Problem without initial conditions for linear and almost linear degenerate operator differential equations
We study the problem without initial conditions for linear and almost linear degenerate operator differential equations in Banach spaces. The uniqueness of a solution of this problem is proved in the classes of bounded functions and functions with exponential behavior as t → –∞. We also establish su...
Збережено в:
| Дата: | 2009 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2009
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3022 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509041187356672 |
|---|---|
| author | Dmytryshyn, Yu. B. Дмитришин, Ю. Б. |
| author_facet | Dmytryshyn, Yu. B. Дмитришин, Ю. Б. |
| author_sort | Dmytryshyn, Yu. B. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:43:35Z |
| description | We study the problem without initial conditions for linear and almost linear degenerate operator differential equations in Banach spaces. The uniqueness of a solution of this problem is proved in the classes of bounded functions and functions with exponential behavior as t → –∞. We also establish sufficient conditions for initial data under which there exists a solution of the considered problem in the class of functions with exponential behavior at infinity. |
| first_indexed | 2026-03-24T02:34:47Z |
| format | Article |
| fulltext |
UDK 517.95
G. B. Dmytryßyn (L\viv. nac. un-t)
ZADAÇA BEZ POÇATKOVYX UMOV DLQ LINIJNYX
TA MAJÛE LINIJNYX VYRODÛENYX OPERATORNYX
DYFERENCIAL|NYX RIVNQN|
We study a problem without initial conditions for linear and almost linear degenerate operator
differential equations in Banach spaces. The uniqueness of a solution of this problem is proved in the
classes of bounded functions and functions with the exponential behavior as t → ∞– . In addition,
sufficient conditions on the initial data are established under which there exists a solution of the
considered problem in the class of functions with the exponential behavior at infinity.
Yzuçaetsq zadaça bez naçal\n¥x uslovyj dlq v¥roΩdenn¥x lynejn¥x y poçty lynejn¥x opera-
torn¥x dyfferencyal\n¥x uravnenyj v banaxov¥x prostranstvax. Dokazana edynstvennost\
reßenyq πtoj zadaçy v klassax ohranyçenn¥x funkcyj y funkcyj s πksponencyal\n¥m povede-
nyem pry t → ∞– . Krome toho, ustanovlen¥ dostatoçn¥e uslovyq na ysxodn¥e dann¥e, pry
kotor¥x suwestvuet reßenye ukazannoj zadaçy v klasse funkcyj s πksponencyal\n¥m povede-
nyem na beskoneçnosty.
Vstup. Nexaj V � dijsnyj refleksyvnyj separabel\nyj banaxiv prostir,
V ′ � sprqΩenyj do V prostir, S � dijsna çyslova vis\ abo promin\ ( – ∞ , T ] ,
T ∈ R . Rozhlqnemo zadaçu bez poçatkovyx umov: znajty funkcig u : S → V
taku, wo
( )( ) ( , ( ))B Au t t u t′ + = f ( t ) , t ∈ S , (1)
de A ( , )t ⋅ , t ∈ S , B � operatory, wo digt\ z V v V ′, a f : S → V ′ � deqka
funkciq. Nas cikavyt\ vypadok, koly operator B [ linijnym, a A ( , )t ⋅ , t ∈ S ,
� sim�q linijnyx abo majΩe linijnyx operatoriv. Taku zadaçu u vypadku B = I
vyvçeno v roboti [1]. Krim toho, zadaçu (1) u vypadku linijnoho operatora B ta
sim�] syl\nonelinijnyx operatoriv A ( , )t ⋅ , t ∈ S , doslidΩeno v roboti [2], de
vstanovleno dostatni umovy na vyxidni dani A ( , )t ⋅ , t ∈ S , B ta f dlq isnuvannq
ta [dynosti rozv�qzku zadaçi (1) bez obmeΩen\ na povedinku rozv�qzku i vyxidnyx
danyx pry t → ∞– . Qkwo Ω operatory A ( , )t ⋅ , t ∈ S , [ linijnymy, to, qk po-
kazugt\ rezul\taty roboty [1], navit\ u vypadku B = I rozv�qzok zadaçi (1) mo-
Ωe buty ne[dynym u klasi funkcij z dovil\nog povedinkog pry t → – ∞ . Tomu
my doslidΩuvatymemo isnuvannq ta [dynist\ rozv�qzku zadaçi (1) u klasax funk-
cij z pevnog povedinkog na neskinçennosti.
Bahato fizyçnyx procesiv, zokrema deqki dyfuzijni procesy u porystyx sere-
dovywax (dyv. [3 – 5]), modelggt\sq rivnqnnqmy ta systemamy dyferencial\nyx
rivnqn\ iz çastynnymy poxidnymy, qki moΩna podaty pry vidpovidnomu vybori
prostoru V u vyhlqdi (1). Motyvovani praktyçnym zastosuvannqm zadaçi dlq
rivnqnnq (1) intensyvno vyvçalysq bahat\ma matematykamy. Zokrema, u vypadku
linijnyx operatoriv A ( , )t ⋅ , t ∈ S = [ 0, T ] , ta B zadaça Koßi dlq nevyrodΩe-
noho rivnqnnq (1) ( )Bv v= ⇔ =0 0 z vidpovidnog poçatkovog umovog vyvça-
lasq u robotax [6 – 9]. Taka Ω zadaça dlq vyrodΩenoho rivnqnnq (1) (operator
B moΩe nabuvaty nul\ovoho znaçennq na nenul\ovyx elementax) vyvçalasq u
robotax [9 – 12]. U vypadku, koly A ( , )t ⋅ , t ∈ S = [ 0, T ] , � sim�q nelinijnyx
operatoriv, zadaça Koßi dlq rivnqnnq (1) vyvçalasq v robotax [11, 13, 14]. Qk
bulo vidmiçeno, zadaça bez poçatkovyx umov dlq rivnqnnq (1) doslidΩuvalasq u
robotax [1, 2]. Krim toho, cq zadaça vyvçalasq u klasi intehrovnyx funkcij na
S = ( – ∞ , 0 ) u robotax [11, 15], koly B = I , a operatory A ( , )t ⋅ , t ∈ S , [ maj-
Ωe linijnymy. U robotax [16 – 19] doslidΩuvalasq zadaça bez poçatkovyx umov
(1) u klasax obmeΩenyx ta majΩe periodyçnyx funkcij pry B = I .
