4 _i Suyra | | A ' | | < 5/2 va ||A|| < 13/5 Suyra | | A ' | | < 5/2 va ||A|| < 13/5 - I E>at 13 13 •' 2 5II (p4 e qi = ( + ! + ( ) ) ; c = q , ; R = 2 2 5 2 ( l - q 0 Ta kidm tra cac gia thiét cua dinh ly 2.1
a - ImA= (ImA* = Ho) - đng b - V X e H (t)(x) e Ho = ImA
c - V e ImA* = lmA= (KerA)^ , V h e Ker A
(t)'(x) h = 2(pi < x,h > + (p3 < (p2, h > = 0 suy ra Ker A c= Ker <|)'(x) Vdri moi x e S (0,R) n ImA* va e du nho
s||<t)'(x)|| <(2||cpi ||R+||(P2|| ||(|)3||)e<C
Dfeu nay chiing to vi du da thoa man cac dieu kien cua dinh ly 2.1
§3 - Phuang phap hieu chinh Gauss - Newton gan dung.
Ta ky hieu:
Mic = F*(Xk) F Xxk) + ak I
(t>k = F'*(Xk) F(Xk) + ak(Xk - ^ )
Thuat toan (1.3) ed thd viét lai nhu sau:
Xk-f-i =Xk-M"'k(t)k (3.1) Trong thuat toan (1.3) tren mdi biioc ta tim nghiem chinh xac
hk = Xk +1 - Xk tu phuang trinh Mkhk=-(t)k
nghiem gan diing Xk ma vSn dam bao su hoi tu cua nghiem g^an diing đ tai x* ( nghiem cua bai toan).
Tren CO so y tuong cua Axelsson [60], chiing toi d^e xuát phuang phap hieu chinh Gauss - Newton gan diing IRGN
Gia sir a bu6c thu* k ta tim dugc C^^
Dat Qk = F * ( i ; k ) F ( Q + akI;
(Pk = F*(Ck)F(Ck) + a k ( C k - 0 XSip XI birde thiJ K + 1 dugc tim tu he thiic :
Ck + i = Ck + P k ( k = 0 , l , . . . . , K ( 6 ) ) (3.2) 6 day 6 > 0 la sai sÓ cho truac, s6' nguyen K (6) se xac dinh sau va budre lap pk dugc
chon tur c6ng thiic :
Pk G 9^k (5) = { p e X : ||QkP + (Pk|| ^ 6||(pk ||} (3.3)
Do - Q'\ (pk e 9ik(6) , nen suy ra 9?k (6 ) ^ ([) So sanh (3.1) , (3.2) ta cd danh gia dO lech giu-a Xk va C,\,
llxk.i - C k . i M l|xk-Ckl|+ ||Pk + M-\(t>kl| (3.4) Tur (3.3) ta cd bát ding thiic
1 1 + 5 1 + 6 llPkll = IJQ'^QkPkll ^ — llQkPkll ^ llcpkjl ^ (II 4)k 11+ Hk-cpkll ak ak ak Ta cd danh gia: |iPk + M-\(|)k||^||M-\||'{l|MkPk + (|)k||} < llM'kll {llQkPk + cpkll + lUk - (Pkll + IK Mk - Qk) Pkll) 1 ^ — {5|lcpk|| + II (|)k - (Pk II + IK Mk - Qk) II ||Pk|| ) 1 1 + 6 ^ — {5||(t)k|| + ( 1 + 6)11 <|)k-cpk||+ llMk-QklKHicll + ll (|)k-cpk||)) (3.5) ttk Otk
Ph~an sau nay ta di chiing minh m6t so h& dang thiic vdi gia thiét rang dieu kien (BI)
i - I U k | | = ||F*(Xk)F(Xk) + ak(Xk-^) II <
< IIF *(Xk) II II F(Xk) - F(x*) II + ak (II Xk - X* II IIX* - ^ II) < • < N ^ | | x k - x * | | + a k | | x k - x * | | + a k | | x * - ^ | |
Dat P = ( 2 | | v | | / N , N 2 ) ' ^ , t a nhan dugc ||xk-x*|| < ek = p a k
va ||x* - ^ II = | | F V ) F ( X * ) V|| < N M | V | | < ( 3 - V 5 ) N,/(4N2) Tir cac bát dang thiic trfin suy ra
