Forced oscillations of an infinite-dimensional oscillator under impulsive perturbations
Existence and uniqueness theorems for the impulsive differential operator equation $$ \frac{d^2}{dt^2}[Au(t)] + Bu(t) = f(t, u(t))$$ are obtained. The operator A is allowed to be noninvertible. The results are applied to differential algebraic equations and partial differential equations, which are...
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| Date: | 2008 |
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| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2008
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509184918814720 |
|---|---|
| author | Vlasenko, L. A. Власенко, Л. А. Власенко, Л. А. |
| author_facet | Vlasenko, L. A. Власенко, Л. А. Власенко, Л. А. |
| author_sort | Vlasenko, L. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:46:54Z |
| description | Existence and uniqueness theorems for the impulsive differential operator equation
$$ \frac{d^2}{dt^2}[Au(t)] + Bu(t) = f(t, u(t))$$
are obtained. The operator A is allowed to be noninvertible. The results are applied to differential algebraic equations and partial differential equations, which are not equations of Kovalevskaya type. |
| first_indexed | 2026-03-24T02:37:05Z |
| format | Article |
| fulltext |
UDK 517.9
L. A. Vlasenko (Xar\kov. nac. un-t)
NESVOBODNÁE KOLEBANYQ BESKONEÇNOMERNOHO
OSCYLLQTORA PRY YMPUL|SNÁX VOZMUWENYQX
Existence and uniqueness theorems for the impulsive differential operator equation
d
dt
Au t
2
2 ( )[ ] +
+ Bu t( ) = f t u t, ( )( ) are obtained. The operator A is allowed to be noninvertible. The results are
applied to differential algebraic equations and partial differential equations, which are not equations of
Kovalevskaya type.
OderΩano teoremy isnuvannq ta [dynosti dlq dyferencial\no-operatornoho rivnqnnq
d
dt
Au t
2
2 ( )[ ] + Bu t( ) = f t u t, ( )( ) z impul\snym vplyvom. Operator A moΩe buty neoborotnym.
Rezul\taty zastosovano do dyferencial\no-alhebra]çnyx rivnqn\ ta dyferencial\nyx rivnqn\ z
çastynnymy poxidnymy ne typu Kovalevs\ko].
1. Vvedenye. Rqd zadaç fyzyky y texnyky pryvodyt k yzuçenyg uravnenyq os-
cyllqtora ˙̇u + ω2u = 0 s ympul\sn¥my vozdejstvyqmy. Takye uravnenyq ys-
sledovan¥ v [1, 2]. Esly kolebanyq ne svobodn¥e, to v pravoj çasty soderΩytsq
nekotoraq funkcyq, voobwe hovorq, nelynejno zavysqwaq ot u. Matematyçes-
kye modely rezonansn¥x πlektryçeskyx cepej [3] v¥z¥vagt ynteres k bolee
ßyrokym klassam dyfferencyal\n¥x uravnenyj, a ymenno, uravnenyj, ne raz-
reßenn¥x otnosytel\no starßej proyzvodnoj. Process¥ v systemax s raspre-
delenn¥my parametramy, mhnovenno menqgwye svoe sostoqnye v opredelenn¥e
moment¥ vremeny, opys¥vagtsq ympul\sn¥my uravnenyqmy s çastn¥my proyz-
vodn¥my. V obwem sluçae πty uravnenyq qvlqgtsq ne razreßenn¥my otnosy-
tel\no starßej proyzvodnoj po vremeny, t.8e. uravnenyqmy ne typa Kovalevskoj
yly typa Soboleva [4]. V abstraktnoj forme uravnenyq ne typa Kovalevskoj
zapys¥vagtsq v vyde neqvnoho dyfferencyal\no-operatornoho uravnenyq, u
kotoroho proyzvodn¥e po prostranstvenn¥m peremenn¥m zamenqgtsq dyffe-
rencyal\n¥my operatoramy.
V dannoj rabote budem rassmatryvat\ polulynejnoe dyfferencyal\no-ope-
ratornoe uravnenye
d
dt
Au t
2
2 ( )[ ] + Bu t( ) = f t u t, ( )( ) dlq poçty vsex t0 ≤ t ≤ t0 + τ0 (1)
s ympul\sn¥my vozdejstvyqmy
∆k Au t( )[ ] = � k k kAu t Au t0 0 0( )( – ), ( ) ( – )′( ),
(2)
∆k Au t( ) ( )′[ ] = � k k kAu t Au t1 0 0( )( – ), ( ) ( – )′( ), k m= …1, , ,
© L. A. VLASENKO, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2 155
156 L. A. VLASENKO
y naçal\n¥my uslovyqmy
( )( )Au t0 = y0, ( ) ( )Au t′ 0 = y1. (3)
Zdes\
∆k tv( )[ ] = v( )tk + 0 – v( – )tk 0 ; (4)
zamknut¥e lynejn¥e operator¥ A, B dejstvugt yz kompleksnoho banaxova
prostranstva X v kompleksnoe banaxovo prostranstvo Y y ymegt oblasty op-
redelenyq DA, DB sootvetstvenno; f t( , )v — funkcyq yz t t0 0 0, +[ ]τ × X v Y ;
� k
j ( , )v ω — funkcyy yz Ωk
1
× Ωk
2
v Y ( AD DA B∩ � Ω Ωk k
1 2, � Y); moment¥
vremeny tk zanumerovan¥ tak: t0 < t1 < … < tm < tm +1 = t0 + τ0. Uravnenye (1) ne
qvlqetsq poln¥m, tak kak ne soderΩyt çlena s pervoj proyzvodnoj. Voobwe
hovorq, uravnenye (1) nel\zq razreßyt\ otnosytel\no proyzvodnoj v sylu
v¥roΩdennosty operatora A (nalyçyq netryvyal\noho qdra). K yssledovanyg
v¥roΩdenn¥x uravnenyj neposredstvenno neprymenyma teoryq kosynus-opera-
tor-funkcyj, kak πto delaetsq dlq qvn¥x uravnenyj s edynyçn¥m operatorom
A = E [5]. Neqvn¥e, a takΩe v¥roΩdenn¥e uravnenyq (1) voznykagt, naprymer,
v πvolgcyonnoj πlektrodynamyke [6]. V ympul\sn¥x vozdejstvyqx (2) y na-
çal\n¥x uslovyqx (3) soderΩytsq operator A v otlyçye ot sootvetstvugwyx
uslovyj dlq ympul\sn¥x v¥roΩdenn¥x uravnenyj yz [7] (podrazdel¥ 6.1, 6.2).
Sm¥sl uslovyj (2), (3) m¥ poqsnym pozΩe. Zdes\ tol\ko zametym, çto dlq qvno-
ho uravnenyq s edynyçn¥m operatorom A = E πty uslovyq sohlasovan¥ s obwy-
my poloΩenyqmy teoryy system s tolçkamy [8].
