Một số các phương pháp nội suy để giải phương trình toán tử và các ứng dụng của chúng
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1
5^Y5
MUC
I. U C
PhSn loo' dSu
SxmsjL
3
Chifo^np: n 5 t : I*i§t sfi ptocme; phap n£i sijy t t ^ e n t i n h
d5 g i a i 6§ja dung phuPcmg t r i n h toan ti^
S 1
M§t vai k h a i nigm vo ty sal phan syy r^ng cho
§ 5 M§t v a i tn^cfr^'hgj? dglc MJ^t cua phop 1$P ( 2 . 9 )
§4
V5 in§t logii phuctog phq? l§p b^c cao thrf h a i
do g i S i gSn dung phifcmg t r i n h to5n ti>. . •
52
37
C^^ctofa: haj. 5 M$t v a i phi:?cmg phap ngi suy tSng qu5t
svQT T§ng d | g i a i gin dung phtfcmg trinli
tofin tiJ
g o D^t v8n etS
45
i 1 M§t v a i kiaai nlga ve cSng thrJo n§i suy Tfiu-tcm
si^y rOng
i^
§2
"VS ffiOt v a i pbucng phSp n§i s^y tong quat s i ^
rgng g i a i gSn dung phu^c?ng t r i n h toan tiJ va
si; h $ l t v cua n6
. • • •
Gfaifgya;;!^ l?a
t Ph8n iJng d^ng
i 1 Mgt s5 T^ag d\wig cho 3191 v a i to^n ti> cy thS
1.1 Phi?c?n6 t r i n h haci thi/c vo^ bi§n s8 thi;c
1^2 H$ phtfc^ng t r i n h dgii sfi p h i tuySn hc$c
siSu v l ^ t
1.5
i 2
^
Phi^c^ng t r i n h t i c h phan p h i txo^Sn . . .
CSc v i d\i bang 06
61
61
62
G5
68
Tai l l § u t r i c b dan
70
o£o
-
p
-
T l f c i5ng dyng g i a i t i c h ham ^to nghien C'\i c i c ph'.^'ng
phap g i a i gan dung cac l o p i phi?o'nc t r i n h n g t c5ch tong q u i t
da dem iQi n h i e u k e t qua v6 cung quan trgni^. '2u» ti^o'nt; cua
g i a i t i c h ham khong nhu^ng cho phep dcn [^Inn hoe each n h i n
cho n h i e u phUHo'ng phfip khac nhau ma eon giup t a ki^m t o a n
m i o u GO' do t i n h t o a n t r o n g nhiou l i n h vy^c khoa hgc k h a c nhau,
•vf dy nhu^ dpi s5 t^iySn t i n h , phu'C^ng t r i n h v i p h a n , phutmg
irinh t l c h p h a n , g i a i t l c h p h i tuyen v . v . . . *
TThieu l o ^ i h a i toan k h a c nhau oou eo che v i 3 t du&i
l?ng chiing
(1)
Ax
=
0,
t r o n g do A l u t o a n t u xac d^nh t r o n g ngz khong g i a n noo do
va CO g i a t r t t r o n g mgt khong i;ian cung l o ^ i .
DS g i a i gan dung phu^c-rig' t r i n h ( 1 ) ngi;?o'i t a dung n h i e u
phi?o'ng phap k h £ c nhau. L.oi phv.Xi^r^'; phap dou oo nhu'ng 'J*ii (Tien
va nhu-ng hgin che n h a t d i n h . 2n?o'ng hg»p A l a t o a n tu' tuy3n
t i n h , cac k e t quo ve g i a i ;;Sn dmux, da duvc :
trj'c^ng hg^ r i e n g , chang hgn l o p toon tu t h a t , loip t o a n tu
(to»n dipu v.^-". ••
Trong nhiJng nam gan dixy nhiou cuSn each chuyen khao
da do cfp o ^ t each dang kt t o i nh5ng phuxmg phSp tong (luat,
Chang h?n, c5c t a i lifni cua Gang to r6 v l c h / " ^ t ^ - . ? *
®latx
^ 1 5 7 > Gratnoxenxki / f I G j , Otcga F"Mj^
Moctmic / " I B J T va
n h i e u cong t r i n h (Tu^g^c a>ng b6 tror_g cue t^^ c h i to.4n hgc k h a c
nhau / " J - 1 ^ 7 , / f 2 1 - > 2 7 .
Trong ban l u y n vun nay chung t o i se t r i n h bay mgt s6
k e t qua t h u g e l i n h vi/c d o .
