On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators
We investigate the problem of the well-posedness of a boundary-value problem for a system of pseudodifferential equations of arbitrary order with nonlocal conditions. The equation and boundary conditions contain pseudodifferential operators whose symbols are defined and continuous in a certain domai...
Збережено в:
| Дата: | 2005 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2005
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3671 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509797720260608 |
|---|---|
| author | Kengne, E. Кенне, Е. |
| author_facet | Kengne, E. Кенне, Е. |
| author_sort | Kengne, E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:01:36Z |
| description | We investigate the problem of the well-posedness of a boundary-value problem for a system of pseudodifferential equations of arbitrary order with nonlocal conditions. The equation and boundary conditions contain pseudodifferential operators whose symbols are defined and continuous in a certain domain $H ⊂ ℝ_{σ}^m$. A criterion for the existence and uniqueness of solutions and for the continuous dependence of the solution on the boundary function is established. |
| first_indexed | 2026-03-24T02:46:49Z |
| format | Article |
| fulltext |
UDC 517.9
E. Kengne (Univ. Dschang, Cameroon)
ON THE PROPER POSEDNESS
OF TWO-POINT BOUNDARY-VALUE PROBLEM
FOR SYSTEM WITH PSEUDODIFFERENTIAL OPERATORS
PRO KOREKTNIST| DVOTOÇKOVO} KRAJOVO} ZADAÇI
DLQ SYSTEM IZ PSEVDODYFERENCIAL|NYMY
OPERATORAMY
The question on the proper posedness of boundary-value problem with nonlocal condition for a system
of pseudodifferential equations of an arbitrary order is investigated. The equation and the boundary
conditions contain the pseudodifferential operators which symbols are defined and continuous in some
domain H ⊂ Rσ
m . The criterion of the existence, uniqueness of solutions and of the continuously
dependence of the solution on the boundary function is established.
Rozhlqnuto pytannq pro korektnist\ krajovo] zadaçi z nelokal\nog umovog dlq systemy
psevdodyferencial\nyx rivnqn\ dovil\noho porqdku. Rivnqnnq ta hranyçni umovy mistqt\ psev-
dodyferencial\ni operatory iz symvolamy, wo vyznaçeni ta neperervni u deqkij oblasti H ⊂
⊂ Rσ
m
. Vstanovleno kryterij isnuvannq ta [dynosti rozv’qzkiv, a takoΩ neperervno] zaleΩnosti
rozv’qzku vid hranyçno] funkci].
1. Introduction. The present paper generalizes and evolves the results of works [1, 2].
It investigates the question of the proper posedness of nonlocal boundary-value
problem for system of equations, containing the pseudodifferential operators that
symbols are defined and continuous in some domain H ⊂ R
m . The solution of the
problem is sought in functional spaces.
Consider in the infinite layer Π = R
m × 0, T[ ] the following two-point boundary
problem
L
t
i
x
u x t
u
t
P i
x
u∂
∂
− ∂
∂
= ∂
∂
+ − ∂
∂
=, ( , ) 0 , (1)
M i
x
u x t A i
x
u x B i
x
u x T x− ∂
∂
= − ∂
∂
+ − ∂
∂
=( , ) ( , ) ( , ) ( )0 ϕ (2)
where
∂
∂
= ∂
∂
∂
∂
… ∂
∂
x x x xm1 2
, , , , u u u um= …( )col 1 2, , , ,
ϕ ϕ ϕ ϕ( ) ( ), ( ), , ( )x x x xm= …( )col 1 2 ;
P i
x
P i
xjk
j k l
− ∂
∂
= − ∂
∂
=, ,1
, A i
x
A i
xjk
j k l
− ∂
∂
= − ∂
∂
=, ,1
,
B i
x
B i
xjk
j k l
− ∂
∂
= − ∂
∂
=, ,1
matrices that elements are pseudodifferential operators with symbols P( )σ , A( )σ , and
B( )σ , respectively, continuous in some domain H ⊂ Rσ
m .
Generally speaking, problem (1), (2) is improperly posed, even if Pjk ( )σ , Ajk ( )σ ,
and Bjk ( )σ are polynomials [3 – 6].
We shall be concerned with the question of the existence and uniqueness of solution
© E. KENGNE, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8 1131
1132 E. KENGNE
of problem (1), (2), and the question of the continuously dependence of this solution on
the boundary function ϕ( )x .