Budem yspol\zovat\ sledugwye oboznaçenyq: L( , )Y X — prostranstvo oh-
ranyçenn¥x lynejn¥x operatorov yz Y v X , L( , )Y Y = L( )Y ; L t t1 0 0( , +
+ τ0; )Y 8— prostranstvo Y -znaçn¥x funkcyj, yntehryruem¥x na t t0 0 0, +[ ]τ ;
W t tm
1 0 0( , + τ0; )Y — prostranstvo Soboleva funkcyj yz L t t Y1 0 0 0( , ; )+ τ , u
kotor¥x obobwenn¥e proyzvodn¥e do porqdka m vklgçytel\no prynadleΩat
L t t1 0 0( , + τ0; )Y ; C I Xp( , ) , p = 0, 1, … , — klass X-znaçn¥x funkcyj, p raz
neprer¥vno dyfferencyruem¥x na I ⊂ R , C I X( , ) = C I X0( , ) .
2. RazloΩenyq prostranstv. S uravnenyem (1) svqzan puçok operatorov
λA + B, kotor¥j opredelen na D = DA ∩ DB ≠ 0{ }. V dal\nejßem budem pred-
polahat\, çto v nekotoroj okrestnosty beskoneçno udalennoj toçky ( )λ > C2
puçok operatorov λA + B ymeet rezol\ventu ( )λA B+ −1 ∈ L( , )Y X y pry neko-
torom celom r ≥ 0 v¥polnena ocenka
( )λA B+ −1 ≤ C r
1 λ , λ > C2. (5)
V sluçae ocenky (5) v lemmax82.1, 2.2 yz [7] utoçnqetsq vozmoΩnost\ pryme-
nenyq metoda spektral\n¥x proektorov typa Ryssa [9]. Spravedlyv¥ prqm¥e
razloΩenyq lyneala D = D1 +̇ D2 y prostranstva Y = Y1 +̇ Y2 takye, çto D2
est\ lyneal sobstvenn¥x y prysoedynenn¥x vektorov puçka µB + A v toçke µ =
= 0, Y2 = BD2, Y1 = AD1, Ker A ∩ D1 = 0{ }, Ker B ∩ D2 = 0{ }, operator¥ A, B
otobraΩagt Dj v Yj, j = 1, 2. Pust\ P1, P2 y Q1, Q 2 — par¥ vzaymno dopol-
nytel\n¥x proektorov na D1, D2 y Y1, Y2 sootvetstvenno. Zamknut¥j lynej-
n¥j operator
G = AP1 + B P2 = Q A1 + Q2B : D → Y, DG = D,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
NESVOBODNÁE KOLEBANYQ BESKONEÇNOMERNOHO OSCYLLQTORA … 157
otobraΩaet Dj v Yj, ymeet ohranyçenn¥j obratn¥j G Y X– ( , )1 ∈L , xaraktery-
zugwyjsq sledugwymy svojstvamy:
G AP−1
1 = P1, G BP−1
2 = P2, AG Q−1
1 = Q1, BG Q−1
2 = Q2,
(6)
( )F r +1
= 0, F = AG Q−1
2 .
3. Razreßymost\ abstraktnoho uravnenyq bez ympul\sn¥x vozdejstvyj.
Budem predpolahat\, çto f t( , )v , kak funkcyq ot t, pry kaΩdom v ∈ X pry-
nadleΩyt klassu L t t1 0 0( , + τ0; )Y . Reßenyem naçal\noj zadaçy (1), (3) naz¥va-
etsq funkcyq u t( ) ∈ L t t1 0 0( , + τ0; )X takaq, çto Au t( ) ∈ W t t1
2
0 0( , + τ0; )Y ,
funkcyq u t( ) poçty vsgdu udovletvorqet uravnenyg (1) y v¥polnen¥ naçal\-
n¥e uslovyq (3). Yz opredelenyq reßenyq u t( ) sleduet, çto u t D( ) ∈ pry
poçty vsex t ∈ t t0 0 0, +[ ]τ . Dlq reßenyq yz klassa L t t1 0 0( , + τ0; )X , m¥, voob-
we hovorq, ne moΩem rassmatryvat\ naçal\n¥e uslovyq vyda
u t( )0 = u0, ′u t( )0 = u1. (7)
Naçal\n¥e uslovyq (3) ymegt sm¥sl, poskol\ku funkcyq Au t( ) ∈ W t t1
2
0 0( , +
+ τ0; )Y qvlqetsq neprer¥vno dyfferencyruemoj na t t0 0 0, +[ ]τ , t. e. Au t( ) ∈
∈ C t1
0[( , t0 + τ0], Y ), posle vozmoΩnoho yzmenenyq na mnoΩestve mer¥ nul\.
Dlq qvnoho uravnenyq s edynyçn¥m operatorom A = E, sohlasno pryvedennomu
v¥ße opredelenyg reßenyq, naçal\n¥e uslovyq prynymagt vyd (7), çto sovpa-
daet s yzvestn¥my postanovkamy naçal\n¥x zadaç v sluçae reßenyj, prynadle-
Ωawyx prostranstvu Soboleva vtoroho porqdka [10] (hl. 3, razdel 8). Naçal\-
n¥e uslovyq na funkcyg Au t( ) dlq psevdoparabolyçeskyx dyfferencyal\-
n¥x uravnenyj, razreßym¥x otnosytel\no proyzvodnoj, takΩe yspol\zovalys\
v8[11].
V prostranstve Y rassmotrym vspomohatel\noe uravnenye
′′v ( )t = W tv( ) + ϕ( )t dlq poçty vsex t0 ≤ t ≤ t0 + τ0, W = − −Q BG1
1, (8)
s yntehryruemoj po Boxneru na t t0 0 0, +[ ]τ vektor-funkcyej ϕ( )t . Pust\
C t( ), S t( ) — kosynus- y synus-operator-funkcyy (razreßagwye operator¥)
uravnenyq (8), kotor¥e opredelqgtsq sledugwymy rqdamy, ravnomerno sxodq-
wymysq po operatornoj norme na kaΩdom kompaktnom otrezke yz – ∞ < t < ∞ [5]
(hl.82):
C t( ) = ch W t1 2( ) = W t
j
j j
j
2
0 2( )!=
∞
∑ , S t( ) = W W t− ( )1 2 1 2sh = W t
j
j j
j
2 1
0 2 1
+
=
∞
+∑ ( )!
