Luận văn Phương trình có nhiều nghiệm

!lwjn 1PU/n~ ~t CHUaNG 1. TRUONG VECTOR I I I. TRU(JNG VECTORTRONGKH6NG GIANHUu HAN CHIEU. 1. Ky hi~uRnla kh6nggianEuc1iden- chi€u. ChoM c Rn .TruiJngvectortrenM la anhx~ cD: M ~ Rn bie"nm6i x E M thanhmQtvectorcDxE Rn ChQnmQth~tQadQtrongRnthlcDc6th€ bi€u di~nduoid~ng: (1.1) <D(x) =«DI (Xl, ...,Xn),...,<Dn(Xl, ...,Xn)) trongd6 Xl"",Xnla to~dQcuax va <Dl,..., <Dnla cachamthanh phftncua<D. . Truong<D lien t~lCtren M ne"ucac thanh phftn<Dl,... , <Dn trong(1.1)la nhil'nghamlientlfctrenM. . Nghi~mcuaphudngtrlnh<Dx=0 gQila diim kYdj cua truongvector<D. . Truongvector<D khongsuy bitn trenM ne"un6 kh6ngc6 di€m ky ditrongM. . Di€m XoduQcgQila diim kYdj co l<zpcuatru'ongvector<D ne"u <Dxo=0 vakh6ngc6di€m kydikhactrongmQtHinc~n cuadi6mXo. . Truong(1.1)duQcgQila {ranne"ucachamthanhphftn - 1- ('{{(fin/IHFlZtltfJC dt <DI,..., <Dnlien t\lckha vi . Di€m x duQcgQila dilm chinhqui cuatruongtrdn<Dne'umatr~n <D'(x) khongsuybie'n;nguQc lc;lix gQila dilm toihfln. . Truongvector<Dla tuytntinhne'u <D(x) =Cx, trongd6C la matr~nvuongcelpn. 2. Phep bie'nd~ngva tru'O'ngdOngluan. 2.1A.nhx~lien t\lc<DduQcxacdinhvai m6i x E M lelygia tfi <D(A,X) E Rn (0 < A <1)gQi la phepbitn dflngcua truong<Dox=<D(0, x) vao<DIX=<D(1, x). Boi khiconn6iphepbie'nd~ngn6i <Pova<Pi. Hai truonglien t\lCbeltky <Pova <Pid~un6i duQcbdi mQt phepbie'nd~ng,nghlala : (1.2) <P (A,X) =(1- A) <Pox+ A <D1x (0<A<1,XEM) Ta gQi(1.2)la phepbie'nd~ngtuye'ntinh. Phepbie'nd~ng<D(A,X)gQila kh6ngsuybitn ne'u<P(A,X)7=0 vai0 <A~1vax E M. Hai truong <Dova <DiduQcgQi la d6ngluiln tren M ne'u chungduQcn6ibdiphepbie'nd~ngkhongsuybie'n. 2.2Tieu chu~ndOngluan. Tieu chuin 1.Cho <Po, <Pila truongvectorkhongsuy bie'ntren M va gia sii'm6i XEM , cacvector <Poxva <Pix khongphaidi€m theohuangd6ixung,nghlala : (1- A) <Dox+A<Dix* 0 vaimQiAE [0,1]. - 2- ftu4n1flantfu"c ~t The'thl <Pova <PId6ngluau. .Iieu chu[n 2. Cho <Po,<PIla hai tniong vector lien t1;lC trenM. Ne'u «PoX,S0 (XE M), trongdoS la matr~nxacdinhdu'dng,thl<Pova<Di d6ng luau trenM. . TrongRn,tichvahu'dngcuahaivectorx=(Xl, ...,Xn) va y =(YI,...,Yn)xac dinhbai (x, y) =XIYl + ...+XnYn , va dQdai haychu:1nIIxllcua x du'Qctinh la:11X II= (X12 +...+Xn 2 ) 1/2 .Ii~lLchuin3. ChoMe Rnva<Po, <Pilahai tru'ongvector lien t1;lcxacdinhtrenM. Ne'u II <Pox - <PIxII <II<Plx II (x EM) thl <Pova<Plkhongsuybie'ntrenM vad6ngluau. 