General time-dependent bounded perturbation of a strongly continuous semigroup
We consider an evolution family whose generator is formed by a time-dependent bounded perturbation of a strongly continuous semigroup. We do not use the condition of the continuity of a perturbation. We prove a formula for a variation of a parameter and the corresponding generalization of the Dyson-...
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| Date: | 2005 |
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Institute of Mathematics, NAS of Ukraine
2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509845493383168 |
|---|---|
| author | Kartashov, M. V. Карташов, М. В. |
| author_facet | Kartashov, M. V. Карташов, М. В. |
| author_sort | Kartashov, M. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T20:02:57Z |
| description | We consider an evolution family whose generator is formed by a time-dependent bounded perturbation of a strongly continuous semigroup. We do not use the condition of the continuity of a perturbation. We prove a formula for a variation of a parameter and the corresponding generalization of the Dyson-Phillips theorem. |
| first_indexed | 2026-03-24T02:47:34Z |
| format | Article |
| fulltext |
UDK 519.21
M. V. Kartaßov (Ky]v. nac. un-t im. T. Íevçenka)
ZAHAL|NE NEODNORIDNE ZA ÇASOM OBMEÛENE
ZBURENNQ SYL|NO NEPERERVNO} NAPIVHRUPY
We consider an evolution family with generator formed by a time-dependent bounded perturbation of a
strongly continuous semigroup. We do not use the condition of the continuity of perturbation. We prove
the formula of variation of a parameter and also the corresponding generalization of the Dyson – Phillips
theorem.
Rozhlqda[t\sq evolgcijna sim’q z heneratorom, utvorenym neodnoridnym za çasom obmeΩenym
zburennqm syl\no neperervno] napivhrupy. Umova neperervnosti zburennq ne zastosovu[t\sq.
Dovedeno formulu variaci] parametra, a takoΩ vidpovidne uzahal\nennq teoremy Dajsona –
Fillipsa.
1. Vstup. Osnovni ponqttq teori] napivhrup navedeno u roboti [1]. Knyha [2,
s.2445 – 462] mistyt\ ohlqd rezul\tativ teori] zburennq syl\no neperervnyx evo-
lgcijnyx simej, wo rozrobleni u [3] ta inßyx robotax. Ci rezul\taty sutt[vo
spyragt\sq na vidnosnu neperervnist\ funkci] zburennq (dyv. teoremu 9.17
u2[2]).
U teori] stijkosti stoxastyçnyx modelej take prypuwennq vyhlqda[ ne duΩe
doreçnym. Napryklad, rozhlqnemo neodnoridne za çasom uzahal\nennq klasyç-
noho procesu ryzyku z heneratorom
At f ( x ) = c f x t f x y dG y t f x
x
′( ) + ( ) ( − ) ( ) − ( ) ( )
− ∞
∫λ λ ,
de c — intensyvnist\ premij, G — rozpodil straxovyx vyplat, a λ ( ⋅ ) — inten-
syvnist\ neodnoridnoho puassonivs\koho potoku straxovyx vymoh. Qkwo my roz-
hlqnemo qk osnovnyj (nezburenyj) odnoridnyj klasyçnyj proces ryzyku, nepe-
rervnist\ zburennq zvedet\sq do neperervnosti intensyvnosti λ ( ⋅ ). Ce prypu-
wennq vyda[t\sq obmeΩuval\nym.
Evolgcijni sim’] z rozryvnymy obmeΩenymy ta neobmeΩenymy heneratoramy
rozhlqdalys\ u roboti [4], ale vidpovidni rezul\taty otrymano lyße dlq separa-
bel\nyx refleksyvnyx banaxovyx prostoriv. Krim toho, zadaça zburennq u [4] ne
rozhlqdalas\.
Osnovy teori] neodnoridnyx za çasom procesiv Markova vykladeno u knyzi [5].
Deqki rezul\taty stijkosti dlq zahal\nyx lancghiv Markova z dyskretnym ça-
som rozhlqnuto u [6].