. (9)
Suwestvugt poloΩytel\n¥e postoqnn¥e C0 0> , ω0 0> takye, çto
C t( ) ≤ C e t
0
0ω , S t( ) ≤ C e t
0
0ω
. (10)
Pryvedem nekotor¥e svojstva operator-funkcyj C t( ), S t( ) [5]:
S t( ) = C s ds
t
( )
0
∫ , ′C t( ) = WS t( ) , ′S t( ) = C t( ), C( )0 = E,
S( )0 = 0, C t( ) = C t( )− , S t( ) = – S t( )− , (11)
2C s S t( ) ( ) = 2S t C s( ) ( ) = S t s( )+ 8+8 S t s( – ) ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
158 L. A. VLASENKO
2C t C s( ) ( ) = C t s( )+ + C t s( – ) , 2WS t S s( ) ( ) = C t s( )+ – C t s( – ) .
Lgboe reßenye v( )t uravnenyq (8) dopuskaet predstavlenye v vyde
v( )t = C t t t( – ) ( )0 0v + S t t t( – ) ( )0 0′v +
+ S t s s ds
t
t
( – ) ( )
0
∫ ϕ dlq poçty vsex t0 ≤ t ≤ t0 + τ0. (12)
Zameçanye 1. Yz predstavlenyq (12), v¥raΩenyj (9) dlq C t( ), S t( ) y opre-
delenyq operatora W v (8) vydno, çto esly v naçal\n¥j moment vremeny t0
ymegt mesto vklgçenyq v( )t0 , ′v ( )t0 ∈ Y1, a takΩe dlq poçty vsex t ∈ t0[ , t0 +
+ τ0] pravaq çast\ ϕ( )t leΩyt v Y1, to y reßenye v( )t leΩyt v Y1 dlq
poçty vsex t ∈ t t0 0 0, +[ ]τ .
Teorema 1. Pust\ v¥polneno ohranyçenye (5); funkcyq f t( , )v : t0[ , t0 +
+ τ0] × X → Y po arhumentu t prynadleΩyt prostranstvu L t1 0( , t0 + τ0; Y )
pry kaΩdom v ∈ X , a po arhumentu v udovletvorqet uslovyg Lypßyca
f t f t w( , ) – ( , )v ≤ M wv – ∀v , w X∈ y poçty vsex t0 ≤ t ≤ t0 + τ0,
(13)
s konstantoj M, ne zavysqwej ot t y takoj, çto
M G Q–1
2 < 1; (14)
funkcyq Ff t( , )v = h t( ) ne zavysyt ot v y F jh t( ) ∈ W t t Yj
1
2 1
0 0 0
+ +( , : )τ , j =
= 0, … , r. Tohda dlq lgb¥x naçal\n¥x vektorov y0, y1 v (3) takyx, çto
Q y2 0 = (– ) ( )1
2
2
0 0
j
j
j
j
t t
j
r
d
dt
F h t[ ] =
=
∑ , Q y2 1 = (– ) ( )1
2 1
2 1
0
0
j
j
j
j
t t
j
r
d
dt
F h t
+
+ =
=
[ ]∑ , (15)
suwestvuet edynstvennoe reßenye u t( ) naçal\noj zadaçy (1), (3). Razreßy-
most\ zadaçy (1), (3) πkvyvalentna razreßymosty yntehral\noho uravnenyq
u t( ) = Φ( )( )u t ≡ G C t t Q y S t t Q y S t s Q f s u s ds
t
t
– ( – ) ( – ) ( – ) , ( )1
0 1 0 0 1 1 1
0
+ + ( )
∫ +
+ G d
dt
F Q f t u tj
j
r j
j
j−
=
∑ ( )[ ]1
0
2
2 21(– ) , ( ) dlq poçty vsex t0 ≤ t ≤ t0 + τ0. (16)
Pry proçyx uslovyqx teorem¥ sootnoßenyq (15) qvlqgtsq neobxodym¥my dlq
razreßymosty naçal\noj zadaçy (1), (3).
Zameçanye 2. Yz svojstva (13) sleduet, çto funkcyq t → f t u t, ( )( ) qvlqet-
sq πlementom prostranstva L t t1 0 0( , + τ0; )Y , esly u t( ) ∈ L t t1 0 0( , + τ0; )X .
Zameçanye 3. Esly ocenka (5) v¥polnqetsq pry r = 0, to operator F v (6)
tryvyalen: F = 0. Poπtomu funkcyq Ff t( , )v = 0 ne zavysyt ot v, a uslovyq
sohlasovanyq (15) na naçal\n¥e vektor¥ y0, y1 prynymagt vyd
Q y2 0 = 0, Q y2 1 = 0.
V πtom sluçae sootnoßenye (16) s uçetom F
0 = 00 = E prynymaet vyd
u t( ) = G C t t y S t t y S t s Q f s u s ds Q f t u t
t
t
– ( – ) ( – ) ( – ) , ( ) , ( )1
0 0 0 1 1 2
0
+ + ( ) + ( )
∫ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
NESVOBODNÁE KOLEBANYQ BESKONEÇNOMERNOHO OSCYLLQTORA … 159
Dokazatel\stvo teorem¥ 1. Prymenenye proektorov Q1, Q 2 k levoj y
pravoj çastqm uravnenyq (1) pryvodyt k πkvyvalentnoj systeme uravnenyj
d
dt
Q Au t
2
2 1 ( )[ ] = W Q Au t1 ( )[ ] + Q f t u t1 , ( )( ) , (17)
d
dt
FQ Bu t
2
2 2 ( )[ ] + Q Bu t2 ( ) = Q f t u t2 , ( )( ). (18)
Yspol\zuq formulu (12) dlq predstavlenyq reßenyq neodnorodnoho uravnenyq
(8), uravnenye (17) s naçal\n¥my uslovyqmy (3) perepyßem v πkvyvalentnoj
forme
Q Au t1 ( ) = C t t Q y( – )0 1 0 + S t t Q y( – )0 1 1 + S t s Q f s u s ds
t
t
( – ) , ( )1
0
( )∫ . (19)
Poskol\ku r — yndeks nyl\potentnosty operatora F (6), uravnenye (18)
preobrazuetsq v uravnenye
Q Bu t2 ( ) = (– ) , ( )1
0
2
2 2
j
j
r j
j
jd
dt
F Q f t u t
=
∑ ( )[ ]. (20)
Otsgda sleduet neobxodymost\ ohranyçenyj (15) na naçal\n¥e dann¥e (3).
Takym obrazom, uravnenye (1) πkvyvalentno systeme uravnenyj (19), (20).
Sledovatel\no, pry sdelann¥x predpoloΩenyqx funkcyq u t( ) ∈ L t t1 0 0( , + τ0;
X ) qvlqetsq reßenyem naçal\noj zadaçy (1), (3), esly y tol\ko esly ona udov-
letvorqet uravnenyg (16).
