x+ (1- )...)y)((l- )...)(x- y)) va f(y) >f()...x+(1- )...)y)+ f'()...x+(1- )...)y)((-)...)(x- y)). Trang15 Vi J'(x)(.) la anhXi:Ltuy§n tfnh nell,'\Ix,YEA, VA E (0,1], Af(x) +(1- A)f(y) > Af(AX+(1- A)Y) + A(1- A)J'(AX + (1- A)Y)(X- y) + (1- A)f(AX + (1 - A)Y) - A(1- A)f'(AX + (1 - A)Y)(X- y) - f(AX + (1- A)Y). V~yf la ham16i. D D!nhly 1.2.5.Giasit(X, II.IDlakh6ngiandinkchutinvaanhxq,f :X ---t R. / ?, NeuXoE X la diemc7jCtieucuaf trenX vaf khavi Gateauxtq,iXothz,VhE X, f'(x)(h) =O. (1.17) H~thlic (1.17)la di@uki~nc~nd@Xola di@mC1!Cti@ucuaanhXi:Lf. 1.2.3 Du'di vi phan Gia sa (X, 11.11)la kh6nggiandinh chuiin. D!nh nghia 1.2.4. Gia sit anhxq,f : X ---t R lahaml6itrenX vaX* lakh6ng gianddingautap6cuaX. V6imtJiXoEX, t4p 8f(xo):={x*E X*/ f(x) > f(xo)+x*(x- xo),'\IxEX}, (1.18) dvx;c99i la du6ivi phancua anhxq,f tq,iXova mtJiphi€n hamx* E 8f(xo) dUr;fC 99i la du6igradientcua anhxq,f tq,iXo. Vi d\l 1.2.1.Giasit(X, II.IDlakh6ngianBanachvaf(x) =Ilxll.Du6iviphan rinhxq,f tq,ix E X dur;fCtinhnhusau. . N€u x -#0 thz 8f(x) = {x*E X*/llx*11= 1,(x*,x)= Ilxll}. . N€u x =0 thz 8f(0)={x*E X* /llx*11< I}. Trang16 Dij,cbi~tneugid siJ:X = R vaf(x) = Ilxllthi 8f(x) = {{llxll-lx}, n~ux #0;(-i,i), neu x - o. Dinh ly 1.2.6. Gid siJ:anhxq,f : x ~ R la hamlei trenX. Khi d6XoE X la 7 7 diem qtc lieu cuaf trenX khi va chi khi Ox E 8f(xo). 1.2.4 D~o ham Clarke Cia S11(X, 1.11)180kh6ng gian dinh chu11n. Dinh nghia 1.2.5. (Xem (23j) Gid siJ:anhxq,f : X ~ R, Xo E X vahEX. Neu gidi hq,n f'(xo)(h)= limsup ~(f(x +Ah)- f(x)), x ~ Xo A ~ 0+ (1.19) ten tq,ithi j' (xo)(h) dU(fCgri la dq,oham Clarke cuaf tq,iXotheohudngh. Neuf c6dq,oham Clarketq,iXo theomri hudnghEX thi f dU(fcgri la khavi Clarke tq,iXo. Vi d\l 1.2.2. Gid siJ:f : R ~ R dU(fCxac dink nhusauf(x) = Ilxll,"Ix E R Khi d6 dq,oham Clarke tq,i0 cuaf theomri hudngh la j'(O)(h) = limsup ~(11x+Ahll - Ilxll)= Ilhll. x~O A ~ 0+ Dinh ly 1.2.7.GidsiJ:f : X ~ R la haml8i, lien t7,lCLipschitztq,iXoEX. Khi d6dq,ohamClarkecuaf tq,iXovadq,ohamtheohudngcuaf tq,iXo la trungnhau. 1.2.5 Non ti~p xuc Dinh nghia 1.2.6. Gid siJ:(X, 11.11)la kh6nggiandink chulin,A c X la t(ipkhac q;vaXoE A. Trang17 (i) Vectdh du(Jc99i la vectdti€p tuytn cuat(j,pA t(LiXontu tiJn t(Liday(xn)c A, day(An)C R+ th6aman Xo= lirnXn,h = lirnAn(xn- xo). n->oo n->oo (1.20) (ii) T(j,ph(JpTxoAcacvetdti€p tuytn cuaA t(LiXodu(Jc99i la non titp xucBouli- ~ gandtheodaycuaA t(LiXoho(icnon contingentcuaA t(LiXo. Dinh ly 1.2.8.Cia s'll(X, 11.11)la kh6nggiandinkchutinvaA c X la t(j,pfiJi, khaccp.Khi dovd'im9iXoE A non titp xuc contingentTxoA la t(j,pl6i. 1.3 , Anh xa da tri. . 1.3.1 MQt 86khai ni~m Cia s11X, Y la cackhanggianvectatapa,L(X,Y) la khanggiancacanhX9- tuy@ntinhlient\lCtti'X vaaY vaF :X ---+2Y la anhX9-da trio Dinh nghia 1.3.1. (i) Mien hi~uquacua anhX(Lda tri F la t(j,p damP:={xE XI F(x) -#cp}. (ii) DiJ thi cua anhX(Lda tri F la t(j,p graphF:= {(x,y)E X x Yly E F(x)}. Dinh nghia 1.3.2. Anh X(Lda tri F du(Jc99i la liJi ntu graphF la t(j,pliJi tren X x Y va F du(JC99i la dongntu graphF la t(j,pdongtrongX x Y. Dinh nghia 1.3.3. (i) AnhX(Ldatri F du(Jc99ila anhX(Ldatri giatri liJi ntuF(x) la t(j,pliJi voim9i x E damP. (ii) AnhX(Ldatri F du(Jc99ila anhX(Ldatri giatri dongntu F(x) la t(j,pdong trongY voim9ix