Representation of solutions of one integro-differential operator equation
We describe solutions of an integro-differential operator equation in the class of linear continuous operators acting in spaces of functions analytic in domains.
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| Datum: | 2007 |
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| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509360724115456 |
|---|---|
| author | Linchuk, Yu. S. Лінчук, Ю. С. |
| author_facet | Linchuk, Yu. S. Лінчук, Ю. С. |
| author_sort | Linchuk, Yu. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:50:22Z |
| description | We describe solutions of an integro-differential operator equation in the class of linear continuous operators acting in spaces of functions analytic in domains. |
| first_indexed | 2026-03-24T02:39:52Z |
| format | Article |
| fulltext |
UDK 517.51
G. S. Linçuk (Çerniv. nac. un-t)
ZOBRAÛENNQ ROZV’QZKIV ODNOHO INTEHRO-
DYFERENCIAL|NOHO OPERATORNOHO RIVNQNNQ
We describe solutions of one integro-differential operator equation in the class of linear continuous
operators which act in spaces of functions analytic in domains.
Opysan¥ reßenyq odnoho yntehro-dyfferencyal\noho operatornoho uravnenyq v klasse ly-
nejn¥x neprer¥vn¥x operatorov, kotor¥e dejstvugt v prostranstvax funkcyj, analytyçeskyx
v oblastqx.
U bahat\ox matematyçnyx doslidΩennqx vyvça[t\sq xarakterystyka operatoriv,
wo zadovol\nqgt\ pevni komutacijni spivvidnoßennq. U klasyçni praci [1] u
klasi linijnyx neperervnyx operatoriv, wo digt\ u prostori cilyx funkcij, do-
slidΩuvalosq operatorne rivnqnnq vyhlqdu
Dn
T = T Dn
, (1)
de D = d
dz
— operator dyferencigvannq, a n — fiksovane natural\ne çyslo.
Zokrema, v [1] stverdΩuvalosq, wo zahal\nyj rozv’qzok rivnqnnq (1) moΩna po-
daty u vyhlqdi
T = T Pk
k
k
n
=
−
∑
0
1
, (2)
de P — linijnyj neperervnyj operator, wo di[ u prostori cilyx funkcij za
pravylom ( P f ) ( z ) = f ( ω z ), ω = exp 2π
i
n
, a koΩen z operatoriv Tk
, k =
= 0 1, n − , [ perestavnym z operatorom D. M. I. Nahnybida [2] vstanovyv, wo
formula (2) opysu[ deqku pidmnoΩynu rozv’qzkiv rivnqnnq (1), a takoΩ znajßov
matryçnym metodom usi rozv’qzky c\oho rivnqnnq u klasi linijnyx neperervnyx
operatoriv, wo digt\ u prostorax funkcij, analityçnyx u kruhovyx oblastqx.
Pizniße rivnqnnq (1) doslidΩuvalosq v inßyx klasax linijnyx neperervnyx ope-
ratoriv [3].
U cij roboti vyvçagt\sq rozv’qzky intehro-dyferencial\noho operatornoho
rivnqnnq vyhlqdu
Dn
T = TJ
n
(3)
u klasi linijnyx neperervnyx operatoriv, wo digt\ u prostorax funkcij, anali-
tyçnyx u oblastqx ( tut J — operator intehruvannq, qkyj di[ u vidpovidnomu
prostori analityçnyx funkcij za pravylom ( J f ) ( z ) = f t dt
z
( )∫0 ). Vstanovleno,
wo dlq rivnqnnq (3) [ pravyl\nym tverdΩennq, analohiçne do sformul\ovanoho
vywe rezul\tatu Del\sarta i Lionsa z [1].
