Two-Sided Approximation of Solutions of a Multipoint Problem for an Ordinary Differential Equation with Parameters
We construct an algorithm for the two-sided approximation of a solution of a multipoint boundary-value problem for a quasilinear differential equation under assumptions that are two-sided analogs of the Pokornyi B-monotonicity of the right-hand side of the equation. We establish conditions for the m...
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| Дата: | 2005 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2005
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3579 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509694428184576 |
|---|---|
| author | Mentynskyi, S. M. Ментинський, С. М. |
| author_facet | Mentynskyi, S. M. Ментинський, С. М. |
| author_sort | Mentynskyi, S. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:59:02Z |
| description | We construct an algorithm for the two-sided approximation of a solution of a multipoint boundary-value problem for a quasilinear differential equation under assumptions that are two-sided analogs of the Pokornyi B-monotonicity of the right-hand side of the equation. We establish conditions for the monotonicity of successive approximations and their uniform convergence to a solution of the problem. |
| first_indexed | 2026-03-24T02:45:10Z |
| format | Article |
| fulltext |
K�O�R�O�T�K�I���P�O�V�I�D�O�M�L�E�N�N�Q
UDK 517.927
S. M. Mentyns\kyj (Nac. un-t „L\viv. politexnika”)
DVOSTORONNQ APROKSYMACIQ ROZV’QZKIV
BAHATOTOÇKOVO} ZADAÇI DLQ ZVYÇAJNOHO
DYFERENCIAL|NOHO RIVNQNNQ Z PARAMETRAMY
We investigate the algorithmof two-sided approximation of a solution of multipoint boundary-value
problem for quasilinear differential equation with many control parameters under the assumption on the
B-monotonicity in the J. Pokornyi sense of the right-hand side of equation. We establish conditions of
monotonicity of successive approximations and of their uniform convergence to the solution of problem.
Pobudovano alhorytm dvostoronn\o] aproksymaci] rozv’qzku bahatotoçkovo] krajovo] zadaçi dlq
kvazilinijnoho dyferencial\noho rivnqnnq z parametramy za prypuwen\, qki [ dvostoronnimy
analohamy V-monotonnosti za G. Pokornym pravo] çastyny rivnqnnq. Vstanovleno umovy mono-
tonnosti poslidovnyx nablyΩen\ ta ]x rivnomirno] zbiΩnosti do rozv’qzku zadaçi.
Stattg prysvqçeno pobudovi i doslidΩenng odnoho sposobu dvostoronn\o] ap-
roksymaci] rozv’qzkiv bahatotoçkovo] zadaçi dlq zvyçajnyx dyferencial\nyx
rivnqn\ z parametramy
x( m ) = H t x x q t xm
m
m
m
( ) − ( ) −( )
=
−
∑, , , , ˙
˜
˜
˜
λ λ ξλ
1
1
, (1)
x ( ti ) = xi , 0 = t1 < t2 < … < tm + k = T, i = 1, 2, … , m + k, (2)
de funkciq H : D = [ 0; T ] × [ a; b ] × [ a; b ] × [ c; d ] × [ c; d ] → C
m
[ 0; T ] neperer-
vna za sukupnistg arhumentiv, a, b, qm̃ ∈ Cm
[ 0; T ] ( Cm
[ 0; T ] — prostir m
raziv neperervno dyferencijovnyx na [ 0; T ] dijsnyx funkcij), λ, ξ, c, d ∈ R
k
.
Dlq m = 1, k = 1 zadaça (1), (2) doslidΩuvalasq v [1, 2]. Procesy poslidov-
nyx nablyΩen\ do rozv’qzku krajovo] zadaçi (2) dlq rivnqnnq
x( m )
( t ) = f ( t, x, ẋ , … , x( m – 1 ), λ1 , λ2 , … , λk )
pry m = 1, k > 1 pobudovano v [3 – 5]. U roboti [6] dlq nablyΩenoho rozv’qzan-
nq zadaçi (1), (2) pry m > 1, k = 1, qm̃ = 0, m̃ = 1, … , m – 1, zastosovano dvo-
storonnij iteracijnyj proces, qkyj dozvolq[ otrymuvaty monotonni poslidovni
nablyΩennq ta zruçni ocinky ßukanoho rozv’qzku za dopomohog vylky. Zasto-
suvannq alhorytmiv takoho typu qk do zadaçi (1), (2), tak i do inßyx krajovyx
zadaç çasto uskladng[t\sq çerez prypuwennq pro monotonnist\ za funkcio-
nal\nymy arhumentamy ta parametramy funkci] H ( t, y , z, η , µ ) (dyv. [ 6 ]).
