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MUC
I. U C
SxmsjL
PhSn
loo'
dSu 3
Chifo^np:
n5t :
I*i§t
sfi
ptocme;
phap
n£i sijy tt^en tinh
d5
giai
6§ja
dung
phuPcmg
trinh toan
ti^
S 1
M§t vai khai nigm
vo ty sal phan
syy
r^ng
cho
§
5
M§t vai tn^cfr^'hgj? dglc MJ^t
cua
phop 1$P (2.9) 52
§4
V5
in§t logii phuctog phq? l§p
b^c
cao
thrf hai
do
giSi gSn
dung
phifcmg
trinh
to5n
ti>.
. •
37
C^^ctofa:
haj.
5 M$t
vai
phi:?cmg
phap
ngi
suy
tSng qu5t
svQT T§ng
d|
giai
gin dung phtfcmg trinli
tofin tiJ
go
D^t
v8n etS
45
i 1
M§t
vai
kiaai nlga
ve
cSng thrJo n§i
suy
Tfiu-tcm
si^y rOng
i^
§2 "VS
ffiOt
vai pbucng phSp n§i
s^y
tong quat
si^
rgng giai gSn
dung
phu^c?ng
trinh
toan tiJ
va
si;
h$l
tv cua n6 .
•
• •
^
Gfaifgya;;!^
l?a
t
Ph8n iJng
d^ng
i
1 Mgt
s5
T^ag
d\wig
cho
3191
vai
to^n ti>
cy
thS
61
1.1
Phi?c?n6
trinh
haci thi/c
vo^
bi§n s8 thi;c
61
1^2
H$ phtfc^ng
trinh
dgii sfi
phi
tuySn hc$c
siSu vl^t
62
1.5
Phi^c^ng
trinh tich phan phi
txo^Sn
G5
i
2
CSc
vi
d\i
bang 06 68
Tai
ll§u tricb
dan 70
o£o
-
p
-
Tlfc i5ng dyng giai tich
ham
^to
nghien
C'\i
cic
ph'.^'ng
phap giai gan
dung
cac
lopi phi?o'nc
trinh
ngt c5ch tong
quit
da
dem iQi
nhieu ket qua v6 cung quan
trgni^.
'2u»
ti^o'nt;
cua
giai
tich
ham khong
nhu^ng
cho phep
dcn
[^Inn
hoe
each
nhin
cho nhieu
phUHo'ng phfip
khac nhau
ma
eon giup ta
ki^m
toan
miou
GO'
do
tinh
toan trong
nhiou
linh
vy^c
khoa hgc khac nhau,
•vf dy
nhu^
dpi
s5
t^iySn
tinh,
phu'C^ng
trinh vi phan,
phutmg
irinh tlch
phan, giai
tlch
phi tuyen
v.v *
TThieu lo^i
hai toan khac nhau
oou
eo che vi3t
du&i
l?ng
chiing
(1) Ax
=
0,
trong do A lu toan tu xac
d^nh
trong
ngz
khong gian noo do
va
CO
gia
trt
trong
mgt
khong
i;ian
cung
lo^i.
DS
giai gan dung
phu^c-rig'
trinh (1)
ngi;?o'i
ta dung nhieu
phi?o'ng
phap
kh£c
nhau.
L.oi
phv.Xi^r^';
phap
dou
oo
nhu'ng 'J*ii (Tien
va
nhu-ng hgin
che nhat
dinh.
2n?o'ng hg»p
A la toan tu'
tuy3n
tinh,
cac ket quo ve giai
;;Sn
dmux,
da
duvc :<Qt
g5n nhu* trgn
vgn.
Vol
A khong
tuyon
tinh,
v§n
do
mc'i
chi
duvc z<6t
cho
timg
trj'c^ng
hg^
rieng, chang
hgn lop
toon tu that,
loip
toan tu
(to»n dipu
v.^-".
••
Trong
nhiJng
nam gan
dixy nhiou cuSn each
chuyen
khao
da do
cfp
o^t
each
dang
kt toi nh5ng phuxmg phSp
tong
(luat,
Chang
h?n,
c5c
tai
lifni
cua Gang to r6
vlch
/"^t^ ?*
®latx
^157> Gratnoxenxki
/fIGj,
Otcga
F"Mj^
Moctmic /"IBJT
va
nhieu cong trinh
(Tu^g^c a>ng
b6
tror_g cue
t^^
chi to.4n
hgc khac
nhau
/"J
-
1^7,
/f21 -
>27.
Trong
ban
luyn vun
nay chung toi se trinh bay
mgt
s6
ket
qua thuge linh vi/c
do.
