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CHUaNG I : B~C TOPO CUA NHUNG TOAN TV L - Hoan toaD lien t.,.c I TOAN TV L - HoAN ToAN LIEN T1)C : 1.1.Tmin tii Fredholm va bai tmin ghi tri bien tuye'ntinh D;nh nghia 1.1 : Cho X, Z la nhung khong gian dinh chuftn th1!c,va ky hi~u \ I chuftn tu'dng ling cua chung la MOt anh x~ tuye'n tinh : L : domL c X ~ Z Voi ;.1 KerL = L {a} va.lmL = L (domL) du'QcgQi la mOt anh x~.Fredholm, ne'u hai di~u ki~n sail xay : (i) KerL co s6 chi~u hOOh~n (ii) ImL la dong va co s6 d6i chi~u hOOh~n (So' d6i chi~u cua ImL la s6 chi~u cua ZhmL, nghla Ia so' chi~u cua d6i h~ch CokerL cua L) Khi L la anh x~ Fredholm, chi s6 cua no ky hi~u IndL la s6 nguyen, du'Qcxac dinh bai IndL = dimKerL - CodimlmL Tinh cha't 1.2 : Tli dinh ilghla tren va til' nhii'ng ke"t qua cd ban cua giai tich ham tuye"n tinh, t6n t~i nhung phep chie'u lien tl}c P : X ~ X, Q:Z~Z SaD cho Imp = KerL, KerQ = ImL VI the' x = KerL EE> Kerp Z = ImL EE> ImQ la t6ng tn!c tie'p tapa Khi d6 slf thu hyp Lp cua L tren domL n KerP la anh x~ - va anh x~ daD (d~i so') Kp : ImL ~ domL n KerP duQc xac dinh Ky hit$u : Kp,Q: Z ~ domL n KerP duQcgQila t6ng quat h6a nguQccua L, dinh nghla bdi Kp,Q = Kp (I - Q) X6t bai toan vi phan thuong (*) d day x'(t) = (d/dt) X1(t)=f(t), tEI=[O,l] { Mx(o) +Nx(l) = C x(t), f ELI (I, Rn) khong gian cua nhii'ng anh x~ kha tich Lebesgue tu vao Rn, cERn, va M, N la nhii'ng ma tr~n thlfc vuong cgp n Ta d~t : X = C (I, Rn) khong gian cua nhii'ng anh x~ lien t1;lctu I vao R n, domL = {x EX: x lien t1;lCtuyt$t do'i tren I}, L: x ~ (x', M XeD) + N xCI)), Z = L (I, Rn) x Rn g = (f, c) Thl bai loan (*) tuong duong vdi phuong trlnh thu gQn Lx Ngoai = g = {x EX: x la anh XC;lh~ng va (M + N) x(0) = O}, d6 dimKerL = dim Ker (M + N) KerL N6i mQt cach khac, phuong trlnh dftu (*) tuong duong vdi t x(t)=x(o)+ ff(s)ds, tEl, VI the- phuong trinh thli' hai (*) duQc vie-t (*.1) (M+N) x(o) =c - N ff(s)ds, , ta suy duQc ImL = {(f'C) EZ:c-N = A-I !fEIm(M+N)} (Im-(M+N)) Vdi A duQc dtnh nghla bdi anh XC;l tie-ptuye-n A : Z ~ Rn A(f,c) =c - f N f(s)ds, Vdi chu~n thuong dung tren L\1, Rn) va Rn va chu~n tich tuong ling tren A, thl A lien tl;1C, ta suy ImL la d6ng Z Ta chli'ngminh L la mQtanh XC;l Fredholm chi s6 zero Ta c6 kerA la d6ng va loan anh tren Rn, VIthe-codim KerA Z =Ker A E9 U (t6ng tOpo) Vdi : dim U = n = n va Do d6, ne'u AU la A thu hyp tren U, thl A-I (Im(M + N)) = Ker A EBAU1(Im(M +N)) = Ker A EBV Voi dim V = dim 1m (M + N) VI the' Codim ImL =n - dimlm (M + N) = dim Ker (M + N) = dim Ker L V~y L la anh x~ Fredholm chi s6 zero Chung ta chi fa cach xfiy dvng nhii'ngphep chi€u P va Q cho ImP = KerL ImL =kerQ va t6ng quat h6a ngu'Qctu'dng ung cua chung f)~t S : Rn ~ Rn la mQt phep chi€u, cho ImS =kef (M + N) va d~t (M + N)s la sv thu hyp cua (M + N) tren Kers, thl (M + N)s la mQt song anh tit KerS vao 1m (M + N) Th~t v~y : Ta c6: dim Ker (M + N) + dim 1m (M + N) =dim KerS + dim ImS, Suy dim KerS =dim 1m (M + N) f)~ chung minh (M + N)s la mQt song anh, chi c~n chung minh (M + N)s la ddn anh Ta co Ker (M + N)s = {x EKerS/ (M + N)Sx = O} = {x/ x EKerS,(M +N)x = O} = {x/ x EKerS,x EKer(M +N)} = -{x/ x EKerS,x ElmS} = {a} Ne'u (f, c) E lmL, thl phuong trlnh (* 1) tu'ong du'ong voi x(oJ = S (X(OJ)+(M+N)S{C-Nlf(SJdS) bi~u thuc tren cho nghit%mcua bai loan gia tri bien sail x(t) =S (x(o)) + If(S)ds+(M+N)S{C-N !f(S)dS) Ngoai fa, ne'u ta dinh