Two-point boundary value problems for differential-operator equations
We study general two-point boundary value problems for a non-homogeneous differential-operator equation of the second order with an unbounded linear operator in a Banach space. The main classical solvability condition is given in terms of the property of the resolvent of the operator at the points,...
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| author | Eidelman, Y. Yakubov, Ya. Eidelman, Y. Yakubov, Ya. |
| author_facet | Eidelman, Y. Yakubov, Ya. Eidelman, Y. Yakubov, Ya. |
| author_institution_txt_mv | [
{
"author": "Y. Eidelman",
"institution": "Tel-Aviv University"
},
{
"author": "Ya. Yakubov",
"institution": "Tel-Aviv University"
}
] |
| author_sort | Eidelman, Y. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
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| datestamp_date | 2018-01-23T01:38:29Z |
| description | We study general two-point boundary value problems for a non-homogeneous differential-operator equation of the second order with an unbounded linear operator in a Banach space. The main classical solvability condition is given in terms of the property of the resolvent of the operator at the points, which are opposite to the eigenvalues of the corresponding ordinary differential operator. At the end of the paper, two particular types of boundary value conditions are treated: periodic and Dirichlet. |
| first_indexed | 2026-08-04T01:04:03Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2016, т. 13, № 1, 58–75
Y. Eidelman, Ya. Yakubov
(Tel-Aviv University, Israel)
Two-point boundary value problems
for differential-operator equations
eideyu@post.tau.ac.il and yakubov@post.tau.ac.il
We study general two-point boundary value problems for a non-
homogeneous differential-operator equation of the second order with
an unbounded linear operator in a Banach space. The main classical
solvability condition is given in terms of the property of the resolvent
of the operator at the points, which are opposite to the eigenvalues
of the corresponding ordinary differential operator. At the end of the
paper, two particular types of boundary value conditions are treated:
periodic and Dirichlet.
1. Introduction
In this paper, we study general two-point boundary value problems
for a non-homogeneous differential-operator equation of the second
order with an unbounded linear operator in a Banach space. We
impose some restrictions on the right-hand side of the equation.
Then, we formulate the conditions of the unique solvability of the
problems in terms of the property of the resolvent of the operator at
the points, which are opposite to the eigenvalues of the correspondi-
ng ordinary differential operator. In fact, we find sufficient conditions
for the unique solvability, but some of them are also necessary. At the
c© Y. Eidelman, Ya. Yakubov, 2016
Two-point boundary value problems... 59
end of the paper, two particular types of boundary value conditions
are treated: periodic and Dirichlet.
We represent the solutions of the problems as a series of vector-
valued functions which include the eigenfunctions of the correspondi-
ng ordinary boundary value problem. To obtain the convergence of
the series, we essentially use the Abel transform of the series. Here
we use an approach suggested by A. V. Knyazyuk in the paper [8]
for the study of the model Dirichlet problem.
In the paper, we study classical solutions of the problems. The
solutions from the Lp spaces and from the Hölder spaces have been
studied by V. Arendt and S. Bu in [1] by using of the technique of
Fourier series and Marcinkiewicz multipliers. The types of boundary
value conditions, which are covered by our results, are essentially
wider than those in the above mentioned papers. In particular, our
paper contains a generalization of the main result in [8].
Problems in Lp spaces, with rather general non-local boundary
value conditions (multipoint, integro-differential, functional), and
for higher order abstract differential-operator equations, have been
studied in a series of papers by A. Favini and Ya. Yakubov [2]-
[5] (paper [2] is joint with V. Shakhmurov). For the results in the
framework of Hilbert spaces, we refer the reader to the monograph
by S. Yakubov and Ya. Yakubov [9] and reference therein.
Solvability of problems of the form (1)-(2), in some weighted
Hölder spaces, has been studied by L. M. Gershtein and P. E.