2. Notations and definitions. For the description of the spaces of solutions, we
introduce the following notation:
WΩ
∞ is the space of vector-functions u x( ) = ( u x1( ) , u x2( ), … , u xl( )), u xj ( ) ∈
∈ L m
2 R( ), such that the Fourier transform ˆ ( )uj σ is compactly supported in Ω ⊂
⊂ Rσ
∞ ; the space WΩ
∞ is invariant relatively to the pseudodifferential operator
Ã
x
− ∂
∂
˜ ( ) ( ) ˜( ) ˆ( ) exp , ( )A i
x
u x A u ix d u x Wm− ∂
∂
= ( ) ∈
− ∞∫2π σ σ σ σ
Ω
Ω
with the matrix ˆ( )A σ continuous in Ω, moreover, Ã i
x
− ∂
∂
: WΩ
∞ → WΩ
∞ is a
continuous application; WΩ
∞( )′ is the space of generalized vector-functions on WΩ
∞ ;
this space is invariant relatively to  i
x
∂
∂
; WΩ
+∞ = WΩ
∞( )′ , WΩ
−∞ = W−
∞( )′Ω , where
– Ω = { σ ∈ R
m : – σ ∈ Ω }; Ck ( 0, T[ ], WΩ
±∞) are spaces of vector-functions that for
every t ∈ 0, T[ ] are functions of space WΩ
±∞ respectively and continuously depend
on t together with the derivatives up to order k.
According to I. G. Petrovski [7], introduce the following definition:
Definition. We say that problem (1), (2) is properly posed in Cn( 0, T[ ], WΩ
±∞)
if, for every boundary function ϕ( )x ∈ WΩ
±∞ , problem (1), (2) should have in
Cn( 0, T[ ], WΩ
±∞) one and only one solution u x t( , ) , continuously depending on
ϕ( )x .
We shall not be concerned with Cauchy problem (the proper posedness of the
Cauchy problem for equation (1) is studied in [7]).
3. On the proper posedness of problem (1), (2). It is easily seen that the
following two applications for every domain Ω ⊂ H are continuous:
L
t
i
x
∂
∂
− ∂
∂
, : Cn( 0, T[ ], WΩ
±∞) → C0( 0, T[ ], WΩ
±∞),
u �
∂
∂
u x t
t
( , )
+ P i
x
− ∂
∂
u x t( , )
and
M i
x
− ∂
∂
: Cn( 0, T[ ], WΩ
±∞) → C0( 0, T[ ], WΩ
±∞),
u � A i
x
− ∂
∂
u x( , )0 + B i
x
u x T− ∂
∂
( , ).
Let us prove the inverse.
If we denote by ˆ( , )u tσ and ˆ ( )ϕ σ the x-Fourier transforms of the solution u x t( , )
of problem (1), (2) and the boundary function ϕ( )x , respectively, it is easily seen that
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
ON THE PROPER POSEDNESS OF TWO-POINT BOUNDARY-VALUE PROBLEM … 1133
ˆ( , )u tσ is a solution of the following boundary-value problem:
L d
dt
u t
du t
dt
P u t, ˆ( , )
ˆ( , )
( ) ˆ( , )σ σ σ σ σ
= + , (3)
M u t( ) ˆ( , )σ σ = A u B u T( ) ˆ( , ) ( ) ˆ( , )σ σ σ σ0 + = ˆ ( )ϕ σ . (4)
Let us find the fundamental matrix of solutions F t( , )σ of system (3). Because
P( )σ is a matrix continuous in H, the roots of characteristic equation
det ( )I Pλ σ+ = 0
are continuous functions of parameter σ. We denote by χ j = χ σj ( ) the multiplicity
of the root λ j = λ σj ( ), j = 1, ν , ν = ν σ( ) . According to the theory of matrices [8],
there exist two polynomial matrices M( , )λ σ and N( , )λ σ = Njk j k l
( , )
, ,
λ σ
=1
such
that M( , )λ σ × L( , )λ σ × N( , )λ σ ≡ Q( , )λ σ ≡ hj jk j k l
( , )
, ,
λ σ δ
=1
, det ( , )N λ σ ≡
≡ N( )σ ≠ 0, where hj ( , )λ σ =
k =∏ 1
ν
(λ – λ σ
k
q jk)
( )
, j = 1, l , where
j
l
jkq=∑ 1
( )σ =
= χ σk ( ) , k = 1, l , and δ jk are the Kronecker symbols.