© G. B. DMYTRYÍYN, 2009
322 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
ZADAÇA BEZ POÇATKOVYX UMOV DLQ LINIJNYX … 323
Zaznaçymo, krim toho, wo zadaça znaxodΩennq rozv�qzku synhulqrnoho evo-
lgcijnoho rivnqnnq
1
h
w w
( )
( ) ( , ( ))( )
τ
τ τ τB A′ + = f ( τ ) , 0 < τ ≤ T0 , (2)
de T0 > 0 � deqke çyslo, a h( )⋅ > 0 � lokal\no intehrovna na ( 0 , T0 ] funk-
ciq taka, wo h d
T
( )τ τ
0
0∫ = + ∞ , za dopomohog zaminy zminnyx t = H ( τ ) , de H ( ⋅ )
� pervisna funkci] h ( ⋅ ) na ( 0 , T0 ] , zvodyt\sq do zadaçi bez poçatkovyx umov
vyhlqdu (1). Tobto funkciq w [ rozv�qzkom rivnqnnq (2) todi i lyße todi, koly
funkciq u ≡ w Ho −1 [ rozv�qzkom zadaçi bez poçatkovyx umov
( ) ( )( ) ( ), ( )B Au t H t u t′ + −1 = f H t( )( )−1 , – ∞ < t ≤ H ( T0 ) .
DoslidΩenng synhulqrnoho evolgcijnoho rivnqnnq (2) (pry B = I ) prysvqçe-
no roboty [11, 15, 20], v qkyx vyvçalasq lyße rozv�qznist\ zadaçi Koßi dlq c\o-
ho rivnqnnq.
Opyßemo budovu ci[] statti. U perßomu punkti navedeno osnovni poznaçennq
ta deqki dopomiΩni fakty, qki vykorystovuvatymemo dali. Zadaçu i osnovni
rezul\taty sformul\ovano v p. 2. Tretij punkt mistyt\ dovedennq osnovnyx re-
zul\tativ. V ostann\omu punkti navedeno prostyj pryklad zastosuvannq otryma-
nyx rezul\tativ.
1. Osnovni poznaçennq ta dopomiΩni ponqttq. Nexaj V � dijsnyj
refleksyvnyj separabel\nyj banaxiv prostir, a S � dijsna çyslova vis\ abo
promin\ ( – ∞ , T ] , T ∈ R . Dali u vypadku, koly S = ( – ∞ , T ] , bez obmeΩennq
zahal\nosti budemo vvaΩaty, wo T ≥ 0 .
Vvedemo poznaçennq, qki vykorystovuvatymemo dali v roboti. Qkwo X �
normovanyj (napivnormovanyj) prostir, to çerez ⋅ X poznaçatymemo normu
(napivnormu) na n\omu. Pid X ′ rozumitymemo sprqΩenyj do X prostir, a pid
〈⋅ ⋅〉, X � kanoniçnyj skalqrnyj dobutok na X ′ × X . Çerez L S X2, ( ; )loc poznaça-
tymemo prostir vyznaçenyx na S zi znaçennqmy v X funkcij, zvuΩennq qkyx na
bud\-qkyj vidrizok [ t1
, t2 ] ⊂ S naleΩyt\ L t t X2 1 2( , ; ). Vidomo, wo prostir
L S X2, ( ; )loc moΩna ototoΩnyty z deqkym pidprostorom prostoru ′� ( ; )S X roz-
podiliv na int S zi znaçennqmy v Xw. Dlq funkci] v z prostoru L S X2, ( ; )loc
pid v ′ rozumitymemo ]] poxidnu v sensi rozpodiliv ′� ( ; )S X [14]. Prostir nepe-
rervnyx funkcij z S v X poznaçatymemo çerez C S X( ; ). Symvolom �( )S my
poznaçatymemo prostir neskinçenno dyferencijovnyx dijsnyx funkcij na S z
kompaktnymy nosiqmy v int S , nadilenyj vidpovidnog topolohi[g (dyv. [14,
c. 41]). Neperervne vkladennq odnoho topolohiçnoho prostoru v inßyj poznaça-
tymemo symvolom �O�, a neperervne ta wil\ne � �
d
O�.
Nexaj B : V → V ′ � linijnyj neperervnyj symetryçnyj (tobto 〈 〉Bv v1 2, V =
= 〈 〉Bv v2 1, V ∀ v1
, v2 ∈ V ) i monotonnyj (tobto 〈 〉Bv v, V ≥ 0 ∀ v ∈ V ) opera-
tor. Todi 〈 ⋅ ⋅ 〉B , V � napivskalqrnyj dobutok, a
⋅ VB
df= 〈 ⋅ ⋅ 〉B , /
V
1 2 � napivnor-
ma na V. Poznaçymo popovnennq prostoru V u cij napivnormi çerez VB .
Oçevydno, wo V
d
O VB i dlq dovil\noho v ∈V ma[mo
v VB
≤
B ⋅ v V , de
B � norma operatora B u prostori �( ; )V V ′ . Napivskalqrnyj dobutok
〈 ⋅ ⋅ 〉B , V moΩna za neperervnistg odnoznaçno prodovΩyty na VB . Çerez oto-
toΩnennq funkcionaliv ma[mo vkladennq ′VB O ′V . Na pidstavi teoremy 3.5
hl. I monohrafi] [9] prostir ′VB [ hil\bertovym. Operator B ma[ [dyne linijne
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
324 G. B. DMYTRYÍYN
neperervne prodovΩennq B : VB → ′VB . Skalqrnyj dobutok na ′VB zadovol\-
nq[ umovu
( , )w VB
B
v ′ = 〈 〉w V, v , w ∈ ′VB , v ∈ V ,
zvidky, pokladagçy w = Bv, ma[mo
B
B
v ′V =
v VB
, v ∈ VB . (3)
Ob©runtuvannq cyx faktiv moΩna znajty v monohrafiqx [9, 11].
2. Formulgvannq zadaçi i osnovnyx rezul\tativ. Prypustymo, wo zadano
sim�g operatoriv A ( , ) :t V V⋅ → ′ , t ∈ S , takyx, wo:
1) dlq dovil\no] vymirno] za Boxnerom funkci] v : S → V funkciq w( )⋅ =
= A ( , ( ))⋅ ⋅v : S V→ ′ [ vymirnog na S ;
2) qkwo
v ∈L S V2, ( ; )loc , to
A ( , ( )) ( ; ),⋅ ⋅ ∈ ′v L S V2 loc .
Rozhlqnemo zadaçu: dlq zadano] funkci] f L S V∈ ′2, ( ; )loc znajty funkcig u ∈
∈
L S V C S V2, ( ; ) ( ; )loc I B , wo zadovol\nq[ rivnqnnq
( )( ) ( , ( ))B Au t t u t′ + = f ( t ) v ′ ′� ( ; )S V . (4)
Dali cg zadaçu nazyvatymemo zadaçeg bez poçatkovyx umov dlq vyrodΩenoho ne-
qvnoho operatornoho dyferencial\noho rivnqnnq (4) abo prosto zadaçeg (4).