Sobolevskii in [6]. The main solvability condition is given in some
implicit form. Moreover, in contrast to our case, the operator A in
[6] is assumed to be bounded weakly positive with a compact inverse
operator A−1. In [7], the authors continue their study for problems
from [6] but with a non-constant operator A = A(t). Some additional
restrictions on A(t) are applied.
60 Y. Eidelman, Ya. Yakubov
2. The statement of the problem
In a Banach space X, we consider the differential-operator equation
d2v
dt2
= Av + f(t), 0 < t < T (1)
with the boundary value conditions{
L1(v) := α11v(0) + α12v
′(0) + β11v(T ) + β12v
′(T ) = 0,
L2(v) := α21v(0) + α22v
′(0) + β21v(T ) + β22v
′(T ) = 0.
(2)
Here, A ia a closed linear unbounded operator with domain D(A),
f(t) is a continuous on [0, T ] vector valued function, the coeffici-
ents αij and βij are complex numbers. It is assumed that the forms
L1(v), L2(v) are linearly independent. By a solution of the problem
(1), (2) we mean a continuously differentiable on [0, T ] function v(t)
which takes the values in D(A), has a continuous on (0, T ) second
order derivative and satisfies (1), (2).
We use the eigenfunctions and eigenvalues of the ordinary di-
fferential operator of the second order defined by
L(y)(t) = −d
2y
dt2
, 0 < t < T,
L1(y) = 0, L2(y) = 0.
(3)
The operator L is defined on continuously differentiable on [0, T ]
functions y(t), which have a continuous on (0, T ) second order deri-
vative y′′(t), with y′′(t) ∈ L2(0, T ), satisfying the boundary value
conditions (2).
3. The conditions
Here, we present the main conditions on the data of the problem
which are used in the paper. Below, C means a positive constant.
The conditions on the ordinary differential operator L in (3) are the
Two-point boundary value problems... 61
following:
(α1) The operator L, treated as an operator in L2(0, T ), is a
symmetric operator.
(α2) The operator L has a complete, in L2(0, T ), orthonormal
system ϕn(t), n = 1, 2, . . . of eigenfunctions.
(α3) The eigenvalues λn, n = 1, 2, . . . of the operator L satisfy
the relation
lim inf
n→∞
|λn|
n2
> 0.
(α4) There is a sequence of numbers sn, n = 1, 2, . . . such that
limn→∞ sn =∞ and
|sn − sn+1| ≤ C,
∣∣∣∣λnsn
∣∣∣∣ ≤ Cn, ∣∣∣∣λn+1
sn+1
− λn
sn
∣∣∣∣ ≤ C, n = 1, 2, . . . .
Note that the typical cases for condition (α4) are λn = s2n or
λn = −s2n.
The common property of the operators −A and L is that their
spectra are disjoint with the following additional condition:
(β) All the numbers −λn, n = 1, 2, . . . are regular points of the
operator A and, moreover, there exists a constant M > 0 such that
the inequalities
‖λn(A+ λnI)
−1‖ ≤M, n = 1, 2, . . .
hold.
To formulate the conditions on the function f(t) in the right hand
side in (1), we define the “Fourier coefficients”
fn =
∫ T
0
f(t)ϕn(t) dt, n = 1, 2, . . . . (1)
The conditions are the following:
(γ) The inequalities
‖fnϕn(t)‖ ≤ C, ‖fnϕ′n(t)‖ ≤ C, n = 1, 2, . . . , 0 ≤ t ≤ T (2)
62 Y. Eidelman, Ya. Yakubov
and
‖
n∑
k=1
skfkϕk(t)‖ ≤ C(δ, γ), n = 1, 2, . . . , δ ≤ t ≤ γ (3)
for any δ, γ, with 0 < δ < γ < T , and with si defined in the condition
(α4), hold.
In particular cases, presented in two last sections, the condition
(γ) turns out to be valid if a more explicit condition holds:
(γ0) The function f(t) has, on [0, T ], the derivative which satisfies
the Hölder condition, i.e., for some C > 0, 0 < α ≤ 1,
‖f ′(t)− f ′(s)‖ ≤ C|t− s|α, ∀t, s ∈ [0, T ].
