Bifurcation of solutions of a linear Fredholm boundary-value problem
We establish constructive conditions for the appearance of solutions of a linear Fredholm boundary-value problem for a system of ordinary differential equations in the critical case and propose an iterative procedure for finding these solutions. The range of values of a small parameter for which the...
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| Дата: | 2007 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2007
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509456533553152 |
|---|---|
| author | Chuiko, S. M. Чуйко, С. М. Чуйко, С. М. |
| author_facet | Chuiko, S. M. Чуйко, С. М. Чуйко, С. М. |
| author_sort | Chuiko, S. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:52:34Z |
| description | We establish constructive conditions for the appearance of solutions of a linear Fredholm boundary-value problem for a system of ordinary differential equations in the critical case and propose an iterative procedure for finding these solutions. The range of values of a small parameter for which the indicated iterative procedure is convergent is estimated. |
| first_indexed | 2026-03-24T02:41:24Z |
| format | Article |
| fulltext |
UDK 517.9
S. M. Çujko (Slavqn. ped. un-t)
VOZNYKNOVENYE REÍENYJ LYNEJNOJ
NETEROVOJ KRAEVOJ ZADAÇY
We obtain constructive conditions for the emergence of solutions of a linear Noetherian boundary-value
problem for a system of ordinary differential equations in the critical case and construct an iterative
procedure for finding these solutions. An estimate is found for the range of values of the small
parameter for which the convergence of the iterative procedure is preserved.
OderΩano konstruktyvni umovy vynyknennq ta pobudovano iteracijnu proceduru dlq znaxod-
Ωennq rozv’qzkiv neterovo] linijno] krajovo] zadaçi dlq systemy zvyçajnyx dyferencial\nyx
rivnqn\ u krytyçnomu vypadku. Znajdeno ocinku oblasti znaçen\ maloho parametra, dlq qkyx
zberiha[t\sq zbiΩnist\ iteracijno] procedury.
1. Postanovka zadaçy. Rassmotrym zadaçu o naxoΩdenyy reßenyq z ( t ) ∈
∈ C1
[ a, b ] system¥ ob¥knovenn¥x dyfferencyal\n¥x uravnenyj
dz
dt
0 = A ( t ) z0 + f ( t ), (1)
udovletvorqgwyx kraevomu uslovyg
l z0 ( ⋅ ) = α, α ∈ Rm
. (2)
Zdes\ A ( t ) — ( n × n )-mernaq matryca y f ( t ) — n-mern¥j vektor-stolbec,
πlement¥ kotor¥x — neprer¥vn¥e na otrezke [ a, b ] dejstvytel\n¥e funkcyy,
l z ( ⋅ ) — lynejn¥j ohranyçenn¥j vektorn¥j funkcyonal vyda l z0 ( ⋅ ) : C [ a, b ] →
→ R
m
. PredpoloΩym, çto ymeet mesto krytyçeskyj sluçaj: PQ* ≠ 0, hde Q =
= l X ( ⋅ ) — ( m × n )-matryca, rank Q = n1 , PQ* — ( m × m )-matryca-ortoproektor
PQ* : R
m → N ( Q*
); ( d × m )-mernaq matryca P
Qd
* sostavlena yz d = m – n1
lynejno nezavysym¥x strok matryc¥-ortoproektora PQ* ; X ( t ) — normal\naq
fundamental\naq matryca ( X ( a ) = In ) odnorodnoj çasty system¥ (1). V πtom
sluçae neterova ( m ≠ n ) zadaça (1), (2) razreßyma dlq tex y tol\ko dlq tex
neodnorodnostej f ( t ) y α, dlq kotor¥x ymeet mesto ravenstvo [1]
P lK f s
Qd
* α − ( ) (⋅){ }[ ] = 0. (3)
Postavym zadaçu o naxoΩdenyy lynejn¥x vozmuwenyj ε A1 ( t ) z y ε l1 z ( ⋅, ε ),
kotor¥e obespeçyvaly b¥ suwestvovanye reßenyj z ( t, ε ) ∈ C ] 0, ε0 ], C1
[ a, b ]
kraevoj zadaçy
dz
dt
= A ( t ) z + f ( t ) + ε A1 ( t ) z, l z ( ⋅, ε ) = α + ε l1 z ( ⋅, ε ) (4)
dlq lgb¥x neodnorodnostej f ( t ) ∈ C [ a, b ] y α ∈ Rm
pry tom, çto uslovye (3),
voobwe hovorq, ne v¥polnqetsq. Zdes\ l1 z ( ⋅, ε ) — lynejn¥j ohranyçenn¥j vek-
torn¥j funkcyonal l1 z ( ⋅, ε ) : C [ a, b ] → Rm; A1 ( t ) — ( n × n )-mernaq matryca,
πlement¥ kotoroj — neprer¥vn¥e na otrezke [ a, b ] dejstvytel\n¥e funkcyy.