2. Oznaçennq. Nexaj X — banaxiv prostir z dual\nym prostorom Y ⊂ X′,
tobto banaxovym pidprostorom sprqΩenoho prostoru X′, ta z dual\nog for-
mog 〈 y, x 〉 = y ( x ), x ∈ X, y ∈ Y, takog, wo
|| x || = sup ( 〈 y, x 〉, || y || ≤ 1, y ∈ Y ). (1)
Poznaçymo çerez L ( X ) banaxiv prostir linijnyx obmeΩenyx operatoriv na X.
Dlq koΩnoho B ∈ L ( X ) poznaçymo çerez B′ sprqΩenyj do B operator na22X′.
Nexaj T = [ 0, a ] — skinçennyj interval u R . Sim’q ( Qt , t ∈ T ) ⊂ L ( X ) [
obmeΩenog napivhrupog, qkwo
Qt + s = Qt Qs ∀t, s, t + s ∈ T, i supt ∈ T || Qt || < ∞. (2)
Cq sim’q [ syl\no neperervnog, qkwo
|| Qs x – x || → 0, s → 0, ∀x ∈ X. (3)
Henerator A tako] napivhrupy vyznaça[t\sq na wil\nij oblasti vyznaçennq
D ( A ) ⊂ X qk
© M. V. KARTAÍOV, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12 1625
1626 M. V. KARTAÍOV
A x = limh → 0 h–
1
( Qh x – x ) ∀x ∈ D ( A ), (4)
de hranycq obçyslg[t\sq u normi prostoru X.
Poznaçymo T2 = {( s, t ), s , t ∈ T, s ≤ t }. Sim’q ( Pst , ( s, t ) ∈ T2 ) ⊂ L ( X ) [
obmeΩenog evolgcijnog sim’[g, qkwo
Psu Put = Pst , s ≤ u ≤ t, s, u, t ∈ T, ta sup( s, t ) ∈ T2 || Pst || < ∞. (5)
Infinitezymal\nu funkcig ci[] evolgci] vyznaçeno qk
At x = limu ↑ t, v ↓ t ( v – u )
–
1
( Pu v x – x ) ∀x ∈ D ( At ) ∀t ∈ T, (6)
de na hranyci ∂T ma[ vykorystovuvatys\ odnostoronnq hranycq.
Oznaçennq. ObmeΩena syl\no neperervna evolgcijna sim’q ( Pst , ( s, t ) ∈
∈ T2 ) [ kvaziodnoridnog, qkwo isnugt\ obmeΩena syl\no neperervna napivhrupa
( Qt , t ∈ T ) z heneratorom ( A, D ( A ) ), obmeΩena operatorna funkciq zburen\
( Dt , y ∈ T ), a takoΩ wil\nyj pidprostir X0 ⊂ X taki, wo
X0 ⊂ D ( A ), Qs X ⊂ X0
, s > 0, X0 ⊂ D ( At ) ∀t ∈ T,
At = A + Dt na X0 ∀t ∈ T. (7)
Napivhrupa ( Qt , t ∈ T ) nazyva[t\sq bazovog dlq evolgci] ( Pst , ( s, t ) ∈ T2 ).
ZauvaΩennq. Umova (7) ekvivalentna umovi
∀t ∈ T ∀x ∈ X0 ∃ limu ↑ t, v ↓ t ( v – u )
–
1
( Pu v x – Qv – u x ) ≡ Dt x ∈ X, (8)
de hranycq obçyslg[t\sq u normi prostoru X.
SprqΩena ( ′Pst , ( s, t ) ∈ T2 ) ⊂ L ( X′ ) evolgcijna sim’q [ syl\no neperervnog,
qkwo
|| ′Puvy – y || → 0, u ↑ t, v ↓ t ∀y ∈ Y, t ∈ T. (9)
Vymirnist\ dijsnoznaçnyx funkcij budemo rozumity qk vymirnist\ za Le-
behom.
