Luận án Bài toán giá trị biên cho phương trình vi phân hàm

CuljnvanCaohqc CHfJdNG II ~traG> . NHUNG BAt rOAN CUA LUA.NVAN GiasaE 1akh6nggianBanachvdiChU~11I. I C =C([- r,O],E)1akh6ng gian Banach cac ham lien t1,lctIeD [ - r,O] VaGE, r ~pvdichuin IIxll=Sup~x(e)1:8 E [- r,O]} f)~tX =C([-f,CO),E) 1akh6nggiantit cacaehamlien Wetren[-[,co) VaGE . '\IxE X va t ~O.B~t XtE C dl.fc;1cdinh bei : . Xt : [ -r,O] ~ E 61-7Xt (8) =x (t+9) Cho<p: C ~ En=E x E x...XE la anhx~tuye'ntinhlientl,lc. Taxetbailoanphuongtrinhvi phan: { X'(t) =f(t, cp(xt))+get,<p(Xt); t ~0 (I) Xo =h h E C chotrudc. (1.1). f: [0,cc)xEn ~ E lien t1,lC,va '\inc:N 3kn>0 saGcho: If(t,u)-f(t,u')1~knllu-u'll '\It E [O,n];u = (UI' U2,..., Un) E En , ( ' , ' ) E o u = U b U 2,...Un E n - IIU-U'I/=2:IUi -u~1 1achuin trenEn i=l (1.2) g:[O,co)xEn ~E la anhX?Compact NghIa1ag lien tl,lCva bienmQtt~pbi cL~ntrong[O,cc~ X En thanh t~ip Compactu'ongd6itrongE. V6inhlingkyhi~uvadint nghlanhu'trentaco : Trant]4 Luqn van CClt)AClC --------- Bini !ly 1: Gia sU'f, g thoacaedi~uki~n(1.1), (1.2)va lin Iget,u)l =: 0 ~u~-)~ llull (1.3) , - deli theot trenmOlt~pbi ch~ncua[0,00), \::11EEn, ilull=[U11+IU21+...+[Unl Kli do bai lOan(I) co nghi~mLIen[[,00) C!ui:ngminh " (1).~ { t t - (I" x(t)=h(O)+jf(s, <p(xJ)ds-+-fg(s,<p(xs))ds, , 0 0 ,xo =h D~jXo= C ([0,00),E) la khonggianFreehetilteacaehamlienWe tren[O,cc)VaGE vdih9lllla chu~n: Pn(x)=Sup ~ I x(t) I : t E [O,n]~ , \::IQ V8- metric -- ooj ) d( =2"~~:-_~_E~~__:::=_X_- ~~- 2' il 1+P (X - y)n-i d n Voi m6i x E Xo, ci~tx: [-r,oo) -0 E ciNc ciinhnghla nhu'sail : - { Xes)+h(O)- xeD);s 2 0 )(s) = h(s) ; --r :;;s :;;0 - 1ehido x lien t1fctren [-f,OO). fa dinh nghlacaelOantli': LJ: Xo-7 Xo X t~Vex) t "(ii U(x)(t) =ff(s,<p(~s»ds; 0 t 2 0 -- --- -~ -------------- Trani] 5 [ucJn l7anCao n(Jc G: Xo-7 Xo x -7 G(x) t voi G(x)(t) =Jg(s,<p(~s))ds+h(O) ; 0 l20 Ta nh~ntha'yding ne'ux E Xo Hi di~mb~tdQngcua U +G tue13 U(x)(t) + G(x)(t) =x(t), Vt 2 0 ! t t -'- x(t) =h(O)+ ff(s,<p(~s))ds+ fg(s,<p(~s))ds r 0 0 . x(O)=h(O) t t . x(t) =h(0)+Sf(s,CP(Xs))ds+ J g(s,cp(~s))ds;t 20 =>~ 0 0 - x(t) =her) ;-r ~t ~0 - Chungto x 1anghit%mcua(1')tren[-r,oo). Ta sechungminhU +G e6di~mba'tdQngb~ngeachsU'd~mgdinh1.9 2Chuang1.TrudetieRtachungminhcaeb6 d~san: B6 d~1: Gia SiTf thoac1i~ukit%ll(1.1),U du'Qed;,nhnghT~nhutren.Khi d6: Vn EN, Vz E Xo Ta c6: ( i i )< 2(n.kn.II<pllrPnDz(x)-Uz(Y)- OJ Pn(x-y)1. Changminh : Ta sechungminhr~ng-: (1) II ' I . I . . 2(tk n. <PJ I IU~(x)(t)-U~(y)(_t)I~ i!" Pn(x-y) Voi ffiQit E [O,n]vaeE [~r,O] Ta c6: 7ranq 6 LuqnvanCaohqc - - - - xsC8)-Ys(8) =x(s+8)-y(s-i 8)= = { X(S+8)-Y(S+8)+Y(O)-X(O);S+82::0 - h(s+8)- h(s+8)=0 ;-r s s+e<0 ~ !xs(8) - Y s(8)1s Ix(s+8)- yes+8)1+Ix(O)- y(O)1= =lex - y)(s+8)1+lex- y)(O)1S 2Pn(x - y) doPn(x-y)=supi Iex - y)(t) I : t E- [0 , n] ~ Tachungminh(1)b~ngquin:iptheoi. Voi i =1: 'IIx , UzCx)(t)=U(x)(t) +let) ! =ff(s,cp(xs))ds+let)" 0 Iuz(x)(t)- Uz (y)(t)1 t t I =I ff(s,cp(~s))ds- ff(8, cp(ys ))dS I0 0 ! t t s flies,cp(~s))ds - J f(s, cp(ys))Idss fknIlcp(~s).- CPcYs )lIds 0 0 0 - --- ! f - k II ( - - ) 1, < I f' 11 1111 - - II -' ~ =0 n1I<rx s - Ys I as - 0Kn <PIlixs - ys as s: t <fknllCPl12Pn(x- y)ds=2kn.t.!lcpll.Pn(x- y) 0 Gilt s11(1) dungvdi i (i2::1). Ta chungminh (1) dung vdi i +1,th~tv~y, Trang 7 Luqnvan Cao hqc IUz i+l(x)(t)- U zi+l(y)(t)1=luz (U / (x»(t) -- U z(U/ (y»(t)l- =!U(Uzi (x»(t) - z(t) +U(U zi (y»(t) +z(t)! =Iu(uzi (x»(t) - U(U zi (y»(d I I =I ff(S, <p[(u~(x)t1ds- J f(s,<p[(u~-(y)TPdS I I 0 0 ~ fknll<p[(u~(x))J-<P[(u~(y))Jds 0 ~ fknl!.<plr[I[(u~(x»)s]- [(u~(X»)s]!dS 0 Ma m6is E [O,n], 8 E [-r,O],taco: (u~(x)t (8)- (u~(y»)s(8) = --. . =U~(x)(s+8)-U~(y)(s+e) = { U;(X)(S +8) - U~(y)(s+8)+U~(y)(O)- u~(x)(O);S-I-G~0 h(s+8)- h(s+8)=0 ;- r ~s+8~a Taco: . . U~(x)(O)- U~(y)(O):=0 i =1, Vi, Uz(x)(O)- Uz(y)(O)=U(x)(O)+zeD)- U(y)(O)- zeD) =U(x)(O)- U(y)(O):=0- 0=a i =2, U;(x)(O)=Uz (Uz(x»(O)=U(U z(x»eo)+z(O) =U[U(x)(O) +zeD)]+zeD)= U[zeo)]+zeD) Tild6: U;(x)(O)-U;(y)(O)=O Bangquin;;tpc6duQcU ~ (x)(O)- U ~ (y)(O)=0 ,Vi --..----- Tranq 8 Luqn van CaDnClc Tild6 : I(u~(x)t(8) - (U~(y))s(8)1~ S;lu~(x)(s+8) - U~(y)(s +8)1::; 2[(s+8).kn.11<pllr S; .xPn(X-Y) 11 u (Dogiathi6tquin~p)- r II II] i 2l(s;kn. <P S;- - .,?<Pn(x - y) ; \f8 E [-r,O]1. \ \ " " suyfa : IU~+1(x)(t)- U~+\(y)(t)!s; S;fkn 11<p11.11((]~(x»)s - (U~(y»)sIrs0 < tf - II II 2(s.kn .JI<pll)i - , - kn <p. ., XPn(x-YJ 0 1. . 2(tkn .jI<pll)i+\- X P (X - 'H) - (i + I)! Xl J V~Y(1)dungchoIIlQilEN, tiIdochungminhdu'<;Jcb6d~1. B6d~2: ToantU'U thoacacdi~ukit$ni, ii, iii cuadinh1y2,ChuangI, Chrlngminh.. . ~. ~(n,kn.II <pI I)1 A' ;, n,k ,11<v11 DochuOl~ or . hQ1 tvvee n "'" i=\ 1. Nen suyfa Jim 2(n.knll<pll)i -i~oo i!--- =0 ,nen ::Jin (inph1,lthuQctheon ) saocho: 2(n.kn .1I<pII)i< 1A= " 1. Vi >i, - n Tild6 : Pn(U~(x)- U~(y»)S;APn(x- y) Trang 9 Luqn vanCaonc;c ------- Vflv U thoaQi~ukieniii..1 . Uz (x)(t)=U(x)(t) +z(t) , '\Ix E Xo -Nen Iuz(x)(t)- V z(y)(t)1=jU(x)(t)- U(y)(t)1= t t =I ff(s, <p(~s»ds - ff(s, <p(ys »ds[~. 