Luận án Về tính Fredholm của toán tử vi phân tuyến tính đối số chậm

CHUdNGI - ~ ?, ? TIEU CHUAN FREDHOLM CUA TOAN TO' VI PHAN TUYEN TINH eol SO CH~M. Trong chudng nay ta se chung minh dieu ki~ncanva duve tfnh Fredholmcua cac loan tv vi phElntuyen Unh d6i 56 chc;\mlac dvng trongkh6nggiancac ham khavi lien tvc tren A. , -" A ? ~1-CAC E>INHNGHIA VA KHAI NleM cd BAN 1- Ky hi~uC - kh6nggiancac hamlientvc bi chi;intren A. x : A ~ An vdi chu§n IIxlic= supllx(.)11 Ky hi~uC1 la kh6ng gian cac ham x E C sac cho x EC vdi chuan IIxlic. = IIxlic +lI~t Co la kh6nggiancac ham lien tvc x : [-h,0] ~ Rn (h > 0) . Cho E, F la 2 kh6nggian Banach. L - loan tv tuyen Unh : E ~ F co mien xac dinh D(L) c E va miengia tri A(L) c LF Toan tv L duQc gQi la dong neu tu Xn~ x, Lxn'~ Y suy ra x E D(L) va Lx =y. Toan tvL duQcgQila giai chu§n lac neu R(L) =R(L). [8] Toan tv tuyenUnhdongL duQcgQila n - chu§nlac neuL la giaichu§n lac va dim(kerL)< 00 L duQCgQi la d- chuan lac neu L la' giai chuan lac va dim(cokerL)< 00 Toan tv tuyentfnhdongL duQcgQila FredholmneuL vuala n- chuantaGvua la d- chuan taGtUGL giai chuan taG,nhan cua L la kh6nggian hG'uh(~lnchieu kh6nggian thudngF/ A(L) la kh6nggian hG'u hc;ln chieu. ' 3 Trang khong gian C1 xet loan tU vi phan tuyen tfnh doi so ch$m d9ng 0 (Lx)(t) =~(t)+ J dsG(t,s)x(t +s) -:;11 (II> 0) 2-Cae gia thiet cd ban Gia sv ma tr$n ham vuong cap n G(t,s) thoa man cac dieu ki$n sau : 1)G(t, -h) =0, G(t, s) =G(t, 0), vdi mQis ;?:o 2) G(t, s) co bien phan hO'uh9n thee s E [-h, 0] deu vt E R 0 tCtclasup v (G(t,s))~M <00 s=-II 3) G(t, 0) lien tl,lc deu thee t, G(t, s) lien tl,lcdeu trung blnh theo t (voi s - co dinh), Wc la' 0 rim sup fIIG(t+ill,s - G(t,s)llcis=0 N-->O I --II Voi gia thiettren,toantv A co d9ng : 0 (Ax)(t) =J dsG(t,s)x(t+s) -II co tfnhchat gioi h9nt9i vo cvc 'J Wc tont9iAdLi<JCxac dinh nhLlsau : (Ax) = klim(SrkAS_rkx)(t) ,x EC deu tren moi khoang hO'uh9n,~co trangdo 'tk ~ CIJ (8, x)(t) =,x(t+ 't) 0 va (SrAS-rx)(t) =JdsG(t +-r,s)x(t+s) -II Do do ta co t$p H(A) ={A }va A co d9ng : - 0 - - (A x )(t) = fcisG(t,s)x(t +s) voi G(t,s)thoamancac dieu ki$n -II tLldngtLj nhLlG(t, s) . 4 ClIngvdiloantli L. Xetloantli L dc;lng: (Lx)(t) ==x(t) +(Ax)(t) ,A EH(A) hay L EH(L) Xet phuongtrlnhthuannhat: x(t) +(Ax)(t)=0 (1) DEi biet [9] neu XOE Co thl ton t<;1iduy nhat nghi~m x(t) eua phuongtrlnhthuannhattrem(0,+00)saocho: x(t)=xo(t), - h ~ t ~ 0 Ky hi~uU(t, s) la loan tli ehuy~ndieh theo quy d<;1oeua phuongtrlnhthuannhat,Wela loantli U(t,1:).D(itXOE Cotuongung vdiham(U(t,1:)xo)(s)=x(t+s),- h ~s ~O. x(t) =x(t, xo) la nghi~meuaphuongtrlnhthuannhattr~n khoang[ 't,-Kb)thoamandiE3Uki~n: x(t)=xo(t- 't) vdi '[ - h ~t ~'t ToantUU(t,'t),t ~1:bieh(inva co trnhchat: U(t,s)U(s,'t)=U(t,'t), 1:~ S ~t Kyhi~uE+(A)lat$pcaehamWComanghj~mtuongu'ngeua phuongtrlnhthuannhatbj ch(in khi t ~O. Trongcaenghi~mcuaphuongtrlnhthuannhat,xacdinhvdi t~0taxetcaenghi~mxacdinhtrenR,bich(inkhit~O. Ky hi~uE_(A)la t$p cae hamXOE Co, ma no la thu h~p nghi~mcuaphuongtrlnhthuannhattren[-h,0].NhO'ngnghi~mnay xacdinhtrenR va bjch(inkhit~O. Tuongtlj dinhnghTacaet$p: E+(1:)=E+(8tA8-t) va E_(1:)= E - (8 t AS -t ) doi vdi phuong trlnh x(t) +(SrAS-rx)(t) =0 , 5 , -' _? ,~-' §2-TINH CHAT NGHII;M CUA PHlJdNG TRINH THUAN NHAT 1- Xet phlidngtrinhthuannhat x(t) + (Ax)(t) = 0 (1) vai G thoa.mangia thietcd ban d §1 Ky hi$u x(t, s, xo) la nghi$mcua phlidng trinh (1) thoa dieu ki$n ban dau x(t,s, xo)=xo(t-s)khis-h :s;t :s;s khi do hamx(t,S, xo)thoaman liac IliQngsau : Ilx(t+o-,,~,xoll,,-:S;e{/(t-s)llxo(o-)!L,t~s poi lIy(o-)1I=max Ily(o-)IIRn0- -"5,0-5,0 (2) a =supV[G(t,s)] t (Ky hi$ugia trjmaxtheocrdliQC Slt dl,mgsau nay) 80 de 1 : Gia suoton t<;1.iday ham kha vi lien tl,lc {Xk(t)}thoa man dieu ki$n: a) Moi hamXk(t)thoamanphlidngtrinh. Xk (t) + (AXk )(t) =0 tren[ak, bk] b) Ton tQiday so (tk),ak<tk<bksao cho Ilxdt)ll>60. max Ilxdt)11 Ok <;,t5,bk 0 <EO <1, EOkh6ngphl,lthuQcVaOk va limItkI=lim(bk - a k ) = lim (t k - ak ) = 00k~oo k-~oo k~oo KhidovaiA E H(A) nao do, phlidngtrinhgiai hQn y(t) +(Ay)(t) = 0 (3) co nghi$m7:- 0 bj ch~ntren R. Chung minh: X ' td "' h " xk(/+lk) e ay am Yk (I) = II II ,ak - Ik ~ 1 ~ bk - Ikxk(lk) Do dieu ki~na), b) cua bo de, hamYk(t)thoaphlldngtrlnh: Ydt)+ (StkAS-tkyd(t) =0 tren doc;ln[ak- tk,bk- tk]va 1 max IIYk(t)II~- "k -tk stsbk,-tk li 0 ;IIYk (0)11= 1 Tv do suy ra ding tren moi khoang hO'uhc;lnday ham {Yk(t)}. (b~hdau tU mQtchi s6 k nao d6) bi ch~ndeu va lien tl,lCdong deu. Theo dinh Iy Arzele, ta co the gia sv chrnhday {Yk(t)}hQitl,ldeu den ham Yo deu tren tUng khoang hO'uhc;ln.Ngoai ra GOnggia sv vdi AEH(A) nao do taco : . (Ax)(t) =Iim(St AS t x)(t)k-)«> k k '\ deu tren tUng khoang hO'uhc;ln. Vi hQ toan tv StkAS_tk la c - lien tl,lCdeu nen ta dong nhat dling thuG : t t Yk (t) = Yk (0) - J (Stk AS-tk Yo )(s)ds +J [StkAS_tJ(Yo - Yk )(s)ds 0 0 Ta suy ra Yothoamanphlldngtrinh: t - Yo(t) =Yo(0) - J (Ayo )(s)ds 0 Dieu do nghTala Yokha vi lien tl,lCva co nghi~mkhac kh6ng, bi ch~ntren R cua phlldngtrinhgidi h(~in(3) .;. ~. 7 2-Uoc hiQcnghi~mbi ch~ncua phlidngtrinh thuannhat trenbantn,lct ~ 0 Ky hi~u E+(T) = EAS t AS -t) la t~pcac ham W Co, la nhG'ng ham ban dau cua nghi~mbi ch~n khi t ~0 cua phLidng trinh : x(t) +(SrAS_-rx)(t)= 0 (4) Bc5de 2 : Gia su tatca phLidngtrinhthuannhat: y(t)-1-(Ay)(t) =0 ,A ell(A) (3) chi co nghi~m kh6ng trong C1. Khi do ton t9i r >O,N>0 saD cho: IIU(t,r)xollco~Ne-r(t-r)IIU(s,r)xolico (5) Xo EOE+('"C) ,t~s ~'"C~O Chl.l'ngminh : Trudchetchungminhdingvdi'mQiXoEOE+(O)thoamand&ng thUGsau : . IimIIU(t,O)xoll=o,xo eE+(0)t--+<1O (6) Ta chung minh bang phan chung: gia su ton t9i Xo EOE+(O), ton tO.k Vi nghi~mx(t,O,xo)bi ch~n: supllx(t,O,xo )11< 00 t;o,O nentheobe de 1 vdiA EOH(A),nao do, phu'dngtrinh giOih9n (5) co nghi~m bi ch~n khac kh6ng. Dieu nay mEW thu~n vdi gia thiel, v~yd&ng thu'C(6) dLi<;1cchung minh. . Vi dieuki~ncua be)de kh6ngthaydeikhita chuyentu loan tvA sang StkAS_tk'T EOR nen W (6) suy ra : \ limllU(t,r)xollc=O,xoEE+(r) vdimoi't codjnhthuQcR. (8)1->00 0 Bay giO ta chung minh tont~i So>0; sao cho voi m,Q1 XoE E+('t)thoa manbatdltngthuG: 1 Ilx(t+s+O"",xo)llus;"2llx(t+O"",xo)!Iu ' s~so;t~,~o (9) Giasu ngLi<;1CI~i,khidotont~icacday('tk), (Sk),(tk),(Xk), Xk E E+('tk), Ilxkllc.=l,sk~ oo,(k~ 00) saocho: . 1 Ilx(tk+Sk+0""k,XdILT~21Ix(tk+0""k,Xk)ILT.' tk~'k~O (10) Cho t'k~tkva Ilx(t'k+O""k,Xk)llu=~~xllx(t+O""k'Xk)t (tont~igia trj 't'k nhLiv~ysuy raW (8)) £) v t ---;-llx(tk +0"" k,Xk)!Iu<;l lIo =Inn k-~ooIIx( t' k+0"" k,xk)1L Co2 kha nEmgxay ra : ao=0 hayao>0 a) ao=0 thlW : Ilx(t'k+0",rk ,xk)1!u s;eQ(t't-tt'IIX(tk+0",'k ,xk)t suyra: t'k - tk ~ 00 khik ~ 00. Dayham x(t +tI k ,r k ,X k) Yk(t) =Ilx(t\+O""k,Xk)IL bi ch~n deu tren -(t'k - tk)~t ~ 00 va hamgidi h~ncua no la nghi$mkhackh6ngbj ch~ncua phLidngtrlnh(3)vdiA E H(A) na-o do, dieunaymauthuanvdigiathietcuabode. b) ao>0, kh6ng giam Unhtong quat co the coi ao la gidi h~n cua mQtday tLidngung. Trong trLiangh<;1pdo day: () x(t+tk+Sk"k'Xk) Yk t = ak ak =IIx(tk+Sk+0",rk'xk)1!u 9 bi ch~ndeu tren [-SkI+ 00). Th~tv~y,voi k kha Ion, qt,lavao dinhnghTaso aova batdAngthuc(10),taco : maxIIYk(t)11=a~1maxllx(t+tk +Sk" k 'Xk )11=-.