Reßenye postavlennoj zadaçy v sluçae l1 z ( ⋅, ε ) ≡ 0 v vyde rqda Lorana b¥lo
dano v monohrafyy [2]. Pozdnee dannaq metodyka v sluçae l1 z ( ⋅, ε ) ≠ 0 b¥la
© S. M. ÇUJKO, 2007
1148 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
VOZNYKNOVENYE REÍENYJ LYNEJNOJ NETEROVOJ KRAEVOJ ZADAÇY 1149
perenesena na kraev¥e zadaçy s nev¥roΩdenn¥m ympul\sn¥m vozdejstvyem [3].
Suwestvenn¥m v dann¥x rabotax b¥lo postroenye reßenyq v vyde rqda Lorana
s yspol\zovanyem metoda Vyßyka – Lgsternyka [4]. V dannoj stat\e budet
postroeno reßenye postavlennoj zadaçy metodom prost¥x yteracyj y najdena
ocenka ε* dlyn¥ promeΩutka ] 0, ε*
], na kotorom soxranqetsq sxodymost\ πtoj
yteracyonnoj procedur¥. Neobxodymoe y dostatoçnoe uslovye razreßymosty
zadaçy (4) ymeet vyd
P l z lK f s A s z s
Qd
* , ,α ε ε ε ε+ (⋅ ) − ( ) + ( ) ( ) (⋅){ }[ ]1 1 = 0.
Esly πto uslovye v¥polneno, to obwee reßenye zadaçy (4)
z ( t, ε ) = Xr ( t ) cr ( ε ) + z(
1
)
( t, ε ),
z(
1
)
( t, ε ) = G f s A s z s l z t[ ]( ) + ( ) ( ) + (⋅ ) ( )ε ε α ε ε1 1, ; ,
predstavymo s pomow\g oboobwennoho operatora Hryna zadaçy (1), (2)
G f s X t Q lK f s K f s tt[ ]( ) [ ] [ ]( ) = ( ) − ( ) (⋅){ } + ( ) ( )+; α α
y ( n × r )-mernoj matryc¥ Xr ( t ), sostavlennoj yz r lynejno nezavysym¥x re-
ßenyj ( r = n – n1 ) odnorodnoj zadaçy (1), (2). Zdes\ Q+
— psevdoobratnaq
matryca po Muru – Penrouzu [1],
K f s t X t X s f s ds
a
t
[ ]( ) ( ) = ( ) ( ) ( )−∫ 1
— operator Hryna zadaçy Koßy dlq system¥ (1). Pry ε ≠ 0, P
B0
* = 0 kraevaq
zadaça (4) ymeet po men\ßej mere odno reßenye, predstavymoe operatornoj sys-
temoj
z ( t, ε ) = Xr ( t ) cr ( ε ) + z(
1
)
( t, ε ),
z(
1
)
( t, ε ) = G f s A s z s l z t[ ]( ) + ( ) ( ) + (⋅ ) ( )ε ε α ε ε1 1, ; , ,
(5)
cr ( ε ) = − − ( ) (⋅){ }+ [ ]B P lK f s
Qd
0
1
ε
α* –
– B P l z lK A s z s P c
Qd
0 1
1
1