Let v(t) be a solution of the problem (1), (2) from section 2, if it
exists. Set
vn =
∫ T
0
v(t)ϕn(t) dt, n = 1, 2, . . . (4)
The following lemma yields the connection between the ”Fourier
coefficients” of the function f(t) in the right hand side of the equation
and of the solution v(t).
Lemma 1. Assume that the condition (α1) holds and that v(t) is a
solution of the problem (1), (2) from section 2 with the given function
f(t).
Then, the equalities
(λnI +A)vn = −fn, n = 1, 2, . . . (5)
hold with vn, fn as in (1), (4), and λn to be the eigenvalues of the
operator L.
Proof. Multiplying the equality (1) from section 2 by ϕn(t) and
integrating from 0 to T we get∫ T
0
d2v
dt2
ϕn(t) dt = Avn + fn, n = 1, 2, . . . .
Two-point boundary value problems... 63
Integrating by parts in the left hand side and using the condition
(α1) and the fact that ϕn(t) is an eigenfunction of the operator L
with the eigenvalue λn, we obtain (5). �
4. The uniqueness theorem
At first, we consider the uniqueness criteria for the problem (1), (2)
from section 2, i.e., for the homogeneous problem
d2v
dt2
= Av, 0 < t < T,
L1(v) = 0, L2(v) = 0.
(1)
Theorem 2. Assume that the conditions (α1), (α2) hold.
The problem (1) has only a trivial solution if and only if each
eigenvalue of the operator L in (3) from section 2 is not an eigenvalue
of the operator −A.
Proof. Assume that λ is a common eigenvalue of the operators
−A and L with the corresponding eigenvector g ∈ X of A and ei-
genfunction ϕ(t) of L. Then, the function v(t) = ϕ(t)g is a nontrivial
solution of the problem (1).
Assume now that the sets of eigenvalues of the operators L and
−A are disjoint. Let v(t) be a solution of the problem (1). Using
Lemma 1, we get
−λnvn = Avn, n = 1, 2, . . . .
Since the operators λnI + A, n = 1, 2, . . . are injective, we get
vn = 0, n = 1, 2, . . . and, therefore, since the system {ϕn(t)} is
complete, we conclude that v(t) = 0, 0 ≤ t ≤ T . �
5. The existence and uniqueness theorems
In this section, we present the basic results of the paper. First, we
show that the existence of the unique solution of the problem for
64 Y. Eidelman, Ya. Yakubov
any admissible function f(t) implies that all the points λn, defined
above, are regular points of the operator −A.
Theorem 3. Assume that the conditions (α1), (α2) hold and the
problem (1), (2) from section 2 has a unique solution for any conti-
nuous f(t) satisfying the condition (γ).
Then, every eigenvalue λn, n = 1, 2, . . . of the operator L is a
regular point of the operator −A.
Proof. For any positive integer m and for any x ∈ X take f(t) =
ϕm(t)x. Let v(t) be the corresponding solution of the problem. Consi-
der the expantion of v(t) in the form
v(t) =
∞∑
n=1
vnϕn(t) (1)
with coefficients vn defined by (4) from section 3. Using Lemma 1,
we obtain the equalities (5) from section 3 with fm = x and fn =
0, n 6= m. From the uniqueness of the solution, using Theorem 2, we
conclude that all the numbers λn, n = 1, 2, . . . are not eigenvalues
of the operator −A. Hence, we get vn = 0, n 6= m and, moreover,
the equation
(λmI +A)vm = −x
has a unique solution vm. Since this holds for any x ∈ X we
conclude that −λm is a regular point of the operator A and
vm = −(λmI + A)−1x. Hence, it follows that v(t) = vmϕm(t) =
−(λmI +A)−1xϕm(t). �
We prove now, under some additional conditions on the resolvent
of the operator A at the points −λn, the unique solvability of the
problem.