0 0 t t I~ ff(s, <p(~s»- ff(s,<p(ys»ds ~ 0 0 I t t ~ fkn 1I<p(~s)- <p(ys )Ids=Jkn Ilcp(xs---ys 0 0 -- ~ Jk,~'P!I-~6',)- (y, i ds~ Jkn 11<p11-2p,(x - y)ds 0 I 0 ~(2t.kn.1I<PIIPn(x - y) ~2kn.n.II<PIIPn(x - y) =k'Pn (x- y) ~ Pn(V(x) - U(y»)~k.pn(x - y). U thoaii) Co th~SHYra tuB6 d~1,voi i =1 cli6i clingtachungminhU thoadi~uki~ni) tile1adi~uki~nA V: Xo-7Xo x t-7Vex) t ; U(x)(t) =ff(s,<p(xs»ds 0 Z E Xo , u z : Xo -7Xo Uz(x}=Vex) +z AI: Vz E Xo , Uz(Xo)c Xo h";? 1.'~Jell lldlen A2: Ta phai chungminh :Vz E Xo , Vpn, :3kz EZ + saacha : Vs >0, :3r E N va (5>0: Vx,y E Xo, a~(x,y) <S+(5 ~ a~(u~(x)~U~(y» <S Trang"10 [wIn van CaDh(JC a day a~(x,y) =max{Pn(U~(x)- U~(y))i i, j =0,..., kz }--- ai(V~(x),U~(y»)=max{Pn(U~H(x)-U~+r(y))/i,j=O,...,kz} Taco: -- Pn(U~+r(x)~ U~+r(y»)=Pn(U~(U~(x»)- U~(lJ~(y)) 2(n.k . ll ml ll) r ( . . \ °, ,- .< . n . 't' V i ( ) - U J (.. ' (B A a/' 1)- f - . x Pn z X - z YJ) 0 ef. . 2(n.kn .11<pllr ( ) - ( --x 8+0(8 r! Khi r duldnva ai (x,y)(s+o. Vf}.yU thoadi~uki~nA. B6d~2duQc hungminhKong. Lzl,anva n CaD ~oc ? ~B'ode3: Gia sli'g rhea(1.2)va Gndlt<;5cdinhnghlanhusail: Gn: Xn ~ Xn =C([O,n].E) t Gn(x)(t)= fg(s,<p(xs)ds+h(O) 0 0::::;t::::;n Khi d6Gnla roantli'CompactIeDkh6nggianBanachXnvoichu~n Ilxll=sup~x(t)1: 0 ::::;t :::;n}. x E X n thl x:[- r,n]~ E - { X(S)+h(O)-X(O);S ~0 xes)= h(s) ;-r ::::; s ::::; 0 xs : [- r,O]~ E eH x(s+e) Chitngminh: TnJ'oc tieDta c6 th§ tha'yGn(x)E Xn, th~tv~y: Vc:)O, . . t t' I IG n(x)(t)-Gn(x)(t')!=I fg(s,<p(xs»ds - Jg(s,<p(xs»dSI 0 0 I t = I fg(s,<p(xs»ds t' t ::::; ~g(s,<p(xs»!ds t' t ::::; fMds::::;let- t')IM t' (Vi g la anhx<:tCompactlienhich?ntIeDt~phi ch?n) Tild6 : I G. (x)(t)- G (x)(t') I <c:')\ 1" E khi It - t'1<8= M Tranc 12 Lzi,arzvanCao ~oc Ta chli'ngminhGn lien tl,lC: La'y(Xk)kladaytrongXnsaccholimxk=Xo k---j>CX) - D~tB ={(Xk)s/s E [O,n],k E Z+} - Khi doB CompactrongC =C([-r,O],E). Th~tv~y,giasa (Cxki)sJi 1aday-trongB, Ta co th~giltsarAnglimsi =s i---j>CX) va limxki=x i---j>CX) (Chu yr~ngx coth~Iii mQttrongcacXkdacho,k E Z+va(xki)ico th€ khongla dayconcua(Xk)k.) Taco: 1/(Xki)Si-Xsllsll(Xki)Si -Xsill+IIXSi-xsIIS2Pn(Xki-x)+IIXSi -Xsil vdim6ix E Xncodin:.lI,faco anhX~ [O,n]~ C =C ([-r,O],E) ; S M Xs lienwc Th~tv~y,V E>0VI x lien tl,lCd~utren[O,n] lien38>0: 'lit,t' E [O,n]: !t-t'I<8 =?lx{t)-x(t')I<Eo Khido '118E[-r,0] 'IIs,s'E[O,n]:ls-s'I<8 thl: Ix(s+8)- x(s'+8)I<E , nay Ixs(8)- xs.(8)I(E dungchom9i8 E [-r,O],dodo : IlKs- Xs'"=sup~xs(8)- xs'(8)1:8E [- r,O])(£ Cholientuba'tding thli'c: II(Xki)Si- xslls2Pn(xki - x)+IlxSi- xsll Taco lim(xki)Si=xs trongC i-to: V~yB CompacttrongC. Trang 13 Lt1,an17anCaohoc --- Do cp: C ~ Entuye'ntinh, lien Wc lien cp(B) CompacttrGRgEn, nen suyra t~p[O,n]X cp(B) Compacttrong[0,00)x En. Vdi mQi c:> a clio trudc, VI g lien Wc trel1t~pCompact [O,n]x cp(B) lien38 >0: V u, V E B. t: IIcp(ut-cp(v)IIIg(s,cp(u))- g(s,cp(v))I<- - - - - 11 Yl- limxk =-Xo trongXn lien 3ko EN: \fk ;;:::ko k-+oo 8 Ilx~ - xoll<21lcpll II(x,),- (;:0 ),11:£211xk- xoll<II:11 Dftnde'n: Suyfa 'itE [O,n].Ta co : t t IGn(Xk Jet) - Gn(Xo)(t),=Ifg(s,cp«~k)s))ds~ fg(s,cp((;Zo)J)ds 0 0 l ~ Ii g(s,cp«~k)s))-g(s,cp«~o)s))ds 0 - ! c: - ~ I- ds~c: r n - Vk;;:::ko (VI Vk;;:::ko,II<p(~k)J-cp(xo)sll " II'I'II-II(:X-k), - (xo t 11<11'1'11-11:11= 8 - Tli do: 110n(xk)- GIi (x 0)11<c:, Vk 2::ko Nghla I} Gnlien h,lc. Bay gid tachungminh Gn13Compact. Giasli'D 13t~pbi chi;introngXn 8~tA = {~sE C: x ED,s E [O,n]}thlA bi ch~ntrongC (d~thay). VI g Compactlien t~pg ([O,n]x cp(An Compacttu'cJngdoi trongE (chuy:A bi ch~ntrongC , cp: C -7Entuyentinh lien tJ,lclien cp(A) bi ch~n trong En,Th~t v~y, Vx E A,llcp(x)11~ Ilcpllllxll~ IlcpilM) Tram] 14 Lwin van CaD h9C ------------ Kbi dotheodinhnghla dla Gntaco Gn(O)Q~nglienh!ctre-n[O,n] Th~tv~y,cp(A) bi ch?n trongEnnen:3 R>0 :cp(A) c B(O,R) Do g l?tanhx~CompactnenWpg ([O,n]x B(G, t6nt~i\1>0: Ig(s,u)~M, \:IsE [0,nJ}iuEBeG,R) : ) / ch?in.Suyra Vdi m9i t,t'E [O,n],taco :_- t t' IG n(x)(t)- Gn (x)(t')1=If g(s,cp(~s»ds - fg(s,cp(~s))ds 0 0 - , t - 1- ~If g(s,cp(~s»dSI~Mlt - t'l Tli do Gn(O)d~nglien tl,lctren[O,n]. B?t K =Co(g([O,n]xcp(A»)U{O}] K CompactrongE. - - - g(s,cp(xs»E K, \:IsE [O,n], \/x E Q nensuyra \:ItE[O,n] taco: { t _1 - Gn (O)(t) = fg(s,cp(xs»ds+h(O)/x E n J c tK +h(O) V~yGn(Q)(t)Compactu'dngd6itrongE. Theodinhly Ascoli- Azela Gn(0) compactu'dngd6i trongXn.V~y Gnl?tanhx~Compact,b6ciS3duQccblIngminh. Tranq15 Lz(anvanCao ~oc Ifo d~4: Tmintu G: Xo -7 Xo X H G(x) t E)inh bdi G(x)(t) = fg(s,<p(X"s))ds+h(O) lil toaD tu'compact(hoan 0 . to~mlien tl,lc)va Pn(G(D))(+oo voiPn(D)(+oo, DcXo Chitngminh : - Giasu(xkhladaytrongXosaorho: limxk=Xo nghHila limPn (Xk - xo):"':0, k-7CC . k~"'" \in Vi (xkhhQitl,ld~uv~Xotrel1[O,n],Vn. Do dotheob6d~3 Gnlien tl,lCtrenXnDenc6 : Gn(Xkl[o,nJ)k-=tOOGn(xol[o,nJ) (*) t maGn(Xk Jet) = fg(s,<p(~k&,))ds+h(O) 0 - G(x Vt\. 