Metog proponovanoho doslidΩennq [ pobudova alhorytmu dlq aproksymaci]
rozv’qzkiv zadaçi (1), (2), qkyj zberiha[ vidomi perevahy dvostoronnix metodiv bez
vykorystannq tyx çy inßyx vlastyvostej monotonnosti pravo] çastyny rivnqnnq
(1). Dlq c\oho, zokrema, vykorystano metodyku, zaproponovanu v [7].
Vykorystavßy konstrukci] vidpovidnyx obernenyx operatoriv iz [6], pobudu-
[mo dvostoronni aproksymaci] do rozv’qzku zadaçi (1), (2) za takyx prypuwen\:
1) ξ > θ, θ ∈ Rk
— nul\-vektor;
2) funkci]
Qp ( t ) = (− ) ( )− ( − )
=
−
∑ 1
1
k p
k
k p
k
p
k p
m
q t C , p = 1, … , m – 1
© S. M. MENTYNS|KYJ, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 125
126 S. M. MENTYNS|KYJ
(Ck
p
— binomial\ni koefici[nty), obmeΩeni i zberihagt\ znak na [ 0; T ] :
0 ≤ Qp ( t ) ≤ mp ;
3) zadano neperervni za sukupnistg arhumentiv nezrostagçi wodo u, η ne-
spadni wodo v , µ dodatni pry t ∈ [ 0; T ], u, v ∈ [ a; b ], η, µ ∈ [ c; d ] funkci]
A1 ( t, u, v ), B1 ( t, u, v ), K1 ( t, η, µ ), N2 ( t, η, µ ), dlq qkyx iz spivvidnoßen\ u ≤ v,
η ≤ µ ( t ∈ [ 0; T ], x, u, v ∈ [ a; b ], λ, µ, η ∈ [ c; d ] ) vyplyvagt\ nerivnosti
– A1 ( t, u, v ) ( v – u ) – K1 ( t, η, µ ) ( µ – η ) ≤ H ( t, v, x, µ, λ ) – H ( t, u, x, η, λ ),
(3)
H ( t, v, x, µ, λ ) – H ( t, x, u, η, λ ) ≤ B2 ( t, u, v ) ( v – u ) + N2 ( t, η, µ ) ( µ – η ).
Poznaçymo
Fm k, [ u, v, η, µ ] = Lm + k [ H ( t, u, v, η, µ ) – W ( t, u, v, η, µ ); H ( t, v, u, µ, η ) +
+ W ( t, u, v, η, µ ) ] + Ψm
k [ u, v ] + x0 ( t ),
Fm k, [ v, u, µ, η ] = Lm + k [ H ( t, v, u, µ, η ) + W ( t, u, v, η, µ ); H ( t, u, v, η, µ ) –
– W ( t, u, v, η, µ ) ] + Ψm
k [ v, u ] + x0 ( t ),
de
W ( t, u, v, η, µ ) = ( A1 ( t, u, v ) + B2 ( t, u, v ) ) ( v – u ) +
+ ( K1 ( t, η, µ ) + N2 ( t, η, µ ) ) ( µ – η ),
a operatory Lm + k , Ψm
k [ u, v ] vyznaçeno spivvidnoßennqmy
Lm + k [ ϕ ( t, x ); ψ ( t, x ) ] =
=
P t
P t
t s
m
s x s ds m
P t
P t
t s
m
s x s ds
m k i
m k i i
i
m
i j
t
t
i
m k
m k i
m k i i
i
m
i j
t
t
i
i
+
+
−
+
=
+
+
+
−
+ −
( )
( )
( − )
( − )
( )
( )
( )
( − )
( − )
( ) +
( )
( )
∫∑ ,
,
,
,
!
, , — ,
!