Nexaj G — dovil\na oblast\ kompleksno] plowyny. Poznaçymo çerez
H ( G ) prostir usix analityçnyx u G funkcij, nadilenyj topolohi[g kompakt-
no] zbiΩnosti [4]. VvaΩatymemo, wo oblast\ G [ opuklog i mistyt\ poçatok
koordynat. Opyßemo spoçatku rozv’qzky intehro-dyferencial\noho operator-
noho rivnqnnq (3) u klasi linijnyx neperervnyx operatoriv T ∈ L ( H ( G ) ), wo
digt\ u prostori H ( G ), u vypadku n = 1.
Oskil\ky oblast\ G [ odnozv’qznog, to systema funkcij { exp ( λ z ) : λ ∈ C }
[ povnog v H ( G ). Tomu koΩen operator T ∈ L ( H ( G ) ) odnoznaçno vyzna-
© G. S. LINÇUK, 2007
136 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
ZOBRAÛENNQ ROZV’QZKIV ODNOHO INTEHRO-DYFERENCIAL|NOHO … 137
ça[t\sq za xarakterystyçnog funkci[g t ( λ, z ) = T ( exp ( λ z ) ). Z umovy nepe-
rervnosti operatora T vyplyva[, wo funkciq t ( λ, z ) [ cilog po λ, analityç-
nog po z v G i zadovol\nq[ umovu
∀K2 ⊂ G ∃ K1 ⊂ G ∃ C > 0 ∀λ ∈ C :
max , exp max Re
z K z K
t z C z
∈ ∈
( ) ≤ ( )( )
2 1
λ λ , (4)
de K1
, K 2 — kompaktni pidmnoΩyny oblasti G. Navpaky, koΩna funkciq
t ( λ, z ), qka [ cilog po λ, analityçnog po z v G i zadovol\nq[ umovu (4), [ xa-
rakterystyçnog dlq deqkoho operatora T ∈ L ( H ( G ) ). Dijsno, zafiksu[mo do-
vil\nu opuklu kompaktnu mnoΩynu K2 ⊂ G i znajdeni dlq ne] K1 ⊂ G ta C > 0
zhidno z (4). Z (4) vyplyva[, wo pry koΩnomu z ∈ K2 indykatrysa hz ( ϕ ) cilo]
funkci] t ( λ, z ) zadovol\nq[ nerivnist\ hz ( ϕ ) ≤ k ( – ϕ ), 0 ≤ ϕ ≤ 2π, de k ( ϕ ) —
oporna funkciq mnoΩyny K1
. Zdijsnyvßy peretvorennq Borelq [5] ostann\o]
funkci] za zminnog λ, oderΩymo, wo pry koΩnomu z ∈ K2 formulog
t1 ( λ, z ) = t z d( ) (− )
∞
∫ µ λµ µ, exp
0
, (5)
v qkij ßlqxom intehruvannq [ promin\ arg µ = ϕ, vyznaça[t\sq funkciq t1 ( λ, z ),
qka [ analityçnog v pivplowyni Re ( λ eiϕ
) > k ( – ϕ ). Tomu funkciq t1 ( λ, z ) [
lokal\no analityçnog na mnoΩyni � G × G [ 4 ]. Nexaj γ — zamknena
sprqmovana Ωordanova kryva, qka mistyt\sq v G i taka, wo mnoΩyna K1 znaxo-
dyt\sq vseredyni oblasti, obmeΩenij γ. Todi dlq dovil\no] funkci] f ∈ H ( G )
pry z ∈ K2 formulog
( T f ) ( z ) = 1
2
0
π
( ) ( ) (− )
∫ ∫
∞
i
f t z d dλ µ λµ µ λ
γ
, exp (6)
vyznaça[t\sq operator T ∈ L ( H ( G ) ) [4]. Oskil\ky t ( λ, z ) = T ( exp ( λ z ) ) pry
λ ∈ C i z ∈ K2
, to t ( λ, z ) [ xarakterystyçnog funkci[g pobudovanoho opera-
tora T.