0<j <) - ]- -kl,-/' _I__c.. ( d"y \ \ 1-0') Densuyfa lim Pn(G(Xk) - G(Xo))=0, \In K-700 Th?tV?y, Pn(G(Xk)- G(XO))=sUP~G(Xk)(t)- G(XO)(t)1:0~t ~n} Tli (*)va(**) suyra: G(Xk)(t) -7 G(xo)(t), Vt E [0,n] k-7OO Do d6 Pn(G(Xk)-G(x.O))-7 0 k-7CO Nghiala G lien tl,lctrenXo GQi[2la t?P bi ch?ntrongXo,tachungminhG(0) Vi compactu'ang doitrongXo.DungBinh ly 3 ChuangIta cbIc§nchungmint d.c di~usau: G(~~)d~nglien tl,lctren[O,n],Vn. Bi~unayc6do : Trang 16 Lz1,an17an CaD ho Co G(X)(t)=Gn(xlxnXt); O~t~n ma Gn(xlXnXt) lien tl.JcdSutren[0,n] (doGn compacttrenXn) 'vE>0, 38>0 : '\It,1'E [O,n] Ie- t'I(8=>IGn(xlXnXt) ~? n(xlXnft' *~ =>IG(x)(t)-G(x)(t' )1<8 . DiSu nay dungcho mQix E Q~V~yG(Q) d~nglien tlfcireDXu. {XCi)/ x E G(O); t E [0, I!J}compacttHongc1cfitrong E . - Di~unaycolado: {x(t)/xE G(Q); t E [O,nJ} ={X(t)/xlxnEG(OnXn); tE[O,n]} T~pnaycompactu'ongd6itrongE VIGncompactrongXn- V?y b6 d~.4duQcchungminh- - .. ? ;I;:Bhde5: Voi toantll'G dinhnghlaa tr&11faco : lim Pn(G(x)) Pn(X)-700Pn(x) =0, \:In Chitngn1inh : Voi E>0ba'tky,do . Ig(s,u)I_Ohm - , IIUII-7CXJ Ilull Ig(s,u)1yM 6nll~II' liencoso'Y>0,y>IIhl!saoeha: '\IsE [0,n] g compactlien: 3M>0:lg(s,u)I~M, '\IuEEn)lIull:::;y,VsE[O,n] M E Ch9nYl> 0 saocha -<- Yl 2n Trant]17 Lzi,a nvanCao hoc -_.----------- --- Vx E Xn ma: IIxlln;:::11 chua'ntIeD Xn Taco: IG(x)(t)1 1 ( n - I. '\ IIxllo ::;N[ jlg(s:cp(Xs»Ids-+Ih(G)1) . ~llxlll ( flg(s,cp(xs»ldS+flg(s,cp(xs»Ids+I n II 12 ? " II =~E [O,n]: Il cp(Xs) ll ::;y}0 day .- -. --I2=[O,n]\II Suy ra-: 'lit E [O,n] IG(x)(t)1<M.n+ fll'l'(;:,)11(S'CP(,,~t~ !!xllo-!!xt 12 "Xlln Ilcp(Xs )11 'lIxlin - rXeS)+h(O)- xeD); s;::0 Dc xes)=1 -Lh(s) ; -[:5;sS;O - - Va Xs(8)~ x(st 8) Neil: IlKs II::;IIxL ::;2/1xlln+Ilhll Th?tv~y,Ixs (9)1=!x(s+8)/ = { Ix(s+e)+h(O)- x(O)I;s ;:::0 Ih(s)1 ;;-r::;s::;O ~ 211xlln +IIhll Suy fa : IIcp(Xs)/I~ flcpllllxsII::;IIcpll(Lllxt+IIhl!) - "' Ig(s,cp(xs)1 's 0 11 - II Ma Ilcp(xs)/1 y (*) Tran9 18 Lzi,an17anCao ~oc f" x ds - 12 Ilxl!n 11<p(~s)1I < E f[ Ilhll JI I£I"\II - 6nll<p112 2 +IlxliuI~ds < Ellffllfn ( 2+it J dS - 6nll<p110 ' Ilxlin .. =~ ( 2+it ] <~(2+1)=~ 6 Ilx!ln 6 ,".. 2 khi Ilxlin auldntilela: , lp(X' )11x Igrs,<p~x,))1ds <-"- 12 Ilxnll '11<p(xs)11 2 (**) Tli (*) va (**) suyra Jim Pn(G(x» =0, Vn , Pn(X)~a) Pn (x) Nhu'v~y,tli B6 d~4,5, G thoadi~uki~niv) v) cuaBinh 112 Chuang ~ , " B~ d~,.... TT G ~ d.~ bK;fA 0 ' 1 . ~ 1 .oJ. h~1va tu 0 e L, m suy ra u + co tern ,at uQng.: '~nnIy uu\:fcCll(ng .. niinh. Chti thich : Pn(G(D» < +00 voi PileD)<+00, D c Xo Th~tv~y,la'yD c Xo : pnCD)<+00 ,11 E N Ta d~tfn =sup~Pn(X): XE D r Suyra IIXI!n~rn, '\IxE D,llt l?lchu{{ntrenC([O,n],E) Ta CO: Pn(G(X»)=sup~G(X)(t)l:t E[0,n]} :0;SUP{~g(S''1'(;[,))Ids+Ih(O)I:tE[0,n]} Do:11<p(~s)11 ~11<pIIII~sII ~1I<p11(21Ixl!n+II wO ~11<p11(2rn+I:hij)=Mn c6dinhvdincbotruck Tram}19 "Cz1,anvan Cao 00C \:! g compactlien:3hn>0: Ig(s,cpC;Zs»1:::;;hn \Is E [O,n], Vx ED.' Suy ra : Pn(G(x» :::;;hnt +Ih(O)1:::;;nhn+Ih(O)[ Vx ED d§:nd€n Pn(G(D» <+00 rtJo*a@rtJo*a@rtJo*a@rtJo*..e.i'*a@ - , , . '\ - - - Gia sli'X =C([-r;-oo);E) la kh6nggianFrechetHitcacachamlien - . lIen [-r,co)vaoE vdihQml'achuin. p'n(x)=sup~X(t)1:t E [-r, n]} A(t) la mQthQcuacacroantli'tuy€n Huhbi ch?ntu En -) E ph1,1 thuQclien t1,1ctheot;:::O. . Voi m?ikE Z+, giasagk: [0,00)x En-) E t~oacacc1i~uki~n (1.2), k 0 . - - (1.3)vag ~. g . Voi day(hkhtrongC thoahmhk =ho . k~oo, Voi m6ik E Z+,-giasaXkla nghi~mcuabailoanvdigia trid~u: _ { XI(t) - . - A . (t)CP(Xt) +gk(t,cp(Xt»; t ;:::0 Ilk Xo = hk Voi nhtinggiathi€t a lIen tacodint 19sail: IJ nhry 2: Ne'uphuongtrmhlIo-!lICphuongtrlnh: { X' (t) =oA(t)CP(Xt)+gO(t,cp(Xt»; t ;:::0 Xo=h - conghi~mduynha'tXO thltaco: lim xk =Xo k~oo --- Tranq 20 Lz(a n17Cln Cao 00 C Nlfanxet: Vdi m6ik phuongtrlnhIlkluonconghi(~nL Th~tv~y,d?tf: [0,00)x En-7E f(t,u)=A(t)(u). / If(t,u)-f(t,u')1=IA(tyu)-ACtXh')1:=~!A(t)(u-u')1Ta co: ~IIA(t)l!X Ilu --:-u'll VI hQ A(t) : En -7E tuy€ntfnh,lienWephl,lthuQclien tlfctheet,nen- . V'n E N, :3kn : IIA(t)1I~kn, V'tE [0,11] Tli Binh 1y1,tacophuongtrlnhIlk co Ta chungminhDinh1y2quacacbucksail: Bude1: Vdi m6in E N, giasaX' n=C([-r,n],E)vdi chua'n IIXii'n=sup~x(t)l:t E [-r,n]} - Bn ={xkh-r,nr k E Z+} Ta eo Bnbi ch?ntrongX' II Chicngminhbui1c 1 : Trucktientachungminhr~ng: Igk(t,u)I~M+~llulln vdi E > 0 cho trude,'v'tE [O,n],Vu E Ell. Vdi m6ik E Z+tadfu co: I p-k C <- ]1 )1. I;::' t, ,-,"hrn -, =0 liuil~oo [lull d~utheet trenm6it~pbieh~nctJa[0,00). ~ Igk(t, u)1 E Neil V'E>0 , :3Nk>0: Illlll <~ V'tE [O,n], V'u E Ell : Ilull>N k Tranq 21 Lwjn win Caonqc ~lgk(t,u)I<~llulln gk~ gO nen3ko>0: 'ifk 2::ko V(t,u)E[O,n]xEll Igk(t,u) - gO(t,u)!<1 Suy ra Igk(t,u)1~Igk(t,u) -gO(t,u)1+Igo(t,u)1<1+Ig0(t,u)/ V~yIgk(t,u)1<1+lgO(t,u)l;~k2::ko - - Theo trenchungtaco: Igk(t,u)1<~IIulln Bung chamQitE[O,n],VuEEll :llull>Nk (N k ph\! thuQcrheak,E). B?t N =max{N0'N 1"'" N ko}. Khi d6'ifu:Ilull>N tacolgk(t,u)1<~IIull, Vk =0,1,...,k0n Vi gk (k =0,1,2,...,kG) compact-m~nt~pgk([0,n]xB(O,N) la compacttu'ongd6i trongE nen3Mk >O:lgk(t,u)I~Mk 'iftE[O,n] Vu:llull~N, (k= 0,1,2;...,ko) ChQn M'= max{Mo,M1,M2,...,Mko} Taco: Tac6:lgk(t,u)I~M'+~llull,'iftE[O,n]n VUE Ell, k=O,1,2,...,ko Tli Igk(t,u)1 ko Taco: Igk(t,u)lkon Va hiSnnhienk =0,1,2,...,ko ba'td~ngthuGy§:ndung V~y Igk(t,U)I<l+NI'+~llull=M+~llulln n' D?t m=Sup~IA(t)11:t E [0,n]} (1IA(t)lichufincuatoaDt1"1'tuye'ntinhbi ch~nACt» Voim6ikEZ,taco VtE[O,n]: Trang- 22 Luqn vanCao hqc t t IXk(t)/:5:/hk(O)/+JIIA(s)lllp(x;)llds+JlgkCS,q>(x;)/ds 0 0 t t ~ I h k (0)[+ m fllcp(x; )!~s+ f(M +~Ilcp(x; )11)ds:5: 0 0 n t t ~Ihk (0)1+mllcpllf x; I~sj- J (M +~Ilcpliiix;il)ds~ 0 0 n t . t .. ~Ihk(O)!+mllcpllfllx;l~s+nM + J~llcp""x;llds~ 0 on t <(N +Mn)+(mll<p11+~11<p11)~Ix~Irs - n 0 N =max{hk(0)1:k E Z +} Tli d6rheab§td~ngthlicGronwalltaco : n mIIIpII ,-+E Ilx ~11.