1+ (⋅ ) − ( ) ( ) (⋅){ } +[ ]*
( ) ( ), ,ε ε ρ ρ , cρ ∈ R
ρ
,
hde B0 = P l X lK A s X s
Q r r
d
* 1 1(⋅) − ( ) ( ) (⋅){ }[ ] — ( d × r )-mernaq matryca, P
B0
* — ( d ×
× d )-mernaq matryca-ortoproektor: Rd → N B( )0
* , PB0
: Rr → N B( )0 — ( r × r )-
matryca-ortoproektor, Pρ — ( r × ρ )-matryca, sostavlennaq yz ρ lynejno ne-
zavysym¥x stolbcov matryc¥-ortoproektora PB0
. Operatornaq systema (5)
prynadleΩyt klassu system, dlq reßenyq kotor¥x prymenym metod prost¥x
yteracyj [1 – 3]. Pervoe pryblyΩenye k reßenyg operatornoj system¥ (5)
ywem kak reßenye kraevoj zadaçy pervoho pryblyΩenyq
dz
dt
1 = A ( t ) z1 , l z1 ( ⋅, ε ) = 0. (6)
Suwestvovanye reßenyq zadaçy (6) harantyrovano odnorodnost\g zadaçy (1),
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1150 S. M. ÇUJKO
(2), samo Ωe reßenye zadaçy (6) predstavymo v vyde z1 ( t, ε ) = Xr ( t ) cr1
( ε ). Vto-
roe pryblyΩenye k reßenyg operatornoj system¥ (5) ywem kak reßenye krae-
voj zadaçy vtoroho pryblyΩenyq
dz
dt
2 = A ( t ) z2 ( t, ε ) + f ( t ) + ε A1 ( t ) z1 ( t, ε ), l z2 ( ⋅, ε ) = α + ε l1 z1 ( ⋅, ε ) (7)
v vyde z2 ( t, ε ) = Xr ( t ) cr2
( ε ) + z t2
1( ) ,( )ε , hde
z t G f s A s z s l z t2
1
1 1 1 1
( ) , , ; ,( ) = ( ) + ( ) ( ) + (⋅ ) ( )[ ]ε ε ε α ε ε .
Pry uslovyy P
B0
* = 0 naxodym pervoe pryblyΩenye k vektoru cr ( ε ) :
cr1
( ε ) = − − ( ) (⋅){ } ++ [ ]B P lK f s P c
Qd
0
1
ε
α ρ ρ* .
Tret\e pryblyΩenye k reßenyg operatornoj system¥ (5) ywem kak reßenye
kraevoj zadaçy tret\eho pryblyΩenyq
dz
dt
3 = A ( t ) z3 ( t, ε ) + f ( t ) + ε A1 ( t ) z2 ( t, ε ), l z3 ( ⋅, ε ) = α + ε l1 z2 ( ⋅, ε ). (8)
Pry uslovyy P
B0
* = 0 naxodym vtoroe pryblyΩenye k vektoru cr ( ε ) :
cr2
( ε ) = − − ( ) (⋅){ }+ [ ]B P lK f s
Qd
0
1
ε
α* –
– B P l z lK A s z s P c
Qd
0 1 2
1
1 2
1+ (⋅ ) − ( ) ( ) (⋅){ } +[ ]*
( ) ( ), ,ε ε ρ ρ.