Theorem 4. Assume that the conditions (α1)-(α4) and (β) hold.
Then, for any continuous f(t), satisfying the condition (γ), the
problem (1), (2) from section 2 has a unique solution.
Two-point boundary value problems... 65
Proof. Take any f(t) satisfying (γ). We consider the expression
v(t) = −
∞∑
n=1
(λnI +A)−1fnϕn(t), 0 ≤ t ≤ T (2)
with fn defined in (1) from section 3, and prove that v(t) is the
unique solution of the problem (1), (2) from section 2. Formally
differentiating, we get
v′(t) = −
∞∑
n=1
(λnI +A)−1fnϕ
′
n(t), 0 ≤ t ≤ T. (3)
The conditions (α3) and (β) and the inequalities in (2) from section 3
imply that the series in (2), (3) converge uniformly on [0, T ]. Hence,
it follows that v(t) is a continuously differentiable on [0, T ] function
with the derivative defined in (3). Moreover, since ϕn(t) satisfy the
boundary value conditions in (3) from section 2, the formulas (2), (3)
imply that the function v(t) satisfy the boundary value conditions
(2) from section 2.
Formally differentiating (2) twice and using the equality ϕ′′n(t) =
−λnϕn(t), we get
v′′(t) =
∞∑
n=1
λn(λnI +A)−1fnϕn(t). (4)
We now prove that the series in (4) converges uniformly on [δ, γ] for
any δ, γ ∈ (0, T ), δ < γ. This will imply that the formula (4) yields
the second derivative of the function v(t) in the interval (0, T ).
Set
an =
λn
sn
(λnI +A)−1, n = 1, 2, . . . .
Then, the series in (4) has the form
∑∞
n=1 ansnfnϕn(t). For any
δ, γ ∈ (0, T ), δ < γ we prove the uniform convergence of this series,
for δ ≤ t ≤ γ, using the Abel transform. To this end, we should
66 Y. Eidelman, Ya. Yakubov
check that
lim
n→∞
(s1f1ϕ1(t) + s2f2ϕ2(t) + · · ·+ snfnϕn(t))an = 0, δ ≤ t ≤ γ.
(5)
Indeed, by virtue of the condition (β), we get
‖an‖ ≤
C
|sn|
, n = 1, 2, . . . ,
which, together with the condition (3) from section 3 and the condi-
tion limn→∞ sn =∞, implies (5). Thus, the Abel transform implies
that the uniform, on [δ, γ], convergence of the series
∞∑
n=1
ansnfnϕn(t)
follows from the uniform convergence of the series
∞∑
n=1
(s1f1ϕ1(t) + s2f2ϕ2(t) + · · ·+ snfnϕn(t))(an − an+1) (6)
on the same segment. So, check the uniform convergence of (6). We
have
an − an+1 =
λn
sn
(λnI +A)−1 − λn+1
sn+1
(λn+1I +A)−1.
Hence, it follows that
an − an+1 = (λnI +A)−1
(
λn
sn
− λn+1
sn+1
)
+
λn+1
sn+1
((λnI +A)−1 − (λn+1I +A)−1).
The resolvent identity yields
(λnI+A)
−1−(λn+1I+A)
−1 = (λn+1−λn)(λnI+A)−1(λn+1I+A)
−1.
Two-point boundary value problems... 67
Substituting this into the previous equality, we get
an − an+1 = (λnI +A)−1
(
λn
sn
− λn+1
sn+1
)
+ (λn+1 − λn)(λnI +A)−1(λn+1I +A)−1
λn+1
sn+1
. (7)
Using (β) and (α3), (α4), and also the equality
λn+1 − λn = sn
(
λn+1
sn+1
− λn
sn
)
+
λn+1
sn+1
(sn+1 − sn),
we get, from (7),
‖an − an+1‖ ≤
C
n2
.