V prostranstve L t t1 0 0( , + τ1; )X , hde çyslo τ τ1 00∈( ], budet opredeleno
nyΩe, rassmotrym otobraΩenye Φ, opredelennoe na funkcyqx u t( ) po formu-
le (16). PokaΩem, çto pry podxodqwem v¥bore τ τ1 00∈( ], otobraΩenye Φ bu-
det sΩymagwym. S pomow\g neravenstv (10), (13) ocenyvaem normu
Φ Φ( ) – ( )u Lv
1
≤
MC G Q e ds M G Q us
L0
1
1
1
2
0
0
1
1
− ⋅ +
∫ ω
τ
– – v .
Sootnoßenye (14) pozvolqet v¥brat\ çyslo τ1 ∈ 0 0, τ( ] tak, çtob¥ otobraΩe-
nye Φ b¥lo sΩymagwym
MC G Q e0
1
1
0 1 1– –⋅ ( )ω τ < 1 1
2 0– –M G Q( )ω .
Poπtomu suwestvuet edynstvennaq nepodvyΩnaq toçka u ∈ L t t1 0 0( , + τ1; )X ,
kotoraq qvlqetsq reßenyem uravnenyq (16), a potomu y zadaçy (1), (3) na t0[ ,
t0 8+ τ1]. Esly τ1 < τ0, to, rassuΩdaq, kak y v¥ße, m¥ prodolΩym reßenye u t( )
na t0 1+[ τ , t0 1 02+ { }]min ,τ τ . Ponqtno, çto za koneçnoe çyslo ßahov m¥ odno-
znaçno prodolΩym reßenye na ves\ otrezok t t0 0 0, +[ ]τ .
Teorema dokazana.
Zameçanye 4. Yz dokazatel\stva teorem¥81 vydno, çto uslovye (14) moΩno
zamenyt\ na uslovye M̃ < 1, hde M̃ qvlqetsq konstantoj Lypßyca funkcyy
G Q f t– ( , )1
2 v : G Q f t– ( , )1
2 v – G Q f t w– ( , )1
2 ≤
˜ –M wv .
4. Razreßymost\ uravnenyq s ympul\sn¥my vozdejstvyqmy. Reßenyem
naçal\noj zadaçy (1), (3) s ympul\sn¥my vozdejstvyqmy (2) na otrezke t0[ ,
t0 8+ τ0] naz¥vaetsq funkcyq u t( ) ∈ L t1 0( , t0 + τ0; X ) takaq, çto Au t( ) ∈
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
160 L. A. VLASENKO
∈ W t t Yk k1
2
1, ;+( ) , k = 0, 1, … , m, funkcyq u t( ) udovletvorqet uravnenyg (1)
dlq poçty vsex t ∈ t t0 0 0, +[ ]τ , ympul\sn¥m vozdejstvyqm (2) y naçal\n¥m us-
lovyqm (3). Yz opredelenyq reßenyq sleduet, çto posle vozmoΩnoho yzmenenyq
na mnoΩestve mer¥ nul\ funkcyq Au t( ) neprer¥vno dyfferencyruema pry
t ≠ t1, … , tm. V toçkax t ≠ t1, … , tm funkcyq Au t( ) y ee proyzvodnaq Au t( )′ ( )
ymegt skaçky. Poπtomu ympul\sn¥e vozdejstvyq (2) ymegt sm¥sl. Pry yssle-
dovanyy neqvnoho uravnenyq v klasse yntehryruem¥x funkcyj m¥, voobwe ho-
vorq, ne moΩem rassmatryvat\ operacyy ∆k (4) nad reßenyqmy v otlyçye ot
qvnoho uravnenyq s edynyçn¥m operatorom A = E, kak, naprymer, v [12], y ot
yssledovanyq neqvn¥x uravnenyj v klase kusoçno-neprer¥vn¥x funkcyj [7]
(podrazdel¥ 6.2, 6.3).
Teorema 2. Pust\ v¥polneno ohranyçenye (5); funkcyq f t( , )v : t0[ , t0 8+
+ τ0] × X → Y po arhumentu t prynadleΩyt prostranstvu L t t Y1 0 0 0( , ; )+ τ
pry kaΩdom v ∈ X , a po arhumentu v udovletvorqet uslovyg Lypßyca (13)
s konstantoj M, ne zavysqwej ot t y udovletvorqgwej neravenstvu (14);
funkcyq Ff t( , )v = h t( ) ne zavysyt ot v y F h tj ( ) ∈ W t t Yj
1
2 1
0 0 0
+ +( , ; )τ , j =
= 0, … , r ; naçal\n¥e vektor¥ y0, y1 v (3) udovletvorqgt ohranyçenyg (15);
dlq ympul\sn¥x vozdejstvyj � k
i w( , )v : Ωk
1 × Ωk
2 → Y v (2) v¥polnen¥ soot-
noßenyq
Q wk
i
2� ( , )v = 0, k = 1, 2, … , m, i = 0, 1, v ∈Ωk
1 , w k∈Ω2 . (21)
Tohda suwestvuet edynstvennoe reßenye zadaçy (1) – (3) na otrezke t0[ , t0 8+
τ0], y πto reßenye udovletvorqet uravnenyg
u t( ) = G C t t Q y S t t Q y S t s Q f s u s ds
t
t
– ( – ) ( – ) ( – ) , ( )1
0 1 0 0 1 1 1
0
+ + ( )
∫ +
+ G d
dt
F Q f t u tj
j
j
j
j
r
– – , ( )1
2
2 2
0
1( ) ( )[ ]
=
∑ +
+
G C t t Au t Au tk k k k
t t tk
– ( – ) ( )( – ), ( ) ( – )1 0 0 0
0
� ′( )
< <
∑ +
+
G S t t Au t Au tk k k k
t t tk
– ( – ) ( )( – ), ( ) ( – )1 1 0 0
0
� ′( )
< <
∑ (22)
dlq poçty vsex t0 ≤ t ≤ t0 + τ0.
Dokazatel\stvo. Pust\ u tk ( ) — reßenye uravnenyq (1) na otrezke [tk ,
tk +1] s naçal\n¥my uslovyqmy Au tk k( )( ) = yk
0
, Au tk k( )′ ( ) = yk
1, k = 0, 1, … , m,
hde
y yi
i0 = , yk
i = Au tk
i
k–
( )
( )1( ) + � k
i
k k k kAu t Au t( )( ), ( ) ( )– –1 1 ′( ), (23)
k = 1, 2, … , m, i = 0, 1.