\.,:5,f<~ f"-.', = a~1maxllx(t + tk' Tk'Xk )11~ a~1maxllx(t + (J" k'Xk )11/,,0 /,,/, tY ~a~1 IIX(t'k+(J, Tk'Xk)t ~2a~'a~'llx(tk+(J"pxk)t ~4a~' f Do do ham gioi hc;lncua day (Yk)bi ch~ntren A, ~ 0 va la nghi$mcua phuongtrinh (3). E>ieunay I~imau thuanvoi gia thiE3't. Vf}.yd~ngthuc (9) dLiQCchungminh.Tu (9)suy ra (6). 3- T~p hQp nghi~mbi chijn khi t ~0 cua cae phu'dngtr1nh gidi h~nIii kh6ng gian huu h~nchh~u Ky hi$uE_(A) la tf}.pcae hamXoE Co, tLidngung voi nghi$m, xacdinhtrenR va bich~nkhit ~0cuaphuongtrinhgioih~n: x(t) +(Ax)(t) =0 ,A E H(A) (11) 86 de 3 : Gia su tatca phuongtrinhthuannhat(3)chico nghi~mkhong trangC1,khido caetf}.ptuyentinhE_(Ao)voi mQi.Ao EH(A) la huu hc;lnchieu. Chungminh: Truochetchungminhrangmoinghi~mx(t)cuaphuongtrinh (11)bjch~nkhit~0thaamandieuki~n: limllx(t)11=0/->--., (12) Thf}.tv~y,gia su x(t) la nghi~mcua (11) xac dinh,trenR va theftman: maxllx(t)1I=1-,supllx(t)1I<co -11:5,1:5,0 f:5,O Neu nhuvoi m(>tday(tk)naodo (tk-+ 00 khi k -+ 00.) thoabat d~ngthuc inf{IIx(tk)11:k=1,2, }>0 thi day ham{Yk(t)=x(t+tkHbi ch~nI deutren(- 00, -tk), h(>i tv deutrentungkhaanghuuh~ndenyo(t) 10 (Yo :;t:.0 bi ch~n).Neu A la loan tu giai hGlncua daycon cua day {StkAS-tk} thi ro rang A E H(A) va yo(t)la nghi~mcua phlidng trlnh : Yo(I) +(Ayo )(/) = 0 mau thUElnnay chung minhdltngthuc (12). Ky hi~uYf (-00,0) la kh6nggiancae hamlien tl,lCbi ch~n: x : (-00,0)~ Ro. Khi do, taco the chung minhrang: Doi vai t~pcae nghi~m cua phtJdng trinh (11) dling thuc (12) GOngthoamandeu theox E Yf(-oo,O).(1Ixl/w(-oo,O):::; 1) E)~tF={x=x(t): xE Yf(-oo ,0) , ~(t)+(Aox)(t) =0 ,!lxll~(-oo,0)~ 1} Chung minh t~pF la t~pcompacttrangYf(-oo,O). Th~tv~y,vi ~(t)+Aox=onen 11~(t)II:::;IIAollllx"w(_oo,O) Do do t~pF lien tl,lCdong deu nen F compacttrangYf(-oo,0). M~tkhac,t~pcaenghi~mcuaphlidngtrinh(11)bi ch~ntren~-oo,a) tc;iOnen kh6ng gian con kin trangYf(-00 ,0)cho nen theodirih Iy Ritz suy ra t~pnghi~mcua phlidng trinh(11) bi ch~ntren (-'00,0) la hO'u hc;inchieu. Chu y : Trangdieuki~ncuabo de 3, t~pcaehghi~mbich~Q khit:::;a cuaphlidngtrinhxuatphat(1)GOngla hO'uhc;inchieu. 4-Huuhc;lnchieucua khonggianthu'dngCoIE+(A) Tli liac IliQC (6) suy ra rang, t~p tuyen trnh E+('t)=E+(StAS_t) lEidong trang Co, Wc no la kh6ng gian con. Khi do kh6ng gian thlidng Col E+('t)la khong gian Banach. Ky hi~uE+(O)la phanbu dc;iiso doi vai kh6nggiancon E+(O)trangkh6nggianCo. 86 de 4 : Gia sv phuongtrinhthuannhat(3)chi co nghi$mkh6ngtrong . C1.Khi do kh6nggian Co/E+(O)la hOuh0 sac cho : IIU(t,O)xollc~ Ne-r(S-t>IIU(s,O)xolie0 0 Chung minh: ChoA E H(A), ton t<;liday (hk),hk--+ookhi k --+00sac cho: ,xo EE+(O) ,s~t~O (13) , Iim(S" AS "x)(t) =(Ax)(t) deu tren m6i khoang hOu hQ,n.GOi m">,,,' . - la chieu cua kh6ng gian con cua nghi$m bi ch~n khi t :::; 0 cua phu'ongtrinh: . x(t) +(Ax)(t) =O. Chungminh ding dimE+(O):::; m. Th{fltv{fly, cho XO1,"" '" XOm1E E+(O) dQc l{flp tuyen tfnh, con ,;L~,;{(XO1"""",XOm1)la kh6nggian con sinh ra tUcac,phan tv do. Khi do, theedinh nghTacua kh6nggian E+(O)va E+(O). " dimU(t,O)9!