5||<t)k|| 3 - V 5 ^ 5 ( Ni + p N ^ + P a k ) < C i 5 (3.6) ttk 4N- 3 - V 5 d day Ci =( Ni + PN^ + p ao ) 4N- n ||(|)k - (pkll = IIF (Xk) F(Xk) - F (Ck) m^d + ak (Xk -1;,\\ <
< IIF*(Ck) II IIF(^k) - F (Xk) II + II [F '(Ck) - F'(Xk)]* II II F(Xk) || +ak || Xk <, || (adsbygoogle = window.adsbygoogle || []).push({});
< N M I Xk -Ck II + N2II F(Xk) II II Xk - Ck II + ak || Xk - Ck 1| (3.7) Di y rang ||F(Xk)|| = ||F(Xk) - F(x*)|| < Nill Xk - x*|| < p N , a k T a c o II (t)k - (Pk II < { N , V ( 1 + pNiNa )ak} || Xk - Ck || ( 3.8) iii - II Mk - Qk II ^ ||F*(Ck) II ||F(Ck) - F(Xk) II + ||F(Xk) || ||F*(Ck) - F ''(Xk) ||
Tur day suy ra :
| | M k - Q k | | ^ 2 N , N 2 | | X k - C k | |
Ngoai ra ta co danh gia tho :
II Mk - Qkll ^ II Mkll + II Qkll ^ 2(N^ + a k ) < 2 (N^ + ao) TCr cac danh gia tren thé vao (3.5) ta co he thiic :
1 + 5 ||Pk + Mk"'(j)k|| < C , 5 + {1 +
2 ( N ^ + ao)
} {N,'+(l+ pN,N2)ak} II Xk-Ck II +
2 C i N i N 2 ( l + 5 )
+ Ilxk-Ckjl
Bait dang thiic (3.4) dugc viét lai duoi dang bát dang thúc sau: ||Xk+l - Ck+l II ^ C i 6 + TUk llxk-f^k II
6 day
tiJk= di + d2ak "^ + dattk"^ ,
di = 1 + ( l + p N i N 2 ) ( l + 5 )
d2 = (1 + 6) N,^ + 2N1N2C1 (1+ 6) +2 ( N i V ao) (1 + 6) (1+ p N i N . ) v a d 3 = 2 ( a o + N i ^ ) ( l + 5) Ni^ v a d 3 = 2 ( a o + N i ^ ) ( l + 5) Ni^
Ta chon t h a m sd' K(6) du Idm sao cho : Vdi ^0 = Xo , 60 = C i 5 , ta cd
II Xk - Ck|| - W k - l ^ k - 2 . " "CTl 60+ tiJk-l ... TU260+ .... tiJk-1 5o + 5o ^
k-1 k-2
hay la
< (nik-i +tiTk-i +.." + Wk-i+ 1) 5o
k 1 tiJ k-1 - 1 tiJ k-1 - 1 | | x k - C k | | ^ C,6 (3.9) tUk-i - 1 T a c o : w k ! -i ^ k = 00 , k h i k —> GO TUk-l - 1
v a y fOn tai duy nhát sÓ K = K ( 6 ) đ cho
k 1 TU k - 1 - 1 TU k - 1 - 1 xa k - i - 1 k 1 in k-1 - t < ( C i ATS )• > ( C i V 5 ) - ^ v 6 i k = 0,l ... K (3.10) vdi k > K V ^ k - i - 1
Tur day suy ra K (6) - > 00 ( 6 ^ 0)
Six dung each (3.9),(3.10) v a (1.4) ta suy ra h e thiic:
Nhu vay ta da chirng minh dugc dinh ly sau:
Dinh ly 3.1: Gia sir dfeu kien (BI) dugc thoa man voi each chon tham so (1.4) thi
phuong phap IRGN hOi tụ Ngoai ra ta cd udre lugng sai s6': ||CK(5)-X*|| < / S + Eoq''^^^
Djnh ly 3.2: Gia str dieu kien (B2) dugc thoa man, tham sÓ aK dugc chon tur he thiic
(1.6) thi phuong phap IRGN hOi tụ Ngoai ra ta cd danh gia: ||CK(6) -X* II < ^ + 0{am)) (adsbygoogle = window.adsbygoogle || []).push({});
Chiing minh:
Tir||F(x„j.)f < 2 a k | | x * - ^ | | ||xejc-x*|| (C6ng thiic 26 [12])
