Nonlinear equations with essentially infinite-dimensional differential operators
We consider nonlinear differential equations and boundary-value problems with essentially infinite-dimensional operators (of the Laplace–Lévy type). An analog of the Picard theorem is proved.
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| Date: | 2010 |
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| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508990252777472 |
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| author | Bogdanskii, Yu. V. Statkevych, V. M. Богданський, Ю. В. Статкевич, В. М. |
| author_facet | Bogdanskii, Yu. V. Statkevych, V. M. Богданський, Ю. В. Статкевич, В. М. |
| author_sort | Bogdanskii, Yu. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:41:38Z |
| description | We consider nonlinear differential equations and boundary-value problems with essentially infinite-dimensional operators (of the Laplace–Lévy type). An analog of the Picard theorem is proved. |
| first_indexed | 2026-03-24T02:33:59Z |
| format | Article |
| fulltext |
UDK 517.986.7
G. V. Bohdans\kyj (In-n prykl. system. analizu Nac. texn. un-tu Ukra]ny „KPI”, Ky]v),
V. M. Statkevyç (Nac. texn. un-t Ukra]ny „KPI”, Ky]v)
NELINIJNI RIVNQNNQ
Z SUTT{VO NESKINÇENNOVYMIRNYMY
DYFERENCIAL|NYMY OPERATORAMY
Nonlinear didderential equations and boundary-value problems with essentially infinite-dimensional
operators (of the Laplace – Levy type) are considered. An analog of the Picard theorem is proved.
Rassmotren¥ nelynejn¥e dyfferencyal\n¥e uravnenyq y kraev¥e zadaçy s suwestvenno besko-
neçnomern¥my operatoramy (typa Laplasa – Levy). Dokazan analoh teorem¥ Pykara.
Klasyçnyj operator Laplasa – Levi bulo vvedeno v roboti [1]. Suçasnyj stan
teori] operatora Laplasa – Levi vykladeno v monohrafi] [2]. Isnu[ velyka kil\-
kist\ publikacij z ci[] tematyky (dyv., zokrema, [3, 4]). V roboti [5] zapropono-
vano sutt[vo neskinçennovymirnyj operator qk uzahal\nennq operatora Lapla-
sa8– Levi, vlastyvostqm c\oho operatora ta zadaçam, wo pov’qzani z nym, prysvq-
çeno roboty [5 – 7]. Na vidminu vid skinçennovymirnoho vypadku cej operator
druhoho porqdku zadovol\nq[ lejbnicevs\ku vlastyvist\ — [ dyferencigvan-
nqm alhebry hladkyx funkcij. Ostannij fakt dav moΩlyvist\ v roboti [8] roz-
hlqnuty linijni sutt[vo neskinçennovymirni rivnqnnq na zrazok zvyçajnyx dy-
ferencial\nyx rivnqn\.
1. Nexaj H — neskinçennovymirnyj separabel\nyj dijsnyj hil\bertiv pros-
tir, B HC ( ) — banaxiv prostir samosprqΩenyx obmeΩenyx linijnyx operatoriv
na H, J — konus nevid’[mnyx linijnyx funkcionaliv na B HC ( ) , j ∈ J — nenu-
l\ovyj funkcional takyj, wo vsi operatory z B HC ( ) skinçennoho ranhu nale-
Ωat\ joho qdru; funkcional, qkyj ma[ navedenu vlastyvist\, zhidno z [5], nazy-
va[mo sutt[vo neskinçennovymirnym.
MnoΩynu D B HC⊂ ( ) nazyva[mo majΩe kompaktnog, qkwo dlq koΩnoho
ε > 0 isnugt\ kompaktna mnoΩyna K B HC⊂ ( ) ta çysla n ∈N ta d > 0 taki,
wo K + Qn d, [ ε-sitkog dlq D (tut Qn d, — mnoΩyna operatoriv z B HC ( ) ,
ranh qkyx ne perevywu[ n, a norma ne perevywu[ d).