nghla Ps la phep chie'u tren X bdi Ps(X) =S(x(o» Voi ve' phcii la anh X~ hang X, nh~n gia tri S(x(o», voi mQi (f, c) E lmL, Taco: (*;2) (Kps (f,C»)(!) = {f(S)ds+ = f(s)ds+ (M + N)S l(c - N ! f(S)dS) (M + N)Sl.A(f,c),(t EI) E)~tT la mQt phep chi€u Rn, cho ImT=Im(M+N) va dinh nghia QTtren Z, bdi QT (f,c) = (0, (I - T)A (f,cn Do (f, c) E ImL ne'u va chI ne'u = Ker QT ImL Ngoai QT la mQt phep chie'u lien tl;lc,tit (* 2) ta co (KPs,QT(f,c»)(t) = f(s)ds+ (M + N)SIT( c - N !f(S)dS] = If(S)ds+(M+N)slTA(f,C), tEl Do sv bi~u thi cua nhii'ng phep chie'u Ps, QT va t6ng qmit hoa nguqc KpS' QT nhung s6 h,:;mgcua nhung phep chie'u Rll vao Ker(M+N) va 1m(M + N) = 0, Khi C ta xet loan tii' triI'u tuqng cho bai tmin vi phan thuong, ta d~t x = {CO,R ll):Mx(o) + Nx(1) = O} Z = Ll(I,Rn) dam L= {x E~ va dinh nghia X : x lien tl;lctuy~t d6i tren I} L bdi ~ Lx =x' Dodo: Ker L = { XEdam L : x la anh x';l h~ng gia tri no ph1;}thuQc vao Ker (M+N)} 10 Luc bai toan (*) voi C = 0, tuang duang voi X(t) = x(o) + {f(S)dS { Mx(o) + Nx(l) =0 Ta c6: Imi:.={f EZ:N If(S)dS EIm(M+N)} = B-1 (1m (M + N)) - ? d ds.y B : Z ~ Rn la anh x~ tuy6n tinh lien t\lC,duQcxac dinh bdi Bf = N 1: f(s)ds VI v~y : - =dim Ker (M + N) dim Ker L la hUllh~n va 1m1- la d6ng, t~i m6i anh x~ f~ 1:f tu Z vaG Rn la to~m anh, Ta c6: Codim1m 1- =dim (lmN/ImN n1m(M + N)), d6 cong thuc L khong nhftt thi6t la chi 86 zero, nhien, n6u del (M + N) "*0, Thl KerL = {O} Va - =dim (ImN limN) Codim 1m L =0 VI v~y [, la kha nghich.Ne'u rank (M, N) - = n L la chi s6 zero, thl [ cling kha nghich Di~u kit$n cu6i cling la di:icbit$t thoa man cho tHrong hc;1pnhG'ng di~u kit$nbien tu~n hoan x(o) - x(1) =0 Vdi - =n =codim 1m L- dim Ker L Ta dinh nghia Ps tren x, vdi S la mQt phep chie'u Rn sac cho ImS = Ker (M + N) Bdi Psx = S (x(o)) -Xc6 gia tri hiing S(x(o)) Ta c6 mQt ? day ve' phai Ja ph~n td' cua phep chie'u lien t1;1csac cho - - 1m Ps = KerL N6i cach khac, Iffy V la mQt phep chie'u Rn sac cho ImV =N-l (1m (M + N)) va dinh nghia toan td' Qv Z bdi QVf r1 = (I - V) \/ - ? - day ve' phai la ph~n td' hiing Z, thl Qv la mQt phep chie'u lien t1;1ctrong Z sac cho : KerQv = {f eZ: £ f(s)ds =V £ f(S)dS} 12 = {f EZ,:£ f(s)ds EW1(Im(M + N))} {f EZ:N If(S)dS Elffi(M+N)} ; , , =ImL Ne'u S la phep chie'utrong Rn duQcdinh nghla nhu tren va ne'u fEZ, thl ta co (Kps'Qv f)(t) = -(M +N)SlNV J~ f(s)ds+ + £(f(S)-(I- V) f(U)dU)ds, t eI Trong truong hQp d~c bi~t cua nhung di~u ki~n gia tri bien tuftn hoan, ta co M=-N=I VI the', ta la'y S = I, V = Ta co , , PIx , , Qof Vii (K~'Qo f)«()= £(f(8)Khi fEe = x(o) r1 = b f(s)ds £ f(U)du}S (1, Rn), ta Iftn lUQtthay Z va Z bdi C (1, Rn) x Rn va C (1, Rn) va thay di~u ki~n lien tl,1Ctuy~t d6i mi~n cua phftn tuye'n tinh bdi tinh khii vi lien tl,1cma khong thay d6i ke'tlu~n va c6ng thllc tren 13 1.2 Tmin tti L - Compact: Dillh Ilghfa 1.3 Cho L : domL c X ~ Z la mQt anh x~ Fredholm, E la mQt khong gian metric va G : E ~ Z la illQt anh x~ Chung ta noi G la L - compact tren E, ne'u nhung anh x~ : QG : E ~ Z va Kp, QG : E ~ X la compact tfen E, nghIa la chung lien tl;1Ctren E va QG(E) va Kp,QG(E) compact tu'dng dot Ta co th~ chung to ding, dinh nghla khong phl;1 thuQc vao sl;1'chQn nhung phep chie'u lien tl;1cP va Q Dinh nghfa 1.4 A.nhx~ G: X ~ Z, du'QcgQila L -: hoan loan lien tl;1C, ne'u no la L compact tren mQit~p bi ch~n E c X Tinh cha't 1.5 : (nhilng di~u kit$ncaratheodory) Gia sa : f : x C ~ RD (t,