Suy ra || F(Xcjc) || ^ CZUK , a day €2= V 2c || x* - ^ || Ta danh gia mOt s6' bat dang thue tuong tu nhu trong dinh ly 3.1 i - II <t)k 11 = 11 F*(Xk)F(Xk) + a k ( X k - ^ ) | | =
= ||F*(Xk) F(Xk) - F*(Xô) F(Xâ) - ak(x„^- ^ ) + ak (Xk - UII V6i gia thiét (B2) ham s6' ||F (x) f + || ăx - ^ || dat cue tri nen suy ra:
F V , ) F ( X a ^ ) + a k ( x „ ^ - O = 0
T a c o :
II <t)k II ^ II F*(Xk) IIII F(Xk) - F(Xak) II + II F*(Xk) - F'(Xak) IIII F(x,j.) II + ak || (Xk - x^) \\ <
^ N M I Xk - Xcjc II + CzNzak II Xk - x^k || + ak || (Xk - Xak) || Sir dung gia thiét (B2) va danh gia ( 1.7) ta co:
||xk-x„jc|| ^ ||xk-x*|| II x„k - X*II < C3ak Suy ra: 5 IkII ^ C3 (Ni^ + C2N2 ttk + ak ) 5 < Ci 5 ttk 2 d day Cx-Cii Ni + C2N2 ao + ao ) i i - II <t.k -(Pkll ^ ||F*(Ck)|||l F(Ck)-F(Xk) II + II F*(Ck) - F ' U ) II llF(x)|| + + akII Xk- ^k II ^ N , ' II Xk - Ck II + N2 ||F(Xk) II II Xk -Ck II + ak I Xk - Ck||
Do he thiic (1.7) ta cd bát dang thúc :
| | F ( X k ) | | < N i | | x k - x * | | < N , C 4 a k
Nhu vay
II <t>k - cPk II < (Ní + N,N2C4ak + a k ) II Xk - Ck || iii - Cud'i Cling ta cd hai bat dang thuẹ
IJMk -Qkll ^ 2 N , N 2 | | x k - C k | | ma ||Mk -Qkll < 2 ( N , ' + ak ) < C5 T u (3.5) suy ra 1 + 5 C5 II Pk + Mk"* (t)k II < C, 6 + (1 + - ^ ) ( N i ' +C4N,N2ak + ak) || Xk -Ck Ok ak 1 + 6 2 N i N 2 C i a k | | x k - C k | | ak vay ta ed : d d a y Va ||Xk+l -Ck-^l II ^ C i 6 + TUk llxk-Ckll
tUk = di + d2ak'^ + d 3 ak"^ d i - ( 1 + 6 ) ( 1 + C 4 N , N 2 ) ;
d 2 = N i ( l + 5 ) ( N i ^ + 2N2Ci + C 5 ( l + C 4 ) N 2 ) ; d^^ €5(1 + 6 ) NT"
Tham s6' K( 6 ) dugc chon thoa man
TUk - 1 1 < k < K ( 5 ) 1/2, tUk -1 ( C , ( 6 ) n cJk' -1 1 > k > K ( 6 ) 1/2 \ lUk - 1 C i ( 6 )
Tijr each chon nen K(6) - > 00 ( 6 - > 0), ta cd danh gia
II Ck(6) - X*|| < |Kk(6) - Xk(6) II + II Xk(5) - X*|| ^ ( 6 ) ' ^ + 0(aK(oj)
54-Phu(yng phap Seilel - Newton hieu chinh va bai toan phi tuyen cong hu&ng.
Trong muc nay chung toi su dung y tudrng cua Bakushinski [12] di phat trien
cac két qua cua Pham Ky Anh [55,57] ve phuong phap Seidel Newton doi voi bai toan c6ng huong.
1-Phuorng phdp Seidel-Newton (SN) vd phuang phdp Seidel- Newton hieu chinh (RSN).
Trudrc hét ta trinh bay ndi dung cua phuong phap Seidel - Newton [57] Cho phuorig trinh toan tur
A x + F ( x ) - 0 (4.1) d day A : X ^ Y la toan tir tuyén tinh Fredholm bi chan tur X va Ỵ X, Y la cac
khOng gian Banach. Vi A la toan tu Fredholm nen khong gian X va Y dugc phan tich thanh tong true tiép cua cac Idiong gian con đng.