Zafiksu[mo R > 0. Nexaj BR = x H x∈{ ≤ R} — kulq radiusa R . Çerez
Z poznaçymo mnoΩynu vsix dijsnoznaçnyx funkcij klasu C H2( ) , nosi] qkyx
naleΩat\ BR , ′′u rivnomirno neperervna na H , a mnoΩyna ′′{u x( ) x BR∈ } [
majΩe kompaktnog. Nexaj X — zamykannq Z za normog sup ( )x BR
u x∈ , X [
komutatyvnog banaxovog alhebrog vidnosno potoçkovyx operacij. Dlq dovil\-
no] funkci] g C∈ ( )R tako], wo g( )0 = 0, vykonu[t\sq g u� ∈ X.
Sutt[vo neskinçennovymirnyj eliptyçnyj operator L X Z: ⊃ → X zada[t\-
sq formulog ( ) ( )Lu x =
1
2
j u x′′( )( ) [5, 6]. Vin dopuska[ zamykannq A L= , vy-
znaçene na D A( ) , qke [ heneratorom ( )C0 -pivhrupy stysku T t( ) u prostori X,
pry c\omu ∀ ≥t 0 ∀ ∈u Z : T t u( ) ∈ Z.
Pivhrupa T t( ) ma[ taki vlastyvosti [6]:
1) isnu[ t0 0> take, wo T t( )0 = 0 (nil\potentnist\ pivhrupy);
2) ∀ u , v ∈X ∀ ≥t 0 : T t u( ) ( )v = T t u T t( ) ( )⋅ v (mul\typlikatyvnist\ piv-
hrupy);
© G. V. BOHDANS|KYJ, V. M. STATKEVYÇ, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11 1571
1572 G. V. BOHDANS|KYJ, V. M. STATKEVYÇ
3) ∀ ∈u X ∀ ≥t 0 ∀ ∈g C( )R tako], wo g( )0 = 0 : T t g u( ) ( )( ) = g T t u( )( ) .
Poznaçymo çerez Y klas takyx poverxon\ v H, qki moΩna podaty u vyhlqdi
S = x H∈{ g x( ) = }1 , de g Z∈ ta inf ( )x S g x∈ ′ > 0. Sε — ε-okil poverxni S.
Zhidno z [7] vidkrytu obmeΩenu oblast\ G H⊂ z meΩeg S klasu Y bude-
mo nazyvaty L-opuklog, qkwo ∀ ∈x Sε ∩ G : g x( ) > 1 ta sup ( ) ( )x S Lg x∈ < 0.
Poznaçymo çerez G ]] zamykannq.
Çerez Z G( ) poznaçymo mnoΩynu vsix dijsnyx funkcij klasu C G2( ) , u
qkyx ′′u x( ) rivnomirno neperervna na G, a mnoΩyna ′′ ∈{ }u x x G( ) [ majΩe
kompaktnog. Nexaj X G( ) — zamykannq Z G( ) za normog sup ( )x S u x∈ .
Zaznaçymo, wo x G ∈ Z G( ) dlq dovil\no] u Z∈ ; x G ∈ X G( ) dlq dovil\no]
u X∈ . Zadamo operator LG : X G Z G( ) ( )⊃ → X G( ) formulog ( ) ( )L xG =
=
1
2
j u x′′( )( ) . Vin korektno vyznaçenyj ta dopuska[ zamykannq A LG G= , vy-
znaçene na D AG( ) . Oçevydno, wo u D A∈( )( ) ⇒ u D AG G∈( )( ) .
U roboti [7] dovedeno, wo isnu[ i do toho Ω [dyna funkciq θ( )x , rivnomirno
neperervna na G , qka zadovol\nq[ umovy: θ( )x > 0 na G, θ( )x = 0 na S,
θ G∈ X G( ) ; ′θ ( )x isnu[ i rivnomirno neperervna na G ; AG Gθ( ) = – 1 skriz\ v
oblasti G (fundamental\na funkciq oblasti G).
