Averaging of initial-value and multipoint problems for oscillation systems with slowly varying frequencies and deviated argument
We prove new theorems on the substantiation of the method of averaging over all fast variables on a segment and a semiaxis for multifrequency systems with deviated argument in slow and fast variables. An algorithm for the solution of a multipoint problem with parameters is studied, and an estimate f...
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| Дата: | 2007 |
|---|---|
| Автори: | , , , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2007
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3316 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509382860603392 |
|---|---|
| author | Danylyuk, I. M. Petryshyn, R. I. Samoilenko, A. M. Данилюк, І. М. Петришин, Р. І. Самойленко, А. М. |
| author_facet | Danylyuk, I. M. Petryshyn, R. I. Samoilenko, A. M. Данилюк, І. М. Петришин, Р. І. Самойленко, А. М. |
| author_sort | Danylyuk, I. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:51:00Z |
| description | We prove new theorems on the substantiation of the method of averaging over all fast variables on a segment and a semiaxis for multifrequency systems with deviated argument in slow and fast variables. An algorithm for the solution of a multipoint problem with parameters is studied, and an estimate for the difference of solutions of the original problem and the averaged problem is established. |
| first_indexed | 2026-03-24T02:40:13Z |
| format | Article |
| fulltext |
UDK 517.929
A. M. Samojlenko (In-t matematyky NAN Ukra]ny, Ky]v),
R. I. Petryßyn, I. M. Danylgk (Çerniv. nac. un-t)
USEREDNENNQ POÇATKOVO} I BAHATOTOÇKOVO} ZADAÇ
DLQ KOLYVNYX SYSTEM IZ POVIL|NO ZMINNYMY
ÇASTOTAMY I VIDXYLENYM ARHUMENTOM
New theorems on substantiation of the method of averaging over all fast variables on a segment and a
semiaxis are proved for multifrequency systems with deviated argument in slow and fast variables. An
algorithm for the solution of a multipoint problem with parameters is investigated and an estimate for the
difference of solutions of the original and averaged problems is obtained.
Dokazan¥ nov¥e teorem¥ obosnovanyq metoda usrednenyq po vsem b¥str¥m peremenn¥m na
otrezke y poluosy dlq mnohoçastotn¥x system s otklonenn¥m arhumentom v medlenn¥x y
b¥str¥x peremenn¥x. Yssledovan alhorytm reßenyq mnohotoçeçnoj zadaçy s parametramy y
ustanovlena ocenka raznosty reßenyj ysxodnoj y usrednennoj zadaç.
Metod userednennq vyqvyvsq plidnym pry doslidΩenni dyferencial\nyx riv-
nqn\ u riznyx funkcional\nyx prostorax. Dlq bahatotoçkovyx system zvyçaj-
nyx dyferencial\nyx rivnqn\, qkym vlastyve qvywe rezonansu, i takyx Ωe sys-
tem z impul\snog di[g cej metod i joho zastosuvannq do rozv’qzannq krajovyx
zadaç dosyt\ povno ob©runtovano v monohrafi] [1]. U vypadku bahatoçastotnyx
system iz zapiznennqm rizni sxemy userednennq vyvçalys\ u pracqx [2 – 5], a use-
rednennq krajovyx zadaç dlq rivnqn\ z peretvorenym arhumentom — u [6, 7].
Analohiçni pytannq doslidΩuvalys\ takoΩ u robotax [8, 9]. U danij statti
oderΩano novi ocinky poxybky metodu userednennq za vsima ßvydkymy zminnymy
na vidrizku ta pivosi v kolyvnyx systemax z povil\no zminnymy çastotamy i vidxy-
lenym arhumentom ta doslidΩeno ]x vykorystannq dlq pobudovy rozv’qzku odni-
[] bahatotoçkovo] zadaçi z parametramy.
Rozhlqnemo systemu n + m dyferencial\nyx rivnqn\ iz vidxylenym arhumen-
tom vyhlqdu
dx
dτ
= a ( x, xλ , ϕ, ϕλ , τ ) ,
d
d
ϕ
τ
=
ω τ
ε
ϕ ϕ τλ λ
( )
( , , , , )+ b x x , (1)
de τ ∈ [ 0, L ] , ε ≤ ε0 << 1 — malyj dodatnyj parametr, x = x ( τ, ε ) ∈ D ⊂ R
n,
ϕ = ϕ ( τ, ε ) ∈ R
m, xλ = x ( λ ( τ ) , ε ) , ϕλ = ϕ ( λ ( τ ) , ε ) , λ = λ ( τ ) — neperervno
dyferencijovna na [ 0, L ] funkciq, qka zadovol\nq[ umovy
λ ( τ ) < τ, λ ( 0 ) = – ∆ < 0, 0 < σ1
1− <
d
d
λ τ
τ
( )
< σ1 = const,
(2)
λ ( τ0 ) = 0, τ0 ∈ ( 0, L ) ,
D — vidkryta obmeΩena oblast\, dijsni funkci] a i b zadovol\nqgt\ umovu
Lipßycq po x, xλ
, ϕ, ϕλ v oblasti G = D
2 × R
2m × [ 0, L ] , obmeΩeni stalog
σ1 i naleΩat\ pevnym klasam majΩe periodyçnyx po ϕ, ϕλ funkcij.
Zadamo dlq (1) poçatkovu umovu
x ( τ, ε ) = f ( τ, ε ) , ϕ ( τ, ε ) = g t dt( , ) ( )τ ε
ε
τ
+ ∫1
0
Ω , τ ∈ [ – ∆, 0 ] , (3)
v qkij funkci] f i g [ neperervno dyferencijovnymy po τ i
d f
d
dg
d
( , ) ( , )τ ε
τ
τ ε
τ
+ ≤ σ1 , τ ∈ [ – ∆, 0 ] , ε ∈ ( 0, ε0 ] . (4)
© A. M. SAMOJLENKO, R. I. PETRYÍYN, I. M. DANYLGK, 2007
412 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
USEREDNENNQ POÇATKOVO} I BAHATOTOÇKOVO} ZADAÇ DLQ KOLYVNYX … 413
Tut i dali pid normog vektora rozumi[mo evklidovu normu, a norma matryci uz-
hodΩena z evklidovog normog vektora.
Nexaj
ω ( τ ) = ( )( ), , ( )ω τ ω τ1 … m ∈ C L
p
[ , ]0
1− ,
(5)
Ω ( τ ) = ( )( ), , ( )ω τ ω τ1 … m ∈ C p
[ , ]−
−
∆ 0
1 , λ ( τ ) ∈ C L
p
[ , ]0
pry deqkomu p ≥ 2 m i poznaçymo çerez A( )τ , A( )τ i B( )τ ( p × m ) -vymirni
matryci
A( )τ =
d
d
l
l
l
p m−
−
=
1
1
1τ
ω τν
ν
( )
,
,
, τ ∈ [ 0, L ] ,
A( )τ =
d
d
d
d
l
l
l
p m−
−
=
1
1
1τ
ω λ τ λ τ
τν
ν
( ( ))
( )
,
,
, τ ∈ [ τ0, L ] ,
B( )τ =
d
d
d
d
l
l
l
p m−
−
=
1
1
1τ
λ τ λ τ
τν
ν
Ω ( ( ))
( )
,
,
, τ ∈ [ 0, τ0 ] ,
a çerez A ( τ ) i B ( τ ) matryci vymiru p × 2 m :
A ( τ ) = ( )( ) ( )A Bτ τ , τ ∈ [ 0, τ0 ] ,
B ( τ ) = ( )( ) ( )A Aτ τ , τ ∈ [ τ0, L ] .
Prypustymo, wo
det ( ) ( )( )A AT τ τ > 0, τ ∈ [ τ0, L ] ,
(6)
det ( ) ( )( )B BT τ τ > 0, τ ∈ [ 0, τ0 ] .
Rozhlqnemo dali oscylqcijnyj intehral [1]
I tk ( ), , ,τ τ ε = F t
i
k l dl dt
t
t
( ) exp ( , ˜ ( ))
ε
ω
τ
τ
∫∫
,
v qkomu τ ∈ [ 0, L ] , τ ∈ [ 0, L ] , t ∈ [ 0, L ] , ε ∈ ( 0, ε0 ] , k — nenul\ovyj 2m-
vymirnyj vektor, i = −1 — uqvna odynycq, ˜ ( )ω τ — 2m -vymirnyj vektor z
koordynatamy
ω τ1( ) , … , ω τm( ), Ω1( ( ))
( )λ τ λ τ
τ
d
d
, … , Ωm
d
d
( ( ))
( )λ τ λ τ
τ
pry τ ∈ [ 0, τ0 ] i
ω τ1( ) , … , ω τm( ), ω λ τ λ τ
τ1( ( ))
( )d
d
, … , ω λ τ λ τ
τm
d
d
( ( ))
( )
pry τ ∈ ( τ0, L ] , ( , ˜ )k ω — skalqrnyj dobutok v R m2 , F — zadana na [ 0, L ] vek-
tor-funkciq.
