On generalized solutions of differential equations with operator coefficients
We prove a theorem on the smoothness of generalized solutions of differential equations with operator coefficients.
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| Date: | 2006 |
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| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509588880621568 |
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| author | Chernobai, O. B. Чернобай, О. Б. |
| author_facet | Chernobai, O. B. Чернобай, О. Б. |
| author_sort | Chernobai, O. B. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:55:42Z |
| description | We prove a theorem on the smoothness of generalized solutions of differential equations with operator coefficients. |
| first_indexed | 2026-03-24T02:43:30Z |
| format | Article |
| fulltext |
UDK 517.9
O.�B.�Çernobaj (Nac. akad. DPS Ukra]ny, Irpin\)
PRO UZAHAL|NENI ROZV’QZKY DYFERENCIAL|NYX
RIVNQN| Z OPERATORNYMY KOEFICI{NTAMY
The theorem on the smoothness of generalized solutions of differential equations with operational
coefficients is proved.
Dovedeno teoremu pro hladkist\ uzahal\nenyx rozv’qzkiv dyferencial\nyx rivnqn\ z operator-
nymy koefici[ntamy.
U perßij çastyni statti rozhlqdagt\sq bil\ß prosti uzahal\neni rozv’qzky ope-
ratornoho rivnqnnq L+
u = 0, koly hil\bertovyj prostir, qkyj mistyt\ znaçennq
vektor-funkcij, [ fiksovanym. U druhij çastyni budemo rozhlqdaty osnawennq
c\oho prostoru. Otrymani rezul\taty moΩna uzahal\nyty na vypadok bil\ß
zahal\nyx linijnyx dyferencial\nyx rivnqn\ z operatornymy koefici[ntamy, v
qkyx isnu[ fundamental\nyj rozv’qzok z vlastyvostqmy, podibnymy do vlas-
tyvostej 1 – 4, wo navedeni nyΩçe. Rezul\taty ci[] roboty pov’qzani z rezul\-
tatamy [1 – 5].
1. Nexaj H — povnyj, separabel\nyj, kompleksnyj hil\bertovyj prostir zi
skalqrnym dobutkom ( ⋅ , ⋅ ) i normog || ⋅ || , L ( H ) — sukupnist\ usix obmeΩenyx
operatoriv u prostori H . Poznaçymo I = ( a , b ) , Ĩ — joho zamykannq.
Rozhlqnemo L2
( H , I ) = L2
( I ) ⊗ H , de L 2
( I ) — prostir L 2, pobudovanyj za
mirog Lebeha dx na intervali I (dyv. [6], hl. 1, § 3).
Dlq dovil\noho k = 0, 1, … viz\memo vidome (dyv. [7]) hil\bertove osnawennq
prostoru L2
( I ) sobolevs\kymy prostoramy
W Ik
2
− ( ) ⊃ L2
( I ) ⊃ W Ik
2 ( )
i pobudu[mo tenzorni dobutky takyx prostoriv:
W Ik
2
− ( ) ⊗ H = W H Ik
2
− ( , ), W Ik
2 ( ) ⊗ H = W H Ik
2 ( , ).
Otryma[mo hil\bertove osnawennq prostoru L2
( H , I ) :
W H Ik
2
− ( , ) ⊃ L2
( H , I ) ⊃ W H Ik
2 ( , ).
U prostori H rozhlqnemo dovil\nyj obmeΩenyj operator A : H � H , A∗ —
joho sprqΩennq. Pobudu[mo dyferencial\nyj vyraz
( L u ) ( x ) = du
dx
+ A u , u ∈ W H Ik
2 ( , ). (1)
Formal\no sprqΩenyj dyferencial\nyj vyraz vidnosno prostoru L2
( H , I ) ma[
vyhlqd
( L+
u ) ( x ) = –
du
dx
+ A∗
u , u ∈ W H Ik
2 ( , ). (2)
MnoΩynu finitnyx na I vektornoznaçnyx funkcij z W H Ik
2 ( , ) poznaçymo çe-
rez W H Ik
2 0, ( , ), mnoΩynu vsix neperervnyx vektornoznaçnyx funkcij Ĩ � x �
� f ( x ) ∈ H — çerez C H I( ), ˜ , a mnoΩynu finitnyx na I , k raziv neperervno
dyferencijovnyx funkcij z C H I( ), ˜ — çerez C H Ik
0 ( ), , k = 0, 1, … , ∞ .
