Naiieeaiei A.A. Iaoa?iau e?aaaua caaea/e aeey nenoai aeeooa?aioeeaeueiuo
e
?aciinoiuo o?aaiaiee.
A a o i ? a o a ? a o
aeena?oaoei? ia caeiaoooy iaoeiaiai nooiaiy
eaiaeeaeaoa oiceei-iaoaiaoe/ieo iaoe
Ee?a 1999
Aeena?oaoei?th ? ?oeiien.
?iaioa aeeiiaia o aiaeaeiei cae/aeieo aeeoa?aioeiaeueieo ?iaiyiue
Iinoeoooo iaoaiaoeee IAI Oe?a?ie
Iaoeiaee ea?iaiee
aeieoi? oiceei-iaoaiaoe/ieo iaoe, i?ioani?
AIE*OE Ieaenaiae? Aiae?ieiae/
Iinoeooo iaoaiaoeee IAI Oe?a?ie
Ioioeieii iiiiaioe:
aeieoi? oiceei-iaoaiaoe/ieo iaoe, i?ioani?
IAO?EOEI ?iiai Iaaiiae/
*a?iiaaoeueeee oiiaa?neoao ii. TH.Oaaeueeiae/a,
caaiaeoth/ee eaoaae?ith
eaiaeeaeao oiceei-iaoaiaoe/ieo iaoe, aeioeaio
AI?ENAIEI Na?aie Aeaieeiae/
Oaoii/iee oiiaa?neoao “EII”, aeioeaio
I?iaiaeia onoaiiaa:
Ee?anueeee iaoeiiiaeueiee oiiaa?neoao iiaii Oa?ana Oaa/aiea, eaoaae?a
iioaa?aeueieo oa aeeoa?aioeiaeueieo ?iaiyiue, i. Ee?a
Caoeno aiaeaoaeaoueny ” 20 ” o?aaiy 1999 ?ieo
i 15-00 aiaeeii ia caniaeaiii niaoeiaeiciaaii? a/aii? ?aaee Ae.26.206.02
i?e
Iinoeoooi iaoaiaoeee IAI Oe?a?ie ca aae?anith:
252601 Ee?a – 4, aoe. Oa?auaieianueea, 3.
C aeena?oaoei?th iiaeia iciaeiieoenue o aiaeiioaoei Iinoeoooo.
Aaoi?aoa?ao ?icineaiee ” 20 ” ea?oiy 1999 ?.
A/aiee nae?aoa?
niaoeiaeiciaaii? ?aaee IAETHO A.I.
CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE
Aeooaeueiinoue oaie.
I?iaeaie iiaoaeiae eiino?oeoeaieo iaoiaeia aiaeico eiiieieo oa neaaei
iaeiiieieo e?aeiaeo caaea/ aeey nenoai cae/aeieo aeeoa?aioeiaeueieo
?iaiyiue,
nenoai c iiioeueniei cao?aiiyi oa ?icieoeaaeo nenoai o?aaeeoeieii
caeiathoue iaeia c oeaio?aeueieo oa i?eioeeiiai aaaeeeaeo iinoeue a
yeiniie oai?i?
aeeoa?aioeiaeueieo oa ?icieoeaaeo ?iaiyiue.
Oea coiiaeaii ia?o ca ana aaaeeeainoth i?aeoe/iiai canoinoaaiiy oai?i?
e?aeiaeo caaea/ aei ?iciiiaiioieo aaeocae ciaiue: oai?i? iaeiiieieo
eieeaaiue,
oai?i? noieeinoi ?ooo, oai?i? oi?aaeiiiy, ?yaeo ?aaeiioaoii/ieo
oa aiieiai/ieo caaea/.
Ua o 1937 ?ioei E.I.Iaiaeaeueooai ienaa: “A?yae ee anoue a ianoiyuaa
a?aiy
iaiaoiaeeiinoue niaoeeaeueii iainiiauaaoue aaaeiia cia/aiea
eieaaaoaeueiuo
i?ioeannia a nia?aiaiiie oeceea e oaoieea”.
