Luận văn Xấp xỉ và khai triển tiệm cận nghiệm của hệ phương trình hàm

Chridng6. " ,./'" A' MOT SOVI DU CU THE VE H~ PHUONG 'TRiNH HAM. Trongchuangnaychungtaxetm9tvaiVIdlJ ClJth€ v~ m9th~ phuongtrinhhamtrongmi~nhaichi~u,g6m:sl!h9itlJ cuadayl~pca'p hailienke'tvdih~phuongtrinhvakhaitri€n ti~mc~nnghi~mcuah~ de'nm9tca'pchotrudctheom9tthallis6be &. . 2 2 gi(x)=llxll;-&Iaij(sijllxlll)2) - Ibij(ISijlllxlll)) )=1 )=1 va aij' bij'Sij la cacs6thl!cchotrudcthoa 2 II[bij]11 =I ~a<xlbijl<1, Isijl s 1.i=1L1-2 Cac ham Sij(x)=sijx, gi(X) lien tlJCnhugiclthie't(H1),(H2). HAng s6 M >0 duQcchQnsaocho: M1 <M <M 2 ' ?, " ? w ,,' 6.1.KHAO SAT THU~T GIAI L~P CAP HAl. Chungta xeth~(1.1)lingvdi m=1,n=p =2. Q={X=(Xl,X2)ER2 :llxlll=IX11+IX2Isl}, 2 2 hex) =&Iaijf} (sijx)+Ibijf)(s ijx)+gi(X)' )=1 )=1 (X E Q ,i =1,2.), trongd6 trongd6 1-II[bij]11- ~(1-II[bij]II)2-12&0 II[aij]ll~g M1 = 6&0II [aij]II ' l-ll[bij]11 + ~(1-II[bij]II)2 -12&0II[aij]ll~g M2 = 6&0 II [aij]II 30 (6.1) (6.2) (6.3) (6.4) (6.5) (1-II[bijJII)2, 1&1~&o, O<co<1211[aij]lI~g (6.6) 2 . . Ilgllx~2+ I (colaijl(sij)2)+lbijllsijlJ)=~g. i,)=1 (6.7) Nghi~mchinhxaccuah~(6.1)la : I (x) =(1Ixlll'llxll~). (6.8) Nhu'trongchu'dng4, thu~tgiai ca'phaichoh~(6.1)Cl;lth~nhu'sail: /i(V) (x) =cIaij [21}V-l)(sijx)ljV)(sijx)- (/}V-l\sijx)) ]J=1 2 + Ibijljv) (sijx)+gi(x), )=1 (6.9) (x EQ; i =1,2;v =1,2,...). Ne'utada chQndu'Qcbu'ocl~pbandgu 1(0) =(ft(O),12(0) saocho 11/(0)Ilx ~M va 13M11/(0)- I Ilx <1, (6.10) voi &11 aukJII >0 PM = l-ll[blJk]II-2& Mil [alJk] II (6.11) Khi do,taco II/(V)- III ~~~MII/(O) - III jV , Vv =1,2,... I x 13M X (6.12) BaygiiJ,fachQnbudcl(ipbanddu 1[0]: Ta Kayd1jngdayl~p{z(17)}c KM xacdinhbdi : zf17)(x)=cIaij(z;17-1)(Sijx))+ Ibijz;17)(sijx)+gi(x) , )=1 )=1 (XEQ;i=1,2;7]=1,2,...) , (6.13) 31 trongdo zeD)=(z}O),z~O) ==(0,0). Khi doday {z(1])}hQitl,ltrongX v~nghi~mI cuah~(6.1)vataco mQtdanhgiasais6 III - z(") Ilx ,; ( II z(O) - Tz(ot ).(1~~)'; 1~CT CT",If '7 = 1,2,... (6.14) voi 2&Mil [aijJII ()= <1 1-11[bijJII . Tli (6.14),(6.15),tachQn7]0EN khaIOnsaocho: PM III - z(1]o)llx~ 7~~(}1]o< 1. V~ytachQn (6.15) (6.16) 1(0) =z(1]o) . 