3.so'quayhayB~ctopocuaTru'ongvector. 3.1.so'quay. GQi ao la biencuat~pmabi ch~n0 c Rn.Vdimoi tru'ongvectorlien t1;lC<Pkhongsuybie'ntrenao chotu'dngling vdi mQts6 nguyen y«P,O). Ta gQi y «P,O )la S(]'quaycua<P trenan. 3.2Nhii'ngHnhchfftcuaso'quay. .I.inh...cllW.Cactru'ongvectord6ngluau ao cos6quay bangnhau. - 3- utjAt/iftU/nthffc d-l Iin.b..c.h!t2.Cho0 Ia t~pIDabi ch~nva( OJ) la daycac t~pIDaroi nhau OJ cO. Giii sii'la tru'onglien t\lc khong 00 suybi€n trenCI 0 \U OJ , CIO Ia baadongcua O.1=1 Khi do y «1>,Oj) khackhongvoi IDQts6hUllh'.lnchI s6j, vataco 00 y«1>,0) =I y «1>,Oj) j=l Iinh..c.b.ill. Cho XoE 0 va x=x- Xothly«1>,0) =1 JJnh...clJiiL4.Cho0 la t~pIDabi ch~n.N€u la tru'ong vectorkhongsuybi€n trenCIO thl y«1>,0)=0 .I.inlL.c.b!L.5.GQi 0 la t~pIDabi ch~nva la tru'ong Vectorlient\lctrenCIO , khongsuybi€n trenao N€u y «1>,0) *-0 thlco it nhatIDQtdi6IDky di trong0 4. Chis6cuadi~mkydicol~p. Cho 0 c RnIa t~pIDabi ch~nva la tnl'ongvectorlien t\lCtrenCIO , khongsuybi€n trenao .Khido,IDQidi€ID ky di (n€u co)d€u thuQcphftntrong cuat~pO. N€u Xola di6IDky di co l~pthls6quaycua tru'ongvector b~ngnhau tren IDQi ID~tcftu,{ x / Ilxll=p },bankinhp du nho.Gia tri chungnay,ky hi~uind (xo,<1»du'QcgQi la chi s(f ciladiim kYdtXo.Khai ni~IDchi s6cuadi6IDky di tu'dngtt;1'nhu' khaini~IDs6bQicuanghi~IDdathuctrongd'.lis6. Ta co nhii'ngHnhchatsail : -4- 7!uwn/IJ(};nlltac (}j' I' . I I ~. Cho0 lat~pmavagiftsli' la tru'ongvector lientlJc,khongsuybi€n trenao . N€u cohUllh£.lndi€m ky di Xl,...,Xstrong0 thl y «1>,0) = ind (Xl, <1»+...+ind (xs,<1» . Chi s6cuadi€m ky di t£.liva h£.ln. Gift sti'du'Qcxac dinh va khong suybi€n t£.limQi X E Rn voichufinduIon.Ta senoidi€m vah£.lnla di€m ky dicol~p cua N€u di€m va h£.lnla di€m ky di co l~pcuathl s6 quay cua bangnhautrenmQim~t cftu {X/ IIxll=r },ban kinh duIon. Gia trichungnay,ky hi~uind ( 00,) va duQcgQila chis(fcuadiim vah(ln. 1luh...cb!t2.Cho la truongvectortrenRn , co hUllh£.ln di€m ky di Xl, ...,Xs. Khi do,taco ind (00,) =ind (Xl, ) ll. TRU(JNGVECTORRoAN ToAN LIEN TUC . 