Ban
lu§.n
van nay gom
ba
chu'o'^
- ;'
-
Chu-^ng
1 trinh bay mgt
Q6
cac
phiAD'ng
phap ngi
cuy
tuyen tfnh
dS glM gSn
d'ms
phuv^ng
trinh toan ti (1).
Thi/c
ra
phuang
trinh
(1)
da
'*u^c
nhiou
t^c
gio
nghion
oS\
\ni
<^xx^n
ra nhieu
phutJ'ng
phop khac nhau
nhu^'
phuvng ph%> Niuto^n
C^^'^J*
phwng
phap
Steffenxen - Aitt:en £
^'y^
^'^ J ^
phu\>'nu:
phap
l^p
clip k tong quat
(phut^ng
phap
Sruede
-
Tsobu'sep) /f
15;
27
J
- • • y
trong s6 do
phuang
phap
riutcn - '-ingtorovlch
t-uy
di.;vo
su'
ct\mg
i^ng
roi,
nh>mg nhuyc
diem chinh la trong
nhiSu tru?o»ng
hgj) vit;c
xac
^nh toi'n ti>
^AxJ"^
(d^o
ham theo
Preso) thu^'n^i
^gp
kho khan
v-o
ph'^e
t^p.
ITgoai
ra trong
ngt
G6
tr-fo'ng
hgp
Cvl
dy khi
t^iei
c<ac
bai toan
bien
phi tuySn
^6i vti h$ ohu'd'ng
trinh
\d.
phfm)
cong
thi?c
hl^
cua
Ax
khong the
biSt tru'iS'c
mo chi
biSt cTuvc thu0t
toan
tr?n
nay
tlnli
difn tu'
vpn
nang
de xac
cT4.nh
gia
tr%
Ax^^
va
v^.
th3
khong the op
dyn^j
dug'c
phv?o^
phap
TUu-to'n.
Di;'a
vao khai
niv'ia
ty sax phan
stjy rQng
cho to5n
ti
trong
nhu'ng nhjsi
gan day cac
con^
tA.nh
cua
/"11-13;
22-237
da lan
lirg^t xuSt hii^n nh?aa khle phyc nhu^ng nhu^c
d-icm
tren,
nhfj?r4g -oSc ilg
hgi ty cua cac
phu'r;ng
phap do
chu*o vxf.gt
qua
tSc
d§
hgi ty cua
phuc?ng phap
T'iiT-tCn.
l^ng
chuttog nkj chx'jig
toi nSu ra mgt
B6
phuNj'ng
phap
CO tinh
chSt ngi suy tuyen
tinh.
Cac
phii'O^ng
phop nay khong
nhu'ng thugt
toan
do?n gian
ma
t5c
dg hgi
t-y
l^i
cao, chung to
ra dgc
bi^^t tign Igl
trong
truVng hg^)
toan tu' A co
dsua haa
phiJc tjip hogic
khong kha vl.
Chi?c^ng
hai trinh
bSy mgt s6 cac
phut)*nc;
phap
ngi
Bvxy
t6ng quat suy rgng cho toon
t&*
PhSn d2u
tien cua
chu'c^ng
nay
la
dUB
TV.
coc khSi nipn
ve
1^
sai phan suy rgng
'oong
quat cho
to
i^n
tl?
va
cac
cong
thi^c
ngi
sy^^
Uiu-ten
tong quat
S"uy
rgng
cho
toSn
tu' tren khong gian
Banach.
PhSn tiep
theo la
x£y dijPng
x^t
Q6
phut^'ng phap ao nh5ng phuang
phap nay bao
gSn ragt
so
5 -
ket
qiTO
c^'ia
ch'.;x»'ni3 I va
cua
coo
toe
gia
khac.
L:5i
phuang
phap 6"
cb^Mn^
T cTuig
nhi,"
& chwng
TT
^eu
dxjt^o
khao
B:OZ
qtia
cac
phSn 5
- ilgi
dung
phittrng
phap
-
S^
hgi ty
-
Gong
th-^c
danh gia sai s6
(Xem
c5e
c^nh
ly :
2.1;
:i:.2;
2.35 2.45
2.2't
2.5';
2.2";
2.5"
§2 chux^ng
I,
5.1;
5-2;
5-;^;
3«4;
5o S 5 cht:c'ng
T,
4.1
g 4 chmng
I 2.1; 2.2; 2.5
0
2
chi.?o'r^'
IT).
ChL?o'*ng
ba (phan
iJng
dyng)
khao cat
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. the;
ITeu
len
thugt
toan, dieu
ki^n
hgi
t\i,
cong thu'c canh gia sai so cho toan tu' dang khao
sat
C-^o^
cac
d^nh
ly
t
Li; 1.2
§1
chuWo^^
ba).
