Luận văn Lời giải chỉnh hóa của phương trình tích phân loại một

CHUCiNGIII -:? AI' "- ? ,? "- DIEM BAT DQNG CUA TOAN TV T- DON DIEU §1.Toan td ddndi~uva diim ba'tdQng TrangchuangniiytavftnxetX Iii khonggianBancahthlJCvoiquanh~thutVsinh Mi nonK. SOA E R duQcgQiIii di€m chinhquycuaroantutuy6ntint A : X -+X n6uAI - A Ii songant,(jdayI Iii roantUd6ngnha'trangX. Ky hi~up (A) Iii t~pmQidi€m chinhquycua, vii () (A) =R \ P (A) duQcgQiIa ph6cuaJoantuA. ToantuA duQcgQiIii compacty6un6uA bi6nthanhmQt~pcompact y6u. A gQiIii lienWcy6un61,lA bi6rim6i dayhQiW y6uthanhdayhQitvy6utrong X. Kyhi~uL (X,X) Iiikhonggiancacroantutuy6ntinttrongX roantu Te-L (~,X) gQiIiiduongn6uT (K)c K voiK IiinontrongX. GiiisaDc X, roantuA :DcX -+X duQcgQiIii T- dondi~un6uA (x)- A (y)~ ? - T (x- y), \;jx ~y. d dayTEL (X,X). Nhuv~yn6uT ==0 thlkhaini~mT dondi~utrathiinhkhain i~mdondi~uthong thuongetabi6t. ' ~? ~ BODE 3.1.1. N6uTEL (X,X) vii -A ~cr(T}thl 1 . (AI +T) - (AA +T) (x)=x ~ A (x)=x. Chungmint:DoAI +T 1ftsongantnen A(x) =x ~ (AI +T)-I (AA+T)x =(A,X+T (x» -I (AX+T (x» =(A +Trl (A +T) (x) =I (x)=x 0 . 20 A? ~ EO DE 3.1.2. Gia slYliO,vo EK valiO-+X IaroantlYT- dondi~uvdiuo:::; Auo,Avo:::;vo.HonmlagiasaT thoadi~uki~n (HI) T duong (H2) :3A E (0,1)saocho:-.A ~cr(T) vaT (x);;::;..Ax suyrax E K KpidoS=(AI+T) -1 ('AA+T) ladondi~utrenvauo:::;S (uo);S (vo):::; YO. Chungminh: Do giathi€t (H2)tacoanhx:;t(AI+T) -1Iii duong. Neux ;;::y tac6A (Ax - Ay) ;;::- AT (x- y) ;;::- T (x - y) ~ AAx +T x;;::AAy +Ty. Tacd~nganhx:;t(AI +T) -1Tenhaiv€ cuaba'td~ngthuctren,taco Sx;;::Sy Ia roantudondi~u. B€ chungrninhuo :::;S (uo); S (vo) :::;vo tachi dn tac dQng(A I +T) -1Ien cacba't d~ngthuc . A uo +T uo :::;A Auo, AAvo +Tvo :::;AVO+T vo. Vab6d~duQchungrninh0 Dinh Ir : 3.1.1. Gia slYKIa nonchinhquy,A : -+X 1aroantlYT-dondi~uvdiuo:::;Auo, Avo:::;va.HonmlaT thoacaedi~uki~n(HI), (H2)trongb6d~3.1.2. Khi doA coit nha'trnQtdi€rn ba'tdQngtren Chungminh S': (AI +T) -1(AA +T)vdi- A ~cr(T). 