Teorema 1 ([dynist\ rozv�qzku). Nexaj:
3) dlq majΩe vsix t ∈ S i dovil\nyx v, w ∈ V , v ≠ w, vykonu[t\sq neriv-
nist\
〈 − − 〉A A( , ) ( , ),t t w w Vv v >
γ ϕ( )t w Vv −( )
B
2 ,
de γ ∈L S1, ( )loc , γ ( )t > 0 dlq majΩe vsix t ∈ S , γ ( )t dt
−∞∫
0
= + ∞ , a funkciq
ϕ ∈ +∞C([ , ))0 taka, wo ϕ( )0 0= , ϕ τ( ) > 0 pry τ > 0 i dτ
ϕ τ( )1
+∞
∫ = + ∞ .
Todi zadaça (4) ne moΩe maty bil\ße odnoho rozv�qzku z prostoru L S V∞( ; )B .
Bil\ß toho, qkwo funkciq Φ−1, obernena do Φ( )
( )
df
s
ds
= ∫ τ
ϕ τ1
, s > 0, dlq bud\-
qkoho a ≥ 0 zadovol\nq[ umovu
Φ− +1( )a b = O b[ ]( )Φ−1 pry b → + ∞ , (5)
to zadaça (4) ne moΩe maty bil\ße odnoho rozv�qzku, wo zadovol\nq[ umovu
u t V( )
B
2 = o d
t
Φ− ∫
1
0
2 γ τ τ( ) pry t → – ∞ . (6)
Naslidok 1. Nexaj v umovax teoremy 1 funkciq ϕ ( τ ) = τ , τ ≥ 0 . Todi za-
daça (4) ne moΩe maty bil\ße odnoho rozv�qzku, wo zadovol\nq[ umovu
u t V( )
B
= o d
t
exp ( )γ τ τ
0
∫
pry t → – ∞ .
Teorema 2 (isnuvannq rozv�qzku). Nexaj vkladennq V O VB [ kompaktnym
ta
4) isnugt\ funkci] α1 ∈ ∞L S, ( )loc i α2 2∈L S, ( )loc taki, wo
A ( , )t Vv ′ ≤ α α1 2( ) ( )t tVv + , v ∈ V , dlq majΩe vsix t ∈ S ;
5) 〈 − − 〉A A( , ) ( , ),t t Vv v v v1 2 1 2 ≥ 0 ∀ v1
, v2 ∈ V , dlq majΩe vsix t ∈ S ;
6) isnugt\ çyslo β1 0> ta funkciq β2 1∈L S, ( )loc , β2 0≥ , taki, wo
〈 〉A ( , ),t Vv v ≥ β β1
2
2v V t+ ( ), v ∈ V , dlq majΩe vsix t ∈ S ;
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
ZADAÇA BEZ POÇATKOVYX UMOV DLQ LINIJNYX … 325
7) dlq majΩe vsix t ∈ S i dovil\nyx v1
, v2 ∈ V dijsnoznaçna funkciq s a
a 〈 + 〉A ( , ),t s Vv v v1 2 2 [ neperervnog na R .
Krim toho, prypustymo, wo dlq deqkoho λ ∈ R , λ B < 2 1β , ma[mo
t
t
V
t
t
f d t d
−
′
−
∫ ∫+
1
2
1
2( ) ( )τ τ β τ ≤ C e t
1
−λ , t ≤ 0, (7)
de C1 > 0 � stala, qka zaleΩyt\ vid f ta β2 .
Todi isnu[ rozv�qzok zadaçi (4), wo zadovol\nq[ ocinku
u t u dV
t
t
V( ) ( )
B
2
1
2+
−
∫ τ τ ≤ C e t
2
−λ , t ≤ 0, (8)
de C2 � dodatna stala, wo zaleΩyt\ lyße vid C1, β1, λ i operatora B .
ZauvaΩennq 1. Dlq toho wob sim�q operatoriv A ( , )t ⋅ : V → V ′ , t ∈ S , za-
dovol\nqla umovu 1, dostatn\o, wob vona zadovol\nqla umovy 6, 7, a funkciq
w( )⋅ = A ( , )⋅ v bula vymirnog na S dlq dovil\noho v ∈ V (dyv., napryklad,
[11]). Umova Ω 2 vykonu[t\sq pry vykonanni umov 1 ta 4.
Teorema 3 (isnuvannq [dynoho rozv�qzku). Nexaj vkladennq V O VB [ kom-
paktnym, sim�q operatoriv A ( , )t ⋅ : V → V ′ , t ∈ S , zadovol\nq[ umovy 4, 6, 7 i
8) isnu[ stala K1 0> taka, wo dlq majΩe vsix t ∈ S i dovil\nyx v , w ∈
∈ V , v ≠ w, vykonu[t\sq nerivnist\
〈 − − 〉A A( , ) ( , ),t t w w Vv v >
K w V1
2
v −
B
.
Todi qkwo dlq deqkoho λ ∈ R , λ βB < 2 1 i λ < 2 1K , vykonu[t\sq neriv-
nist\ (7), to isnu[ [dynyj rozv�qzok zadaçi (4) u klasi funkcij v ∈C S V( ; )B , wo
zadovol\nqgt\ umovu
v( )t VB
= o e K t−[ ]1 pry t → – ∞ . (9)
Bil\ß toho, cej rozv�qzok takoΩ zadovol\nq[ ocinku (8).
3. Dovedennq osnovnyx rezul\tativ. Dovedennq teoremy 1. Nexaj u1,
u2 � dva rizni rozv�qzky zadaçi (4). Todi dlq w
df= u u1 2− z rivnqnnq (4) otry-
mu[mo
( )( ) ( , ( )) ( , ( ))B A Aw t t u t t u t′ + −1 2 = 0 v � ′ ( S; V ′ ) . (10)
Zvidsy ta z umovy 2 vyplyva[, wo
( ) , ( ; )Bw L S V′ ′∈ 2 loc , a tomu na pidstavi lemy
2.1 roboty [2] ma[mo
1
2
2d
dt
w t V( )
B
= 〈 〉′( )( ) , ( )Bw t w t V dlq majΩe vsix t ∈ S . (11)
PomnoΩyvßy (10) skalqrno na w, dlq majΩe vsix t ∈ S oderΩymo
〈 ′ 〉 + 〈 − − 〉( ( )) , ( ) ( , ( )) ( , ( )), ( ) ( )B A Aw t w t t u t t u t u t u tV V1 2 1 2 = 0. (12)
Z (11) i (12) otryma[mo
1
2
2
1 2 1 2
d
dt
w t t u t t u t u t u tV V( ) ( , ( )) ( , ( )), ( ) ( )
B
A A+ 〈 − − 〉 = 0 (13)
majΩe skriz\ na S.