Vkladennq poverxni S v H induku[ rimanovu metryku na poverxni S. Nexaj
∇ — zv’qznist\ Levi – Çyvity, qka vidpovida[ cij metryci. Dlq koΩnoho x S∈
prostir H rozklada[t\sq v ortohonal\nu sumu H = T S T Sx x⊕ ⊥( ) . Operatoru
∇2u x( ) , vyznaçenomu na dotyçnomu prostori T Sx , vidpovida[ operator
∇ ⊕2 0u x( ) ∈ B HC ( ) , qkyj teΩ poznaçymo çerez ∇2u x( ) . MnoΩynu dijsnyx
funkcij na poverxni S, u qkyx ∇2u isnu[ i rivnomirno neperervna na S ta
∇ ∈{ }2u x x S( ) [ majΩe kompaktnog mnoΩynog, zamknemo za normog
sup ( )x S u x∈ ; zamykannq poznaçymo çerez X S( ) .
Nexaj na poverxng S klasu Y dodatkovo nakladeno umovy: g x Z( ) ∈ , ′′′g
isnu[ ta rivnomirno neperervna na H, mnoΩyna ′ ⋅( )′′{ g z y( ), ( ) 8 z H∈ , z ≤ 1,
y S∈ } [ majΩe kompaktnog mnoΩynog, a ϕ ∈X S( ) . Zhidno z [7] isnu[ i do toho
Ω [dyna funkciq v, vyznaçena na G , taka, wo v G ∈ X G( ) , AG Gv( ) = 0 ta
v S = ϕ, ]] moΩna vyznaçyty za formulog v( )x = T x xθ ϕ( ) ( )( )( ) , de ϕ — do-
vil\ne prodovΩennq ϕ na H. U roboti [7] dovedeno, wo dovil\nu funkcig
u X G∈ ( ) moΩna prodovΩyty do funkci] u X∈ ; neperervne prodovΩennq u
na S da[ funkcig ˆ ( )u X S∈ .
2. Poznaçymo çerez F ( )Q banaxovu alhebru usix dijsnoznaçnyx obmeΩenyx
funkcij, vyznaçenyx na dovil\nij mnoΩyni Q (vidnosno potoçkovyx operacij, z
sup-normog).
Nexaj X — zamknena pidalhebra v F ( )Q , T t( ) — ( )C0 -pivhrupa stysku v X
z heneratorom A = ′T ( )0 . Nexaj pivhrupa T t( ) [ nil\potentnog ta mul\typli-
katyvnog (dyv. p. 1). Todi dlq bud\-qko] g C∈ ( )R tako], wo g( )0 = 0, dlq
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
NELINIJNI RIVNQNNQ Z SUTT{VO NESKINÇENNOVYMIRNYMY … 1573
bud\-qkoho t ≥ 0, u ∈X ma[mo T t g u( ) ( )( ) = g T t u( )( ) , zokrema T t u( ) ( ) =
= T t u( ) .
Nexaj F : X → X — nelinijne vidobraΩennq, wo zadovol\nq[ umovu: ∃ >C 0
∀ u , v ∈X : F u F( ) ( )− v ≤ C u − v (cg umovu nadali pryrodno nazyvaty
„umovog Lipßycq”).
Teorema 1 (abstraktnyj variant teoremy Pikara). Za danyx umov rivnqnnq
Au Fu= (1)
ma[ i do toho Ω [dynyj rozv’qzok v X.