<::;(N + Mil )e II<pll=(N +Mn )emn+s \7'tE [O,n] ,Vk E Z+ Suy ra B n bi eh~ntrongX n (xong bu'oc1) Bu'oc2: Vdi m6i k E Z +' tadinhnghiacaeroantU' U'va Ck : X -+ X =C([-r,co),E) nhu'sail : { fACS)<P(XS)dS;t;::::0 D' (x)(t) = 0 0 ;-r.<::;t.<::;O ! }gk(S,<P(XJdS+hk(O) ; t;::::0 Ck(x)(t) = 0 hk(t) ;-r.<::;t.<::;O ~ vi gk -+g° va limhk =h° trongC =C([-r,O)],E)k~co lien ta c6 (C kh hQi tl,l d~uv~Co tren X' n=C([-r,n], E) - ( fgO(S,<PCXJdS+hkCO) ;t;::::0 Co(x)(t) = 0 ho(t) :- r::; t::; 0 Trant;! 23 Lwjn van CaD h(JC Vi {A(t)}la tuye-nHnb, bi ch~ntu En-7 ~ lien U' clingla tmlnttr tuye-ntinh va hon mla 11(0')'11',,;(mono!I<pII)'\fiEN1. * Chung minh U' tuye'nHnh: Vx, Y EX, Vt 2:0 t t U' (x +y)(t)=fA(s)cp[(x+y)s]ds=fA(s)cp(xs+ys)ds 0 0 t t =fA(s)cp(xs)ds + fA(s)cp(ys )ds =U' (x)(t) + u'(y)(t) 0 0 / Tu'ong tl1 U'(ax)(t)=aU(x)(t) Chuy: x:[-r,CX)~E ;t2:0,xt :[-r,O]~E ; Taco (x+Y)t=Xt+Yr va (ax)t=axt Th~tv~yV8E[-r,O] (X+Y\ (8)=(x+y)(t+8)=x(t+8)+y(t+8) =Xt(8)+Yr(8)=(Xt +Yt)(8) (ax\ (8)=(axXt+8)=a.x(t+8)=axt(8) 8H x(t+8) Voi phgnchliynaytadtichungminhU' tuye-ntinh: Ta chungminh: 'v'iE N thi 1(U'r(x)(t)- (U')'(y)(t)l-:;(m.II~II.t)iIlx- yll'n1. 'v'x,yE X~,'v'tE [O,n] i =1: t t I iU' (x)(t) - U' (y)(t)1=If A(s)cp(xs)ds- fA(s)<p(ys)dS Iu 0 t t I =lfA(s)l<p(xs)-<p(ys)!ds=fA(s)cp[(xs)-(ys)]ds! () iO I t , ~ fIIA(s)II.II<pil.llxs- ysllds~m.llcpl/.t.llx- ylln () , (dol-r.O]c[-r.nlnen x\-Ys!l~ilx-yl'n) Tranq 24 LU{jnvanCaDhqc Gia sadi~ukh~ngdinhtrendUligyoi i :2:1 (1U')i+1(x)(t) - (U')i+l (y)(t)1 =1U'[(U')i(x)](t) - U,[(U')i (y)](tJI I I IjA(S)<P(((U')i(x),)]ds - jA(S)<P[((U')i(y),)]dS ,', I ~ ~IA(s)!I.II<plll(U')i(x», - ((U')i (y)~llds 0 Ta c6: 1((U')i(X)\(8)-((U')i(y»,(8)1 = I . I [m.II<pII.(s +8)]' II II ' / (U')I(x»(s+8)-(U')'(y»(s+8) ~ " . X -y n (gia thiet qui nc:).p) 1. (m.II<plj.s)i II II ' ~ .x- y n.,l. Suyra: II((U')'(x», - ((U,)i(y»,/I~(m."~".s;;.IIX- d" . I (mII II S)i , Tli d6:1(U')'+I(X)(t)_(U')'+I(y)(t)I~m.II<pII,f 'I~I' ,IIX-ylln,ds 0 1. (m.II<pII.S)i+1.IIX - II'" (i+1)! y V~ybatd~ngthuctrenduQcchungminh.Tli d6suyra: /I(U')i(X)_(U')i(y)l(n ~ (m.n.-!'<pll), .IIx-yl(n1. / . ' (m.n.II<pII)', VI U' tuyen tlnh nen: II(U')'(X)IIn ~ " .!lxll nl. D~n de'n:II(U');/I'" ~(m.n..!!<pllr1. (Thu h~pVI trenX~ '-.C([-r,n],E) Bu'oc3 : Tren kh6nggian X~=C([-r, n],E) ..:r. I xetchin moi Ilxli n = 2:11(0' Y(x~1n, '\Ix E X~ i=O Chuoi ('jvephai hQitl,lVI theobuGe2 taco : Trang 25 CuqnvanCaDhqc 11(U')'(x~ln~ (mo~o:I<pIf)iollxll~ nensuyfa i: II(U')' (x ~I' ~ Ilx/L i: (mono!lcpllJ'=emnll.II.llxll~i=O n i=O 1. ~--- --Hi~nnhle-n-Ilxt:s;!lxll: Mijt khactaco: II(U')'(x~1:=~II(U')'[U'(x)( =~II(U')' (x~In =fll(UIY(x~l~-/lxt =/iX/l:-IIXII~~llx!l:-e-m.n.II~11xllx[1=0 . =(1- e-m.n.lI~ilJlxll: =>Ilu'l!: :s; (1- e-m.n.II~II)=a <1 (voi IIU'II:la chufincuaanhx~tuyentinhtrenkh6nggian(X~,11-11:). 0"___'-- ~--~ -~' .'--- Bude4 : Giii saX la dqdophicompactKuratowskitrenIdlonggianBanach (x: ,11.11:)du<,1cdinh nghianhusau: yoi ba'tky A trongx~ leA) =inf{d>O/Adu<,1CphilbdimqtsO'hii'uh~ncact~pco duongkfnh ~d} Khi dosO'leA) conhungtlnhcha'tsail: +leA) =0A compactu'ongd6itrongX~ +A c B thlleA) <X(B) +x(A u B) :s;X(A)+x(B) +X(tA) =!tlx(A) TITcacdinhnghlav~U' va Ck'Xklanghi~mcuallknen taco: xk =U'(xk )+Ck (xk) Vk E Z Tranq 26 CwJn vanCao nc;c Tac6:xk -xo =V'(Xk -xO)+[CO(Xk)-CO(xO~+[Ck(Xk)-colxk~ (1) f)~t A ={(Xk- xo),[- r,n]: k E Z+} ~A =~Xk-xo)j[-r,n] :k E Z+} Khi d6A bi ch~ntrong(X~,11.11:).Th~tv~y,~tibu'Dc1tac6: .lIx~11~ (N +Mn)em.n+E,t E [O,n] =>IIx~- x~11~2(N +Mn)em.n+E,t E [O,n],kE Z+ =>A bi ch~ntrong(X~,11.11:)vlll.lI~-11.11:nenA bi ch~ntrong (X'n,11.11:) Tti cacgiathi6tCk compact,A bich~ntrangX~ JimCk =Co.trongX~,tac6:Ck =+Cok-+a:> X[{CJXk )-Co (xo)/k E Z+}]=0 X[{Ck(Xk)- Co(xo)/k E Z+}]=0 (vI hai t~pnay Ia compacttu'dngdoi) Theobu'Dc3:IIV'II:~a <1 tucV'la anhX(;lCllh~soaE (0,1). D~nde'nV ciingla anhX(;lk =a-cod~c. Tti d6X(V'(A))~ax(A). Tti (1) trongbu'Dc4 naytac6 : X(A)~x(V'(A)) ~ax(A) VI 0< a <1nen X(A) = 0 tuc A Ia t~pcompacttu'dngdoi trong , , * X~vDichuffnilt(VIlit -lit) Tli d6 t6n t(;liday con (xki)icua(xk)ksaccho Frnxki I =ytrong(X~,11.11') 1-+00 ,[-r.nl n Ta co Xki=V'(Xki)+Cki(Xki) Cho i ~+:o VI Ck ~ Co trongX~nentaco: y=V' (y)+Co(y) trongX;1 Tranq 27 [.wJn van CaD hqc f)i~unaychotha"y y 1affiQtnghi~fficuanotren[-[,n]. VI bailoan(IIo)co I nghi~mduynha"tXOlien: (xo) 1 =y [-r,n] Nhu' v~y miday con hQiW (Xki)d[-r,nl-~ua_~~_k)I[=r.nl--~i~u--~o_gi~ihc;J (Xo)I[-r,n] VI A 1acompact tu'ongd6i trong X~lien 1im(xk) 1 =x° I I trongdoX~"iinE N k~oo [-r,n] [-r,n] Tli dosuyra 1imxk =X°trongX . k~oo Giasa E,C,X nhu'(jph~ntru'oc(dinh1y2) ~aX~{oai t6,al: (IIIJ[X(t)-A(t)X(t-r)]' =g(t,<p(Xt);t~O L Xo =h voi h E C =C([-r,O],E) ehotru'oc g va {A(t)}th6acacdi~uki~nsau: (III. I) {A(t)}1a hcaeloanta tuye'ntinhbi chiJ.nlaE..) F ph1,1thuQclien Wctheot ~0 (III.2) g: [0,00)x En -1-E th6acacdi~uki~n(1.2),(1.3). Binhly3: Ntu {A(t)}va gth6acaedi~ukit%n (IlL 1),(III.2) thlbairoanIII conghit%mtren[-f, 00) Ch((ngminh:'- Bairoan(III) tu'ongdu'ongyoiphuongtrinh { X(t)=A(t)x(t - r) - A(O)x(-f) +h(G)+ £g(s,<p(xs))ds Xo=h ;t Tranq 28 Cuqn vanCaohqc £)~tZ,G: X ~ X =C([-r,oo),E} Xacdinhnhu'sail: Z(xXt)= { A(t)X(t-r) - A(O)x(-r)E;t ~0 0 -r~t~O G(xXI)={~(:r<p(~:j)d~+h({))-~If?:~~::---- Ta chungminhdinh1)'3quacacb6d~sailday: B6 d~1: Toantti'Z tuy€n tinhva Zk(x)(t)=0,\:it:-r ~t :::;(k - l)r ChUngminh : '\ix,Y EX, '\it~0 Ta co: =Z(x +y)(t) =A(t)(x +y)(t ~r) - A(O)(x + y)(-f) =A(t)[x(t - r) - yet- r)]- A(O)[x(-f) +y(-r)] . =A(t)x(t - r) -A(O)x(-f) +A(t)y(t - r) - A(Q)y(-f) Z(x)(t)+Z(y)(t) Tu'ongt1;1'Z(AX)(t)=AZ(X)(t), A ER V~yZ tuy€n tinh Ta chungminhZk (xXt)=0, '\it: -r:::;t ~(k -1)r B~ngqui n(,lptoanhQcnhu'sail : Vdik=l, Zk(X)(t)=O,'Vt:-r:::;t~(k-l)r dungtheodinh nghlacuaZ vdi k =2 .Ta co: Z2(XXt)=Z(Z(x)(t)= = { A(t)Z(XXt- r) - A(O)Z(xX-r) ;t ~0 0 -r~t~O = { A(t)Z(xXt-r) ;t~O 0 -r~t~O Tram] 29 Luqnvan CaD hC(JC (ViZ(x)(-r )=0 vaA(O)tuye'ntinhnenA(0)Z(x)(- r)=0) Ta l£;tico: { A (t)[Z(t - r)x(t- 2r)- A(0)x(-r)] ;t - r ~( A(t)Z{x)(t- r)=. 0 ;-r:::;t-r:::;C hay A(t)Z(xXt- r)=0; 0::;t::;r =(2-l)r V~y Z2(XXt)=0 '\It: -r::;t::;r GiasU' Zk(X)(t)=O'\Ik~2: -r::;t::;(k-l)r Ta coZk+l(xXt)=Z(Zk(x)(t) =jA(t)Zk(xXt-r)-A(O)Zk(xX-r) ;t:2:0 (.0 -r::;t::;O = { A(t)Zk(xXt-r) ;t~O 0 -r::;t::;O ( Vi rheagiathie'tquin~pZ\x)( -r)=O) Ne'u+0:::;t :::;kr -r :::;t - r :::;(k -l)r thlrheagiathietquin Zk(x)(t-r)=O suyraA(t)Zk(x)(t-r)=O; '7t:-r:::;t:::;kr V~yZk+l(x)(t)=0 '\it: -r:::;t :::;kr B6d~1chungminhxang B,,?d;, 'J.0 e-. Toanti'!Z th6acacdi~uki~ni)-iii) cuaBinh 1y2Chu'ung1 Chl'rngminh: Theabe)d~1tac6: 'v'n EN, 3kn EN: 'v'k>kn ,x E X~ : Zk (xXt)=O;-r<t<(kn-1)r <n =>p'n(Zk(x))=0.Trongd6X~ =C([-r,n],E) Voi chu{}nIIxt =sup~x(t~:-r < t ~n} p~(x)=sup~x(t~:-r ~t <n}. Tranq 30 Luqn vanCao hc;c KigmtraZ th6adi~ukit$nA (AI): \ia E X, Za(X) C X hi~nnhien , _.- - u_.- - ---- ._- (A2): \ia EX vap~,3kaEZ+voi tinhchat: \iE>0, 3roEN va(3>0:\ix,YEX . , a:n (x,y) a:n (Z:o(x),Z:o(y)) <E Taco: a:~(Z~)(x),Z:o(y))=max{p~(Z~(Z:o(x)- Z~(Z:' (y))):i, j =0, ka} (Chli yk~Ia sonho nhatsaGcho ZKa(xXt)=0\it: -r::;t::;(ka-I)r::; n) Voi mQit E [- r,nJta co: Z~(Z~)(x)Xt)=Z:o(Z~(x)Xt) \ik EZ+,\ix,y EX Ta co: Z; (x)- Z~(y)=Zk(x)- Zk(y) Th~tv~y,Z;(x)= Za(Za(x))=Z(Za(x))+a=Z(Za(x)+a.)+a =Z2(x)+Z(a)+a (vIZ tuye'ntinh) Z~(x)=Za(Z; (a))=Z(Z~(a))+a=Z(Z2(x)+Z(a)+a)+a =Z3(x) + Z2(a)+ Z(a)+ a Trant] 31 Cuqn van CaD AflC B~ngquin'.ipSuyfa: Z~(x)=Zk(x)+Zk-I(a)+...+Z(a)+a di~unaydungchomQix EX. Tli d6 Z~(x)- Z~(y)=Zk(X)- Zk(y) Z~(Z:o(x))- Z~(Z:o(y))= ~_:~_(~~i~))~_~t(Z~i0)-, ,- .---- Zro(Z~(x))- Zro(Z~(y)) ChungtachQn8=8 vafodliIonsaGcho:\:Ix Zro(xXt) =0;- f < t <n Suy fa Z ro (Z ~(x)) =Z ro (Z ~ (y))=0 Tli d6n€u a:n(x,y) <8+8 thlhi€n nhien a:~(Z:o(x),Z:o(y))<E ii) Voi ba'tky P~taco: P~(Z(x)- Z(y))=p~(Z(x - y)) (Z -tuyentinhbuGe1) , IZ(x-y)(t~ ~IA(tXx-yXt-r)-A(O)(x-y)(-r~ ::;IA(tXx - yXt - r~+IA(OXx - yX- r~ ::;IIA(t~I.(x- yXt - r)+ IIA(O~I.(x- yX- r)::;2mllx- yll~ Trangd6 m=sup~IA(t~I:0::;t::;n} Tli d6suyfa p~(Z(x- y))::;2mp~(x- y) Hay p~(Z(x)- Z(y))::;2mp~(x- y) iii) \:IxoEX, p~.La'yraEN saGcho Z[o(x)(t)=O;-r::;t::;(ra-l)r::;n Tranq 32 Luq.nvanCaohqc Khi do p~(z~o(x) - Z~oo(y))= p~(zro(x)- Zro(y)) (chungminhaphftni) =0::;Ap~(x- y) vdi0~_~~<l~,~_~~-..- - M~tkhac,chungminhtu'dngn;rnhuadinh1:91taco: G compactvath6a lim p~(G(x))=0 p~(x)-+co p~(x) Khi dotuB6 d~2 tacoZ, G th6acac di~uki~nd Binh 192ChuangI. Do do co di~mba'tdQng. f)i~mba'tdQngnayla nghi~mcuabai tminIII tren[-r,co). Tranq 33 Luqn vanCae Aqc Voi nhunggiathi~tnhuddinh1:93v~X,E,C. k EZ+taxetbairoanvoigiatridftu III { [x . (t)-A(t)x(t-r)]' =gk(t,<P(Xt)tt~O k k Xo =h Voi hk E C,chotntoc{A(t)},gkth6aca~'.d~~'u.~ki~~Tili:i),(III.2 Tli nhunggia thi~tnayrheaDinh 1:93phuongtrlnhInk conghi~mxk tren[-r,00),'v'kE Z+ Dinh ly4 : Voi nhunggiathi~trenvane'u(gk)khQitl,ld~uvago'hk hQitl,lv~h° phuongtrlnhIlloconghi~mduynhatx0.Thi taco: lim xk =XO trongX =C([-r,00),E) k~CXJ Chungminh: Ta chungminhdinh1:94quacacbuocsail: Buoc1: Voi bat ky a~0,d~tX a =C([-r, a],E) la kh6nggianBanachvoichua'n Ilxll;=sup~x(t~:t E [- r,a]} Za :Xa ~ Xa Za(x)(t)= { A(t)x(t- r)- A(O)x(i- rt 0~t ~a 0 -r~t~O Theab6d€ 1dinh193tasuyfac6se;ka E Z+nhonha'tsaDcha[Za(x)]ka=0 Trang 34 Cuf!nvanCaeh(Jc TadinhnghiachufintuangduangII.II~trenXa bdi: ~ ka IIxll~=I"z~(x~1 =I"z~(x~1 ,VXEXa i=O a i=O a ka Khido: Ilxlia~llxll~~Illz~II.IIxlia =Aallxlla ,(*) --- i=O (ChtiY za tuye"ntfnh,rheab6d~1,dinhIy3) ka Aa = IIIZ~II~l i=O IIZa (x~l~ =tllz~(Za(x)~1=tjlz~(x~1=IIxll~-llxt ~(1- A~lJlx[ i=O a i=l a ~llxll~-IIZa(x~l:~A~l.lIxt ~ IIx --Z~(x~l~~A~I .lIxt X E Xa (**) Buck2: Wi m6in EN, dij.tB~: kI [-c,nJ:k E Z.) Ta coB~bi ch~ntrongX~ =C([-r,n],E) voichufinIIxt =sup~X(t~:tE[-r,n]} Voim6iE>O,3M>O:lgk(t,u~~M+~.llulln citingchoyoi mqi t E [O,n],u E En, k E Z~ - (Chungminhdi~unaygi6ngnhubuoc1,dinhIy 2, chuangIII). Voi m6ik E Z+, (Ink)tuangduang: Tranq 35 LUeJnvan Cao h(Jc { X(I)- ~(I)x(1- r)+ A(O)x(- r)~ hk (0)+.( gk (I, <p(X,))d1; t -" 0 Xo =h VI xk la nghit%meuaphuongtrinhtrennenco: J x k(t)- A(t)Xk (t~r)+A(O)xk(-;)~-hk-(O)~1~-k(s,~(;f )~~;O~t ~ n ---- lX~=hk \it E[0,n],taco: Ixk(t) - A(t)xk (t- r) +A(O)xk(- r~ ~ Ihk (0~ + £Ig k (s,cp(x~ )~sl ~ Ih k(O~+ i(M +: Ilcp(xnl}s~Ihk(O~+Mn +: IlcpHllx~l~s ~Mn +N +~11cpll.£ IIx~I~sn N =sup {hk(0~ :k E Z + } D~nd€n : ~~~~{Xk(0)- A(s)xk(s- r)+A(O)Xk(- r~} ~Mn+N +~llcpll.