Hence, using (3) from section 3, we conclude that the series (6)
converges uniformly on t in [δ, γ]. Thus, v(t) has a continuous, on
(0, T ), second derivative which is defined by (4).
Now, using the formula
A(λnI +A)−1 = I − λn(λnI +A)−1,
we get, from (2), for 0 < t < T ,
Av(t) =
∞∑
n=1
λn(λnI +A)−1fnϕn(t)−
∞∑
n=1
fnϕn(t) = v′′(t)− f(t),
which implies that v(t) is a solution of the equation (1) from section
2. �
6. The periodic boundary value conditions
Consider a problem of the form (1), (2) from section 2 with periodic
boundary value conditions
d2v
dt2
= Av + f(t), 0 < t < 2π,
v(0)− v(2π) = 0, v′(0)− v′(2π) = 0.
(1)
68 Y. Eidelman, Ya. Yakubov
The corresponding operator L in (3) from section 2 has
a complete, in L2(0, 2π), orthonormal system of eigenfunctions
ϕn(t) =
1√
2π
eint, n = 0,±1,±2, . . . with eigenvalues λn = n2. The
sequence sn in (α4) is defined by sn = in, n = 0,±1,±2, . . . .
Assume that the operator A satisfies the condition (β). This
means that the numbers −n2, n = 0, 1, 2, . . . are regular points
of the operator A and the inequalities
‖n2(A+ n2I)−1‖ ≤M, n = 0, 1, 2, . . . (2)
hold.
Assume that the function f(t) in (1) satisfies the condition (γ0).
We check that the condition (γ) holds. The inequalities
‖fnϕn(t)‖ ≤ C, n = 0,±1,±2, . . . , 0 ≤ t ≤ 2π
are obvious. Integrating by parts in (1) from section 3, we get
fn =
1√
2π
∫ 2π
0
f(t)e−int dt =
1√
2π
(
i
f(2π)− f(0)
n
+
1
in
∫ 2π
0
f ′(t)e−int dt
)
. (3)
>From here, using the fact that the functions f(t) and f ′(t) are
bounded, we obtain the inequalities
‖fnϕ′n(t)‖ ≤ C, n = 0,±1,±2, . . . , 0 ≤ t ≤ 2π.
So, (2) from section 3 has been proved. It remains to check (3) from
section 3, i.e.,∥∥∥∥∥
n∑
k=−n
ikfke
ikt
∥∥∥∥∥ ≤ C, n = 0, 1, 2, . . . , 0 < δ ≤ t ≤ γ < 2π. (4)
Set g(t) = f ′(t). The Fourier coefficients of the function g(t) are
gk =
1√
2π
∫ 2π
0
f ′(t)e−ikt dt, k = 0,±1,±2, . . . .
Two-point boundary value problems... 69
The formula (3) implies
ikfk = gk −
f(2π)− f(0)√
2π
, k = 0,±1,±2, . . .
and, therefore,
n∑
k=−n
ikfke
ikt =
n∑
k=−n
gke
ikt+
f(0)− f(2π)√
2π
n∑
k=−n
eikt, n = 0, 1, 2, . . . .
or
‖
n∑
k=−n
ikfke
ikt‖ ≤ ‖
n∑
k=−n
gke
ikt‖+ ‖f(0)− f(2π)‖√
2π
∣∣∣∣∣
n∑
k=−n
eikt
∣∣∣∣∣ . (5)
The condition (γ0) implies that the Fourier series of the function
g(t) converges to g(t), uniformly on any [δ, γ] ⊂ (0, 2π). Hence,∥∥∥∥∥
n∑
k=−n
gke
ikt
∥∥∥∥∥ ≤ C, 0 < δ ≤ t ≤ γ < 2π, n = 0, 1, 2, . . . . (6)
Further, we have
n∑
k=−n
eikt =
sin (2n+1)t
2
sin t
2
and, therefore,∣∣∣∣∣
n∑
k=−n
eikt
∣∣∣∣∣ ≤ C, 0 < δ ≤ t ≤ γ < 2π. (7)
>From the relations (5)-(7), the relations in (4) follow.
