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Одна новая теорема существования решения нелинейного уравнения в банаховых пространствах.

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y,n:K 519.853
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3YIOT JIu60 Te )Ke rrapaMeTpbl, JI1160 qaCTb 113 H11X, rr03TOMY He MoryT ,n;aBaTb JIy'-IIll11X ou,eHOK,
qeM
dl.
IIpe,n;cTaBJI5IeT 11HTepec, HaCKOJIbKO OH11 xY)Ke, '-IeM
dl.
~JI5I y,n;o6cTBa cpaBHen115I C TA rrpHBe,n;eM '-IaCTb TM (LITM) OTHOCHTeJIbHO CYlU,eCTBO­
naUH5I 11 y,n;aJIeHHOCT11 peIlleUH5I
[4,
1]
TeopeMa
00
qTM. IIo.lw;)ICu.M H(h):= :Ek=O (h/2))
B uaIll11X 0603HaqeH115IX *) .
2k 1
­
- U d M := fogoH(h). IIycm'b 6'btnO.lI:J-te'H:bt
YC.IW6UJt 1) - 3) meope.M'bt A U, 1CpO.Me m02o; 4) h:= fMfoLgo < 2; 5) S(xo,d M) cD.
T02Ja pewe'J-tue ypa6'J-te'J-tuJt (1) npu'J-taJ.fI,e;)ICum S(xo,d M).
B [4] yCJIOBIre 2) HeMuoro C11JlbHee: IIJ/(x)11 ~ L Vx E D. IIepexo,n; K JI11rrIllHI~eBocT11,
[1].
o,n;HaKo, ne CJIO)KeH. Ero MO)KnO naHTH, uanpHMep, B BepcHH TM y OpTer11
YCJIOn11e
ne,n;eH115I
4)
fogo
LITM Tpe6yeT MaJIOCTH 3an11CHMoro OT BbI60pa na'-IaJIbHOH TOqK11 rrpo113­
== h/(fML),
qTO rrp11,n;aeT LITM rrOJIyrJI06aJIbHbIH xapaKTep. TA B 3TOM
CMbICJIe - rJI06aJIbHa5I.
d1 11 d M rrpH BbII10JIHeH1111 YCJIOB11H
._ fM/r'o - 1 _ z- 1
T M'LfM
- gOf o - 11
h
CpanH11M Terrepb KOJI11qeCTneHlIO ou,eUKH y,n;aJIeIIHOCT11
TM . 0603Ua'-I11M
dK
=
fM/fO
'-Iepe3
z.
PL
Tor.n;a
1 - \1'1 - 2PL P
fogo
-
= fogoZ
= h/z,
1 - V1 - 2h/ z
h
=
L
OTcro,n;a rrp11 YCJIonH5IX
dK
z;:: 1, P L
< 1/2
~
z
> 2h
2fogo
~-~=======;= 1 + .)1- 2h/z
CJIe,n;yeT
~ TM ~ Z (1- \11 - 2h/z) ~ z -1 ~ zV1- 2h/z;::
~ Z2 - 2hz - 1 ;::
°~
z;:: h +
~
1
J1+h2.
IIocJIe,n;nee yCJIOBHe C11JIbHee ,n;ByX rrepBblx. TaKHM 06Pa30M, TpH YCJIOBH5I B rrepeMeHHblx
PL , z
CBO,n;5ITC5I K o,n;nOMy B rrepeMeHHblx
_ -
d1 - gofo
qTO B ,n;l1arra30He
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h, Z,
H ou,eHKa
+ V1- 2h/z),
[2hz _ (z -1)2]/(2h),
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dl
rrpe,n;cTaeT B B11,n;e
z ;:: h + \1'1 + h2 ,
1 ~ z < h + \1'1 + h 2 ,
rrpHno,n;l1T K paCqeTHbIM ¢opMYJIaM
d M _ {(1 + V1 - 2h/ z )H(h)/2,
2hH(h)/[2hz - (z - 1)2],
d; -
rpa¢l1K 3TOH 3anHCHMOCT11 rrpe,n;CTaBJIeU Ha pHC.
z ;:: h + y'1.th2,
1~z<h+\I'l+h2.
1.
JI11HH5I CTbIKOBK11 ,n;ByX rroBepxHo­
cTeH, COOTBeTCTBYIOlU,l1X ,n;ByM CJIyqa5IM paCqeTHOH ¢OpMyJIbI, rrOKa3aJIa Ha HeM )KHPHOH JI11­
HueH. IIoCKOJIbKY
P5I,n; H(2)
pacxo,n;HTC5I, rpa¢l1K rrOCTpoeH ,n;JI5I ,n;l1arra30I-Ia
h E [0, 1,95).