Ban lu§.n van nay gom ba chu'o'^
-
;'
-
Chu-^ng 1 t r i n h bay mgt Q6 cac phiAD'ng phap ngi cuy
tuyen tfnh dS g l M gSn d'ms phuv^ng t r i n h t o a n ti ( 1 ) . Thi/c
r a phuang t r i n h ( 1 ) da '*u^c nhiou t ^ c g i o nghion oS\ \ni <^xx^n
ra nhieu phutJ'ng phop k h a c nhau nhu^' phuvng ph%> Niuto^n C^^'^J*
p h w n g phap S t e f f e n x e n - Aitt:en £ ^'y^ ^'^ J ^ phu\>'nu: phap l^p
clip k tong q u a t (phut^ng phap Sruede - Tsobu'sep) /f 1 5 ; 27 J - • • y
t r o n g s6 do phuang phap r i u t c n - ' - i n g t o r o v l c h t-uy di.;vo su'
ct\mg i^ng r o i , nh>mg nhuyc diem chinh l a t r o n g nhiSu tru?o»ng
hgj) v i t ; c xac ^ n h toi'n ti> ^ A x J " ^ (d^o ham theo P r e s o ) thu^'n^i
^gp kho khan v-o ph'^e t ^ p . ITgoai ra t r o n g n g t G6 tr-fo'ng hgp
Cvl dy k h i t^iei c
mo chi b i S t cTuvc t h u 0 t t o a n t r ? n nay t l n l i d i f n tu' vpn nang
de xac cT4.nh g i a tr% Ax^^ va v^. th3 khong t h e op dyn^j dug'c
phv?o^ phap TUu-to'n.
Di;'a vao k h a i niv'ia t y sax phan stjy rQng cho to5n ti
t r o n g nhu'ng nhjsi gan day cac con^ tA.nh cua / " 1 1 - 1 3 ; 2 2 - 2 3 7
da l a n lirg^t xuSt hii^n nh?aa k h l e phyc nhu^ng nhu^c d-icm t r e n ,
nhfj?r4g -oSc ilg hgi t y cua cac phu'r;ng phap do chu*o vxf.gt qua t S c
d§ hgi t y cua phuc?ng phap T'iiT-tCn.
l ^ n g chuttog nkj chx'jig t o i nSu ra mgt B6 phuNj'ng phap
CO t i n h chSt n g i suy tuyen t i n h . Cac phii'O^ng phop nay khong
nhu'ng t h u g t t o a n do?n g i a n ma t 5 c dg hgi t-y l ^ i c a o , chung t o
r a dgc bi^^t t i g n I g l t r o n g truVng hg^) t o a n tu' A co dsua haa
phiJc tjip hogic khong kha v l .
Chi?c^ng h a i t r i n h bSy mgt s6 cac phut)*nc; phap n g i Bvxy
t 6 n g q u a t suy rgng cho toon t&* PhSn d2u t i e n cua chu'c^ng nay
l a dUB TV. coc k h S i n i p n ve 1^ s a i p h a n suy rgng 'oong q u a t cho
t o i^n tl? va cac cong thi^c ngi sy^^ U i u - t e n tong q u a t S"uy rgng
cho toSn tu' t r e n khong g i a n Banach. PhSn t i e p theo l a x£y dijPng
x^t Q6 phut^'ng phap ao nh5ng phuang phap nay bao gSn ragt so
5 k e t qiTO c^'ia ch'.;x»'ni3 I va cua coo t o e g i a k h a c .
L:5i phuang phap 6" cb^Mn^ T cTuig nhi," & c h w n g TT --^eu
dxjt^o khao B:OZ qtia cac phSn 5
- ilgi dung phittrng phap
- S^ hgi t y
- Gong th-^c danh g i a s a i s6
(Xem c5e c^nh l y : 2 . 1 ; :i:.2; 2.35 2.45 2 . 2 ' t 2 . 5 ' ; 2 . 2 " ;
§2
g 4
chux^ng I , 5 . 1 ; 5 - 2 ; 5-;^; 3«4; 5 o
chmng I
2 . 1 ; 2.2; 2.5
0 2
S 5
2.5"
cht:c'ng T, 4 . 1
chi.?o'r^' I T ) .
ChL?o'*ng ba (phan iJng dyng)
khao c a t ngt; oS v'nx: ^yng cua cac pht!t?ng phnp o' di^x^nz 1 va
chi't5»ng TI cho m^t so to5n ti*? cy. t h e ; ITeu l e n t h u g t t o a n , d i e u
k i ^ n hgi t \ i , cong thu'c canh g i a s a i so cho t o a n tu' dang khao
s a t C-^o^ cac d^nh l y t L i ; 1.2
§ 1 chuWo^^ b a ) .
Phan cu^i cung ci^o ch*:'c?ng nay l a nigl v a i v l dy bang
28 do k i e n nghii^m cac phuc?ng phap neu r a .
C/Xic k e t qua ohlnh. c»^a Ir^/n v l n dn dirg'c r a i r o c boo cao
t r o n g cac b u o i sinh h c p t x6-iai-na "Cac phuti*ng ph&p gl.al gan
dung ^.ihu^i'ng t r i n h toan ti" vh nhu*ng ftxg dyng cu.a chnng" cua
to bO ^5n 'Doan hgc vinh t o a n , khoa 'Joan ir.>?o'ng Bjii h g c l o n g
hfi'p "^a-ngi l';?76 va ia$t oh'&n da dug^c cong b6 t r o n g / " 5 5 - 5 5 7 »
Tac g i a x i n chan thanh can o'n dor^g chi -"oang dll'c ITgijye]
da dgc ky lu'^ng ban tiiao va cho n h i e u y k i e n dong gop q u i ban
Dgic bi^^t, t a c g i a x i n chan thanli c^:a o'n d3ng chi Phan van ^^p
da da x u a t ^phucfxux; hu'o'ng, da cho n h i e u y k i ^ n dong gop q u i b a
t r o n g c5 qua t r i n h hoan thanh ban l u g n v2.n n a y .
