Hệ phương trình hàm

[ing n m p(1) := LLla~~1~p<1 . i,)=lk=l Do dinhly 2.3,t6nt:;tiduynhcft p[l] = (Pl[l],..., poll]) E C(I;RO) la Wi giai cuah~. (3.4) (3.5) (3.6) (3.7) (3.8) n m ~[lJ(X) =L2:a&~FpJ(Sijk(X»+g;(x), j=lk=l 1 ::;;i ::;;n, Hon mla,dotinhduynh:1t,Wi giai nayclingtIlingvdi d~oham e=(f/, ...,in/)cua f, . P [IJ- f / '- 11.e. i -i,l-,...,n. XE I, Tu'onghI, n€u f E Cr(I; Rn) l?tWi giai cuah~(2.11)tu'ongling vdi g E CreI;Rn).Ta c6saudaykhi d~ohamcua(3.1)d€n r -11~n. n In J;(r) (x) =LLGijkb;kfY) (Sijk(x» +gY\x), j=1k=1 1::;; i ::;;n, Tli (2.10),(2.13)tasuyra n m jJ(r) :=LLIGijkb~kl::;;jJ <1. i,j=Ik=1 Do d6h~phu'ongtrlnhsau n m ~[rJ(x) =LLaijkb~kFY](Sijk(X)) +g;r)(x), j=lk=l 1::;; i ::;;n, t6nt~iduynh:1tmQtWi giai p[r] = (PIle],...,purr])E C(I;Rn) 12 XE 1. XE I, vaWigiainaytrungvoid~ohamca'pr (f) = (f1(f),...,fn(f») cuaWigiai f. Dod6,tac6dinhly san DinhIf 3.1: Gidsa/ =[-b, bJ vacacsffthl!C a!jk'bUlaCijknhutrangd;nhIy 2.3. Chog E Cr(/;Rn). Khi do t6nt()i f E Cr (I; Rn) vaF[rJ E C(/; Rn) Ian Iu(ftIa cac Iai gidi duynh{i'tcua cac h~phudngtrlYlhham(2.11)va (3.7),d6ngthai F[r] trungvaid()ohamcap r cuaf Chu thich 3.1: (3.9) (3.10) Trongtru'ongh<;1pI =R, tagia thiStthemr~ngcacsa thljc aUk'bijb Cijkthoa di~uki~n n m maxI I !aijkb;kI <1 . O:::;s:::;ri,j=lk=l. Khi do,nSu g E Cbf (I; Rn) =={g E Cb (I; Rn) / g', gOO,...,g(f) E Cb (I; Rn) }, thlkStlu~ncuadinhly 3.1viincondung,trongd6cackhonggian 13 hamC(I; Rn)va Cr(l;Rn)xua"thi~ntrongdinhIy 3.1du'Qcthaytheboi Cb(I;Rn)va Cbr(l;Rn),I:1nIu'Qt.Chungminhket quanaytu'dngtV'nhu' chungminhcuadinhIy 3.1 Bay giGtro I<;litru'onghQp1=[-b,b].Gia sag E CP(I;Rn)va gQi f E CP(I;Rn)Ia Wi giai duynha"tcua (2.11)tu'dngungvdi g. Vdi m6i 1~ r ~ P, tacop[r]nhu'trongdinhIy 3.1 Khi dotheocongthucMaclaurintaco (3.11) p-l 1: (r) (0) 1 x f(x)= I i , xr+ ,f(X-ty-lf(P)(t)dt,r=O r. (p-l).o l~i~n. M(ttkhac,taco: (3.12) p[r]=rr) , V r =1,...,P vad(tt Fro]=f. Tacotil' ,:3.11),(3.12)ding (3.13) p-l F[r] (0) I x fJx) =I i , xr + I f(X-t)P-l f';[P](t)dt,r=O r. (p - I). 0 l~i~n. Bao I<;li,gia samQtham f =(f;,...,fJ E C(I; Rn) choboi congthuc (3.14) ~ p-l F[r] (o) 1 x /;(x)=I i I xr+ ,f(X-ty-1F}P](t)dt,r=O r. (p - 1).0 l~i~n. Khi do,til' (3.12),(3.14)taco 14 (3.15) ];(x) =I J;<r);o)xr + 1 I f(X-t)P-l t;<P)(t)dtr=O r. (p - 1).0 = fj(x), i=l,.. .,n,XE I. V~y f IaWigiiiicua(2.11). Dod6,tac6dinhIy sauday. DinhIf 3.2: Giasa / =[-b, bJ vacacsa thT!c a!ik'bijbCijk nhu trangdtnhly 2.3. Chog E Cpr/;Rn).Khido,liJi giaif E cpr/;Rn)cua(2.11)du(fcbiiu diln dutJi dqng (3.13),trangdo Ffr] E C(/; Rn) la liJi giai duynhdt cua (3.7).Dao lqi, mQihamf E