Nexaj sim’q ( mt , t ∈ T ) ⊂ X′ taka, wo funkci] 〈 mt , x 〉 : T → R obmeΩeni ta
vymirni dlq vsix x ∈ X. Vyznaçymo slabkyj intehral m duus
t
∫ = I ms
t( ) qk takyj
element X′, wo
〈 〉( ) = 〈 〉∫I m x m x dus
t
u
s
t
, , ∀x ∈ X.
Dlq sim’] ( ft , t ∈ T ) ⊂ X dual\nyj slabkyj intehral I fs
t( ) [ takym elementom
X′′, wo
〈 〉( ) = 〈 〉∫y I f y f dus
t
u
s
t
, , ∀y ∈ X′.
3. Rezul\taty.
Teorema 1 (formula variaci] parametra). Nexaj kvaziodnoridna evolgcijna
sim’q ( Pst , s, t ∈ T, s ≤ t ) ma[ bazovu napivhrupu ( Qt , t ∈ T ) i vidpovidnu obme-
Ωenu operatornu funkcig zburen\ ( Dt , t ∈ T ). Todi dlq vsix y ∈ Y ma[ misce
rivnqnnq
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
ZAHAL|NE NEODNORIDNE ZA ÇASOM OBMEÛENE ZBURENNQ … 1627
′ = ′ + ( ′)− −∫P y Q y P D Q y dust t s su u t u
s
t
, (10)
de slabkyj intehral u pravij çastyni vyznaçeno korektno.
Teorema 2 (rqd Dajsona – Fillipsa). Nexaj ( Pst , s , t ∈ T, s ≤ t ) — kvaziod-
noridna evolgcijna sim’q z bazovog napivhrupog ( Qt , t ∈ T ) ta z obmeΩenog
operatornog funkci[g ( Dt , t ∈ T ). Qkwo sprqΩena evolgcijna sim’q syl\no
neperervna, to dlq vsix y ∈ Y ta dlq koΩnoho fiksovanoho s ∈ T ma[ misce
zobraΩennq
′ = ( )
≥
∑P y U s t yst n
n
,
0
, (11)
de rqd syl\no zbiha[t\sq, a joho skladovi vyznaçagt\sq rekurentno
U s t y Q yt s0( ) = ′−, , (12)
U s t y D Q U s u y dun u t u n
s
t
+ −( ) = ( ′) ( )∫1 , , , n ≥ 0. (13)
Teorema 3 (dual\na formula variaci] parametra). Nexaj ( Pst , s, t ∈ T, s ≤ t )
— kvaziodnoridna evolgcijna sim’q z bazovog napivhrupog ( Qt , t ∈ T ) ta z
obmeΩenog operatornog funkci[g ( Dt , t ∈ T ). Qkwo sprqΩena evolgcijna
sim’q syl\no neperervna, to dlq vsix x ∈ X
P x Q x Q D P x dust t s u s u ut
s
t
= +− −∫ . (14)
4. Dovedennq. Vyznaçymo
p ( t ) = supu ≤ v ≤ t || Pu v || < ∞,
q ( t ) = sups ≤ t || Qs || < ∞, ε ( t ) = sups ≤ t || Ds || < ∞, (15)
de skinçennist\ vyplyva[ z oznaçennq obmeΩeno] evolgcijno] sim’] ta z kvaziod-
noridnosti.
Lema 1. Spravedlyvi taki tverdΩennq:
a) Qt x ∈ D ( A ) ta A Qt x = Qt A x ∀x ∈ X0 ∀t ∈ T;
b) dlq bud\-qkoho x ∈ X funkciq Qt x : T → X [ syl\no neperervnog;
c) dlq bud\-qkoho x ∈ X funkciq P uv x : T2 → X [ syl\no neperervnog u
toçci ( t, t ) : Pu v x → x, u ↑ t, v ↓ t, ∀t ∈ T;
d) dlq bud\-qkoho x ∈ X funkciq Pu v x : T2 → X [ syl\no neperervnog zliva
po u ta sprava po v u koΩnij toçci ( s, t ) ∈ T2.