" 6
Cn'^^^n^-^; I
IE Cg-AT GAN DTTITG PHHONG 'H^lTii lOAN l^T
PJxao s a t phi?o»ng tri.nh to5n t\^ (1) •
Ax
^
0,
t r o n g ro A l a toan tu" p h i IrL^on anh x^ ingt inien l o i ^TL
nho
do c\ia khong g i a n Banaeh X vao n g t khong g i a n cung l o ^ i
i.
Gia t h i e t r a n g A kha v i th^»o P'rOsG va TL lo nghicm
dune cua ( 1 ) t r o n g l a n ojn iTIi .
§ ^ • ^'-9t v a i k h a i n i g n ve t y s a l phC^n suy r ^ n g ^'-]0
t o a n tu" ( : ^ m ^ 5 l 7 ) * ^
Xot n g t ham tr\i
t':Vng Ax, han nhy chTij^'n khong g i a n
d5.n!i chuan X vao khong g i a n dj.nh ehu*n ! . ,.-^,-"
:tiu^T^''*^
Hi hi^u khong gi-.n cna riii?ng toan tiFVx v a o r
fX-^
I J.
Thanh Igp khor^- g i a n t l c h
E - == ^ ' ^
iZv
*"
X . A.. A ,
:
( t l c h 3:^-Cac)
. . . ,
t » -,
t r o n o i^o k l a n $ t so tr/ nhien.
~
la
Phan tS ciia B2 d-j'ge v:5.et di:'ol d^^ng (x^,, x-> TOI X^. ^
*
X,
( i " 0 , 1 ) . iltWig ty:, phsn ti> cua B^.^ di^vc -^rio-i; dL^'c'l djing
(XQ,
XT_,...,
Gia ^
:5C^„-()»
"^^ ^ 1 ^ ^ " '
^^ --^ c T l ^ ) .
tBn t f i ngu h^a tr*a: ui^ijr'ng A(x^, :^:^), han nay
chuyen nhiJng ph3n tu' cua khong g i a n Z-, vho nhSn;: phan txf cua
khong g i a n f x - ^ l j ,
tron^
Co >^ e X,
(i ^ 0 , 1 ) .
D6i v c i nhu'ng -xyYian x,-^^ x^ , ( i = 0 , 1 ) ,
o5 d3.rih tn?ng X
t h i A(XQ, 0:^) l a ni$t t o a n tu tuySn t i n h .
OJoan tu' t-uyen txnh A(::^, oi-j ) thoa man d i e u k i p n
(1.1)
AC-,, x^) ( x . - 2 „ ) =
A::^ - A ^
,
difg'e Qg± l a ty s a i ph;'i^ui bye nhSt ei\a h-an tru'^u tu'o'ng Ax l 3 y
^ t ? i cac phSn tt> x^ e X,
(i = 0,1).
Gia BU ton t ? i n g t ^an trv^ni tu'g'ng A(x^, x^ , 00,)
ham
nay chuySn nhJng phSn tV cua khong i-:;3.an Sx vao nhSng phnn
cua khong g i a n ^ X ^
f^
-^Ij
7, tronu
do r^ e X,
tu
( i = oT^
j o i v(?i c5c phSn ti: x ^ , ( i = 0 , 2 ) o5 ^^nh t^x^ng X t h i
A(x , X | , Xp) l a n g t t o a n tu LzonQ tuySn l a n h .
Toan tl? so.ng- tuy-Sn t i n h
ACX^, :20: , s>^) t h o a nan *iou
kign ;
(1.2)
ACx^, x ^ , x^) (x;^ - 3ii) ^ A ( X Q , X;^) - A ( X Q , X . ) ,
di^c ^ i
l a t y s a i phan b-v^c h a i cua han trOYi tJ'g'ng Ax i S y
cac phSn tir x^ ^ X,
( i = 0,2 ) .
tgi
B^a vao ( 1 . 0 ,
t'^ ('1.2) (cau k ^ i •-?, d-g^c -coc dyng ^:ji
vS l e n (:<>, - ^Q^ ^ "^^ -^-^^7 ''^^ •
(1.2)'
A(XQ, x p
x - ) (x^ - x ^ J
(:o. - ::^) =
= Ax- - A:.:^ - - A ( x ^ , :.-) ( x . - : : . J .
"^•^oon toon tux^'ng ti; t a co zho d^nh nghia t y s a i ph-'ln b.;.c k :
( 1 . 5)
A(x^,
Xp . . . ,
Xv_)
( x i . - X|. _,^ )
^
= A ( X Q , X-, , . . • , ^}-_p » --1-) " •^- "^^"^Q' 'H ' • * • * "V-1
Feu A t o n t g i cac r^go han l i u n t y e don bgc 1: ( t h o o r--eae) t h i
cac t y col phan xac d^n^ nho' coc t i c h phan 'Urjan tru^i tii'g'ng,
t o CO 'fc^c lu^j'ng ;
(1.4)
(I A(x^, : c . ) | U
0
^
V'^J
6 ^
»
^ -
:.^ . ^
(x.