Lema)1. Qkwo vykonugt\sq nerivnosti (6) i F ( τ ) ma[ na [ 0, L ] kuskovo-
neperervnu poxidnu, to isnugt\ taki dodatni stali ε1 i σ0
, ne zaleΩni vid τ,
τ , t , k, ε i F, wo
I tk ( ), , ,τ τ ε ≤ σ ε τ τ
τ0
1
0 0
1
1 1/
[ , ] [ , ]
sup ( ) sup
( )p
L Lk
F
k
dF
d
+
+
(7)
dlq vsix τ ∈ [ 0, L ] , τ ∈ [ 0, L ] , t ∈ [ 0, L ] , k ≠ 0, ε ∈ ( 0, ε1 ] .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
414 A. M. SAMOJLENKO, R. I. PETRYÍYN, I. M. DANYLGK
Qkwo Ω F ( τ ) [ kuskovo-hel\derovog na [ 0, L ] z pokaznykom α ∈ ( 0, 1 ] i
stalog Hel\dera σ̃0, to
I tk ( ), , ,τ τ ε ≤ σ ε τ σα
0
1 1 1
0
0
1 1/ ( )
[ , ]
sup ( ) ˜p
p
Lk k
F+ +
+
. (8)
Dovedennq. Na pidstavi nerivnostej (6) ocinku (7) oderΩano u monohrafi]
[1, c. 18]. Dovedemo ocinku (8). Zhidno z oznaçennqm kuskovo-hel\derovo] funk-
ci] [10, c. 340] vidrizok [ 0, L ] moΩna rozbyty na skinçenne çyslo vidrizkiv toç-
kamy τ0 = 0 < τ1 < … < τn0
= L i na koΩnomu z vidrizkiv [ ],τ τr r+1 funkciq
F zadovol\nq[ nerivnist\
F F( ) ( )′ − ′′τ τ ≤ σ̃ τ τ α
0 ′ − ′′ , ′ ′′ ∈ +τ τ τ τ, ,[ ]r r 1 , r = 0 10, n − .
Pry c\omu pid znaçennqmy funkci] na kincqx vidrizka rozumi[mo hranyçni zna-
çennq F r( )τ + 0 i F r( )τ + −1 0 . Nexaj [ ],τ τr r+1 ⊂ [ , ]τ τ . Vyberemo dosyt\ male
dodatne çyslo h, qke oznaçymo nyΩçe, i podamo [ ],τ τr r+1 u vyhlqdi
[ ],τ τr r+1 =
[ ]˜ , ˜α αν ν
ν
+
=
1
0
q
∪ ,
de α̃0 = τr , ˜ ˜α αν ν+ −1 = h pry ν < q, α̃q+1 = τr+1, q — cila çastyna çysla
( )τ τr r h+
−−1
1, q ≤ Lh−1. Todi
F t l t t dtk
r
r
( ) ( , , )ε
τ
τ +
∫
1
=
ν
ν
α
α
α ε
ν
ν
=
∑ ∫
+
0
1q
kF l t t dt( ˜ ) ( , , )
˜
˜
+
+ ( )( ) ( ˜ ) ( , , )
˜
˜
F t F l t t dtk−
+
∫ α εν
α
α
ν
ν 1
,
de
l t tk ( , , )ε = exp ( , ˜ ( ))
i
k l dl
t
t
ε
ω∫
. (9)
Vraxovugçy ocinku oscylqcijnoho intehrala [1, c. 81]
l t t dtk ( , , )ε
τ
τ
∫ ≤ σ ε0
1 1 1/ p
pk k
+
, (10)
qka pokrawu[ ocinku (7) pry F ( t ) ≡ 1 dlq vypadku k → ∞ , oderΩu[mo ne-
rivnist\
F t l t t dtk
r
r
( ) ( , , )ε
τ
τ +
∫
1
≤ 2
1
1
1
0
1 1L h
k
pσ
α
ε+
+
−/ +
+ 1
0
0k
F t hp
L
+
sup ( ) ˜
[ , ]
σ α .
Najkrawyj porqdok po ε ostann\o] ocinky bude v tomu vypadku, koly h
α =
= ε1 1/ p h−
abo h = ε α1 1/ ( )p + . Zvidsy vyplyva[ nerivnist\ (8) zi stalog σ0
1 =
= 2 10 0
1Ln ( )( )σ α+ + − .
Lemu dovedeno.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
USEREDNENNQ POÇATKOVO} I BAHATOTOÇKOVO} ZADAÇ DLQ KOLYVNYX … 415
Prypustymo, wo ( n + m ) -vymirna vektor-funkciq c y( , , )θ τ = ( ( , , )a y θ τ ,
b y( , , ))θ τ , de y = ( , )x xλ , θ = ( , )θ θλ , naleΩyt\ klasu majΩe periodyçnyx po
θ funkcij, qki rozkladagt\sq v rivnomirno po θ zbiΩnyj rqd Fur’[
c y( , , )θ τ = c y es
i k
s
s( , ) ( , )τ θ
=
∞
∑
0
, (11)
v qkomu k0 = 0, ks ≠ 0 pry s ≥ 1, ( , )ks θ — skalqrnyj dobutok v R m2 .
Rozhlqnemo userednenu za vsima ßvydkymy zminnymy ϕ, ϕλ systemu
dx
dτ
= a x x0( , , )λ τ ,
d
d
ϕ
τ
=
ω τ
ε
τλ
( )
( , , )+ b x x0 , τ ∈ [ 0, L ] , (12)
de
( )( , ), ( , )a y b y0 0τ τ = c y0( , )τ = lim ( , , )
T
m
T T
mT c y d d
→∞
− ∫ ∫… …2
0 0
1 2θ τ θ θ ,
i vidpovidnu ]j poçatkovu umovu
x ( τ, ε ) = f ( τ, ε ) , ϕ ( τ, ε ) = g t dt( , ) ( )τ ε
ε
τ
+ ∫1
0
Ω , τ ∈ [ – ∆, 0 ] . (13)
U nastupnij teoremi x ( τ, ε ), ϕ ( τ, ε ) i x ( τ, ε ), ϕ ( τ, ε ) poznaçagt\ roz-
v’qzky zadaç (1), (3) i (12), (13).
Teorema)1. Nexaj: 1) kryva x = x ( τ, ε ) leΩyt\ u D razom iz svo]m ρ-
okolom pry τ ∈ [ – ∆, L ] , ε ∈ ( 0, ε0 ] ; 2) vykonugt\sq umovy (2), (5) i nerivnosti
(4), (6); 3) funkciq c y( , , )θ τ ma[ obmeΩeni v G stalog σ1 çastynni poxidni
perßoho porqdku po y, θ, τ, rozklada[t\sq v rqd (11), pryçomu
s s G
s
s G
s
k
c y
k
c y
y=
∞
∑ +
+ ∂
∂
1
1
1 1
1 1
sup ( , ) sup
( , )τ τ
+
+ sup
( , )
G
sc y
1
∂
∂
τ
τ
≤ σ1 , G1 = D
2 × [ 0, L ] .
Todi pry dosyt\ malomu ε0 > 0 dlq vsix τ ∈ [ 0, L ] , ε ∈ ( 0, ε0 ] spravdΩu-
[t\sq ocinka
x x( , ) ( , ) ( , ) ( , )τ ε τ ε ϕ τ ε ϕ τ ε− + − ≤ σ ε2
1/ p (14)
zi stalog σ2 , ne zaleΩnog vid ε .
Dovedennq. Iz vyxidno] ta useredneno] system dlq funkci] v( , )τ ε =
= ( )( , ) ( , ), ( , ) ( , )x xτ ε τ ε ϕ τ ε ϕ τ ε− − dista[mo nerivnist\
v( , )τ ε ≤
σ ε λ ε ε θ ε
τ τ
1
0 0
( )( , ) ( ( ), ) ˜( ( , ), ( , ), )v vt t dt c y t t t dt+ +∫ ∫ , (15)
v qkij y x x( , ) ( ( , ), ( ( ), ))τ ε τ ε λ τ ε= , θ τ ε ϕ τ ε ϕ λ τ ε( , ) ( ( , ), ( ( ), ))= , ˜( , , )c y θ τ =
= c y c y( , , ) ( , )θ τ τ− 0 .
Oskil\ky v( ( ), )λ τ ε ≡ 0 pry τ τ≤ 0 i
v( ( ), )λ ε
τ
t dt
0
∫ ≤
σ ε
λ τ
1 v( , )
( )
t dt
−
∫
∆
=
σ ε
λ τ
1
0
v( , )
( )
t dt∫ ≤
σ ε
τ
1
0
v( , )t dt∫ ,
to z (15) oderΩu[mo nerivnist\
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
416 A. M. SAMOJLENKO, R. I. PETRYÍYN, I. M. DANYLGK
v( , )τ ε ≤
σ σ ε
τ
1 1
0
1( ) ( , )+ ∫ v t dt +
s
s
i k t
kc y t t e l t dts
s
=
∞
∑ ∫
1
0
0
( ( , ), ) ( , , )( , ( , ))ε τ εψ ε
τ
,
(16)
de ψ τ ε ψ τ ε ψ τ ε( , ) ( ˜ ( , ),
˜
( , ))= — 2m-vymirnyj vektor,
˜ ( , )ψ τ ε = g t dt b x t x t t dt( , ) ( ) ( ( , ), ( ( ), ), )0
1
0
00
0
ε
ε
ω ε λ ε
ττ
+ + ∫∫ , τ ≥ 0,
˜
( , )ψ τ ε = g( ( ), )λ τ ε , τ ∈ [ 0, τ0 ] ,
˜
( , )ψ τ ε = g b x t x t t dt( , ) ( ( , ), ( ( ), ), )
( )
0 0
0
ε ε λ ε
λ τ
+ ∫ , τ > τ0
.