X=Xi 0X2 , Y = Y i © Y 2
oday Yi = I m A c Y ; X 2 = KerĂ=Xva
d i m X2 = d i m Y2 < + 00 (adsbygoogle = window.adsbygoogle || []).push({});
Ngoai ra A la han ché cua A tren Xi cd nghich dao gioi noị Cac toan tir chiéu tuyén tinh P va Q trong khong gian Y xac djnh nhu sau:
P : Y - > Y i ; Q : Y - ^ Y 2
Tir day suy ra ImP = Ker Q = Yi , ImQ = KerP2 va P+Q = I la toan tu don vi trong Ỵ
Gia su 6 budfc lap thii n nghiem g~an dung da biét
Xn = Un + Vn ( Un e Xi , Vn e X2)
Tren budfc lap thii n + 1 ta xac dinh thanh phan thii nhát theo cong thiic
Un+i = - A - ' P F ( X n )
Dat T„ = Un+i + v„
Khi đ thanh phan con lai dugc tinh nhu sau:
6 day ky hieu [ Q F ' j l a ban ché cua Q F (x) tren X2. Cud'i cung dat Xn+l = Un+i + Vn+l
T.
Nhi^eu cai bien cua phuong phap Seidel - Newton da dugc vao trinh bay trong [55,57,69]. Trong muc nay ta gia thiét rang, toan tu [QF '(Xn)l suy bién, cd nghia la
nd khOng cd nghich dao bi chan, do đ phuomg phap Seidel - Newton Idiong ap dung dugc.
Gia sir X, Yla hai khdng gian Hilbert thue, y tudtig hieu chinh cua Tikhonov da dugc ap dung trong [12], df day chung toi trinh bay phuong phap Seidel - Newton hieu chinh RSN nhu sau:
Gia sur d xáp xi thir n
Xn = Un + Vn ( Up e Xi , Vn G X2, n > 0) da biét (4.2) Khiđ Un+l=-A"^PF(Xn) (4.3) Dal Tn = Un+1 + Vn (4.4) T u n Vn ^ 1 = Vn -{ [QF '(3rn)]*[QF '(Xn)] + Onl ) ' ( [QF(Xn)]*[QF(Xn)] + an ( Vn - V^'^ ) } ^ ^ ^2 ^ (4.5) Cud'i cung Xn+l = Un+1 + Vn+l
d day Ix la toan tur don vi trong X2, v^°^ la phan tu* cua X2 duoc chon mot each dac 2
biet va {an} la day so duong.
Ta se sur dung gia thiét cua Bakushiski dÓi voi v* - v^^^ ,0 day v* la thanh ph"^an thii hai ciia nghiem x* = u* + v* cua phuong trinh (4.1). Do X2 huu han chi^eu nen dieu kien Bakushinski (BI) de kidm tra hon.
/ / - Su hoi tu dia phuang cua phuang phdp RSN:
Trong phan nay ta van gia thiét phuong trinh (4.1) cd nghiem x* va toan tu F kha vi lien tuc tren tap ehiia hinh cau đng S"(x*, r).
Ngoai ra ta con gia thiét rang
V x , y e S ( x V ) : ||PF(x)|| <p,|lQF(x)|| <m va||QF(x) - QF'(y)||^ L||x - y||
Kihieu
C = [QF(x*)]* [QF'(x*)]
' ^ ' ^
Xet tnrcmg hcfp C ?!: 0 va Ker C ^ {e)
Do X2 = ImC © Ker C - - Thi C la han ché cua C tren ImC ed nghich dao bi chan.
