On the Solvability of a Class of Operator Differential Equations of the Second Order on the Real Axis
Solvability of the operator differential equation of the second order with variable coe cients on the real axis in a certain weight space is studied. The main part of the equation is an abstract elliptic equation in Hilbert space. We note that sufficient conditions on operator coeffiients of the per...
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| Опубліковано в: : | Журнал математической физики, анализа, геометрии |
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| Дата: | 2006 |
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| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
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| Цитувати: | On the Solvability of a Class of Operator Differential Equations of the Second Order on the Real Axis / A.R. Aliev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 4. — С. 347-357. — Бібліогр.: 5 назв. — англ. |
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Aliev, A.R. 2016-10-02T18:26:58Z 2016-10-02T18:26:58Z 2006 On the Solvability of a Class of Operator Differential Equations of the Second Order on the Real Axis / A.R. Aliev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 4. — С. 347-357. — Бібліогр.: 5 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106674 Solvability of the operator differential equation of the second order with variable coe cients on the real axis in a certain weight space is studied. The main part of the equation is an abstract elliptic equation in Hilbert space. We note that sufficient conditions on operator coeffiients of the perturbed part, preserving ellipticity of the equation, are found in the paper, and estimations of the norms of intermediate derivative operators via the main part of the operator differential equation in a certain weight space are also obtained. The Author is thankful to Acad. of NAS of Azerbaijan M.G. Gasimov and Prof. S.S. Mirzoyev for discussion of the paper. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии On the Solvability of a Class of Operator Differential Equations of the Second Order on the Real Axis Article published earlier |
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On the Solvability of a Class of Operator Differential Equations of the Second Order on the Real Axis |
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On the Solvability of a Class of Operator Differential Equations of the Second Order on the Real Axis Aliev, A.R. |
| title_short |
On the Solvability of a Class of Operator Differential Equations of the Second Order on the Real Axis |
| title_full |
On the Solvability of a Class of Operator Differential Equations of the Second Order on the Real Axis |
| title_fullStr |
On the Solvability of a Class of Operator Differential Equations of the Second Order on the Real Axis |
| title_full_unstemmed |
On the Solvability of a Class of Operator Differential Equations of the Second Order on the Real Axis |
| title_sort |
on the solvability of a class of operator differential equations of the second order on the real axis |
| author |
Aliev, A.R. |
| author_facet |
Aliev, A.R. |
| publishDate |
2006 |
| language |
English |
| container_title |
Журнал математической физики, анализа, геометрии |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
Solvability of the operator differential equation of the second order with variable coe cients on the real axis in a certain weight space is studied. The main part of the equation is an abstract elliptic equation in Hilbert space. We note that sufficient conditions on operator coeffiients of the perturbed part, preserving ellipticity of the equation, are found in the paper, and estimations of the norms of intermediate derivative operators via the main part of the operator differential equation in a certain weight space are also obtained.
|
| issn |
1812-9471 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/106674 |
| citation_txt |
On the Solvability of a Class of Operator Differential Equations of the Second Order on the Real Axis / A.R. Aliev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 4. — С. 347-357. — Бібліогр.: 5 назв. — англ. |
| work_keys_str_mv |
AT alievar onthesolvabilityofaclassofoperatordifferentialequationsofthesecondorderontherealaxis |
| first_indexed |
2025-11-25T15:34:09Z |
| last_indexed |
2025-11-25T15:34:09Z |
| _version_ |
1850516908397297664 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2006, vol. 2, No. 4, pp. 347�357
On the Solvability of a Class of Operator Di�erential
Equations of the Second Order on the Real Axis
A.R. Aliev
Baku State University
23 Z. Khalilov Str., Baku, AZ-1148, Azerbaijan
Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan
9 F. Agaev Str., Baku, AZ-1141, Azerbaijan
E-mail:alievaraz@yahoo.com
Received December 15, 2005
Solvability of the operator di�erential equation of the second order with
variable coe�cients on the real axis in a certain weight space is studied. The
main part of the equation is an abstract elliptic equation in Hilbert space.