Phan
cu^i
cung
ci^o ch*:'c?ng
nay la
nigl
vai vl dy bang
28
do
kien nghii^m
cac
phuc?ng
phap neu ra.
C/Xic
ket
qua
ohlnh.
c»^a Ir^/n vln
dn
dirg'c rai
roc boo cao
trong
cac buoi sinh
hcpt x6-iai-na
"Cac
phuti*ng ph&p
gl.al gan
dung
^.ihu^i'ng
trinh toan
ti"
vh nhu*ng
ftxg dyng
cu.a
chnng"
cua
to
bO ^5n 'Doan
hgc vinh toan, khoa 'Joan
ir.>?o'ng Bjii
hgc long
hfi'p "^a-ngi
l';?76
va
ia$t oh'&n
da
dug^c
cong b6 trong
/"55-557»
Tac
gia
xin
chan thanh can
o'n
dor^g chi -"oang dll'c ITgijye]
da dgc ky
lu'^ng
ban
tiiao
va cho nhieu y kien dong gop qui
ban
Dgic
bi^^t,
tac gia xin chan
thanli
c^:a
o'n
d3ng chi
Phan van
^^p
da
da
xuat
^phucfxux;
hu'o'ng, da cho nhieu y
ki^n
dong gop qui ban
trong
c5
qua trinh hoan thanh ban
lugn v2.n
nay.
"
6
Cn'^^^n^-^;
I
IE
Cg-AT
GAN
DTTITG PHHONG 'H^lTii lOAN
l^T
PJxao
sat
phi?o»ng tri.nh
to5n
t\^
-
(1)
• Ax
^
0,
trong ro A la toan
tu"
phi
IrL^on
anh
x^ ingt inien
loi
^TL
nho
do
c\ia
khong gian
Banaeh
X vao
ngt
khong gian
cung lo^i
i.
Gia thiet rang A kha vi
th^»o
P'rOsG
va
TL
lo
nghicm
dune cua
(1) trong lan
ojn iTIi .
§ ^
•
^'-9t
vai khai
nign
ve
ty
sal
phC^n
suy
r^ng
^'-]0
toan
tu"
(:^m^5l7)*^
Xot ngt
ham tr\i
t':Vng
Ax, han
nhy chTij^'n
khong gian
d5.n!i
chuan X vao khong gian
dj.nh ehu*n !. , ^,-"
:tiu^T^''*^
Hi hi^u
khong
gi n
cna
riii?ng
toan
tiFVx vaor
la
fX-^
I
J.
Thanh
Igp
khor^-
gian tlch :
E-
== ^'^
(tlch
3:^-Cac)
iZv *"
X.
A A , . . . ,
t»
-, ~
trono
i^o
k
la
n$t
so
tr/
nhien.
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_, ,
:5C^„-()»
"^^
^1^^"'
^^ ^
cTl^).
Gia
^ tBn tfi ngu
h^a
tr*a:
ui^ijr'ng
A(x^,
:^:^),
han
nay
chuyen
nhiJng
ph3n
tu'
cua khong gian
Z-,
vho
nhSn;:
phan
txf
cua
khong gian
fx-^lj,
tron^
Co
>^
e
X, (i
^
0,1).
D6i vci
nhu'ng
-xyYian
x,-^^
x^
, (i
=
0,1),
o5 d3.rih tn?ng
X
thi
A(XQ,
0:^)
la
ni$t
toan tu
tuySn
tinh.
OJoan
tu'
t-uyen
txnh
A(::^,
oi-j
) thoa man dieu
kipn
(1.1)
AC-,,
x^)
(x.
-2„) =
A::^
-
A^
,
difg'e Qg±
la
ty
sai
ph;'i^ui bye nhSt ei\a
h-an
tru'^u tu'o'ng
Ax l3y
^ t?i
cac
phSn tt> x^ e
X,
(i
=
0,1).
Gia
BU
ton
t?i ngt ^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
tu
cua khong gian
^X ^
f^
-^Ij
7,
tronu
do
r^
e
X,
(i
=
oT^
joi v(?i c5c phSn
ti:
x^,
(i
=
0,2 )
o5 ^^nh
t^x^ng
X thi
A(x ,
X|,
Xp)
la ngt toan tu
LzonQ tuySn
lanh.
