On solvability of inhomogeneous boundary-value problems in Sobolev—Slobodetskiy spaces
We investigate the most general class of Fredholm one-dimensional boundary-value problems in the Sobolev—Slobodetskiy spaces. Boundary conditions of these problems may contain a derivative of the whole or fractional order. It is established that each of these boundary-value problems corresponds to...
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Видавничий дім "Академперіодика" НАН України
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irk-123456789-1704042020-07-16T01:28:41Z On solvability of inhomogeneous boundary-value problems in Sobolev—Slobodetskiy spaces Mikhailets, V.A. Skorobohach, T.V. Математика We investigate the most general class of Fredholm one-dimensional boundary-value problems in the Sobolev—Slobodetskiy spaces. Boundary conditions of these problems may contain a derivative of the whole or fractional order. It is established that each of these boundary-value problems corresponds to a certain rectangular numerical characteristic matrix with kernel and cokernel having the same dimension as the kernel and cokernel of the boundary- value problem. The sufficient conditions for the sequence of the characteristic matrices of a specified boundary-value problems to converge are found. Досліджено найбільш широкий клас нетерових одновимірних крайових задач у просторах Соболєва—Слободецького. Крайові умови в них можуть містити похідні розв'язку цілого або дробового порядку. Встановлено, що кожній із таких крайових задач відповідає деяка прямокутна числова характеристична матриця, вимірність ядра і коядра якої збігаються відповідно з вимірністю ядра і коядра крайової задачі. Знайдені достатні умови збіжності послідовності характеристичних матриць розглянутих крайових задач. Исследуется наиболее широкий класс нетеровых одномерных краевых задач в пространствах Соболева—Слободецкого. Краевые условия в них могут содержать производные решения целого или дробного порядка. Показано, что каждой из таких краевых задач соответствует некоторая прямоугольная числовая характеристическая матрица, размерность ядра и коядра которой совпадают соответственно с размерностью ядра и коядра краевой задачи. Найдены достаточные условия сходимости последовательности характеристических матриц рассмотренных краевых задач. 2020 Article On solvability of inhomogeneous boundary-value problems in Sobolev—Slobodetskiy spaces / V.A. Mikhailets, T.V. Skorobohach // Доповіді Національної академії наук України. — 2020. — № 4. — С. 10-14. — Бібліогр.: 7 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2020.04.010 http://dspace.nbuv.gov.ua/handle/123456789/170404 517.927 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Математика Математика |
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Математика Математика Mikhailets, V.A. Skorobohach, T.V. On solvability of inhomogeneous boundary-value problems in Sobolev—Slobodetskiy spaces Доповіді НАН України |
| description |
We investigate the most general class of Fredholm one-dimensional boundary-value problems in the Sobolev—Slobodetskiy
spaces. Boundary conditions of these problems may contain a derivative of the whole or fractional
order. It is established that each of these boundary-value problems corresponds to a certain rectangular numerical
characteristic matrix with kernel and cokernel having the same dimension as the kernel and cokernel of the boundary-
value problem. The sufficient conditions for the sequence of the characteristic matrices of a specified boundary-value
problems to converge are found. |
| format |
Article |
| author |
Mikhailets, V.A. Skorobohach, T.V. |
| author_facet |
Mikhailets, V.A. Skorobohach, T.V. |
| author_sort |
Mikhailets, V.A. |
| title |
On solvability of inhomogeneous boundary-value problems in Sobolev—Slobodetskiy spaces |
| title_short |
On solvability of inhomogeneous boundary-value problems in Sobolev—Slobodetskiy spaces |
| title_full |
On solvability of inhomogeneous boundary-value problems in Sobolev—Slobodetskiy spaces |
| title_fullStr |
On solvability of inhomogeneous boundary-value problems in Sobolev—Slobodetskiy spaces |
| title_full_unstemmed |
On solvability of inhomogeneous boundary-value problems in Sobolev—Slobodetskiy spaces |
| title_sort |
on solvability of inhomogeneous boundary-value problems in sobolev—slobodetskiy spaces |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| publishDate |
2020 |
| topic_facet |
Математика |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/170404 |
| citation_txt |
On solvability of inhomogeneous boundary-value problems in Sobolev—Slobodetskiy spaces / V.A. Mikhailets, T.V. Skorobohach // Доповіді Національної академії наук України. — 2020. — № 4. — С. 10-14. — Бібліогр.: 7 назв. — англ. |
| series |
Доповіді НАН України |
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AT mikhailetsva onsolvabilityofinhomogeneousboundaryvalueproblemsinsobolevslobodetskiyspaces AT skorobohachtv onsolvabilityofinhomogeneousboundaryvalueproblemsinsobolevslobodetskiyspaces |
| first_indexed |
2025-07-15T05:35:37Z |
| last_indexed |
2025-07-15T05:35:37Z |
| _version_ |
1837689983027118080 |
| fulltext |
10 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2020. № 4: 10—14
Ц и т у в а н н я: Mikhailets V.A., Skorobohach T.B. On solvability of inhomogeneous boundary-value problems in
Sobolev—Slobodetskiy spaces. Допов. Нац. акад. наук Укр. 2020. № 4. С. 10—14. https://doi.org/10.15407/
dopovidi2020.04.010
Introduction. Boundary-value problems for systems of ordinary differential equations arise in
many problems of analysis and its applications. Unlike Cauchy problems, the solutions to such
problems may not exist or may not be unique. Thus, it is interesting to investigate the nature of
the solvability of inhomogeneous boundary-value problems in the functional Sobolev and So-
bolev—Slobodetskiy spaces and the dependence of their solutions on the parameter. For Fredholm
boundary-value problems, similar issues have been investigated in papers [1-5]. The case of un-
derdefined or overdefined boundary-value problems in Sobolev spaces was investiga ted in
paper [6].