Thus, for any operator A, satisfying the condition (2), and for
any f(t), satisfying the condition (γ0), the problem (1), by Theorem
4, has a unique solution.
70 Y. Eidelman, Ya. Yakubov
7. The Dirichlet boundary value conditions
Consider now a problem of the form (1), (2) from section 2 with
Dirichlet boundary value conditions
d2v
dt2
= Av + f(t), 0 < t < π,
v(0) = 0, v(π) = 0.
(1)
The corresponding operator L in (3) from section 2 has a
complete, in L2(0, π), orthonormal system of eigenfunctions ϕn(t) =√
2
π sinnt, n = 1, 2, . . . with eigenvalues λn = n2. The sequence sn
in (α4) is defined by sn = n, n = 1, 2, . . . .
Assume that the operator A satisfies the condition (β). This
means that the numbers −n2, n = 1, 2, . . . are regular points of
the operator A and the inequalities
‖n2(A+ n2I)−1‖ ≤M, n = 1, 2, . . . (2)
hold.
Assume that the function f(t) in (1) satisfies the condition (γ0).
We check that the condition (γ) holds. Integrating by parts in (1)
from section 3, we get
fn =
√
2
π
∫ π
0
f(t) sinnt dt =
√
2
π
(−f(π) cos(nπ) + f(0)
n
+
1
n
∫ π
0
f ′(t) cosnt dt
)
. (3)
>From (3), using the fact that the functions f(t) and f ′(t) are
bounded, we obtain (2) from section 3. It remains to check (3) from
section 3, i.e.,∥∥∥∥∥
n∑
k=1
kfkϕk(t)
∥∥∥∥∥ ≤ C, n = 1, 2, . . . , 0 < δ ≤ t ≤ γ < π. (4)
Two-point boundary value problems... 71
The relation (3) implies that
n∑
k=1
kfkϕk(t) =
2
π
n∑
k=1
(
f(0)− f(π) cos kπ
+
∫ π
0
f ′(s) cos ks ds
)
sin kt,
i.e.,
n∑
k=1
kfkϕk(t) =
2
π
n∑
k=1
f(0) sin kt+
2
π
n∑
k=1
f(π)(−1)k+1 sin kt
+
2
π
∫ π
0
f ′(s)
n∑
k=1
sin kt cos ks ds. (5)
Set
Dn(t) =
n∑
k=1
sin kt. (6)
The formula
n∑
k=1
sin kt =
hn(t)
sin t
2
, (7)
with hn(t) = sin n+1
2 t sin nt
2 , implies
|Dn(t)| ≤
C
| sin t
2 |
. (8)
The first entry in (5) has the form
2
π
n∑
k=1
f(0) sin kt =
2
π
f(0)Dn(t)
and, hence, the uniform boundedness of this entry on t ∈ [δ, γ] ⊂
(0, π), n = 1, 2, . . . follows from (8). The second entry in (5) has the
72 Y. Eidelman, Ya. Yakubov
form
2
π
n∑
k=1
f(π)(−1)k+1 sin kt = − 2
π
f(π)
n∑
k=1
sin(kt+ kπ)
= − 2
π
f(π)Dn(t+ π)
and, using (8), we get
‖ 2
π
n∑
k=1
f(π)(−1)k+1 sin kt‖ ≤ C
| sin t+π
2 |
=
C
| cos t2 |
and, hence, the uniform boundedness of this entry on t ∈ [δ, γ], n =
1, 2, . . . follows. Finally, consider the third term in (5). We have
2
π
∫ π
0
f ′(s)
n∑
k=1
sin kt cos ks ds =
1
π
∫ π
0
f ′(s)Dn(t+ s) ds
+
1
π
∫ π
0
f ′(s)Dn(t− s) ds. (9)
Using formula (7), we get
Dn(t+ s) =
hn(t+ s)
sin t+s
2
.