1.Truonglient.,ctrongkhonggianYOh~nchi~u. ChoE la khonggianBanachthlfc.T~pM c E va :M ~ E la loantit Toan tti'gQi la hi chtJnn€u t6n t£.liA > 0 saGcho 11(x)11< A voi mQiX E M. N€u loan tti'lien tl:lcthl tagQila tnt(Jnglien tf:lc. - 5- flutfn 1JiM~lltq,c J,t Truonglien Wc du<jcgQila khongsuybie'ntrenM n6u x -:f.0 voix EM; du<jcgQila khongsuybie'ndJu n6ucos6 a> 0 saGcho II xII >a voimQix E M 2. Toan tii hoan toaDlien tl.1C. Cho E1,E2 la cackh6nggianBanachthlfc,gQiA la loan ttrtil El vaGE2(khiEl =E2 =E, tanoiA taed()ngtrongE); Ky hi~uqj)(A)la mi€n xacdinhcuaA va[!R(A)la anhcuaA. A du<jcgQila loantucompactrent~pM c Qj)(A)n6uanh m6it~pconbich~ncuaM lacompactu'ongd6i. MQtloantu du<jcgQila hoantoclnlien t~{en€u no vila compacvila lien tl.;lc. 3. Nhii'ngHnhcha'tph6 cua toaDtii tuye'nHnhboan toaDlien tl.1c. 56 thlfCAodu<jcgQila trj riengcualoan tti'tuy6ntinhB : E ~ E (E lakhonggianBanach),n€u phuongtrlnh( B - AoI)x=0 coitnhfitmQtnghi~mkhackhongXoE E. M6i nghi~mnaygQila vector iengcuaB lingvoi tririengAo. T~ph<jpcacvectorrieng(k6 ca vector0) ling voi AogQi la khonggianriengTI(Ao;B).Cho Ao* 0la mQttririengcualoan ttrtuy6ntinh hoanroanlien tl.;lCB. Voi m6i n = 1, 2,...,ta xet phuongtrlnh (1.3) (B - AoI/ x =0 -6- utfn1ftrPntltqc~t Theoly thuy€tcuaRiesz- 5chauderc6 s6 nosaochovoi n<nocacnghi~mcua (B - AoI)nx =0 la t~pconriengcuat~p hc;1pcacnghi~mcua(B - AoI)n+lx =0, trongkhi voi n ~not~p cacnghi~mcua (B - AoI/ x =0 bAng t~pcac nghi~mcua (B - AoI)ox =o.Honnlia,cacnghi~mcuamoiphuongtrlnhd(;lng (1.3)la khonggianvectorhUllh(;lnchi€u. Voi n=nokhonggianE(Ao;B) cacnghi~mcua (1.3)duQc gQila khonggiannghi~mcuaB lingvoi tri riengAo. 56 chi€u S(Ao)cuakhonggiannghi~mE(Ao;B)duQcgQila s6'b()icuatri riengAo, va reAD) =ngQilahc;mgcuaAo Khonggiannghi~mE(Ao;B) la khonggianconbatbi€n voiB; ngoairaBE (Ao;B)=E (Ao;B) Tli ly thuye'tcuaRiesz_5chauder,tac6 t6ngtn!ctie'p E =E(Ao ;B) EBN (Ao;B) cua khong gian nghi~mE(Ao;B)va khonggiancon bli vo h(;lnchi€u N(Ao;B); khong gian con bli N(Ao;B) clingbatbie'nvoi B. H(;lnch€ BocuaB trenN(Ao;B) la roantti'hoanroanlienWe;s6Aokhongla tri riengcuaBo.Moi ph~ntti'x E E c6bi€u di~nduynhatduoid(;lng x=u +v , trongd6 u E E (Ao; B) vav E N(Ao;B) Congthlicnayxacdinhhaiphepchi€u tuy€ntinhP(Ao) vaQ(Ao)tuonglingtenE(Ao;B),N(Ao;B)c6tinhchat P(Ao)X=u ; Q(Ao)x=v (x EE) va P(Ao)+Q(Ao)=I ; P(Ao)Q(Ao) =Q(Ao)P(Ao)=0 - 7- I(l{upt/lJitn tlzqc dt Ne'us6'A* 0khongphaila tri riengeualoantti'tuye'ntinh hoanroanlien t\IeB thl : R ('A;B)=(B - "Jr1 la roantti'lient\IexaedinhlIenkhong gianE, va dtfQegQila gidi thac(Resolvent)euaB. 4.D~ohamFrechet. ChoA