-:c„),
1
|A(.:^, ::,, x.)|| ^ (-^) II A \ | 1 .
O
i.
(k)
A ( ^ . ^^1
^ ) l l ^ C-jV)
' ^ l = ' ^ - 1 -^ ® k - 1 ^ r ^lc~1^'
II .^fZi^ll
,
0 - ^ ©0' ^ l " - - ' ^1:_1 ^
IWo'ns t y nhu' tnPtfns- hi55> Han th6nG thuc'nc, co t h '
1-
chi?no .-rdnb
5uvc rane» t y s a i phan b.Jc k cua ham tr>ii tuvnjj Ax l a ham
-^«
<^
-9 De den gian, t^:- day vo sau se ky hi^^u :
^01...k
^
''^^^o' ^^•••» -^1:^
^Jo1...k = ^^^' ^^o' ^^l'---' ^1:^
§ 2 . Ve mot ohUD^'A ohao 1 5 P bg.c k do f:±a± ;-an dnj\p;
Phu^tyng t r i n h toan ti? Cl).
Xu3t phat tiJP gia t r i ban dau yP du gan x^, t a x§y di/ng
c§c phSn t i theo each sau s
(2.1)
z^ = 4
^ fAA^l,
4
= 2:5 - x °
x° := x ° ,
0
<:M.
4=
24_^
-xj_2.
-^1.
Tu' (2.1) suy ra :
...,
(4-^0)-
JL-^(,o_^^).
T-JP (2.2) suy ra :
(2.3)
A(xg, x ^ , . . . , ^^) ( : ^ » x j ) ( x | - ^ ) . , ,
:= ( ^ " '^ ),!
,k-l
t r o n s d<$
ACT^O
^ ^ o »
O
ON
^ ' • • • '
^''
^^JO
( x j - ^^^^) =
O >^
<^2&^C " ^ ^
'
( 4 - x°)^:= (:^ - x ° ) . . . ( ^ - xgj
k lan
Dya vao djnh nghia ty s a i phan suy i^ng bf.c k cho toan tu"
( ( 1 . 1 ) - (1.3) ) vo dieu k i f n ( 2 . 2 ) , ( 2 . 3 ) ta suy ra :
- 10 k
(2.^) A 4 - A x ° - T , , ^ ( x ° - x ° ) = - ^ . . o 1 , . . . k ^ ( 4 -x°y
,
trong do
Aol^.-.k^ ^
(2.4)'
A (x^, X? , . . . , :^)
I,,^ = I? ^-'^' 4-1 "-^ ^-^^0^
(^
=
k(k-l)... (k-i M )
i!
L'^t khac, cung d^a vao dj.nh nGhia cua ty s a i phan suy r9ns
b | c k dio toan ti? va dieu ki^>n ( 2 . 2 ) t a I p i co :
(2.5)
Ax - /.x° - l b , k (x - x°) - 'lx° (x - x ° ) g ° =
X
=
AZIQSQ
. . . ( k - D ^ (x - x*^) (x - - i ^ ) . - . (x - 2g_.^)
trong dc5
k-S
'te° = Z (-1)^ cf. o Aoi„(if-l),, Ho'
'O
i=1
+
k-3
>• (-1)^ C^_, A0i„(i+1)„ ( i ^ 2 ) ^ ( ^ - ^ % ) +
i=1
+ . . . f-
A01Q
.. (k-D^ ( x - x ? ) . . .
(x-sg_.^).
Ti> ( 2 . 4 ) va ( 2 . 3 ) t a suy r a rang vo'i \ / x € i Q g ^ C i i ^ ^
trcng fid 5
^ g j = i X : l | ' £ x ° ( x - x ^ ) ^ ° | / ^ |)A2t>1„...(k-1)J( n ^ ))x-x°|| f
- 11 -
k
(ic) B§t dang thi5c t r o n g LQ* co nang t i n h ch3-c d}.nh t i n h n h i e u
hd»n l a djnh lijg'ng. VSn do d-^t ro lt>, co t^n t ? l cae phSn
tii
X ^ u 2J gS
, sao cho thoo n a n ( 2 . 6 ) khon^:^* ? 'Tay
n o i each k h a c can chgn [fh^o
l a k h a c t r S n g . l l i g t vgy, do
nhnn du'g^c ( 2 . 6 ) t a cSn cd :
(1*) 5-x°(x-x°)&° = - ^ ^ ^ A ^ l . . . ( k - l ) „ ( x - x ° ) "
k
k-1
- Axc1^...(k-1)^ n
(x - x p
0
C x-O
-
^
Ta thSy (1^) se dung k h i x =- x° va x - r^. (di^a vao ( 2 . 5 )
va each d g t 6 ^ * -^^ " ^k ^'^ ^^^'•^* "^-^^ "^"^ '^''"^^ b*ng vo p h '
v s bang k h S n g ) .
Khi X ?^ x° va X j^ x ^ , d^-a vao each d g t
£°,
t a nhgn ;
(2*)
q.&f
^ Cj^-l^fc "^ ^ - - - ^ ^ ^ x > Co = 0 .