Dovedennq. Ma[mo
a) A Qs x ≡ limh → 0 h–
1
( Qh Qs x – Qs x ) = Qs limh → 0 h–
1
( Qh x – x ) = Qs A x;
b) || Qs + h x – Qs x || = || Qs ( Qh x – x ) || ≤ q ( s ) || Qh x – x || → 0, h ↓ 0,
|| Qs – h x – Qs x || = || Qs – h ( Qh x – x ) || ≤ q ( s ) || Qh x – x || → 0, h ↓ 0.
TverdΩennq c) dlq x ∈ X0 [ oçevydnym naslidkom (6) ta (7).
Dlq koΩnoho x ∈ X isnugt\ xn ∈ X0 taki, wo xn → x. Todi
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1628 M. V. KARTAÍOV
lim lim, ,u t t u u t t u n nP x x P x x↑ ↓ ↑ ↓− ≤ −v v v v +
+ ( 1 + p ( a ) ) || x – xn ||,
de perßyj dodanok u pravij çastyni dorivng[ nulg, a druhyj moΩna zrobyty qk
zavhodno malym vidpovidnym vyborom n.
TverdΩennq d) [ naslidkom tverdΩennq c) ta rivnosti
Puv x – Pst x = Put ( Ptv – I ) x + ( Pus – I ) Pst x (16)
dlq u ≤ s ≤ t ≤ v.
Lema 2. Dlq vsix x ∈ X, y ∈ Y ta ( t, s ) ∈ T2 dijsna funkciq
Fu = Fu ( x, y ) ≡ 〈 y, Psu Qt – u x 〉, u ∈ [ s, t ], (17)
[ neperervnog.
Dovedennq. Nexaj x ∈ X, y ∈ Y.
Dlq dovedennq neperervnosti sprava vyberemo u ↑ v dlq fiksovanoho v.
Todi
Fv – Fu = 〈 y, Psv Qt – v x 〉 – 〈 y, Psu Qt – u x 〉 =
= 〈 y, Psu ( Qt – v – Qt – u ) x 〉 + 〈 y, ( Psv – Psu ) Qt – v x 〉 =
= 〈 ′P ysu , Qt – v ( I – Qv – u ) x 〉 + 〈 ′P ysu , ( Pu v – I ) Qt – v x 〉. (18)
Za lemog 1
| Fv – Fu | ≤
≤ || y || p ( t ) ( q ( t ) || ( I – Qv – u ) x || + || ( Pu v – I ) Qt – v x || ) → 0, u ↑ v. (19)
Dlq dovedennq neperervnosti zliva zafiksu[mo u ta obçyslymo dlq v > u
riznycg Fv – Fu :
Fv – Fu = 〈 y, Psv Qt – v x 〉 – 〈 y, Psu Qt – u x 〉 =
= 〈 y, Psv ( Qt – v – Qt – u ) x 〉 + 〈 y, ( Psv – Psu ) Qt – u x 〉 =
= 〈 yPs′v , Qt – v ( I – Qv – u ) x 〉 + 〈 ′P ysu , ( Pu v – I ) Qt – u x 〉. (20)
Za lemog 1
| Fv – Fu | ≤
≤ || y || p ( t ) ( q ( t ) || ( I – Qv – u ) x || + || ( Pu v – I ) Qt – u x || ) → 0, v ↓ u. (21)
Lemu 2 dovedeno.
Lema 3. Dlq vsix x ∈ X0 , y ∈ Y ta ( s, t ) ∈ T2 funkciq Fu ( x, y ) u (17) [
dyferencijovnog po u ∈ [ s, t ] ta
d
du
Fu ( x, y ) = 〈 y, Psu Du Qt – u x 〉 ≡ fu ( x, y ). (22)
Dovedennq. ZauvaΩymo, wo vnaslidok (8) dlq vsix x ∈ X0 ma[ misce zobra-
Ωennq
|| ( Pu v – Qv – u – h Dt ) x || = o ( || h || ), h ↓ u, (23)
pry u ↑ t, v ↓ t, v – u = h.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
ZAHAL|NE NEODNORIDNE ZA ÇASOM OBMEÛENE ZBURENNQ … 1629
Dovedemo, wo liva ta prava poxidni Fu isnugt\ ta zbihagt\sq z fu
.