la roantii'tuE1vaoE2,xaedinhtrongmQtHine~neua di~mXOEE1.Gia sii' A(xo +h) - Axo=Bh +ro(Xo,h) , trongdoB la roantti'tuye'ntinh(tu E1vao E2 ) va w(xo,h) =0 (IIhll ), nghia lit J~ ill (~;II h) = 0 Khi do A gQi la khd vi Frechet t~iXo; loan tti'B du<;1egQi la d(lo hamFrecheteuaA t~iXo, kyhi~ubdi A'(xo). DjnhIy 1.1 ChoA la loantii'hoanloanlien t\Ie,khavi Freehett~iXo.The'thld~ohamA'(xo)la roantti'tuye'ntinhhoan roanlien We. £hllng"minh. Gia sti'khAngdinhlIen kh6ngdung.V~y kh6ngcot~peompaehuagiatrieuaA'(XO)lIenm~te~udonvi { x : IIxll =1}. Khido,codaycaevectorddnviemvas6 8>0saocho II A'(XO) (ei - ej)II>30 (i,j =1,2,...; i*j) Giasii' llro(xo,h) II ~ 0 II h II khi II h II ~ p (p >0) B~t Xm =Xo+pem(m=1,2,...).Taco AXi- Axj =pA'(xo)(ei- e) +ro(xo,pej)- ro(xo,pej) suyra -8- --j,{pn'Iftan thqcJe 1/Axj - Axj 1/>P IIA'(xo)(ei - ej)1I- llro(xo,pei)11 - Ilro(xo,pej )11. VI the- IIAxj - Axj II>po> 0 ( i, j = 1,2, ...;i :;tj ) Nhungdi€u nay co nghIaday Axmkh6ngco day con hQi t\1. Mau thuftnvoi tinhhoanloanlien t\1ccuaA . 0 5.Dfing luaucuatrtiongvectorhoantoaDlien tl.lc. 5.1.Cacdinhnghia. Truongvector<Ptrenkh6nggianBanachE (duQc xacdinhtrenroanbQE ho~ctrent~pconM c E) gQila hoan roanlien t\1C,ne-u (1.4) <Px =x - Ax , trongd6A la roantli'hoanroanlien t\1CtrenE. Truong vectorhoan roan lien t\1C(1.4) gQi la hUll h~n chi€u, ne-uroantli'A hUllh~nchi€u. ChoM la t~pconbich~ncuaE. Taxetham (1.5) <p ( A,X) =x - A (A,X ) duQcgiasli'lien t\1Ctren[0,1]x M. Ntu A(O,x)=AoxvaA(l,x) =Alx thl (1.5)du'QcgQila phep bitn d~ngcua tru'ong<Pox=x - Aox ( X E M) vao truong <PIX=X - Alx (x E M). Phepbitn d~ng(1.5)duQcgQi la Compacntu loan tli'A(A ,,) hoanloan lien t\1Cvoi m6i A , vahonnuat~p ,oil(A) ={A(A, x): 0::;A<1 , x EM } -9- !fuq,n 1ftaniltffC {}1 la Compactu'dngd6i . Phep bien d~ngCompac(1.5) gQi l?t compaclienthongcactru'ongvectorhoanroanlien t1.JCI - Aova I - Al trenM. Ta thu'ong~pphepbiend~ng(1.5)du'did~ng <D (A,X)=x - AAIx - (I-A)Aox , (0 :::;;A :::;; 1, x E M) n6i cac tru'onglienWcI - AovaI - Al ,vagQila phepbiend~ngtuyen tinh. Phepbiend~ng(1.5)du'QcgQila kh6ngsaybienlieUham <D(A,X):;t: O. Hai tru'onghoan roan lien t1.JC0=I - Ao va 1=I - Al du'QcgQila dangluautrenM lieUchungdu'Qcn6i bdi phepbien d~ngkh6ngsaybien. 