Di viec trinh bay dugc don gian ta ki hieu cac loan tu
G(x) = QF (X) ; G'o(x) = QF \Xn)U I2 - I ; M(Xn) = [G'o(£)]*G'o (Xn) + anh- (adsbygoogle = window.adsbygoogle || []).push({});
2 \
Khi do (4.5) cd ihi viét lai dudfi dang
Vn+l = Vn - [ M ( X n ) ] - ' [ G o* (Xn) G'o(Xn) + On ( Vn - V^'^ ) ]
Djnh ly 4.1: Gia su cac dieu kien sau thoa man i - 2 p ||A-*||+ T n m + 2 p ||A-'||)' < 1 ii - V* - v^^^ e ImC ||v<°'. iii - -v*|| an = 1 - 2 P IIA < ||C'||[1+ r = — q " 'II-ram 2TDL(1 + + 2PI|A-'||: 2 P | | A - ' | | ) ] adayq=2p||A-'||+Tum+2p||A-'||)^+[l+2Tiim+2p||A-'||)]||v(°V||||C'|| (4.6) Thi phirang phap RSN hoi tu vdri moi xo trong tap
va ta cd danh gia
| | x n - x * | | < rq" (4.7)
» Chiing minh: Ta chung minh bang qui nap he thiic saụ
Xn,XneS(xV) Vn>0 (4.8) ||un-u*|| <r/2q" , ||Vn-v|| <r/2q" (n > 0 ) (4.9) Til cac dieu kien ( i - iii) suy ra q < 1
Vdri k = 0 cac he thu-c (4.8), (4.9) hidn nhien dung, tru dieu kien Xo e S (x*, r). Tir: ||xo-x*|l < |lui-u*||+ |lvo-v*|| < HA"^ ||PF(xo) - PF (x*)|| +||vo-v*||<
< P | | A"' ||r + r/2<r Nhu vay "x0 e S (x*,r).
Gia sur (4.8), (4.9) diing vdi moi k < n ta se chung minh chung dung vdi k = n + 1 Tu (4.3) ta cd u* = - A "^ PF(x*)
Ta cd danh gia
II u „ . , - u* II = IIA-' [PF (Xn) - PF(x*)] II < P II A" IIII x„ - X* II ^
<P|| A ' l l r q " ^ q".r/2.
Do ||xn-x*|| < | | u „ . , - u * | | + | | v „ - v * | |
< p||A-' | | r q " + r / 2 q " < r q " < r Suy ra Xn e S (x*,r).
Dg dang tháy QF(x*) = 0 va tir (4.5) ta co
Vn+l = Vn - [ M ( X „ ) ] " ' (G'*(X„) [G(Xn) - G ( X * ) ] + ttn (V„ - v'°^} = = Vn - [ M ( X n ) ] ' ' {G'*(Xn) G'(Xn) (Xn " X*) + gn + a „ ( V n - V*) + a „ ( V n - v ' ° ' ) I 6 d a y g n = G'o*(Xn) J [ G ' ( X * + t ( I n " X*)) - G'(Xn ) ] (X„ - X*) dt 0 tnL Suyra ||gn 1 1 ^ — ||Xn-x*f (4-10) 2
Do v„+, = Vn -[M(x„)]"' {[G'o*(Xn) G'o(Xn) + a,h ](v„ - v*) +
+ G'o*(Xn) G ' ( X n ) (Un+1 - U*) + gn + ttn ( v * - V^°>) } Ta c6: _ _ _
Vn+l - V* = - [ M ( X „ ) ] - ' G'o*(Xn) G ' ( X „ ) ( U „ + , - U* ) - [ M ( X n ) ] - ' g „ -
- a n [ M ( X n ) r ' ( v ' - v < ° ' ) (4.11)
Ta danh gia m6t s6 hfe thiic rifing biet.