Nexaj v fiksovane ta u ↑ v. Poznaçymo h = v – u. Todi za oznaçennqm (22)
Fv – Fu = h fv + h 〈 y, ( Psu – Psv ) Dv Qt – v x 〉 +
+ 〈 ′P ysu , ( Pu v – Qv – u – h Dv ) Qt – v x 〉 =
= h fv + h o ( 1 ) + o ( h ), h ↓ 0, (24)
de vykorystano neperervnist\ z lemy 1(d), vklgçennq Qt – v x ∈ X0 ta zobraΩen-
nq (23).
Zvidsy otrymu[mo livu poxidnu
d
du
Fu
−
= fu .
Oskil\ky X0 ⊂ X [ wil\nym, to dlq koΩnoho y ∈ Y ta x ∈ X z (23) vyplyva[
zobraΩennq
〈 y, ( Pu v – Qv – u – h Dt ) x 〉 = o ( h ), h ↓ 0. (25)
Nexaj u [ fiksovanym, v ↓ u ta h = v – u. Todi z vklgçennq x ∈ X0 ⊂ D ( A )
ta z oznaçennq (22) otrymu[mo
Fv – Fu – h fu = 〈 y, Psu ( Pu v – Qv – u ) Qt – v x 〉 – h fu =
= 〈 ′P ysu , ( Pu v – Qv – u – h Du ) Qt – u x 〉 +
+ 〈 ′P ysu , ( Pu v – Qv – u ) Qt – v ( I – Qv – u + h A ) x 〉 +
+ h ( 〈 y, Psu Qt – u A x 〉 – 〈 y, Psv Qt – v A x 〉 ). (26)
Tomu za oznaçennqm (17)
| Fv – Fu – h fu | ≤ o ( h ) + || y || p ( t ) ( 1 + q ( t ) ) || ( I – Qv – u + h A ) x || +
+ h | Fu ( A x, y ) – Fv ( A x, y ) | = o ( h ), h ↓ 0, (27)
de perßyj dodanok u pravij çastyni dorivng[ o ( h ) zhidno z (25), druhyj — za
oznaçennqm heneratora (4) ta ostannij — za lemog 2.
OtΩe, prava poxidna takoΩ dorivng[ fu
:
d
du
Fu
+
= fu ,
tomu funkciq Fu [ dyferencijovnog z poxidnog fu .
Lemu 3 dovedeno.
Lema 4. Nexaj mu : T → Y — syl\no neperervna funkciq. Qkwo sprqΩena
evolgcijna sim’q syl\no neperervna, to dlq vsix x ∈ X , t ∈ T dijsna funkciq
〈 mu , Du Qt – u x 〉 : [ 0, t ] → R [ vymirnog ta obmeΩenog po vidnoßenng do normy
|| x ||.
Dovedennq. Nexaj x ∈ X0
. Todi Qt – u x ∈ X0 ta zhidno z (8) ma[ misce riv-
nist\
〈 mu , Du Qt – u x 〉 = limh ↓ 0 〈 mu , h–
1
( Pu, u + h – Qh ) Qt – u x 〉. (28)
Vymirnist\ livo] çastyny [ naslidkom neperervnosti funkci] pid znakom hra-
nyci. Cq neperervnist\ vyplyva[ iz zobraΩennq
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1630 M. V. KARTAÍOV
〈 mu + ε , Pu + ε , u + ε + h Qt – u – ε x 〉 – 〈 mu , Pu , u + h Qt – u x 〉 =
= 〈 mu + ε – mu , Pu + ε, u + ε + h Qt – u – ε x 〉 + 〈 mu , Pu + ε, u + ε + h ( Qt – u – ε – Qt – u ) x 〉 +
+ 〈 ′+ +Pu u hε, mu , ( Pu + h, u + ε + h – I ) Qt – u x 〉 + 〈 ( Pu, u + ε – I ) ′ mu , Pu + ε, u + h Qt – u x 〉, (29)
de perßyj dodanok prqmu[ do nulq pry ε → 0 vnaslidok neperervnosti mu ,
druhyj — zhidno z syl\nog neperervnistg Qs, tretij — za lemog 1(d) ta ostan-
nij — vnaslidok syl\no] neperervnosti sprqΩeno] evolgcijno] sim’] (9).