ProdolΩaq rassuΩdenyq, pryxodym k yteracyonnoj procedure
z1 ( t, ε ) = Xr ( t ) cr1
( ε ),
cr1
( ε ) = − − ( ) (⋅){ } ++ [ ]B P lK f s P c
Qd
0
1
ε
α ρ ρ* ,
………………………………………………………
zk + 1 ( t, ε ) = Xr ( t ) crk
+
1
( ε ) + z tk + ( )1
1( ) , ε , (9)
z t G f s A s z s l z tk k k+ ( ) = ( ) + ( ) ( ) + (⋅ ) ( )[ ]1
1
1 1
( ) , , ; ,ε ε ε α ε ε ,
crk
+
1
( ε ) = − − ( ) (⋅){ }+ [ ]B P lK f s
Qd
0
1
ε
α* –
– B P l z lK A s z s P c
Q k k
d
0 1 1
1
1 1
1+
+ +(⋅ ) − ( ) ( ) (⋅){ } +[ ]*
( ) ( ), ,ε ε ρ ρ, k = 1, 2, … .
DokaΩem sxodymost\ πtoj procedur¥ k yskomomu reßenyg zadaçy (4). Ope-
ratornaq systema (5) πkvyvalentna zadaçe o postroenyy reßenyq operatornoho
uravnenyq z ( t, ε ) = Φ z ( t, ε ), hde
Φ z ( t, ε ) = − ( ) − ( ) (⋅){ }+ [ ]1
0ε
αX t B P lK f sr Qd
* –
– X t B P l G f s A s z s l zr Qd
( ) { ( ) + ( ) ( ) + (⋅ ) (⋅)+ [ ]0 1 1 1* , ; ,ε ε α ε ε –
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
VOZNYKNOVENYE REÍENYJ LYNEJNOJ NETEROVOJ KRAEVOJ ZADAÇY 1151
– lK A s G f A z l z s1 1 1( ) ( ) + ( ) ( ) + (⋅ ) ( )[ ](⋅)}[ ]τ ε τ τ ε α ε ε, ; , +
+ X t P c G f s A s z s l z tr( ) + ( ) + ( ) ( ) + (⋅ ) ( )[ ]ρ ρ ε ε α ε ε1 1, ; , .
Operator Φ z ( t, ε ) — neprer¥vn¥j, ohranyçenn¥j, dejstvugwyj yz prostran-
stva neprer¥vn¥x na otrezke [ a, b ] y promeΩutke ] 0, ε0 ] dejstvytel\n¥x vek-
tor-funkcyj z ( t, ε ) ∈ C1
[ a, b ], C ] 0, ε0 ] v sebq. Dlq ocenky dlyn¥ promeΩut-
ka, na kotorom soxranqetsq sxodymost\ yteracyonnoj procedur¥ (9) k yskomo-
mu reßenyg zadaçy (4), ocenym dlynu promeΩutka, na kotorom operator Φ z ( t,
ε ) qvlqetsq sΩymagwym. Normu vektor-funkcyy ϕ ( t ) ∈ C [ a, b ] polahaem ta-
kovoj [5, 6]:
ϕ ϕ( ) = ( )
≤ ≤
( )t t
i n
imax
1
, ϕ ϕi
a t b
it t( ) = ( )
≤ ≤
( )max .
Normoj ( m × n )-matryc¥ A ( t ) = aij ( t ), aij ( ⋅ ) ∈ C [ a, b ], budem naz¥vat\ çyslo
A t a t
i n
ij
j
n
( ) = ( )
≤ ≤ =
∑max
1 1
.
V sylu teorem¥ Ryssa vektorn¥j funkcyonal, opredelenn¥j na prostranstve
neprer¥vn¥x vektor-funkcyj x ( t ) ∈ C [ a, b ], predstavym v vyde
l x ( ⋅ ) = d t x t
a
b
Ω( ) ( )∫ ,
hde Ω ( t ) — ( m × n )-matryca, πlement¥ kotoroj — funkcyy ohranyçennoj na
[ a, b ] varyacyy. Zdes\ ymeetsq v vydu yntehral Rymana – Styl\t\esa. Pry πtom
lx(⋅) = Ω( )t . Pust\ x ( t, ε ), y ( t, ε ) — vektor-funkcyy yz maloj okrestnosty
nulq, pryçem x ( ⋅, ε ), y ( ⋅, ε ) ∈ C1
[ a, b ], C ] 0, ε0 ]. Ocenym normu raznosty
Φ Φx t y t q x y( ) − ( ) ≤ ( + ) +[ ] −, ,ε ε λ λ µ µ ε1 2 .