Heu c o i i^x ^^ ^^ ^^9 <^^ *
°o = - - % i f ^ Axo1,...(k-l)^ ( 4 - X^)^ .
JC
.• Axo1^...(k-l)^ jT (.4 - 4 ) ,
k-2
i=1
(Tlop trang 12)
tir ( l * :
- 12 t o 00 :
( 2 . 6 ) Ax - Ax^ - To,k ( x - x ^ ) = " ^ — ^ Ax^1 . . . (k-^ )^ Cx-x^) .
XSp x i Ax bang da th'?c taiyen t i n h <:io(x) :
( 2 . 7 ) Ax ^%M
= Ax^ ^• To,k i^ - :^^)
-"•
X
—
*
1
Gia su QoC^*) * 0 "^'^ t o a n t;' nghich dao TQ I; t 2 n t g i , t^j ( 2 . 7
Bvy r a t
1 - 0
-x
-
y:
„~^
. o
- 'j<5^1: Ax
long q n a t , neu t o d g t 2
xg := --c^,
Tigp (±)
n = 0,1,
.(^^-x^)(^^-:^^^^)....^- Ao1^...(k-l)^(4-x^)(x^-x|),
k-1
...
(:4"^k-1^
i-...^r A x o 1 ^ . . . ( k - l ) Q
;
irZ
-:^),
. . . .
t h i ( 2 * ) se l a mgt phitVng t r i n h d g i s8 bC^c k .
Vol
i^g^du b e , t a cc5 t h o t h a y ( 2 ^ ) bang phu'cng t r i n h
t r o n g do 0^ nhgn d^rg'^ tu G^ bung each thay cac s8 hgng
CO chu^a Axo1-Q . . . (k-1
o lI -Q, . . . k^
k^.
( k - D)_^ bar^b?ir^- A O
Tic (5^) t a se x a c djnh duVc cZ
.
- 15 t h i l y l u g n hoan t o a n tit^ng tg' nhti* t r e n t o GO C6 :
(2.8) A x - A x " - T n , k (x-x")-"'
^\^^'
ir,A
k--^
A x n 1 „ . . . (k-1 )^^'^'""
n
n
t r o n g do :
a'n,k
=
/, ( - 1 ) ^ G^ ^ An ( k - i ) n
i=1
^"^
iixnl ^ . . . (k-1 )ji = A(x, Xo^ , x-f , . . . , :xg_^ ) .
Tl) ( 2 . 8 ) sijy ra qna t r i n h Igp :
( 2 . 9 ) :^'^^ - x^^ - Tn,], Ax^^ ,
n - 0,1,...
S\? h g i t y ciis qua trin^i Igp ( 2 . S ) ^u<^c t h e hiC^n o^ cac d^nh
l y dUol day :
JtiO do 1
Gila su* toSn t^jr A ton t g i dgo ham don cap k , Ichl do t a
CO bSt dh.n^j thu*c t
(2.11)
\ Tn.kCx^- ^ ; ^
t r o n g d6 ^
(-^,
. 5 , ^TT Ax-^ (X-"- -- ^ ) ^ II ^
st5> Ii A ( > . t(x^^- >:^) i . 5 ^ II % * | | ) II x " - x*|l
o^t^l'^
duVc xoc d^nb ti^'o'nG ti;' nhir ^
tronij ( l . ' i - ) .
i^p dyng cone; thu'c 'lay-lo co p h a n dir cho t o a n tiV £"1v_7
(2.12)
Ax^ = A(x* + x " - x^) =
= Ax* ^ ^ iL^(x-^ _ ^f
1=1 i-
, ^(^^^ ^ ) ,
- 14
t r o n g d6
(2.12)' |hv(x*, x-")!! 4 ' - ^
=^^ M A t(x^-x^)l""^
•
04.t£:1
•Ja (2.12) ta suy ra :
k-1
(i)
(2.15) Ax^ - >; - f r A^ (.^ - ^^y
i=1
l'-
= \
( 3 ^ , x^-).
^
COns tCrng ve ( 2 . 8 ) ( s a u k h i da thay x = x^) cho ( 2 . 1 3 )
(chu
y ( 2 . 1 2 ) ' ) t a nh0n difn'c :
(2.14) I K (X--X*) - ^
-J^ AJ? (x^ - x^)' II 4
1=^1
- '
k
/ _L«
3^p
} A xi^+ t ( x - x"^) n i»
0 ; ^ t <;1
, , _ ( ^ : 1 L - |lAx^n1....(k-l)Jlll^'-^""
Sau k h i ap dyng ( 1 . 4 ) dS danh \~Lk |]A3i*^n1,^ . . .
( k - D j ^ 1(
tl? ( 2 . 1 4 ) suy r a ( 2 . 1 1 ) .
a i e u p h a i chi^n^j a i n h .
r4nh l y
2^
Keu xap XI ban '^au x° thoa niin cac d i e u ki^in :
1 ^ / l o a n tu* AO1Q t o n t a i
toan
uJ nijhjch dao Aol^^ va
IIAOIOII 4 Boj
2°/
||Ao(k-i)^|i
<. L^,
(^i = 1 , k - 2 ) ;
3°/
Cac hans s6 Bo, L,, th.oa nlin b a t dSnfj thifc
- 15 k-2
i=-
-1
t h i t8n t^i toan ti> nghich dao To.it va
(2.15)
K]J|
< - ^
^o
Chimg ninh
TU* ( 2 . 4 ) ' Guy ra :
V—-^
1=1
= (-1)
k-1
iAo^
^ - ^ , ,i-(v-0
,
r L (-1)
C^^-^ Ao(k-l)Q
i=1
1^ _r>
k-1
( 2 . 1 6 ) ai,,ic = C - ^ r "
.