We note that su�cient conditions on operator coe�cients of the perturbed
part, preserving ellipticity of the equation, are found in the paper, and
estimations of the norms of intermediate derivative operators via the main
part of the operator di�erential equation in a certain weight space are also
obtained.
Key words: operator di�erential equation, selfadjoint operator, discon-
tinuous coe�cient, regular solution, Hilbert space.
Mathematics Subject Classi�cation 2000: 47E05, 34B05.
Let A be a selfadjoint positive de�nite operator in the separable Hilbert
space H.
We denote by L2;�(R;H), R = (�1; +1) Hilbert space ofH-valued functions
de�ned in R with the norm
kuk
L2;�(R;H) =
0@ +1Z
�1
ku(t)k2
H
e��tdt
1A1=2
; � 2 R:
We denote by W 2
2;�(R;H) the space of H-valued functions, such that
d2u(t)
dt2
2 L2;�(R;H); A2u(t) 2 L2;�(R;H);
c
A.R. Aliev, 2006
A.R. Aliev
with the norm
kuk
W
2
2;�
(R;H) =
d2udt2
2
L2;�(R;H)
+
A2u
2
L2;�(R;H)
!1=2
; � 2 R:
It is obvious that for � = 0 we have the spaces L2;0(R;H) = L2(R;H),W 2
2;0(R;H)
= W 2
2 (R;H) described in details in [1, Ch. 1].
Here and below derivatives are considered in the sense of the theory of gene-
ralized functions.
We introduce the following notations: L(X;Y ) is the set of linear bounded
operators acting from Hilbert space X to another Hilbert space Y , and L1(R;B)
is the set of B-valued essentially bounded operator functions in R; where B is a
Banach space.
Now let us formulate the statement of the problem to be studied. Consider
the operator di�erential equation of the second order of the form
�
d2u(t)
dt2
+A2u(t) +A1(t)
du(t)
dt
+A2(t)u(t) = f(t); t 2 R; (1)
where f(t) 2 L2;�(R;H), u(t) 2 W 2
2;�(R;H), A1(t) and A2(t) are, in general,
linear unbounded operators, de�ned for almost all t 2 R.
De�nition 1. A vector function u 2 W 2
2;�(R;H); satisfying equation (1) al-
most everywhere for a given f(t) 2 L2;�(R;H); is called a regular solution of
equation (1).
De�nition 2. If a regular solution of equation (1) exists for any f(t) 2
L2;�(R;H); and the inequality
kuk
W
2
2;�
(R;H) � const kfk
L2;�(R;H)
holds, then equation (1) is called regularly solvable.
In this paper we suggest conditions on the operator coe�cients of equation
(1) to provide its regular solvability. We note that the main part of the studied
equation (1) is the abstract elliptic equation [2] in the Hilbert space. Taking into
account the above, the aim of this paper is to �nd conditions on the operator
coe�cients of the perturbed part of equation (1) that preserve ellipticity of this
equation.
We should note that equation (1) was considered in [3�5], where su�cient
conditions of its solvability in Hilbert spaces without weight were obtained. Ana-
logous problems in the weight space, when the operators of the perturbed part
348 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Solvability of a Class of Operator Di�erential Equations...
of the equation were the powers of the operator A, multiplied by the complex
numbers, were investigated in [2].
Consider the operator di�erential equation
�
d2u(t)
dt2
+A2u(t) = f(t); t 2 R; (2)
where f(t) 2 L2;�(R;H); u(t) 2W 2
2;�(R;H):
Denote by �0 the operator acting from the space W 2
2;�(R;H) to L2;�(R;H) in
the following way:
�0u(t) = �
d2u(t)
dt2
+A2u(t); u(t) 2W 2
2;�(R;H):
Then the following theorem showing a link of solvability of equation (2)* with
the lower bound of the spectrum of operator A is valid.
Theorem 1. Let A be a selfadjoint positive de�nite operator with the lower
bound �0 of the spectrum, i.e., A = A� � �0E (�0 > 0), E is a unit operator and
the number � 2 R satis�es the condition j�j < 2�0. Then �0 is an isomorphism
between W 2
2;�(R;H) and L2;�(R;H).