_ _ 1 1
a) II [ M(Xn)r' II = ||[G'o*(Xn) + a„l2l-' II < Sup = (4.12) A,^ 0 ^ + Un an
b) VI X la kh6ng gian Hilbert va G'o(Xn) = [ G'(Xn) ] nfin chiing ta cọ
^ _ _ A, II [M(x„)]-'G'o*(x„)G'(x„) II < II [G'*(x„)G'(Xn)+anI ] ' G'*(Xn)G'(Xn) II < Sup < 1
X>0 A, + an
Suy ra: _ _ _
II [ M ( X n ) ] - ' G o * ( X „ ) G ( X n ) ( u „ + , - U* ) || < || Un+, - U* || ( 4 . 1 3 ) (adsbygoogle = window.adsbygoogle || []).push({});
Do gia thiét (ii) nSn ta co th^ viét
an [M(x„)]-' (v* - v^°') = an [ M(x„)]-' C h d day h = C ' (v* - v<°^) e ImC
C) II a n [ M ( X n ) ] - ' C h II < II a n [ M ( X n ) r ' G ' o * ( X n ) G ' o ( X „ ) h | | +
+ ||a„ [ M(x„)]-' [G'oV)G'o(x*) - G'o*(Xn)Go(x„) ] ||
< a„ ||h|| + a„ II [ M(x~)]-'II ||Go*(x*)|| || Go(x*) - Go(Xn) || ||h|| + a n | | [ M ( X n ) ] - ' | | | | G ' o V ) - G ' o ( X n ) ] ' | | | | G o ( X n ) | | | | h | |
< a „ | | h | | + 2 m L | | x „ - x * | | | | h | | Cu6'i ciing
d) an ||[M(x'n)]-^ ( / - v^^^) II < (an + 2TIT L ||Xn - X* || ) || C ^ l ||(v^'^ - V*) II (4.14) Iffít hop cac d ^ g thu-c (4.10) (4.12) - (4.14) vdfi (4.11) ta cd
IKVn^l - V*)||<||(Un.l - U*)||+ 1| ( Xn - X*) ||+(2TnL|| Xn - X*||+ an) |1 C' \\v^'^ - V*) ||
2an
TIT gia thiét quy nap va bát dang thúc euoi suy ra TUL 1 ||(Vn.i-v*)||<p||A-^||rq" + r V " ( — + [3|| A ' | | ) ^ + 2an 2 1 + [ 2 m L r q " ( — + p | | A - ' | | ) + a n ) ] | | C ^ | | v ^ ° > - v * | | 2 r
Do an = — qn va q dugc dinh nghia boi (4.6) ta cd danh gia 2 r
| | ( V n . i - v * ) | | < — q"-^
2
Nhu vay he thiJc (4.8), (4.9) dugc chung minh vdfi moi n di^eu đ cd nghia la la cd danh giạ
| | x „ - x * | | < r q - Dinh ly chung minh xong []
/ / / Bdi todn bien tudn hodn cho phuang trinh Duffing - Van der pol
Xet bai toan bien tu^n hoan saụ
/ X + X + sf (t, X, X, X) = 0 0 < t < 1% (4.15a)
1 X (0) = X (2TC), x(0) = X (271) (4.15b)
6 day ham f dugc gia thiét lien tuc theo bién t , kha vi lien tuc doi vcd cac bién con l a i , e > 0 la tham so bẹ
Gia su:
Y-L2[o,27c] ^
X = {x(t) e C ^ : x(t) tuyet doi lien tuê x e L2[0,27c]. J [o,2n]
x(0) = x(27t) , X (0) = X (In)
2K
V x,y e Y <y,z> ^ J y(s) z (s) ds
V ^,i; e X [^ ,^ ] = s <^^'^c^'^>
i = 0
Cac chudn ciia Y va X la:
||y||=<y,y>^/2 (Y G Y) |||x|||=[x,x]'^ ( x e X ) (adsbygoogle = window.adsbygoogle || []).push({});
Bai loan (4.15a) va (4.15b) dugc viél dudfi dang loan lu Ax + £F(x) = 0
Trong đ A,F : X - ^ Y , A x = x + x, F(x) - f (t, x, x, x,)
Gia su:
Tacd
Dal
X2== Y2 = Span ({ci, e2)) , o d a y e] =
<ei, ej> = 6ij
X i = { X e X : < x , ei> = 0 ( i = i , 2 ) } sint VTT , ^2 = cost VTT Y , - { y e Y : < y , e i > - 0 (i=l,2)}
Qy = < y, ei> ci + <y, e2> e2 , Py = y - Qy (y e Y).
De dang tháy P,Q la cac toan tu chiéu tuyén tinh, bi chan trong khong gian Y, ngoai ra ta cd danh gia,
| | Q y i r = l < y , e , > P + |y,e2l< 2||y||^ llQyll^ V 2 " | | y | |
Suyra || Q|| < V2 va || P|| < 1 + V2 hon nOa dS dang tháy rang
ImP = KerQ = Y, , ImQ = KerP = Y2,
Bd tfe (4.1): A : X ^ Y la toan tur Fredholm tuyén tinh bj chan, ngoai ra Ker A = X2 ,1mA = Yi , thi han chS' A cua Atrfin X co nghich dao bi chan.
1 2n
y = CiCi + C2e2 + — J sin |t -s | y(s)
2 °
ds (4.16) ^ 2 . 1 2 .
6 day Ci = J I y(t) cost dt, C2 = - J I y(l) sint dt.