Na pidstavi prypuwennq 3 teoremy 1 i ocinky (7) dlq
F ( t ) = c y t t es
i k ts( ( , ), ) ( , ( , ))ε ψ ε
iz (16) otrymu[mo
v( , )τ ε ≤ σ σ ε
τ
1 1
0
1( ) ( , )+ ∫ v t dt + σ σ σ ε0 1
2
1
12( ) /+ p , ( τ, ε ) ∈ [ 0, L ] × ( 0, ε0 ] .
Zvidsy z uraxuvannqm nerivnosti Hronuolla – Bellmana dista[mo ocinku (14) zi
stalog σ2 = e Lσ σ σ σ σ1 11
0 1
2
12( ) ( )+ + . Malyzna çysla ε0 vyznaça[t\sq lemog 1
i nerivnistg σ ε2 0
p ≤ ρ / 2 , qka razom z ocinkog (14) i umovog λ ( τ ) < τ haran-
tu[, wo rozv’qzok x ( τ, ε ) , ϕ ( τ, ε ) zadaçi (1), (3) vyznaçeno dlq vsix τ ∈ [ – ∆,
L ] , εT∈ ( 0, ε0 ] .
Teoremu dovedeno.
Vidmovymos\ teper vid prypuwennq 3 teoremy 1 i vvaΩatymemo, wo funkciq
c ( y, θ, τ ) zadovol\nq[ umovu Lipßycq
c y c y( , , ) ( ˜, ˜ , ˜ )θ τ θ τ− ≤ σ θ θ τ τ1 y y− + − + −( )˜ ˜ ˜ , (17)
( , , )y θ τ ∈ G, ( ˜, ˜ , ˜ )y θ τ ∈ G,
a koefici[nty c ys( , )τ rivnomirno zbiΩnoho po θ rqdu (11) spravdΩugt\ neriv-
nist\
s
s
s s
n
c y
k k=
∞
∑ +
1
1 1
( , )τ ≤ σ1 , ( y, τ ) ∈ G1 . (18)
Qk i pry dovedenni lemyT1, rozib’[mo vidrizok [ 0, τ ] na skladovi za formu-
log
[ 0, τ ] =
[ ]˜ , ˜β βν ν
ν
+
=
1
0
q
∪ ,
de β̃0 0= ,
˜ ˜β βν ν+ − =1 h pry ν < q , β̃ τq+ =1 , q — cila çastyna çysla τh−1
.
Todi
˜( ( , ), ( , ), )c y t t t dtε θ ε
τ
0
∫ ≤
ν β
β
τν
ν
ε ψ ε
ε
ω
=
∑ ∫ ∫
+
+
0
1
0
1q t
c y t t l dl t
˜
˜
˜ ( , ), ( , ) ˜ ( ) , –
– ˜ ˜ , , ˜ , ˜ ( ) , ˜( ) ( )c y l dl dt
t
β ε ψ β ε
ε
ω βν ν
τ
ν+
∫1
0
+
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
USEREDNENNQ POÇATKOVO} I BAHATOTOÇKOVO} ZADAÇ DLQ KOLYVNYX … 417
+
s
s kc y l t dt
s
=
∞
∑ ∫
+
1
0
1
( ( ))˜ , ( , , )
˜
˜
β ε τ εν
β
β
ν
ν
.
Vraxovugçy nerivnosti (10), (17), (18) i
y t y( , ) ˜ ,( )ε β εν− ≤ σ σ1 11( )+ h , ψ ε ψ β εν( , ) ˜ ,( )t − ≤ σ σ1 11( )+ h ,
t ∈ [ ]˜ , ˜β βν ν+1 ,
oderΩu[mo
˜( ( , ), ( , ), )c y t t t dtε θ ε
τ
0
∫ ≤ σ ε3
1 1( )/h hp+ − ,
de σ3 = Lσ σ σ σ1 0 1 12 1 1( ( ) )+ + + .
Teper z (15) otrymu[mo nerivnist\
v( , )τ ε ≤
σ σ ε σ ε
τ
1 1
0
3
1 11( ) ( , ) ( )/+ + +∫ −v t dt h hp ,
qka pryvodyt\ do ocinky
v( , )τ ε ≤ σ εσ σ
3
1 1 11 1e h hL p( ) /( )+ −+ , τ ∈ [ 0, L ] , ε ∈ ( 0, ε0 ] .
}] najkrawyj porqdok po ε bude pry h = ε1 2/ p .
OtΩe, dovedeno taku teoremu.
Teorema)2. Qkwo vykonugt\sq umovy 1, 2 teoremy T1 i nerivnosti (17),
(18), to isnugt\ taki stali σ4 > 0 i ε2 > 0, wo pry ε0 ≤ ε2
x x( , ) ( , ) ( , ) ( , )τ ε τ ε ϕ τ ε ϕ τ ε− + − ≤ σ ε4
1 2/ p
dlq vsix τ ∈ [ 0, L ] i ε ∈ ( 0, ε0 ] .
Nexaj systemu (1) zadano pry τ ∈ [ 0, ∞ ) = R + . Vyvçymo dali metod usered-
nennq na pivosi. Rozhlqnemo userednenu systemu dlq povil\nyx zminnyx
dx
dτ
= a x x0( , , )λ τ , τ ∈ R + , (19)
i prypustymo, wo dlq deqko] neperervno dyferencijovno] na [ – ∆, 0 ] funkci]
f ( τ ) rozv’qzok x = x0( )τ systemy (19), qkyj spravdΩu[ umovu x0( )τ = f ( τ )
pry τ ∈ [ – ∆, 0 ] , vyznaçeno dlq vsix τ ∈ [ – ∆, ∞ ) .
VvaΩatymemo, wo normal\na fundamental\na matrycq Q ( τ, t ) linijno] sys-
temy
dz
dτ
= H z1( )τ , H1( )τ =
∂
∂
a x x
x
0
0 0( )( ), ( ( )),τ λ τ τ
,
zadovol\nq[ nerivnist\
Q t( , )τ ≤ Ke t− −γ τ( ) , τ ≥ t ≥ 0, (20)
z deqkymy stalymy K ≥ 1 i γ > 0, a çyslo
σ0 = sup ( )
τ
τ
∈ +R
H2 , H2( )τ =
∂
∂
a x x
x
0
0 0( )( ), ( ( )),τ λ τ τ
λ
,
nastil\ky male, wo
γ – K σ0 > 0. (21)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
418 A. M. SAMOJLENKO, R. I. PETRYÍYN, I. M. DANYLGK
Prypustymo takoΩ, wo funkciq a ( y, θ, τ ) rozklada[t\sq v G̃ = D
2 ×
× R Rm2 × + u rqd Fur’[
a ( y, θ, τ ) =
s
s
i ka y e s
=
∞
∑
0
( , ) ( , )τ θ ,
koefici[nty qkoho spravdΩugt\ nerivnist\
s s G
s
s G
s
k
a y
k
a y
y=
∞
∑ +
+ ∂
∂
1
1
1 1
1 1
sup ( , ) sup
( , )
˜ ˜
τ τ
+
+ sup
( , )
G̃
sa y
1
∂
∂
τ
τ
≤ σ1 , G̃1 = D R2 × + . (22)
Teorema)3. Nexaj:
1) funkci]
d
d
l
lτ
τνΩ ( ) [ neperervnymy na [ – ∆, 0 ] , a funkci]
d
d
l
lτ
ω τν( ( )) i
d
d
d
d
l
lτ
ω λ τ λ τ
τν( ( ))
( )
— rivnomirno neperervnymy na R + pry l = 0 1, p − , ν =
= 1, m i
( )( ) ( ) ( )A A AT Tτ τ τ−1 ≤ σ̃1 = const, τ ∈ [ τ0, ∞ ) ,
det ( ) ( )( )B BT τ τ > 0, τ ∈ [ 0, τ0 ] ;
2) isnu[ rozv’qzok x x= 0( )τ useredneno] systemy (19), x f0( ) ( )τ τ= pry
τ ∈ [ – ∆, 0 ] , qkyj vyznaçenyj dlq vsix τ ∈ [ – ∆, ∞ ) i leΩyt\ u D razom iz
svo]m ρ1 -okolom;
3) funkciq
∂
∂
a y
y
0( , )τ
rivnomirno neperervna po ( , )y Dτ ∈ × +
2
R , a funk-
ciq c ( y, θ , τ ) ta ]] çastynni poxidni perßoho porqdku neperervni v G̃ i obme-
Ωeni stalog σ1
;
4) vykonugt\sq umovy (2) pry τ ∈ R + i nerivnosti (20) – (22).