Vektornoznaçnu funkcig ϕ ( x ) ∈ W H Il
2
− ( , ) , l = 1, 2, … , nazvemo uzahal\ne-
nym rozv’qzkom rivnqnnq L+
u = 0 vseredyni intervalu I , qkwo
© O.<B.<ÇERNOBAJ, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5 715
716 O.<B.<ÇERNOBAJ
( , )
( , )
ϕ Lv
L H I2 = 0 ∀u ∈ C H I0
∞( ), . (3)
Pid fundamental\nym rozv’qzkom budemo rozumity (dyv. [7]) operatorno-
znaçnu funkcig
Ĩ × Ĩ � ( x , ξ ) � E ( x , ξ ) ∈ L ( H ) ,
wo ma[ taki vlastyvosti:
1) pry koΩnomu fiksovanomu ξ ∈ Ĩ , x ≠ ξ isnugt\ çastynni poxidni
( Dx E ) ( x , ξ ) qk zavhodno vysokoho porqdku, neperervni po ( x , ξ ) v koΩnomu z
trykutnykiv
{ ( x , ξ ) ∈ Ĩ × Ĩ | x ≤ ξ } , { ( x , ξ ) ∈ Ĩ × Ĩ | x ≥ ξ } ; (4)
2) pry ξ ∈ I
E ( ξ + 0 , ξ ) – E ( ξ – 0 , ξ ) = 1;
3) spravdΩu[t\sq rivnist\
L E x f d
I
( , ) ( )ξ ξ ξ∫
= f ( x ) ,
de x ∈ Ĩ , f ( x ) — vektornoznaçna funkciq z prostoru C ( H , I ) ;
4) fundamental\nyj rozv’qzok zavΩdy isnu[, i joho moΩna zapysaty u
vyhlqdi
E ( x , ξ ) =
1
2
1
2
e x
e x
A x
A x
( )
( )
, ,
, .
ξ
ξ
ξ
ξ
−
−
<
− >
Teorema 1. Bud\-qkyj uzahal\nenyj rozv’qzok ϕ ( x ) ∈ W H Il
2
− ( , ) , l = 1, 2, … ,
rivnqnnq L+
u = 0 v dijsnosti vxodyt\ u W H Ip
2 ( , ) dlq bud\-qkoho p = 1, 2, … .
Dovedennq (uzahal\ng[ na dyferencial\ni vyrazy z operatornymy koefici-
[ntamy vidpovidne dovedennq z [7] hl. 16, §<6, p.<1)). Spoçatku pokaΩemo, wo dlq
koΩno] toçky x0 ∈ I isnu[ okil U ( x0 ) = ( x0 – ε , x0 + ε ) ⊆ I takyj, wo ϕ ( x ) ∈
∈ W H U xp
2 0, ( ), ( )loc , de indeks loc oznaça[ lokal\ne vxodΩennq u prostir.
Zafiksu[mo x0 ∈ I i vyberemo ε > 0 dosyt\ malym tak, wob ( x0 – 3ε , x0 + 3ε ) ⊆
⊆ I .