Statement of the problem. Let a finite interval ( , )a b ⊂ and parameters
, , (0{ ) \ 1} , ,m l s p∈ ∞ <⊂ ∞ � ,
be given. By ([ , ]; ): n
p p
nW W a b= , we denote a complex Sobolev space and set 0 :p pW L= . By
( ) ([ , ]; ):s sm m
p pW W a b= and ( ) ([ , ]; ):m m nn m m
p pW W a b× ×= , we denote the Sobolev spaces of
https://doi.org/10.15407/dopovidi2020.04.010
UDC 517.927
V.A. Mikhailets, T.B. Skorobohach
Institute of Mathematics of the NAS of Ukraine, Kyiv
NTU of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”
E-mail: mikhailets@imath.kiev.ua, tetianaskorobohach@gmail.com
On solvability of inhomogeneous boundary-value
problems in Sobolev—Slobodetskiy spaces
Presented by Corresponding Member of the NAS of Ukraine A.N. Kochubei
We investigate the most general class of Fredholm one-dimensional boundary-value problems in the Sobolev—Slo-
bodetskiy spaces. Boundary conditions of these problems may contain a derivative of the whole or fractional
order. It is established that each of these boundary-value problems corresponds to a certain rectangular numerical
characteristic matrix with kernel and cokernel having the same dimension as the kernel and cokernel of the boun da-
ry-value problem. The sufficient conditions for the sequence of the characteristic matrices of a specified bounda ry-
value problems to converge are found.
Keywords: inhomogeneous boundary-value problem, Sobolev—Slobodetskiy space, Fredholm operator, index of
operator.
11ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2020. № 4
On solvability of inhomogeneous boundary-value problems in Sobolev—Slobodetskiy spaces
vector functions and matrix functions, respectively, with elements from the function space n
pW .
By ⋅
,n p
, we denote the norms in these spaces. They are defined as the sums of the correspon-
ding norms of all elements of a vector-valued or matrix-valued function in n
pW . The space of
fun ctions (scalar functions, vector functions, or matrix functions) in which the norm is intro-
duced is always clear from the context. For 1m = , all these spaces coincide. It is known that
n
pW are separable Banach spaces.
We denote, by ([ , ]; ):p p
s sW W a b= , where 1 p < ∞� and 1s > is not integer, the Sobolev—
Slobodetskiy space of all complex-valued functions belonging to Sobolev space [ ]
p
sW and sa-
tis fying the condition
+
⎛ ⎞
−⎜ ⎟
= + < +∞⎜ ⎟
−⎜ ⎟
⎝ ⎠
∫ ∫
[ ]
1/
[ ]
[ ], 1 { },
( ) ( )
:
s
pp
sb b
s p s p
a a
n p
f x f y
f f dxdy
x y
,
where [s] is the integer part, and { }s is the fractional part of the number s. Here, we recall that
⋅
[ ],s p
is the norm in the Sobolev space [ ]
p
sW . This equality defines the norm
,s p
f in the space s
pW .
Consider a linear boundary-value problem on a finite interval ( , )a b for the system of m first-
order scalar differential equations
( ) ( ) : ( ) ( ) ( ) ( ), ,( )Ly t y t A t y t f t t a b′ ∈= + = , (1)
By c= , (2)
where the matrix function ( )A ⋅ belongs to the space ×( ) m
p
s mW , the vector function ( )f ⋅ belongs
to the space ( )m
p
sW , the vector c belongs to the space l , аnd B is a linear continuous operator
1: ( )s m l
pB W + → . (3)
The boundary condition (2) consists of l scalar boundary conditions for the system of m dif-
ferential equations of the first order. We represent vectors and vector functions in the form of
columns. A solution to the boundary-value problem (1), (2) is understood as a vector function
1( )ms
py W +∈ satisfying Eq. (1) for > +1 1/s p everywhere and, for +1 1/s p� , almost everywhere
on ( , )a b and equality (2) specifying l scalar boundary conditions. The solutions to Eq. (1) fill
the space 1( )p
s mW + , if its right-hand side ( )f ⋅ runs through the space ( )m
p
sW . Hence, the boun-
dary condition (2) is the most general condition for this equation and includes all known types of
classical boundary conditions, namely, the Cauchy problem, two- and multipoint problems, in-
tegral and mixed problems, and numerous nonclassical problems. The last class of problems
may contain derivatives of integer or fractional order k of required vector — functions, where
0 1k s< < + .