Todi isnugt\ taki dodatni stali ε2
, σ5 i ρ ρ2 1< , wo pry ε ε0 2≤ :
a) dlq dovil\no] neperervno dyferencijovno] po τ ∈ [ – ∆, 0 ] funkci] g ( τ, ε ) ,
qka zadovol\nq[ nerivnist\
dg
d
( , )τ ε
τ
≤ σ1 , τ ∈ [ – ∆, 0 ] , ε ∈ ( 0, ε0 ] ,
rozv’qzok x ( τ, ε ) , ϕ ( τ, ε ) systemy (1) z poçatkovog umovog
x ( τ, ε ) = f ( τ ) , ϕ ( τ, ε ) = g t dt( , ) ( )τ ε
ε
τ
+ ∫1
0
Ω , τ ∈ [ – ∆, 0 ] , (23)
[ vyznaçenym dlq vsix τ ∈ [ – ∆, ∞ ) , ε ∈ ( 0, ε0 ] i vykonu[t\sq nerivnist\
x x( , ) ( )τ ε τ− 0 ≤ σ ε5
1/ p, τ ∈ [ 0, ∞ ) , ε ∈ ( 0, ε0 ] , (24)
b) dlq dovil\nyx neperervno dyferencijovnyx po τ ∈ [ – ∆, 0 ] funkcij h ( τ, ε )
i g ( τ, ε ) , qki zadovol\nqgt\ nerivnosti
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
USEREDNENNQ POÇATKOVO} I BAHATOTOÇKOVO} ZADAÇ DLQ KOLYVNYX … 419
h f( , ) ( )τ ε τ− ≤ ρ2 ,
∂
∂
+ ∂
∂
h g( , ) ( , )τ ε
τ
τ ε
τ
≤ σ1 , τ ∈ [ – ∆, 0 ] , ε ∈ ( 0, ε0 ] ,
povil\na komponenta x ( τ, ε ) rozv’qzku x ( τ, ε ) , ϕ ( τ, ε ) systemy (1) z poçat-
kovog umovog
x ( τ, ε ) = h ( τ, ε ) , ϕ ( τ, ε ) = g t dt( , ) ( )τ ε
ε
τ
+ ∫1
0
Ω , τ ∈ [ – ∆, 0 ] , (25)
rivnomirno obmeΩena pry τ ∈ [ – ∆, ∞ ) , ε ∈ ( 0, ε0 ] .
Dovedennq. Nexaj [ 0, T1 ) , T1 = T1 ( ε ) , — maksymal\nyj pivinterval, dlq
qkoho komponenta x ( τ, ε ) rozv’qzku x ( τ, ε ) , ϕ ( τ, ε ) zadaçi (1), (25) zadovol\nq[
nerivnist\
x x( , ) ( )τ ε τ− 0 < ρ3 , τ ∈ [ 0, T1 ) ,
z dodatnog stalog ρ3 , qku bude oznaçeno nyΩçe. Todi dlq funkci] z ( τ, ε ) =
= x ( τ, ε ) – x0
( τ ) ma[mo zobraΩennq
z ( τ, ε ) = h ( τ, ε ) – f ( τ ) , τ ∈ [ – ∆, 0 ] ,
(26)
z ( τ, ε ) = Q h f Q t H t z t( , )[ ( , ) ( )] ( , ) ( ) ( ( ), )τ ε τ λ ε
τ
0 0 0
0
2− + [∫ +
+ F z t z t t dt I1( )( , ), ( ( ), ), ( , )ε λ ε τ ε] + , τ > 0 ,
de
I ( τ, ε ) =
0
τ
τ ε θ ε∫ Q t a y t t t dt( , ) ˜ ( , ), ( , ),( ) ,
y ( t, ε ) = ( x ( t, ε ) , x ( λ ( t ) , ε )) , θ ( t, ε ) = ( ϕ ( t, ε ) , ϕ ( λ ( t ) , ε )) ,
˜ ( , , )a y tθ = a ( y, θ, t ) – a0 ( y, t ) ,
F1 ( z, zλ , t ) = a0 ( z + x0
( t ) , zλ + x0
( λ ( t )) , t ) –
– a0 ( x0
( t ), x0
( λ ( t )) , t ) – H1 ( t) z – H2 ( t) zλ .
Zhidno z navedenymy vywe prypuwennqmy i nerivnistg
sup ( ( ), )
[ , )τ
λ τ ε
0 1T
z ≤ sup ( , )
[ , )0 1T
z τ ε ,
otrymu[mo
0
2
τ
τ λ ε∫ Q t H t z t dt( , ) ( ) ( ( ), ) ≤ K e h t f t dttσ λ ε λ
τ
γ τ0
0
0
∫ − − −( ) ( ( ), ) ( ( )) ≤ σ σ ρ0
6 2
pry τ ∈ [ 0, τ0 ] i
0
2
τ
τ λ ε∫ Q t H t z t dt( , ) ( ) ( ( ), ) ≤ σ σ ρ τ ε
τ
τ
γ τ0
6 2
0
0 1
+
∫ − −K e z dtt
T
( )
[ , )
sup ( , ) ≤
≤ σ σ ρ σ
γ
τ ε0
6 2
0
0 1
+ K
z
T
sup ( , )
[ , )
pry τ > τ0 . Tut σ6 = K eγ γτ− −1 0 1( ).
Oskil\ky
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
420 A. M. SAMOJLENKO, R. I. PETRYÍYN, I. M. DANYLGK
F z t1(˜, ) =
0
1
0
0
0
0
∫ ∂ +
∂
− ∂
∂
a y lz t
y
a y t
y
dl z
( ) ( )˜ ˜, ˜ ,
˜ ,
de z̃ = ( z ( t, ε ) , z ( λ ( t ) , ε )) , ỹ0 = ( x0
( t ) , x0
( λ ( t ))) , a funkciq
∂
∂
a y t
y
0( , )
rivnomirno neperervna po ( , )y t D∈ × +
2
R , tobto dlq dovil\noho çysla µ̃1 > 0
isnu[ take dodatne µ̃2 = ˜ ˜( )µ µ2 1 , ne zaleΩne vid ỹ0
i t, wo
∂ +
∂
− ∂
∂
a y lz t
y
a y t
y
0
0
0
0( ) ( )˜ ˜, ˜ ,
< µ̃1, t ∈ R + ,
pry
z̃ ≤ z t z t( , ) ( ( ), )ε λ ε+ < µ̃2 ,
to
F z t1(˜, ) ≤ ˜ ( , ) ( ( ), )( )µ ε λ ε1 z t z t+ .
Çyslo µ̃1 oznaçymo nyΩçe.
U zv’qzku z cym
0
1
τ
τ ε λ ε∫ Q t F z t z t t dt( , ) ( , ), ( ( ), ),( ) ≤ σ ρ µ σ µ τ ε6 2 1 7 1
0 1
˜ ˜ sup ( , )
[ , )
+
T
z ,
de σ7 = 2 1Kγ − .
Todi z (26) znaxodymo
sup ( , )
[ , )0 1T
z τ ε ≤
σ
σ γ σ µ
ρ8
0 1
7 1
21− −−K ˜
+
1
1 0 1
7 1 0 1
− −−K
I
Tσ γ σ µ
τ ε
˜
sup ( , )
[ , )
.
(27)
Tut σ8 = K + +σ σ µ σ0
6 1 6˜ . Zhidno z prypuwennqm (21) çyslo σ̃0 = 1 0 1− −Kσ γ
[ dodatnym, tomu poklademo µ̃1 =
σ̃
σ
0
72
.
Podamo dali I ( τ, ε ) u vyhlqdi
I ( τ, ε ) =
s
q
s
q
sr t dt r t dt
=
∞
=
− +
∑ ∑ ∫ ∫+
1 0
1 1
0
0
0ν ντ
ν τ
τ
τ
τ ε τ ε
( )
( , , ) ( , , ) , (28)
de
r ts( , , )τ ε = Q t a y t t e l ts
i k t
k
s
s
( , ) ( ( , ), ) ( , , )( , ( , ))τ ε τ εεΨ
0 ,
y ( t, ε ) = ( x ( t, ε ) , x ( λ ( t ) , ε )) , θ ( t, ε ) = ( ϕ ( t, ε ) , ϕ ( λ ( t ) , ε )) ,
q — cila çastyna çysla ττ0
1− , funkciq l tks
( , , )τ ε0 oznaçena formulog (9), a
2m-vymirnyj vektor Ψ ( t, ε ) = ( )˜ ( , ),
˜
( , )Ψ Ψt tε ε ,
˜ ( , )Ψ t ε = g d b y d
t
( , ) ( ) ( ( , ), ( , ), )0
1
0 0
0
ε
ε
ω τ τ τ ε θ τ ε τ τ
τ
+ +∫ ∫ , t ≥ 0 ,
˜
( , )Ψ t ε = g ( λ ( t ) , ε ) , t ∈ [ 0, τ0 ] ,
˜
( , )Ψ t ε = g b y d
t
( , ) ( ( , ), ( , ), )
( )
0
0
ε τ ε θ τ ε τ τ
λ
+ ∫ , t ≥ τ0 .
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USEREDNENNQ POÇATKOVO} I BAHATOTOÇKOVO} ZADAÇ DLQ KOLYVNYX … 421
Na pidstavi prypuwennq 1 teoremy 3 ocinka (7) oscylqcijnoho intehrala [
spravedlyvog i v tomu vypadku, koly zamist\ vidrizka [ 0, L ] vzqty dovil\nyj
vidrizok [ ξ, ξ + τ0 ] , ξ ∈ R + , dovΩyny τ0 [1, c. 18]. Pry c\omu stala σ0 zale-
Ωyt\ vid τ0
, ale ne zaleΩyt\ vid ξ . Tomu, vraxovugçy, wo
dQ t
dt
( , )τ
= – Q t H t( , ) ( )τ 1 ,
dQ t
dt
( , )τ
≤ K e tσ γ τ
1
− −( ), τ ≥ t > 0,
d
dt
tΨ( , )ε ≤ σ σ1 11( )+ , t ∈ R + , ε ∈ ( 0, ε0 ] ,
ma[mo nerivnist\
ντ
ν τ
τ ε
0
01( )
( , , )
+
∫ r t dts ≤ ε σ τγ τ ντ1
9
0
1
1
1/ ( )
˜
sup ( , )p
s G
se
k
a y− − +
+
+
1
1 1
k
a y
y
a y
s G
s
G
ssup
( , )
sup
( , )
˜ ˜
∂
∂
+ ∂
∂
τ τ
τ
zi stalog σ9 = K eσ σ σγ τ
0 1 1
20 1( )+ + . Taku Ω nerivnist\ zadovol\nq[ intehral
vid funkci] r ts( , , )τ ε po vidrizku [ q τ0, τ ] .