5.2 Tieu chu~nd6ngluan. JiSn chuftn1. ChoM c E la t~pbi ch~nva I - Ao, I - Al lanhii'ngtru'ongvectorhoanroanlientl;1Ckh6ngsaybien, anht~im6ix khongphaidi€m rheahu'dngd6ixung: 1/x - AoxwI(x- Aox)=1= - 1/ x - Alx wI(x- AIx) Khido,cactru'ongI - AovaI - Al dangluautrenM. I. " h ~ 2leu C nan . ChoM c E la t~pbi ch~nvaI - Ao, I - Al la cac tru'ongvectorhoanroanlien t1.JC,kh6ngsay bien trenM. Neu 1/Aox - Alx 1/:::;; 1/ x - Aoxl/ (x E M) thl tru'ongI - Ao vaI - Al dangluautrenM. 6.86quaycuatruongvectorhoantoaDlien t1}.c. - 10- (P,(q;n"~an tltf!C (j,i' 6.1 86 quaycuatrtiongvO'itoaDtit hUll h~nchi~u. Trongphftnnay,taxettn1ongvectorhoanroanlien tl.Jc. (1.6) x=x-Ax trenbiena0 cuat~pmabich~n0 c E. GiasttA la tmlntii'hUllh£.lnchi€u.Ky hi~uEola kh6nggian canhUllh£.lnchi€u chuaA(a 0 ) va it nhatmQtdi€m trongcua o. D~t00=0 (1 Eovaa 00la biencua00 . Ky hi~uAola h£.lnche'cuaA trena00. s6quay1«1>,0)cuatru'ongkh6ngsuybie'n(1.6)voiroan ttthUllh£.lnchi€u A du'Qcxac dinhtu s6 quay1 «1>0,00)cua tru'dngvectorlien t\lCox=x - Aox ( X E a00) xemnhu'tntdng vectortrenkh6nggianconhii'llh£.lnchi€u Eo.s6 quay1«1>,0) kh6ngph\lthuQcvaocachchQnkh6nggiancanEo. 6.2Binh nghIatrongtrtionghqpt6ngquat. Gia sii'tru'ongvectorhoanroanlien t\lC(1.6)kh6ngsuy bie'ntrenbiena0 cuat~pmabi ch~n0 c E. Tronglopd6ng luancuatru'ongvectorlien tl.JC,kh6ngsuybie'ntrena 0 chua (1.7) IX=X - A1x (x E a0 ) ,... , ~)H.K~I.TlJNHIEN TUUVIEN f ~ -. . 00356 tru'ong(1.6), t6n t£.litru'onghii'llh£.lnchi€u s6 quay1 «1>1,Q) cua (1.7)dfi du'Qcxac dinh trongphftn tru'oc.s6 quay1 «1>,0) cua (1.6) trena 0 bay gio du'Qcxac dinhbaid~ngthuc1«1>,0)=1«1>I,Q). - 11- (Pawn/IHln iliac ('yt. . £>6nha'nm~nhdangxettn.tongvectortrongkh6nggianE, tavie'ty «1>1,0;E) thayVIy «D,O). Ne'uloantii'beanloanlien tlJCA du'c;lcxacdinhtrongC10 va kh6ngco di6mbfftdQngtren 8 0, thl s6 quayy ( I-A, 0) trlingvoib~cLeray- SchaudercuaI - A trong0 d6ivoidi6m g6c. s6 quaycuatntongvectorhoanlOanlien tl.;lcla mQt s6nguyen. 6.3 CaeHoheha'teuas6quay. Iinb...c.hill.Nhungtru'ongvectorbeanloanlientlJc trenbienao cuat~pmabi ch~n0 c E mad6ngluanthlco clings6quay. Tlnhcha't2.Cho0 c E la t~pmabi ch~nvaOJ la cact~pmaroinhauOJcO. Ne'u <D=I - A la tru'ongvectorbeanloanlienWc, khongsuybie'n~renCIO \ UOj thls6quayy«1>,Qj)khackh6ng voimQts6hUllh~nchi s6j, va y «1>,0)=y «D,01)+...