P r o o f. In the equation �0u(t) = f(t), u(t) 2W 2
2;�(R;H), f(t) 2 L2;�(R;H)
we substitute u(t) = #(t)e
�
2
t, then #(t) = u(t)e�
�
2
t 2 L2(R;H). Since
�
d2u(t)
dt2
+A2u(t) = �
�
d
dt
+
�
2
�2
#(t)e
�
2
t +A2#(t)e
�
2
t = f(t);
we obtain
�
�
d
dt
+
�
2
�2
#(t) +A2#(t) = f(t)e�
�
2
t: (3)
Having g(t) = f(t)e�
�
2
t 2 L2(R;H); we can write (3) in the form
�
�
d
dt
+
�
2
�2
#(t) +A2#(t) = g(t) (4)
in the space L2(R;H), i.e., #(t) 2W 2
2 (R;H), g(t) 2 L2(R;H).
*Note that equation (2) is of the type considered in [2, p. 17], however our theorem does not
follow from Th. 5.5, since in our paper we provide a certain weight index and the problem is
considered on the axis R, while in [2] the problem is studied on the semi-axis R+. Besides, the
conditions on the right-hand side of the equation and the proof are di�erent.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 349
A.R. Aliev
Now let us denote by
�0;�#(t) = �
�
d
dt
+
�
2
�2
#(t) +A2#(t); #(t) 2W 2
2 (R;H):
In this case equation (4) can be rewritten in the form �0;�#(t) = g(t) where
#(t) 2 W 2
2 (R;H), g(t) 2 L2(R;H). To solve this equation we do the Fourier
transformation in (4)�
�
�
�i� +
�
2
�2
E +A2
� b#(�) = bg(�); (5)
where b#(�); bg(�) are Fourier transforms of the vector functions #(t); g(t) respec-
tively. Let us prove that for j�j < 2�0 the operator pencil
�0;�(�i�;A) = �
�
�i� +
�
2
�2
E +A2 (6)
is invertible. Really, let � 2 �(A) (� � �0), then characteristic polynomial of (6)
has the following form:
�0;�(�i�;�) = �
�
�i� +
�
2
�2
+ �2 = �2 + i���
�2
4
+ �2:
Hence we obtain that
j�0;�(�i�;�)j =
�����2 + i���
�2
4
+ �2
����
=
"�
�2 �
�2
4
+ �2
�2
+ �2�2
#1=2
� �2 �
�2
4
+ �2
� �2 �
�2
4
� �20 �
�2
4
> 0;
i.e., it follows from the spectral decomposition of the operator A that the operator
pencil �0;�(�i�;A) is invertible for j�j < 2�0. So we can �nd b#(�) from (5):
b#(�) = ����i� + �
2
�2
E +A2
�
�1 bg(�): (7)
We obtain
#(t) =
1
p
2�
+1Z
�1
�
�
�
�i� +
�
2
�2
E +A2
��1 bg(�)ei�td�:
350 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Solvability of a Class of Operator Di�erential Equations...