Aeienii, caaaeyee eeane/iei ?iaioai Eaa?aiaea, Eaieana, Ioaiea?a,
Eyioiiaa, E?eeiaa oa Aiaiethaiaa, ooai?eany
iaoaiaoe/iee aia?ao, yeee aeei?enoiao?oueny aeey aea/aiiy eieeaaiue
aeinoaoiuei
aeecueeeo aei eiiieieo, oiaoi aeey yeeo aiaeiiaiaeii aeeoa?aioeiaeueii
?iaiyiiy
oi/ i yaeythoueny iaeiiieieie, aea iinoyoue aeayeee iaeee ia?aiao? ,
yeee aoiaeeoue a oei ?iaiyiiy oae, ui i?e ioeueiaiio cia/aiii
aiie ae?iaeaeothoueny a eiiieii aeeoa?aioeiaeueii ?iaiyiiy.
Oe?iea ?iciianthaeaeaiiy i?e aea/aii
iaeiiieieo eieeaaiue iaaoa iaoiae ona?aaeiaiiy M.I.E?eeiaa oa
M.M.Aiaiethaiaa, iniiae yeiai aoee iioaeieiaaii a iiiia?aoi?
“Aaaaeaiea a iaeeiaeioth iaoaieeo” (1937 ?.).
Oeae iaoiae io?eiaa iiaeaeueoee ?icaeoie a ?iaioao
TH.O.Ieo?iiieuenueeiai,
A.I.Naiieeaiea, Ae.I.Ia?oeithea, I.I.Ia?anothea,
A.I.A?aaaiieiaa, TH.O.?yaiaa, A.I.Aieiniaa, I.I.Oeiey
i?e aeineiaeaeaiii
aeeoa?aioeiaeueieo ?iaiyiue c aaeeeei oa iaeei ia?aiao?aie, c
iiaieueieie oa
oaeaeeeie ciiiieie, ?iaiyiue c iiioeueniith aei?th, ?iaiyiue c
caiiciaiiyi.
A inoaiii ?iee iioaineaii ?icaeaa?oueny oai?iy e?aeiaeo caaea/ aeey
?iaiyiue,
yei ? i?aaeiaoii aea/aiiy eeane/ii? oai?i? iaeiiieieo eieeaaiue –
A.I.Naiieeaiei, I.E.?iioi, A.TH.Eo/ea,
I.A.Aie/oe, ?.I.Iao?eoei.
Niaoeeoiea oaeeo e?aeiaeo caaea/ iieyaa? a oii, ui a aaaaoueio aeiaaeeao
?o
eiiieia /anoeia ? iia?aoi?oi, yeee ia ia? iaa?iaiiai, ui ia aeicaiey?
aaciina?aaeiuei canoiniaoaaoe o?aaeeoeieii iaoiaee aeineiaeaeaiiy
e?aeiaeo caaea/,
yei caniiaaii ia canoinoaaiii i?eioeeio ia?ooiii? oi/ee. Iaiaa?iaiinoue
eiiieii? /anoeie iia?aoi?a ? ianeiaeeii oiai, ui eieueeinoue m
e?aeiaeo
oiia ia niiaiaaea? c ?icii?iinoth n aeoiaeii? aeeoa?aioeiaeueii? aai
?icieoeaai? nenoaie. Oaei caaea/i aeey nenoai aeeoa?aioeiaeueieo oa
?icieoeaaeo
?iaiyiue ? iaoa?iaeie e iaeaaeaoue aei neeaaeieo oa iaeiaeineiaeaeaieo
iaaeiaecia/aieo oa ia?aaecia/aieo e?aeiaeo caaea/. Aieueoinoue ?iaio,
i?enay/aieo aea/aiith oaeeo caaea/, aeeiiaii a i?eiouaiii ?o
o?aaeaieueiiainoi, oiaoi o i?eiouaiii, ui m=n ( I.A. Acaae?a,
Ae.Aaenea?, A.I.