ObmeΩenist\ 〈 mu , Du Qt – u x 〉 [ oçevydnog:
supu ≤ t | 〈 mu , Du Qt – u x 〉 | ≤ supu ≤ t || mu || ε ( t ) q ( t ) || x ||.
Lemu dovedeno.
Dovedennq teoremy 1. TverdΩennq teoremy dlq vsix x ∈ X, y ∈ Y ekviva-
lentne takij rivnosti:
〈 y, Pst x 〉 = 〈 〉 + 〈 〉− −∫y Q x y P D Q x dut s su u t u
s
t
, , . (30)
ZauvaΩymo, wo funkciq pid znakom intehrala zbiha[t\sq z poxidnog fu ( x, y )
u (22). Za lemamy 2 ta 3 cq funkciq [ vymirnog qk hranycq neperervnyx funk-
cij ( Fu – Fv ) ( u – v )
–
1
ta obmeΩenog:
sups ≤ u ≤ t | fu | ≤ || y || p ( t ) ε ( t ) q ( t ) || x ||. (31)
Nexaj x ∈ X0
, y ∈ Y. Za lemog 3 dlq majΩe vsix ( s, t ) ∈ T2
Ft ( x, y ) – Fs ( x, y ) = f x y duu
s
t
( )∫ , . (32)
Obydvi çastyny ci[] rivnosti neperervni po s, t. Tomu vona vykonu[t\sq dlq vsix
( s, t ) ∈ T2
. OtΩe, za oznaçennqm (17) rivnist\ (30) spravdΩu[t\sq dlq vsix x ∈
∈ X0
, y ∈ Y.
U zahal\nomu vypadku, koly x ∈ X, vyberemo xn ∈ X0 tak, wob xn → x. Todi
Ft = Ft ( x, y ) = limn → ∞ Ft ( xn
, y ), wo dovedeno u (21), ta
sups ≤ u ≤ t | fu ( x, y ) – fu ( xn
, y ) | = sups ≤ u ≤ t | fu ( x – xn
, y ) | ≤
≤ || y || p ( t ) ε ( t ) q ( t ) || xn – x || → 0, n → ∞, (33)
rivnomirno po x, y vnaslidok (31). OtΩe, rivnist\ (30), qk i (32), vykonu[t\sq
dlq vsix x ∈ X, y ∈ Y.
Teoremu 1 dovedeno.
Dovedennq teoremy 2. Zafiksu[mo s ∈ T. Poznaçymo çerez Cs banaxiv
prostir syl\no neperervnyx funkcij m = ( m ( u ), u ∈ T ) : [ s, a ] → X ′ z supre-
mum-normog || m || = supu ∈ T || m ( u ) ||.
Rozhlqnemo operator � : Cs ( Y ) → Cs ( Y ) z di[g
〈 ( � m ) ( t ), x 〉 = 〈 〉( ) −∫ m u D Q x duu t u
s
t
, , x ∈ X. (34)
Za lemog 4 funkciq pid znakom intehrala [ vymirnog ta obmeΩenog stalog
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
ZAHAL|NE NEODNORIDNE ZA ÇASOM OBMEÛENE ZBURENNQ … 1631
vyhlqdu K || x ||. Syl\na neperervnist\ po t obrazu ( � m ) ( t ) vyplyva[ z syl\no]
neperervnosti Q t – u x ta z obmeΩenosti pidintehral\no] funkci]. Tomu � [
linijnym obmeΩenym operatorom na Cs
.