-
V- r ^ r ^
Ao1^(I•^Ao1o ./^^-'^^
'^r.i
^ ^
Ao(l^-i)
t r o n g do I l a t o a n tt> den v ^ .
The nhu^ng, di;a \^ao c5c g i a t h i o t c^ja dxlrin Tj ta co
11 A?!,"? (-O'"^^-"^ 4.^ Ao(^-i)o
"
i=1
Dya vao d5.nh l y Banach, tu' uSt dang thu^c cu.6i cun;j suy ra
t o n t ? i t o a n ti? nghich dao :
.
- 1 ^Zf
i-(k-1)
.
^
v-1
( l + A0I3 £. ( - 1 )
-i^^T -'io(i^-^)J)
^
V'^
-1 ^ J ^
l-(k-1)
.
-1,1
( | ( l + A o 1 o } _ ' (-1)
t.5_^ A o ( k - i ) o ) \\ ^
^
- 1 -
•
- IG -1
Do do t o a n ti
.1 -1
nghi eh ^^^o l-o,!: ton t ? i va
l|l'o,k \\ 4:
- ^
•^-^io
Die^a p h a i chu'ng n i n h .
Gia su' :
1 ° / ifin t j i toan f'^ nGhich dao in,!:: va 1| % , ! : |l 4 B;
2''/ l u i ^ ^ l U ^
^k '
x"€..^
v.'l
Vn.
Lhl do qu5 t r i n h Igp ( 2 . y ) GO !^;i "cy t o i n-ghl^^n x"^ cri.a (1 )
l o e dg hgi ty d-u<;^c :-*anh .gi.a bang b S t dang thu'c :
1 K
^'^ - =--* I U
i
f. II '^^- - ^*
ii
il"
t r o n g do
^'
Ch'nmg minh
-^ d;ing cor.£: th'j»c l^ay-lo co 'phTm du cho t o ' n tu t a co
(2.17)
A xn'
_
=
, / _ i s. . n
A(x'^
X
-
__±:
X
y
=
k-1
/ .
1=1
*"7^ ^ x ^
Uv
- X ^
-I- i,„ t x
^•
t r o n g do
(2.18)
||r^^(x*, x^) II 4 - L - i,^ II x^ _ ^ f
I^,
=
cup IIA O^ . t (X^ - X^^^) II
o^t<1
IVJ' ca h a i ve ( 2 . 9 ) cho ^
t a di;Vc s
,
, X ;,
(2.19)
^^'^-^^J'-^-C^i^^
;
nV^l^/j^l
Thay (2.17) vao (2.19) ta nh^n :
k_-1
x ^ ^ ' ' - X* I k II Tn'k 1! l l ' I n , k ( x ^ - ^ ) - i : - ^ 7 A x - ( ^ - ^ ) ' l l +
i=1 1 '
-1
* II I^ik II II \ ( ^ . ^ ) I) •
Dya vao cac dieu kipn ciia d^nh l y va c5c bSt dang thu*c ( 2 . 1 8 ) ,
( 2 . 1 1 ) , tii* b a t dong thu'c cu5i cung suy ra dieu p h a i c^iihig
minh. I^u khdng gi5 t h i e t trx^&o ve ay.' tSn t ^ i n g h i ^
cua
phi?cmg t r i n h ( 1 ) , ta co cac dinh l y sau day «
Dinh l y 2 . ?
Gih sv? X thoa nan cac dieu kiyn :
1^/ Toan tiJ To,k tSn t g i toan tu' nghidi dao
-1
va
fJTo,k ll
2^/
-4
To,k
Bo ;
\\Ak
iLcL^ 4 I^i
f
(i = ^ ) »
trong do l a n cgn liTj ^ duyc >:ac dinh ti^ ( 2 . 2 4 ) .
3°/ II To"'k Ax°|U >'^, )i:^.i - ^ _ J U >o' 4^/ Cac hang s8
(2.21)
/ ^ , L^ th6a man :
k-2
0 ^ p^ := 1 . ( T o ^ r ,
G^.^ T], ) <
(2.22)
O^q^
=
Bo,
^ - <
1,
It
- o
t r o n g do / v dt?g»c xac djnh tiV nghipn d:i»cmg cua phi?o'ng t r i n h :
- 18 k-2
(2.23)
(1V&i
' ^ (1 . • : ^ ^ ^ ) )A
7^
=
v i
J^.k
- r i , k A - Y^^j, = o.