The main purpose of this work is to establish whether the boundary-value problem (1), (2)
has the Fredholm property; to find its index and the dimension of the cokernel and the kernel
of the operator of an inhomogeneous boundary-value problem in terms of the properties of a
special rectangular numerical matrix and to investigate its stability. In the case of Sobolev spaces
of in teger order, similar results were obtained in [6].
12 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2020. № 4
V.A. Mikhailets, T.B. Skorobohach
Main results. We rewrite the inhomogeneous boundary-value problem (1), (2) in the form
of a linear operator equation
)( , () ,L B y f c= ,
where ( ),L B is a linear operator in the pair of Banach spaces
1)( , ) : ( ( )m m ls s
p pL B W W+ → × . (5)
Let Х and Y be Banach spaces. Recall that a linear continuous operator :T Х Y→ is called a
Fredholm operator, if its kernel kerT and cokernel (/ )Y T Х are finite-dimensional. If the ope-
rator is a Fredholm one, then its range ( )T Х is closed in Y , and the index
)ind : dim ker (dim )/(T YT T Х−=
is finite (see, e.g., [7], Lemma 19.1.1).
Theorem 1. The linear operator (5) is a bounded Fredholm operator with index m l− .
Denote, by )( () m ms
pY W ×⋅ ∈ , the unique solution to a linear homogeneous matrix equation
( ) ( ) ( ) , ,( )mY t A t t atY O b′ + ∈= , (6)
with the initial condition
( ) mY a I= . (7)
Here, mO are zero matrices, and mI are identity ( )m m× matrices. The unique solution to
the Cauchy problem (6), (7) belongs to the space 1( )p
s m mW + × .
By [ ]BY , we denote a numerical matrix of dimension ( )m l× whose і-th column is a result of
the action of the operator B from (3) on і-th column of the matrix function ( )Y ⋅ , {1, , }і m∈ .
Definition 1. A rectangular numerical matrix
×= ∈, [) ]( m lM L B BY , (8)
is called the characteristic matrix for the inhomogeneous boundary-value problem (1), (2).
Here, m is the number of scalar differential equations of system (1), and l is the number of
scalar boundary conditions.
Theorem 2. The dimensions of the kernel and cokernel of operator (5) are equal to the dimen-
sions of the kernel and cokernel of the characteristic matrix (8), respectively:
)dim ker , dim ker( ) ( ( ),L B M L B= ,
)dim , dimcoker( ) coker ( ( ),L B M L B= .
A criterion for the invertibility of the operator ( ),L B follows from Theorem 2, i.e., the condi-
tion under which problem (1), (2) possesses a unique solution, and this solution continuously
depends on the right-hand sides of the differential equation and the boundary condition.
Corollary 1. Operator (5) is invertible, if and only if l m= , and the square matrix ,( )M L B is
nondegenerate.
13ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2020. № 4
On solvability of inhomogeneous boundary-value problems in Sobolev—Slobodetskiy spaces
Application. In addition to problem (1), (2), we consider the sequence of inhomogeneous
boundary-value problems
( ) ( , ) : ( , ) ( , )) ( , ), () ( , ,L k y t k y t k A t k y t k f t k bt a′ ∈= + = , (9)
)( ) ),( , (B k y k kc k⋅ = ∈ , (10)
where the matrix functions ,( )A k⋅ , the vector functions ,( )f k⋅ , the vectors ( )c k and linear
continuous operators ( )B k satisfy the above conditions for problem (1), (2).
With the boundary-value problem (9), (10), we associate a sequence of linear continuous
operators 1( ), (( )) : ( ) ( )m m ls s
p pL k B k W W+ → ×
and a sequence of characteristic matrices
×= ⋅ ⊂( ), ( ) : [( ( ) ( , )]) m lM L k B k B k Y k
depending on the parameter k∈ .
We now formulate a sufficient condition for the convergence of the characteristic matrices
)( ), ( )(M L k B k to the matrix ,( )M L B .
Theorem 3. If the sequence of operators )( ),( ( )L k B k converges strongly to the operator
( ),L B for k→∞ , then the sequence of characteristic matrices )( ), ( )(M L k B k converges to the
matrix ,( )M L B .