Dali, n-j stupin\ operatora � ma[ vyhlqd
〈 ( �n
m ) ( t ), x 〉 = … ( ) … …〈 〉− −∫∫∫ m u D Q D Q x du duu u u u t u
s
u
s
u
s
t
nn n
n
1 11 2 1
2
, . (35)
Tomu
|| �n
m || ≤ || m || ( ε ( a ) q ( a ) )
n
an
/ n! → 0, n → ∞,
i �n
[ operatorom styskannq dlq deqkoho n.
Z teoremy 1 vyvodymo, wo dlq koΩnoho y ∈ Y neperervna za lemog 1 funk-
ciq m ( u ) = ′P ysu zadovol\nq[ rivnqnnq
m ( t ) = ′−Q yt s + ( � m ) ( t ). (36)
Ostatoçno tverdΩennq teoremy 2 vyplyva[ z teoremy pro vidobraΩennq
styskannq, oskil\ky z oznaçen\ (12), (13) v rezul\tati rekurentnyx obçyslen\
U0 ( s, t ) y = ′−Q yt s ,
Un + 1 ( s, t ) y = � ( Un ( s, ⋅ ) ) ( t ) y = �n
+
1
( U0 ( s, ⋅ ) ) ( t ) y (37)
[dynyj rozv’qzok (36) otrymu[t\sq qk suma rqdu Nejmana:
′P yst = m ( t ) =
n≥
∑
0
�n
( U0 ( s, ⋅ ) ) ( t ) y =
n≥
∑
0
Un ( s, t ) y.
Teoremu 2 dovedeno.
Dovedennq teoremy 3. Z (35) ta (37) dlq vsix x ∈ X, y ∈ Y ta n ≥ 1 vyply-
va[ zobraΩennq
〈 Un ( s, t ) y, x 〉 = �n
( U0 ( s, ⋅ ) ) ( t ) y =
= … ( ) … …〈 〉− −∫∫∫ U s u y D Q D Q x du duu u u u t u
s
u
s
u
s
t
nn n
n
0 1 11 2 1
2
, , =
= … … …〈 〉− − −∫∫∫ y Q D Q D Q x du duu s u u u u t u
s
u
s
u
s
t
nn n
n
,
1 1 2 1
2
1 . (38)
Dlq fiksovanoho t ∈ T vyznaçymo prostir Ct ( X′′ ) syl\no neperervnyx fun-
kcij f ( u ) : [ 0, t ] → X′′ z supremum-normog ta linijnyj operator ¥ na Ct ( X′′ ) z
di[g
〈 y, ( ¥ f ) ( s ) x 〉 = 〈 〉−∫ y Q D f x duu s u u
s
t
, ∀y ∈ Y. (39)
Qk i v lemi 4, moΩna dovesty obmeΩenist\ ta vymirnist\ pidintehral\no]
funkci] ta obmeΩenist\ linijnoho operatora ¥.
Vyznaçymo V0 ( t, s ) x = Qt – s x ta rekursyvno
Vn + 1 ( t, s ) x = ¥ ( Vn ( ⋅, t ) ) ( s ) x = ¥n
+
1
( V0 ( ⋅, t ) ) ( s ) x.
Tomu za oznaçennqm (39)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1632 M. V. KARTAÍOV
〈 y, Vn ( t, s ) x 〉 = 〈 y, ¥n
( V0 ( ⋅, t ) ) ( s ) x 〉 =
= … … ( − ) …〈 〉−∫∫∫ y D Q D V t u s x du duu u u u n
s
u
s
u
s
t
nn
n
, ,
1 2 1
2
0 1 =
= … … …〈 〉− − −∫∫∫ y Q D Q D Q x du duu s u u u u t u
s
u
s
u
s
t
nn n
n
,
1 1 2 1
2
1 = 〈 Un ( s, t ) y, x 〉
vnaslidok (38).