It is obvious that #(t) satis�es equation (4) almost everywhere. Now we prove
that #(t) 2W 2
2 (R;H). Really, by the Plancherel theorem it su�ces to show that
A2b#(�) 2 L2(R;H) and �2b#(�) 2 L2(R;H). It is clear that
k#k2
W
2
2
(R;H) =
d2#dt2
2
L2(R;H)
+
A2#
2
L2(R;H)
=
�2b#(�)
2
L2(R;H)
+
A2b#(�)
2
L2(R;H)
:
Since
A2b#(�)
L2(R;H)
=
A2
�
�
�
�i� +
�
2
�2
E +A2
��1 bg(�)
L2(R;H)
� sup
�2R
A2
�
�
�
�i� +
�
2
�2
E +A2
�
�1
kbg(�)kL2(R;H) ;
we can estimate the norm
A2
�
�
�
�i� + �
2
�2
E +A2
��1
for � 2 R. It follows
from the spectral theory of selfadjoint operators that
A2
�
�
�
�i� +
�
2
�2
E +A2
��1
= sup
�2�(A)
������2
�
�
�
�i� +
�
2
�2
+ �2
��1�����
= sup
�2�(A)
�2��
�2 � �
2
4
+ �2
�2
+ �2�2
�1=2
� sup
�2�(A)
�2
�2 � �
2
4
+ �2
� sup
�2�(A)
�2
�2 � �
2
4
�
�20
�20 �
�
2
4
:
Hence,
A2
�
�
�
�i� +
�
2
�2
E +A2
��1 bg(�)
L2(R;H)
� b0(�) kbg(�)k
L2(R;H) ;
where
b0(�) =
4�20
4�20 � �2
: (8)
Similarly, we have
�2b#(�)
L2(R;H)
=
�2
�
�
�
�i� +
�
2
�2
E +A2
��1 bg(�)
L2(R;H)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 351
A.R. Aliev
� sup
�2R
�2
�
�
�
�i� +
�
2
�2
E +A2
�
�1
kbg(�)kL2(R;H) :
Then for � 2 R and j�j < 2�0 we obtain that
�2
�
�
�
�i� +
�
2
�2
E +A2
�
�1
= sup
�2�(A)
������2
�
�
�
�i� +
�
2
�2
+ �2
�
�1
�����
� sup
�2�(A)
�2
�2 � �
2
4
+ �2
�
�2
�2 + �20 �
�
2
4
� 1:
We conclude
�2b#(�)
L2(R;H)
� kbg(�)k
L2(R;H) :
Consequently, we have found #(t) 2 W 2
2 (R;H): It is obvious that the vector
function #(t)e
�
2
t 2 L2;�(R;H) and it is a regular solution of equation (2).
On the other hand, the operator �0 is bounded fromW 2
2;�(R;H) into the space
L2;�(R;H). Really,
k�0uk2
L2;�(R;H) =
�d2udt2 +A2u
2
L2;�(R;H)
� 2 kuk2
W
2
2;�
(R;H) :
Hence, the operator �0 : W 2
2;�(R;H) ! L2;�(R;H) is one-to-one and bounded.
Then it follows from the Banach theorem on the inverse operator that the operator
��10 : L2;�(R;H) ! W 2
2;�(R;H) is bounded. Therefore �0 is an isomorphism
between the spaces W 2
2;�(R;H) and L2;�(R;H). Theorem 1 is proved.
The theorem shows that the norm k�0uk
L2;�(R;H) is equivalent to the norm
kuk
W
2
2;�
(R;H) in the space W 2
2;�(R;H) and, since it is known that the operators of
intermediate derivatives
A2�j d
j
dtj
: W 2
2;�(R;H)! L2;�(R;H); j = 0; 1;
are continuous, the norms of these operators can be estimated via k�0uk
L2;�(R;H) :
The following theorem is true.
Theorem 2. The following inequalities are valid for any u(t) 2 W 2
2;�(R;H)
and j�j < 2�0 :
A2u
L2;�(R;H)
� b0(�) k�0uk
L2;�(R;H) ; (9)
Adudt
L2;�(R;H)
� b1(�) k�0ukL2;�(R;H) ; (10)
352 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Solvability of a Class of Operator Di�erential Equations...