6.2.KHAO SATHt (6.1)KHI [; =0 Khi & =0,h~(6.1)la h~tuye'ntinhsail: 2 hex)=Ibijlj (sijx)+gi(x) , j=l xEQ={X=(XI,X2)ER2 :llxIII=lxII+lx21~1},i=1,2. (6.17) Ta xetcactru'ongh<;1psaildaycuaham gi(x), a/ HAM gi(X) LA DA THUC. Giasa gi(X) la dathuctheohaibie'ncob~cnhohdnhayb~ngr gi(X)=I diyxY = I diyx{lx/2 , i =1,2. Iyl::;r y=(Yl,Y2)eZ; Yl +Y2::;r Theo mQtke'tqua trong[3] cua cac lac gia N.T Long, N.H. Nghla, nghi~mcua h~(6.17)cling la cac da thuc.Ta Hmnghi~mcua (6.17) theod(;lng hex)=I CiyXY = I Ciyx{lx/2 , i =1,2. Iyl::;r y=(Yl,Y2)EZ; YI +Y2::; r (6.18) (6.19) 32 Thay hex) vao (6.17)tathuduQc(C1r,C2r)1anghi~mcuah~phuong trlnhtuye'"ntinh 2 C. -" b..sl.rlc . =d. 1.=12 Iyl<rzr L..J l} l} lr zr ' " - . )=1 (6.20) Giaih~(6.20),taduQc: (1- b22S~)d1r+b12S~~ld2rc -1r- (1 b Irl )(1 b Irl ) b b Irl Irl- l1sl1 - 22s22 - 12 2]s12 s2] b2]st1ld1r+(1- b11s1~1)d2rc -2r - (1 b Irl)(1 b IrJ ) b b Irl Irl- l1sl1 - 22s22 - 12 2]s12s21 (6.21) , ? A K K b./HAM gJx) KHA VI LIEN T1JCDEN CAP q. Gia sa g=(gl,g2)Ecq(Q,R2). GQi 1=(fi,12) la nghi~mda thliccuah~(6.17)tuonglingvoi g =(gl,g2),trongdo = I 2 r=(rl,r2)EZ+ Yl +Y2~q-l gi(X) = I ~Dr gJO)xr Irl~q-1 y. 1 alrl ( ) ( gi (00) rl r2) . , xl x2 ' 1=1,2. Y1!Y2' axrlax~2 (6.22) Theoke'"tquatrong[3]cuacaclacgiaN. T. Long,N. H. Nghla tacosail~chgiuahainghi~mf, I cuah~(6.17)l§n luQttudngling voi g,g , duQcchobdi danhgia: Ilf - 711x ~(1-11;by]11)('r~)lnrglJ) , (6.23) trongdo lex)= I CirXr,(i =1,2.), Irl~q-1 (6.24) 33 (1- b22S~h~Dr gl(0)+b12S1;11-Dr g2 (0) r! r!c -lr - (1 b Irl)(1 b Irl) b b Irl Irl- llsll - 22s22 - 12 21S12S2l b21S~11~Dr gl (0) +(1- bllsl~I)~Dr g2(0) r! r!c -2r - (1 b Irl)(1 b Irl) b b Irl Irl- llsll - 22s22 - 12 21S12S21 , Irl~q - 1. (6.25) / ~ cNOr gJx) C1J THE. Ta xetmQtVId1;lvdi hamg =(gbg2) nhtisail: 1 9+i gj(X) =gJxl,x2) = .' 