21 Do b6d~3.1.1.tachic~nchungminhS coit J;1hatmQtdi€m batdQngtren<uo, vo> -. , Nhob6d~3.2.1.,S :~1adondi~u.;\p dl,mgh~qua2.1.2,S co di€mbatdQngtrenvadinh19duQcchungminh0 Dink if 3.1.2. Gia sii'K 1anonchuc1n.A ~ X 1aloantii'T dondi~u,couo::;A (uo),A (vo)::;vo,T th?acacdi€u ki~n(HI) (H2)trongb6d~3.1.2.Ne'uX 1akh6nggianphanX<;l thlAcolt nha~mQtdi€m batdQngtien. Chungminh . Ta d€ bill, dodinh191.2.1,khi.K 1anonchuc1nthlt~p1at~p16idongva bich~n.VI X 1aphanX1acompactye"u.Theob6 d~3.1.2thl loantii'S = (AI+T) -1(AA +T) bie'nvaova dondi~u.Ap d\;mgh~qua2.1.3.ta tha'yS co di€m batdQnglIen ,theob6 d~3.1.1.A co di€m batdQnglIen <uo, vO>.D ~ 22 § 2.NguyenIy anhx~co tren 16'pcacphftntii'so sanhduQc Chox lakhonggianBanachthvcdu'Qcs~pboinonK.A laroantatrenX. Xet phuongtrinhroanta Ax=x . (1) Nghi~in(1)thu'ongdu'Qctimdu'oid,;mgioih:;mtuadayl?p : . )(n =A (~II-I) (2)(n~1)voigiatIi band§uxotuy9.Ke'tquadabie'ttronggiai tichhamdolanguyen19anhx~co. Du'oidayla chungmQtke'tquatu'dngtvnguyen19anhx~co,songsvdanhgia chidvatrencacph§ntU'sosanhdu'Qc. Dinh 1'£3.2.1. Giasa: 1. K Ianonsinh,chuffn 2. A 12.roantatrongX thaadi~uki~n Ne'ux ~y thi : - B (x- y)~A (x)- A (y)~B (x- y) (3) (j dayB laroantatuye'ntinhdu'dngvoibankinhph6r (B)<1. Khi doA cotrangX di€m ba'tdQngduynha't,voikhoid§uxotuy9trongX. . Chungminh: Ta dabie'tvoim6iloantatuye'ntinhB trongX coth€ xetmQt chu~ntu'ongdu'dngvoichuffnbandftusaGchoII B II var (B)saikhacnhaudunha.V~y tugiathie'tr (B<1tacoth€ xemla II B II <1 . K lanonsinhnenvoim6ix E X, coth€ bi~udi<§ndu'oid~ngx=u(x)- vex)voi . u(x);vex)E K. Nghla.lavoi x E.K coungvoimQty E K saGcho- y ~x ~y (ch~ngh~ntacoth€ la'yy =u(x) +vex)tUkhai tri€n tIeD). TrenX tadinhnghlachuffnmojo 23 ,1/xllo=imllyll ,.(4) .' . 1~~~~ '" (Vi~cki€m ITa II . 11~lachu§ndvatrentinhcha'tII " II vadinhnghlainfimum). VI K la nonsinhnenvoix E X ~oth€ chQnu.v E K saocho II u II, II v II II ~ a II x II, ,0daya lamQth~ngso'khongph1,1thuQcvao:x(dinhly 1.2.5).V~yV x EX: "x II 0 =II u- v II o. ~ II u II 0 +11v II 0 ~ II u II + II v II ~2a II x II. M~tkhacK lanonchu§nnencoh~ngs6chuinN saochotU 0~x ~y =>II x II ~N II y II. Dov~ytaco II x II ~(2N +1) II x II o' V~yII . II~ II . II o' T~xetx,ytuyYEX. Giasii'-u~x-y~u voiUEK ", R6rangtudaytacox~'!2(x+y-u) , y~ Yz (x+y-u) Tugiathie't(3)suyra : , -B(~ -y +u)~A(x)-A(x -Y +u)~B{x-Y +u) 2, 2' 2 -B(Y -x +u) ~A(y)-A(x +Y- Uf:-B(Y-x +u) 2 2 2 Truve'voi ve'ba'td~ngthuGtrenchoba'td~ngthuGduoi taduQc: -B(ll) ~A(x) - A(y) ~B(u) V?y II