2 ^ ' 2^ '
Ngoai ra
||A-'||<co= V2 71 +(1+V2~"7c +371V2)'''
Chiing minh: De tháy X = Xi © X2 , Y = Yi © Y2 va Ker A = X2
Vdfi moi y e InA tbn tai x e X sao cho y = Ax. Tich phan lung phan la dugc.
<y,ei> == <x + x, ei> = x.ei j - < x,e2 > + < x, ei> = -x.e2 I - < x,ei> + < x,ei> = 0
0 0
2n 2jt
<y,e2> = <x + X , e2> = x.e2 I + < x,ei > + < x, e2> = x.co | - < x,e2> + < x,e->> = 0
0 0
Tu daysuy ra y e Yi (adsbygoogle = window.adsbygoogle || []).push({});
1 2 .
Ngirac lai néu y e Yi va x = J sin 11 - s | y(s) ds e X
2 °
Thi Ax = y dieu nay cd nghia la y e ImA,
suy ra Yi = ImA, ngoai ra de dang thay Yi đng va KerA = X2 la khong gian huu han
chieụ Vi
dim K3rA= Codim ImA= 2,
Suy ra Ala toan tir Fredholm lien tuc va de dang danh gia dugc || A || ^ 1. D a t X = Xi + X2 , xi = C I e i + C2 62,
1 2n
X2 = J sin 11 - s I y(s) ds, cac he so Cj C-> chon sau:
2 '
Bay gicr ta chon Ci, C2 sao cho x e Xi , ed nghia la <x,ei> = 0 ( i = 1,2)
Sau khi bién d6i ta tim dugc
1 271 Ci = - < X2, ei > = J ty (t) cost dt. Ci = - < X2, ei > = J ty (t) cost dt. 2 ^ ' 1 2n C2 - - < X2, e2 > = 2^' t y(t)sint dt
Til day suyra (4.16)
2n 2n
Tu ||xi f = C^i + C?2 ^ — {jt^ sinl^ dt + jt^eos' t dt ) | | y f ^'iiyir-
471
Taco [ | | x , r < 3 | | x , p < 2 7 r ^ | | y f Suyra |||x,||| < n V2 ||y||
Do I X2 I < I — J sin 11 - s I y(s) ds j < — ( | sin' (t - s) ds)"' ||y || = 2 "
n
Nen||x2|| < — l i y
VT
1 2n
Mat khac 1 x 2 ^ — IJ cos (t - s) y(s) ds | + — | J cos (t - s)y(s)ds | <
2 0 2 " 1 2n 1 2n "VTT 1 2n 1 2n "VTT Dodo < — j | c o s ( t - s ) | |y(s)|ds< — ( l c o s ' ( t - s ) d s ) ^ ^ ||y|| < — i | y 2 ° 2 ° 2 Iix2||^ ||y|| Tur X2 + X2 = y t a c o 7t I|x2|l^||x2||+||y||<(i + — ) | | y | |
_ 3n'
Suyra |||x2||| < (1+TT V 2 + ^^ \\y\\
2
va ll|A-'y||Nl||x||| <l||x,|||+|||x2||I <co||y|| (adsbygoogle = window.adsbygoogle || []).push({});
- - 371'
6 day CO = 7:V2 + (1+7rV2 + ) 2
B6 d& duoc chung minh[]
Dat S = S(0,R), A = K t , x ) : 0 < t < 2 7 i , Ui! < R , ( i = U ) ) Bd tfe (4.2). af _
Gia sit ham f( t, Xi, X2 ,X3) co cac dao ham rieng , i = 1,3
axi di
lifintuc v6ita[tcăt, x) e Avavoimoi (t, x) ,(t, x) e A j (t, x) | <a 3f af _ 3 _ axi I (t,x) - (t, x) I < L Z I Xj - Xj Ị Khi do toan tur
axi axi J='
F : S cr X -> Y kha vi lifin tuc, hon nira ||F'(x) || < aV 3
||F '(X) - F '(y) II ^ 3L 111 X - y III ngoai ra h = Ci e, + C. cj e X2 [QF'(x)] h = Til e, +11262 e Y2, "^ or day( TI , TI 2)'^ = (l)(x) (c, ,02^ (4.17) (j)(x) = (< Ci, (pj >)\j=, va (pi(x) = af af ei + e2 axi ax2 af af ei - e2 - axi ax2 af t i ax3