Here, it is clear that |hn(t+ s)| ≤ 1 and that
1/| sin t+ s
2
| ≤ C, 0 ≤ s ≤ π, 0 < δ ≤ t ≤ γ < π.
Since f ′(t) is a bounded function, we conclude that the first term in
(9) is uniformly bounded on t ∈ [δ, γ], n = 1, 2, . . . . Now, consider
the second term. We have
1
π
∫ π
0
f ′(s)Dn(t− s) ds =
1
π
∫ π
0
(f ′(s)− f ′(t))Dn(t− s) ds
+
1
π
f ′(t)
∫ π
0
Dn(t− s) ds. (10)
Two-point boundary value problems... 73
By the condition (γ0), we have
‖f ′(s)− f ′(t)‖ ≤ C|t− s|α, ∀t, s ∈ [0, π].
Using (8), we get
|Dn(t− s)| ≤
C
| t−s2 |
, t 6= s.
Thus, we conclude that
‖ 1
π
∫ π
0
(f ′(s)− f ′(t))Dn(t− s) ds‖ ≤ C.
Since f ′(t) is bounded, it is enough to check the boundedness of the
last integral in (10). Using (6), we get∫ π
0
Dn(t− s) ds =
n∑
k=1
∫ π
0
sin k(t− s) ds
=
n∑
k=1
cos(kt− kπ)− cos kt
k
= −2
p∑
m=0
cos(2m+ 1)t
2m+ 1
.
On the other hand,
∑n−1
m=0 cos(2m + 1)t = sin 2nt
2 sin t . Then,
limn→∞
∑n−1
m=0 cos(2m+1)t· 1
2n−1 = 0, 0 < δ ≤ t ≤ γ < π. Therefore,
by the Abel transform, the uniform convergence of
∑∞
m=0
cos(2m+1)t
2m+1 ,
0 < δ ≤ t ≤ γ < π, follows from uniform convergence (on the same
segment) of
∞∑
n=1
n−1∑
m=0
cos(2m+1)t·
( 1
2n− 1
− 1
2n+ 1
)
=
∞∑
n=1
sin 2nt
(2n− 1)(2n+ 1) sin t
which is true by the Weierstrass M -test. Thus, we conclude that
‖
∫ π
0
Dn(t− s) ds‖ ≤ C, n = 1, 2, . . . , 0 < δ ≤ t ≤ γ < π,
74 Y. Eidelman, Ya. Yakubov
which completes the proof of (4).
So, for any operator A, satisfying the condition (2), and for any
f(t), satisfying the condition (γ0), the problem (1), by Theorem 4,
has a unique solution.
Consider now the Dirichlet problem for the homogeneous equation
with non-homogeneous boundary value conditions
d2v
dt2
= Av, 0 < t < π,
v(0) = x0, v(π) = x1
(11)
with x0, x1 ∈ D(A). By a standard way, the problem (11) is reduced
to the problem (1) with f(t) = π−t
π Ax0+
t
πAx1. Obviously, the linear
function f(t) satisfies condition (γ0).
Лiтература
[1] Arendt V., Bu S. The operator-valued Marcinkiewicz multiplier
theorem and maximal regularity // Mathematische Zeitschrift. —
2002. — 240.— P. 311–343.
[2] Favini A., Shakhmurov V., Yakubov Ya. Regular boundary value
problems for complete second order elliptic differential-operator
equations in UMD Banach spaces // Semigroup Forum. — 2009. —
79. — P. 22–54.
[3] Favini A., Yakubov Ya. Regular boundary value problems for elliptic
differential-operator equations of the fourth order in UMD Banach
spaces// Scientiae Mathematicae Japonicae. — 2009. — 70. — P. 183–
204.
[4] Favini A., Yakubov Ya. Irregular boundary value problems for
second order elliptic differential-operator equations in UMD Banach
spaces // Mathematische Annalen. — 2010. — 348. — P. 601–632.