It follows from the equality M( , )λ σ × L( , )λ σ × N( , )λ σ ≡ Q( , )λ σ and
condition det ( , )N λ σ ≡ N( )σ ≠ 0 that M( , )λ σ × L( , )λ σ ≡ Q( , )λ σ × N−1( , )λ σ ,
and after using the transformation y t( , )σ = N d
dt
u t−
1 , ˆ( , )σ σ , system (3) takes the
form
Q d
dt
y t, ( , )σ σ
= 0,
that is,
h d
dt
y tj j, ( , )σ σ
= 0, j = 1, l . (5)
The fundamental system of solutions of each of equations (5) reads as
y t d
d
tj k
k
, ( , ) exp( )α
α
λ λ
σ
λ
λ=
−
=
1
, α σ= 1, ( )qjk , k = 1, ν .
Therefore, the fundamental matrix of solutions of system (3) reads as
F t( , )σ = exp( ) ( )!
! !
( , )
, , , , ,
, ,
λ α
ρ λ
λ σ
α
β
λ λ β α
ν ρ α
β
k
s
s p
j
j l q
k s
t t
s
d
d
N
k k
× − ×
=
−
= = =
= = − −
∑1 1
0
1
1 1
1 1
.
Using the theorem of dependence of solutions on the parameter [9], we conclude that
F t( , )σ is continuous with respect to σ.
Therefore, the solution of system (3) reads as ˆ( , )u tσ = F t C( , )σ , where C =
= col (C1, C2 , … , Cl ). Using the boundary condition (4), we find that C is a solution
of the following linear algebraic system
A F B F T C( ) ( , ) ( ) ( , ) ˆ ( )σ σ σ σ ϕ σ0 +( ) = ,
whose determinant is
∆( ) det ( ) det ( ) ( , ) ( ) ( , )σ σ σ σ σ σ= = +D A F B F T0 ,
where
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1134 E. KENGNE
D A F B F T( ) ( ) ( , ) ( ) ( , )σ σ σ σ σ= +0 .
Let Ω = H N\ ∆ , N∆ = {σ ∈ R
m : ∆( )σ = 0}. For every σ ∈ Ω , C = C( )σ =
= D−1( )σ × ˆ ( )ϕ σ and the solution of problem (3), (4) reads as ˆ( , )u tσ = F t( , )σ ×
× D−1( )σ ˆ ( )ϕ σ ∀ ∈( )σ Ω . Let W t( , )σ = F t( , )σ × D−1( )σ .
Associate with matrix W t( , )σ the pseudodifferential operator W i
x
t− ∂
∂
, that
acts continuously from WΩ
±∞ respectively to Cn( 0, T[ ], WΩ
±∞), that is,
W i
x
t− ∂
∂
, : WΩ
±∞ → Cn( 0, T[ ], WΩ
±∞), u x( ) � W i
x
t− ∂
∂
, u x( )
is a continuous application on WΩ
±∞ .
Theorem. In order that problem (1), (2) should be properly posed in Cn( 0, T[ ],
WΩ
±∞), it is necessary and sufficient that Ω = H N\ ∆ .
Proof. It is clear that Ω ⊂ H . We prove the theorem in the case of Cn( 0, T[ ],
WΩ
+∞) (the case of Cn( 0, T[ ], WΩ
−∞) can be done by analogy with [8]).
Necessity. If σ0 ∈ Ω ∩ N∆ , the homogeneous problem
L d
dt
u t, ˆ( , )σ σ0 0 0
= ,
M u t A u B u T( ) ˆ( , ) ( ) ˆ( , ) ( ) ˆ( , )σ σ σ σ σ σ0 0 0 0 0 00 0= + =
possesses more than one solution. Consequently, the solution (if it exists) of the
homogeneous problem
L
t
i
x
u x t∂
∂
− ∂
∂
=, ( , ) 0,
M i
x
u x t A i
x
u x B i
x
u x T− ∂
∂
= − ∂
∂
+ − ∂
∂
=( , ) ( , ) ( , )0 0
is not unique in Cn( 0, T[ ], WΩ
+∞), and this implies the nonuniqueness of solutions of
problem (1), (2) in Cn( 0, T[ ], WΩ
+∞). The ill-posedness of problem (1), (2) in
Cn( 0, T[ ], WΩ
+∞) follows from the nonuniqueness of its solution in Cn( 0, T[ ],
WΩ
+∞).