£llx~l~s,\iSE[O,t]n Tli (*) va (**) dbuDe1taco : \ik E Z+ , A~2l1xkllt~ A~2l1xkllt , ~A~lllxk _ZtCXk)I! t ~ Ilxk - Zt (x k)t II t ~(Mn+N)+~llcpllfllx~llds ;tE[O,nl, n 0 t Suyra:\\xk\\\ ~A~(Mn+N)+A~ ~\\<p\\J\\X~\\ds0 ;tEl 0,n\. k E Z + Trang- 36 Cwjn van CaD hc;c Giasa kn Ia sOnguyendl1dngnhonha'tsaocho: [Zn(x)]kn=O/x E Xn =c([-r,n],E) Khi,-J6k >k k IasOnguyen~ElH'ona-benha't--n- t' t /::) saocho[Zt(x)]kt=0 0::;t ::;n kn kt An =L/lZ~/I ~At =L IIZ ~ II 0::;t ::;n i=O n i=O t Tli do:llxk/lt::;A~(Mn+N)+A~.~.llcpll£llx;l~s :::? I!x~/I~A~(Mn+N)+ A~.~.llcpll£/Ix;I~sn Theo ba'td~ngthlicGronwall taco: t E [0,n],k E Z+ Ilx k /I ~ A~(Mn + N)e E.A~.11<p1!t , t E [0,n], k E Z+ ChungtoB~bich~ntrongX'n=C([-r,n],E) Bu'dc3: Vdi m6i k E Z+,tadinhnghlacaeroantu Z va Ck :X ~ X =C([-r,co),E)nhu'sau: Z(xXt)= { A(t)x(t-r)-A(O)x(-r) ;t~O 0 -r::;t~O ( X )- { 1 gk(s,cp(xJ)cis +hk(0) ;t ~0CK X t - hk(t) ;-r~t~O -7. ~ VI gk~go va hk ~ hO lien coCk ~O trenX~=C([-r,n],E), vdillxt =sup~x(q;-r~t~n} ( )( - { £go(s,cp(XJ)ciS+hO(O);t~OCo x t- hO(t) ;-r~t~O Theobu'dc1,t6nt~ikn Ii songuyendu'dngbenhat k .,.1( ) , saocho: Z n X =0, X E X n Tranq 37 Luqn vanCaohc;c D?t chua:nmoitrenX~nhu'sail: kn ' \ix E X~, Ilxll:=~=J(zy(x~ln Taco: Ilxll~~llxll~(1+IIZII+/lz211+...+llznll)lxt IIxll~ IIxll:~Anllxll~ VOiAn=1+lIzII+IIZ211+...+/lZn/l2:1 M?t khac * k". "kn+l. ' IIZ(x~ln=IllZi (Z(x)~1= IliZi (Z(x)~1i=O n i=\ n =IIxl!: -llxll~~IIxl!:-A~lllx[ =(1-A-~Jlxll:=allxll:~ IIZII:~a <1 (XemZ:(X~,II.":)~(X~,II.II:)) Bu'oc4: Hoantoangi6ngnhu'bu'oc4,djnhly2chuongII vachungnllnhduQcxk~xO Trang 38 Cuqn van CaD nqc Voi nhunggiathie'tnhliddinhly '. v~X, E, C Voi m6i k E Z+,xet baitoan v~giatri d~u (IV) [X(I)- ~(I)c(t;--r)]' =gk(I, 'Pk(x,)); t"' 0 lxo =h hk E C.--chotrlioc {A(t)},.gkthOa.cacd.i~lLki~IL.(III.l),~III.2) TiI caegiathie'tnayrheadinhly 3phlidngtrinhIV k co nghi~mx k lien [-r, 00),'v'kE Z+ Binh ly 5 : Voi nhunggia thie'tlien va ne'u(gk)khQit1,ld~uv~go va go lien U:IC d~urheabie'nthu 2, hkhQit1,1v~ho trongC, <PkhQit\1v~<Potrongol(c,En ~phlidngtrinhIVo conghi~mduy nha'tx0.Thl taco: lim Xk =XO trongX =C([-r,oo),E)k-->oo Chungminh: Ta chungminhdinhly5quacaebliocsail: Blioc1: Voi ba'tky a~0,d~tXa =C([-r,a,E]) thonggianBanachvoichua:n Ilxlla=sup~X(t~:tE[-r,a]} Za : Xa ~ Xa Z (x)(t)= { A(t)x(t-r)-"-A(O)x(-r);0:::;t:::;a ,I 0 ;-r :::;t :::;0 Nhu(jbuGC1dinh1:94chuangIII, cochuffntu'angduangtrenX a nhu sail : ,'" k" Ilxlla=2]Z:(x~la=IIIZ~ (x~la,x E Xa ;=0 ;=0 ka la songuyenduangbenhathoaZ~~I(x)=0 Tadaco: k Ilxll" :::;Ilxll~ :::;2]Z: II"!Ixlla=A"IIxlla i=O Ilx - Z"(xl ;::i.~1.llxll.. ' Vx E x" Tranq 39 Cuqn van CaD hc;c Budc2: Wi m6inEN co'dinh,d~tB~=klh",]: kE Z+) Clingnhubudc2,dinhly4,chuangII chang - minhdu<;$cB ~ bi ch~ntrongX~~=:=~c{D=-('J)lEL ---- vdichu~nIlxt=sup~x(t~: E [-r,n]} ChicffnchtiY k~ 0trong oC(C,E-n)nen ~IkID bi ch~n Blidc 3: Vdi m6i k E Z+, tadinhnghlacaeloan tli'Z va Ck :X-+X=C([-r,n),E)nhusau: ( X \ - { A(t)x(t - r) - A(0)x(- r) ;t ~0 Z x tJ- 0 ;-r~t ~0 ( X )- { 1 gk(s,k(xJ)cIs + hk(0~ t ~0 Ck X t - I hk(t). ;-r~t~O Khidodilbie'tcos6kn nguyenduangnh6nha't k +\( ) , saochoZ n X =0, '\Ix E Xn X6t chu~n11.11:trenX~khacnhaunhusail: . OJ ,kn - . "xII:=~"(zY (x~fn=~."(zY(x~ln Taco "Z[ ~a<1 Ta changminhCk=;Co trenX~=c([-r,n],E) ,. ( X ) { 1g0(S,CPO(XJ)cIS+hO(OXt~OVal Cox t = hO(t) ;-r~t~O ~ Th~tv~y,'\IE>0 chotrudcvi gk-+gO nen3k1EN: '\Ik~k I' '\I(s,u) E [0,n]x En Igk(S,U)-go(s,u~<~30 Tranq 40 Lu~nvan Cao hQc V'x E X~,'\ft E [O,n] Ta co: ICk(xXt)- Co(xXt~~s:Igk(S'k(xJ)- go(S,o(xJYps + +[hk (0)-h0(O~~ f~/gk(s,~~~~))~~_?Ql?~<Pk~~~~))ld~ +f~Igo(s,<Pk(Xs))- go(s,<Po(Xs)~ds+ihk(0)- h0(0)/ VI go lienWed6utheabie'ntha 2nen38>0 :Vu,v EEn: Ilu-vllE Igo(s,u)-go(s, v)1<~ n 3n VI <Pk--+CPotrang ,:,t(C,En )nen 3k2 EN: Vk 2::k 2 II<pk(XJ-- <PO(xJII< 8 dungehamQix E X~ Vs E [0,11: Tu dolgo (s,<Pk(xs))- go(s,<Po(xs))1< 3: Vk 2::k2 h k ~ h 0 trang C =ctl- r,O],E), nen3k:>E N Vk 2::k 3 I hk (0)- h0(0~ <~ Khi do : Vk 2::k 0 =max{kI ' k 2 ' k 3}taeo ICk(x)(t)-Co(x)(t)1~rt~ds +rt~ds +~~~+~+~=8Jo3n - Jo3n 3 3 3 3 dungchomQix E Xn, mQit E (O,nJ SoyraliCk(x)-CO(X)I:1~E, V'k~ko,V'xE X;l -7 . Tile 1aCk ~()tren XI!' Bu'oc4: Hoanroangi6ngbu'oc4d djnh1y2ehu'dngII ' h ' , h A k ()va c ling mIll uu'Qcx ~ x Trang 41 Lu~nvan Cao hQe X6t phu'dngtrlnh: V { xt(t)=A(t)cp(xt)+get,cp(xt));t ~0 Xo=h vdi hE C =C([-r,O],E) V.I V.2 cp: C ~ Entuyfuidnh lien fijc~---'---'-'---'" ~._,_. A(t) la hQcaeroantU'tuye'ntinh, bi eh~ntitEn -7E ph1,lthuQc lien tl,letheot, tu~nhoanehuky cotheot. g : [O,co)x En ~ E tu~nhoanvdi ehuky COtheot va thoacae dieu ki<$n1.2,1.3. V.3 VA Voim6i h E C clIotru'dephu'dngtdnh Va trene6nhi~ul~m la m(>tnghi<$mx(h)tren[-r,co)thoaxo(h)=h. Vdi nhungdi~uki~n V.l de'nVA, rheadinhly 2,Chu'dngII, phu'dng tdnhV e6 nghi<$mduy nh3'tx(h) tren [-r,co) thoaxo(h)=h. va anhX<;1 hH x(h) la lien tl,1e. Gia sa Vet,s),s ~0, t ~ -r la h9 caeloanta tuye'ntinh, bi ch$ntuE vaoE' lientl,lem<;1nhrhea(t,s)vathoaphu'dngtdnhsailday: 8 - Vet,s)=A(t)cp(Vt(.,s));t ~s~08t { I ; t=s Vcr,s)= 0 ; s-r::;t<s ( V(t,S)=fA(u)cp(Vu(,.s))du+I ;Hay s Vet,s)=0 Vt(.,5)(6)=Vet+6,s) Vt(.,s):E~C t~s~O s-r::;t<s Trangd6 6 E [-r,O] Vdi s~t~O,clIoSet,s):C -7 C lahQcaeroantatuye'ntinh, bich~n xacdinhhdiS(t,s)(h)=Yt(h),trangd6yell)la