Dovedennq. Dlq f ∈X rivnqnnq Au f= ma[ ([dynyj) rozv’qzok u =
= −∫ T t f dt
t
( )
0
0
. Rivnqnnq (1) ekvivalentne rivnqnng
u = − ∫ T t Fu dt
t
( ) ( )
0
0
= g u( ) , (2)
g u g um m( ) ( )1 2− ≤ T t F g u F g u dt
t
m m( ) ( ) ( )
0
1
1
1
2
0
∫ − −( ) − ( )( ) ≤
≤ C dt T t s F g u F g u d
t t
m m
0 0
2
1
2
2
0 0
∫ ∫ + ( ) − ( )( )− −( ) ( ) ( ) ss ≤ …
… ≤ C dt dt T t t F u F um
t
m m
t
− ∫ ∫… +…+ −( )1
1
0
1
0
1 2
0 0
( ) ( ) ( ) ≤
≤
C t
m
F u F u
m m−
−
1
0
1 2
!
( ) ( ) ,
zvidky g u g um m( ) ( )1 2− ≤
C t
m
u u
m m
0
1 2
!
− ta isnu[ m , dlq qkoho gm
[
styskom v X . ToΩ rivnqnnq (2) (a tomu i rivnqnnq (1)) ma[ i do toho Ω [dynyj
rozv’qzok.
Naslidok 1. Nexaj X = X — funkcional\na alhebra z p. 1, T t( ) — ( )C0 -
pivhrupa z heneratorom ′T ( )0 = A . Nexaj f x p( , ) — funkciq na H × R , qka
ma[ nastupni vlastyvosti: dlq bud\-qkoho p ∈R f p( , )⋅ ∈ 8X ta f [ lipßy-
cevog za druhym arhumentom (rivnomirno vidnosno perßoho): ∃ >C 0 ∀ ∈x H :
f x u f x( , ) ( , )− v ≤ C u − v . Todi rivnqnnq
( ) ( ) , ( )Au x f x u x= ( )
ma[ i do toho Ω [dynyj rozv’qzok v X.
Dovedennq. Dosyt\ pokazaty, wo dlq koΩno] funkci] u X∈ funkciq
f x u x, ( )( ) naleΩyt\ do X, a takoΩ lipßycevist\ (v sensi teoremy81) vidobra-
Ωennq F : X � u � f u⋅ ⋅( ), ( ) ∈ X, pislq çoho skorystatys\ teoremog81.
Lema 1 (uzahal\nena teorema Stouna – Vej[rßtrassa). Nexaj Y — zamkne-
na pidalhebra F ( )Q , 1∈Y ; T — xausdorfiv kompakt; C T Y( ; ) — alhebra vsix
neperervnyx funkcij na T zi znaçennqmy v Y; W — pidalhebra v C T Y( ; ) , wo
mistyt\ totoΩno odynyçnu funkcig ta podilq[ toçky: dlq bud\-qkyx t1 ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
1574 G. V. BOHDANS|KYJ, V. M. STATKEVYÇ
t T2 ∈ isnu[ g W∈ , dlq qko] g t( )1 – g t( )2 — oborotnyj element v Y . Todi
W [ wil\nog v C T Y( ; ) .
Dovedennq analohiçne dovedenng klasyçno] teoremy Stouna – Vej[rßtras-
sa z vykorystannqm strukturnyx vlastyvostej alhebry Y.
ProdovΩymo dovedennq naslidku81. Pry[dna[mo do alhebry X odynycg,
tobto budemo rozhlqdaty alhebru Y funkcij vyhlqdu c ⋅ 1 + u, de u X∈ . Za
lemog81 koΩnu funkcig z C a b Y, ;[ ]( ) moΩna nablyzyty mnohoçlenamy vyhlq-
du q p( ) = h pkk
m k
=∑ 0
, de h Yk ∈ .
Vyberemo teper u X∈ , a b;[ ] = inf ; supH Hu u[ ] . Todi dlq dovil\noho ε > 0
isnu[ funkciq h ukk
m k
=∑ 0
, dlq qko] sup , ( )H f x u x( ) – h x u xkk
m k
=∑ 0
( ) ( ) ≤ ε.
Tomu f x u x, ( )( ) ∈ Y. Oskil\ky, za umovog naslidku81, sup , ( )p f x u x( ) [ obme-
Ωenym, to f x u x, ( )( ) ∈ X .