A?aaaiieia, TH.I. ?yaia, A.I.Aeaieaeiae/,
Ae.I. Ia?oeithe
, A.I. Iaetho, A.Oaeaiae, ).
Aieueo oiai, cia/ia /anoeia
?acoeueoaoia aeeiiaia a i?eiouaiii, ui iia?aoi? eiiieii? /anoeie
aeoiaeii?
e?aeiai? caaea/i ia? iaa?iaiee ( iae?eoe/iee aeiaaeie), ui a
aeena?oaoei? ia
i?eionea?oueny.
Canoiniaoth/e oai?ith ocaaaeueiaii-iaa?iaieo iao?eoeue oa iaoiae
Eyioiiaa-Oiiaeoa,
a aeena?oaoei? aeineiaeaeaii
iaoa?iai e?aeiai caaea/i aeey aeeoa?aioeiaeueieo nenoai c iiaieueieie oa
oaeaeeeie ciiiieie
( ?icaeie I ) oa aeey neii/aiii-?icieoeaaeo nenoai ( ?icaeie II).
Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie,
oaiaie.
?iaioa i?iaiaeeeanue caiaeii c caaaeueiei ieaiii aeineiaeaeaiue
aiaeaeieo cae/aeieo
aeeoa?aioeiaeueieo ?iaiyiue Iinoeoooo iaoaiaoeee IAI Oe?a?ie.
Iaoa i caaea/i aeineiaeaeaiiy.
Iaoa oei?? ?iaioe —
aeineiaeaeaiiy eiiieieo oa neaaei iaeiiieieo e?aeiaeo caaea/ aeey nenoai
cae/aeieo aeeoa?aioeiaeueieo ?iaiyiue oa ?iaiyiue c iiioeueniei
cao?aiiyi c
iiaieueii ciiiieie /anoioaie, oa aeey nenoai ?icieoeaaeo ?iaiyiue.
?icaeyaeathoueny e?aeiai caaea/i, eiiieia /anoeia yeeo ? iaoa?iaei
iia?aoi?ii.
Iaoeiaa iiaecia iaea?aeaieo ?acoeueoaoia.
Iniiaii ?acoeueoaoe, ui aecia/athoue iaoeiao iiaecio i aeiinyoueny ia
caoeno, oaei:
[1.]
Io?eiaii oiiae inioaaiiy ?ica’yceia aaaaoi/anoioieo neaaei iaeiiieieo
e?aeiaeo caaea/ aeey nenoai cae/aeieo aeeoa?aioeiaeueieo ?iaiyiue oa
?iaiyiue
c iiioeueniei aieeaii,
ui iinoyoue iiaieueii oa oaeaeei ciiiii, a ieiei ?ica’yceia
ona?aaeiaii? e?aeiai? caaea/i.
[2.]
Iiaoaeiaaii ?iaiyiiy aeey ii?iaeaeoth/eo aiieiooae ?icieoeaai? e?aeiai?
caaea/i,
ca aeiiiiiaith yeiai aeiaaaeaia oai?aia i?i iaiaoiaeio oiiao inioaaiiy
?ica’yceo neaaei iaeiiieii? ?icieoeaai? e?aeiai? caaea/i a e?eoe/ieo
aeiaaeeao.
[3.]
Io?eiaii aeinoaoii oiiae inioaaiiy ?ica’yceia neaaei iaeiiieieo
?icieoeaaeo
e?aeiaeo caaea/ c iaoa?iaith eiiieiith /anoeiith a e?eoe/ieo aeiaaeeao
ia?oiai oa ae?oaiai ii?yaeeo. Cai?iiiiiaaii caiaeii ioa?aoeieii
aeai?eoie
iiaoaeiae oaeeo ?ica’yceia.