OtΩe, vraxovugçy nerivnist\ (22), iz (28) otrymu[mo ocinku
I( , )τ ε ≤ σ ε10
1/ p, τ ∈ [ 0, T1 ) , ε ∈ ( 0, ε0 ] , (29)
v qkij σ10 = σ σ
γ τ
γ τ1 9
0
0 1
e
e −
.
Povernemos\ do nerivnosti (27) i poklademo v nij
ρ2 =
σ̃ ρ
σ
0
3
86
, ρ3 = min , ˜ ( ˜ )
1
2
1
41 2 1ρ µ µ{ },
a dodatne ε0 vyberemo nastil\ky malym, wo
2 10
0
1σ
σ
ε
˜
/ p ≤
1
3 3ρ .
Todi z (27) i (29) oderΩu[mo nerivnist\
sup ( , ) ( )
[ , )0
0
1T
x xτ ε τ− ≤
2
3 3ρ .
Qkwo prypustyty, wo pry deqkomu ε ∈ ( 0, ε0 ] çyslo T1 = T1 ( ε ) [ skinçen-
nym, to na pidstavi ostann\o] nerivnosti i nerivnosti
min ( )
[ , ]
( )
τ
τ λ τ
∈ +
−
0 11T
≡ ∆1 ( ε ) > 0
rozv’qzok x ( τ, ε ) , ϕ ( τ, ε ) zadaçi (1), (25) metodom krokiv z dovΩynog kroku
∆1 ( ε ) moΩna prodovΩyty na deqkyj promiΩok [ ], ˜0 1 1T + ∆ , de ∆̃1 = ˜ ( )∆1 ε > 0,
pryçomu
sup ( , ) ( )
[ , ˜ ]0
0
1 1T
x x
+
−
∆
τ ε τ ≤
3
4 3ρ .
A ce supereçyt\ oznaçenng çysla T1 . OtΩe, T1 = ∞ dlq vsix ε ∈ ( 0, ε0 ] i
x( , )τ ε < ρ τ3
0+
+
sup ( )
R
x , τ ∈ [ 0, ∞ ) , ε ∈ ( 0, ε0 ] .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
422 A. M. SAMOJLENKO, R. I. PETRYÍYN, I. M. DANYLGK
Ocinka (24) zi stalog σ5 = 2 10
0 1σ σ( )˜ − vyplyva[ z nerivnostej (27) pry
ρ2 = 0 i (29).
Teoremu dovedeno.
Vykorysta[mo oderΩani vywe rezul\taty ta rozroblenu v [1] metodyku dlq
doslidΩennq rozv’qznosti bahatotoçkovo] zadaçi z parametramy vyhlqdu
dx
dτ
= a x x( , , , , , )λ λϕ ϕ ξ τ ,
d
d
ϕ
τ
=
ω τ
ε
ϕ ϕ ξ τλ λ
( )
( , , , , , )+ b x x , τ ∈ [ 0, L ] , (30)
x ( τ, ε ) = f x x r( ( , ), , ( , ), , )τ ε τ ε ξ τ1 … ,
(31)
ϕ ( τ, ε ) = g x x t dt Ar
k
r
k k( ( , ), , ( , ), , , ) ( ) ( , )τ ε τ ε ξ τ ε
ε
ϕ τ ε
τ
1
0 1
1… + +∫ ∑
=
Ω ,
τ ∈ [ – ∆, 0 ] ,
η ϕ ϕ ξ τ τλ λ( , , , , , )x x d
L
0
∫ = 0, (32)
v qkij – ∆ ≤ < < … < ≤τ τ τ1 2 r L , r ≥ 1, ξ = ( ), ,ξ ξ1 … l ∈ D̃ ⊂ R
l
— nevido-
myj vektor parametriv, D̃ — obmeΩena vidkryta oblast\, η — l -vymirna
vektor-funkciq, Ak , k = 1, r , — stali ( m × m ) -matryci.
Prypustymo, wo ( n + m + l ) -vymirna vektor-funkciq d ( y , θ , ξ , τ ) = ( a ( y , θ ,
ξ , τ ) , b ( y , θ , ξ , τ ) , η ( y , θ , ξ , τ )) ma[ neperervni obmeΩeni stalog σ1 çastynni
poxidni po vsix zminnyx do druhoho porqdku vklgçno na mnoΩyni D
2 × R
2m ×
× D̃ × [ 0, L ] i rozklada[t\sq v rqd Fur’[
d ( y , θ , ξ , τ ) = d y es
i k
s
s( , , ) ( , )ξ τ θ
=
∞
∑
0
,
de, qk i vywe, k0 = 0 i ks ≠ 0 pry s ≥ 1, koefici[nty ds ( M ) , M = ( y , ξ , τ )T∈
∈ D
2 × D̃ × [ 0, L ] = B1 , qkoho zadovol\nqgt\ nerivnist\
s
s
s B
s
s B
sk
k
d M
k
d M
M≥
∑ +
+ +
∂
∂
1
1 1 1
1 1
sup ( ) sup
( )
+
+
1
1 1 1
2 2
1
2 2
k
d M
y
d M d M
y ys B
s
B
s
j
n
B
s
j
sup
( )
sup
( )
sup
( )∂
∂ ∂
+ ∂
∂ ∂
+ ∂
∂ ∂
=
∑τ ξ τ
+
+ sup
( )
B
s
j
d M
y
1
2∂
∂ ∂
ξ
≤ σ1
. (33)
VvaΩatymemo takoΩ, wo pry koΩnomu ε ∈ ( 0, ε0 ] funkci] f ( µ, ξ , τ ) i g ( µ,
ξ , τ , ε ) , de µ = ( µ1, … , µr ) ∈ D
r, µk ∈ D , k = 1, r , magt\ neperervni çastynni
poxidni po µ, ξ, τ takoΩ do druhoho porqdku vklgçno i
f M
f M
M
g M
M
f M
( ˜ )
( ˜ )
˜
( ˜ , )
˜
( ˜ )+ ∂
∂
+ ∂
∂
+ ∂
∂ ∂
ε
µ τ
2
+
(34)
+
∂
∂ ∂
+ ∂
∂ ∂
+ ∂
∂ ∂
2 2 2f M g M g M( ˜ ) ( ˜ , ) ( ˜ , )
ξ τ
ε
µ τ
ε
ξ τ
≤ σ1
,
M̃ = ( µ , ξ , τ ) ∈ D
r
× D̃ × [ – ∆, 0 ] , ε ∈ ( 0, ε0 ] .
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USEREDNENNQ POÇATKOVO} I BAHATOTOÇKOVO} ZADAÇ DLQ KOLYVNYX … 423
Zapyßemo userednenu po ϕ, ϕλ zadaçu
dx
dτ
= a x x0( ), , ,λ ξ τ , τ ∈ [ 0, L ] , (351 )
x( )τ = f x x r( )( ), , ( ), ,τ τ ξ τ1 … , τ ∈ [ – ∆, 0 ] , (352 )
0
0
L
x x d∫ η ξ τ τλ( ), , , = 0, (353 )
d
d
ϕ
τ
=
ω τ
ε
ξ τλ
( )
, , ,( )+ b x x0 , τ ∈ [ 0, L ] , (354 )
ϕ τ ε( , ) = g x x t dtr( )( ), , ( ), , , ( )τ τ ξ τ ε
ε
τ
1
0
1… + ∫ Ω +
k
r
k kA
=
∑
1
ϕ τ ε( , ), τ ∈ [ – ∆, 0 ] ,
(355 )
v qkij ( a0 , b0 , η0 ) = d0 . Oçevydno, wo zadaça (351 ) – (355 ) znaçno prostißa,
niΩ zadaça (30) – (32), nasampered tomu, wo perßa z nyx rozpada[t\sq na dvi za-
daçi. Zadaçu (351 ) – (353 ) moΩna rozv’qzuvaty nezaleΩno vid zadaç (354 ), (355 ),
pislq çoho rozv’qzok zadaçi (354 ), (355 ) budu[t\sq alhebra]çnym metodom.
Qkwo rozhlqnuty poçatkovu zadaçu
dx
dτ
= a x x0( ), , ,λ ξ τ , τ ∈ [ 0, L ] ,
x( )τ = f ( , , )µ ξ τ , τ ∈ [ – ∆, 0 ] ,
i prypustyty, wo ]] rozv’qzok x( )τ ≡ x( , , )τ µ ξ [ vyznaçenym dlq vsix τ ∈ [ – ∆,
L ] , ξ ∈D̃, µ ∈Dr
0 ( D0 — deqka vidkryta pidoblast\ oblasti D ) , to para
x ( ), ,τ µ ξ , ξ bude rozv’qzkom zadaçi (351 ) – (353 ) pry umovi, wo systema rivnqn\
µ1 = x( , , )τ µ ξ1 , … , µr = x r( , , )τ µ ξ ,
(36)
0
0
L
x x d∫ η τ µ ξ λ τ µ ξ ξ τ τ( )( , , ), ( ( ), , ), , = 0
ma[ rozv’qzok µ = µ = ( ), ,µ µ1 0… ∈r
rD , ξ = ξ ∈ D̃ .