Dali, za teoremog 1
〈 y, Pst x 〉 – 〈 y, Qt – s x 〉 =
n≥
∑
1
〈 Un ( s, t ) y, x 〉 =
n≥
∑
1
〈 y, Vn ( t, s ) x 〉 =
=
n≥
∑
0
〈 y, ¥ ( Vn ( ⋅, t ) ) ( s ) x 〉 = y V t s x
n
n, ¥ ,
≥
∑ (⋅ )
( )
0
=
= 〈 y, ¥ ( P⋅ t ) ( s ) x 〉 = y Q D P x du y Q D P x duu s u ut
s
t
u s u ut
s
t
, ,− −∫ ∫= 〈 〉 .
Teoremu 3 dovedeno.
1. Kato T. On linear differential equations in Banach spaces // Communs Pure and Appl. Math. –
1956. – 9. – P. 479 – 486.
2. Engel K.-J., Nagel R. One-parameter semigroups for linear evolution equations // Grad. Texts
Math., – 2000. – 194. – 586 p.
3. Raebiger F., Rhandi A., Schnaubelt R. Perturbation and an abstract characterization of evolution
semigroups // J. Math. Anal. and Appl. – 1996. – 198. – P. 516 – 533.
4. Hockman M. The abstract time-dependent Cauchy problem // Trans. Amer. Math. Soc. – 1968. –
133, # 1. – P. 1 – 50.
5. Gikhman I. I., Skorokhod A. V. Theory of random processes. – Moscow: Nauka, 1973. – Vol. 2. –
640 p.
6. Kartashov N. V. Strong stable Markov chains. – Utrecht: VSP, 1996. – 138 p.
OderΩano 12.01.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
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| id | umjimathkievua-article-3714 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:47:34Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0d/1148771a239a67a386427396c3f30a0d.pdf |
| spelling | umjimathkievua-article-37142020-03-18T20:02:57Z General time-dependent bounded perturbation of a strongly continuous semigroup Загальне неоднорідне за часом обмежене збурення сильно неперервної напівгрупи Kartashov, M. V. Карташов, М. В. We consider an evolution family whose generator is formed by a time-dependent bounded perturbation of a strongly continuous semigroup. We do not use the condition of the continuity of a perturbation. We prove a formula for a variation of a parameter and the corresponding generalization of the Dyson-Phillips theorem. Розглядається еволюційна сім'я з генератором, утвореним неоднорідним за часом обмеженим збуренням сильно неперервної напівгрупи. Умова неперервності збурення не застосовується. Доведено формулу варіації параметра, а також відповідне узагальнення теореми Дайсона - Філліпса. Institute of Mathematics, NAS of Ukraine 2005-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3714 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 12 (2005); 1625–1632 Український математичний журнал; Том 57 № 12 (2005); 1625–1632 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3714/4153 https://umj.imath.kiev.ua/index.php/umj/article/view/3714/4154 Copyright (c) 2005 Kartashov M. V. |
| spellingShingle | Kartashov, M. V. Карташов, М. В. General time-dependent bounded perturbation of a strongly continuous semigroup |
| title | General time-dependent bounded perturbation of a strongly continuous semigroup |
| title_alt | Загальне неоднорідне за часом обмежене збурення сильно неперервної напівгрупи |
| title_full | General time-dependent bounded perturbation of a strongly continuous semigroup |
| title_fullStr | General time-dependent bounded perturbation of a strongly continuous semigroup |
| title_full_unstemmed | General time-dependent bounded perturbation of a strongly continuous semigroup |
| title_short | General time-dependent bounded perturbation of a strongly continuous semigroup |
| title_sort | general time-dependent bounded perturbation of a strongly continuous semigroup |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3714 |
| work_keys_str_mv | AT kartashovmv generaltimedependentboundedperturbationofastronglycontinuoussemigroup AT kartašovmv generaltimedependentboundedperturbationofastronglycontinuoussemigroup AT kartashovmv zagalʹneneodnorídnezačasomobmeženezburennâsilʹnoneperervnoínapívgrupi AT kartašovmv zagalʹneneodnorídnezačasomobmeženezburennâsilʹnoneperervnoínapívgrupi |