where b0(�) is de�ned in (8), and
b1(�) =
8>><>>:
�0
21=2(2�20 � �2)1=2
; if 0 �
�2
4�20
<
1
3
;
2�0j�j
4�20 � �2
; if
1
3
�
�2
4�20
< 1:
(11)
P r o o f. If we denote by #(t) = u(t)e�
�
2
t, then inequality (9) has the following
form:
A2#
L2(R;H)
� b0(�) k�0;�#k
L2(R;H)
: (12)
It follows from equality (7) that (bg(�) = �0;�
b#(�)) and
A2b#(�)
L2(R;H)
� b0(�)
�0;�
b#(�)
L2(R;H)
;
which is equivalent to inequality (12). That's why inequality (9) is true. Let
us prove inequality (10). Substituting u(t) = #(t)e
�
2
t; inequality (10) can be
rewritten in the equivalent form
A� d
dt
+
�
2
�
#
L2(R;H)
� b1(�) k�0;�#k
L2(R;H)
: (13)
We show that (13) is true. Substituting �0;�#(t) = g(t) and applying the Fourier
transformation, we obtain
A��i� + �
2
�
��10;�(�i�;A)bg(�)
L2(R;H)
=
A��i� + �
2
��
�
�
�i� +
�
2
�2
E +A2
��1 bg(�)
L2(R;H)
� sup
�2R
A��i� + �
2
��
�
�
�i� +
�
2
�2
E +A2
�
�1
kbg(�)kL2(R;H) : (14)
Thus we estimate the following norm for � 2 R
A��i� + �
2
��
�
�
�i� +
�
2
�2
E +A2
��1
= sup
�2�(A)
��������i� + �
2
��
�
�
�i� +
�
2
�2
+ �2
�
�1
�����
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 353
A.R. Aliev
= sup
�2�(A)
�
�
�2 + �
2
4
�1=2
��
�2 + �2 � �
2
4
�2
+ �2�2
�1=2
� sup
�2�(A)
�
�
�2 + �
2
4
�1=2
�2 + �2 � �
2
4
� sup
�2�(A)
�
�
2
�
2 +
�
2
4�2
0
�1=2
�
2
�
2 + 1� �
2
4�2
0
� sup
r�0
�
r + �
2
4�2
0
�1=2
r + 1� �
2
4�2
0
:
We denote
�(r) =
�
r + �
2
4�2
0
�1=2
r + 1� �
2
4�2
0
; r � 0; j�j < 2�0:
Then,
�0(r) =
1
2
�
r + �
2
4�2
0
��1=2 �
1� r � 3�2
4�2
0
�
�
r + 1� �
2
4�2
0
�2 :
It is clear that if
1
3
�
�2
4�20
< 1;
then �0(r) < 0; because �(r) attains its maximum at zero, i.e., if
1
3
�
�2
4�20
< 1;
then
max �(r) = �(0) =
2�0j�j
4�20 � �2
:
If 0 � �
2
4�2
0
< 1
3
; then the function �(r) attains its maximum at r0 = 1 � 3�2
4�2
0
: In
this case
max �(r) = �(r0) =
�0
21=2(2�20 � �2)1=2
:
Taking into account the expressions in (14), we obtain
A��i� + �
2
�
��10;�(�i�;A)bg(�)
L2(R;H)
� b1(�) kbg(�)k
L2(R;H) ;
which is equivalent to inequality (13) and which, in its turn, is equivalent to the
inequality
Adudt
L2;�(R;H)
� b1(�) k�0uk
L2;�(R;H) ;
354 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Solvability of a Class of Operator Di�erential Equations...
where
b1(�) =
8>><>>:
�0
21=2(2�20 � �2)1=2
for 0 �
�2
4�20
<
1
3
;
2�0j�j
4�20 � �2
for
1
3
�
�2
4�20
< 1:
Theorem 2 is proved.
Remark 1. The operator �0 is not invertible for � = �2�0.
Before obtaining the exact conditions of regular solvability for the operator
di�erential equation (1), we formulate the following lemma.
Lemma 1. Suppose that the conditions of Th. 1 are satis�ed and the operators
S
j
(t) = A
j
(t)A�j 2 L1(R;L(H;H)), j = 1; 2. Then the operator �, correspon-
ding to the left-hand side of equation (1), is a continuous map from the space
W 2
2;�(R;H) into L2;�(R;H).
P r o o f. Since for any u(t) 2W 2
2;�(R;H)
k�uk
L2;�(R;H) � k�0uk
L2;�(R;H) +
2X
j=1
A
j
(t)
d2�ju
dt2�j
L2;�(R;H)
� k�0uk
L2;�(R;H) +
2X
j=1
sup
t
kS
j
(t)k
H!H
Aj
d2�ju
dt2�j
L2;�(R;H)
;
then, taking into account Th. 1 and the theorem on the intermediate derivatives
[1, Ch. 1], we have from the last inequality
k�uk
L2;�(R;H) � const kuk
W
2
2;�
(R;H) :
Lemma is proved.