PokaΩem, çto v¥polnen¥ sootnoßenyq
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
NESVOBODNÁE KOLEBANYQ BESKONEÇNOMERNOHO OSCYLLQTORA … 161
Q yk
i
2 = −( ) [ ]
+
+
=
=∑ 1
2
2
0
j
j i
j i
j
r
j
t t
d
dt
F h t
k
( ) , i = 0, 1. (24)
Tohda v sylu teorem¥81 suwestvuet edynstvennoe reßenye u tk ( ), kotoroe udov-
letvorqet yntehral\nomu uravnenyg
u tk ( ) = G C t t Q y S t t Q y S t s Q f s u s dsk k k k k
t
t
k
– ( – ) ( – ) ( – ) , ( )1
1
0
1
1
1+ + ( )
∫ +
+ G d
dt
F Q f t u tj
j
j
j
r
j
k
– , ( )1
2
2
0
21−( ) ( )[ ]
=
∑ dlq poçty vsex t t tk k≤ ≤ +1. (25)
Poskol\ku v¥polnen¥ ohranyçenyq (15), sootnoßenyq (24) spravedlyv¥ pry
k = 0. Sledovatel\no, suwestvuet edynstvennoe reßenye u t0( ). Yz (25) pry
k = 0 poluçaem
Q Au ti
2 0 1( )( )( ) = −( ) [ ]
+
+
=
=∑ 1
2
2
0
1
j
j i
j i
j
r
j
t t
d
dt
F h t( ) , i = 0, 1.
Otsgda, a takΩe yz predpoloΩenyj (21) y opredelenyj (23) pry k = 1 sleduet
spravedlyvost\ ravenstv (24) pry k = 1. Tohda suwestvuet edynstvennoe reße-
nye u t1( ) , kotoroe udovletvorqet yntehral\nomu uravnenyg (25) pry k = 1.
RassuΩdaq analohyçn¥m obrazom, posledovatel\no odnoznaçno naxodym reße-
nyq u t2( ), … , u tm( ) y ubeΩdaemsq, çto ony qvlqgtsq reßenyqmy yntehral\n¥x
uravnenyj (25) sootvetstvenno pry k = 2, … , m. Reßenye u t( ) zadaçy (1) – (3)
sovpadaet s u tk ( ) poçty vsgdu na t tk k, +[ ]1 , k = 0, 1, … , m.
Ubedymsq v spravedlyvosty formul¥ (22). Yz (23), (25) pry k = 0 poluçaem,
çto u t( ) udovletvorqet uravnenyg (22) pry poçty vsex t0 ≤ t ≤ t1. S pomow\g
(22) naxodym y1
0
, y1
1
. Ymeem
Q y1 1
0
= C t t Q y( – )1 0 1 0 + S t t Q y( – )1 0 1 1 + S t s Q f s u s ds
t
t
( – ) , ( )1 1
0
1
( )∫ +
+ �1
0
1 10 0( )( – ), ( ) ( – )Au t Au t′( ),
(26)
Q y1 1
1 = WS t t Q y( – )1 0 1 0 + C t t Q y( – )1 0 1 1 + C t s Q f s u s ds
t
t
( – ) , ( )1 1
0
1
( )∫ +
+ �1
1
1 10 0( )( – ), ( ) ( – )Au t Au t′( ) .
Yz (23), (25) pry k = 1 s yspol\zovanyem predstavlenyj (26) y svojstv kosynus-
y synus-operator-funkcyj C t( ) y S t( ) (11) ustanavlyvaem, çto funkcyq u t( )
udovletvorqet uravnenyg (22) pry poçty vsex t1 ≤ t ≤ t2. Provodq analohyçn¥e
rassuΩdenyq posledovatel\no dlq otrezkov [ , ]t t2 3 , … , [ , ]t tm m +1 , poluçaem
trebuem¥j rezul\tat.
Teorema dokazana.
Zameçanye 5. Ohranyçenyq na funkcyy � k
i w( , )v (21) qvlqgtsq ne tol\ko
dostatoçn¥my dlq razreßymosty zadaçy (1) – (3), no y neobxodym¥my pry v =
= ( )( – )Au tk 0 , w = ( ) ( – )Au tk′ 0 . Ohranyçenyq (21) xaraktern¥ tol\ko dlq v¥-
roΩdennoho uravnenyq s ympul\sn¥m vozdejstvyem, kohda Q2 ≠ 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
162 L. A. VLASENKO
Dejstvytel\no, reßenye u t( ) pry poçty vsex t t tk k≤ ≤ +1, k = 1, … , m,
udovletvorqet uravnenyg (1) y naçal\n¥m uslovyqm
( ) ( )( )Au ti
k + 0 = ( ) ( – )( )Au ti
k 0 + � k
i
k kAu t Au t( )( – ), ( ) ( – )0 0′( ), i = 1, 2. (27)
Prymenqq teoremu81 na otrezkax [ , ]t tk k +1 , poluçaem neobxodym¥e ohranyçenyq
na reßenye u t( ) zadaçy (1) – (3):
Q Au ti
k2 0( ) ( )( ) + = Q Au ti
k2 0( ) ( – )( ) = −( ) [ ]
+
+
=
=∑ 1
2
2
0
j
j i
j i
j
r
j
t t
d
dt
F h t
k
( ) .
Otsgda y yz (27) sleduet trebuem¥j rezul\tat.
Pry yssledovanyy zadaçy (1) – (3) v vewestvenn¥x prostranstvax X, Y sle-
duet perejty k kompleksn¥m oboloçkam prostranstv X, Y y kompleksn¥m ras-
ßyrenyqm operatorov A, B, kak, naprymer, v [13].
5. PryloΩenyq. Rassmotrym pryloΩenyq poluçenn¥x rezul\tatov k urav-
nenyqm v koneçnomern¥x prostranstvax — dyfferencyal\no-alhebrayçeskym
y k uravnenyqm v çastn¥x proyzvodn¥x, ne razreßenn¥m otnosytel\no starßej
proyzvodnoj po vremeny, — ne typa Kovalevskoj.