Nexaj operator T ∈ L ( H ( G ) ) z xarakterystyçnog funkci[g t ( λ, z ) zado-
vol\nq[ rivnist\
D T = T J. (7)
Podiqvßy oboma çastynamy rivnosti (7) na funkcig exp ( λ z ), λ ∈ C, oderΩy-
mo, wo funkciq t ( λ, z ) pry λ ∈ C \ { 0 } i z ∈ G zadovol\nq[ dyferencial\ne
rivnqnnq
∂
∂
= ( ) − ( )( )t
z
t z z1
λ
λ ϕ, , (8)
de ϕ ( z ) = T1 — funkciq z prostoru H ( G ). Rozv’qzavßy rivnqnnq (8), oderΩy-
mo, wo pry λ ∈ C \ { 0 }, z ∈ G
t ( λ, z ) = exp expz d c
z
λ λ
ϕ τ τ
λ
τ λ
− ( ) −
+ ( )
∫1
0
. (9)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
138 G. S. LINÇUK
Oskil\ky funkciq t ( λ, z ) [ cilog po λ, to z (9) vyplyva[, wo c ( λ ) — cila
funkciq, pryçomu c ( λ ) = λ ϕk k
k
( )
=
∞
( )∑ 0
0
, a ϕ( )z — deqka funkciq z klasu
1
2
, ∞
. ZauvaΩymo, wo symvolom [ ρ, σ ), 0 < ρ < ∞, 0 < σ ≤ ∞, poznaçagt\
klas cilyx funkcij, porqdok qkyx menßyj za ρ abo Ω dorivng[ ρ, ale typ
menßyj za σ. Todi pry λ ∈ C i z ∈ G
t ( λ, z ) = λ ϕk k
k
z( )
=
∞
( )∑
0
, (10)
pryçomu cila funkciq ϕ ( z ) taka, wo funkciq t ( λ, z ) zadovol\nq[ umovu (4).
Vidnovlggçy za xarakterystyçnog funkci[g operator T, oderΩu[mo neobxid-
nist\ umov nastupno] teoremy.
Teorema 1. Nexaj G — dovil\na opukla oblast\ kompleksno] plowyny, qka
mistyt\ poçatok koordynat. Dlq toho wob operator T ∈ L ( H ( G ) ) buv roz-
v’qzkom rivnqnnq (7), neobxidno i dostatn\o, wob vin mav vyhlqd (6), de ϕ ( z ) —
deqka funkciq z klasu
1
2
, ∞
, a t ( λ, z ) vyznaça[t\sq formulog (10) i zado-
vol\nq[ umovu (4).
Dostatnist\ umov teoremy 1 vstanovlg[t\sq bezposerednim obçyslennqm.
U vypadku, koly G = { z : | z | < R }, tobto H ( G ) = AR
, 0 < R ≤ ∞, umova (4)
dlq t ( λ, z ) rivnosyl\na tomu, wo funkciq ϕ ∈
1
2
2, R
i formula (6) naby-
ra[ v c\omu vypadku vyhlqdu
( T f ) ( z ) = f zk k
k
( ) ( )
=
∞
( ) ( )∑ 0
0
ϕ .
Rivnqnnq (3) dlq dovil\noho natural\noho n rozv’qzu[t\sq za ti[g Ω sxe-
mog, wo i rivnqnnq (7).
Teorema 2. Nexaj G — dovil\na opukla oblast\ kompleksno] plowyny, qka
mistyt\ poçatok koordynat i [ invariantnog vidnosno povorotu navkolo toçky
z = 0 na kut 2π / n. Zahal\nyj rozv’qzok rivnqnnq (3) u klasi operatoriv T ∈
∈ L ( H ( G ) ) zada[t\sq formulog (2), de Tk
, k = 0 1, n − , — deqki operatory
z klasu L ( H ( G ) ), wo zadovol\nqgt\ rivnqnnq (7), a P ∈ L ( H ( G ) ) i di[ za
pravylom ( P f ) ( z ) = f ( ω z ), de ω = exp 2π
i
n
.