Z (13) ta z umovy 3 distanemo dyferencial\nu nerivnist\
dy t
dt
t y t
( )
( ) ( ( ))+ 2 γ ϕ ≤ 0 dlq majΩe vsix t ∈ S , (14)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
326 G. B. DMYTRYÍYN
de y ( t ) =
u t u t V1 2
2( ) ( )−
B
, t ∈ S , � absolgtno (lokal\no) neperervna funkciq.
Qkwo y ≡ 0 na S, to z (13) ta z umovy 3 vyplyva[, wo u t1( ) = u t2( ) dlq majΩe
vsix t ∈ S , a ce supereçyt\ tomu, wo u1 ta u2 � dva rizni rozv�qzky zadaçi (4).
Tomu vnaslidok neperervnosti funkci] y isnu[ toçka t0 ∈ S taka, wo y t( )0 > 0.
Oskil\ky
dy t
dt
( )
≤ 0 dlq majΩe vsix t ∈ S , to funkciq y ne zrosta[ na S .
Tomu y ( t ) ≥ y ( t0 ) > 0 pry t ≤ t0 . Rozhlqnemo nerivnist\ (14) na ( – ∞ , t0 ] . Po-
dilymo cg nerivnist\ na ϕ ( y ) i zintehru[mo po t vid t1 do t0 , de t1 ( t1 < t0 ) �
dovil\ne çyslo. Pislq neskladnyx peretvoren\ otryma[mo
d d
y t y t
τ
ϕ τ
τ
ϕ τ( ) ( )
( ) ( )
1 1
1 0
∫ ∫− ≥ 2
1
0
γ ( )t dt
t
t
∫ . (15)
Prypustymo, wo u1, u2 ∈ L S V∞( ; )B , todi funkciq y [ obmeΩenog na S .
Zvidsy i z nerivnosti (15), vraxuvavßy, wo γ ( )t dt
t
−∞∫ 0 = + ∞ , otryma[mo supereç-
nist\, qkwo viz\memo t1 dostatn\o menßym za t0 . Tomu u t1( ) = u t2( ) dlq maj-
Ωe vsix t ∈ S . Perßu çastynu teoremy dovedeno.
Dovedemo druhu çastynu teoremy. Nexaj funkciq Φ−1 (obernena do funk-
ci] Φ ) zadovol\nq[ umovu (5), a dlq rozv�qzkiv u1 ta u2 vykonu[t\sq umova (6).
Todi funkciq y zadovol\nq[ umovu
y ( t ) = o d
t
Φ− ∫
1
0
2 γ τ τ( ) pry t → – ∞ . (16)
Viz\memo t2 ≤ min{ , }0 0t take, wo Φ( ( )) ( )y t d
t
t
0 2
2
0+ ∫ γ τ τ ≥ 0. Z (5) i (16) ma[mo
y ( t ) = o d
t
t
Φ− ∫
1 2
2
γ τ τ( ) pry t → – ∞ . (17)
Z nerivnosti (15) dista[mo
Φ( ( ))y t1 ≥ Φ( ( )) ( )y t t dt
t
t
0 2
1
0
+ ∫ γ ∀ t1 ≤ t2 .
Zvidsy vnaslidok toho, wo funkciq Φ−1 monotonno zrosta[, otrymu[mo
y ( t ) ≥ Φ Φ− +
∫1
0 2
0
( ( )) ( )y t d
t
t
γ τ τ ≥ Φ− ∫
1 2
2
γ τ τ( )d
t
t
∀ t ≤ t2 . (18)
Ale (18) supereçyt\ (17). Tomu u t1( ) = u t2( ) dlq majΩe vsix t ∈ S .
Teoremu dovedeno.
Dovedennq teoremy 2 rozib�[mo na try kroky.
Krok 1 (aproksymaciq rozv�qzku). Pobudu[mo poslidovnist\ funkcij, wo v
pevnomu sensi aproksymugt\ rozv�qzok zadaçi (4). Vyznaçymo Sk
df= S tI { :∈R
t k≥ − } , k ∈ N . Dlq koΩnoho k ∈ N rozhlqnemo zadaçu znaxodΩennq funkci]
ˆ ( ; )u L S Vk k∈ 2 , B Bˆ ( ; )u C S Vk k∈ ′ , tako], wo
( )ˆ ( ) ( , ˆ ( ))B Au t t u tk k′ + = f ( t ) v ′ ′� ( ; )S Vk ,
(19)
lim ˆ ( )
t k
ku t
→ −
B = 0 v ′VB .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
ZADAÇA BEZ POÇATKOVYX UMOV DLQ LINIJNYX … 327
Isnuvannq [dynoho rozv�qzku zadaçi (19) vyplyva[ z naslidku III.6.3 [11]. Pro-
dovΩymo funkcig ûk na ves\ promiΩok S , poklavßy ]] rivnog nulg na ( – ∞ ,
– k ] , i poznaçymo ce prodovΩennq çerez uk . Oçevydno, wo funkciq uk dlq
koΩnoho k ∈ N [ rozv�qzkom zadaçi bez poçatkovyx umov
( )( ) ( , ( ))B Au t t u tk k′ + = fk ( t ) v ′ ′� ( ; )S V , (20)
de fk ( t ) = f ( t ) na Sk i fk ( t ) = A ( t, 0 ) na ( – ∞ , – k ] .
Krok 2 (ocinky aproksymugçyx rozv�qzkiv). Teper dlq koΩnoho k ∈ N znaj-
demo ocinky rozv�qzku uk zadaçi (20). Z (20) vyplyva[, wo
( ) , ( ; )Bu L S Vk ′ ′∈ 2 loc ,
a tomu na pidstavi lemy 2.1 [2] ma[mo, wo u C S Vk ∈ ( ; )B i
1
2
2d
dt
u tk V( )
B
= 〈 ′ 〉( ( )) , ( )Bu t u tk k V (21)
dlq majΩe vsix t ∈ S .
Nexaj τ1 , τ2 ∈ S , τ1 < τ2 , � dovil\ni dijsni çysla. Pidstavymo uk v riv-
nqnnq (20) i otrymanu rivnist\, pomnoΩyvßy skalqrno na exp( ) ( )λt u tk , de λ ∈
∈ R λ βB <( )2 1 � poky wo dovil\ne çyslo, zintehru[mo po t vid τ1 do τ2 .