Nexaj dali ϕ τ ε( , ) ≡ ϕ τ µ ξ ν ε( ), , , , — rozv’qzok poçatkovo] zadaçi
d
d
ϕ
τ
=
ω τ
ε
τ µ ξ λ τ µ ξ ξ τ( )
( , , ), ( ( ), , ), ,( )+ b x x0 , τ ∈ [ 0, L ] ,
ϕ τ µ ξ ν ε( ), , , , = g t dt A
k
r
k k( ), , , ( )µ ξ τ ε
ε
ν
τ
+ +∫ ∑
=
1
0 1
Ω , τ ∈ [ – ∆, 0 ] ,
de ν = ( ν1, … , νr ) ∈ R
mr
. Dlq toho wob funkciq ϕ τ µ ξ ν ε( ), , , , bula rozv’qz-
kom krajovo] zadaçi (354 ), (355 ), dosyt\, wob systema rivnqn\
ν1 = ϕ τ µ ξ ν ε( ), , , ,1 , … , νr = ϕ τ µ ξ ν ε( ), , , ,r ,
abo, wo te same, linijna alhebra]çna systema
( )A E A Ar r1 1 2 2− + + … +ν ν ν = – α1
,
A A E Ar r1 1 2 2ν ν ν+ − + … +( ) = – α2
,
. . . . . . . . . . . . . . . . . . . . . . . (37)
A A A Er r1 1 2 2ν ν ν+ + … + −( ) = – αr
,
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424 A. M. SAMOJLENKO, R. I. PETRYÍYN, I. M. DANYLGK
v qkij E — odynyçna matrycq,
ϕ τ µ ξ ν ε( ), , , , = g A t dt
k
r
k k( ), , , ( )µ ξ ε ν
ε
ω
τ
0
1
1 0
+ +
=
∑ ∫ +
+
0
0
τ
µ ξ λ µ ξ ξ∫ b x t x t t dt( )( , , ), ( ( ), , ), , , τ ∈ [ 0, L ] ,
αs = ϕ τ µ ξ ν ε ν( ), , , ,s
k
r
k kA−
=
∑
1
, s = 1, r ,
mala rozv’qzok ν = ν = ( ), ,ν ν1 … r . Dlq isnuvannq [dynoho rozv’qzku systemy
(37) dosyt\ prypustyty, wo
det E Ak
k
r
−
=
∑
1
≠ 0. (38)
Todi
νs = α αs k
k
r
k k
k
r
E A A+ −
=
−
=
∑ ∑
1
1
1
, s = 1, r .
OtΩe, spravedlyvog [ nastupna lema, v qkij x( )τ ≡ x ( ), ,τ µ ξ , ϕ τ ε( , ) ≡
≡ ϕ τ µ ξ ν ε( ), , , , .
Lema)2. Qkwo systema rivnqn\ (36) ma[ rozv’qzok µ , ξ , dlq qkoho kryva
x = x( )τ leΩyt\ u D dlq vsix τ ∈ [ – ∆, L ] , i vykonu[t\sq nerivnist\ (38),
to userednena zadaça (351 ) – (355 ) ma[ rozv’qzok x( )τ , ϕ τ ε( , ), ξ .
Poznaçymo çerez P ( )rn l+ -vymirnu kvadratnu matrycg
∂
∂
− ∂
∂
X
E
X
P P
( , ) ( , )µ ξ
µ
µ ξ
ξ
1 2
,
de
X( ),µ ξ = ( )( , , ), , ( , , )x x rτ µ ξ τ µ ξ1 … ,
P1 =
0
0 0
L x x
x
x x x
x
x
d∫
∂
∂
∂
∂
+
∂
∂
∂
∂
η ξ τ
µ
η ξ τ
µ
τλ λ
λ
λ( , , , ) ( , , , )
,
P2 =
0
0 0 0
L x x
x
x x x
x
x x x
d∫
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
η ξ τ
ξ
η ξ τ
ξ
η ξ τ
ξ
τλ λ
λ
λ λ( , , , ) ( , , , ) ( , , , )
,
x x= ( , , )τ µ ξ , x xλ λ τ µ ξ= ( ( ), , ) .
Nexaj x( , , , , )τ µ ξ ν ε , ϕ τ µ ξ ν ε( , , , , ) — toj rozv’qzok systemy (30), qkyj pry
τ ∈ [ – ∆, 0 ] zbiha[t\sq z funkci[g
f ( , , )µ ξ τ , g t dt Ak k
k
r
( , , , ) ( )µ ξ τ ε
ε
ν
τ
+ +∫ ∑
=
1
0 1
Ω .
Dlq pobudovy rozv’qzku krajovo] zadaçi (30) – (32) potribno ocinyty funk-
cig
v = ( )( , , , , ) ( , , ), ( , , , , ) ( , , , , )x xτ µ ξ ν ε τ µ ε ϕ τ µ ξ ν ε ϕ τ µ ξ ν ε− −
ta ]] çastynni poxidni perßoho porqdku po µ , ξ , ν . Oçevydno, wo v ≡ 0 pry
τT∈ [ – ∆, 0 ] . Dlq τ > 0 ma[mo rivnosti
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USEREDNENNQ POÇATKOVO} I BAHATOTOÇKOVO} ZADAÇ DLQ KOLYVNYX … 425
x f a x x t dt( , , ) ( , , ) ( , , , )τ µ ξ µ ξ ξ
τ
λ= + ∫0
0
0 ,
x f a x x t dt( , , , , ) ( , , ) ( , , , , , )τ µ ξ ν ε µ ξ ϕ ϕ ξ
τ
λ λ= + ∫0
0
,
ϕ τ µ ξ ν ε µ ξ ε ν
ε
ω ξ
τ τ
λ( , , , , ) ( , , , ) ( ) ( , , , )= + + +
=
∑ ∫ ∫g A t dt b x x t dtk k
k
r
0
1
1 0 0
0 ,
ϕ τ µ ξ ν ε µ ξ ε ν
ε
ω ϕ ϕ ξ
τ τ
λ λ( , , , , ) ( , , , ) ( ) ( , , , , , )= + + +
=
∑ ∫ ∫g A t dt b x x t dtk k
k
r
0
1
1 0 0
,
v qkyx x x t= ( , , )µ ξ , x x tλ λ µ ξ= ( ( ), , ), x x t= ( , , , , )µ ξ ν ε , x x tλ λ µ ξ ν ε= ( ( ), , , , ) ,
ϕ ϕ µ ξ ν ε= ( , , , , )t , ϕ ϕ λ µ ξ ν ελ = ( ( ), , , , )t .
Vykorystovugçy ci rivnosti, ocinku (7) oscylqcijnoho intehrala, obmeΩennq
(33) na koefici[nty Fur’[ ta metod dovedennq teoremyT1, za analohi[g z [1, c. 30]
vstanovlg[mo nastupne tverdΩennq.
Teorema)4. Qkwo vykonugt\sq umovy (2), (5), nerivnosti (6), (33), (34) i
kryva x x= ( , , )τ µ ξ leΩyt\ u D razom iz svo]m ρ-okolom pry τ ∈ [ – ∆, L ] ,
µ ∈Dr
0 , ξ ∈D̃ , to pry dosyt\ malomu ε0 0> spravdΩu[t\sq ocinka
v
v v v+ ∂
∂
+ ∂
∂
+ ∂
∂µ ξ ν
≤ σ ε11
1/ p (39)
dlq vsix ( , , , , ) [ , ] ˜ ( , ]τ µ ξ ν ε ε∈ × × × ×0 00 0L D D Rr mr
z deqkog stalog σ11,
ne zaleΩnog vid ε .
Zhidno z lemogT2 x D( , , )τ µ ξ ∈ pry τ ∈ [ – ∆, L ] . Oskil\ky D — vidkryta
oblast\, to isnu[ take dodatne çyslo ρ , wo kryva x x= ( , , )τ µ ξ leΩyt\ u D
razom iz svo]m 2ρ-okolom. Vyberemo dali ρ1 0> nastil\ky malym, wob dlq
vsix τ ∈ [ – ∆, L ] i µ ∈Dr , ξ ∈D̃ , qki zadovol\nqgt\ nerivnist\
µ µ ξ ξ− + − < ρ1
,
kryva x x= ( , , )τ µ ξ mistylasq v D razom iz svo]m ρ-okolom.
Perejdemo do rozv’qzannq zadaçi (30) – (32). Poznaçymo ]] rozv’qzok x ( τ , µ ,
ξ , ν , ε ) , ϕ ( τ , µ , ξ , ν , ε ) , ξ, a nevidomi parametry µ i ξ pidberemo tak, wob
µk = x ( τk , µ , ξ , ν , ε ) , k = 1, r ,
0
L
x x d∫ η ϕ ϕ ξ τ τλ λ( , , , , , ) = 0
abo
hk = x x xk k k
( , , ) ( )τ µ ξ µ τ τ− + − = , k = 1, r , (40)
P h P z1 2+ = –
0 0
0 0
L L
x x d x x x x d∫ ∫− −[ ]˜ ( , , , , , ) ( , , , ) ( , , , )η ϕ ϕ ξ τ τ η ξ τ η ξ τ τλ λ λ λ –
–
0
0 1 2
1
L
x x
L
P h P z d∫ − +
η ξ τ τλ( , , , ) ( ) , (41)
de x = x ( τ , µ , ξ , ν , ε ) , ϕ = ϕ ( τ , µ , ξ , ν , ε ) , x = x( , , )τ µ ξ , η̃ = η η− 0 , µk =
= µk kh+ , ( h1, … , hr ) = h, ξ = ξ + z .