+y «D,On)+... .Iin.h...c.haL.. Cho0 la t~pmabi ch~nva XoE Q. The'thls6quaycuatru'ongvectorx=x- Xotren80 b~ng1. ~. Cho 0 c E la t~pma bi ch~n.Ne'u tru'ongvectorbean loan lien tlJC =I - A xacdinhtrongCIO vakh6ngsuybie'ntren80.N€u y «D,O)7=0 thlco it nhfftmQt di6m ky di cua - 12- aq;n //)an ihfIC {jet .TinlLchiL5.Cho0 c E la t~pmabi ch~nvagii sli' trl1ongvector hoan roan lien t\IC=I - A xacdinhva khong suybie'ntrongCIO thly «1>,0)=0. N€u y«1>,0)*- 0thlcoitnhfftmQtdi€m kydicuatrongO. Binhnghladi€m ky di co l~pXocuatniongva chIso ind(xo,<1»khong thayd6i khi chuy€n tu trliong trongkhong gianhUllh~nchi€u sangtrlionghoanroanlien t\lCtrongkhong gianBanach. IinlLcll!L6... Cho0 c E Ia t~pmabich~nva cD=I - A la trliongvectorhoanroanlien t\ICduQcxac dinh trongCIOvakhongsuybie'ntrenaO. Ne'ucohUllh~ndi€m ky diXl, ...,Xk trong0 thl y«1>,0)=ind (Xl, ) +... + ind (Xk , <1» Th.h..cllill. Gii sli' trliong vector hoan toaD lien wc =I - A xacdinhtrenE chIco hUllh~n(ji€m ky di Xl,...,Xk. The'thl ind (00,) =ind (Xl, ) + ...+ind (Xb <1» 7.T6ngquathoadfnhIy Hopf. TrongkhonggianhUllh~nchi€u, mi€n bi ch~n0 c E dliQcgQila mienJordan,ne'ut~pE\ CIO lienthong.Chftngh~n, hinhc~ula mi€n Jordan. - 13- mJn 1fl();nt~ (U Dinhly 1.2 ChoQ c E Ia mi€n Jordanva <D=I - A, \\f=I - B Iahaitru'ongvectorhoanloanlienWc,khongsuybie'n trenaO. Ne'uy (O,<D);j:.y (\V,O) thl <Dva \Vd6ngluan trenaQ. Dang rninh: D~ddngianta chi xettru'onghejptrongd6 anlam~tcftu.Khongmattinhtangquat, gia sti'0 la hlnh cftu{x/llxll <I} dod6aOIa m~tcftu{x: Ilxll=I}. Ta c6 th~gia sti' va \11la cac tru'onghuuh~n chi€u.GQiEoIa khonggianconhuuh~nchi€u chaaA(oO)va B(an).Ky hi~uAovaBoIa h~nche'cuaA vaB trena 00 , trongd6 00 =Q n Eo. Tli dinhnghlav€ sf)quayva gia thie't cuadinhly, tac6: y (I -Ao, 00 ; Eo) =y (I - Bo, , 00 ; Eo ). Chonentheodinhly Hopf,suyra cactru'ongox= -Aox va \\fox=x- Box,khixettrongEo,thld6ngluantrena00nghlala t6nt~iloantti'lien tvcCo(A,x), 0 ~A ~1, X E 8 00 voi gia tri trongEo saochoCo(O,x)=Aox, CoO,x)=Boxvax ;j:.CoCA, x) voi0~A<1va x E a00. GQiPola phepchie'utrenkhonggianconEo,du'ejcxacdinhtren E.SadvngCoCA,x)vaPo,taxacdinhphepbie'nd~ngcompac x - "Poxll Co (A, IIPoxW1Pox) XCA-,x) = ne'u0 ~A < 1,x E8 00, Pox;j:.0 x ne'uPox=0 - 14- 'wJn1JanfY~ fPt Phepbiend~ngnaykh6ngsuybien,n6i caetru'ong x (O,x)=x - IIPox IIA( IIPoxwI Pox) va x (l,x) =x - IIPox IIB( IIPoxwI Pox) maVIv~ydangluau.N6i eachkhac,suyra tru'ongX(O,x)dang luautrenan voi,vaX(1,x)dangluauvoi \V. - 15-