=dim9!=m11mQit ~o M~t khac, lien hanh IV lu{flntu'ongtV'vi$c chung minh"bo de 2, sv dl,mg tfnh hOu hc;inchieu cua kh6ng gian £E,ta se thiet l{flp du0, N>O kh6ng phl,l thuQc XoE 9!. ChQn Xj(k) E £E, I = 1, ,m1sac cho ham: y/k) =U(Izk,O)X/k) IIU(Izk ,O)Xi (k)11 ,(i =1, ,1111) (14) thoa mancac batdl1ngthUG: 12 i~lfllY2 (k) +a1Y1(k)11~ ~ ~~Jy3(k) +a1Y,/k)+a2y 2(k)II ~~ ................ in" II (k)+a (k)+ +a (k) ll >1 a a a Ym1 1Y1 ... ml-lYml-l --I, 2,---rnl~1 2 (15) Dayham: zY')(t)= x(lzk+t,O,X~k» IIU(lIII ,0, )X1'I)II (i =1,2, ,m.),/(=1,2,.... I thoamandieuki~n: Z~k)(t)=y}k)(t)khi-hs t s 0va do(12)bi ch~ndeutrenmoikhoanghiJuhf;tn,vadoditngthUG: (k) - (k) Zi (t)+(S"kAS_"kZi )(t)=O lientvcdongdeutrenmoikhmlnghiJuhf;tn.Dodoco thE§coi day{zYc)(t)}hQitv deutrenmoikhoanghiJuhf;tndenZj(t) (i=1,..,m1) HamZj(t)la nghi~mcuaphlidngtrinh(11)vadolioc"li<Jng: Ilx(l+a,O,x;(k)ILsNe-Y(S-I)lIx(s+a,O,xYl,.,(s~t~O) sethoamancacbatditngthUGsau: IIZi(t+O")t ~Ne-y(S-t)lIzi(s+O")t ,s ~t (16) Ngoaira, do cac batditngthl1c(15), cac ham Zj(t)dQc l<%ip tuyenUnhtren[-h,0] nen chung dQc'~ptuyenUnhtrenkhoang (- 00, 0].TLi(16)suyra hamZj(t)bi ch~ntren(- 00,0], cho nen m1s mdochungdQc'~ptuyenUnh.Tv dosuyradimE+(O)s mva u'ocIli<Jng(13).Bodedli<Jchungminh. Nh~nxet : Thaytrongdieuki~ncuabo de4 loantll A bpi loan tll A( 1:)= St AS. t ( 1:> 0).Ta nh~ndli<JCkef qua: khonggian thlidngCo/E+(1:)hiJuhf;tnchieuva voi1:2> 1:1~ 0 thoamanbatditng thUGsau : dim CO/E+(-t1)s dim Co/EA "(2)s m 13 Th~tv~y,U(11,12)EA11)c EA12)va anhXc;lthlidngtlidng l1ng,: 0(12,11); Co/E+(11)~ CO/E+(12)la ddn cau. 5- Uoc lu<;ingnghi~mkhongbi chcjncua phudngtrinh thw)nnhat , Tli nh~nxetsaube>de4, guyratontGli1+ saochovoi1 ~ 1:+ cae khonggianconCo/E+(1:)co cOngmOtso chieu.Gia sa E+(1:+)la phanbOthiingcuakhonggianE+(1:+): Co=E+( 1+)EB E+(1+) Di;lt E+(-r)=U( 1, 1:+)E+(1+), 1:> 1:+ khi d6 Co=E+(1:+)EB E+(1:)vdi 1:> 1:+. Bieudiennaycua Co sinh ra hQcae loan tu chieuP+(-r) len khonggianconE+(1). TheodinhnghTacua khonggiancon E+(t),loan tu P+(1:)thoa mandiingthu'c: U(a', a) P+(1)= P+(-r')U(1',1:) , 1:+~t ~1:' 80 de5 : Giasutatcaphlidngtrlnhthuannhat: yet)+(Ay)(t)=0 ,A E H(A) (3) chico nghi~mkhongtrongC1.Khidotontc;liN>0, r >0 sao cho: IIU(t,r)xoll::;Ne-r(s-t>IIU(s,r)xoll 'Xo E E+(r) ,r+::;r ::;t ::;s (17) Chungminh: Tlidnghi nhlibe>de4, taco : IIU(t,r+)yoll::;Ne-r(s-t>IIU(s,r+)yoil'Yo E E+(r+) ,s c.t ~r+ (18) 14 Voi mQi Xo E E+(-r), (t ~t+).Do dilng thuG E+(t)=U(t,t+)E+(t+). Tim duQcYOEE+(t+)sao cho xo=U(t,t+)Yo. Luuy tfnhchat: U(t,t+)=U(t,t)U(t,t+), (t~t ~t+). Tu (18)suyra(17),be>deduQcchungminh. 6- U'ocIu'Qngnghi~mphu'dngtrinh thuan nhat tren khoang t:::;;O Donh$nxetsaube>de3,caekh6nggiancanE.('t)cuakh6ng gianEolahCi'uh<;lnchieu.ThemVEtOdo,dodinh1'1duynhatvephra phai,taco : dim E.(t) :::;;dim E.(t') :::;;mo , ( t' :::;;t:::;;0), mo la mQt sonaodo. Tu batdilngthu'cnaysuy ratontC;lit. <0saocho,Voit<t.co dimE.('t)=dim E.(t.). . Voi 'tda chQn,moinghi~mbjch~nx(t)trent :::; 0 cua phudng trjnh(1)du'QcxacdinhduynhatboihambandauduQcchotren doC;ln[ t- - h, 'r.]. Do do co dilng thuc : U(-r,l')E_(l') = E_(l) , ('r' :::;;t :::;l.) Ky hi~uF(t.) la phanbuthilngcuakh6nggianE.(t.). (19) Ky hi~uP.(l_)la loantuchieulenkh6nggianE.(t.)duQcsinh ranha : Co=E-(l-) EB E-(l) £)~t P-(t) =U(t,t-)P. (l.) U(t.,t) , ( t < t- ) d dayU(t,t-): E.(t.)-+E.(t) la anhXC;lnguQccuaanh'XC;l, U('t.,'t) : E.