5.1. PryloΩenye k ympul\sn¥m dyfferencyal\no-alhebrayçeskym urav-
nenyqm. Dyfferencyal\no-alhebrayçeskye yly v¥roΩdenn¥e uravnenyq v
koneçnomernom prostranstve v poslednee vremq qvlqgtsq oblast\g yntensyv-
noho yssledovanyq (sm. monohrafyg [14], hde pryveden obzor sootvetstvugwyx
rezul\tatov). Zdes\ m¥ rassmotrym systemu dyfferencyal\n¥x uravnenyj
′′u t2( ) + u t1( ) = f t u t u t1 1 2, ( ), ( )( ) , ′′u t2( ) + u t2( ) = f t u t u t2 1 2, ( ), ( )( ) (28)
dlq poçty vsex t t tm0 1≤ ≤ +
s ympul\sn¥my vozdejstvyqmy
∆k u t2( )[ ] = u tk2 0( )+ – u tk2 0( – ) = d u t u tk k k2 20 0( – ), ( – )′( ),
(29)
∆k u t′[ ]2( ) = ′ +u tk2 0( ) – ′u tk2 0( – ) = e u t u tk k k2 20 0( – ), ( – )′( ) , k = 1, … , m,
y naçal\n¥my uslovyqmy
u t2 0( ) = a, ′u t2 0( ) = b. (30)
Funkcyy f t x yi( , , ) ; [ , ]t tm0 1+ × C × C → C, i = 1, 2, pry fyksyrovann¥x x, y ∈
∈ C qvlqgtsq πlementamy kompleksnoho prostranstva L t tm1 0 1( , )+ y dlq poç-
ty vsex t t tm0 1≤ ≤ + udovletvorqgt uslovyqm Lypßyca
f t x y f t x yi i( , , ) – ( , , )1 1 2 2 ≤ M x x y yi 1 2
2
1 2
2– –+ , i = 1, 2,
∀ ∈x y x y1 1 2 2, , , C ,
s konstantamy Mi , ne zavysqwymy ot t; funkcyy d x yk ( , ) , e x yk ( , ) , k =
= 1, … , m, dejstvugt yz C
2
v C. V prostranstve C
2
zadaça (28) – (30) zapy-
s¥vaetsq v abstraktnoj forme (1) – (3):
A =
0 1
0 1
, B =
1 0
0 1
, f t( , )v =
f t
f t
1 1 2
2 1 2
( , , )
( , , )
v v
v v
, y0 =
a
a
,
y1 =
b
b
, � k w0 ( , )v =
d t w
d t w
k
k
( , , )
( , , )
v
v
2 2
2 2
, � k w1 ( , )v =
e t w
e t w
k
k
( , , )
( , , )
v
v
2 2
2 2
.
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NESVOBODNÁE KOLEBANYQ BESKONEÇNOMERNOHO OSCYLLQTORA … 163
Reßenye u t1( ) , u t2( ) zadaçy (28) – (30) ponymaem v sm¥sle reßenyq abstrakt-
noj zadaçy (1) – (3): u t1( ) , u t2( ) ∈ L t tm1 0 1( , )+ ; u t2( ) ∈8 W t tk k1
2
1( , )+ , k = 0, 1, … m;
funkcyy u t1( ) , u t2( ) udovletvorqgt uravnenyg (28) dlq poçty vsex
t t tm0 1≤ ≤ + ; funkcyq u t2( ) udovletvorqet ympul\sn¥m vozdejstvyqm (29) y
naçal\n¥m uslovyqm (30). Uslovye (5) v¥polneno s r = 0. Naxodym
D1 = Y1 = Lin
1
1
, D2 = Y2 = Lin
1
0
, Q1 = P1 = A,
Q2 = P2 =
1 1
0 0
–
, G = G–1 =
1 0
0 1
, W = – A,
C t( ) =
1 1
0
cos –
cos
t
t
, S t( ) =
t t t
t
sin –
sin0
.
PredpoloΩym, çto M1 + M2 < 1. S uçetom zameçanyq 4 uslovyq teorem¥82 v¥-
polnen¥. Poπtomu zadaça (28) – (30) ymeet edynstvennoe reßenye u t1( ) , u t2( ),
pry πtom
u t2( ) = a t tcos ( – )0 8+8 b t tsin ( – )0 8+8 sin ( – ) , ( ), ( )t s f s u s u s ds
t
t
2 1 2
0
( )∫ +
+ cos ( – ) ( – ), ( – ) sin ( – ) ( – ), ( – )t t d u t u t t t e u t u tk k k k k k k k
t t tk
2 2 2 20 0 0 0
0
′( ) + ′( )[ ]
< <
∑ ,
u t1( ) = u t2( ) 8+8 f t u t u t1 1 2, ( ), ( )( ) 8–8 f t u t u t2 1 2, ( ), ( )( ).
5.2. PryloΩenyq k dyfferencyal\n¥m uravnenyqm v çastn¥x proyzvod-
n¥x ne typa Kovalevskoj s ympul\sn¥my vozdejstvyqmy. Pry yssledova-
nyy πvolgcyonn¥x reΩymov πlektromahnytnoho polq v cylyndryçeskom volno-
vode s dyspersnoj sredoj [6] voznykaet odnorodnoe uravnenye ne typa Kova-
levskoj. Zdes\ m¥ yssleduem nelynejnoe vozmuwenye πtoho uravnenyq
ε ∂
∂
∂
∂
2
2
2
2
2t
u t x
x
u t x
( , )
( , )+
+ –
( , )
( , )
∂
∂
2
2
2u t x
x
k u t x+
=
= g t x u t x, , ( , )( ) dlq poçty vsex t t tm0 1≤ ≤ + , 0 ≤ x ≤ π, (31)
hde ε2
, k2
— poloΩytel\n¥e postoqnn¥e, g t x z( , , ) — funkcyq yz t0[ , tm + ]1 ×
× 0, π[ ] × C v C. Oboznaçym
l u0[ ] = l u t x0[ ]( )( , ) =
∂
∂
2
2
u t x
x
u t x
( , )
( , )+
,
l u1[ ] = l u t x1[ ]( )( , ) = ∂
∂
∂
∂t
u t x
x
u t x
2
2
( , )
( , )+
.
Lgbug funkcyg u : t, x → u t x( , ) budem takΩe rassmatryvat\ kak funkcyg
ot t so znaçenyqmy v prostranstve funkcyj ot x y zapys¥vat\ kak u t x( )( ).
Dlq uravnenyq (31) rassmatryvaem kraev¥e uslovyq
u t( , )0 = u t( , )π = 0 dlq poçty vsex t t tm0 1≤ ≤ + , (32)
naçal\n¥e uslovyq
l u x0 0[ ]( )( , ) = y x0( ), l u x1 0[ ]( )( , ) = y x1( ) dlq poçty vsex 0 ≤ x ≤ π, (33)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
164 L. A. VLASENKO
y ympul\sn¥e vozdejstvyq
∆k il u x[ ]( )[ ]( , )0 = l u t xi k[ ]( ) +( , )0 – l u t xi k[ ]( )( – , )0 =
= J x l u t x l u t xk
i
k i k, ( – , ), ( – , )0 0 0[ ]( ) [ ]( )( ) , k = 1, … , m, i = 0, 1, (34)
dlq poçty vsex 0 ≤ x ≤ π,
hde J x z zk
i ( , , )1 2 — funkcyy yz 0, π[ ] × C × C v C. Budem predpolahat\, çto
funkcyq g t x z( , , ) pry fyksyrovannom z prynadleΩyt klassu g t z x( , )( ) ∈
∈8 L t1 0( , tm +1; L2 0( , )π ) , hde L2 0( , )π — prostranstvo yntehryruem¥x s kvadra-
tom funkcyj; funkcyy J x z zk
i ( , , )1 2 pry fyksyrovann¥x z1, z 2 prynymagt
znaçenyq v L2 0( , )π . Pust\ v¥polnqgtsq uslovyq Lypßyca
g t x z g t x z( , , ) – ( , , )1 2 ≤ M z z1 2– , (35)
z1, z2 ∈C , dlq poçty vsex t t tm∈[ ]+0 1, , x ∈[ ]0, π ,
s konstantoj M, ne zavysqwej ot t, x, y
J x z z J x z zk
i
k
i( , , ) – ( , , )1 2 1 2′ ′ ≤ M z z z zk
i
1 1 2 2– –′ + ′( ), (36)
z1, ′z1, z2, ′ ∈z2 C , dlq poçty vsex x ∈[ ]0, π ,
s konstantamy Mk
i
, ne zavysqwymy ot x.