Nexaj k ( t ) ∈ C∞
( R ) anulg[t\sq pry | t | ≥ ε i dorivng[ odynyci v deqkomu
okoli nulq. Za vektornoznaçnog funkci[g ω ∈ C H U x0 0
∞( ), ( ) pobudu[mo vek-
tornoznaçnu funkcig na I :
v ( x ) = k x E x d
I
( ) ( , ) ( )−∫ ξ ξ ω ξ ξ =
= k x E x d E x d
U x U x
( ) ( , ) ( ) ( , ) ( )
( ) ( )
− −[ ] +∫ ∫ξ ξ ω ξ ξ ξ ω ξ ξ1
0 0
, x ∈ I . (5)
Cq vektornoznaçna funkciq anulg[t\sq pry | x – x0 | ≥ 2ε , tomu [ finitnog vid-
nosno I . Vona hladka — vxodyt\ u C H I0
∞( ), (ce vyplyva[ z dyferencigvannq
pid znakom intehrala ta naqvnosti poxidnyx ( Dx E ) ( x , ξ ) pry x ≠ ξ dovil\noho
porqdku) i neperervna v obox trykutnykax (4). ( ZauvaΩymo, wo taku Ω hlad-
kist\ magt\ dva intehraly rivnosti (5).) OtΩe, funkcig v ( x ) moΩna pidstavyty
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
PRO UZAHAL|NENI ROZV’QZKY DYFERENCIAL|NYX RIVNQN| … 717
u rivnist\ (3).
Vraxovugçy tretg vlastyvist\ fundamental\noho rozv’qzku, ma[mo
( L v ) ( x ) =
Lx
U x
k x E x d x( ) ( , ) ( ) ( )
( )
− −[ ]( ) +∫ ξ ξ ω ξ ξ ω1
0
, x ∈ Ĩ . (6)
Rozhlqnemo qdro K ( x , ξ ) = Lx k x E x( ) ( , )− −[ ]( )ξ ξ1 x , ξ ∈ Ĩ . Vraxovugçy
anulqcig mnoΩnykiv k x( )− −ξ 1 v okoli diahonali x = ξ ta navedeni vlastyvos-
ti fundamental\noho rozv’qzku, robymo vysnovok, wo isnugt\ poxidni dovil\no-
ho porqdku ( Dx Dξ K ) ( x , ξ ) dlq vsix x , ξ ∈ Ĩ , pryçomu neperervni po ( x , ξ ) ∈
∈ Ĩ × Ĩ .
U prostori L2
( H , I ) vyznaçymo operator
( B u ) ( x ) = K x u d
I
( , ) ( )ξ ξ ξ∫ , u ∈ L2
( H , I ) , x ∈ Ĩ . (7)
ZauvaΩymo, wo funkciq ( B u ) ( x ) neskinçenne çyslo raziv dyferencijovna.
Operator (7) moΩna rozßyryty po neperervnosti do operatora, wo di[ nepererv-
no z prostoru W H Ip
2
− ( , ) u prostir W H Ip
2 ( , ) , de p = 1, 2, … — dovil\ne fik-
sovane.
Dlq dovedennq zafiksu[mo β = 0, … , p i vvedemo qdro
Lβ ( x , ξ ) = ( )( , )D K xx
β ξ , x , ξ ∈ Ĩ .
Takym çynom,
( D
βB u ) ( x ) = Lβ ξ ξ ξ( , ) ( )x u d
I
∫ , u ∈ L2
( H , I ) , x ∈ Ĩ .
Dlq u ∈ L2
( H , I ) i bud\-qkoho f ∈ H ma[mo
D Bu x f x u f d u x f d
H H
I
H
I
β
β βξ ξ ξ ξ ξ ξ( )( ) = ( ) = ( )∫ ∫ ∗( ), ( , ) ( ), ( ), ( , )L L ≤
≤ u x f d b a u x
L H I
I
W H I W H Ip p( ), ( , ) ( ) ( , )
( , ) ( , ) ( , )
ξ ξ ξβ βL L∗ ∗( ) ≤ − ⋅∫ || || || ||−2
2 2
. (8)
Oskil\ky Lβ ( x , ⋅ ) — hladke qdro, to i Lβ ξ∗ ( , )x = ( )( , )Lβ ξx ∗ bude takym. Todi z
deqkog konstantog cβ > 0 budemo maty
|| || || ||∗ ⋅ ≤Lβ β( , )
( , )
x f c f
W H I Hp
2
.
Takym çynom, z (8) dlq bud\-qkoho f ∈ H otrymu[mo
D Bu x f u c f
H W H I Hp
β
β( )( ) ≤ || || || ||−( ),
( , )2
, x ∈ Ĩ .