Corollary 2. Under the assumptions from Theorem 3, the following inequalities hold
)dim ker ( ), ( ) (dim ( ker ,)L k B k L B� ,
)dim ( ), ( )coker( ) cokedim r( ,L k B k L B� .
for sufficiently large k.
In particular:
1) If l m= and the operator ( ),L B is invertible, then the operators )( ),( ( )L k B k are also
invertible for large k;
2) If the boundary-value problem (1), (2) has a solution for any values of the right-hand
sides, then the boundary-value problems (9), (10) also have a solution for large k;
3) If the boundary-value problem (1), (2) has a unique solution, then problems (9), (10)
also have a unique solution for each sufficiently large k.
REFERENCES
1. Kodliuk, Т. I. & Mikhailets, V. А. (2013). Solutions of one-dimensional boundary-value problems with a pa-
rameter in Sobolev spaces. J. Math. Sci. (N.Y.), 190, No. 4, pp. 589-599.
2. Gnyp, E. V., Kodliuk, Т. I. & Mikhailets, V. A. (2015). Fredholm boundary-value problems with parameter
in Sobolev spaces. Ukr. Math. J., 67, No. 5, pp. 658-667.
3. Hnyp, Y., Mikhailets, V. & Murach, A. (2017). Parameter-depent one-dimensional boundary-value problems in
Sobolev spaces. Electron. J. Differ. Equat., No. 81, 13 pp.
4. Atlasiuk, O. M. & Mikhailets, V. A. (2019). Fredholm one-dimensional boundary-value problems in Sobolev
spaces. Ukr. Math. J., 70, No. 10, pp. 1526-1537.
5. Atlasiuk, O. M. & Mikhailets, V. A. (2019). Fredholm one-dimensional boundary-value problems with a pa-
rameter in Sobolev spaces. Ukr. Math. J., 70, No. 11, pp. 1677-1687.
14 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2020. № 4
V.A. Mikhailets, T.B. Skorobohach
6. Atlasiuk, O. M. & Mikhailets, V. A. (2019). On the solvability of inhomogeneous boundary-value problems in
Sobolev spaces. Dopov. Nac. akad. nauk Ukr., No. 11, pp. 3-7. https://doi.org/10.15407/dopovidi2019.11.003
7. Hörmander, L. (1985). The analysis of linear partial differential operators. III: Pseudo-differential operators.
Berlin: Springer.
Received 30.01.2020
В.А. Михайлець, Т.Б. Скоробогач
Інститут математики НАН України, Київ
НТУ України “Київський політехнічний інститут ім. Ігоря Сікорського”
E-mail: mikhailets@imath.kiev.ua, tetianaskorobohach@gmail.com
ПРО РОЗВ’ЯЗНІСТЬ НЕОДНОРІДНИХ КРАЙОВИХ ЗАДАЧ
У ПРОСТОРАХ СОБОЛЄВА—СЛОБОДЕЦЬКОГО
Досліджено найбільш широкий клас нетерових одновимірних крайових задач у просторах Соболєва—
Слободецького. Крайові умови в них можуть містити похідні розв’язку цілого або дробового порядку.
Встановлено, що кожній із таких крайових задач відповідає деяка прямокутна числова характерис-
тична матриця, вимірність ядра і коядра якої збігаються відповідно з вимірністю ядра і коядра крайової
задачі. Знайдені достатні умови збіжності послідовності характеристичних матриць розглянутих кра-
йових задач.
Ключові слова: неоднорідна крайова задача, простір Соболєва—Слободецького, нетерів оператор, індекс
оператора.
В.А. Михайлец, Т.Б. Скоробогач
Институт математики НАН Украины, Киев
НТУ Украины “Киевский политехнический институт им. Игоря Сикорского”
E-mail: mikhailets@imath.kiev.ua, tetianaskorobohach@gmail.com
О РАЗРЕШИМОСТИ НЕОДНОРОДНЫХ КРАЕВЫХ ЗАДАЧ
В ПРОСТРАНСТВАХ СОБОЛЕВА—СЛОБОДЕЦКОГО
Исследуется наиболее широкий класс нетеровых одномерных краевых задач в пространствах Соболева—
Слободецкого. Краевые условия в них могут содержать производные решения целого или дробного
порядка. Показано, что каждой из таких краевых задач соответствует некоторая прямоугольная числовая
характеристическая матрица, размерность ядра и коядра которой совпадают соответственно с размер-
ностью ядра и коядра краевой задачи. Найдены достаточные условия сходимости последовательности
характеристических матриц рассмотренных краевых задач.
Ключевые слова: неоднородная краевая задача, пространство Соболева—Слободецкого, нетеров опе ра-
тор, индекс оператора.
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