Terrepb rrp11Be,n;eM qaCTb TeopeMbI KaHTOpOBH'-Ia (LITK) OTHOC11TeJIbHO ou,eHK11 y,n;aJIeH­
HOCTH pellIeHH51 ypaBHeH115I
(1)
OT UaqaJIbHOH TOqK11, CJIe,n;Y5I
[8].
qTK. IIycm'b o6.1Lacm'b 3aJa'J-tuJt D 'J-tenpep'bt6'J-t020 omo6pa;)ICe'J-tuJt g ecm'b 3a.M1C'J-tym'btU
wap S(xo, R) u g U.Meem npou36oJ'J-ty'lO J 6 e20 6'J-tympe'J-t'J-tux mO"{,1Cax, .ftunwu'4e6Y c
1CO'J-tcmamnou L;
IIJ-1(xo)g(xo)11 ~ "10;
P L := foL"IO ~ 1/2;
S(xo,R) ~ S(xo,d K ) ,
(
1 - \1'1 - 2PL
2Je d K :=
"10· T02Ja ypa6'J-te'J-tue (1) U.Meem pewe'J-tue 6 wape S xo,d K ).
PL *) YCJIOBHR TM H ~TM COBrra,D,aIOT.
42
5
1,7
1,6
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1,4
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h
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K
OTHOlliemUI ou;eHKH
dK
K
3aMe"l,a'HUe 1. B npl1Be,n;eHHoll <P0PMYJIl1pOBKe c,n;eJIaHO He60JIbillOe YTO"fHelIl1e cpaJ3IIl1­
T eJIbHO C
[8]
11.3), r,n;e Hl1"ferO He rOBopHTC5I 0 3 aMKHYTOCTl1 HcnO.ITb3yeMblx IIIapOB,
12.2) T e )f(e 0603Ha"feHH5I OTHOC5ITC5I K OTKPbITOMY IIIapy. ,n:aHlIOe YTO"f­
Henocpe,n;C'l'BeHHO H3 ,n;OKa3aTeJIbCTBa T eopeMbI 11.3. Y caMo r o KaHTOpo­
(TeopeMa
XOT51,n;aJIee (TeopeMa
H eI-Il1e H3BJIe"feHO
BWIa 06a IIIapa 3aMKHYTbI 11 eIlJ;e BbICTaBJIe HO ,n;onOJIIlHTeJIbHOe T pe60naHl1e CYIIJ,e CTBOBaHH5I
np0113Bo,n;HOll Ha rpaH111~e
as(xo, R).
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HY)f(,n;aeTC5I.
3aMe"l,a'HUe
2.
MHHaHl1e 0 IIIape
IIplme,n;er-ma51 <P0PMYJIHpOBKa LITK n03BOJI5IeT JIerKO BH.n;eTb, "fTO yno­
S(xo, R)
MO)f(HO Bo06uJ;e 113'b5lTb , He YCHJll1B H He OCJIa6HI3 TeopeM y.
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D,
HMeroru;eMY JIHnIIIHIJ;eBY npoH3llo.n;uyro B
1
1
H ou;eHKaMl1 B Ha"faJIbHOll TO"fKe:
TO
~
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C KOHCTaHTOll
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L,
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K TeopeMe
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a HMellHO HCnOJIb30Ba H H 5I BMeCTO
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z
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'rio
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•
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PL
g,
:::;
1/ 2,
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B
TK CpaBHl1TeJIbHO C TA 11 TM He XBaTaeT o,n;Horo Bxo,n;HorO napaMeTpa:
T M. IIpo"fl1e
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PL
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11
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(3))
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1
IIJ- (x )1I
Kor.n;a
d1
:::;
dK • Henocpe.n;CTBeHI-IO)f(e 113 <POPMYJI Bl1MO, "fTO Kor,n;a
d 1 = d K • 3TO 06'b5lCH5IeTC5I TeM,
,n;OCTaTO"fHO BeJIl1KO, l1MeeT MeCTO
J- 1 (x)
B MaJIOll OKpeCTlIOCTH I-Ia'-laJIbIIOll TO"fKl1, BeJIH"fl1Ha
He ,n;OCTl1raeT CBoero noporOBoro 3I-1a"feHH5I T M
PL
> 1/ 2, rapaHTl151
•
cYlu;eCTBOBaHl151 peIIIeHH5I H ou;eHKa ero y,n;aJIeHHOCTl1 eCTb B
TA, HO HeT B LITK (npH BbIllOJIHeHHH OCTaJIbHbIX YCJIOBHll B 3Tl1X TeopeMax).