Zavdqky dovil\nosti f ∈ H ce oznaça[, wo
D Bu x c u
H W H Ip
β
β( ) ≤ || || −( )
( , )2
, x ∈ Ĩ . (9)
Slid zaznaçyty, wo dlq neskinçenno dyferencijovno] vektornoznaçno]
funkci] Ĩ � x � v ( x ) ∈ H , oçevydno, vykonu[t\sq nerivnist\
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
718 O.<B.<ÇERNOBAJ
|| || || ||≤ ∈
=
∑v v
W H I x I H
p
p c D x
2 0
( , ) ˜max ( )( )β
β
, (10)
de c > cβ — deqka stala.
Z nerivnostej (9) i (10) vyplyva[
|| || || || || ||≤ ≤∈
=
∑ −Bu c D Bu x d u
W H I x I H
p
W H Ip p
2 20
( , ) ˜ ( , )
max ( )( )β
β
, d > 0. (11)
Nerivnist\ (11) oznaça[, wo operator B di[ neperervno z prostoru W H Ip
2
− ( , ) u
prostir W H Ip
2 ( , ) , wo j potribno bulo dovesty.
Ale todi sprqΩenyj vidnosno L 2
( H , I ) operator B+ di[ neperervno z pro-
storu W H Ip
2
− ( , ) u prostir W H Ip
2 ( , ) . Vykorystovugçy rivnosti (7) ta (6),
ma[mo
L v = B ω + ω , ω ∈ C H U x0 0
∞( ), ( ) .
Pidstavymo ostanng rivnist\ u spivvidnoßennq (3):
0 = ( , ) ( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , ) ( , )
ϕ ϕ ω ϕ ω ϕ ω ϕ ωLv
L H I L H I L H I L H I L H I
B B2 2 2 2 2= + = ++ .
Takym çynom, dlq bud\-qkoho ω ∈ C H U x0 0
∞( ), ( )
( , ) ( , )
( , ) ( , )
ϕ ω ϕ ω
L H I L H I
B2 2= − + ,
de – B+ϕ ∈ W H Ip
2 ( , ) , a ce oznaça[, wo
ϕ ( x ) ∈ W H Ip
2, ( , )loc . (12)
Pozbudemos\ indeksu loc u vklgçenni (12). Zhidno z teoremamy pro vkla-
dennq W H Ip
2, ( , )loc ⊂ C ( H , I ) , tomu funkciq ϕ ( x ) naleΩyt\ osnovnomu prosto-
ru. Zafiksu[mo c ∈ ( a , b ) i poznaçymo çerez ϕ rozv’qzok ω zadaçi Koßi na Ĩ =
= ( a , b ) :
( L+
ω ) ( x ) = 0, x ∈ Ĩ , ω ( c ) = ϕ ( c ) .
Zhidno z klasyçnymy teoremamy cej rozv’qzok isnu[ i vxodyt\ u W H Ip
2 ( , ) .
Z<inßoho boku, funkciq ϕ ( x ) takoΩ [ rozv’qzkom ci[] zadaçi Koßi v deqkomu
okoli toçky c . Vnaslidok [dynosti rozv’qzku zadaçi Koßi ϕ ( x ) = ω ( x ) , x ∈ I ,
otΩe, ω = ϕ ∈ W H Ip
2 ( , ) i [ rozv’qzkom rivnqnnq ( L+
ϕ ) ( x ) = 0, x ∈ Ĩ .
ZauvaΩennq 1. Oskil\ky ϕ ∈ W H Ip
2 ( , ) z qk zavhodno velykym p = 1, 2, … ,
to ce oznaça[, wo ϕ ∈ C∞
( H , I ) .
ZauvaΩennq 2. NevaΩko perekonatysq, wo teorema 1 [ pravyl\nog i dlq
bil\ß zahal\nyx linijnyx dyferencial\nyx rivnqn\ na R1 z operatornymy koe-
fici[ntamy i nenul\ovog pravog çastynog, dlq qkyx isnu[ fundamental\nyj
rozv’qzok E ( x , ξ ) z vlastyvostqmy, vidpovidnymy do vlastyvostej 1 – 4.