1- xl +x2 9+1- Xl - x2 9+i X =(xl,x2) E Q.,i =1,2. Ta vie'tl~i gJx) nhti sail : ( ) - 1 _~(Xl+X2 ) ) g. X - - L. I 1- xl +x2 )=0 9+i 9+i q-l ( ) ) 00 ( ) ) = I Xl +~2 +I Xl +~2 . )=0 9+1 )=q 9+1 Ta chuyr~ng = (6.26) (6.27) . ., Irl! (X +X )J =" J. X Yl X Y2 =" - xr .12 L. ,,12 L.,Yl+Y2=)rl.r2' Irl=)r. q-l ( J ) q-l 1 Irl ' q-l 1 Irl'I Xl +~2 = I . . I ~ xr = I I - . xr )=0 9+1 )=0 (9+l)J Irl=) r! )=0 Irl=)r! (9+Olrl 1 == I-Dr gJO)xr . Irl:S;q-l r! (6.28) D~t q-l 1 II ' ~[q](x)=I I - r. xr, )=0 Irl=)r! (9+Olrl (6.29) 34 taco Dodo TagQi Igi(X)-~[q](x)1=If(XI +~2J j j=q 9+z ,; i: Ilxll{j ,; i: - I . J=q(9+z) j=q(9+i)J < 1 - (8+i)(9+i)q-I ' '\I x=(xI,x2)EO. (6.30) 2 1 1 ji g - p[q] 11 =supI l g,(x) - p[q](x) 1 ~ +- x '- Z z 910q-I 1011q-IxEQz-I " 1 . ~ 1 ~ 0 khz q~ 00.10q- (6.31) J[q] =(fr[q],J2[q]) Ia nghi~mdathlieeuah~(6.17)tu'dnglingvoi V~y g =p[q] =(p,[q]p [q])1 , 2 . J[q] =(fr[q],J2[q]),f(x)= I CirXr ,(i=1,2.), Irl~q-I trongdo,caeh~s6 (Clr,C2r)du'Qetlnhtheoeongthlie(6.25)voi DYg,(O)= Irl'II' IrlS:q-l,(i=1,2,),(9+i)r tlieIa (6.32) (6.33) (1 b Irl) b Irl- 22s22 12s12 - - - +- Irl! 10lrl 111rlC --x Ir -, b Irl)( b Irl ) b b Irl Irlr. (1- Iisil 1- 22s22 - 12 2ls12s21 b Irl (1 b Irl )2Is21 - llsll+---- Irl' 10lrl I11rl c2r=,x Irl ) b. Irl ) b b Irl Irl 'r. (1- bllsll (1- 22s22 - 12 2ls12s21 , Irl ~q - 1. (6.34) 35 M~tkhac,tli'cacht%sailday: f =Bf +g , 7[q] =B7[q] +p[q] , , tasuyrading f - 7[q] =B(f - 7[q]) +g - p[q] . V~y IIi - 7[q]llx~liBel - 7[q])llx+jig- p[q]llx ~IIBllllf- 7[q]llx+llg-p[q]llx ~11[bij]llllf-7[q]llx+llg-p[q]llx' (6.35) Suyra Ilf-7[q] 11 < 1 II -p[q] 11 x -1-II[bij]11 g x 10I-q ~ II II ~ 0 khiq~ 00 (theo (6.31)). 1- [bijk] (6.36) ,? A A A? A 6.3. KHAI TRIEN TI~M C!N NGHI~M CUA H~ (6.1)THEO [;. Trongph~nnaychungtasesadl;lngcaccongthlic(5.1)-(5.3)d chu'dng5d~xacdinhcacthanhph~ntrongkhaitri~ntit%mc~nnghit%m. Ta giasar~ngaij,bij,sij la cacs6thlfcchotru'octhoa(6.3), cachamSij(x)=sijx thoagiathie't(HI), Giasa gi(x) la dathlictheo haibiSnchotru'ocdQcl~pvoi [; nhu'sau: gJx) =L diyxY = L diyx{jX/2 ,(i =1,2.). Iyl:::;r y=(yj,Y2)EZ1 n+Y2::>r (6.37) Ap dl;lngcongthlic(6.18),(6.19),(6.21),nghit%mcuaht%(6.1) ling