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426 A. M. SAMOJLENKO, R. I. PETRYÍYN, I. M. DANYLGK
Oskil\ky a0 i f magt\ neperervni obmeΩeni çastynni poxidni do druhoho
porqdku vklgçno, to spravdΩugt\sq rivnosti
x( , , )τ µ ξ = x
x
h
x
z R u( , , )
( , , ) ( , , )
( , )τ µ ξ τ µ ξ
µ
τ µ ξ
ξ
τ+ ∂
∂
+ ∂
∂
+ 1 ,
(42)
∂
∂
x( , , )τ µ ξ
µ
=
∂
∂
+x
R u
( , , )
( , )
τ µ ξ
µ
τ2 ,
∂
∂
x( , , )τ µ ξ
ξ
=
∂
∂
+x
R u
( , , )
( , )
τ µ ξ
ξ
τ3 ,
v qkyx u = ( h, z ) , τ ∈ [ – ∆, L ] , u < ρ1 , R u1( , )τ ≤ σ12
2u , R u2( , )τ +
+ R u3( , )τ ≤ σ12 u , a σ12 — stala, ne zaleΩna vid τ, u.
Z rivnostej (41) oderΩu[mo
∂
∂
−
+ ∂
∂
X
E h
X
z
( , ) ( , )µ ξ
µ
µ ξ
ξ
= R u4( , , )ν ε , (43)
de
R u4( , , )ν ε = – R u x x R u x xr r1 1 11
( , ) ( ) , , ( , ) ( )τ ττ τ τ τ+ − … + −( )= = .
Ob’[dnugçy (41) i (43) ta prypuskagçy, wo matrycq P [ nevyrodΩenog,
ma[mo rivnqnnq
u = F2 ( u, ν , ε ) , (44)
v qkomu
F2 ( u, ν , ε ) = P R u R u−1
4 5( )( , , ), ( , , )ν ε ν ε ,
a R u5( , , )ν ε poznaça[ pravu çastynu rivnosti (41).
Na pidstavi rozkladu
η τ0( ),N N+ = η τ η τ τ0
0
6( ) ( )
,
,
( , )N
N
N
N R N+ ∂
∂
+ ,
de
N x x( )( , , ), ( ( ), , ),τ µ ξ λ τ µ ξ ξ , N = ( , , )( )N N zτ λ τ ,
Nτ =
∂
∂
+ ∂
∂
+x
h
x
z R u
( , , ) ( , , )
( , )
τ µ ξ
µ
τ µ ξ
ξ
τ1 ,
R6 ( τ, N ) =
0
1
0 0∫ ∂ +
∂
− ∂
∂
η τ η τ( ) ( ), ,N Nt
N
N
N
dt N ,
R N6( , )τ ≤ σ̃12
2u ,
ocinky (39) i nerivnosti
0
L
x x d∫ ˜ ( , , , , , )η ϕ ϕ ξ τ τ∆ ∆ ≤
˜
/σ ε12
1 p,
qka vyplyva[ z ocinky (7) oscylqcijnoho intehrala, otrymu[mo nerivnist\
F u2( , , )ν ε ≤ σ ε13
2 1( )/u p+ , ε ∈ ( 0, ε0 ] , ν ∈ R
mr
, u < ρ1
, (45)
zi stalog σ13 , ne zaleΩnog vid u, ν, ε. Zvidsy robymo vysnovok, wo pry
koΩnyx fiksovanyx ν ∈ R
mr
i ε ∈ ( 0, ε0 ] F u2( , , )ν ε vidobraΩa[ mnoΩynu K1 =
= { }, /u u R unr l p∈ ≤+ 2 13
1σ ε v sebe, de
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USEREDNENNQ POÇATKOVO} I BAHATOTOÇKOVO} ZADAÇ DLQ KOLYVNYX … 427
ε0 < min ; ( )
ρ
σ
σ1
13
13
2
2
2
−
p
p .
Doslidymo dali vidobraΩennq F K K2 1 1: → na stysk. Ma[mo rivnosti
∂
∂u
R u4( , , )ν ε =
∂
∂
∂
∂
− ∂ + +
∂
∂ + +
∂
X X X h z X h z( , ) ( , ) ( , ) ( , )µ ξ
µ
µ ξ
ξ
µ ξ
µ
µ ξ
ξ
–
–
∂
∂
− … −( )= =u
x x x x
r
( ) , , ( )τ τ τ τ1
,
∂
∂u
R u5( , , )ν ε = –
0
L
y
y y
u u y
y
u u∫ ∂
∂
∂ −
∂
+ ∂
∂
∂ −
∂
+ ∂
∂
∂
∂
+ ∂
∂
∂
∂
η η
θ
θ θ η η
θ
θ( ) ( ) ˜ ˜
+
+
∂
∂
+ ∂
∂
− ∂
∂
∂
∂
+ ∂
∂
− ∂
∂
+
˜ ( , , ) ( , , ) ( , , ) ( , , ) ˜ ˜( )η η ξ τ η ξ τ η ξ τ η ξ τ τ
u
y
y
y
y
y
u
y
u
y
u
d P P0 0 0 0
1 2 ,
v qkyx y = ( , )x xλ , y = ( , )x xλ , θ = ( , )ϕ ϕλ , θ = ( , )ϕ ϕλ , η = η θ ξ τ( , , , )y ,
η̃ = ˜ ( , , , )η θ ξ τy ,
P̃1 = P
x x
x
x x x
x
x
d
L
1
0
0 0−
∂
∂
∂
∂
+
∂
∂
∂
∂
∫
η ξ τ
µ
η ξ τ
µ
τλ λ
λ
λ( , , , ) ( , , , )
,
P̃2 = P
x x
x
x x x
x
x x x
d
L
2
0
0 0 0−
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∫
η ξ τ
ξ
η ξ τ
ξ
η ξ τ
ξ
τλ λ
λ
λ λ( , , , ) ( , , , ) ( , , , )
.
Zastosovugçy ocinku vyhlqdu (7) do intehrala
0
L
y
y
u u u
d∫ ∂
∂
∂
∂
+ ∂
∂
∂
∂
+ ∂
∂
˜ ˜ ˜η η
θ
θ η τ
ta vraxovugçy ocinku (39), dista[mo nerivnist\
∂
∂
+ ∂
∂u
R u
u
R u4 5( , , ) ( , , )ν ε ν ε ≤ σ ε14
1( )/u p+ (46)
dlq vsix u < ρ1 , ν ∈ R
mr
, ε ∈ ( 0, ε0 ] z deqkog stalog σ14 , ne zaleΩnog vid
ε, ν, u. Todi
∂
∂u
F u2( , , )ν ε ≤
1
2
, u ∈ K 1 , ν ∈ R
mr
, ε ∈ ( 0, ε0 ] , (47)
pry umovi, wo
ε0 < 2 1 214
1
13σ σP
p− −
+( )( ) .
Takym çynom, rivnqnnq (44) ma[ [dynyj rozv’qzok u = u ( ν, ε ) = ( h ( ν, ε ) , z ( ν,
ε )) ∈ K 1 , qkyj moΩna vyznaçyty metodom poslidovnyx nablyΩen\ za dopomohog
formul
u0 ( ν, ε ) ≡ 0, uk + 1 ( ν, ε ) = F2 ( uk ( ν, ε ), ν, ε ) , k ≥ 0,
(48)
u ( ν, ε ) = lim ( , )
k
ku
→∞
ν ε .
Oskil\ky
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428 A. M. SAMOJLENKO, R. I. PETRYÍYN, I. M. DANYLGK
∂
∂ν
ν εR u5( , , ) = –
0
0 0
L
y
y y
A A d∫ ∂
∂
∂ −
∂
+ ∂
∂
∂ −
∂
+ ∂
∂
+ ∂
∂
η
ν
η
θ
θ θ
ν
η
ϕ
η
ϕ
τ
λ
( ) ( )
,
∂
∂ν
ν εR u4( , , ) = –
∂
∂
− … −( )= =ν τ τ τ τ( ) , , ( )x x x x
r1
, A0 =
A
A
Ar
1
2
�
,
to
∂
∂ν
ν εF u2( , , ) ≤ σ ε15
1/ p , u ∈ K 1 , ν ∈ R
mr
, ε ∈ ( 0, ε0 ] . (49)
Z (48) oderΩu[mo rivnist\
∂
∂
+uk 1( , )ν ε
ν
=
∂
∂
∂
∂
+ ∂
∂
F u
u
u F uk k k2 2( ( , ), , ) ( , ) ( ( , ), , )ν ε ν ε ν ε
ν
ν ε ν ε
ν
,
qka zhidno z (47) i (49) pryvodyt\ do nerivnosti
∂
∂
+uk 1( , )ν ε
ν
≤
1
2 15
1∂
∂
+uk p( , ) /ν ε
ν
σ ε
abo
∂
∂
uk ( , )ν ε
ν
≤ 2 15
1σ ε / p, ν ∈ R
mr
, ε ∈ ( 0, ε0 ] , k ≥ 0.
C\oho dosyt\ [11], wob hranyçna funkciq u ( ν, ε ) zadovol\nqla umovu Lipßycq
u u( , ) ( ˜ , )ν ε ν ε− ≤ 2 15
1σ ε ν ν/ ˜p − (50)
dlq vsix ν ∈ R
mr
, ν̃ ∈ R
mr
, ε ∈ ( 0, ε0 ] .