Toan
tl?
so.ng-
tuy-Sn tinh
ACX^,
:20:
, s>^)
thoa nan
*iou
kign ;
(1.2)
ACx^,
x^,
x^)
(x;^
-
3ii) ^
A(XQ,
X;^)
-
A(XQ,
X.),
di^c ^i
la ty sai phan
b-v^c
hai
cua
han
trOYi tJ'g'ng
Ax
iSy
tgi
cac
phSn tir x^ ^
X, (i
=
0,2 ).
B^a
vao
(1.0,
t'^
('1.2)
(cau
k^i
•-?,
d-g^c
-coc dyng ^:ji
vS
len
(:<>,
-
^Q^
^ "^^ -^-^^7 ''''^^ •
(1.2)'
A(XQ,
xp
x-)
(x^
-x^J
(:o.
- ::^) =
=
Ax-
-
A:.:^
A(x^,
: )
(x.
-
::.J.
"^•^oon
toon
tux^'ng
ti;
ta co
zho
d^nh
nghia ty sai
ph-'ln
b.;.c
k :
(1.
5)
A(x^,
Xp , Xv_)
(xi.
-
X|.
_,^
)
^
=
A(XQ
,
X-,
, •
, ^}-_p » 1-) " •^-
"^^"^Q'
'H ' •
*
• * "V-1
Feu
A ton tgi cac
r^go
han
liun
tye don bgc 1: (thoo
r eae)
thi
cac ty
col
phan xac
d^n^
nho'
coc tich phan 'Urjan tru^i
tii'g'ng,
to
CO 'fc^c lu^j'ng ;
(1.4)
(I
A(x^,
:c.)|U
V'^J
» ^- :.^ .^
(x.
-:c„),
0
^
6
^
1
|A(.:^,
::,,
x.)||
^
(-^)
II
A\|1.
i.
O
(k)
A(^.
^^1 ^)ll
^
C-jV)
II .^fZi^ll
,
'^l = '^-1 -^ ®k-1^r
^lc~1^'
0-^
©0'
^l" '
^1:_1 ^
1-
IWo'ns ty
nhu'
tnPtfns-
hi55> Han th6nG
thuc'nc,
co
th'
chi?no rdnb
5uvc rane» ty
sai phan
b.Jc
k cua ham
tr>ii tuvnjj
Ax
la 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
15P
bg.c
k do
f:±a±
;-an
dnj\p;
Phu^tyng
trinh toan
ti?
Cl).
Xu3t
phat
tiJP
gia tri ban dau
yP
du gan
x^,
ta
x§y di/ng
c§c phSn ti
theo
each
sau s
(2.1)
z^
= 4 ^
fAA^l,
4
=
2:5
-x° 4=
24_^
-xj_2.
x° :=
x°,
0
<:M.
-^1.
Tu' (2.1) suy ra :
,
(4-^0)-
JL-^(,o_^^).
T-JP
(2.2)
suy ra :
(2.3)
A(xg,
x^, , ^^)
(:^»xj)
(x|-^).,,
(xj - ^^^^) =
:=
(^
" '^
),!
ACT^O
O
ON
^^JO
O
>^
,k-l ^^o»
^'•••' ^''
<^2&^C
" ^^ '
trons d<$ (4
-
x°)^:=
(:^
-
x°)
(^
-
xgj
k lan
Dya
vao
djnh
nghia
ty
sai phan suy
i^ng
bf.c
k cho toan tu"
( (1.1) - (1.3) ) vo dieu
kifn
(2.2), (2.3) ta suy ra :
-
10 -
k
(2.^)
A4-Ax°-T,,^(x°-x°)=-^ o1, k^
(4
-x°y
,
trong do
Aol^ k^
^
A
(x^,
X?
, ,
:^)
(2.4)'
I,,^
= I? ^-'^'
4-1
"-^
^-^^0^
(^
=
k(k-l)
(k-i
M)
i!
L'^t
khac, cung
d^a
vao
dj.nh nGhia
cua
ty
sai phan suy
r9ns
b|c
k
dio
toan
ti?
va
dieu
ki^>n
(2.2)
ta
Ipi
co :
(2.5)
Ax - /.x°
-
lb,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),,
H-
i=1
k-3
o'
'O
+
>•
(-1)^
C^_,
A0i„(i+1)„
(i^2)^
(^-^%)
+
i=1
+
.
. .
f-
A01Q
(k-D^
(x-x?) (x-sg_.^).
Ti>
(2.4)
va
(2.3)
ta
suy
ra
rang vo'i
\/x €
iQg^Cii^
^
trcng
fid 5
^gj = i X :
l|'£x°(x-x^)^°|/^
|)A2t>1„ (k-1)J(
n^
))x-x°||
f
-
11 -
k
(ic)
B§t
dang
thi5c
trong
LQ*
co
nang
tinh ch3-c d}.nh
tinh nhieu
hd»n
la
djnh
lijg'ng.
