SCIENTIFIC ABSTRACT LYUBICH, YU.I. - LYUBIMOV, M.L.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R001031210005-8
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: August 31, 2001
• Sequence Number: 5
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

Mat.Sbornik, n. Ser. 38, 183-202 (1956) CARD 2/2 PG - 456
constants C,>O, q >0 (depending only on a(x)) such that for every solution of
1
fa(x)f(x+t)clx - Lf(t)
0
there holds the estimation
x q
X. C max f f(s)d,l1
For m >0 the existence theorem demands the satisfaction of further assumptions
which are given. The solution of a homogeneous regular equation of positive
rank is developed into a generalized Fourier series. The proof bases on
the properties of the introduced characteristic numbers.

L'@ @ i--@ i @ kA, @ U @ I
SUBJECT USSR/VATHEMATICS/Differential equations CARD 1/1 PG - 419
AUTHOR Lj-uBid Ju. I.
TITLE On the fundamental solutions of linear partial differential
equations of elliptic type.
PERIODICAL Mat. Sbornik, n. Ser. 19,_ 23-36 (1956)
reviewed 12/1956
The author proves some theorems on the existence of the fundamental solution
(in the large) for the equation - n @2u/%x2 + C(X) u _ 0, n-2,3. These
theorems mostly are special cases of the well known theorems (;ompare e.g.
F.John, Proc. of the symp. on spectral theory..., Oklahoma-Sillwoter (1951)
113-175). Finally the author proves the equivalence of the existeuce in
spAerical regions of the fundamental solution of the equation
3 21 2u + cu -A u, -00 4 )@.4 oo
-5 7
i
with the assertion of Courant's theorem on the decrease of eigenvalues with
the increase of the region.

AUTHOR: LTUBICH,YU.I. PA - 2908
TITLEa Some-TIUMTs- Theorem for Generalized YOURILR@Tranaformations..
(Nekotoryye Taiaberovy teoremy dlya obobehchenVkh preobrazovanly
ftr1ye, Russian)
PERIODICAL3 Doklady Akademii Nauk SSSR, 1957, Vol 113, Nr 1, prp 32 - 35
N.S.S.R,)
Received, 5 / 1957 Reviewed: 6 / 1957
A.BSTRACTs The following problem arisest Should parameters be introduved in the
case of kernels T(N,@-) of the general type (which occur in the integral

co
T(N,X)da@(k), (N >0)) in relation to which it would be possible
i
0o
to speak of a critical value? The critical value sorts out that
method from the scale of the single-type summ tion-methods, whi-ah
corresponds as closely as Possible to the divergence velocity of the
@ , "10
integral@ doeO-), The maximum number k of derivatives of the kernel

inrelation to X serves as such a parameter and the last of these
derivatives has a limited variation.
In the case of the transition to non-integer n (if BOCE[NER-STJELTJES-
transformationa of the En + 2]-th order are considered), it is
natural to introduoe derivatives of a non-integer order. The results
Card 1/2 found by the author generalize several results found earlier_ The

PA - 2908
Some TAUBER's Theorems for Generalized FOURIER-Transformations,
difficulties encountered here depend on the non-local. character
of the differentiation of a non-integer order and on the mse@=stry
of the differentiation with respect to a change of sidn of the
argument. The author confines his investigations, for the sake of
simplicit to formulating kernels of the type T(N),) 9 T(N2,),
whers T(;@Odenotes a finite function. Two hypotheses are given for
TM.
Finally, thres theorems are written down and pro-red, With the help
of these theorems it is possible to compute the critical value of
the parameter k. (Ho illustration),
ASSOCIATIONt Polytechnical Institute Charkow
PRESENED BYsS.K.BERNSFITEYN, Member of the Academy
SUBMITTEDs 5-10A956
IVAILOM Library of Congress
Card 2/2

16(l)
AUTHOR: Lyubich, Yu.I. SOV/140-59-4-13/26
TITLE: On Eigen- and Adjoint Functions of the Differential Operator
PERIODICAL: Izvestiya vyeshikh uchebnykh zavedeniy. Matematika, 1959,
Nr 4, PP 94 - 103 (USSR)
ABSTRACT: Let 6(x) be of bounded variation on -t, -iij and is
assumed to grow at the ends-of L-V,fll . Let Da, be the
differential operator Dr f(x) = i fl(x), generated in
C by the boundary condition
f(x)d6'(x) = 0
Jr
The eigenvalues are determined from S(A) e dT(x) - 0
-iA x
The eigenfunction a n and the adjoint functions
x 0 n n e n correspond to a -told, 'li >
Card 1/4

15
On Eigen- and IdJoint Functions of the Differential 307/140-59-4-13/26
Operator
e'.genvalue A n- Because of (1) the system E., of the eigen-
and adjoint function of Da- is not complete in the space
Let E@, be the orthogonal complement of E., in
the space V(- it , 71 ) of the linear functionals on C(- 11-1 70.
def ET - dim E;,, is denoted as the defeat of ET . In order
that 'C(x) G V(- W,V) belongs to E@- it is necessary and
sufficient that all the roots of S(A) 0 are also roots of
T(A) Z' e- iAx 4C(X) = 0 and that they occur in T(A) with
V
at least the same multiplicities as in S(A). Let Q,5- be the
linear manifold of those quotients Q(A) - T(A) s 3(,A) which
N
7' k
are entire functions. Let QkA.)EQ,, and Q(A)= / qkA , N oo
0
Card 2/4