,
1
1
1
1
1
1
ξ
ξ
qkwo neparne
∫∫∑
∫∑
=
+
+
−
+
= +
+
( )
( )
( − )
( − )
( )
( )
i
j
m k i
m k i i
i
m
i j
t
t
i j
m P t
P t
t s
m
s x s ds m
i
1
1
1 1
!
, , — ,,
,
ξ qkwo parne
Ψm
k m k i
m k i i
i
m k p
p i j m k p
p
m
t
t
i
j
u t t
P t
P t
t s
m k p
Q s w s ds
i
[ ]( ) ( ) =
( )
( )
( − )
( + − − )
( ) ( )+
+
+ − −
+ + + +
=
−
=
∑∫∑,
!
,
,
v
1
0
1
1 1
+
+
P t
P t
t s
m k p
Q s w s dsm k i
m k i i
i
m k p
p i j m k p
p
m
t
t
i j
m k
i
+
+
+ − −
+ + + + +
=
−
= +
+ ( )
( )
( − )
( + − − )
( ) ( )∑∫∑ ,
, !
1
1
0
1
1 1
,
de
ξi* =
ϕ
ψ
, ,
, ,
*
*
qkwo – parne
qkwo – neparne
i
i
wi* =
u i
i
, — ,
, — ,
*
*
qkwo parne
qkwo neparnev
t ∈ [ tj , tj + 1 ],
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
DVOSTORONNQ APROKSYMACIQ ROZV’QZKIV BAHATOTOÇKOVO} ZADAÇI … 127
Pm k i+ , = ( − )
=
≠
+
∏ t ti j
i
j i
m k
1
, x0 ( t ) =
P t
P t
xm k i
m k i i
i
i
m k
+
+=
+ ( )
( )∑ ,
,1
.
Krim toho, nexaj
Sm k, [ t, u, v, η, µ ] = Sm, k [ H ( t, u, v, η, µ ) – W ( t, u, v, η, µ ); H ( t, v, u, µ, η ) +
+ W ( t, u, v, η, µ ) ],
Sm k, [ t, u, v, η, µ ] = Sm, k [ H ( t, v, u, µ, η ) + W ( t, u, v, η, µ ); H ( t, u, v, η, µ ) –
– W ( t, u, v, η, µ ) ],
Sm, k [ ϕ ( t, x, λ ); ψ ( t, x, λ ) ] =
=
ξ−
+=
+
+ −
+ + +( + − ) ( )
( − ) ( ) +
∑ ∫
1
1
1
1
0
1
1
m k P t
t s q s ds x
k m i ii
k m
i
m k
m k i i
ti
! ,
,
de qi* =
ϕ
ψ
, ,
, .
*
*
qkwo – parne
qkwo – neparne
i
i
Oznaçymo poslidovni nablyΩennq do rozv’qzku zadaçi (1), (2) za formulamy
yn + 1 = T y zm k n n, ,( ),
(4)
zn + 1 = T y zm k n n, ,( ) ,
de
yn* =
u
n
n
*
*η
, zn* =
v
n
n
*
*µ
,
(5)
T y z
F
Sm k
m k
m k
,
,
,
,[ ] =
, T y z
F
Sm k
m k
m k
,
,
,
,[ ] =
.
Teorema 1. Nexaj spravdΩugt\sq prypuwennq 1 – 3 i zadaça (1), (2) ma[ v
oblasti D0 = [ a; b ] × [ c; d ] xoça b odyn rozv’qzok w* =
x*
*λ
. Todi dlq posli-
dovnyx nablyΩen\ (4), (5) z nerivnostej
y0 ≤ y1 ≤ w* ≤ z1 ≤ z0
vyplyvagt\ spivvidnoßennq
yn – 1 ≤ yn ≤ w* ≤ zn ≤ zn – 1 , n = 1, 2, … . (6)
Dovedennq. Zaznaçymo, wo umovy 1 – 3 ta specyfika pobudovy operatoriv
Lm + k , Ψm
k , Sm, k zabezpeçugt\ te, wo z prypuwennq pro vykonannq nerivnostej
(6) pry n = ñ – 1 vyplyva[ ]x vykonannq pry n = ñ . A oskil\ky vykonannq
nerivnostej (6) pry n = 0 postulg[t\sq umovamy teoremy, to dovedennq moΩna
provesty, vykorystovugçy pryncyp matematyçno] indukci].