Lipßycevist\ vidobraΩennq F oçevydna.
3. Nexaj G — L-opukla obmeΩena oblast\ v H z meΩeg S, wo zadovol\nq[
umovy hladkosti klasu C 3
, sformul\ovani v p.81; u G: → R . Postavymo
krajovu zadaçu
A uG G( ) = F u G( ) , (3)
u X SS = ∈ϕ ( ) (4)
i dovedemo ]] korektnist\ za pevnyx umov na nelinijne vidobraΩennq F : X G( ) →
→ X G( ) . Dlq c\oho nam znadobyt\sq qvnyj vyhlqd rozv’qzku tako] krajovo]
zadaçi:
A uG G( ) = v ∈X S( ) , (5)
u X SS = ∈ϕ ( ) . (6)
Lema 2. Rozv’qzok zadaçi ( 5 ), (6 ) ma[ vyhlqd u x( ) = T x xθ ϕ( ) ( )( )( ) –
– T t x dt
x
( ) ( )
( )
v( )∫0
θ
(tut ϕ , v — prodovΩennq funkcij ϕ, v na ves\ pros-
tir H, isnuvannq qkyx poqsngvalos\ v p.81).
Dovedennq. Qk bulo dovedeno v [7] ta nahadano v p.81, dlq funkci] u x1( ) =
= T x xθ ϕ( ) ( )( )( ) ( x G∈ ) vykonugt\sq spivvidnoßennq A uG G( )1 = 0, u S1 = ϕ.
Oskil\ky θ S = 0, to dosyt\ dovesty, wo dlq funkci] u x2( ) =
= − ( )∫ T t x dt
x
( ) ( )
( )
v
0
θ
A uG G( )2 = v . (7)
Cej fakt dovodyt\sq za nastupnog sxemog (analohiçno dovedenng lemy81 z
[9]). Vyberemo v ∈8 D AG( ) , a takoΩ poslidovnosti ηn Z G{ } ⊂ ( ) , vn Z G{ } ⊂ ( ) ,
dlq qkyx ηn → θ G , LG nη → – 1 pry n → ∞ ta vn → v , LG nv → AGv pry
n → ∞. Çerez ηn , vn poznaçymo vidpovidni prodovΩennq funkcij ηn , vn na
ves\ prostir H. Rozhlqnemo poslidovnist\ F xn ( ) = − ( )∫ T t x dtn
xn
( ) ( )
( )
v
0
η
. Ne-
skladni obçyslennq ′Fn , ′′Fn , LFn ta nastupnyj hranyçnyj perexid dovodqt\,
wo A uG G( )2 = v.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
NELINIJNI RIVNQNNQ Z SUTT{VO NESKINÇENNOVYMIRNYMY … 1575
Wil\nist\ D AG( ) v X G( ) dozvolq[ stverdΩuvaty, wo rivnist\ (7) vyko-
nu[t\sq dlq vsix v ∈8 X G( ) .
ZauvaΩymo, wo procedura prodovΩennq funkci] u ∈ 8 X G( ) do funkci]
u X∈ [7] [ homomorfizmom alhebr zi zbereΩennqm sup-normy.
Teorema 2. Nexaj vidobraΩennq F : X G( ) → X G( ) zadovol\nq[ umovu Lip-
ßycq. Todi zadaça (3), (4) ma[ i do toho Ω [dynyj rozv’qzok.