2Bo T2 '^ 0 <. 1
^-k
v'''
- 2l^-1i,k 3^k-^
k-1 '
=_,
2,1,
;L_____
i F ^ - 1
j^k
Khi do t r o n g i S n oan :
( 2 . 2 4 ) u2.^
^
=
X :
k
[jx ~ x ^ | | : ^
C
'
"^
^-Po^o
^
phiJo^ng t r i n h (1) sa co n^'^hifxi J:* va qua t r i n h l^p ( 2 . 9 ) se
hgi t y t(H n g h i ^ d o . 'i6c dO h^i t;i difg'c danh s i ^ t a n g b 5 t
dang thi?c 8
k^-:^i
n-1
( 2 . 2 5 ) IJ x ^ -x*^ij
^~'^
^
W^
k - 1
^ - Po%
^o •
'-'
Chfeg minh
Chu'^ minh rang k h i chuyen ti? ph&n ti5= x*^ song x
thi
cac d i e u k i g n cua dinh l y v^m cT>-?g'c bac t o a n . Do dang a-^y r o
r a n g cac ph§n tfr s ^ ,
x\
^ LT/^*
( i - o,k-*l).
Kiem t r a d i e u k i e n 1 .
^
^
^
^
-1
l'rL?o'c t i o n t a can ch'Jng n i n h t e n t ^ l t o a n tu* n g h i c h dao T^ ^
Thft v g y , x e t t
—1
(2.26)
||TO,V
(To,:.c-2:i,k) II 4
—1
II To^k |l II I o , k " Ti ,3, ([ =
- 19 k-1
-1
To,k
2Z (-1) i
1=1
,.i
^ . - 1 (A
^^•o(k-i),
- A^ci
Kk-i)^);:
=
k-1
''^x4li^,'^-'^ ^-1':^o(k-i)^-A(k-i)„i "
+ ^(k-i)Qi --^Kk-i)^ ) 11 ^
k-1
^
2 Be 1:2 • o (
(^ .
i=1
)
=
1
-Pc-..l-
'heo dinh l y Banach, ti? ( 2 . 2 6 ) suy ra t 8 n t ^
"1
dao r i - ^o^i; (Tc,k - T i , k ) 7 ''=
^^r^t
t r o n g do
I
t o a n tir n g h i c h
l a toan tu" dc^'n v i va
—J-
Vgy t o a n tu'
T-^ ^j^ =
('I?o,k % )
Bo
II
_ _
-1
t o n t g i va
. 0 ,
Dieu kiOn 1 da kiem t r a xong.
KiSm t r a d i § u k i p n 3«
CSn dsnh g i a fl C, ,]r Ax^||
V ^ n = 0 va X = X , d'^a \'ao ( 2 . 9 ) , tt? ( 2 . 8 ) t a ST:y ra :
Ax
Do d6
•^
^ - ^ ^Ic ^
-^
o^
Ck
o
- 20
(2.27)
j j < i , Ax-^ll
-
<, - ^ \ ^ ,
k>1
^^1^'
-o
Qua ngt vai phep biSn doi dc?n gian, t'J* (2.27) si:ty ra !
(2.28) K ; , , Ax^ii 4 ?;^(A,^^
^
(
1,k
^ ^2,k A
'1.k :o>
) Po
.0
0
o''
"o^
.
De dang chi?ng lainh du^c ??lnn;t nsu / \ l a n g h i ^ ciia phtJWng
trinii (2.23) t h i
A= ('^'^ ^j, ^ ^\^-^
y\""^) - V
Do do t'Jf (2.28) suy ra :
(^.29)
ll ^ \ Ax^ ^^ X, ^ lY^ .r^
=
\
Dieu kign 5 da diro'c chu'ng minh xong.
Cung t'JP (2.29) ta 3uy ra :
(2.50)
f^
:-
2B^ L^
^
=
~»ifi--'*o
Kiem t r a dieu kipn 4.
TiJP (2.50) suy ra
f^
^i^C'o^
k-2
Pi = i - ( i - J;^ C ^ )
•!
. 1,
^1 ^1 " ^ *'l "^^ ^o - ^^
W-Ou k i g n 4 da dxfg'c ki&a t r a xon^;.
KiSm t r a dieu k i ? n 2 .
De kl^m t r a diou klgn nay, ta chi can ch'ing minh nlng
iQ 2
C- 3>--->|» trong do «
- 21
»a2=
-
, ^ • .i^-^'Mi ^ — - —
^
1 i
'
1 -Piq^
LSy mOt phSn t5 bSt ky x* e
g*
^^ ^^* ^ ^
^^^'^ ^^^
^^^
va cac dieu ki^n 5, 4, cua dinh l y ta suy ra :
llx' -x^W
^ l l x ' -x'^ll
+ 11 x'^ - x ^ l i ^
>i
—^—
''. + '
>
0
Dieu klgn 2 d i dxJg'c kiem t r a 2X>ng.