Wodo umovy 3 slid zauvaΩyty, wo nerivnosti (3) vykonugt\sq, napryklad,
qkwo funkciq H ( t, u, v, η, µ ) ne spada[ wodo u, η i ne zrosta[ wodo v , µ . V
c\omu vypadku za A1 , B2 , K1 , N2 dostatn\o vzqty dovil\ni nevid’[mni stali, zo-
krema pry A1 = B2 = 0, B1 = N2 = 0 otryma[mo dvostoronnij alhorytm iz [6].
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
128 S. M. MENTYNS|KYJ
Nexaj
4) funkciq H ( t, u, v, η, µ ) v oblasti D obmeΩena,
| H ( t, u, v, η, µ ) | ≤ M,
i pry c\omu
a
P t
P t
t t
m k
M M t b
t T
m i
m i i
i
m k
p i
p
m
i
m k
+
( )
( )
( − )
( + )
+ ( )
∈[ ]
+
=
−
=
+
∑∑max
!;
,
,
,
0
0
0
1
1
≤ x0 ( t ) ≤
≤ b
P t
P t
t t
m k
M M t b
t T
m i
m i i
i
m k
p i
p
m
i
m k
−
( )
( )
( − )
( + )
+ ( )
∈[ ]
+
=
−
=
+
∑∑max
!;
,
,
,
0
0
0
1
1
, (7)
c
MP
m k m k
d
MP
m k m k
+
( + − )( + − )
≤ −
( + − )( + − )
− −ξ ξ1 1
1 1 1 1
, (8)
de
P =
t
P t
i
m k
k m i ii
m k + −
+=
+
( )∑
1
1 ,
, b0 = max ( | a |, | b | ), Mp, i ( t ) =
( − )
( − )
−t t
m p
mi
m p
p!
.
V c\omu vypadku, poznaçyvßy
x t
x t P t M t P t M t
m k
x t P t M t P t M t
i j
i
i j
i
i j
m k
i
j
i j
i
i j
i
i j
0
0
1
0
1
1
11
0
1
0 1
1 1
1 1
( ) =
( ) + (− ) ( ) ( ) + (− ) ( ) ( )
+
( ) + (− ) ( ) ( ) + (− ) ( ) ( )
+ − + −
= +
+
=
+ − +
= +
∑∑ ,
– ,
,
qkwo parne
111
m k
i
j
m k
+
=
∑∑
+
qkwo neparne– ,
(9)
x t x t x t0 0 02( ) = ( ) − ( ),
de
t ∈ [ tj ; tj + 1 ], j = 1, 2, … , m + k, Pi ( t ) =
P t
P t
m k i
m k i i
+
+
( )
( )
,
,
,
M0 ( t ) =
( − )
( + )
+t t
m k
Mi
m k
!
+ M t bp i
p
m
, ( )
=
−
∑ 0
0
1
,
M1 ( t ) =
( − )
( + )
+ (− ) ( )
+
=
−
∑t t
m k
M M t bi
m k
p
p i
p
m
! ,1 0
0
1
,
za poçatkovi nablyΩennq moΩemo vzqty
y0 =
x
c
0
, z0 =
x
d
0
. (10)
Dlq doslidΩennq zbiΩnosti alhorytmu (4), (5) prypuska[mo, wo
5) zadano neperervni za sukupnistg arhumentiv dodatni pry t ∈ [ 0; T ], u, v ∈
∈ [ a ; b ], η, µ ∈ [ c ; d ] funkci] A2 ( t, u, v ), B1 ( t, u, v ), K2 ( t, η, µ ), N1 ( t, η, µ ),
dlq qkyx iz spivvidnoßen\ u ≤ v, η ≤ µ ( t ∈ [ 0; T ], x, u, v ∈ [ a ; b ], λ , η, µ ∈ [ c ;
d ] ) vyplyvagt\ nerivnosti
H ( t, x, v, λ, µ ) – H ( t, x, u, λ, η ) ≤ A2 ( t, u, v ) ( v – u ) + K2 ( t, η, µ ) ( µ – η ),
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
DVOSTORONNQ APROKSYMACIQ ROZV’QZKIV BAHATOTOÇKOVO} ZADAÇI … 129
– B1 ( t, u, v ) ( v – u ) – N1 ( t, η, µ ) ( µ – η ) ≤ H ( t, v, x, µ, λ ) – H ( t, u, x, η, λ ).