nghit';mduynh§teuaphlMng tdnh. { yl(t) =A(t)cp(yt) Y =h. s ; t;::::s ~0 Trang 42 Lu~nvan Cao hc Set,s)dinhnghlanhutrendungla mQtlOantU'tuye'ntinhvabich~n. Th~tv~y,Vh,k E C, tachungminh: S(t,s)(h+k)=S(t,s)(h)+S(t,s)(k) Q Yt(h+k)=Yt(h)+Yt(k) y(h+k)(t+8)=y(h)(t+8) +y(k)(f+8),--- V8-E [::r,O] .Q Voi m6i h E C phuongtrInh : { yl(t)=A(t)q>(Yt) Vo Ys =h conghit$mduy nha"ty(h)nentaco : ; t2::s2::0 . t y(h)(t)=h(O)+fA(u)q>(yu(h»du ;t2::s2::0 Voi kE C phuongtrInhVoco nghit$mduynha"tnen : t y(k)(t)=k(O)+fA(u)q>(yu(k»du :t2::s2::0 Va h +k E C nenco : t y(h+k)(t)=(h +k)(O)+fA(u)q>(yu(h+k»du Taco: .. '., t. t y(h)(t)+y(k)(t)=h(O)-+k(O)+fA(u)q>[yu(h)}iu+fA(u)q>[yu(k)}iu s t =h(O)+k(O)+fA(u)q>[yu(h)+yn(k)}iu s t =h(O)+k(O)+j'A(u)q>[(y(h)+y(k»)u}iu s Chung toy(h)+y(k)dIng la nghit$mciiaphuongldnh : { yl . (t) =A(t)q>(Yt) Ys =h+k ; l2::s2::0 Trang 43 Lu?n van Ca(! hQc VI phu'dngtrinh~6nghit$mduynha'tlien : yell)+y(k) =y(h-:.k). Tu'dngtl,ichungminhGttQcyeAh)=Ay(h),AE R Tli do sur fa Set,s)tuy€n tinh . -_.- t M?t khac: yet)=h(O)-j-J A(u)(j')(Yll)d~ t => ly(t)1 ~ Ih(o)I"+ JIIA(u)lI.[[cpll.IIYulldu t s;Ih(O)1+n./lcpll.Jllyulid'] mt 1 K 2 th'~ G 11 'u:-":'::O0 : (.(..ng uC fOnWa ta co : Ily II (11)11:::;Ih(O)le 1l1114>11(t-S) :::>Ily II (h)11~[Ihlle~il4>iin =>IIS(t,S)(h)ll:::;ell1ii<piinllhll t,s E [O,n] Set,S)bi ch?n tfen C voi t,s E [O,n] ; nchotnioc, t,SE[O,ll] n clIo t:Lfocco din!1. M~l1J) d'~1 : S(t+co,O)=S(t,O). S«(o,O) Ch«l1gmint: \1hE:C : S(t+co,O)(h)::::Yt-;-0)(11)S(tO)[S(lO,O)(l1)]:S(t,O)[yO)(h)]=Yt(yw(h») Ta cLen;;m~nn: Yw-j(h)::::Yt(yw(h» Tll~tv~.y,S(~-H.v,O~(k)==Yt+o~(h)vo: y(h)1:\fBhi~mcua p'::":JhgL:nh : { y'(t'; CD)::":AU +C0)(p(Yt+OJ) ; t -: u:?:0 Yo =h hay tu'dngc1l(OL~: { y' (t + CD)=A(t)cp(y t+<,J Yo =h ,.+CD2:0 (Vi A tuftnho~l:1voi dIu ky CD) Trang 44 Lu~nvanCaohQc Suyra : { yl(h)(t +co)=A(t)<p(yt+ro(h)) Yo=h ;t+co;:::O M~t khacvdih E C phltdngtrlnh: {Y'(~=A(t)<p(Yt) LYo - h . . co nghi<$mduy nhffty(h). Do do taco : y(h)(t)=y(h)(Hro) hay: Yt+ro(h)=Yt(h) Cho t=0 co : Yro(h)=yo(h)=h Til do: Yt(Yro(h))= Yt(h) = Yt+ro(h)Tuc Ia : S(t,O)S(co,O)(h)=S(t+co,O)(h) V?y S(t+ro,O)== S(t,O)S(co,O) Menh d~2 : ;t;:::0 x(h)la mQtnghi<$mcuaphu'dngtrlnh: , \x'(t)=A(t)q>(xt)+get,q>(Xt»;t ~0 1Xo=h NK , h~ K (h) h ? euva C 1neux t oa : t Xt (h) =S(t,O)(h)+fVt (.,s)g(s,<p(xs(h)))ds 0 Chung minh : t ~Ne'u Xt(h) =S(t,O)(h)+fVt (.,s)g(s,<p(xs(h)))ds 0 Suy ra : t x(h)(t) =S(t,n)h(O)+fVet, s)g(s,<p(Xs(h)))ds 0 ~ [x(h)(t) J. ~ [S(t,O)h(O)J, +([V(t,s)g(s,'p(x, (h)»dS). . ! 0 =A(t)<p(yt (h))+f -(vet, s)g(s,<p(Xs(h) Xis+ Vet, t)g(t, (p(xs (h») 0 at Trang 45 Lu~nvan Cao hQc Dftnden: t x'(h)(t)=A(t)<p(yt (h»+fA(t)<p[Vt(.,s)g(s,<p(xs(h»)}is+get,<p(xt(h») 0 . t =A(t)<p(yt (h» +A(t)f<p[Vt(.,s)g(s,<p(Xs(h))}is +get,<p(xt(h») 0 - =A(t{<p(y,(h))+I <p[v,(0,s)g(s,<p(x,(h)))};IS]+g(t,<p(x,(h))) =A(t)({(x,(h)))dS]+g(t,q>(xI (h))) [ t l =A(t)<pS(t,O)(h)+fvt(.,s)g(s,<p(xs(h»)dsj+g(t,Q)(xt(h»)0 ~ =A(t)<p(xt(h» +g(t,<p(xt(h») V~yx'(h)(t) =A(t)<p(xt(h» +get,<p(xt(h») Bay lfl di~uphaichungminh. =>Neux(h)Ia nghi~mcuaphu'dngtdnh: { X'(t) =A(t)<p(xt(h)+get,<p(Xt(h») Xo=h Ta phaichungminh: . t X t (h) =S(t,O)(h)+f Vt(.,s)g(s,<r(xs(h»)ds 0 , Tac6: [S(t,O)(h(O»+[V(t,S)g(S,<P(X,(h»)dS] =A(t)<P[y,(h)+[v, (.,s)g(s,<p(xJh))dS] +g(t,<p(x, (h)) , Suyra: [X(h)(t)- S(t,O)h(O)- [V(t, s)g(s,<p(x,(h)))] =A(t)q>rXI (h)-Yt (h)-IV! (.,S)g(S,CP(Xs(h»)dS ]L 0 (*) Trang 46 Lu~n van Lao hQc L~ico : 0 KoCh)- Yo(h) - fVo(.,s)g(s,q>(xs(h»)ds 0 (**) =h-h-O=O va vi phridngtrinh {:~(~~A(t)q>(:L _;_I:~ - n - co nghi<$mduynhfftz=O.Nen tu(*) va(**) taco : t Xt (h) =S(t,O)(h)+fVt(.,s)g(s,q>(xs(t»)ds 0 M<$nhde 2 du'<;1cchungminh Kong. M<$nhde3saildaylah<$quacuaM<$nhde2 . Menh d~3 : co xco(h) =S(co,O)(h)+fVco(.,s)g(s,q>(xs(h»)ds 0 Voi x(h)la nghi<$mcuaphu'dnEtdnl.: { X1(t)=A(t)q>(Xt)+g(t,K(Xt» ; t~O ~xo=h Lu~nvanCaa hQc Vdi t~0 Bi[ttT(t) : C-7C h H T(t)(h) =Xt (h) Trangdo x(h) la nghi~mcuaphu'dngtrinhV, hayphu'dngtrinhvila neud tren. Menh d~4 : -------- T(t+co)=T(t)T(co) ; t~0 ChT1ngminh: Cachchungminhgi6ngnhu'cachchungminhm~nhd~2 . Th~tv~y,\1hE C taco : T(t+co)(h)=Xt+rotrongdox(h)Ia nghi~mcuaphu'dngtrlnh: { X r (t + co) =A (t + co)<p(x t +co) + g( t + co, <pCX t +0) ) ) Xo =h VI A(t),g tu~nhaanvdi chuky corheat nenphu'dngtrlnhtrentu'dngGltdng { X'(t + co)=A(t)<p(xt+co)+ get,<p(Xt+ro) Xo =h ' ~ { X'(h)(t +co).=A(t)<p[xt+oJh)]+get,<p[Xt+O)(h)b Xo=h VI vdim6ih phu'dngtrlnhV coduynha'tmQtnghi~mla x(h)nensuyfa : Xt(h)=Xt+ro(h) La'y t =0co: xro(h)=xo(h)=h T(t)T(co)(h)=T(t)[T(co)(h)] =T(t)[xro(h)]=T(t)(h)=xt(h)=xt+(r)(h) =T(t+co)(h) \1hE C. V~yco : T(t+co)=T(t)T(co) Trang 48 Lu~nvan Cao hQc 8~t F C-7C ro hH F(h) =fvro(.,s)g(s,<p(xs(h)))ds 0 s C-7C hH S(h).=S(co,O)(h)=Y(J)(h) T =T(co) Taco: T ==S +F. NSu T co diemba'tdQngh E C thix(h)Ia nghi~mtugnhofmchuky cocua V. Th~tv~y,T(h) == h =>T(co)(h) ==h - =>T(t+co)(h)==T(t)T(co)(h)==T(t)(h) =>xt+w(h)==xt(h) V~Y x(h) tugn hofm chu ky co [,u~nva'nCao J1Qc HQ {A(t)}du'O,b>O saocho: IIV(t,s)":::;be -a(t-s) j "i1t~s~O - -- DjnhIf 6: Xetphu'dngtrlnhva trenclingvoicacdieuki~nV.I, V.2, V.3,VA vahQ{A(t)}la5ndinhti~mc~ndeli. Khi dophu'dngtrinhV conghi~mtu~nhoanchuky co. Voi di~uki~nvilanh~nxetatrentaChIc~nchungminh T =S +F codi~mbit dQnghE C . . Changminh: Bu'oc1 : Qua cac bu'ocsail TmintU'F xacdinha trenla compact. Gia sa (hn)nE C saochohn >ho.Khi do,rheaBinh ly 2 Ph~n1 limx(hn)=x(ho)trongC([-r,co],E)rheaChlj~T1,nghIc.la day (x(hn))nn~ao hQi tl;1deli v~x(ho)trendo~n[-r,co]. 