Takym obrazom, pry 0 < ε < ε* = q( + ) +[ ]−λ λ µ µ1 2
1 ≤ ε*
operator Φ z ( t, ε ) qvlq-
etsq sΩymagwym, pry πtom, sleduq pryncypu Kaççyoppoly – Banaxa [6], y
uravnenye z ( t, ε ) = Φ z ( t, ε ), y operatornaq systema (5) ymegt reßenye, dlq na-
xoΩdenyq kotoroho prymenyma yteracyonnaq procedura (9). Zdes\
q = X t B Pr Qd
( ) +
0 * , λ1 = l G A s l1 1 1( )[ ](⋅)*; * ,
λ2 = l K A1 1( )[ ](⋅)τ * , µ = G A l s1 1( )[ ]( )τ *; * .
Teorema. Pust\ kraevaq zadaça (4) predstavlqet krytyçeskyj sluçaj
P
Q* ≠ 0, pry πtom uslovye (3) razreßymosty nevozmuwennoj zadaçy (1), (2) ne
v¥polnqetsq pry proyzvol\n¥x neodnorodnostqx f ( t ) ∈ C [ a, b ] y α ∈ Rn
. Toh-
da pry uslovyy P
B0
* = 0 zadaça (4) ymeet po men\ßej mere odno reßenye z ( t,
ε ) ∈ C1
[ a, b ], C ] 0, ε0 ], hde
B0 = P l X lK A s X s
Q r r
d
* 1 1(⋅) − ( ) ( (⋅){ }[ ]
— ( d × r )-mernaq matryca. ∏to reßenye moΩno opredelyt\ s pomow\g sxodq-
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1152 S. M. ÇUJKO
wehosq pry ε ∈ ] 0, ε* ] yteracyonnoho processa (9).
Prymer. Uslovyq teorem¥ v¥polnqgtsq v zadaçe
dz
dt
= ( t2 – t ) z + et2
–
t + ε z, l z ( ⋅, ε ) = z ( 0, ε ) – z ( 1, ε ) = 0. (10)
Sohlasno prynqt¥m oboznaçenyqm m = n = 1, A ( t ) = t2 – t, f ( t ) = et
2
–
t
, l z ( ⋅ ) =
= z ( 0, ε ) – z ( 1, ε ) = 0, α = 0, l1 z ( ⋅ ) ≡ 0, A1 ( t ) = 1. Normal\naq fundamental\-
naq matryca odnorodnoj çasty dyfferencyal\noho uravnenyq (10) sut\ funk-
cyq X ( t ) = et
2
–
t
. Poskol\ku Q = 0, ymeet mesto krytyçeskyj sluçaj; pry πtom
P
Qd
* = PQ r
= 1. Obwee reßenye nevozmuwennoj zadaçy dlq (10) ymeet vyd z0 ( t,
c ) = c et
2
–
t
. Nevozmuwennaq zadaça dlq (10) nerazreßyma, pry πtom slabovozmu-
wennaq zadaça ymeet reßenye, ybo B0 = 1. Na pervom Ωe ßahe yteracyonnoj
procedur¥ (9) naxodym z1 ( t, ε ) = – ε–
1
X ( t ), pry πtom vse posledugwye prybly-
Ωenyq sovpadagt s perv¥m; πto obæqsnqetsq tem, çto z1 ( t, ε ) — toçnoe reße-
nye zadaçy (10). Sohlasno dokazannoj teoreme πto reßenye opredeleno dlq ε ∈
∈ ] 0, ε* ], ε* = 0,340977.