A(x)- A(y) II 0 ~ II B(u) II ~qJ!ullvoiq<l (dogiathie'tvanh~nxetliB11<1). => IIA(x)-A(y) II o~qinf II u II =:=qII x-y II 0 -u~x-y~u V?yAla anhX<;l'COtheochuinII -I/o. VIX IakhongianBanachnenA co'di€m ba'tdQngduynha't. Chu yr~ngdinh ly 3.2.1v~ndungne'u(3) duQcthayboi di~uki~n-Bl(x-y) ~ A(x)-A(y)~B2(x-y);x ~y,V'x,yEX(3'),0dayB1,132lacactoantii'tuye'ntinhlienWc du'ongvoir(B1+B2)<1. § 3.Phuongtrinh v8i toan til nguQcduong 3.1XetphuongtrlnhF(x)=z (1) Voi F la toantii't.ll'khonggianBanachXl duQCs~pboi nonKl V8.0khonggian Banach,X2duQcs~pboinon K2" z la'ph~ntii'co'dinhtrongX2.- , , 24 Gia sttF thoadi~uki~n.: BI(x-y) ::;F(x) -F(y) ::;B2(x-y) (2) ddayBI,B2la caetoanttt.tuye'ntinhtuXl VaGX2. Ta codinh19sari: Binhly 3.3.1 Giastt1)Xlla nonsinh,chu§'n 2)ToantttF:XI -+X2thoadi~uki~n(2) BI vaBI +B2cocactoantttnguQcduong.. Khi dophuongtrlnh(1)cotrongXI nghi~mduongduynha'tvoim6i ZEX2. Chungmin~ Ta d~tD =Y2 (~I+B2)khido(1)cotheduav6d<;lngy=AytrongX voito<lnttt Ay =Y -FDOly + Z (3) Nghienx*cua(1)duQcxacdinhquanghieny* cua(3)bdih~thUGx* =DOI(y*). . . Tlr gia thie'tdoi voi BI,B2suyfa 151latoan ttttuye'ntinhduong,VI v~yvoi u,V EX2vau ~v tacoDOI(u)~Dol(v).Va tugiathie't(2)tacodanhgia: BI D-I (u-v)::;F Dol(u)- F D-I(v)::;B2Dol(u-v) Buav~danhgiacochuaA, vatudinhnghlaD taduQc - (I-BIDOl)(u-v)::;A (u)- A (v)::;(I-BID-I)(u-v) (4) ',. X . >VOl U,V E 2,u- V. Ba'td&ngthUG(4)cothexemnhu(3)trong2 doivoi toan tttB=I-BI Dol. Theodinh193.2.1dehoanthanhphepchungminhta chic$nchirar( I-BI Dol)<1 tS) (5)Tu (4) taclingconh~nxet : I-B] Do] la toantli'tuye'ntinh duongtu X2 VaGX2.VI the'I-B ]Dol lien t\lC( VI toanttttuye'ntinh duongtunonsinhVaGnonchu§'nthllien We). SongkhidoB] Do]toan tttnguQccuanoD BI -Ila lien t\lc.Bieu thUG(5)seduQcchungmiJih ne'utacoduQcdanhgia: ltU').< It(G\)- ~-tlt(P)' 25 (6) d day P =I-BI D-I; Q ~DB1-1- I (7) Cacroantii' P,Qtrang(7)la duongvachunglienh~nhaubC'Jih~thuc: Q =pel-F) -1=per-F) -1-1=(I-P) -lp (8) Gis.sii'£>0var(8Q))<I. Khi dOI-'£QcOroantii'nguQclient\lC (I -£Q)"1=I +£Q+ +£nQn+.... Honn~atli tint duongcuaQ suyratint duong'cua(I - 8Q) -1.Tli dangnhuc1- (I +£)P =(I - P) (I - £ Q) suyraroantii'1- (1+£)P cOroantii'nguQclien t\lCvavdim6i n ~1cO: n ,L (1 +8)j =[I - I (1.