566
ISSN 1064-5624, Doklady Mathematics, 2016, Vol. 94, No. 2, pp. 566–568. © Pleiades Publishing, Ltd., 2016.
Original Russian Text © S.S. Mirzoev, A.R. Aliev, G.M. Gasimova, 2016, published in Doklady Akademii Nauk, 2016, Vol. 470, No. 5, pp. 511–513.
Solvability Conditions of a Boundary Value Problem
with Operator Coefficients and Related Estimates of the Norms
of Intermediate Derivative Operators
S. S. Mirzoev
a,b
*, A. R. Aliev
b,c
**, and G. M. Gasimova
a
Presented by Academician of the Russian Academy of Sciences V.A. Sadovnichii November 20, 2015
Received January 12, 2016
Abstract—Sufficient conditions for the proper and unique solvability in the Sobolev space of vector functions
of the boundary value problem for a certain class of second-order elliptic operator differential equations on a
semiaxis are obtained. The boundary condition at zero involves an abstract linear operator. The solvability
conditions are established by using properties of operator coefficients. The norms of intermediate derivative
operators, which are closely related to the solvability conditions, are estimated.
DOI: 10.1134/S1064562416050239
1. Let be a separable Hilbert space, and let be
a normal operator with completely continuous inverse
whose spectrum is contained in the angular sector
, .
If , , are the eigenvalues of
and , , form a complete orthonormal sys-
tem of eigenvectors, then, for , we have
and the operator under consideration can be repre-
sented in the form , where
and
H A
-1
A
{
ε
= λ λ≤ε : arg } S
π
≤ε< 0
2
ϕ
λ =μ
n
i
n n
e = , ,... 12 n A
n
e = , ,... 12 n
∈ Dom( ) x A
∞
ϕ
=
= λ , , λ =μ ,
ϕ ≤ ε, μ ≥μ ≥ ... > ,
∑
1
1 2
( )
0
n
i
n n n n n
n
n
Ax xe e e
= A UC
∞
=
= μ , , ∈ = ,
∑
1
( ) Dom( ) Dom( )
n n n
n
Cx xe e x A C
Obviously, is a self-adjoint positive definite
operator on and is a unitary operator on .
The domain of the operator ( ) is a Hilbert
space with respect to the inner product
, . For , we
assume that .
Let denote the Hilbert space of all func-
tions defined on almost everywhere,
taking values in , and Bochner square integrable
with norm
Following monograph [1], we introduce the Hil-
bert space
with norm
Here and in what follows, we understand deriva-
tives in the sense of the theory of distributions on a
Hilbert space [1].
∞
ϕ
=
= , , ∈ .
∑
1
( )
n
i
n n
n
Uy e ye e y H
C
H U H
γ
C γ≥ 0
γ
H
γ γ
γ
, = , ( ) ( ) xy CxCy
γ
, ∈ Dom( ) xy C γ= 0
=
0
H H
+
;
2
( ) L R H
() ft
+
= ,+∞ (0 ) R
H
+
/
+∞
;
= .
∫ 2
12
2
( )
0
()
L R H
f ft dt
+
+ +
;
= ∈ ; , ∈ ;
2
2
2
2 2
( )
{: '' ( ) ( )}
W R H
uu L R H Cu L R H
+ + +
/
; ; ;
= . + 2
2 2 2
12
2 2
2
( ) ( ) ( )
( '' )
|| ||
W R H L R H L R H
u u
Cu
MATHEMATICS
a
Baku State University, Baku, AZ 1148, Azerbaijan
b
Institute of Mathematics and Mechanics, National Academy
of Sciences of Azerbaijan, Baku,
AZ1141 Azerbaijan
c
Azerbaijan State Oil and Industry University, Baku,
AZ 1010 Azerbaijan
*e-mail: mirzoyevsabir@mail.ru
**e-mail: alievaraz@yahoo.com