Sufficiency. Let Ω = H N\ ∆ . For every ϕ( )x ∈ WΩ
+∞ ,
u x t( , ) = W i
x
t− ∂
∂
, ϕ( )x
is a solution of problem (1), (2) in Cn( 0, T[ ], WΩ
+∞) and continuously depends on
ϕ( )x , which implies the existence of solution of problem (1), (2) in Cn( 0, T[ ], WΩ
+∞)
that continuously depends on ϕ( )x . In order to prove the uniqueness of solution of
problem (1), (2), let us notice that if u x t( , ) ∈ Cn( 0, T[ ], WΩ
+∞), its x-Fourier
transform ˆ( , )u tσ will be a solution of problem (3), (4). Under the condition of the
theorem, this problem (3), (4) possesses one and only one solution for every σ ∈ Ω . If
σ ∈ R
m \ Ω , then ˜( , )u tσ ≡ 0. This implies the uniqueness of solution of problem (1),
(2) in Cn( 0, T[ ], WΩ
+∞), and the theorem is proved.
4. Application. Consider an infinite set of point-like particles, disposed on a string
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
ON THE PROPER POSEDNESS OF TWO-POINT BOUNDARY-VALUE PROBLEM … 1135
at the same distances l from each other; the mass of each particle is m, while F is
the tension of the string. The values of F and m are supposed to be constant every-
where and independent of time. Particles are supposed to have only one degree of
freedom. At each given time t, the motion of the j-th particle is completely defined in
terms of the position of its adjacent particles, i.e., the ( j – 1)-th and ( j + 1)-th ones
( j = 0, ± 1, ± 2, … ). Thus, the fundamental law of dynamics is given by
˙̇ ( ) ( ) ( ) ( )v v v vn n n nt a t t t= + −( )+ −
2
1 14
2 , n = 0, ± 1, ± 2, … , (6)
being a = 2 F
ml
. We associate with (6) the boundary conditions
v vn n nT( ) ( )0 + = ϕ ,
(7)
˙ ( ) ˙ ( )v vn n nT0 + = ψ .
Consider the differentiable function v( , )x t that takes the values of vj at the
nodes of the lattice, i.e., v( , )jl t = vj t( ) . Therefore, system (6) acquires the form of
the difference-differential equation
∂
∂
= + − + −[ ]
2
2
2
4
2
v
v v v
( , )
( , ) ( , ) ( , )
x t
t
a x l t x t x l t , 0 < t < T. (8)
After using the equality e x t
i i
x
− ∂
∂
2
v( , ) = v( , )x l t+ , equation (8) takes the form of
the pseudodifferential equation:
∂
∂
+ − ∂
∂
=
2
2
2 2 0v
v
t
a i
x
sin , 0 < t < T. (9)
By setting u = col (u1, u2 ) with u1 = v and u2 =
∂
∂
v
t
, equation (9) and the boundary
conditions (7) become
∂
∂
+ − ∂
∂
=u
t
P i
x
u 0 , (10)
u x u x T x1 1 10( , ) ( , ) ( )+ = θ ,
(11)
u x u x T x2 2 20( , ) ( , ) ( )+ = θ ,
respectively, where
P i
x a i
x
− ∂
∂
=
−
− ∂
∂
0 1
02 2sin
and θ1( )x and θ2( )x are two differentiable functions that take the values of ϕn and
ψn , respectively, at the nodes of the lattice. For this example, F t( , )σ reads as
F t( , )σ =
cos sin sin sin
sin sin sin sin cos sin
at at
a at a at
σ σ
σ σ σ σ
( ) − ( )
− ( ) − ( )
and
∆( ) sin cos sinσ σ σ= − + ( )[ ]2 1a aT .
Therefore,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1136 E. KENGNE
N k
k
aT
aT
k
aT
kk∆ = { } + − + ≤ ≤ + ∈
∈π π π
π
π
πZ
Z∪ arcsin
( )
, ,
2 1
2 2
.
