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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