V rezul\tati otryma[mo
τ
τ
λ
1
2
∫ 〈 ′ 〉 + 〈 〉{ }( ( )) , ( ) ( , ( )), ( )B Au t u t t u t u t e dtk k V k k V
t =
τ
τ
λ
1
2
∫ 〈 〉f t u t e dtk k V
t( ), ( ) . (22)
Z (22), vykorystavßy (21), matymemo
τ
τ
λ
τ
τ
λ
1
2
1
2
2 2∫ ∫+ 〈 〉d
dt
u t e dt t u t u t e dtk V
t
k k V
t( ) ( , ( )), ( )
B
A = 2
1
2
τ
τ
λ∫ 〈 〉f t u t e dtk k V
t( ), ( ) . (23)
Zintehruvavßy perßyj dodanok u livij çastyni rivnosti (23) za formulog inteh-
ruvannq çastynamy, distanemo
u e t u t u t e dtk V k k V
t( ) ( , ( )), ( )τ λτ
τ
τ
λ
2
2 2
1
2
2
B
A+ 〈 〉∫ =
=
λ τ
τ
τ
λ
τ
τ
λ λτ
1
2
1
2
12
1
22∫ ∫+ 〈 〉 +u t e dt f t u t e dt u ek V
t
k k V
t
k V( ) ( ), ( ) ( )
B B
. (24)
Ocinymo, vykorystavßy umovu 6, druhyj dodanok u livij çastyni rivnosti (24):
2
1
2
τ
τ
λ∫ 〈 〉A ( , ( )), ( )t u t u t e dtk k V
t ≥ 2 21
2
2
1
2
1
2
β β
τ
τ
λ
τ
τ
λ∫ ∫−u t e dt t e dtk V
t t( ) ( ) . (25)
Oskil\ky
v VB
2 ≤ B v V
2 ∀ v ∈ V , to ma[mo ocinku
τ
τ
λ
1
2
2∫ u t e dtk V
t( )
B
≤
B
τ
τ
λ
1
2
2∫ u t e dtk V
t( ) . (26)
Dali, zastosuvavßy nerivnist\ Gnha z ε > 0, ocinymo druhyj dodanok u pravij
çastyni rivnosti (24):
2
1
2
τ
τ
λ∫ 〈 〉f t u t e dtk k V
t( ), ( ) ≤ ε
ε
τ
τ
λ
τ
τ
λ
1
2
1
2
2 21
4∫ ∫+ ′u t e dt f t e dtk V
t
k V
t( ) ( ) . (27)
Z rivnosti (24), vykorystavßy (25) � (27) z ε = β λ1 2− B , qkwo λ > 0, i ε =
= β1 v inßomu vypadku, otryma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
328 G. B. DMYTRYÍYN
u e K u t e dtk V k V
t( ) ( )τ λτ
τ
τ
λ
2
2
2
22
1
2
B
+ ∫ ≤
≤
C f t e dt t e dt u ek V
t t
k V3
2
2 1
2
1
2
1
2
12
τ
τ
λ
τ
τ
λ λτβ τ∫ ∫′ + +( ) ( ) ( )
B
, (28)
de K2 > 0 i C3 > 0 � deqki stali, wo zaleΩat\ lyße vid β1 , operatora B i,
moΩlyvo, λ .
Poklademo
M
df=
e C C
K
λ
1 3
2
2 1( )+ +
B
.
Teper pokaΩemo, wo dlq dovil\nyx τ1 , τ2 ∈ Sk takyx, wo τ1 < τ2 ≤ 0 i τ2 –
– τ1 ≤ 1, ta λ z umovy teoremy vykonu[t\sq nerivnist\
u ek V( )τ λτ
2
2 2
B
≤
max ( ) ,u e Mk Vτ λτ
1
2 1
B
{ } . (29)
Qkwo
u ek V( )τ λτ
2
2 2
B
≤
u ek V( )τ λτ
1
2 1
B
, to nerivnist\ (29) vykonu[t\sq. Tomu
prypustymo, wo
u ek V( )τ λτ
2
2 2
B
>
u ek V( )τ λτ
1
2 1
B
. Todi z (28) vyplyva[ nerivnist\
K u t e dtk V
t
2
2
1
2
τ
τ
λ∫ ( ) ≤ C f t e dt t e dtk V
t t
3
2
2
1
2
1
2
2
τ
τ
λ
τ
τ
λβ∫ ∫′ +( ) ( ) . (30)
Oskil\ky τ2 – τ1 ≤ 1, to z (7), vraxuvavßy, wo max
[ , ]t
te
∈ −
{ }
τ τ
λ
2 21
≤ e eλ λτ2 , otry-
ma[mo
τ
τ
λ
1
2
2∫ ′f t e dtk V
t( ) ≤
τ
τ
λ
2
2
1
2
−
′∫ f t e dtk V
t( ) ≤ e e f t dtk V
λ λτ
τ
τ
2
2
2
1
2
−
′∫ ( ) ≤ C e1
λ (31)
ta
τ
τ
λβ
1
2
2∫ ( )t e dtt ≤
τ
τ
λβ
2
2
1
2
−
∫ ( )t e dtt ≤ e e t dtλ λτ
τ
τ
β2
2
2
1
2
−
∫ ( ) ≤ C e1
λ . (32)
Zvidsy ta z nerivnosti (30) distanemo
τ
τ
λ
1
2
2∫ u t e dtk V
t( ) ≤ e
C C
K
λ 1 3
2
2( )+
. (33)
Z nerivnostej (26), (33) ta neperervnosti funkci] t a
u t ek V
t( )
B
2 λ na [ , ]τ τ1 2
vyplyva[ isnuvannq toçky τ τ τ3 1 2∈[ , ] tako], wo
u ek V( )τ λτ
3
2 3
B
=
τ
τ
λ
1
2
2∫ u t e dtk V
t( )
B
≤
B
τ
τ
λ
1
2
2∫ u t e dtk V
t( ) ≤
e C C
K
λ
1 3
2
2( )+ B
.
(34)
Teper zastosu[mo nerivnist\ (28) dlq τ1 = τ3 . Vykorystavßy pry c\omu (31) �
(34), budemo maty
u ek V( )τ λτ
2
2 2
B
≤
e C C
K
λ
1 3
2
2 1( )+ +
B
= M .