43
Kor,Ll;a
T M < dK ,
d K / d1 B
OTHOllleHHe
a
PL
:::;
1/2,
OU;eHKa B TA MeHhllle OU;eHKH B LITK . PaccMoTpHM
,Ll;aHHOM CJIy~ae. I1Me.H llpe,Ll;CTaBJIeHH.H
d1 = ( Z
nOJIy~aeM ,Ll;BynapaMeTpH~ecKyIO
(Z-I)2)
2zP
-
_
Togo,
L
3aBHCHMOCTh
Z ~ (1 - 2PL )-1/2 ,
Z < (1 - 2PL )-1/2.
rpa<pHK :no:i1: 3aBHCHMOCTH npe,Ll;CTaBJIeH Ha pHC.
B
2.
cpaBHeHHH C TeopeMo:i1: raByplma (Tr) HHTepec npe,Ll;CTaBJI.HeT TaK)Ke cO,Ll;ep)KaIlI,H:i1:c.H
B <P0PMYJIHPOBKax MeTO,Ll; nOJIy~eHH5I pellleHH5I , l103TOMY H3JIO)KHM ee llOJIHOCThIO, CJIe,LJ;y.H
[5] (TeopeMa 1), HO C y~eToM 3aMe~aHH5I 2 K LITK.
Tr. IIycm'b 6'btnOJt1te'H'bt YCJt06U.R 1),3) meope.M'bt A U) 'KpOMe moco)
2) npOU360a'Ha.R ramo J' Ocpa'Hu"te'Ha 6 'He'Komopoii o'Kpecm'Hocmu 'Ka::HCaoii mo"t'KU XED';
4) S(xo,d r ) c D) cae dr := TMgO.
x=
Tocaa 3aaa"ta Kowu
-J-1(x(t))g(x(t)),
onpeaeJte'H'Hoe aJt.R 6cex t ~ 0 U x(t) E S(xo,d r );
.R6Jt.RemC.R pewe'HUe.M ypa6'He'HU.R (1).
LIacTHhI:i1: cJIy~a:i1: TA, KOr,Ll;a
yCJIOBH5I
2)
= TO ,
eCTh He6oJIhlllOe YCHJIeHHe Tr. ,n:e:i1:cTBHTeJIhHO, H3
Tr H <P0PMYJIhI KOHe'-:lHhIX npHpalIJ;eHH:i1: CJIe,LJ;yeT JIHlllllHu;eBOCTh
oKpecTHocTH Ka)K,Ll;o:i1: TO'-:lKH H3
To~eK H3
TM
=
x(O)
xo) UMeem pewe'Hue x(t))
npeaM limt~oo x(t) cy~ecm6yem U
D'
D'.
Ho JIHnlllHu;eBOCTh
J
B HeKoTopo:i1:
TOJIhKO H Tpe6yeTC5I B TA .
B cJIy~ae,
KOr,Ll;a
TA ,Ll;OCTaBJI.HeT
Z
E
(1, h + vI + h2 )
H H3BeCTHa ,Ll;JI5I
JIy~lllyIO ou;eHKY, ~eM Tr: d r
Ou;eHKa y,Ll;aJIeHHOCTH OT TO~KH
Xo
,Ll;O
llOJIy~aeTC.H H3 aHaJIOrH~Ho:i1: OU;eHKH
dM
dr
-
d1
=
J
KOHCTaHTa JIHlllllHll,a B
(TM/TO - 1)2
2T ML
- pellleHH5I ypaBHeHH.H
B LITM 3aMeHo:i1:
H(h)
Puc . 3. 3aBHCHMOCTb
z, h.
OT
44
J
B HeKoTophIX OKpeCTHOCT.HX Bcex
-
(z - 1)2
= goTo 2h
(1) Ha
D,
> O.
B Tr <pOPMaJIhHO
z.