Cpr/;Rn)du(fCbiiu diendutJidqng(3.14)diu la liJi giai cua (2.11). Chu thich 3.2: Tru'onghQpI =R va cacs6thlfc (j'iik'bijk.Cjjkthoamandi€u kic$n(3.9). N€u g E CbP(I;Rn)thlk€t ku~ncuadinhIy 3.2v~ndung,trongd6 cac thonggianhamC(I; Rn)va CP(I;Rn)xuftthic$ntrongdinhIy 3.2du'Qc thaybdi Cb(I;Rn)vaCbP(I;Rn)I~nIu'Qt. 15 TrdvStru'onghQp1=[-b,b],titdinhly 3.2,tacoht$quasau: He Qua3.1: GidsitI =[-b, b] vacacs6 th1!CaUk'bjjk,CjjknhlltrangdjnhIy 2.3. Khi do, ne'ugJ, ..,;gnIa cac da thacbqc ~r -1, thlliJi gidi f cua (2.11) ding Ia da thacbqc ~ r -1. Chungminh: Taco (3.16) gj(r)(x)= 0, 1 :::;;i:::;;n, '\j xE [-b, b] . Khi dop[r](x)=0 la Wi giai duynh1tcuaht$(3.7). Ap dvng(3.13)voip=r,taco (3.17) f(x) =f F;[S] (0) Xl s=o s! . . DinhIf 3.3: GidsitI =[-b, b], vacacs6 th1!C aiik'bUbC!jknhll trangdjnh Iy 2.3. Chog E CP(I;Rn).GQi f E CP(I;Rn)cua(2.11) zIngvai g vagQi f Ia liJi gidi da thac bqc .~ p -1 cua h~(2.11)zIngvai da thac g=(gl' ,gJ ' trangdo 16 (3.18) (3.19) (3.20) (3.21) (3.22) ~ p-l (r) g;(X) ="g; (0)L xr r=O r! . Khi d6 tac6 ~ 1 bP II (p) 11 III - It ~1- P p! g x' Chungminh: Khai trienMaclaurinchogj(x)taco gi(X) =gJx)+(p ~I)! r(x -. t)p-l g;p) (t)dt . Ap dl;mgdanhgia (2.7)yoi a =~taco III - It ~1~pjig- gllx . Do (3.20)taduQc n JIg- gllx ~supIlgi(X) - gJx)1 . !xl:;;bi=l Ta danhgia t6ngcu6iclingcua(3.22), Truoc he't,yoi 0 ~ x ~ b: 17 (3.23) (3.24) tlr(x - tV-1g;p)(t)dtl:s;r(x - t)p-ll1g(p)(t)lldt P :s;IlgCP)!Ix.r (x - tV-1dt =IIg(p)!Ix.xp bP :s;IIg(p)!Ix.p Tu'dnghf,b1td~ngthuc (3.23)clingv~nluonluondung voi -b :s;x :s;o. Dodotu (3.22),(3.23)taco jig- gl!x:s;IIg(pjll ~ x p! vatu (3.21)taco(3.19) . ;HeQua3.2: (3.25) Gid sa 1= [-b, bj, vacacs{fth1!caUk'bi}",CijknhutrongdjnhIy 2.3. Cho g E CCO(I;Rn) saDcho 3d> 0.. IIg(pjllx::;;dP, 'Vp2:0 GQif Ia liJi gidicuah~(2.11)lingWJigvagQi ][P], Ia liJi gididathac b(ic ::;;p -1cuah~(3.11)lingWJidathac g, nhutrongdjnhIy3.3. Khido Hrnl l ! - ][PJ II =0 . p~CIJI X 18 (3.26) HCInnila, taconcodankgia Ilf-][p]11 ~~ (bd)Px 1- fJ '~'p. 'rip=1,2, ... Chungminh: Hi€n nhien tu (3.19)va (3.25), tad~nden(3.26). . He gmi3.3: (3.27) Gidsa / =[-b, b], vacacsa th1!CaUk'bijktCijknhutrangdinkly 2.3. Chog E C(/;Rn)va! la liJi gidicuah~(2.11)angvaig Khi do,t6ntc;zimQtdaycacdathacbqc ::;;p -1: ][p] =if[p], ,J!p]) saDcho p~roIlf- 7[P]!Ix =0 . Chungminh: Theodinhly Weierstrass,m6ihamgiduQcxa'pXlbdimOtdaycac dathuchQitv d€u p/p]khi b~cp-1 ~ 00 Do d6 p[p]=(Pl[P],...,Pn[p]) hQitv trongC(l; Rn)v€ g khi p~ 00 GQi ][P] laWigiaidathuccua(2.11)ungvoig=p[p]. 19 (3.28) Thea danhgill (2.7)vdi a =ptaco: 111[p]- it ~1~fJ .llp[p]- gllx~ 0 20 khi P-7 00. . ._.