VSn
do
d-^t
ro
lt>,
co
t^n
t?l cae phSn
tii X ^ u
2J
gS
, sao cho
thoo
nan (2.6)
khon^:^*
?
'Tay
noi
each
khac
can
chgn
[fh^o
la
khac trSng.
lligt
vgy,
do
nhnn du'g^c
(2.6) ta
cSn cd
:
(1*)
5-x°(x-x°)&°
=
-^^^
A^l
(k-l)„
(x-x°)"
-
k
k-1
-
Axc1^ (k-1)^
n (x
-
xp
0
C
x-O
^
Ta
thSy (1^)
se
dung
khi x
=-
x°
va x -
r^.
(di^a
vao (2.5)
va
each
dgt
6^
* -^^ " ^k ^'^ ^^^'•^* "^-^^ "^"^ '^''"^^
b*ng
vo ph'
vs bang
khSng).
Khi
X
?^
x°
va
X
j^
x^,
d^-a
vao
each
dgt
£°,
tir
(l*:
ta
nhgn
;
(2*)
q.&f
^ Cj^-l^fc "^ ^ ^ ^ ^x > Co =
0.
Heu
coi
i^x
^^ ^^ ^^9
<^^
*
°o =
%if^
Axo1, (k-l)^
(4
-
X^)^
.
JC
.•
Axo1^ (k-l)^
jT
(.4
-4),
k-2
i=1
(Tlop
trang 12)
-
12 -
to
00
:
(2.6)
Ax -
Ax^
-
To,k (x-x^) = "^—^ Ax^1
(k-^
)^
Cx-x^) .
XSp
xi Ax
bang
da
th'?c taiyen tinh
<:io(x)
:
(2.7)
Ax
^%M =
Ax^
^• To,k
i^
- :^^)
-"•
X
—1
*
Gia
su QoC^*) * 0 "^'^
toan
t;'
nghich
dao
TQ
I;
t2n
tgi,
t^j
(2.7
Bvy
ra
t
1-0
„~^
.
o
-x
-
y:
-
'j<5^1:
Ax
long qnat,
neu to dgt
2
xg
:=
c^,
n = 0,1,
Tigp
(±)
.(^^-x^)(^^-:^^^^) ^-
Ao1^ (k-l)^(4-x^)(x^-x|),
k-1
(:4"^k-1^
i ^r
Axo1^ (k-l)Q ; irZ
-:^),
thi
(2*) se la
mgt
phitVng
trinh
dgi s8
bC^c
k.
Vol
i^g^du
be, ta
cc5
tho
thay
(2^)
bang phu'cng trinh
trong
do
0^
nhgn
d^rg'^
tu
G^
bung each thay
cac
s8
hgng
Q
(k-D^
b?ir^-
AOIQ
k^
CO chu^a
Axo1-
(k-1
)_
bar^-
Aol-,
k^.
Tic (5^)
ta se xac
djnh duVc cZ
.
- 15
-
thi ly
lugn
hoan toan
tit^ng
tg'
nhti*
tren to
GO
C6
:
(2.8)
Ax-Ax"-Tn,k
(x-x")-"-
^\^^'
Axn1„
(k-1
)^^'^'""
' ir,A
n
n
k ^
trong 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 trinh
Igp
:
(2.9)
:^'^^ - x^^ -
Tn,],
Ax^^
, n
-
0,1,
S\?
hgi ty
ciis
qua
trin^i Igp
(2.S)
^u<^c
the
hiC^n
o^
cac
d^nh
ly
dUol
day :
JtiO
do 1
Gila
su*
toSn
t^jr
A ton tgi
dgo
ham don cap k,
Ichl
do ta
CO
bSt
dh.n^j thu*c t
(2.11)
\
Tn.kCx^-
^;-
.5,
^TT
Ax-^
(X-"-
^)^
II ^
^
(-^,
st5> Ii
A(>.
t(x^^-
>:^)
i .
5^
II
%*||
)
II x"-
x*|l
o^t^l-
'^
trong
d6 ^ duVc xoc
d^nb
ti^'o'nG
ti;'
nhir ^ tronij
(l.'i-).
i^p
dyng
cone;
thu'c
'lay-lo
co phan
dir
cho toan
tiV
£"1v_7
(2.12)
Ax^
=
A(x*
+
x"
-
x^)
=
=
Ax*
^ ^
iL^(x-^
_ ^f ,
^(^^^
^),
1=1
i-
- 14
trong 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^
- >; -fr
A^
(.^
- ^^y = \
(3^,
x^-).
i=1
l'-
^
COns
tCrng
ve (2.8) (sau khi da thay x
=
x^)
cho (2.13) (chu
y
(2.12)')
ta
nh0n
difn'c
:
(2.14) IK (X X*) - ^
-J^
AJ?