[4.]
Ciaeaeaii oiiae iiyae ?ica’yceo neaaeicao?aii? eiiieii? e?aeiai?
caaea/i aeey nenoai ?icieoeaaeo ?iaiyiue o i?eiouaiii, ui
ii?iaeaeoth/a e?aeiaa caaea/a ia ia? ?ica’yceia i?e aeiaieueieo
iaiaeii?iaeiinoyo.
I?aeoe/ia cia/aiiy iaea?aeaieo ?acoeueoaoia.
?acoeueoaoe, io?eiaii a ?iaioi, iiaeooue aooe aeei?enoaii a eeane/iie
oai?i?
iaeiiieieo eieeaaiue, a oaeiae i?e
aeineiaeaeaiii caaea/ oai?i? noieeinoi ?ooo oa oai?i? oi?aaeiiiy.
Iniaenoee aianie caeiaoaa/a.
Ii oaii aeena?oaoei? iioaeieiaaii
5 naiinoieieo ?iaio aaoi?a oa iaeia niieueia c iaoeiaei ea?iaieeii.
Iai?yiie aeineiaeaeaiue oa iinoaiiaea caaea/ iaeaaeaoue iaoeiaiio
ea?iaieeo.
?acoeueoaoe aeena?oaoei?, yei aeiinyoueny ia caoeno, iaea?aeaii aaoi?ii
naiinoieii.
Ai?iaaoeiy ?acoeueoaoia aeena?oaoei?.
Iniiaii ?acoeueoaoe aeena?oaoei? aeiiiaiaeaeenue oa
iaaiai?thaaeenue:
ia caniaeaiii naiiia?o
aiaeaeieo cae/aeieo aeeoa?aioeiaeueieo ?iaiyiue
Iinoeoooo iaoaiaoeee IAI Oe?a?ie;
ia Anaoe?a?inueeie iaoeiaie eiioa?aioei?
“Aeeoa?aioeiaeueii ?iaiyiiy oa ?o canoinoaaiiy” ( i.*a?iiaoei, 1996
?.);
ia iaoeiaie eiioa?aioei? “Iaeeiaeiua i?iaeaiu aeeooa?aioeeaeueiuo
o?aaiaiee e
iaoaiaoe/aneie oeceee” ( i.Iaeue/ee, 1997?.);
ia Iiaeia?iaeiie iaoeiaie eiioa?aioei? “Fourth
ternational Con-ference on Difference
Equations and Applications” ( i.Iiciaiue, Iieueoa, 1998?.).
Ioaeieaoei?.
Ca oaiith aeena?oaoei? iioaeieiaaii 6 ?iaio, ic ieo aeai – a i?iaiaeieo
oaoiaeo
ia?iiaee/ieo iaoeiaeo aeo?iaeao,
aeai – a cai?ieeo iaoeiaeo i?aoeue Iinoeoooo iaoaiaoeee IAIO,
iaeia – o cai?ieeo iaoeiaeo i?aoeue
Iiaeia?iaeii? iaoeiai? eiioa?aioei? oa iaeia – o cai?ieeo oac
Anaoe?a?inueei?
eiioa?aioei?.
No?oeoo?a i ianya aeena?oaoei?.
Aeena?oaoeieia ?iaioa neeaaea?oueny ic anooio, aeaio
?icaeieia, aeniiaeo i nieneo
oeeoiaaii? eioa?aoo?e, ui iinoeoue 89 iaca. Ianya ?iaioe neeaaea?
109 noi?iiie iaoeiiieniiai oaenoo.
INIIAIEE CIINO ?IAIOE
O anooii
iaa?oioiao?oueny aeooaeueiinoue oaie, oi?ioeth?oueny iaoa
aeineiaeaeaiiy,
aea?oueny ei?ioeee aiaeic no/aniiai noaio i?iaeai, yei aea/athoueny a
aeena?oaoei?, oa iaaiaeeoueny aiioaoeiy iaea?aeaieo ?acoeueoaoia.