('t)-+E_('t-). N6 ton tC;lido (19). R6 rang P.(t)('t<'t.) laloantuchieu. . Xac dinhkh6nggianF(t) boidilngthuG: E'(t) =(I - P.(t))Co. 15 Toan tuU('t,'t')lac dl,lngtU E_('t')va F('t') tLidngung VaGE('t) vaE-('t).DieunaysuyratadltngthuG: U('t,'t')P_('t')= P-('t)U('t,'t') 80 de6 : Gia su tatca phlidngtrlnhthuannhat(3) chI co.nghi~m kh6ngtrongC1,Khidotontc;lir >0 , N >0 saDcho: IIU(t,r)xollco~Ne-r(t-s)IIU(s,r)xollco IIU(t,r)xollcu~Ne-Y(H)IIU(s,r)xollco Chu'ng minh : ,r~s~t~ 0 ,xoE E-(r) ~20) ,r~s~t~O ,xoEE_(r) (21) De chungminh(20), trLioc het ta chung minh ton tc;liSo> 0 saDcho thoa man: . IIU(r+s,r)xo"c,,~~ ,xoEE-(r) ,IIxollco=l ,s~so ,r~r+,\'o~O (22) Gia SUonglieJc Ic;li (22) kh6ng dung, khi do ton tc;liday ham' (Xk), Xk E F('tk), Ilxkll=l, ton tc;liday so ('tk),(Sk).(Sk--+00 khi k --+00) va Sk + 'tkS 0 saDcho: 1 IIU(rk+sk,rk)x,.lIxo>2 (23) GQi tk E ['tk,0] saDcho: ak=IIU(tk,rk)xk,IIc=maxllU(t ,rk)xkll 0 Tk5t50 Co Co 2 khanangxayra: a) Day ,( Itk I) bi ch~n b) Day ( Itk I) kh6ng bi ch~n Xet trliong heJpa) : , VI 'tk s -Sk , Sk --+00 nen day (Yk)nghi~mcua phLidngtrlnh(1) 16" Yk(I) =a;1x(/, rk ,Xk) ,!lxll=1 ,k=1,2,..... bi ch~fJdeu va lien tt)c dong deu tren moi khoEmghO'uh~;m Qua(-00,0]batdau tu mQtchi so k nao d6. C6 the coi chrnhday(Yk) hQitl,lden Yodeu tren tung khoEmghO'uh9n, con day (tk)hoi tl,l den to ::::;O. Vi IIYk(/k+o-)!Iu=a;lllx(/k+o-,rk,xk)1!u=1 nen ham yo(t)la nghi~mkhac khangQuaphuongtrlnh(1), bi ch~nkhi t::::;O. Do d6 ham xo(o-)=Yo(r- +0-) ,(-h:::;0-:::;0) khac khang va thuQCvao E-('L). M~t khac, ham Xk(cr)= x('t-+ cr,'tk,Xk), (Xk E F('tk)) voi k kha IOn thuQc khang gian con F('t_). 8ieu nay mauthuan dieu ki~nIlxk - xollco~ 0 ,k ~ 00 , v~y 109itruong h<;1pa). b) Gia su tk~ -00.Gia su day (ak)bi ch~n,c6 the'coi day (-'tk- Sk)c6 gioi h9n (hO'uh9n hayva h9n)bo~O. . x( 1+rk +SI.,rk,Xk) 0 ay ham: YIi(I) =Ilx(r k +S k +0-, r k ,Xk )110. Do (23)thoabatd~ngthUG: max IIYk(/+o-)11 :::;2maxllx(/+0-,rk,xk)11 :::;2supak <00 -Sk:S:t:S:-(Tk+Sk) CT Tk:S:t:S:O CT Do d6 c6 the coi chrnhday (Yk)hQitt).den Yo deu tren tUng khoanghO'uh9n Qua(-oo,bo). Neu bo hO'uh9n thl ham zo(t)= Yo(t- bo) la nghi~mcua phuongtrlnh(1) bi ch~ntren (-00,0]. Tien hanh cac buoc tuongtt!, Iy lu~ntuong tLjnhu trongtruongh<;1pa), ta dU<;1Cdieu mau thuan voi hamzo(cr)=z('t-+ cr), (-h ::::; cr::::;0) thuQcF('t_). Neu bo va h9n (bo = 00)thl Yo la nghi~mbi ch~n Qua mQt phuongtrlnhgioi h9n dc;mg(3) nao d6. Mau thuanvoi dieu ki~nQua bo de. . 17 Gis.su cuoi clIng (ak)kh6ngbi ch~n.Kh6nggis.mtlnhte>ng quat,co thecoi Hma"=r:fJ ,,~oo (24) Xetdayham yk(t)=a;lx(t+tk,Tk,Xk) nothos.mandieuki$n: max lIy"(t +o-)llcrS 1 ,IIYdo-)t =1 Tk -tk 5,t<;,-tk . (25) TO'batdiing th(k (2), (24)suy ra Hm(t"-1",,)=00 k~oo . TO'do va tU (25) (chu y : tk~ 00, k ~ 00) tanh~nduQcham gidi h9n cua day (Yk)xac dinh va bi ch~ntren R chfnh la nghi$m cua mQtphudngtrlnhd9ng(5)naGdo, h;timEwthuanvai dieuki$n cua be>de. Nhu v~yta chung minhdu'QCbat d~ngthuG(22).Tv (22) suy ra(2O),(21)chu'ngminhtudngtlJ, 8e>de (6)du'Qchungminhxong. 7- Uoc hlQngchu§n cuacaeloan tll chieuP_('t),P+('1:) 80 de 7: Gis.su cacdieuki$ncuabe>de2 duQcthos.man.Khido : supIIp-(1")11<00 , supllp+(1")11<00 T5,T- T~T+ (26) Chung minh: Ta chung minh supIIP_(r}11<00 . T<;'T- Cho XoE CosaGcho P.(s)xo-j:.0, P.