OtΩe, nerivnist\ (29) dovedeno.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
ZADAÇA BEZ POÇATKOVYX UMOV DLQ LINIJNYX … 329
Oskil\ky u tk ( ) = 0 dlq majΩe vsix t ≤ – k , to z (29) vyplyva[ ocinka
u tk V( )
B
2 ≤ Me t−λ ∀ t ≤ 0. (35)
Poklademo v (33) τ1 = t – 1, τ2 = t, de t ≤ 0 � dovil\ne çyslo. Zvidsy,
vraxuvavßy, wo min
[ , ]τ
λτ
∈ −
{ }
t t
e
1
≥ e e t− λ λ , otryma[mo
t
t
k Vu d
−
∫
1
2( )τ τ ≤ e e u e dt
t
t
k V
λ λ λττ τ−
−
∫
1
2( ) ≤
e C C
K
e t
2
1 3
2
2λ
λ( )+ − . (36)
Z ocinok (28) pry λ = 0 i (35) pry t = 0, vybravßy τ1 = 0, τ2 = t > 0, de t �
dovil\ne çyslo, distanemo
u t K u dk V
t
k V( ) ( )
B
2
2
0
2+ ∫ τ τ ≤ C f d d M
t
k V
t
3
0
2
0
22∫ ∫′ + +( ) ( )τ τ β τ τ . (37)
Z ocinok (35) � (37) i obmeΩenosti poslidovnosti fk k{ } =
+∞
1 v L S V2, ( ; )loc ′ vyply-
va[, wo
poslidovnist\ uk k{ } =
+∞
1 [ obmeΩenog
u prostori
L S V L S V∞, ,( ; ) ( ; )loc locB I 2 , (38)
a z (38) i umovy 4 � wo
poslidovnist\ A ( , ( ))⋅ ⋅{ } =
+∞uk k 1 [ obmeΩenog u prostori L S V2, ( ; )loc ′ . (39)
Oskil\ky operator B : V → V B [ linijnym, to joho realizaciq B :
L S V2, ( ; )loc →
L S V2, ( ; )loc B takoΩ [ linijnym i neperervnym operatorom (dyv.,
napryklad, [11]). Tomu z (3) i umovy (38) vyplyva[, wo
poslidovnist\ { }Buk k=
+∞
1 [ obmeΩenog u prostori
L S V∞ ′, ( ; )loc B . (40)
Krok 3 (hranyçnyj perexid). Oskil\ky V � refleksyvnyj banaxiv prostir,
a ′VB � hil\bertiv prostir, to z (38) � (40) vyplyva[, wo z poslidovnosti { }uk k=
+∞
1
moΩna vybraty pidposlidovnist\ (qku my znovu poznaçatymemo çerez { }uk k=
+∞
1 )
taku, wo
uk ( )⋅ k →∞ → u( )⋅ slabko v L S V2, ( ; )loc , (41)
Buk ( )⋅ k →∞ → ζ( )⋅ * - slabko v
L S V∞ ′, ( ; )loc B , (42)
A ( , ( ))⋅ ⋅uk k →∞ → χ( )⋅ slabko v L S V2, ( ; )loc ′ . (43)
Z (41), vraxovugçy, wo operator B, buduçy neperervnym, [ slabkoneperervnym,
ma[mo
Buk ( )⋅ k →∞ → Bu( )⋅ slabko v
L S V2, ( ; )loc ′B . (44)
Perejdemo v (20) do hranyci pry k → ∞ , vraxuvavßy pry c\omu (43), (44) i
oznaçennq funkci] fk . V rezul\tati otryma[mo
( ( )) ( )Bu t t′ + χ = f t( ) v ′ ′� ( ; )S V . (45)
Zvidsy i z lemy 2.1 roboty [2] vyplyva[, wo u C S V∈ ( ; )B . Krim toho, z (42) i (44)
ma[mo Bu t( ) = ζ( )t dlq majΩe vsix t S∈ . Tomu zi zbiΩnosti (42) vyplyva[, wo
Buk ( )⋅ k →∞ → Bu( )⋅ * - slabko v
L S V∞ ′, ( ; )loc B . (46)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
330 G. B. DMYTRYÍYN
OtΩe, teoremu bude dovedeno, qkwo my pokaΩemo, wo
χ( )t = A ( , ( ))t u t v ′V dlq majΩe vsix t S∈ . (47)
Dovedemo (47), vykorystavßy metod monotonnosti. Viz\memo dovil\nu funk-
cig ψ ≥ 0 z �( )S i dlq bud\-qkoho v ∈L S V2, ( ; )loc poznaçymo
Ek =
S
k k Vt u t t t u t t t dt∫ 〈 − − 〉A A( , ( )) ( , ( )), ( ) ( ) ( )v v ψ , k ∈N .
Z umovy (5) vyplyva[, wo Ek ≥ 0.
PomnoΩyvßy (20) skalqrno na ψ uk ta zintehruvavßy otrymanu rivnist\ po t
v S, otryma[mo
S
k k V k k Vu t u t t u t u t t dt∫ 〈 ′ 〉 + 〈 〉{ }( ( )) , ( ) ( , ( )), ( ) ( )B A ψ =
S
k k Vf t u t t dt∫ 〈 〉( ), ( ) ( )ψ . (48)
Z (48), vykorystavßy (21), oznaçennq funkci] fk i formulu intehruvannq ças-
tynamy, distanemo
S
k k Vt u t u t t dt∫ 〈 〉A ( , ( )), ( ) ( )ψ =
1
2
2
S
k V
S
k Vu t t dt f t u t t dt∫ ∫′ + 〈 〉( ) ( ) ( ), ( ) ( )
B
ψ ψ . (49)
Nexaj t1, t2 � dovil\ni çysla taki, wo supp ′ψ ⊂ [ , ]t t1 2 ⊂ S . Z (41) ma[mo
slabku zbiΩnist\ do u poslidovnosti { }uk k=
+∞
1 u prostori L t t V2 1 2( , ; ). Zvidsy na
pidstavi kompaktnosti vkladennq V O VB ta lemy 2.2 roboty [2] vyplyva[, wo z
poslidovnosti { }uk k=
+∞
1 moΩna vydilyty pidposlidovnist\, za qkog my zbereΩemo
te same poznaçennq:
uk ( )⋅ k →∞ → u( )⋅ syl\no v L t t V2 1 2( , ; )B . (50)
Z (50) ma[mo
S
k Vu t t dt∫ ′( ) ( )
B
2 ψ k →∞ →
S
Vu t t dt∫ ′( ) ( )
B
2 ψ . (51)
Vykorystavßy (41) ta (51), perejdemo v (49) do hranyci
S
k k Vt u t u t t dt∫ 〈 〉A ( , ( )), ( ) ( )ψ k →∞ →
1
2
2
S
V
S
Vu t t dt f t u t t dt∫ ∫′ + 〈 〉( ) ( ) ( ), ( ) ( )
B
ψ ψ .