(x^
-
x^)'
II 4
1=^1
- '
k
/
_L«
3^p
}
A
xi^+
t(x -
x"^)
n
i»
0;^t
<;1
,,_(^:1L-
|lAx^n1 (k-l)Jlll^'-^""
Sau khi
ap
dyng (1.4)
dS
danh
\~Lk
|]A3i*^n1,^
(k-Dj^
1(
tl?
(2.14) suy ra (2.11).
aieu
phai
chi^n^j
ainh.
r4nh
ly
2^
Keu
xap
XI
ban
'^au
x°
thoa
niin
cac dieu
ki^in
:
1^/
loan
tu*
AO1Q
ton tai 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
bat
dSnfj
thifc
-
15
-
k-2
i=-
-1
thi 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
k-1
^-^ ,
,i-(v-0
,
= (-1)
iAo^
r L
(-1)
C^^-^
Ao(k-l)Q
i=1
1^
_r>
k-1
. -
V-
r
^r
^ '^r.i
(2.16)
ai,,ic =
C-^r"
Ao1^(I•^Ao1o ./^^-'^^ ^^ Ao(l^-i)
trong do I la toan
tt> den
v^.
The
nhu^ng,
di;a \^ao c5c
gia thiot
c^ja
dxlrin Tj
ta co
11 A?!,"?
(-O'"^^-"^
4.^
Ao(^-i)o
"
i=1
Dya
vao
d5.nh
ly
Banach,
tu'
uSt
dang
thu^c cu.6i cun;j
suy ra
ton
t?i
toan
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+Ao1o}_'
(-1)
t.5_^
Ao(k-i)o)
\\
^
-1-
•
-
IG
-
-1 .1
-1
Do do toan
ti nghi
eh
^^^o
l-o,!:
ton
t?i
va
l|l'o,k
\\
4:
-^
Die^a
phai
chu'ng
ninh.
•^-^io
Gia
su'
:
1°/
ifin tji
toan
f'^
nGhich
dao
in,!::
va
1|
%,!:
|l
4
B;
2''/
lui^^lU ^ ^k '
x"€ ^
v.'l
Vn.
Lhl
do
qu5
trinh
Igp (2.y)
GO
!^;i "cy toi
n-ghl^^n
x"^ cri.a
(1 )
loe
dg hgi ty
d-u<;^c
:-*anh .gi.a
bang
bSt
dang thu'c :
^'^
-
= *
IU
f.
II
'^^-
-
^*
ii
il"
1
K
i
trong do
^'
Ch'nmg minh
-^
d;ing
cor.£:
th'j»c
l^ay-lo
co 'phTm
du
cho
to'n
tu ta co
n
_ ,/_i
.
.n
__±:
(2.17)
Ax'
=
A(x'^
s
X -
X
y
=
k-1
/ .
*"7^
^x^
Uv
-
X
^
-I-
i,„
tx
,
X
;,
1=1
^•
trong 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 hai ve (2.9) cho
^
ta
di;Vc s
(2.19)
^^'^-^^J'-^-C^i^^
; nV^l^/j^l
Thay (2.17) vao (2.19) ta
nh^n
:
k_-1
x^^''-
X*
Ik II Tn'k
1!
ll'In,k(x^-^)-i:
-^7 A x-
(^-^)'ll
+
i=1
1'
-1
*
II
I^ik
II II
\
(^.
^) I)
•
Dya
vao cac dieu
kipn ciia
d^nh
ly va
c5c bSt
dang
thu*c
(2.18),
(2.11),
tii*
bat
dong thu'c
cu5i
cung suy ra dieu phai
c^iihig
minh.
I^u
khdng
gi5
thiet
trx^&o
ve ay.'
tSn
t^i
nghi^
cua
phi?cmg
trinh (1), ta co cac dinh ly sau day
«
Dinh
ly
2.?
Gih sv? X
thoa nan cac dieu
kiyn
:
1^/
Toan tiJ To,k tSn
tgi toan tu'
nghidi
dao
To,k
-1
va
fJTo,k ll
-4 Bo ;
2^/
\\Ak
iLcL^
4
I^i
f (i = ^)»
trong do lan
cgn liTj ^ duyc
>:ac
dinh
ti^
(2.24).