2. Perejdemo do vykladu rezul\tativ u bil\ß zahal\nomu vypadku. Zamist\
prostoru H budemo rozhlqdaty joho fiksovane osnawennq
H– ⊃ H ⊃ H+ (13)
i pry k = 0, 1, … — lancgΩok prostoriv
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
PRO UZAHAL|NENI ROZV’QZKY DYFERENCIAL|NYX RIVNQN| … 719
W H Ik
2
−
−( , ) = W Ik
2
− ( ) ⊗ H– ⊃ L2
( H , I ) =
= L2
( I ) ⊗ H ⊃ W Ik
2 ( ) ⊗ H+ = W H Ik
2 ( , )+ . (14)
Prypuska[mo, wo zvuΩennq operatora A z (1) na H + di[ u prostori H+ i
[ obmeΩenym operatorom. Todi vsi rezul\taty z p.<1 zberihagt\sq pry zamini H
na H+ i A n a A Û H+ . Ale nam potribna vidpovidna teorema dlq inßoho typu
uzahal\nenoho rozv’qzku rivnqnnq L+
u = 0, pov’qzanoho z lancgΩkom (14).
A same, vektornoznaçnu funkcig ϕ ( x ) ∈ W H Il
2
− ( , ) , l = 1, 2, … , nazvemo
uzahal\nenym rozv’qzkom c\oho rivnqnnq vseredyni intervalu I , qkwo dlq bud\-
qkoho v ∈ C H I0
∞
+( ), vykonu[t\sq spivvidnoßennq
( , )
( , )
ϕ Lv
L H I2 = 0. (15)
Teorema 2. Pry vkazanomu prypuwenni na operator A bud\-qkyj uzahal\-
nenyj rozv’qzok u sensi (15) ϕ ( x ) ∈ W H Il
2
− ( , ) , l = 1, 2, … , rivnqnnq L+
u = 0 v
dijsnosti vxodyt\ u W H Ip
2 ( , )− dlq bud\-qkoho p = 1, 2, … .
Nasampered zauvaΩymo, wo pokrawannq uzahal\nenoho rozv’qzku „v naprqm-
ku H ” nema[: vin vxodyt\ u W H Ip
2 ( , )− , a ne v W H Ip
2 ( , )+ , wo [ pryrodnym.
Dovedennq. PokaΩemo, qk z teoremy 1 pry zamini H i A n a H + i A Û H+
vyplyva[ teorema 2.
Rozhlqnemo porqd z lancgΩkamy (13) i (14) taki dva lancgΩky:
W Ik
2
− ( ) ⊃ L2
( I ) ⊃ W Ik
2 ( ), (16)
W H Ik
2
−
+( , ) = W Ik
2
− ( ) ⊗ H+ ⊃ L2
( H+ , I ) =
= L2
( I ) ⊗ H+ ⊃ W Ik
2 ( ) ⊗ H+ = W H Ik
2 ( , )+ , k = 1, 2, … . (17)
Nexaj I — standartnyj operator, pov’qzanyj iz lancgΩkom (13), Ik , Kk , L k —
taki Ω operatory, pov’qzani vidpovidno z lancgΩkamy (16), (17), (14). Oskil\ky
prostory (17), (14) utvoreni qk tenzorni dobutky, to ma[mo (dyv. [6])
Kk = Ik ⊗ 1, Lk = Ik ⊗ I , k = 1, 2, … . (18)
Vykorystovugçy lancgΩky (14), (17), v qkyx pozytyvni prostory odnakovi, i
formuly (18), dlq ϕ ∈ W H Ik
2
−
−( , ) i u ∈ W H Ik
2 ( , )+ otrymu[mo
( , ) ( , ) ( , )
( , ) ( , ) ( , )
ϕ ϕ ϕu u u
L H I k W H I k k L H Ik2
2
2
1= =
+
−
L K L . (19)
Ale zhidno z (18)
K Lk k
−1 = ( Ik ⊗ 1 )–1
( Ik ⊗ I ) = ( Ik
−1 ⊗ 1 ) ( Ik ⊗ I ) = Ik
−1
Ik ⊗ I , = 1 ⊗ I ,
tomu (19) perejde u rivnist\
( , ) ( ) ,
( , ) ( , )
( )ϕ ϕu u
L H I L H I2 21= ⊗ I , ϕ ∈ W H Ik
2
−
−( , ), u ∈ W H Ik
2 ( , )+ . (20)
Nexaj teper ϕ ( x ) ∈ W H Il
2
−
−( , ) [ uzahal\nenym rozv’qzkom rivnqnnq L+
u = 0 v
sensi (15). Todi zhidno z (20) v ∈ C H I0
∞
+( ), , oskil\ky
0 =
( , ) ( ) ,
( , ) ( , )
( )ϕ ϕL Lv v
L H I L H I2 21= ⊗ I .