Dovedennq. Qk vyplyva[ z lemy82, krajova zadaça (3), (4) ekvivalentna riv-
nqnng v X G( ) : u x( ) = T x xθ ϕ( ) ( )( )( ) – T t F u x dt
x
( ) ( ) ( )
( ) ( )∫0
θ
. ToΩ dostatn\o pe-
reviryty, wo pevnyj stepin\ vidobraΩennq g : X G( ) → X G( ) , wo vyznaçeno za
formulog g u x( ) ( )( ) = T x xθ ϕ( ) ( )( )( ) – T t F u x dt
x
( ) ( ) ( )
( ) ( )∫0
θ
, [ styskom. Nexaj
u1 , u2 ∈ X G( ) , todi
g u g um m( ) ( )1 2− ≤ T t F g u F g u dtm m
G
( ) ( ) ( )
( )
− −
⋅
−( )∫ 1
1
1
2
0
θ
≤
≤ T t F g u F g u dtm m
G
( ) ( ) ( )
( )
− −
⋅
−( )∫ 1
1
1
2
0
θ
≤
≤ C T t g u g u dtm m
G
( ) ) )
( )
− −
⋅
−( )∫ 1
1
1
2
0
θ
≤
≤ C dt T t T s F g u F g u
t
m m
G
0
2
1
2
2
0
0
∫ ∫ − −
⋅
−( )( ) ( ) ( ) ( )
( )θ
dds
G
≤
≤ C dt T t s F g u F g u ds
t
m m
t
G0
2
1
2
2
0
0 0
∫ ∫ + −( )− −( ) ( ) ( ) ≤ …
… ≤
C t
m
u u
m m
0
1 2
!
− ,
zvidky j vyplyva[ isnuvannq m, dlq qkoho gm
[ styskom v X G( ) .
Naslidok 2. Nexaj f x p( , ) — funkciq na G × R , qka ma[ taki vlasty-
vosti: dlq bud\-qkoho p ∈R f p( , )⋅ ∈ X G( ) ta f [ lipßycevog za druhym
arhumentom (rivnomirno vidnosno perßoho): ∃ >C 0 ∀ ∈x G : f x u( , ) –
– f x( , )v ≤ C u − v . Todi krajova zadaça
A u xG G( ) ( )( ) = f u xG ( )( ) , u X SS = ∈ϕ ( )
ma[ i do toho Ω [dynyj rozv’qzok.
Dovedennq. Za analohi[g z dovedennqm naslidku81 slid zastosuvaty lemu81
do alhebry X G( ) .
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OderΩano 29.04.10
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
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| id | umjimathkievua-article-2981 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:33:59Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/19/e277269f04b78cdbbb131c115806c219.pdf |
| spelling | umjimathkievua-article-29812020-03-18T19:41:38Z Nonlinear equations with essentially infinite-dimensional differential operators Нелінійні рівняння з суттєво нескінченновимірними диференціальними операторами Bogdanskii, Yu. V. Statkevych, V. M. Богданський, Ю. В. Статкевич, В. М. We consider nonlinear differential equations and boundary-value problems with essentially infinite-dimensional operators (of the Laplace–Lévy type). An analog of the Picard theorem is proved. Рассмотрены нелинейные дифференциальные уравнения и краевые задачи с существенно бесконечномерными операторами (типа Лапласа - Леви). Доказан аналог теоремы Пикара. Institute of Mathematics, NAS of Ukraine 2010-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2981 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 11 (2010); 1571–1576 Український математичний журнал; Том 62 № 11 (2010); 1571–1576 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2981/2712 https://umj.imath.kiev.ua/index.php/umj/article/view/2981/2713 Copyright (c) 2010 Bogdanskii Yu. V.; Statkevych V. M. |
| spellingShingle | Bogdanskii, Yu. V. Statkevych, V. M. Богданський, Ю. В. Статкевич, В. М. Nonlinear equations with essentially infinite-dimensional differential operators |
| title | Nonlinear equations with essentially infinite-dimensional differential operators |
| title_alt | Нелінійні рівняння з суттєво нескінченновимірними диференціальними операторами |
| title_full | Nonlinear equations with essentially infinite-dimensional differential operators |
| title_fullStr | Nonlinear equations with essentially infinite-dimensional differential operators |
| title_full_unstemmed | Nonlinear equations with essentially infinite-dimensional differential operators |
| title_short | Nonlinear equations with essentially infinite-dimensional differential operators |
| title_sort | nonlinear equations with essentially infinite-dimensional differential operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2981 |
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