Poznaçymo
MA B =
max
;
, ;
t T
u a b
∈[ ]
∈[ ]
0
v
( 2 ( A1 ( t, u, v ) + B2 ( t, u, v ) ) + A2 ( t, u, v ) + B1 ( t, u, v ) ),
M K t N tKN
l
t T
c d
l l l l l l
l l l l
2 2 2 2 2 2 2
0
1 2
2 2 2 2
2= ( ) + ( )
∈[ ]
∈[ ]
( ) ( ) ( ) ( ) ( ) ( )( ( )max , , , ,
;
, ;η µ
η µ η µ +
+ K t N tl l l l l l
2 1
2 2 2 2 2 2( ) ( ) ( ) ( ) ( ) ( )( ) + ( )), , , ,η µ η µ ,
Q =
P M T
m k
T
m k p
m
M T P
m k
M T P
m k
P M T
m k k
M T P
m k k
M T
AB
m k m k p
p
p
m
KN
m k
KN
k m k
AB
m k
KN
m k
KN
k
0
0
1 1
0 0
1
1
1
1
1
+ + −
=
− ( ) + ( ) +
+ ( ) + ( )
( + )
+
( + − ) ( + ) ( + )
( + ) ( + )
∑! ! ! !
! !
�
�
ξ ξ
mm k
AB
m k
k
KN
m k
k
KN
k m k
k
P
m k k
P M T
m k k
M T P
m k k
M T P
m k k
+
+ ( ) + ( ) +
( + )
( + ) ( + ) ( + )
1
1
1
1
1 1
!
! ! !
ξ
ξ ξ ξ
� � � �
�
,
de
P0 = max
;
,
,t T
m k i
m k i ii
m k P t
P t∈[ ]
+
+=
+ ( )
( )∑
0 1
, P1 = 1
1 P tm k i ii
m k
+=
+
( )∑
,
.
Teorema 2. Nexaj spravdΩugt\sq umovy 1 – 5 ta nerivnist\
|| Q || ≤ Q0 < 1. (11)
Todi isnu[ [dynyj v D0 rozv’qzok w*
zadaçi (1), (2), do qkoho rivnomirno wodo
t ∈ [ 0; T ] zbihagt\sq poslidovnosti { yn }, { zn }, vyznaçeni za formulamy (4),
(5), (9), (10), i pry c\omu spravdΩugt\sq spivvidnoßennq (6).
Dovedennq. Pry vykonanni v oblasti D umovy 5 otryma[mo ocinky
|| vn + 1 ( t ) – un + 1 ( t ) || ≤
≤ T
m k
P M t u t M
m k
AB n n KN
l
n
l
n
l
l
k+
+ +
( ) ( )
=( + )
( ) − ( ) + ( − )
∑! 0 1 1
1
2 2 2
2
v µ η +
+
T
m k p
m t u t
m k p
P n n
p
m + −
+ +
=
−
( + − )
( ) − ( )∑ !
v 1 1
0
1
=
=
P M T
m k
T
m k p
mAB
m k m k p
P
p
m
0
0
1+ + −
=
−
( + )
+
( + − )
∑! !
×
×
vn n
KN
l m k
n
l
n
l
l
k
t u t
M T P
m k+ +
+
( ) ( )
=
( ) − ( ) +
( + )
( − )∑1 1
0
0
2
2 2
2
!
µ η ,
µ ηn
l
n
l
+
( )
+
( )−1 1
2 2 ≤
≤
PT
m k k
M t u t M
m k
l
AB n n KN
l
n
l
n
l
l
k
1
1 1
12
2 2 2
2
+
+ +
( ) ( )
=( + )
( ) − ( ) + ( − )
∑!ξ
µ ηv =
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
130 S. M. MENTYNS|KYJ
=
PT M
m k k
t u t
PT M
m k k
m k
AB
l
n n
m k
KN
l
l
n
l
n
l
l
k
1
1 1
1
12
2
2
2 2
2
+
+ +
+
( ) ( )
=( + )
( ) − ( ) +
( + )
( − )∑! !ξ ξ
µ ηv ,
de
P0 = max
;
,
,t T
m k i
m k i ii
m k P t
P t∈[ ]
+
+=
+ ( )
( )∑
0 1
, P1 = 1
1 P tm k i ii
k m
+=
+
( )∑
,
.