Pry sdelann¥x predpoloΩenyqx funkcyq g : x → g t x x, , ( )v( ) pry poçty
vsex fyksyrovann¥x t y funkcyy Jk
i : x → J x x w xk
i , ( ), ( )v( ) qvlqgtsq πlemen-
tamy prostranstva L2 0( , )π , esly v( )x , w x( ) ∈ L2 0( , )π . V prostranstve X =
= Y = L2 0( , )π smeßannaq zadaça (31) – (34) zapys¥vaetsq v abstraktnoj forme
(1) – (3). Dyfferencyal\n¥e operator¥ A, B opredelqgtsq kak
Av = ε2
2
2
d x
dx
xv
v
( ) ( )+
, Bv = – ( ) ( )d x
dx
k x
2
2
2v
v+
,
D = DA = DB =
�
W2
2 0( , )π =
v v v( ) ( , ), ( ) ( )x W∈ = ={ }2
2 0 0 0π π ,
hde W2
2 0( , )π — prostranstvo Soboleva porqdka 2 funkcyj yz L2 0( , )π . Pola-
haem f t( , )v : t, v( )x → g t x x, , ( )v( ), � k
i w( , )v : v( )x , w x( ) → J x x w xk
i , ( ), ( )v( ) .
Reßenye smeßannoj zadaçy (31) – (34) budem ponymat\ v sm¥sle reßenyq abs-
traktnoj zadaçy (1) – (3).
Dlq vsex
λ ≠ λn
2 =
n k
n
2 2
2 2 1
+
ε ( – )
, n = 2, 3, … ,
suwestvuet rezol\venta
( ) ( )–λ A B x+ 1v =
vn
n
nx
k n n
sin
( – )2 2 2 2
1 1+ +=
∞
∑ λε
, vn =
2
0
π
π
v( ) sinx nx dx∫ ,
dlq kotoroj v¥polnena ocenka (5) s r = 0. V dannom sluçae
Y2 = D2 = Ker A = sin x{ }, Y1 = Ker A( )⊥ , D1 = Ker A( )⊥ ∩ D,
P x2v( ) = Q x2v( ) = v1 sin x , P1 = Q1 = E – P2,
G xv( ) = ε
2 ′′ +( )v v( ) ( )x x +
1 2
1+( )k xv sin ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
NESVOBODNÁE KOLEBANYQ BESKONEÇNOMERNOHO OSCYLLQTORA … 165
G x– ( )1v =
v1
21 + k
xsin +
n
n
n
nx
=
∞
∑
2
2 21
v
ε ( – )
sin , W xv( ) =
n
n n nx
=
∞
∑
2
2λ v sin ,
C t x( ) ( )v =
n
n n t nx
=
∞
∑
2
v ch λ sin , S t x( ) ( )v =
n
n n
n
t
nx
=
∞
∑
2
v sh λ
λ
sin .
V¥polnqetsq neravenstvo
G Q f t x G Q f t w x
L
– –( , )( ) – ( , )( )1
2
1
2
2
v ≤
M
k
x w x L1 2 2+
v( ) – ( ) .
Budem predpolahat\, çto
M < 1 + k2
. (37)
Pust\ dlq vsex i = 0, 1, k = 1, … , m, v( )x , w x L( ) ( , )∈ 2 0 π v¥polnqgtsq soot-
noßenyq
y x x dxi( ) sin
0
π
∫ = 0, J x x w x x dxk
i , ( ), ( ) sinv( )∫
0
π
= 0. (38)
S pomow\g zameçanyq 4 y teorem¥82 moΩno sformulyrovat\ sledugwyj re-
zul\tat.
UtverΩdenye. Pust\ znaçenyq g t z x( , )( ) , kak funkcyy ot t, pry fyksy-
rovann¥x z prynadleΩat L t tm1 0 1, +( ; L2 0( , )π ) ; funkcyy J x z zk
i ( , , )1 2 , k =
= 1, … , m, i = 0, 1, pry fyksyrovann¥x z1, z2 prynymagt znaçenyq v L2 0( , )π ;
spravedlyv¥ uslovyq Lypßyca (35), (36) s konstantamy M , Mk
i
, ne zavysq-
wymy ot t, x, y neravenstvo (37); dlq vsex i = 0, 1, k = 1, … , m, v( )x , w x( ) ∈
∈ L2 0( , )π v¥polnqgtsq sootnoßenyq (38). Tohda smeßannaq zadaça (31) –
(34) ymeet edynstvennoe reßenye u t x( , ) takoe çto, u t x( )( ) ∈ L t tm1 0 1, +( ;
L2 0( , )π ) , u t x( )( ) ∈ 8
�
W2
2 0( , )π dlq poçty vsex t y l u t x0[ ]( )( ) ∈ W t tk k1
2
1, +( ;
L2 0( , )π ) , k = 0, … , m. Pry poçty vsex t t tm0 1≤ ≤ + y 0 ≤ x ≤ π reßenye
u t x( , ) udovletvorqet uravnenyg
u t x( , ) =
y
n
t t nxn
n
n
0
2 2
2
01ε
λ
( – )
( – ) sin
=
ch
∞
∑ +
+
y
n
t t nxn
nn
n
1
2 2
2
01ε λ
λ
( – )
( – ) sin
=
sh
∞
∑ +
+
g s u
n
t s nx dsn
nn
n
t
t
( , )
( – )
( – ) sin
ε λ
λ2 2
2 1
0 =
sh
∞
∑∫
+
+
g t u
k
x1
21
( , )
sin
+
+
J u
n
t t nxkn
n
n k
t t tk
0
2 2
2 1
0
( )
( – )
( – ) sin
ε
λ
=
ch
∞
< <
∑∑ +
+
J u
n
t t nxkn
nn
n k
t t tk
1
2 2
2 1
0
( )
( – )
( – ) sin
ε λ
λ
=
sh
∞
< <
∑∑ ,
hde
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
166 L. A. VLASENKO
yin = 2
0
π
π
y x nx dxi( ) sin∫ ,
J ukn
i ( ) =
2 0 00 1
0
π
π
J x l u t x l u t x nx dxk
i
k k, ( – , ), ( – , ) sin[ ] [ ]( )∫ ,
g t un( , ) = 2
0
π
π
g t x u t x nx dx, , ( , ) sin( )∫ , k = 1, … , m, i = 0, 1, n = 1, 2, … .