+8)P j -1 [P (1 +8)n+ 1P n+2 1-0 = [I - 8Q) -1(I - P) -1P [ 1- (1 +8)n+I P n+I =(I - 8 Q) -I Q [ I - (1+8)n+1 P n+I] VI v~yroantii'duongP thoadi€u ki~n (1 +£) n P n+1(u) :$(I - £ Q) - 1Q (u) (u E K2,n ~ I ) (9) Giasax ladiemtuy trongX2.VI K2la nOnsinhnenx =u- v, u, V E K2.Tli cac ba'td~ngthuc -v<x<uva(9)suyra ~(I- EQ)-I Q (v):$(1 +Et P n- I (x) :$(1 - EQ) -I Q (u) VIK2lanOnchu£nnendays6 11(1+E)npn-I (x) IlIa bich~n.VakhidOds.y (1+E)n II pn-I II bi ch~n,VI the', (1 + e) r (P):$ 1. . 1 - I( Q) 0 V~y: r(P):$ mf 1+8-l+r(Q) Ta cOnh~nxet : Trangcacdi€u ki~ncliadint 193.3.1nghi~mx (z) cua phuongtrinh(1)ph\lthuQcdondi~uVaGz.Ne'u1 :$~thlx (Zl):$x (Z2)' 26 3.2.BaygiGtax6tphuongtrlnh Tx =Gx (10) (10) T, G la{oantlYtacdQngtuXI vaoX2;T la roantlYtuye'ntint, G la roantlYphi tuye'nthoadieuki~n: -B(x- y)~G ex)- G (y)~B (x- y) (11) (\ix, Y E X1.~~~y) ? () dayB la roantiYtuye'ntinh.TuongtVdinhly 3.3.1tachungminhke'tquasail. Dink 1£3.3.2. Gia slY1)K21anonsinh,chuffn: . 2) Cac roantiYT,G thoadi~uki~n(11)themnlYacacroantlYT, T - B co roan tlYnguQc duong. Khi dophuongtrlnh(10)congqi~mduynha't Chungminh: Phuongtrlnh(10)tuongduongvoi T(x) -G(x) =8 D~tF(x)=T(x)- G(x),taco (T-B) (x-y)~F(x) - F(y)~(T+B)(x-y) V~yF thoadieuki~ndinhly 3.3.1voiBI =T-BvaB2=T+B, nenphuongtrinh T(x) -'-G(x)=8.conghi~mduynha't0 ' 3.3 Trangph~nnaytav~nxetphucmgtrinh(10).Ne'ugiathie'tT coroantlY nguQcthlphuongtrlnh(10)tuongduongvoiphuongtrlnhsailtrongX2 y~ Grl y (12) Eli di 3.3.1Giii slY 1)~ ~i.~Bi~-ti~I ZoE K2 va HI :Xj -7X2tuye'ntint duong. 2)T, T- BI coroantlYnguQcduong. 27 . Khi dophftntU'Uo=T(T-Blyl ZOthuQcK2vatoantii'arl bie'n°vao chinhno . . Chungmillh Tir d6ngnha'tthucT( T-Blyl =I+Bl( T-B1yl tatha'yUoE K2.Do tinhdu'ongcua toantii'rl ta c6 V y E 8 $ Ty-l $ rl (uo) =>8 $ arl y$ BI rly +Zo$ BlTluo +Zo Nhu'ngBlTl(uo) +Zo=Bl(T-Blylzo +Zo=uo' Nentacodi€u dn chungminh0 Billh If 3.3.3 aia sii'cactoantii'T.a thoacacdi€u ki~n1)2)cuab6d€ 3.3.1.vacacdi€u ki~n sail. (i) K2 la nonchinhqui arl( ) 13t~pcompactu'ongd6i. (ii) X2lakh6nggianphanx~. Khido phu'ongtrinh(10)conghi~mtren. Chungminh VI ddondi~u,Tl tuye'ntinh,du'ongnenaTl clingdondi~uvabie'nVaG .Ap dt,lngcach~qua2.1.1,2.1.2,2.1.3,tatha'yaTl codi~mba'tdQngtr~~<8, Uo>hay(10)c6nghi~mtren .0 5\J