'T~p B={xs(hk)/sE[O,co],kEZ+yla t~pcompact trang C (chung minhnhu'trongB5 de3 , dinh ly 1). Vi <plien tt,lCnen<p(B)={<p(xs(hk))/sE[O,co],kE Z+} compacttrang En.Suy ra day (g(s,<p(xs(h0)))khQi t1,1deli v6 g(s,<p(xs(ho)))tren [O,co]. Tildo lir~lF(hn)=F (h0)n~ao V~yF lien t1;c. Gia sa Q la t~pbich~ntrangC. Khi do t~pA ={xs(h)/sE[O,CO],h E O} la t~pbich~ntrangC. Tli do <peA)la t~pbich~ntrangEn(Vi <p:C -7'Entuye'nttnhlient1,lc: nen 11<p(xs(h)11:::;11<pllllxs(h)11) Trang50 Lu~nvan Cao hc VI g compactIH~nt~pg([O,co]x <peA))compactu'dngd6itrongE. d~n de-nt~p D=Co{VO)(.,s)g(s,<p(xs(h)))/sE[O,CD];hEO)la t~pcompact trongC. VI VO)(., s)g(s,<p(xs(h))) E D, \is E [0,co]hE 0 Taco: (» F(h) =fV0) (.,s)g(s,~(xs(h)))dsE coD dungchomih E Q 0 ChungtoF(O) compactu'dngd6itrongC. v~yF compact. Lu~nvan CaDhQC Bu'oc2 : ? I/F(h)1/ Tlfa chuan IF/ = Jim sup 1/ 1/ =0IIhll~oo h Changminh: ~ -. ~ ~-- ---._--- \iE >0,tli cachchungminh(j dinh1:92, ph~n1,t6ntOsaDcho: [get,U)[~N +EllUl/,\it E [O,m],\iU E En Bat m =sup~IV(t,s)lI,t E [-r,m],sE [0,0)]~ . M =sup~IS(t,s)ll,t E [-r,m],s E [O,m]} t Tli Xt(h)=S(t,O)(h)+fVt (.,s)g(s,cp(x~(h)))ds 0 Taco: t I/Xt (h)11~Mllhl/+mfllg(s,cp(xs(h)))l/ds 0 t ~ Mllh!1 +mf{N+ EI/cpllllxs(h)l/~s a t t ~MI/hl/+mf Ndt +mEl/cpl/fI/xs(h)I/ds ~ a a . t ~Ml/hl/+m.N.m+mEllcpl/fllxs(h)llds, \it E [O,m] 0 Theaba'td~ngthucGronwalItaco : Trang52- Lu~n van Cao hQc IIXt(h)1I~(Mllhll+m.N.co~m.s.tlq>lloo Taco: 00 . IIF(h)1I~f/lVoo(.,s)g(s,<p(xs(h»)llds 0 00 ~mfllg(s,<p(xs(h»llds 0 -- . 00 ~ mf(N +811<pllllxs(h)II}is 0 00 ~m.co.N+m.8.1I<pllf/lxs(h)llds 0 00 < ' N + ~II Ilf (Mllhll N \_m.s.;;q>llood- m.w. m'~'II<P\ +m. .coIV s 0 ~m.co.N+m.8.11lloo Chung to r~ng: IFI ~ m.8.11lIoo = (mllcpllMcoeITLSiiq>ilw} diSunaydungchomQi8> 0 nenlFI=0 Bu'oc3 : Ta bi€t r~nghQ {A(t)} 6n dinh ti~mc~nkhi va chi khi co cac s6 a,b>OsaGcho IIS(t,O)(h)1I~bllhlle-at vdi mQih E C va t 2:0 Suy ra IIS(n.co,O)11=Ilsn(CO,o)/i~IISIlII~be-anw '\inE N Ta chungminh S thoacacdiSu ki~ni,ii"iii cua dinh ly 2, chu'dng1. Iisn(h)II~/lsn1!llh"~be-aI1h" '\inEN, '\ihE C E>~t A =be-anW jno saGcho A=be-an~l E>iSunay cho tha'ydiSuki~niii) duQcthoa. M~tkhac, '\itt'E C va vdi ffiQi8>0,taco : IIS:~(h)-S:~(k)"=lis11(h)-SIl(k)1I ~Ilsllllllh-kll Trang 53 Lu?n van Cao hQc Suy ra :Ils~v(h) - S~(k)11~be-an~h- kll Ta phai chQnr EN va 8 >0saGcho lib- kll<£+8 =>b.e-an~h- kll<£ khi lIb- kll<£+8 =>b.e-anlfh- kll<b.e-an~£+8) Taco: b.e-an~£+8) <£Q £+8 <£.eanw~ . b ( an"'" JQ8<£ eb -1 arw Ta co thS chon r saG cho ~ -1 >0 va do do ta chon . . b . ( "rw \ a<8<eleb -1j dieunaychUngtoS thOadieu ki~n A , tue dieu ki~ni) cua dinh ly 1. IIS(h)- S(k)1I:s;Ilsll.llh- kll ~ be -atfth- kll,\ih, k E C S thoaii) Tom l(;li,S,F thoacaedi~uki~ncua 8inh 1y2, Chu'dng1. Suy ra , T co diSmba'tdQng h E C. Lu~nvan Cao hQc Xet phtidngtrinh : VI { [X(t)- A(t)x(t - r)]' =get,cp(xt»;t ~o Xo =h h E C chotru'oc VI.l cp:C -7 Entuye"ntinh lien tvc VI.2 {A(t)}la hQcac to<:lntU'tuye"ntlnhbi ch?n tli'E -7 E phlf thuQc lien tvc theot, tu~nhoanvoi chuky corheat. VI.3 g: [O,co)x En-7 E thoacacdi6uki~n(1.2),(1.3)vatugnho~m chuky CDrheat. VIA Voi m6ih E echo tru'oc, phlTdngtrinhVI co nhi6ul~mla ffiQt nghi~mx(h) lIen [-r,co)thOC'xo(h)=h Vdi cae diSu ki~n tU VI.l de"nVL3, rhea Dinh 194 Chu'dngII VOl m6i h E C cho tru'ocphu'dngtrinh VI co duy nha'tmQtnghi~mx(h) lIen [- r,co) thoaxo(h)=h vaanhx~ht-7x(h)la lien tLJc. Gia SltWet,s):E -7 E la mQthQcacloantlttuy€n t ~-r lien tvcm~nhrhea(t,s)vathoaphu'dngtrinh: r~[Wet,s)- A(t)W(t - r,s)]=-0 1 - { o ;s-r::;;t<s Wet,s)= I ;s=t Hay: { [Wet,s)- A(t)W(t- r,s)]=I Wet,s)=0 s-r ::;;t <s >ch~m,s ~ 0, ;0::;;s::;;t ;0::;;S ::;;t VOl 0::;;s ::;;t, gia sa'i'o(t,s): C -7 C la mQthQcac loan at tuye"ntinh bi ch~nau'QcdinhnghIanhu'sail : To(t,s)(h)=Yt(h)voi yell)la nghi~mcua : { [yet)- A(t)y(t - r)J =o Ys=h (ChungminhTo(t,s)tuye'ntlnh, bich~ngic3ngnhu'ph§ntruoc) O::;;s::;;t Trang55 Lu~nvan Cao hQc ;:;Menh de5 : To(t+co,O)=To(t,O).To(co,O) Chungrninhnhu'm~nhd~1. Menh d~6 : ,x(h)la mQtnghi~mcuaphu'dngtrinh { [X(t)- A(t)x(t- r)J =get,<p(xt)) t ?:0 Xo =h neuvachineux(h)thoa: t Xt (h)=To(t,O)(h)+fWt(.,s)g(s,<p(xs(11)))ds 0 Chungrninhnhu'm~nh062. Voi t?:0,d~tT1(t):C -7 C voi T1(t)(h)=xt(h) 'dh E C Voi x(h)la nghi~mcuaphu'dngtdnhVI Menh d~7: T](t+ co)=T1(t)T1(co) Chung rninhgi6ngm~llhd6 4. £)~tFl : C-7C 0) h H F] (h) =fw,0)(., s)g(s, Xs(h))ds 0 C-7C h H 5] (h) =To (co,O)(h) = YO)(11) Tl =Tdw) 5] : Taco: TI =51 + FI NeuTI co di€m ba'tdQngh EC thlx(h)Ia nghi~mtugnboanchuky cocua VI. To(lll tU'D(t,h) =h(O)- A(t)(h) du'QcgQi Ia 6n GirthlieUnghi~m zerocuaphu'dngtdnh D(t,Yt)=0 ]a6ndinhti(;mc~no0.u,tue]3: 3a,b>0: IIW(t,s)!1~be-a(l-s) voi mQi t?: s ?:0 Trang56 Lu~nvanCao hQc Dinh Iv 7 : Xet phuongtrinh VI (j tren cling vdi cac di~uki~nVI. J . '11.2.'11.3. VIA va roantii' D(t,h) la 6n dinh. Khi do phuongtrinh VI co m(>tnghi~m tu~nhoanvdi chuky co. Changminh: Hoanroangi6ngvdicachchungminh(j Dinh 196. if:y>*~ if:y>*~if:y>..~~