V svqzy s yzuçenyem nestacyonarn¥x system y processov predstavlqet ynte-
res yssledovanye uravnenyq (1) s nestacyonarn¥my operatoramy A t( ), B t( ) .
Dlq yssledovanyq v¥roΩdennoho uravnenyq (1) v stat\e predloΩeno v¥polnyt\
specyal\n¥e razloΩenyq prostranstv X, Y (p. 2) y sootvetstvugwee πtym raz-
loΩenyqm razbyenye uravnenyq na systemu dvux uravnenyj (17), (18). ∏tot
metod dopuskaet rasprostranenye na sluçaj nestacyonarn¥x operatorov A t( ),
B t( ) [15].
1. Samojlenko A. M., StryΩak T. H. O dvyΩenyy oscyllqtora pod dejstvyem mhnovennoj sy-
l¥8// Tr. sem. po mat. fyzyke y nelynejn¥m kolebanyqm. – Kyev, 1968. – V¥p. 4. – S. 213 –
218.
2. Samojlenko A. M., Perestgk N. A. Dyfferencyal\n¥e uravnenyq s ympul\sn¥m vozdejst-
vyem. – Kyev: Vywa ßk., 1987. – 288 s.
3. Mytropol\skyj G. A., Molçanov A. A. Maßynn¥j analyz nelynejn¥x rezonansn¥x ce-
pej.8– Kyev: Nauk. dumka, 1981. – 240 s.
4. Sobolev S. L. Zadaça Koßy dlq çastnoho sluçaq system, ne prynadleΩawyx typu Kova-
levskoj // Dokl. AN SSSR. – 1952. – 82, # 2. – S. 205 – 208.
5. Fattorini H. O. Second order linear differential equations in Banach spaces // North-Holland Math.
Stud. Notas Mat. – 1985. – 99. – 313 p.
6. Rutkas A., Vlasenko L. Implicit operator differential equations and applications to electrodyna-
mics // Math. Meth. Appl. Sci. – 2000. – 23, # 1. – P. 1 – 15.
7. Vlasenko L. A. ∏volgcyonn¥e modely s neqvn¥my y v¥roΩdenn¥my dyfferencyal\n¥my
uravnenyqmy. – Dnepropetrovsk: System. texnolohyy, 2006. – 273 s.
8. M¥ßkys A. D., Samojlenko A. M. System¥ s tolçkamy v zadann¥e moment¥ vremeny // Mat.
sb. – 1967. – 74, # 2. – S. 202 – 208.
9. Rutkas A. H. Zadaça Koßy dlq uravnenyq Ax t′( ) + Bx t( ) = f t( ) // Dyfferenc. uravne-
nyq.8– 1975. – 11, # 11. – S. 1996 – 2010.
10. Lyons Û.-L., MadΩenes ∏. Neodnorodn¥e hranyçn¥e zadaçy y yx pryloΩenyq. – M.: Myr,
1971. – 372 s.
11. Haevskyj X., Hreher K., Zaxaryas K. Nelynejn¥e operatorn¥e uravnenyq y operatorn¥e
dyfferencyal\n¥e uravnenyq. – M.: Myr, 1978. – 336 s.
12. Samojlenko A. M., Ylolov M. Neodnorodn¥e πvolgcyonn¥e uravnenyq s ympul\sn¥my voz-
dejstvyqmy // Ukr. mat. Ωurn. – 1992. – 44, # 1. – S. 93 – 100.
13. Rutkas A. G., Vlasenko L. A. Existence, uniqueness and continuous dependence for implicit semili-
near functional differential equations // Nonlinear Anal. TMA. – 2003. – 55, # 1-2. – P. 125 – 139.
14. Samojlenko A. M., Íkil\ M. I., Qkovec\ V. P. Linijni systemy dyferencial\nyx rivnqn\ z
vyrodΩennqm. – Ky]v: Vywa ßk., 2000. – 296 s.
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Poluçeno 11.09.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
|
| id | umjimathkievua-article-3145 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:37:05Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/aa/0c1222eb1e90f68e4aa87de20e150faa.pdf |
| spelling | umjimathkievua-article-31452020-03-18T19:46:54Z Forced oscillations of an infinite-dimensional oscillator under impulsive perturbations Несвободные колебания бесконечномерного осциллятора при импульсных возмущениях Vlasenko, L. A. Власенко, Л. А. Власенко, Л. А. Existence and uniqueness theorems for the impulsive differential operator equation $$ \frac{d^2}{dt^2}[Au(t)] + Bu(t) = f(t, u(t))$$ are obtained. The operator A is allowed to be noninvertible. The results are applied to differential algebraic equations and partial differential equations, which are not equations of Kovalevskaya type. Одержано теореми існування та єдиності для диференціально-операторного рівняння $$ \frac{d^2}{dt^2}[Au(t)] + Bu(t) = f(t, u(t))$$ з імпульсним впливом. Оператор А може бути необоротним. Результати застосовано до диференціально-алгебраїчних рівнянь та диференціальних рівнянь з частинними похідними не типу Ковалевської. Institute of Mathematics, NAS of Ukraine 2008-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3145 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 2 (2008); 155–166 Український математичний журнал; Том 60 № 2 (2008); 155–166 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3145/3038 https://umj.imath.kiev.ua/index.php/umj/article/view/3145/3039 Copyright (c) 2008 Vlasenko L. A. |
| spellingShingle | Vlasenko, L. A. Власенко, Л. А. Власенко, Л. А. Forced oscillations of an infinite-dimensional oscillator under impulsive perturbations |
| title | Forced oscillations of an infinite-dimensional oscillator under impulsive perturbations |
| title_alt | Несвободные колебания бесконечномерного осциллятора при импульсных возмущениях |
| title_full | Forced oscillations of an infinite-dimensional oscillator under impulsive perturbations |
| title_fullStr | Forced oscillations of an infinite-dimensional oscillator under impulsive perturbations |
| title_full_unstemmed | Forced oscillations of an infinite-dimensional oscillator under impulsive perturbations |
| title_short | Forced oscillations of an infinite-dimensional oscillator under impulsive perturbations |
| title_sort | forced oscillations of an infinite-dimensional oscillator under impulsive perturbations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3145 |
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