OtΩe,
|| zn + 1 – yn + 1 || ≤ || Q || || zn – yn ||, (12)
|| zn + 1 – yn + 1 || ≤ Qn
0
1+ || z0 – y0 ||. (13)
Spivvidnoßennq (6) ta (11) – (13) zabezpeçugt\ isnuvannq spil\no] hranyci dlq
poslidovnostej { yn } ta { zn } : lim
n
ny
→∞
= lim
n
nz
→∞
= w*
. Perexodqçy do hranyci v
(8) i dyferenciggçy otrymanu rivnist\ po t m raziv, perekonu[mosq, wo w* =
=
x*
*λ
[ rozv’qzkom rivnqnnq (1), qkyj zadovol\nq[ krajovi umovy (2). Skorys-
tavßys\ metodom vid suprotyvnoho, moΩna dovesty [dynist\ c\oho rozv’qzku.
Teoremu dovedeno.
Zaznaçymo, wo u vypadku nul\ovyx operatoriv A1 ( t, u, v ), B2 ( t, u, v ) ta K1 ( t,
η, µ ), N2 ( t, η, µ ), vzqvßy za A2 ( t, u, v ), B1 ( t, u, v ), K2 ( t, η, µ ), N1 ( t, η, µ ) stali
Lipßycq funkci] H ( t, u, v, η, µ ) zi zminnymy u, v, η, µ vidpovidno, z navede-
nyx vywe rezul\tativ otryma[mo rezul\taty iz [6]. Otrymani rezul\taty [
blyz\kymy takoΩ do vidpovidnyx rezul\tativ iz [8]. Vidmitymo takoΩ, wo alho-
rytm (4), (5) [ zruçnym dlq praktyçno] realizaci] z ohlqdu na pevnu dovil\nist\
vyboru A1, B2 ta K1 , N2 , qka dozvolq[, napryklad, vraxovuvaty vplyv poxy-
bok zaokruhlen\ na dvostoronnist\ ta monotonnist\ otrymuvanyx poslidovnyx
nablyΩen\ i t. p.
Podibni rezul\taty moΩna vstanovyty i pry zastosuvanni do inßyx krajovyx
zadaç dlq rivnqn\ z parametramy. Tomu perspektyvnymy u c\omu naprqmku
moΩna vvaΩaty rozßyrennq klasiv zadaç, dlq aproksymaci] rozv’qzkiv qkyx
moΩna zastosovuvaty doslidΩenyj dvostoronnij alhorytm, a takoΩ pobudovu
sxem dyskretyzaci], prydatnyx dlq joho realizaci] za dopomohog suçasnyx ob-
çyslgval\nyx zasobiv.
1. Kybenko A. V., Perov A. Y. O dvuxtoçeçnoj kraevoj zadaçe s parametrom // Dokl. AN USSR.
– 1961. – # 10. – S. 1259 – 1261.
2. Marusqk A. H. Ob odnom dvustoronnem metode reßenyq kraevoj zadaçy s parametrom // Mat.
fyzyka. – 1980. – V¥p. 27. – S. 39 – 45.
3. Sobkovyç R. Y. Ob odnoj kraevoj zadaçe dlq dyfferencyal\noho uravnenyq pervoho po-
rqdka s neskol\kymy parametramy // Ukr. mat. Ωurn. – 1982. – 34, # 6. – S. 796 – 802.
4. Kurpel\ N. S., Marusqk A. H. Ob odnoj mnohotoçeçnoj kraevoj zadaçe dlq dyfferency-
al\noho uravnenyq s parametrom // Tam Ωe. – 1980. – 32, # 2. – S. 223 – 226.