(s)xo-j:.Xc. Khi do tv (2) vai 't<t <'tota co : V(t )[ P_(r)xo (I-P_(r»xO] < a(t-T)P_(r)xo + (I-P_,(r»xo ,r IIP_(r)xoll+II(I-P-(1"»xoll_e IIP_(r)xoll II(I-P_(r»xoll Dlja VaGdo tU(20)va (21)taco : 18 P- (T)XO + (/ - P- (T»XO >e-a(t-,){U(t ) P- (T)XO U(t ) (/ - P- (T»XO II} > IIp-(T)Xoll 11(/-P-(T»Xoll - ,T IIp-(T)Xoll ,T 11(/-P_(T»Xoll - > -a(t-,) {I r(t-,) 7\.T -r(t-,) }_e -e -HeN Tu' d ' 0" 4111N 4111Na v I 1 < 1.- , t=T +-r r '. P_(T)Xo (/ -P_(T»Xo Ta ca . "11 I-' +- 11- 11"~5">0P- (T)Xo (/ - P- (T»Xo vdi 80kh6ngphl) thu9C1.Tli (27)vabatd~ngthUG: (27) 11:11+11;11I!max{lIxlI, lIyll} s zllx+yll (X,y :I:0) "'Ii T <T- - 4111N r ta nh$ndLi<1C: max{IIP_(T)xoll,II(/-p-(T»xoll}S50-1zllxoll (28) Neu TE [T-- 4111N ,T_] thl Slj bi chi;incua P-(1)suy ra tli tfnh r lientl)c. TLiongW, SU'dl)ng(5),(bode 2) va (17),(bode 5) ta chung minhdLi<1C: sUpIlP+(T)1I<00 . '~'+ 80 de 7 dL1<1Cchu'ngminhxang. 8- Nhungbatd~ngthuc cd ban Trangml)C,naytatongkefcac LidcILiQngdoi vdinghi~mcua phL1ongtrlnhthuannhat(1). Trang IIU(t,T)xollS Ne-r(t-s)IIU(t,T)xollco (6) Di;it1 =S, Xo= P+(S)Yo,Yo E Co,taco: IIU(t,s)P+(s)yolls Ne-r<t-s)IIU(s,s)P+(s)yollco (t~S~T+) hay IIU(t,s)P+(s)yollS Ne-r(t-s)llp+(s)yollc0 (t~S~T+) . (29) 19 Vi U(s,'t)E+(-t)=E+(s), S 2't 2 't+nen tli (17) IIU(t,r)xoll~Ne-y(S-t)IIU(s,r)xoll (17) suy ra : IIU(t,r)(I - P+(s»Yoll ~ Ne-y(s-t)III ~P+(s)Yollco (30) Tli (26), (29),(30)ta nh~'lndU~ccac bat ditngthtlcco ban sau: IIU(t,s)P+(s)Yolic~N1e-y(t-S)IIYollc0 0 (t~s~r+) (31) IIU(t,s)(I- P+(s»Yolic ~Nte-y(.H)IIYolic0 0 (s~t~r+) (32) TUongtv'nhuv$ytu'(20),(21)va (26)taco : IIU(t,s)P-(s)Yollc ~ N1e-y(S-t)IIyoIIc0 0 (t~s~r_) (33) IIU(t,s)(I-P-(s)Yolico ~Nte-y(t-S)IIYollco . (s ~t ~r ) (34) .:.>:::: ft" ,ft §3- BIEU DIEN NGHIt;:MCUA PHUdNG TRINH KHONG THUAN NHAT Xet phu'ongtrlnhkh6ngthuannhat : x(t)+(Ax)(t)=f(t) , f EC (35) Dinh nghTaham as nhU sau : l l-&-ta- Idzi -&~a-~O ac(a-)= 0 khi -h ~a-~-& O<&<h I,(a-) =f(r +a-) (-.It~a-~0) Ky hi$u: sao clto I, E Co (r E R) Df;it: (acf,)(a-)=ac(a-)/,(a-) Slj phl,J thuQc cua ham U(t,'t)xovaa bien crE[-h,O]dLi~cviet du'didc;lng[U(t,'t)xo](cr).Tli dinh nghTacua ham U(t,1:)xota suy ra cac ditng thu'csau : 20 {[U(t+O-'1")XO](O)kId L~t+o- [U(t,1")xo](o-)= xo(t+o--1") klti 1"-ltst+o-S1"st (36) , d 0 va-[U(t, 1")xo](O)=- fdsG(t,s)[U(t,1")xo](s) , t ~1" dt -II (37) Cho xo,YoE Co,ta li;tp2 ham: ,- z; (t) =[U(t,1"-)P- (1"-)xo](0)- f[U(t,1" )P- (1")a&f, ](O)d1"+ I (38)t f [U(t,1")(l-P_(1" »a&f,](O)d1" , t S 1"- -0() t Z; (t) =[U(t, 1"+)P+(1"+)Yo](0)- f[U(t, 1")P+(1")a&f, ](O)d1"- '+ (39) +0() f [U(t,1")(1 -P+(1"»a&f,](O)d1" , t ~1"+ t Vi cac hamdtJaidau Uchphanphl,JthuQclien tl,Jc\tao 1: va thoa mancac tJac ItJQng(31)- (34) chonen cac hamZ;(t) va Z;(t) voi 8 E (0,h)se xac dinh,lientl,Jcva bi chi;indeuttJdngung . tren cac khoang(-0011:-)va (1:+,+00). 86de8: Cachamz; (t)vaZ;(t)thoamancacphtJdngtrlnhsau: dZ;(t)+(AZ;)(t)=f(t)+rp&(t) , t s"-dt +, dZ&(t) +(AZ+)(t) =f(t)+rp&(t) , t ~1"+dt & (40) (41) \ 0 0 vai rp,,(t)=JdsG{t,s)[Ja,,(O")f(t+O")dO"] ,-I. s Chungminh: Layd<;iOhamhamZ;(t) khi t<1:- taco: dZ- (t) d ~ =[P_(t)a&f,](O)+[(l- P_(t»a&f,](O)+-d [U(t, 1"-)P~(1"_)xo](O)-t ' t '- d t d - f-[U(t,1" )P_(1" )a&f,](O)d1"+f -[U(t,1" )(l-P_(1" )a&f,] (0)d1" t dt -0()dt . " 21 Su dl)ngbatdltngthuc(37)vadltngthucsau: [P_(t)aEft](O)+ [(I - P-(t))aJt](O)=f(t)taco : dZ- ( ) 0 ~tt =f(t)- fdsG(t,s)[U(t,-z-)P_(-z-_)xo](S)+-II 0 ~ t +fdsG(t,s){f [U(t,-z-)P-(-z-)aSfT](s)d-z-- f[U(t, -z-)(I- P- (-z-»aSfTJ(s)d-z-} -II t ' -00 