Ce<oznaça[, wo (( )1 ⊗ I ϕ [ uzahal\nenym rozv’qzkom c\oho rivnqnnq v sensi (3),
de H i A zamineno na H + i A Û H+ . Qk vΩe zaznaçalosq, zhidno z teoremog<1
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
720 O.<B.<ÇERNOBAJ
(( )1 ⊗ I ϕ ∈ W H Ip
2 ( , )+ = W Ip
2 ( ) ⊗ H+ , p = 1, 2, … , tobto ϕ ∈ W Ip
2 ( ) ⊗ H – =
= W H Ip
2 ( , )− , = 1, 2, … .
ZauvaΩennq 3. ZauvaΩennq 2 [ spravedlyvym i dlq uzahal\nenyx rozv’qzkiv
typu (15) pry pevnyx umovax na vyraz L .
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hyl\bertovom prostranstve // Tam Ωe. – 1981. – 33, #<4. – S.<513 – 518.
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– Kyev, 1981. – 18<s.
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6. Horbaçuk+M.+L., Horbaçuk+V.+Y. Hranyçn¥e zadaçy dlq dyfferencyal\n¥x operatorn¥x
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OderΩano 06.06.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
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| id | umjimathkievua-article-3489 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:43:30Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c6/4c29a0f60f907e40b2e5a966c99b41c6.pdf |
| spelling | umjimathkievua-article-34892020-03-18T19:55:42Z On generalized solutions of differential equations with operator coefficients Про узагальнені розв'язки диференціальних рівнянь з операторними коефіцієнтами Chernobai, O. B. Чернобай, О. Б. We prove a theorem on the smoothness of generalized solutions of differential equations with operator coefficients. Доведено теорему про гладкість узагальнених розв'язків диференціальних рівнянь з операторними коефіцієнтами. Institute of Mathematics, NAS of Ukraine 2006-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3489 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 5 (2006); 715–720 Український математичний журнал; Том 58 № 5 (2006); 715–720 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3489/3715 https://umj.imath.kiev.ua/index.php/umj/article/view/3489/3716 Copyright (c) 2006 Chernobai O. B. |
| spellingShingle | Chernobai, O. B. Чернобай, О. Б. On generalized solutions of differential equations with operator coefficients |
| title | On generalized solutions of differential equations with operator coefficients |
| title_alt | Про узагальнені розв'язки диференціальних рівнянь з операторними коефіцієнтами |
| title_full | On generalized solutions of differential equations with operator coefficients |
| title_fullStr | On generalized solutions of differential equations with operator coefficients |
| title_full_unstemmed | On generalized solutions of differential equations with operator coefficients |
| title_short | On generalized solutions of differential equations with operator coefficients |
| title_sort | on generalized solutions of differential equations with operator coefficients |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3489 |
| work_keys_str_mv | AT chernobaiob ongeneralizedsolutionsofdifferentialequationswithoperatorcoefficients AT černobajob ongeneralizedsolutionsofdifferentialequationswithoperatorcoefficients AT chernobaiob prouzagalʹnenírozv039âzkidiferencíalʹnihrívnânʹzoperatornimikoefícíêntami AT černobajob prouzagalʹnenírozv039âzkidiferencíalʹnihrívnânʹzoperatornimikoefícíêntami |