(52)
DomnoΩymo (45) skalqrno na ψ u ta zintehru[mo otrymanu rivnist\ po t v S. V
rezul\tati distanemo
S
Vt u t t dt∫ 〈 〉χ ψ( ), ( ) ( ) =
1
2
2
S
V
S
Vu t t dt f t u t t dt∫ ∫′ + 〈 〉( ) ( ) ( ), ( ) ( )
B
ψ ψ . (53)
Z (50) ta (53) ma[mo
S
k k Vt u t u t t dt∫ 〈 〉A ( , ( )), ( ) ( )ψ k →∞ →
S
Vt u t t dt∫ 〈 〉χ ψ( ), ( ) ( ) . (54)
Vykorystovugçy (41), (43) ta (54), znaxodymo
0 ≤ lim
k
kE
→∞
=
S
Vt t t u t t t dt∫ 〈 − − 〉χ ψ( ) ( , ( )), ( ) ( ) ( )A v v . (55)
Poklademo v (55) v = u – sw, de s > 0, a w L S V∈ 2, ( ; )loc � dovil\na funkciq.
V rezul\tati otryma[mo
S
Vt t u t sw t w t t dt∫ 〈 − − 〉χ ψ( ) ( , ( ) ( )), ( ) ( )A ≥ 0. (56)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
ZADAÇA BEZ POÇATKOVYX UMOV DLQ LINIJNYX … 331
Sprqmuvavßy v (56) s do 0, na pidstavi umovy 7 distanemo
S
Vt t u t w t t dt∫ 〈 − 〉χ ψ( ) ( , ( )), ( ) ( )A ≥ 0.
Zvidsy vnaslidok dovil\nosti ψ ≥ 0 z �( )S i w z L S V2, ( ; )loc otryma[mo riv-
nist\ (47).
Ocinka Ω rozv�qzku (8) vyplyva[ bezposeredn\o z ocinok (35), (36), vlastyvos-
ti (3), zbiΩnostej (41), (46) ta teorem 1 i 9 monohrafi] [21, c. 173, 179].
Dovedennq teoremy 3. Isnuvannq [dynoho rozv�qzku u zadaçi (4) u klasi
funkcij, wo zadovol\nqgt\ ocinku (8), vyplyva[ z teoremy 2. Krim toho, z ocin-
ky (8) i toho, wo λ < 2 K1 , ma[mo
u t V( )
B
2 ≤ C e t
2
−λ = C e eK t K t
2
2 21 1− −( )λ = o e K t[ ]−2 1 pry t → – ∞ . (57)
Zvidsy ta z naslidku 1 vyplyva[, wo u � [dynyj rozv�qzok zadaçi (4) u klasi
funkcij, wo zadovol\nqgt\ umovu (9).
4. Pryklad. Nexaj Ω � obmeΩena oblast\ v Rn, n ∈ N , S = R . Poznaçy-
mo çerez
o
H1( )Ω prostir Sobol[va, otrymanyj v rezul\tati zamykannq prostoru
�( )Ω wodo normy
w H
o 1( )Ω = w w dxxi
n
i
2 2
1
1 2
+( )( )=∑∫Ω
/
, a çerez H −1( )Ω �
sprqΩenyj do
o
H1( )Ω prostir. Z teoremy Rellixa � Kondraßova (dyv. [11,
c. 57]) vyplyva[, wo vkladennq
o
H1( )Ω O L2( )Ω [ kompaktnym.
Poklademo V =
o
H1( )Ω , todi ′V = H −1( )Ω . Vyznaçymo sim�g operatoriv
A( , )t ⋅ : V V→ ′ , t ∈ R , za pravylom
〈 〉A ( , ),t u Vv =
a x t u x x a x t u x x dxij x x
i j
n
i j
( , ) ( ) ( ) ( , ) ( ) ( )
,
v v+
=
∑∫
1Ω
, u, v ∈ V,
de aij , a L∈ ×∞( )Ω R , i, j = 1, n , � deqki zadani funkci], wo zadovol\nqgt\
umovu: dlq majΩe vsix ( , )x t ∈ ×Ω R
a x t a x tij i j
i j
n
( , ) ( , )
,
ξ ξ ξ+
=
∑ 0
2
1
≥ a i
i
n
ξ2
0=
∑ ∀ ∈ξ R
n , ξ0 ∈R,
de a = 0 � deqka stala.
TakoΩ vyznaçymo operator B : V → V ′ za pravylom
〈 〉B ( ),u Vv = b x u x x dx( ) ( ) ( )v
Ω
∫ , u, v ∈ V,
de b L∈ ∞( )Ω � deqka funkciq taka, wo b ( x ) ≥ 0 dlq majΩe vsix x ∈ Ω . Za-
uvaΩymo, wo b moΩe dorivngvaty nulg na mnoΩyni dodatno] miry.
Prypustymo, wo b df= b L∞ ( )Ω > 0. Z oznaçennq operatora B , nerivnostej
Koßi � Bunqkovs\koho i Hel\dera vyplyva[, wo B ≤ b . Krim toho,
v VB
2 ≤
≤ b Lv
2
2
( )Ω dlq dovil\noho v ∈L2( )Ω . Zvidsy i z kompaktnosti vkladennq v O
O L2( )Ω ma[mo kompaktnist\ vkladennq V O VB .
Nexaj λ < 2a b/ � dovil\ne çyslo, f L L∈ 2 2, ( ); ( )loc R Ω � funkciq taka,
wo
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
332 G. B. DMYTRYÍYN
t
t
f x dxd
−
∫ ∫
1
2( , )τ τ
Ω
≤ C e t
4
−λ , t ≤ 0,
de C4 > 0 — zaleΩna vid f stala, f x t( , )
df= f t x( )( ). Todi s teoremy 3 vyply-
va[ isnuvannq [dynoho uzahal\nenoho rozv�qzku
u L V C V∈ 2, ( ; ) ( ; )loc R RI B zadaçi
bez poçatkovyx umov dlq linijnoho eliptyko-paraboliçnoho rivnqnnq
∂
∂
− ( ) +
=
∑t
b x u a x t u a x t u
i j
n
ij x xi j
( ( ) ) ( , ) ( , )
, 1
= f x t( , ), ( , )x t ∈ ×Ω R,
z odnoridnog krajovog umovog Dirixle u klasi funkcij, wo zadovol\nqgt\
umovu
b x u x t dx( ) ( , )2
Ω
∫ = o
at
b
exp −
2
pry t → – ∞ ,
de u x t( , ) df= u t x( )( ).
1. Bokalo N. M. O zadaçe bez naçal\n¥x uslovyj dlq nekotor¥x klassov nelynejn¥x parabo-
lyçeskyx uravnenyj // Tr. sem. ym. Y. H. Petrovskoho. � 1989. � V¥p. 14. � S. 3 � 44.