E damF. Dinh nghia 1.3.4. Cia S'U:F : X ---+2Y la anhX(Lda trio Trang18 (i) Anh xg, F dUf/C99i la mJ:a lien t'l),Ctren (vi~t tiit la u.s. c) tg,i Xo E damP n~u vdi m9i lan c~nN cila F(xo), tOntg,i[(in c~nM cila Xo saD rho F(M) c N. (ii) Anh xg,F duQc99i la mJ:alien t'l),Cdudi (vi~ttiit la l.s.c) tg,iXo E damP n~u vdim9it~pmdU c Y thoamanUn F(xo) =Fcp,tOn tg,i lan c~nM cila XosaD rho Un F(x) =F cp, \:Ix E M. ~ (iii) Anh xg,F dUf/C99i la nita lien t'l),Cdudi theoday (vi~ttiit la s.l.s.c) tg,iXodamP, n~uvdi m9i Y E F(x), m9i day (xn) c damP vaXn -+ x, tOntg,iday(Yn)C F(xn) saorhoYn -+ y. (iv) Anh xg,da tri F duf/C99i la hemi lien t'l),Ctren (vi~ttiit la u.h.c) tg,iXoE damP n~uvdi m9i x EX, anh xg,da tri a I---tF( ax + (1 - a)xo) la anh xg,nita lien t'l),C tren tg,i0+. (v) Cia sit t~pA c X va T : A -+ 2L(X,Y) la anh xg, da trio Anh xg, T dUf/C99i la hemi lien t'l),ctren suy r{)ng(viti tiit la g.u.h.c) tg,iXo E A n~uvdi m9i x E A, vdi m9i a E [0,1], anh xg, da tri a I---t(T(ax +(1- a)xo, x - xo) la anh xg,U.S.C tg,i 0+, trong do (T(x), z) la gia tri cila anh xg,tuy~ntinh T(x) E L(X, Y) tg,idi€m z E X. (vi) Cia sit t~pA c X vaT : X -+ 2L(X,Y) la anh xg,da trio Anh xg,T duf/C99i la hemi lien t'l),Cdudi suy r{)ng(vi~ttiit la g.l.h.c) tg,iXo E A n~uvdi m9i x E A, vdi m9i a E [0,1],anh xg, da tri a -+ (T(ax +(1- a)xo, x - xo) la anh xg,l.s.c tg,i 0+. Dinh nghia 1.3.5. Cia sit t~pA eX, Y la khonggian tuy~ntinh dUf/Cslip bdi non th(Ctv:C trongd6 C : Y -+ 2Y la anh xg,da tri thoa man, vdi moi Y E Y, Cry}la nonlOi,dong,intO =Fcp. (i) Anh xg,ddntri f : A -+ L(X, Y) duf/C99i la ddndi~u(monotone)trenA n~u vdi m9i x E A, z E A (j(z) - f(x), z - X)E Y\( -intC(x)). (1.21) (ii) Anh xg,da tri T : A -+ 2L(X,Y) duf/c99i la ddn di~u (monotone) tren A n~u vdim9i x E A, z E A, vdi m9i tx E T(x), tz E T(z) (tz - tx, z - X)E Y\( -intC(x)). (1.22) Trang 19 (iii) Anh X(Lda tri T : A ---+2L(X,Y)du(Jc99i la gid ddn difU (p8eudomonotone) trenA neuv{jim9ix E A, z E A [~sE T(x), (s, Z - X)E Y\( -intC(x))] =* [ViE T(z), (t,z - X)E Y\( -intC(x))]. (1.23) (iv) AnhX(Ldatri T : A ---+2L(X,Y)du(Jc99i la gidddndifUyen(weakp8eudomono- tone)trenA neuv{jim9ix E A, z E A [~sE T(x), (s,z - X,)E Y\( -intC(x))] =* [~tE T(z),(t,z - X)E Y\(-intC(x))]. (1.24) (iv) Gid 811T : A ---+2L(X,Y)la anhX(Lda tri va f : A x A ---+Y la anhX(Lddntrio C~p(T,f) du(Jcg9ila c~pgidddndifUtrenA neu,v{jim9ix E A, z E A, [~sE T(x), (s,z - x)+f(z, x) E Y\( -intC(x))] =* [ViE T(x),(t,z - x)+f(z,x)EY\( -intC(x))]. (1.25) (v) Gid 811T : A ---+2L(X,Y)la anhX(Ldatri vaf : A x A ---+Y la anhX(Lddntrio C~p(T, f) du(Jc99i la c~pgid ddndifU yentrenA neu,v{jim9ix E A, z E A, [~sE T(x),(s,z - x)+f(z,x)E Y\( -intC(x))] =* [~tE T(x), (t,z - x)+f(z, x) E Y\( -intC(x))]. (1.26) Dtnh Iy 1.3.1. (i) Neu anhX(Lda tri F : X ---+2Y la anhX(Lda tri gia tri dongva n11alien t7j,C trenthi F la anhX(Ldong. (ii) Neu F(A) la t(ipcompactv{jim9iA c domF la t(ipcompactva F la anhX(L dongthi F la anhX(Ln11alien t7j,Ctren. 1.3.2 ., .. Cae dtnh Iy diem bat dQng Cia 811X la khanggianvectatapa. Dtnhnghla1.3.6.(Xem[3J)Gid 811A c X vaF : A ---+2A la anhX(Ldatrio Diim XoE A du(Jc99i la diembatd(Jngcuaanhxq,datri F trenA neuXoE F(xo). D~cbift neuF la anhX(LddntTi thi Xo=F(xo). Trang 20