If we take H = R, we conclude from the above theorem that problem (10), (11) is
properly posed in Cn( 0, T[ ], WΩ
±∞) with Ω = R \ N∆ .
1. Borok V. M., Fardigola L. V. Nonlocal properly posed boundary problem in the layer // Math. Note
Acad. Sci. USSR. – 1990. – 48. – P. 20 – 25.
2. Kengne E. Criterion of the regularity of boundary problems with an integral in the boundary
condition. – Moscow: VINITI, 1992. – Vol. 92. – 20 p.
3. Antepko I. I. On the boundary problem in an infinite layer for system of linear partial differential
equations // News Kharkov Univ. – 1971. – Issue 67 (36). – P. 62 – 72.
4. Borok V. M., Antepko I. I. Criterion of the proper posedness of boundary problem in the layer //
Teor. functs., Funkts. Analys i Pril. – 1976. – 26. – P. 3 – 9.
5. Kengne E., Pelap F. B. Regularity of two-point boundary-value problem // Afr. Math. – 2001. –
12, # 3. – P. 61 – 70.
6. Kengne E. Properly posed and regular nonlocal boundary-value problems for partial differential
equations // Ukr. Math. J. – 2002. – 54, # 8. – P. 1135 – 1142.
7. Petrovskii I. G. On the Cauchy problem for system of linear partial differential equations in the
domain on nonanalytical functions // Bull. Moscow State Univ. Sect. A. – 1938. – 1. – P. 1 – 72.
8. Gantmakher F. R. Theory of matrices. – Moscow: Nauka, 1988. – 552 p.
9. Petrovskii I. G. Lecture on the theory of ordinary differential equations. – Moscow: Nauka, 1970. –
279 p.
Received 22.01.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
|
| id | umjimathkievua-article-3671 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:46:49Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b4/1d2f9c8785e150a515713dc4893320b4.pdf |
| spelling | umjimathkievua-article-36712020-03-18T20:01:36Z On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators Про коректність двоточкової крайової задачі для систем із псевдодиференціальними операторами Kengne, E. Кенне, Е. We investigate the problem of the well-posedness of a boundary-value problem for a system of pseudodifferential equations of arbitrary order with nonlocal conditions. The equation and boundary conditions contain pseudodifferential operators whose symbols are defined and continuous in a certain domain $H ⊂ ℝ_{σ}^m$. A criterion for the existence and uniqueness of solutions and for the continuous dependence of the solution on the boundary function is established. Розглянуто питання про коректність крайової задачі з нелокальною умовою для системи псевдодиференціальних рівнянь довільного порядку. Рівняння та граничні умови містять псевдодиференціальні оператори із символами, що визначені та неперервні у деякій області $H \subset \mathbb{R}^m_{\sigma}$. Встановлено критерій існування та єдиності розв'язків, а також неперервної залежності розв'язку від граничної функції. Institute of Mathematics, NAS of Ukraine 2005-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3671 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 8 (2005); 1131 – 1136 Український математичний журнал; Том 57 № 8 (2005); 1131 – 1136 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3671/4070 https://umj.imath.kiev.ua/index.php/umj/article/view/3671/4071 Copyright (c) 2005 Kengne E. |
| spellingShingle | Kengne, E. Кенне, Е. On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators |
| title | On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators |
| title_alt | Про коректність двоточкової крайової задачі для систем із псевдодиференціальними операторами |
| title_full | On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators |
| title_fullStr | On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators |
| title_full_unstemmed | On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators |
| title_short | On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators |
| title_sort | on the well-posedness of a two-point boundary-value problem for a system with pseudodifferential operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3671 |
| work_keys_str_mv | AT kengnee onthewellposednessofatwopointboundaryvalueproblemforasystemwithpseudodifferentialoperators AT kennee onthewellposednessofatwopointboundaryvalueproblemforasystemwithpseudodifferentialoperators AT kengnee prokorektnístʹdvotočkovoíkrajovoízadačídlâsistemízpsevdodiferencíalʹnimioperatorami AT kennee prokorektnístʹdvotočkovoíkrajovoízadačídlâsistemízpsevdodiferencíalʹnimioperatorami |