A ia?oiio ?icaeiei
?icaeyaea?oueny aaaaoi/anoioia neaaei iaeiiieia e?aeiaa caaea/a aeey
eieeaieo
nenoai aeaeyaeo:
Nw33
dx/dt = P(t)x+A(x,,t,) + f(t),
d/dt = (t)/ + B(x,,t, ),
lx() = R^m,
l_1() = _1+J(x(),(), ) R^k.
Ii?yae c nenoaiith (Nw33) ?icaeyaea?oueny ona?aaeiaia ii anio eooiaeo
ciiiieo
[0,2]
e?aeiaa caaea/a:
N25
dbar x /dt=P(t)bar x + f(t),
N26
dbar / dt = (t)/ + bar B(bar x,t, ),
N27
lbar x()= R^m,
N28
l_1bar()=_1 + bar J(bar x( ), R^k,
t(0,b].
I?eionoeii, ui:
x=col(x_1,…,x_n)
D R^n, =col(_1,…,_k) R^k, k
2; t (0,b], ]0, _0] –
iaeee ia?aiao?; D – iaiaaeaia iaeanoue, R^j
(j=n,m,k) – j – aeii?iee aaeeiaeiaee i?inoi?; l, l_1 oa J,bar J –
eiiieii oa
iaeiiieii iaiaaeaii aaeoi?-ooieoeiiiaee ?icii?iinoi m oa k,
aiaeiiaiaeii;
(t)=col(_1(t),…,_k(t))
C^s (0,b] , s
k-1.
Aaeoi? – ooieoeiy c(x,,t,) =[A(x,,t, );
B(x,,t, )]- iaeei ?ac iaia?a?aii aeeoa?aioeieiaia ii
(x,,t) D R^k (0,b]
i?e eiaeiiio oieniaaiiio
(0,_0], 2 –
ia?iiaee/ia ii eiaeiie ic ciiiieo
_, ( = 1,…,m)
?iaiiii?ii aiaeiinii
(x,t, ) D (0,b](0, _0] = G oa
caaeiaieueiy? oiiae, yei caaacia/othoue
aeeceinoue ?ica’yceia caaea/ Eioi aeoiaeii? nenoaie (Nw33) oa
ona?aaeiaii? nenoaie (2)-(5):
Nw34w
a)
sup_G | c_0 | + sup_G| c_0/ x | + sup_G
| c_0/ t | +
+_| k | > 0
[sup_G | c_k | + 1/ | k |(sup_G| c_k/
x | +
+ sup_G| c_k/ t |)]
_1=const;
b)
det[W_k^T(t) W_k(t)] not=0,
Eiao?oe??ioe Oo?’? i?e aa?iiiieao
expi(k,) ?iceeaaeo ooieoei? c(x,,t,)
a ?yae
Oo?’?; k=(k_1,…,k_m)- aaeoi? c oeiei/enaeueieie eii?aeeiaoaie;
W_k(t) =[ matrix
w_1(t) … w_k(t)
… … …
w_1^(p-1)(t) … w_k^(p-1)(t) ].
Iiae ii?iith iao?eoei ooo i iaaeaei aoaeaii
?icoiioe noio iiaeoeia ?? aeaiaioia.
Ona?aaeiaiiy ii
aecia/a?ii o?aaeeoeieiei /eiii;
bar A(x,,t,) = 0;
bar J(x,,) = bar J(bar x,) .