1. Boichuk A. A., Samoilenko A. M. Generalized inverse operators and Fredholm boundary-value
problems. – Utrecht; Boston: VSP, 2004. – XIV + 317 p.
2. Bojçuk A. A. Konstruktyvn¥e metod¥ analyza kraev¥x zadaç. – Kyev: Nauk. dumka, 1990. –
96 s.
3. Bojçuk A. A., Ûuravlev V. F., Samojlenko A. M. Obobwenno-obratn¥e operator¥ y netero-
v¥ kraev¥e zadaçy. – Kyev: Yn-t matematyky NAN Ukrayn¥, 1995. – 318 s.
4. Vyßyk M. Y., Lgsternyk L. A. Reßenye nekotor¥x zadaç o vozmuwenyqx v sluçae matryc y
samosoprqΩenn¥x y nesamosoprqΩenn¥x dyfferencyal\n¥x uravnenyj // Uspexy mat.
nauk. – 1960. – 15, v¥p. 3. – S. 3 – 80.
5. Hantmaxer F. R. Teoryq matryc. – M.: Nauka, 1988. – 552 s.
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Poluçeno 06.06.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
|
| id | umjimathkievua-article-3378 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:41:24Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/bf/5086342ba9ac566f6a5d4629a45d0dbf.pdf |
| spelling | umjimathkievua-article-33782020-03-18T19:52:34Z Bifurcation of solutions of a linear Fredholm boundary-value problem Возникновение решений линейной нетеровой краевой задачи Chuiko, S. M. Чуйко, С. М. Чуйко, С. М. We establish constructive conditions for the appearance of solutions of a linear Fredholm boundary-value problem for a system of ordinary differential equations in the critical case and propose an iterative procedure for finding these solutions. The range of values of a small parameter for which the indicated iterative procedure is convergent is estimated. Одержано конструктивні умови виникнення та побудовано ітераційну процедуру для знаходження розв'язків нетерової лінійної крайової задачі для системи звичайних диференціальних рівнянь у критичному випадку. Знайдено оцінку області значень малого параметра, для яких зберігається збіжність ітераційної процедури. Institute of Mathematics, NAS of Ukraine 2007-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3378 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 8 (2007); 1148–1152 Український математичний журнал; Том 59 № 8 (2007); 1148–1152 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3378/3499 https://umj.imath.kiev.ua/index.php/umj/article/view/3378/3500 Copyright (c) 2007 Chuiko S. M. |
| spellingShingle | Chuiko, S. M. Чуйко, С. М. Чуйко, С. М. Bifurcation of solutions of a linear Fredholm boundary-value problem |
| title | Bifurcation of solutions of a linear Fredholm boundary-value problem |
| title_alt | Возникновение решений линейной нетеровой краевой задачи |
| title_full | Bifurcation of solutions of a linear Fredholm boundary-value problem |
| title_fullStr | Bifurcation of solutions of a linear Fredholm boundary-value problem |
| title_full_unstemmed | Bifurcation of solutions of a linear Fredholm boundary-value problem |
| title_short | Bifurcation of solutions of a linear Fredholm boundary-value problem |
| title_sort | bifurcation of solutions of a linear fredholm boundary-value problem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3378 |
| work_keys_str_mv | AT chuikosm bifurcationofsolutionsofalinearfredholmboundaryvalueproblem AT čujkosm bifurcationofsolutionsofalinearfredholmboundaryvalueproblem AT čujkosm bifurcationofsolutionsofalinearfredholmboundaryvalueproblem AT chuikosm vozniknovenierešenijlinejnojneterovojkraevojzadači AT čujkosm vozniknovenierešenijlinejnojneterovojkraevojzadači AT čujkosm vozniknovenierešenijlinejnojneterovojkraevojzadači |