Nareßti pidberemo ν = ( ν1, … , νr ) ∈ R
mr
tak, wob
ϕ ( τk , µ , ξ , ν , ε ) = νk , k = 1, r , (51)
de µ = µ ν ε+ h( , ), ξ = ξ ν ε+ z( , ). Poklademo ν = ν γ+ ˜ , γ̃ = ( )˜ , , ˜γ γ1 … r .
Vraxovugçy rivnist\
ϕ ( τ , µ , ξ , ν , ε ) = ϕ µ ξ ν ε γτ( ), , , , ˜+
=
∑As s
s
r
1
,
perepysu[mo (49) u vyhlqdi
s
r
s s kA
=
∑ −
1
˜ ˜γ γ = – β γ εk (˜ , ), k = 1, r , (52)
de
β γ εk (˜ , ) = ϕ τ µ ξ ν ε ϕ τ µ ξ ν ε ϕ τ µ ξ ν ε ϕ τ µ ξ ν ε( , , , , ) ( , , , , ) ( , , , , ) ( , , , , )k k k k− + − .
Pry vykonanni prypuwennq (38) z systemy rivnqn\ (52) otrymu[mo rivnqnnq
γ̃ = δ γ ε(˜ , ), (53)
v qkomu
δ γ ε(˜ , ) = δ γ ε1(˜ , ) , … , δ γ εr (˜ , ),
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USEREDNENNQ POÇATKOVO} I BAHATOTOÇKOVO} ZADAÇ DLQ KOLYVNYX … 429
δ γ εk (˜ , ) = β γ ε β γ εk s
s
r
s s
s
r
E A A(˜ , ) ( ˜ , )+ −
=
−
=
∑ ∑
1
1
1
.
Na pidstavi nerivnostej (39), (50) i
µ µ ξ ξ− + − ≤ 2 13
1σ ε / p (54)
robymo vysnovok pro isnuvannq tako] stalo] σ16 , wo
δ γ ε(˜ , ) ≤ σ ε16
1/ p, δ γ ε δ γ ε(˜ , ) ( , )− ≤ σ ε γ γ16
1/ ˜p −
dlq vsix γ̃ ∈Rmr , γ ∈Rmr , ε ∈ ( 0, ε0 ] . Tomu pry ε0 ≤ ( )2 16σ − p
rivnqnnq (53)
ma[ v Rmr
[dynyj rozv’qzok γ̃ = ˜ ( )γ ε , pryçomu ˜ ( )γ ε ≤ σ ε16
1/ p.
OtΩe, pobudovano rozv’qzok x ( τ, ε ) , ϕ ( τ, ε ) , ξ ( ε ) zadaçi (30) – (32), de
x ( τ, ε ) ≡ x ( τ, µ ( ε ), ξ ( ε ), ν ( ε ), ε ) , ϕ ( τ, ε ) = x ( τ, µ ( ε ), ξ ( ε ), ν ( ε ), ε ) ,
µ ( ε ) = µ ν ε ε+ h( ( ), ), ξ ( ε ) = ξ ν ε ε+ z ( ( ), ), ν ( ε ) = ν γ ε+ ˜ ( ).
Zaznaçymo, wo z nerivnostej (39), (52) i ν ε ν( ) − ≤ σ ε16
1/ p
vyplyvagt\ neriv-
nosti
x x( , ) ( )τ ε τ− ≤ x x( , ) ( , ( ), ( ))τ ε τ µ ε ξ ε− +
+ x x( , ( ), ( )) ( , , )τ µ ε ξ ε τ µ ξ− ≤ σ ε17
1/ p ,
ϕ τ ε ϕ τ ε( , ) ( , )− ≤ ˜ /σ ε17
1 p
dlq vsix τ ∈ [ – ∆, L ] , ε ∈ ( 0, ε0 ] , v qkyx
σ17 = σ σ τ µ ξ
µ
τ µ ξ
ξ11 132+ ∂
∂
+ ∂
∂
sup
( , , )
sup
( , , )x x
,
σ̃17 = σ σ σ11 13 162+ +( ) ×
× sup
( , , , , )
sup
( , , , , )∂
∂
+ ∂
∂
+
=
∑ϕ τ µ ξ ν ε
µ
ϕ τ µ ξ ν ε
ξ k
r
kA
1
,
a supremum beretsq po τ, µ, ξ, ε iz mnoΩyny
τ ∈ [ – ∆, L ] , µ µ ξ ξ− + − < ρ1 , ε ∈ ( 0, ε0 ] .
Nareßti, qkwo vvaΩaty dodatni çysla ρ1 i ε0 nastil\ky malymy, wo
ρ1 ≤ min ;
1
2
1
413
1
14σ σP−
, ε0 ≤ min ;ρ
σ0
2
1
14
1
4
p
p
P−
−
,
to na pidstavi nerivnostej (45) i (46) znajdene vywe znaçennq u = u ( ν, ε ) [ [dy-
nym rozv’qzkom rivnqnnq (44) v kuli u < ρ1 . Zvidsy robymo vysnovok, wo ko-
ly zadaça (30) – (32) ma[ takoΩ rozv’qzok ˜ ( , )x τ ε , ˜ ( , )ϕ τ ε , ˜ ( )ξ ε , qkyj ne zbiha-
[t\sq z rozv’qzkom x ( , )τ ε , ϕ τ ε( , ) , ξ ε( ) , to pry koΩnomu ε ∈ ( 0, ε0 ] xoç v
odnij toçci τ ∈ [ – ∆, L ] vykonu[t\sq nerivnist\
˜( , ) ( , ) ˜ ( ) ( )x xτ ε τ ε ξ ε ξ ε− + − ≥
ρ1
2r
.
Takym çynom, ma[ misce taka teorema.
Teorema)5. Nexaj vykonugt\sq umovy (2), (5), prypuwennq lemyT2, nerivno-
sti (6), (33), (34) i matrycq P [ nevyrodΩenog. Todi isnugt\ taki dodatni
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
430 A. M. SAMOJLENKO, R. I. PETRYÍYN, I. M. DANYLGK
stali ε0 i σ18 , wo pry koΩnomu ε ∈ ( 0, ε0 ] , ε0 ≤ ε0, v malomu okoli roz-
v’qzku x( )τ , ϕ τ ε( , ), ξ useredneno] zadaçi (351 ) – (355 ) isnu[ [dynyj rozv’qzok
x( , )τ ε , ϕ τ ε( , ) , ξ ε( ) zadaçi (30) – (32), pryçomu
x x( , ) ( ) ( , ) ( , ) ( )τ ε τ ϕ τ ε ϕ τ ε ξ ε ξ− + − + − ≤ σ ε18
1/ p
dlq vsix τ ∈ [ – ∆, L ] , ε ∈ ( 0, ε0 ] .
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ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
|
| id | umjimathkievua-article-3316 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:40:13Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/35/84d81fd0086c34c2db3919940de56d35.pdf |
| spelling | umjimathkievua-article-33162020-03-18T19:51:00Z Averaging of initial-value and multipoint problems for oscillation systems with slowly varying frequencies and deviated argument Усереднення початкової і багатоточкової задач для коливних систем із повільно змінними частотами і відхиленим аргументом Danylyuk, I. M. Petryshyn, R. I. Samoilenko, A. M. Данилюк, І. М. Петришин, Р. І. Самойленко, А. М. We prove new theorems on the substantiation of the method of averaging over all fast variables on a segment and a semiaxis for multifrequency systems with deviated argument in slow and fast variables. An algorithm for the solution of a multipoint problem with parameters is studied, and an estimate for the difference of solutions of the original problem and the averaged problem is established. Доказаны новые теоремы обоснования метода усреднения по всем быстрым переменным на отрезке и полуоси для многочастотных систем с отклоненным аргументом в медленных и быстрых переменных. Исследован алгоритм решения многоточечной задачи с параметрами и установлена оценка разности решений исходной и усредненной задач. Institute of Mathematics, NAS of Ukraine 2007-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3316 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 3 (2007); 412–430 Український математичний журнал; Том 59 № 3 (2007); 412–430 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3316/3377 https://umj.imath.kiev.ua/index.php/umj/article/view/3316/3378 Copyright (c) 2007 Danylyuk I. M.; Petryshyn R. I.; Samoilenko A. M. |
| spellingShingle | Danylyuk, I. M. Petryshyn, R. I. Samoilenko, A. M. Данилюк, І. М. Петришин, Р. І. Самойленко, А. М. Averaging of initial-value and multipoint problems for oscillation systems with slowly varying frequencies and deviated argument |
| title | Averaging of initial-value and multipoint problems for oscillation systems with slowly varying frequencies and deviated argument |
| title_alt | Усереднення початкової і багатоточкової задач для коливних систем із повільно змінними частотами і відхиленим аргументом |
| title_full | Averaging of initial-value and multipoint problems for oscillation systems with slowly varying frequencies and deviated argument |
| title_fullStr | Averaging of initial-value and multipoint problems for oscillation systems with slowly varying frequencies and deviated argument |
| title_full_unstemmed | Averaging of initial-value and multipoint problems for oscillation systems with slowly varying frequencies and deviated argument |
| title_short | Averaging of initial-value and multipoint problems for oscillation systems with slowly varying frequencies and deviated argument |
| title_sort | averaging of initial-value and multipoint problems for oscillation systems with slowly varying frequencies and deviated argument |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3316 |
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