Eaaei aa/eoe, ui ona?aaeiaia e?aeiaa caaea/a (2)-(5) ? cia/ii i?inoioith
aeey aeineiaeaeaiiy, iiae aeoiaeia caaea/a (Nw33), ineieueee ona?aaeiaia
caaea/a
?iciaaea?oueny ia e?aeiai caaea/i (N25),(N27) oa
(N26),(N28). sse iia’ycaii iiae niaith ?ica’ycee e?aeiaeo caaea/
(Nw33) oa (2)-(5)? Aeineiaeaeaiith oeueiai ieoaiiy e i?enay/aiee oeae
?icaeie.
Ni?aaaaeeeaa ianooiia
Oai?aia 1.3.
Iaoae rank [Q=lX(,a)] = n_1 n oa inio? (l_1E)^-1,
oiaei e?aeiaa caaea/a (N25)-(N28) ?ica’ycia oiaei e oieueee oiaei,
eiee
N29
P_Q_d^* –
_0^b
K(,)f()d=0 (d=m-n_1),
oa ?? ?ica’ycie ia? aeaeyae
N30
bar x(t,c_ r)=X_r(t)c_r+(Gf)(t)+X(t)Q^+,
r=n-n_1,
c_r R^r,
N31
bar
(t,c_r)=
_0+
_0^t ()
+bar B(bar x(,c_r),,
) d ,
aea
_0=(l_1E)^-1
_1 + bar J(bar x(,c_r),)-l_1
_0^[ ()/ +bar B(bar
x(,c_r),,)] d .
(Gf)(t) –
ocaaaeueiaiee iia?aoi? A?iia e?aeiai? caaea/i (1)-(3);
X(t,a) –
ii?iaeueia ooiaeaiaioaeueia iao?eoey ?iaiyiiy
d/bar x /dt=P(t)bar x ;
Q^+ – (n m) — aeii?ia inaaaeiiaa?iaia aei Q iao?eoey,
X_r(t) = X(t,a)P_Q_r,
P_Q_r – (n r)- aeii?ia iao?eoey,
neeaaeaia c iiaii? nenoaie r eiiieii-iacaeaaeieo noiaaoeia iao?eoei
P_Q: R^n N(Q)
;
P_Q_d^* – iao?eoey ?icii?iinoi d m, ceeaaeaia c iiaii? nenoaie
d – eiiieii-iacaeaaeieo ?yaeeia iao?eoei P_Q^* yea i?iaeoo? R^m
ia ioeue-i?inoi? N(Q^*) iao?eoei Q^*.
I?e aeineiaeaeaiii aeoiaeii? caaea/i (1) nii/aoeo ?icaeyiooi iae?eoe/iee
aeiaaeie,
eiee aiaeiiaiaeia ona?aaeiaia e?aeiaa caaea/a ia? ?aeeiee ?ica’ycie.
Oai?aia 1.5( iae?eoe/iee aeiaaeie ).
Iaoae aeeiiothoueny cacia/aii aeua oiiae,
oae ui
rank Q = n_1 = n = m, rank(l_1 E) = k
oa caaea/a (2)-(5) ia? ?aeeiee ii?iaeaeoth/ee ?ica’ycie. Oiaei
e?aeiaa caaea/a (Nw33) ia? ?aeeiee ?ica’ycie x(t, ,), (t,x, ), ui
iaa?oa?oueny a ii?iaeaeoth/ee ?ica’ycie
bar x(t),bar
(t) i?e ona?aaeiaiii ii
cenoaie (Nw33).
Oeae ?ica’ycie iiaeia ciaeoe ca aeiiiiiaith caiaeiiai (a aeinoaoiuei
iaeiio ieiei
ii?iaeaeoth/iai ?ica’yceo) ioa?aoeieiiai i?ooeano.
I?e aeineiaeaeaiii e?eoe/iiai aeiaaeeo, eiee aiaeiiaiaeia ona?aaeiaia
e?aeiaa caaea/a (2)-(5)
ia? nii’th ii?iaeaeoth/eo ?ica’yceia, aeiaaaeaii:
Oai?aia 1.6 (iaiaoiaeia oiiaa).
Iaoae
rank Q=n_1
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