2. Bokalo M., Dmytryshyn Yu. Problems without initial conditions for degenerate implicit evolution
equations // Electron. J. Different. Equat. – 2008. – 2008, # 4. – P. 1 – 16.
3. Clark G. W., Showalter R. E. Fluid flow in a layered medium // Ouart. Appl. Math. – 1994. – 52,
# 4. – P. 777 – 795.
4. Clark G. W., Showalter R. E. Two-scale convergence of a model for flow in a partially fissured
medium // Electron. J. Different. Equat. – 1999. – 1999, # 2. – P. 1 – 20.
5. Showalter R. E., Visarraga D. B. Double-diffusion models from a highly-heterogeneous medium //
J. Math. Anal. and Appl. – 2004. – 295. – P. 191 – 210.
6. Showalter R. E. Existence and representation theorems for semilinear Sobolev equation in Banach
space // SIAM J. Math. Anal. – 1972. – 3, # 3. – P. 527 – 643.
7. Showalter R. E. Equations with operators forming a right angle // Pacif. J. Math. – 1973. – 45,
# 1. – P. 357 – 362.
8. Lagnese J. Existence, uniqueness and limiting behavior of solutions of a class of differential
equations in Banach space // Ibid. – 1974. – 53, # 2. – P. 473 – 485.
9. Showalter R. E. Hilbert space methods for partial differential equations // Monogr. and Stud. Math.
– London etc.: Pitman, 1977. – 1. – vi + 212 p.
10. Showalter R. E. Degenerate evolution equations and applications // Indiana Univ. Math. J. – 1974.
– 23, # 8. – P. 655 – 677.
11. Showalter R. E. Monotone operators in Banach space and nonlinear partial differential equations //
Math. Surv. and Monogr. – Providence: Amer. Math. Soc., 1997. – 49. – xiv + 278 p.
12. Mel\nykova Y. V. Zadaça Koßy dlq vklgçenyq v banaxov¥x prostranstvax y prostranstvax
raspredelenyj // Syb. mat. Ωurn. � 2001. � 42, # 4. � S. 892 � 910.
13. Showalter R. E. Nonlinear degenerate evolution equations and partial differential equations of
mixed type // SIAM J. Math. Anal. – 1975. – 6, # 1. – P. 25 – 42.
14. Haevskyj X., Hreher K., Zaxaryas K. Nelynejn¥e operatorn¥e uravnenyq y operatorn¥e
dyfferencyal\n¥e uravnenyq. � M.: Myr, 1978. � 336 s.
15. Showalter R. E. Singular nonlinear evolution equations // Rocky Mountain J. Math. – 1980. – 10,
# 3. – P. 499 – 507.
16. Levytan B. M., Ûykov V. V. Poçty peryodyçeskye funkcyy y dyfferencyal\n¥e uravne-
nyq. � M.: Yzd-vo Mosk. un-ta, 1978. � 204 s.
17. Pankov A. A. Ohranyçenn¥e y poçty peryodyçeskye reßenyq nelynejn¥x dyfferencyal\-
n¥x operatorn¥x uravnenyj. � Kyev: Nauk. dumka, 1985. � 184 s.
18. Bahaj M., Sidki O. Almost periodic solutions of semilinear equations with analytic semigroups in
Banach spaces // Electron. J. Different. Equat. – 2002. – 2002, # 98. – P. 1 – 11.
19. Hu Z. Boundedness and Stepanov’s almost periodicity of solutions // Ibid. – 2005. – 2005, # 35. –
P. 1 – 7.
20. Freedman M. A. Existence of strong solutions to singular nonlinear evolution equations // Pacif. J.
Math. – 1985. – 120, # 2. – P. 331 – 344.
21. Yosyda K. Funkcyonal\n¥j analyz. � M.: Myr, 1967. � 624 s.
OderΩano 23.05.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
|
| id | umjimathkievua-article-3022 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:34:47Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cb/8892bd7083f38c149829d025f6127fcb.pdf |
| spelling | umjimathkievua-article-30222020-03-18T19:43:35Z Problem without initial conditions for linear and almost linear degenerate operator differential equations Задача без початкових умов для лінійних та майже лінійних вироджених операторних диференціальних рівнянь Dmytryshyn, Yu. B. Дмитришин, Ю. Б. We study the problem without initial conditions for linear and almost linear degenerate operator differential equations in Banach spaces. The uniqueness of a solution of this problem is proved in the classes of bounded functions and functions with exponential behavior as t → –∞. We also establish sufficient conditions for initial data under which there exists a solution of the considered problem in the class of functions with exponential behavior at infinity. Изучается задача без начальных условий для вырожденных линейных и почти линейных операторных дифференциальных уравнений в банаховых пространствах. Доказана единственность решения этой задачи в классах ограниченных функций и функций с экспоненциальным поведением при t → –∞. Кроме того, установлены достаточные условия на исходные данные, при которых существует решение указанной задачи в классе функций с экспоненциальным поведением на бесконечности. Institute of Mathematics, NAS of Ukraine 2009-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3022 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 3 (2009); 322-332 Український математичний журнал; Том 61 № 3 (2009); 322-332 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3022/2793 https://umj.imath.kiev.ua/index.php/umj/article/view/3022/2794 Copyright (c) 2009 Dmytryshyn Yu. B. |
| spellingShingle | Dmytryshyn, Yu. B. Дмитришин, Ю. Б. Problem without initial conditions for linear and almost linear degenerate operator differential equations |
| title | Problem without initial conditions for linear and almost linear degenerate operator differential equations |
| title_alt | Задача без початкових умов для лінійних та майже лінійних вироджених операторних диференціальних рівнянь |
| title_full | Problem without initial conditions for linear and almost linear degenerate operator differential equations |
| title_fullStr | Problem without initial conditions for linear and almost linear degenerate operator differential equations |
| title_full_unstemmed | Problem without initial conditions for linear and almost linear degenerate operator differential equations |
| title_short | Problem without initial conditions for linear and almost linear degenerate operator differential equations |
| title_sort | problem without initial conditions for linear and almost linear degenerate operator differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3022 |
| work_keys_str_mv | AT dmytryshynyub problemwithoutinitialconditionsforlinearandalmostlineardegenerateoperatordifferentialequations AT dmitrišinûb problemwithoutinitialconditionsforlinearandalmostlineardegenerateoperatordifferentialequations AT dmytryshynyub zadačabezpočatkovihumovdlâlíníjnihtamajželíníjnihvirodženihoperatornihdiferencíalʹnihrívnânʹ AT dmitrišinûb zadačabezpočatkovihumovdlâlíníjnihtamajželíníjnihvirodženihoperatornihdiferencíalʹnihrívnânʹ |