3°/
II To"'k Ax°|U
>'^,
)i:^.i
-
^_JU
>o'
<i =
^^)'
4^/
Cac hang
s8
Bo,
/^,
L^
th6a man :
k-2
(2.21) 0
^
p^
:=
1
. (To ^
r,
G^.^
T],
)
<
1,
(2.22)
O^q^
= ^-< It
-
o
trong do
/v dt?g»c
xac
djnh tiV nghipn
d:i»cmg
cua
phi?o'ng
trinh :
- 18 -
k-2
(2.23)
(1-
'^ (1 .•:^ ^^) )A -ri,kA
-
Y^^j,
=
o.
V&i
7^
= 2Bo T2 '^ 0
<.
1
vi ^-k
v'''
=_,
;L_____
J^.k
- 2l^-1i,k
3^k-^
k-1
'
2,1,
iF^-1
j^k
(2.24)
u2.^
=
X : [jx
~
x^||:^
Khi do trong
iSn
oan :
k
' "^
^ C ^-Po^o ^
phiJo^ng
trinh (1) sa co
n^'^hifxi
J:*
va
qua
trinh
l^p
(2.9) se
hgi ty
t(H
nghi^
do. 'i6c
dO
h^i
t;i
difg'c
danh
si^
tang
b5t
dang
thi?c 8
k^-:^i
n-1
^~'^
(2.25) IJ x^
-x*^ij<i-
^ ^
W^
^o •
k - 1
^ -
Po%
'-'
Chfeg
minh
Chu'^
minh rang khi chuyen
ti? ph&n ti5= x*^
song
x
thi
cac dieu
kign
cua dinh ly
v^m
cT>-?g'c
bac toan.
Do
dang
a-^y
ro
rang cac
ph§n tfr
s^,
x\
^
LT/^*
(i
-
o,k-*l).
Kiem tra dieu kien 1.
^ ^ ^ ^
-1
l'rL?o'c
tion
ta can
ch'Jng
ninh
ten
t^l
toan
tu*
nghich dao
T^
^
Thft
vgy,
xet t
—1
—1
(2.26)
||TO,V (To,:.c-2:i,k)
II
4
II
To^k
|l II
Io,k " Ti
,3,
([
=
-
19 -
-1
To,k
k-1
2Z
(-1)
1=1
i
,.i
(A
^ 1
^^•o(k-i),
-
A^ci
Kk-i)^);:
=
k-1
''^x4li^,'^-'^
^-1':^o(k-i)^-A(k-i)„i "
+ ^(k-i)Qi
^Kk-i)^
)
11 ^
^ 2 Be
1:2
• o (
k-1
i=1
(^
.
)
=
1
-Pc l-
'heo
dinh
ly Banach,
ti?
(2.26) suy ra
t8n
t^
toan
tir
nghich
"1
dao
ri -
^o^i;
(Tc,k - Ti,k)7 ''= ^^r^t
trong do I la toan
tu"
dc^'n vi
va
—J-
-1
Vgy
toan tu'
T-^
^j^
=
('I?o,k
%) ton
tgi va
Bo
II
<lc
__
.0,
Dieu
kiOn
1 da kiem tra
xong.
KiSm
tra
di§u kipn 3«
CSn dsnh
gia fl
C,
,]r
Ax^||
V^
n
=
0 va
X = X
,
d'^a
\'ao
(2.9),
tt?
(2.8) ta
ST:y
ra :
Ax
•^ ^-^ ^Ic ^ -^
Ck
o^
o
Do d6
- 20 -
k>1
(2.27)
jj<i,
Ax-^ll
<,
-^\^, -o
^^1^'
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 :o>
o''
"o^
^ (
1,k
^ ^2,k A
)
Po .0
0 .
De dang
chi?ng lainh du^c ??lnn;t nsu /\
la
nghi^
ciia phtJWng
trinii
(2.23) thi
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 tra 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
kign
4 da
dxfg'c ki&a
tra
xon^;.
KiSm
tra dieu
ki?n
2.
De
kl^m
tra
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
ly ta suy ra
:
llx'
-x^W
^llx'
-x'^ll + 11 x'^
-x^li^
—^— ''. + ' >
>i
0
Dieu klgn 2 di
dxJg'c
kiem tra
2X>ng.
Th;/c
hifn phep
ch^Jng
minh hoan toan
ti?o'*ng
tg?,
ta 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)
^ = Al2i£L
^
n
1^0
^^-52) 'n =A
U-T^
""n-l
TU-
(2.31) va (2.32) ta
BVQT
ra :
n-1
k-1
B^
vi
jjx^^''-
x^JUr^,
non
n-1
^ 11 x^*-"-
x«||^
ZZ ^n.J ^
P?"'