Dovedennq. Nexaj operator T ∈ L ( H ( G ) ) z xarakterystyçnog funkci[g
t ( λ, z ) zadovol\nq[ rivnqnnq (3). Todi pry λ ∈ C \ { 0 } i z ∈ G funkciq t ( λ, z )
zadovol\nq[ rivnqnnq
∂
∂
= ( ) − ( )
=
−
∑
n
n n n
k
k
k
n
t
z
t z
k
z1 1
0
1
λ
λ
λ
λ ϕ,
!
, (11)
de ϕk ∈ H ( G ), pryçomu ϕk ( z ) = T zk
, k = 0 1, n − . Rozv’qzavßy dyferencial\ne
rivnqnnq (11) metodom variaci] stalyx, oderΩymo, wo zahal\nyj rozv’qzok riv-
nqnnq (11) zada[t\sq formulog
t ( λ, z ) = t zk
k
k
n
( )
=
−
∑ ω λ,
0
1
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
ZOBRAÛENNQ ROZV’QZKIV ODNOHO INTEHRO-DYFERENCIAL|NOHO … 139
de koΩna z funkcij tk ( λ, z ) [ xarakterystyçnog funkci[g deqkoho operatora
Tk ∈ L ( H ( G ) ), qkyj [ rozv’qzkom rivnqnnq (7). Vidnovlggçy za xarakterys-
tyçnog funkci[g t ( λ, z ) operator T , oderΩu[mo, wo T ma[ vyhlqd (2). Toj
fakt, wo koΩen operator T, qkyj vyznaça[t\sq formulog (2), zadovol\nq[
rivnqnnq (3), vstanovlg[t\sq bezposeredn\og perevirkog.
1. Delsartes J., Lions J. L. Transmutations d’oprateurs differentieles dans le domaine complexe //
Comment. math. helv. – 1957. – 32, # 2. – P. 113 – 128.
2. Nahnybida M. I. Klasyçni operatory v prostorax analityçnyx funkcij. – Ky]v, 1995. –
297Ps.
3. Korobejnyk G. F. Operator¥ sdvyha na çyslov¥x semejstvax. – Rostov-na-Donu: Yzd-vo
Rostov. un-ta, 1983. – 156 s.
4. Köthe G. Dualität in der Funktionentheorie // J. reine und angew. Math. – 1953. – 191, # 1 – 2. –
S. 30 – 49.
5. Leont\ev A. F. Rqd¥ πksponent. – M.: Nauka, 1976. – 536 s.
OderΩano 10.07.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
|
| id | umjimathkievua-article-3297 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:39:52Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/14/370c6b98fe3da24691b9c5501a852e14.pdf |
| spelling | umjimathkievua-article-32972020-03-18T19:50:22Z Representation of solutions of one integro-differential operator equation Зображення розв'язків одного інтегро-диференціального операторного рівняння Linchuk, Yu. S. Лінчук, Ю. С. We describe solutions of an integro-differential operator equation in the class of linear continuous operators acting in spaces of functions analytic in domains. Описаны решения одного интегро-дифференциального операторного уравнения в классе линейных непрерывных операторов, которые действуют в пространствах функций, аналитических в областях. Institute of Mathematics, NAS of Ukraine 2007-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3297 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 1 (2007); 136–139 Український математичний журнал; Том 59 № 1 (2007); 136–139 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3297/3339 https://umj.imath.kiev.ua/index.php/umj/article/view/3297/3340 Copyright (c) 2007 Linchuk Yu. S. |
| spellingShingle | Linchuk, Yu. S. Лінчук, Ю. С. Representation of solutions of one integro-differential operator equation |
| title | Representation of solutions of one integro-differential operator equation |
| title_alt | Зображення розв'язків одного інтегро-диференціального операторного рівняння |
| title_full | Representation of solutions of one integro-differential operator equation |
| title_fullStr | Representation of solutions of one integro-differential operator equation |
| title_full_unstemmed | Representation of solutions of one integro-differential operator equation |
| title_short | Representation of solutions of one integro-differential operator equation |
| title_sort | representation of solutions of one integro-differential operator equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3297 |
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