5. Korol\ I. I. Pro odyn pidxid do intehruvannq parametryzovanyx krajovyx zadaç // Nauk. vis-
nyk UΩhorod. un-tu. Ser. mat. i inform. – 2002. – Vyp. 7. – S. 70 – 78.
6. Sobkovyç R. Y. Dvustoronnyj metod yssledovanyq nekotor¥x kraev¥x zadaç s parametra-
my. – Kyev, 1981. – 36 s. – (Preprynt / AN USSR. Yn-t matematyky; 81.52).
7. Íuvar B. A. Dvustoronnye yteracyonn¥e metod¥ reßenyq nelynejn¥x uravnenyj v polu-
uporqdoçenn¥x prostranstvax // Vtoroj symp. po metodam reßenyq nelynejn¥x uravnenyj y
zadaç optymyzacyy. – Tallyn: Yn-t kybernetyky AN ∏SSR, 1981. – 1. – S. 68 – 73.
8. Rabczuk R. Elementz nierownosci rozniczkowzh. – Warshawa: PWN, 1976. – 276 p.
OderΩano 18.03.2004
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
|
| id | umjimathkievua-article-3579 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:45:10Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/39/4d838a8578ad315be793c58a9cc55339.pdf |
| spelling | umjimathkievua-article-35792020-03-18T19:59:02Z Two-Sided Approximation of Solutions of a Multipoint Problem for an Ordinary Differential Equation with Parameters Двостороння апроксимація розв'язків багатоточкової задачі для звичайного диференціального рівняння з параметрами Mentynskyi, S. M. Ментинський, С. М. We construct an algorithm for the two-sided approximation of a solution of a multipoint boundary-value problem for a quasilinear differential equation under assumptions that are two-sided analogs of the Pokornyi B-monotonicity of the right-hand side of the equation. We establish conditions for the monotonicity of successive approximations and their uniform convergence to a solution of the problem. Побудовано алгоритм двосторонньої апроксимації розв'язку багатоточкової крайової задачі для квазілінійного диференціального рівняння з параметрами за припущень, які є двосторонніми аналогами B-монотонності за Ю. Покорним правої частини рівняння. Встановлено умови монотонності послідовних наближень та їх рівномірної збіжності до розв'язку задачі. Institute of Mathematics, NAS of Ukraine 2005-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3579 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 1 (2005); 125–130 Український математичний журнал; Том 57 № 1 (2005); 125–130 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3579/3891 https://umj.imath.kiev.ua/index.php/umj/article/view/3579/3892 Copyright (c) 2005 Mentynskyi S. M. |
| spellingShingle | Mentynskyi, S. M. Ментинський, С. М. Two-Sided Approximation of Solutions of a Multipoint Problem for an Ordinary Differential Equation with Parameters |
| title | Two-Sided Approximation of Solutions of a Multipoint Problem for an Ordinary Differential Equation with Parameters |
| title_alt | Двостороння апроксимація розв'язків багатоточкової задачі для звичайного диференціального рівняння з параметрами |
| title_full | Two-Sided Approximation of Solutions of a Multipoint Problem for an Ordinary Differential Equation with Parameters |
| title_fullStr | Two-Sided Approximation of Solutions of a Multipoint Problem for an Ordinary Differential Equation with Parameters |
| title_full_unstemmed | Two-Sided Approximation of Solutions of a Multipoint Problem for an Ordinary Differential Equation with Parameters |
| title_short | Two-Sided Approximation of Solutions of a Multipoint Problem for an Ordinary Differential Equation with Parameters |
| title_sort | two-sided approximation of solutions of a multipoint problem for an ordinary differential equation with parameters |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3579 |
| work_keys_str_mv | AT mentynskyism twosidedapproximationofsolutionsofamultipointproblemforanordinarydifferentialequationwithparameters AT mentinsʹkijsm twosidedapproximationofsolutionsofamultipointproblemforanordinarydifferentialequationwithparameters AT mentynskyism dvostoronnâaproksimacíârozv039âzkívbagatotočkovoízadačídlâzvičajnogodiferencíalʹnogorívnânnâzparametrami AT mentinsʹkijsm dvostoronnâaproksimacíârozv039âzkívbagatotočkovoízadačídlâzvičajnogodiferencíalʹnogorívnânnâzparametrami |