-lo ^^
^o'
j=o
+ +
- 22 -
n-i
•^"1
k-1
,
i
^~
(P
q ^-'- )
•
Khi V
-^ '
t^j?
bSt
dang
thu'c cu6i
cung suy ra bSt dang
tht5»c
(2.25).
TSn
de con
Igi
la
c§n chiJng mirJi x*
Iv^
nghifm cua
phifCng
trinh (1).
agt
v^y,
tu*
(2.9) ta
syy
ra :
AX-
-
T,,^
(x-^^-
x^)
•
Do do
(2.54)
IJAx^/k illn,kli
li^''-^l!
^
Ii2n,k'
"n-
Dv'a
vao cac dieu
ki^n
cua dinh ly suy ra
\\Tj^
^\{
la
mOt
dgi
li^g'ng gl^l
ngi, do do
t
0
=
lim
Ax^
=
Ax*
n—»
:JO
Binh
ly da
du'o'c ch'?ng
minh*
Gia
siV x*^
thoa man cac dieu kifn cua dinh ly
2^3,
ngoai
ra con thoa man thom :
Bjj.<,B
v^
Vnf
.
k-2
J^—
\ *'" ~i
(i-i>««ao) ^
-
25 -
Ehi do trong lan
c?.n
J
se
cht?a
duy
nhSt m§t
nghl^
ar
cua
phirc^
trinh
(1)*
Bang
phifc^ng
phap phan
chi'hag,
ta gia
su*
rang trong lan
c§n
,
^^
ngoai x ,
ph:/o*ng
trinh (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^"''
-
%,i,x Ax-
^t
%,k(^''-^^)-
Dypa
vao
(2.59)
va
Ax**
= C (vi
x**
la
n^+nigm
cua
phifcug
tsinh
(1) theo gia
thist),
ta suy ra :
%,kCx-"' '^)=
Vk=^-^-'-^.k^^^'
=
AJ^
-
Ax-
-
T^^^
i^
-
X-).
Do do
:
(2.40)
y^^^^
X**
=
iQl^
fA:^
.
Ax^
-
T^^^
(x*^
. :,^) J
M|lt
khic,
d\fQ
vao c6ng
th'^c Tay-lo cd phSn diP
cho
toan
ti5 ts oc5
:
(2.41)
Ax-
=
A
(X**
>
X-
-
z^)
=
=
Ax^
.
A'x**
(X-
•-
x^)
.
1^2
(^^*)'
trong do :
||T^2(x-
x**^)
|(
^
sup
11
A"x**.
t(x x^)il
11
X
^*
11'
o^t4l
Tt:^
(2.41) suy ra :
Ax^
-
Ax^'
.
A'X**X=
"
^2^^
-^^
Do do :
(2.41)»
ii
Ax^
-
Ax^
^
\'x^
(x^-
sup
I
A".:**
X
t(x^-x^)il
llx^
O^ti^l
Dya
vao (2.11) va
(2.41)',
t'>
(2.i^)
suy ra :
^"^-
y^Hl
E^^ic
:
l|Ax^-i^x-
-
a^,v(x^-
X-)
ii
<
ll
<k !i ^
K,k^^-
^> - -^ ^^ ^^-
^>'
^' -
i=1
k-1
(i)
+
i=2
^
l^|j/^>)|
i|x x^f'4.
j[Ax**-A.:-i-A'x*=^
(X X**)
.
k-1
>
A,
-fr
11
A^ii
li^-
^li
"^^
3UP
ii
AV->-t(x x^)
ljx x**illjix x**i
*^
=^
- 25 -
k-1
4
^[^^•^hn^-'^lX^*^,-^^"^-^"
Do do :
(2.41)";;
x-^^- x^
il .^ B
(-^
. i)
L,
ll
X
x**!!''"'
-H
lc*
k
+ •
-jT^i:^-^^;.^^
L2jjx x**/M X
i=2 -•
X jj
X^
- X**|j
Dya vao
t,l3
t'-ilt
cua dinh ly,
t*'
(2.41)" suy
r-a
:
f|:^'
-^1!
4
^oii
i-1
X
~
,*>.
u
i
k-1
|:.^-x"|i,B
(^*-1j)T,^,*-
0
^
(^-Po'^o''
k-1 , r i-1
! ^i 'O
••.1-IPc^).
'
(l-""''-''^l<rJ|x x^i(.^
rfl
,x°-
Khi cho n
-ro^,
tu'
bSt dong th'jc cu6i cunc suy ra :
X = lim X ' - X *• .
t^inh
ly da dug'c chu'ng
min^