On Eigen- and Adjoint Functions of the Differential 90040-59-4-11,,26
Operator
the magnitude V. - I + max N is denoted as the order of
Q(A ),c go,
Theorem The defect of the system E., is equal to the order
Of Q 6-
Theorem In order that def E,,, 1 < oo holds it
is necessary and sufficient :
1.) 5(x) possesses an absolutely continuous derivative of
the order @'-2 and T(6 -1 ) (x) is of bounded variation.
2.) If K > 2 , then it is 0- (k) (� -,/-) = 0, k-1 , .. ., 8- -2
3.) If is a function of bounded variation which
is equal to 6- (x) in those points of (- where
V(15 -1)(x) exists, and which vanishes for x then
it is def E = 1 -
Theorem i If for certain p>, 0, q;@,O there exist the limits
Card 5/4

16
On Eigen- and Adjoint Functions of the Differential SOV/140-59-4-13/26
Operator
L = lim h) 1 = lim 6*(- @,-' + h)
h--,o hp h-* o hq
and if these limits are different from zero, then it is
defE., !!@ min 1[P1+' I [qj+1L.
j
B.Ya. Levin is mentioned in the paper.
There are 3 references, 1 of which is Soviet, I German, and
I American.
ASSOCIATION: Kharlkovskiy gosudarstvennyy universitet imeni A.M. Gor1kogo
(Kharlkov State University imeni A.M. Gorlkiy)
SUBMITTED: May 8, 1958
Card 4/4

LYUBICH, -Yaj.
Inequalities between the degrees of an linear operator. Izv.
AN SSSR. Ser. nat. 24 no. 6:825-864 H-D 160. (MIRA 14:1)
1. Predstavlano akademikom A.H. Kolmogorovym.
(Operators (Mathematics))

LYUDIcii., YU.-,
Theoren of vie unic:-,@err:s3 of ti @ ::clution 0@, @-,-n abzt -t
Cauchy problem. Us-. i:,,t. nLzf@ 16 no,5:1FI-11'.2 S-C.
e- z:.tions, P-L
rtia-L)

685 95
4-6@ 6. L14 oo S/020/60/130/05/004/061
AUTHORt "Lyubich, Yu.I.
TITLEs Conditions for the Uniqueness of the Solution to Cauchy's
Abstract Problem
\\Ii
PERIODICALs Doklady Akademii nauk SSSR,1960,Vol 130,Nr 5,pp 969-972 (USSR)
ABSTRACT: A vector function x(t) (t@!;O) is sought in the Banach space
which satisfies the equation
(1) dx(t) Ax(t) (t> 0)
dt
and the initia.3 condition x(O) = X0 , where A is a linear operator.
X(t) (t:@.O) is called weak (strong) solution of the problem
(1)-(2), if 1.) it is weakly (strongly) absolutely continuous
and almost everywhere on t >0 weakly (strongly) differentiable,
2.) it satisfies (1) almost everywhere, 3.) it is weakly (strong-
ly) continuous for t - 0, 4.) it satisfies (2).
Theorem 1: If on a ray L of the positive semiaxis the spectrum of
the operator A is absent, and if the resolvent R,\ of A satisfies
the condition
Card 1/ 2

4
61@595
Conditions for the Uniqueness of the Solution tc S/020/160,'1z:@,/,-1, L4/06!
Cauchy's Abstract Problem
ln R
(3) lim < 00
)k4 +00
then (1) (2) cannot possess two different weak sclutions.
Theorem 2s Let g(N) (A > 0) be a positive continuous function,
t ln
i
t
Hilb
t
l
h
th
f
s
s a
e
en
ere ex
er
or + oo
+ oo . T
space and there a linear operator A, such that i
a.) the positive semiaxis possesses no spectrum of A and it is
(7) 11 R X 11 0
b.) the problem (1) - (2) possesses a nontrivial atrong solution
with X = 0 . - There are 3 references, 1 of which is Soviet,
I Dutes, and I German.
ASSOCIATIONs Khailkovskiy gosudaretvannyy universitet imeni A.M.Gor'kago
(Kharlkov State University imeni A.M. Gorlkiy)
PRESENTEDs October 12,1959, by S.R. Bernshteynt Academician,
SUBkTITTEDs October 11,1959
CarQ 2/2

b
68795
S/020/60/131/01/004/C60
AUTHORS: Lyubich, Yu.1., Matsayev, V.I.
TITLE: On the Spectral Theory of Linear Operators in Banach 'pace
PERIODICALt Doklady A-kademii nauk 9SSR,1960,Vol 131,Nr 1,pp 21-23 (USSR)
ABSTRACT: A linear operator A in a Banach space is called S-operator,
if 1.) its spectrum lies on the real axis, 2.) to every finite
interval & of the real axis there corresponds a subapace L(,&)
invariant under A which has the properties : a.) A is every-
where defined and bounded in L(A) b.) the spectrum of the part
of A induced on L(A) consists of the intersection of the spectrum
of A with 46 and possibly of the ends Of A c.) every in-
variant subspace, on which A is everywhere defined and bounded
and possesses a part of.& as the spectrum, lies in L(A) ;
3.) the system of the invariant subspaces L(4) which corresponds
to an arbitrary covering of the real axis by finite in-cervals
is complete.
A is called locally correct, if 1.) A is a closed operator with
a dense domain of definition D A ; 2.) the Cauchy problem
Card 1/ 4

68795
On the Spectral Theory of Linear Operators in 9/020/60/131/01/004/0060
Banach Space
i Ax(t) (- co