Th;/c hifn phep ch^Jng minh hoan toan ti?o'*ng tg?, t a co
the Chiang minh di?g'c cho tr^.?o»ns hgp tSng quat khi chi:Qren ti?
phSn t^ x^"" sang phSn tiJ x^. Tif phep chiJng minh do ta xac
dinh ^<^c cac d^i li?g»ng sau day :
(2.31)
^
^^-52)
^ n
'n
= Al2i£L
1^0
= A U-T^ ""n-l
TU- (2.31) va (2.32) ta
B ^ vi
ra :
n-1
k-1
BVQT
jjx^^''- x ^ J U r ^ ,
non
+...+
n-1
^ 11 x^*-"- x « | | ^ ZZ
j=o
^n.J ^ P ? " ' -lo ^ ^
^o'
•^"1
^ ~
Khi V -^
'
(P
q
- 22
-
n-i
k - 1
, i
^-'-
)
•
t^j? b S t dang thu'c cu6i cung suy r a b S t dang tht5»c
(2.25).
TSn de con I g i l a c§n chiJng mirJi x * Iv^ nghifm cua
phifCng t r i n h ( 1 ) .
a g t v ^ y , tu* ( 2 . 9 ) t a syy r a :
AX-
-
T , , ^ ( x - ^ ^ - x^) •
Do do
(2.54)
IJAx^/k illn,kli
li^''-^l!
^
Ii2n,k'
Dv'a vao cac d i e u k i ^ n cua dinh l y suy r a
"n\\Tj^ ^\{
l a mOt dgi li^g'ng g l ^ l n g i , do do t
0
=
lim
n—»
Ax^
=
Ax*
:JO
Binh l y da du'o'c ch'?ng minh*
Gia siV x*^ thoa man cac d i e u k i f n cua dinh l y 2 ^ 3 , ngoai
r a con thoa man thom :
Bjj.<,B
v^
Vnf
. k-2
(i-i>««ao) ^
J^—
\ *'"
~i
- 25 Ehi do trong l a n c?.n J
se cht?a duy nhSt m§t n g h l ^ a r cua p h i r c ^ t r i n h (1)*
Bang phifc^ng phap phan chi'hag, t a gia su* rang trong l a n
c§n , ^^ ngoai x , ph:/o*ng t r i n h (1) eon ton t$± mgt nghigm
khac x*^. Can c^'5'ng inin^ rang x ^ trTing vt3l xT.
TV ( 2 . 9 ) ta suy ra 5
Ax^^^ T^^^ ( x ^ ^ ' - x ^ )
-0
Do do :
(2.59)
It.k^"''
^t
-
%,i,x--Ax-
%,k(^''-^^)-
Dypa vao (2.59) va
Ax** = C
(vi x** l a n^+nigm cua phifcug
t s i n h (1) theo g i a t h i s t ) , ta suy r a :
%,kCx-"'--'^)=
=
AJ^
V k = ^ - ^ - ' - ^ . k ^ ^ ^ '
- Ax- - T^^^ i ^
- X-).
Do do :
(2.40)
y^^^^
X** =
iQl^ fA:^
. Ax^ - T^^^ (x*^ . :,^) J
M|lt k h i c , d\fQ vao c6ng th'^c Tay-lo cd phSn diP cho
toan ti5 t s oc5 :
*^ =^
(2.41)
Ax-
=
A (X** > X- - z^)
=
=
A x ^ . A'x** (X- •- x ^ )
. 1^2 ( ^ ^ * ) '
trong do :
||T^2(x- x**^) |( ^
sup
o^t4l
11 A"x**. t ( x - - x ^ ) i l 11 X - . ^ * 11'
Tt:^ ( 2 . 4 1 ) suy ra :
A x ^ - Ax^' .
A'X**X=
"
^2^^
- ^ ^
Do do :
(2.41)»
ii A x ^ - Ax^ ^ ..\'x^ ( x ^ sup
I A".:**
X t(x^-x^)il
llx^
O^ti^l
Dya vao ( 2 . 1 1 ) va ( 2 . 4 1 ) ' , t'> (2.i^)
^"^-
y ^ H l
suy ra :
E^^ic : l | A x ^ - i ^ x - - a ^ , v ( x ^ - X-) ii
< ll < k !i ^ K,k^^- ^ > - i=1
-^
k-1
+
i=2
(i)
l ^ | j / ^ > ) | i|x--x^f'4.
^
^ ^ ^^- ^ > ' ^' -
j[Ax**-A.:-i-A'x*=^ ( X - - X * * )
.
>A,
- f r 11 A ^ i i
li^- ^ l i
ljx--x**illjix--x**i
"^^
3UP
k-1
ii A V - > - t ( x - - x ^ )
- 25 k-1
4
^[^^•^hn^-'^lX^*^,-^^"^-^"
i-1
Do do :
(2.41)";; x-^^- x ^ il .^ B ( - ^ . - - i ) L, ll X-- x**!!''"' -H
k
lc*
+•
- j T ^ i : ^ - ^ ^ ; . ^ ^
i=2
L2jjx--x**/M X
-•
X jj X^ - X**|j
Dya vao t,l3 t ' - i l t cua dinh l y , t*' ( 2.41) " suy r-a :
f|:^' - ^ 1 !
4 ^oii X
~
,*>.
u
i
k-1
|:.^-x"|i,B
(^*-1j)T,^,*^
0
(^-Po'^o''
k-1
,
r
i-1
! ^ i 'O
••.1-IPc^). '
(l-""''-''^l
X =
,x°-
tu' bSt dong th'jc cu6i cunc suy ra :
lim
X '
-
X *• . t^inh l y da dug'c chu'ng min^