Now we prove the main theorem of the paper, i.e., the theorem on the regular
solvability of equation (1). It should be noted that conditions of solvability are
expressed only in terms of the operator coe�cients of (1).
Theorem 3. Suppose that the conditions of Lem. 1 are satis�ed, and
2X
j=1
c
j
(�) sup
t
kS
j
(t)k < 1;
where the numbers c
j
(�) = b2�j(�), j = 1; 2, are de�ned in Th. 2. Then equation
(1) is regularly solvable.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 355
A.R. Aliev
P r o o f. We present equation (1) in the form of the following operator
equation
�0u(t) + (�� �0)u(t) = f(t); (15)
where f(t) 2 L2;�(R;H); u(t) 2W 2
2;�(R;H):
The equation �0u(t) = f(t) is regularly solvable by Th. 1. After substitution
�0u(t) = w(t), equation (15) can be written in the form (E+(���0)�
�1
0 )w(t) =
f(t): Then for any w(t) 2 L2;�(R;H) we have by Th. 2:
(�� �0)�
�1
0 w
L2;�(R;H)
= k(�� �0)uk
L2;�(R;H)
=
2X
j=1
A
j
(t)
d2�ju
dt2�j
L2;�(R;H)
�
2X
j=1
Aj
(t)
d2�ju
dt2�j
L2;�(R;H)
�
2X
j=1
sup
t
kS
j
(t)k
H!H
Aj
d2�ju
dt2�j
L2;�(R;H)
�
2X
j=1
sup
t
kS
j
(t)k
H!H
c
j
(�) k�0uk
L2;�(R;H)
=
2X
j=1
c
j
(�) sup
t
kS
j
(t)k
H!H
kwk
L2;�(R;H) :
Since
2P
j=1
c
j
(�) sup
t
kS
j
(t)k
H!H
< 1, the operator E+(���0)�
�1
0 is invertible in
the space L2;�(R;H): Then, u(t) can be de�ned by the following formula
u(t) = ��10 (E + (�� �0)�
�1
0 )�1f(t):
It follows that
kuk
W
2
2;�
(R;H) � const kfk
L2;�(R;H) :
Theorem 3 is proved.
Corollary 1. Assume that � = 0 and the inequality
1
2
sup
t
A1(t)A
�1
+ sup
t
A2(t)A
�2
< 1
holds. Then the operator � is an isomorphism between the spaces W 2
2 (R;H) and
L2(R;H) (see [4, 5]).
The Author is thankful to Acad. of NAS of Azerbaijan M.G. Gasimov and
Prof. S.S. Mirzoyev for discussion of the paper.
356 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Solvability of a Class of Operator Di�erential Equations...
References
[1] J.-L. Lions and E. Madjenes, Nonhomogeneous Boundary-Value Problems and their
Applications. Mir, Moscow, 1971. (Russian)
[2] Yu.A. Dubinskii, Certain Di�erential Operator Equations of Arbitrary Order. �
Mat. Sb. 90(132) (1973), No. 1, 3�22. (Russian)
[3] S.Ya. Yakubov, Linear Di�erential Operator Equations and their Applications.
Baku, Elm, 1985. (Russian)
[4] S.S. Mirzoyev, On the Correct Solvability of Boundary-Value Problems for the
Operator-Di�erential Equations in Hilbert Space. � Dep. VINITI, 03.06.91,
No. 26708-B91, 1991, 46 p. (Russian)
[5] S.S. Mirzoyev, Questions of Solvability Theory of the Boundary-Value Problems
for the Operator-Di�erential Equations in Hilbert Space and Spectral Problems,
Connected with them. Diss. ... d. ph.-m. sci., BSU, Baku, 1994, 229 p. (Russian)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 357
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