Dotfnhchat(36)cuahamU(t,1:)xodltngthuccuoiclingco th~ viet19idlloid9ng(40). TlldngtLjco th~chungminh(41).V$Y be>de 8 dLiQcchu'ng minhxong. Tli (40),(41)suy ra ranghQcac ham {Z;(t) : O<&<It}va {Z;(t) : 0<&<It}lien tl)c dong deu tlldng ung tren cae khoang (-00 ,'L) va(1:+,+00).Ngoaira,tont9icaegioih9nsau: T- Z~(CT)=limZ; (,- +CT)=[P- ('- )Xo]{CT)+Hmf[U( '-' -z-)(I- P-(-z-»asj~J(o-)d,s~o s~o, -00 . -I"'- Z;«T) =IimZ;('r+ +<T)=(P_(r+)Yo]«T)-lim f[U(r+,r)(I -P+(r»a£/r](ci)dr '£-)0 £-)0 T. HrnqJs(t) =0 s~o chonendosLjchQnILja1:-va1:+tont9igioi h9nsau : Z-(t) =HrnZ;(t) s~o Z+(t) =HrnZ;(t)s~o , -z-+~t<+oo (42) (43) , -oo<t<-z-- NhOnggioih9nnaythoamanphlldngtrlnhkhongthuannhat (35) tlldngung tren khoang(-00,'t-)va ('t+,+oo).R6 rang, bat ky nghi$mbi ch~ntren(-00,'t-)(('t+,+oo))cuaphlldngtrinh(35)deu co the bieudien dlloi d9ng Z-(t)(Z+(t))voi hambandautlldngun,g XOE CO (Yo E Co). ~ ft ~ ,.,.:! ,., - ft' §4- £>IEUKII;N CAN VA £>U£>ETOAN TO'VI PHAN TUYEN TINH £>01SO CHAM LA FREDHOLM Giasucaedieuki$n§1vanthoaman. 22 £>inhIV 1 : Cac m$nhde sau day la tl1ongdl1ong: AI Toantv L la n-chuanHic. - 81 PhuongtrlnhthuannhatLx =0 , \I L E HI (A) (44) chi co nghi$mkh6ngtrongC1. CI :3so M>O,N>Osao cho : IlxllclS;M(IILxllc +maxllx(t)ll) ,\Ix E Cl Ifl~N (45) Chung minh: Ta se chung minhAI ~ 8/. Xet kh6nggiancan: Eo ={ X Eel, x(t) = 0 khi ItI S N } Do nhancua L hliu hq.nchieu,cho nen co th~chQnN kha Ian sao cho Eo kh6nggiaovai KerL. VI anh cua LEocua kh6nggiancan EodongtrongC, nen :3d>Osac cho thoaman l1ac111Qng[8]. IILxllc~dllxllc' ,\Ix E Eo (46) Cho LE/f(L) vaday(1k) (Irkl~ex) khi k~ex)) sao cho - Srk LS-rk x ~ Lx , \Ix Ecl cogiatrihliu hq.n(suppx-compact).Ham (S-Tkx)(t) ==x(t - r k) thuQcEovai k khaIonchonentli (46)taco : IISTkLS-TAXllc=IILS-rkXllc ~dIlS-TkXllcl =dllxllc1 chok ~ ex)tadl1Qc: IlLxii ~dllxllcl c (47) Gia sv x tuyY thuQcc1. Lay hama(t) 0 s a(t)s 1 [ 1 kId ItIs1 act) = 0 khi Itl;:::2 khavi lientl,lCsac cho 23 Khido tu(47)suy ra : IIL(akx)11Z dllakxllcl t c ;. (48) , ",'", Trang do (akx)(I) = a(~)X(I) "L cho k ~ 00 I tU(48)ta co : Lxii zdllxllcJ c , 'dXECl Tu daysuyram~nhde B. Bay gia chu'ngminhtUB/ ~ C/, Gia su batdang thuG(45) kh6ng dung, khi do ton t~iNk~ 00, Xli ECI ,lIxllc1=1 saoc/to Iim(IILxlillc+maxllxk(t)II)=O li~oo Itl:O:;Nk. (49) Gia su tkERsao cho IIXk(lk)II~!.R6 rang Ilkl~Nk.£Hit:2 . Uk(t)=Xk(t+tk),Kh6ng giamtinhtong quat co the coi day (Uk) cLIngvdi d~oham cua no hQitlJ den UOEC1deu tren tung khoang hO'uh~;lnva lIuo (0)11~!,2 M~tkhac ta co : (S'kLS-tk ud(l) ==Uk (I) +(Stk AS-tk ud(l) (50) Kh6ng giam tinh tong quat, ta co the coi chinh day (S'kAS_tkUk)(t) hQitlJ den (Ax)(t) deu tren tung khoanghO'uh~nk~i k ~ 00. Do (49), lay giai hc;ln2 ve khik ~ 00 I ta dU<;1C: CLUo)(t):=Uo(t)+(AUo)(t)=O mau thuan vdi gift thiet. Cuoi cLIngta chung minhtUC/ ~ A/. TrUdc het chung minhKerL hO'uh~nchieu. Do b6 de Ritz, ta chi can chung minh m~tcau don vi cua kh6ng gian con KerL la compact.Th~tv~y,neuday (Xk)c KerL va IIXkllcl=1 thiday(Xk)bi ch~ndeuva lientl,lCdongdeu.Do do, kh6nggiamtinhtongquat, co thecoi chinhday(Xk(t))hQitlJ den xo(t) E C deu tren tUng khoanghO'uhc;lntrongC. 