[5] Favini A., Yakubov Ya. Regular boundary value problems for ordi-
nary differential-operator equations of higher order in UMD Banach
spaces // Discrete and Continuous Dynamical Systems. — 2011. — 4,
No. 3. — P. 595–614.
Two-point boundary value problems... 75
[6] Gershtein L. M., Sobolevskii P. E. Coercive solvability of general
boundary-value problems for second-order elliptic equations in
Banach space // Differentsial’nye Uravneniya. — 1974. — 10,
No. 11. — P. 2059–2061.
[7] Gershtein L. M., Sobolevskii P. E. Coercive solvability of general
boundary-value problems for second-order elliptic equations in
Banach space. II // Differentsial’nye Uravneniya. —1975 — 11,
No. 7. — P. 1335–1337.
[8] Knyazyuk A. V. The Dirichlet problem for second-order differential
equations with operator coefficients// Ukrain. Math. Zh. — 1985. —
37, No. 2. — P. 256–260.
[9] Yakubov S., Yakubov Ya. Differential-Operator Equations. Ordinary
and Partial Differential Equations. — Boca Raton: Chapman and
Hall/CRC, 2000. — 568 p.
|
| id | oai:trim.imath.kiev.ua:article-207 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-08-04T01:04:03Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/53/1222310b082c87c380f2983e4afe7753.pdf |
| spelling | oai:trim.imath.kiev.ua:article-2072018-01-23T01:38:29Z Two-point boundary value problems for differential-operator equations Two-point boundary value problems for differential-operator equations Eidelman, Y. Yakubov, Ya. Eidelman, Y. Yakubov, Ya. We study general two-point boundary value problems for a non-homogeneous differential-operator equation of the second order with an unbounded linear operator in a Banach space. The main classical solvability condition is given in terms of the property of the resolvent of the operator at the points, which are opposite to the eigenvalues of the corresponding ordinary differential operator. At the end of the paper, two particular types of boundary value conditions are treated: periodic and Dirichlet. We study general two-point boundary value problems for a non-homogeneous differential-operator equation of the second order with an unbounded linear operator in a Banach space. The main classical solvability condition is given in terms of the property of the resolvent of the operator at the points, which are opposite to the eigenvalues of the corresponding ordinary differential operator. At the end of the paper, two particular types of boundary value conditions are treated: periodic and Dirichlet. Інститут математики НАН України 2016-05-16 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/207 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 13 No. 1 (2016): Differential equations and related problems of analysis; 58-75 Сборник Трудов Института математики НАН Украины; Том 13 № 1 (2016): Диференціальні рівняння і суміжні питання аналізу; 58-75 Збірник Праць Інституту математики НАН України; Том 13 № 1 (2016): Диференціальні рівняння і суміжні питання аналізу; 58-75 3083-7529 1815-2910 uk en https://trim.imath.kiev.ua/index.php/trim/article/view/207/184 Авторське право (c) 2016 Y. Eidelman, Ya. Yakubov |
| spellingShingle | Eidelman, Y. Yakubov, Ya. Eidelman, Y. Yakubov, Ya. Two-point boundary value problems for differential-operator equations |
| title | Two-point boundary value problems for differential-operator equations |
| title_alt | Two-point boundary value problems for differential-operator equations |
| title_full | Two-point boundary value problems for differential-operator equations |
| title_fullStr | Two-point boundary value problems for differential-operator equations |
| title_full_unstemmed | Two-point boundary value problems for differential-operator equations |
| title_short | Two-point boundary value problems for differential-operator equations |
| title_sort | two-point boundary value problems for differential-operator equations |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/207 |
| work_keys_str_mv | AT eidelmany twopointboundaryvalueproblemsfordifferentialoperatorequations AT yakubovya twopointboundaryvalueproblemsfordifferentialoperatorequations AT eidelmany twopointboundaryvalueproblemsfordifferentialoperatorequations AT yakubovya twopointboundaryvalueproblemsfordifferentialoperatorequations |