voi [; =0 (tucla h~(6.17))clingla cacdathlic: f[O]=(11[0],f}O])=(I - B)-I g , 36 voi f[O](x)=I CirXr = I CirX/' X/2 ,(i =1,2.), Irl~r r=(r"r2)EZ,; YI+Y2~r trongd6 (C1r'c2r) chobai (1- b22stJ)d1r+b12slild2r c1r= Irl )( b Irl ) b b Irl Irl'" (1- b11s11 1- 22s22 - 12 21S12S21 b21stld1r+(1- b11Sl~1)d2rC =2r (1 b Irl)(1 b Irl) b b Irl Irl'- l1sl1 - 22s22 - 12 2lS12S21 Irl~r. G9i f[l] la nghi~mcuah~(6.17)lingvoi g =Af[O],tlicla: f[1] =(f/1],f2[1])=(I - B)-l Af[O], voi Af[O]=((Af[O])l'(Af[O]h) , trongd6 (AfIOJ);{x)=Iaij(J}OJ(Sijx»2 =Ia/2: cJrs)rlxr ) 2 . j=l j=l llrl~r Ta l~ic6 l } 2 Irl r - la+fJl a+fJI CjrSij x - I CjaCjfJSij x Irl~r lal~r,lfJl~r - " ( " ) Irl r- L. L.CjfJCjr-fJ Sij x . Irl ~2r fJ :0;r (AfIOJ)i(X) =fa/ I ciys)rlxy } 2 j=l llrl:o;r 2 =" a" " ( "C ,pC, fJ )S"Irl xrL. 1] L. L. J Jr- 1] j=l Irl:o; 2r . fJ :0;r 2 = " ( " a" S"Irl "c 'fJ C' fJ ) xrL. L. 1] 1] L. J Jr- Irl:O;2r j=l fJ:O;r 37 (6.38) (6.39) (6.40) (6.41) (6.42) (6.43) - " d~l)xr L..,. lr ' Irl ::;2r (6.44) trongdo,tada:d?t : 2 d(1)- " Irl" I I 2ir -L..,.aijsij L..,.c)fJc)r-fJ' r::; r. )=1 fJ ::;r Tli (6.21)taco bi€u thuccua 1[1]= (//1],12[1])chobdi congthuc: (6.45) h[I](x) = I cg)xr = I Irl::;2r r=(rl,r2)EZ~ n+n:S:2r (1)x YIx r2Cir I 2 , (6.46) d" ( (1) (1)) h b ?" h" (621) ". ( ) ' trong 0 clr' c2r COOl cong t tic . , Val Clr 'C2r va (d d ) 1 /1: 1 h b?' ( (1) (1)) , (d(1) d(1)) ". I I 2Ir' 2r an uQtt ay 01 Clr' C2r va Ir' 2r ' VOl r::; r, nhu sail : ( b Irl)d (1) b Irld (1) c(1) - 1- 22s22 Ir + 12s122rIr - (1 b Irl )(1 b Irl ) b b Irl Irl- lIS II - 22s22 - 12 2lS12 S21 b Irl d(1) (1 b Irl )d(1) c(l) = 21S21Ir + - IISII 2r2r (1 b Irl)(1 b Irl ) b b Irl Irl '- lIS II - 22s22 - 12 21S12S21 ,Irl::;2r. (6.47) Theok€t quacuadinhIy 5.2,chuang5,taco ffiQtdanhgia ffiQt khaitri€n ti~ffic~ndip 2theoE: dunhonhusail: Ih(x)- h[O] (x)- £ h[1](x)1 ='h(x)- I CirXr - £ I cg)xrl::;2Cllu- B)-III £2. Irl::;r Irl::;2r (6.48) voi ffiQix EQ, i =1,2vavoi £ dunho,C >0 Ia h~ngs6dQcI~pvoi x va £. 38