IIo3ToMY Ta )Ke
OTHOUleHH~ o~eHKH
dr
K
d1
dM j d1
3aMeHa B OTHOllleHI1l1:
h E (0 , 00)
.:mana30He
dr
d1
_
-
{
npHBO,ll,HT K paC'leTHbIM ¢OpMYJIaM ,ll,JIR OTHOllleHHR
dr j d1
B
BH,ll,a
(1 + \11 - 2hjz)zj2,
2hzj[2hz - (z - 1)2],
z ;:: h + vl1 + h 2 ,
1 ~ z < h + vl1 + h2 •
(10)
JIHHHR CTbIKOBKH ,ll,BYX nOBepxHocTeii, COOTBeTcTBylOlll,HX ,ll,BYM CJIY'laRM pac'leTlIOii ¢op­
VYJIbI
(10),
OTMe'leHa Ha pHC.
3
)KupHoii JIHHHeii.
TIOMHMO oqeHoK Y1I,aJleHHocTH pelllenm1 ypaBHeHHR
Hbrn
cropOHbI HHTerpHpOBaHHe 3a,l1,a'lH
(4)
H
Tf
YKa3bIBaIOT B03MO)K­
[0,+00).
(PeI1leUHe ypanHeuHR
C
npaKTH'leCKOii
He ,ll,OJIbI1le, 'leM ,ll,O KOHe'lHOrO MOMeHTa
Dpe,l1,llO'lTHTeJIbIleii HlfTerpHpOBaImR 3a,ll,a'lH KOI1lH H3
Ke
(1), TA
cnoco6 ero HaXO)K,ll,eHHR nYTeM peI1leHHR HeKoTopoii 3a,ll,a'lH KOI1lH.
(1)
eCTb
d1 ,
BblrJIR,ll,HT
Tr na nOJIy6ecKone"Y:HOM npOMe)KYT­
x(+oo).)
Summary
Miheev S. E.,
Banach' spaces.
I Pozniak L. T.I
A new theorem of existence of nonlinear equation solution in
For nonlinear equation g(x) = 0, 9 : D C U -t W in Banach' spaces U, W an existence of
the solution in a ball from D under assumption partly unknown before is proved. The radius of
the ball is expressed via known quantity characteristics of the reflection 9 in D and in the center
of the ball. It is shown that in the definite sense the gained results can not be improved. They are
compared with the analogical results of the known theorems of L. V. Kantorovich, 1. P. Mysovskih,
M. K. Gavurin theorems about Newton's method.
1. Opmeza ,ll,~., Peu1i6o,//,Jm B. IhepaIVioIlllble MeTO,[!pI pellIemUI HcmmeiiHbIX CUCTeM ypaB­
BeHHH co MHorliMH HeH3BeCTHbIMH. M.: MHP, 1975. 558 c.
2. Schwetlick H. Numerische Losung nichtlinearer Gleichungen. Berlin: Deutscher VerI. der
Wiss. (Mathematik fur Naturwissenshaft und Technik. Bd 17), 1979. 346 S.
3. KaHmOp06U"l, JI. B., A'K:u,//,o6 r. II. <DYHKIT,HOHa.J1bHbIH aHa.J1H3. M.: HaYKa, 1977. 744 c.
4. M'b6C06C'K:UX H. n. 0 CXO,ll,HMOCTH MeTO,l1,a JI. B. KaHTopoBH"Y:a pellIeHH}l <PYHKlT,HOHaJIbHbIX
YPaBHeHHH Hero rrpHMeHeHH}lX / / ,IJ;OKJI. AH CCCP. 1950. T. 70, N~ 4. C. 565-568.
5. ra6YPuH M. K . HeJIHHeHHble <PYHKlT,HOHaJIbIlhle ypaBHeHH}l H HeopepbIBHble aHa.J10rH HTepa­
n;HOHHbIX MeTO,l1,OB / / H3B. BY30B. 1956. N~ 5 (6). C. 18- 31.
6. Muxee6 C. E. HeJ1HHem-Ible MCTOAbI B orrTHMH3alT,HH. CIT6.: H3,l1,-BO C.-ITeTep6. YH-Ta, 2001.
276 c.
7. KapmaH A. ,IJ;H<p<pepelIlT,Ha.J1bHOe HC':!HCJleHHe, ,ll,Hql<pcpeHIIHa.J1bHbIe <pOPMbI. M.: MHP, 1971.
392 c.
8. KpaCHOCeA'bC'K:UU M. A., BaUHU'K:'liJO r. M., 3a6peu'K;o n. n. 11 ,l1,p . ITpH6J1H)KCHIlOe pellIeHHe
onepaTopHbIX YPaBHeHHH. M.: HaYKa, 1969. 455 c.
CTaTb.H nOCTynHJIa B pe~aKQHJO
6
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