24 Taco :XoE C1 vaxo E KerL. Th$tv$y, lay gioi h~;ln2 ve ditngthuG: Xk (t) +(Axk)(t) =0 ta I du'<;5C Xo(t) +(Axo )(t) =0 Wc XoE c1 va XoE KerL. Bay giOchung minhanh LC1cua tOElntv L dongtrongC1. Gia SU' ngu'<;5cIc;li, thl :3 day (fk) E Lc1 saD cho 11.f~llc-70 Idli I( ~ 00 nhLing: inftllxllct:Lx =.t~}=0 LayXk E C1 SaD cho LXk =fk , Ilxkllcl=1 Kh6nggiamtlnhtongquat,cothegiasv chfnhday(Xk)hQitv denXotrentUngkhoanghO'uhc;lnvaXoE c1. Do(45),Ilxk-Xollct ~O klli k~oo. M~tkhac,vi : L(Xk- xo)=fk va IIIkllc ~ 0 klli k ~ 00 nen tU (51) ta c6 : . Ilxk-xollct21. Mau thuan do, chung minh anh cua LC1 dong. Binh Iy dLi<;5Cchu'ngminhxong. (51) E>inhIy 2 : Neu loantv L la n-chuanlac thlL la Fredholm. Chung minh: Theo dinh Iy 1, mQiphLidngtrlnh thuan nhat dc;lng(44) hay dc;lng(3) chi co nghi~mkh6ngtrongC1. Do do, thoa man dieu ki~n cua cac bo de 2 - 7, nh$ndLi<;5Ccac LiocILiQngtrongm,Vc8 va thiet I$p dLi<;5Ccac hamtrongbo de 8. Nghi~mT(t) cua phLidngtrlnh(35)dLiQCxac dinh bdi ditng thu'c(42)dLiQckeodaimQtcachduynhattrenkhoangt 2 'to Be nghi~mT(t) bi ch~ntrenR thldieuki~ncan va du la voi mQt nghi~mnaodoZ+(t),dLiQCxacdinhbdiditngthuG(43)phaithoa dieuki~nsau : ' T(t++cr)=Z+(tk+cr) Kh6ng kho khan,thaydLiQcrang: (52) 25 r+ Z- (r + +a) =HmflU(r+'r)(I - P-(r»aEf. ](a)dr +E~O -00 r+ +IU( r+,r - )P- (r- )xo Iea-)+Jim f[U(r+,r)acf. ](a)drE~O r- Chonen(52)tuongduongvoidltngthuc: r+ IP+(r+)Yo](a) - IU( r+'r - )P- (r - )xo](a) =Jim flU( r+'r)aEf. ](a)dr +E~O'- 00 ~ +lim{fIU(r+,r)(I - P+(r»ar.fr J(a)dr+ fIU(r+,r)(l- P_(r»aEf.](a)dr) ==tp(a)E~O ~ -00 (53) Tli (53) suy ra rang, phuong trinh kh6ng thuan nhat (35) co nghi~mbj ch~ntren toan tn,lcso R khi va chI khi ham <p(cr)thuQc tong E+('t+)+ U('t+,'t)E_('t_)cua cac kh6ng gian con E+('t+)va U('t+,'t_)E_('t_). . Tli do ta co : dimC/LC1 = dimCo/ (E+('t+)+ U('t+,'t_)E_('t_» V~y L la d- chuan tacoDo do, L la toan tv Fredholm. E>inhIV 3 : Neutoantv L lad- chuantacoLucdophuongtrinh: Lx=f , VLEH(L) , fEC co it nhatmQtnghi~mtrongC1. Chung minh: Vi LC1 la kh6ng gian con dong va dim C/LC1 <00'cho nen phanbutn,icgiao(LC1)L c C* la kh6nggianconhOuhGlnchieuva Lc1co phanbuhOuh<;lnEo : C = Eo EBLC1. I Gia sv g1,g2, ..., gmE C, <P1,<P2, , <PmE phantv dQcI~ptuyentinhsac cho : . <Pi (gj) = OijVa C = Lc1 EBZ(g1,g2, ..., gm) Vi toan tv L la d- chuantac nen phuongtrinh: (LC1)L la nhOng ; III Lx =f - L/P j(f)gj j=1 (54) 26 conghi~mxthoamanbatdilngthUG: IIxllc1~N.llfllc (55) I Trong do N1kh6ngphl,lthu9Cf E C. Lay day {'tk}( ITkl~ 00 klti k ~ 00) saocho: STkLS-TkX ~ Lx vdix EC1co giacompact. Ky hi~uXklEinghi~mcua phuongtrinh(54) thee.man bat dilng thUG(55)vdi f(t)=go(t-'tk),trongdo goE C co gia compact. TLido suyratont<;lihamXoEc1thoamanphlidngtrlnh - Lxo =go va lidc IliQngIIxollc1~N.lIgolic Baygio,giasu f E C la hamtuyV. Khido phlidngtrlnh: LXk =akf vdi (akf)(t) =a(~)f(t) , k co nghi~mXktrongc1. Gidi hc;lncuaday{Xk}thu9Cc1va,Ia nghi~m cua phlidng trlnh Lx =f . DinhIy3 dliQCchungminhxong. TLidinhIV1- 3,suyradinhIVcdbansau : E>inhIV 4 : De loantUL la Fredholmthldieu ki~ncanva'dula tont<;1i loantungliQcbichi[ln: - -1 L : C ~ C1 ,VLEH(L) Chungminh: 1- Thuc%1n , I NeuL la FredholmthlloantuL la n-chuanlac. Dodo theo - dinh IV 1 thl phlidng trlnh : Lx =0 ,VL E H(L) chI co nghi~m kh6ng. Mi[ltkhac VIL GOngla d- chuan lacnentheodinh IV 3, phlidng- - trlnh: Lx=f ,VLEH(L) ,fEC 27 co It nhatmQtnghi~mtrongC1.Nhu v~yton t~iloan tunguQc bichi;}n - -1 L : C ~ c1 . 2- DEw NguQc I~i: - Neu L co loan tungu<;1Cbi chi;}nthl theo dinh Iy1 loantu L la n- chuan tElC(m~nhde A va B trongdinh Iy 1 la tudngdudn,g). ' Con tfnhFredholmcua loan tu L duQcsuy tv djnh Iy 2. Dinh Iy 4 duQcchu'ngminhxong. 28I