SCIENTIFIC ABSTRACT KRASNOSELSKIY, M.A. - KRASNOPOLSKIY, V.A.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R000826120014-7
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: January 3, 2017
• Document Release Date: July 31, 2000
• Sequence Number: 14
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

On a Principle of the Existence of Bounded, Periodic and SOV/20-123-2-6/5ri
Almost-.Periodic Solutions of a System of Ordinary
DiffQrential Equations
let be valid the non-local theorems of existence and uniqueness.
Let every solution x(t), t 6[TiIT,:~] of (1) satisfying the initial
condition x(T1),e r , satisfy the condition x(t) ~ x(T, )I
t e(T,,T21. Then (1) has at least one solution x*(t) for which
x*(T 2) - x*(T1)4-G.
There are 12 references, 9 of which are Soviet, I American, and
2 Polish.
ASSOCIATIONtVoronezhskiy gosudarstvennyy universitet (Voronezh State University)
PRESENTED& June 9, 1956, by P.S.Aleksandrov, Academician
SUBMITTED3 May 10, 1958
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KRASMOSELISKIT, N.A., red.; HOISEYEV, N.N.; SMOKWSEV, Ye.D., red.;
tekhn.red,
[Theory of surface waves; collection of translational Teoriia
poverkhnostnykh voln; abornik parevodov. Pod red. M.A.Kramno-
sellskogo i N.N.Hoiseeva. Hoskva, Izd-vo inostr.lit-ry, 1959.
366 p. (MIRA 12:11)
(Wavea)

110
?WE I BOOK EXPLOITAT:031 SCV/3177 Z-1
Hatemati" v SSSR za norck: let, 1917-19:57.-m 1: Obzornyyo stat I
(Nathaimatito In the USSR for Party Yearv. 19IT-1957).Val 1:
Review Articles) soncow, Pizzatgiz. 1959, 1002 P. 5,5W caples
printed.
Eder A. 0. Xuroah, (Chief Ed.)$ V. X. Bltyjtakov, V. 0. DcU kly
11
,a
B Dynkin
Shilova
0
Ye
and A
Tushmovicb
Ed
.
.
.
.
.
.
,
,
;
booll); A. P. Lapkol Tech. Ed.: S. X. Akhlazov.
Ftmpont This book 1: lnt:nded for mathematIclans and historians
Of sathematIcs Int Met d In Soviet coo.-r-'butlon* to the field,
h
COVERUX: T I& book Is Volume I of a major 2-volume work on the

So
history Of vtet mathematics. Volu" I surveys the chief can-
tr1butLona made by Soviet mAthtnaticlama during the period 1947-
1957; Volume 11 -111 contain a blbllograpimy of major woma since
191T &Ad biographic sketches or *ova or t::* loading mat
tic Lan . This worJ1 follows the tradition 11 t- by two earl--or
o
kaz
Xatematlka v 333R to pyatnadtoat- M (HAthematica in
the MSR for, JJ_UAZ9J_
_&nd_yAt emAt I Ica v 533A to tridt"t- I
t
.
_
(XiLtheizatLes in the ma for
30 Years). The book In 41vid x
into the major divisions of the field. I.e.. algebra, topolW.
theory of probabilttlas~ functional LrAl"Is, ate., and c=-
tributiona and outstanding problems in each discussed. A list-
ing or some 1400 Soviet mathematicians is :zciudod with refer-
ence# to their contributions In the field.

jnkhljR_J~A~ Linear Integral Equations
T.-Prodholm equations
2. Co"Ietsly Continuous Operators
3. Xornals dependent on the parameter 654
4. Ons dimensional singular Integral oquatIons 6s0
Squatlons With difference kernels 1665
Mult1dimanalonal singular Integral equations 669 1
T. rategro-Wforentlal equations 673
9. A X. A. Saymark, and 0. To. ShIlov.
------------ 675
1. banach end ftlbortspac*4 52
30AL-ordered space and Spaces with coo*
.
?
_
_
- - ---------
698
3. Marm*4 rings
4. Reprosentations of rings and groups 704
Di.frereati'al equations in abstract spaces M3
Equations a1tb nonlinear continuous *Porsta. 72a
7- Spectral analysis of self-conjugat* d.Cfv"Qt1al
146
operotors
a. Spectral analysis or non-s*ir-conjusate operators 763
9. Llmo&r topological spaces, gonorallmwd 'unctions T73
--galmagospaw.-A-I.Probabillty Theory 781
1 Distr-butions. Random functions and processed 782
2: Station&" Processes and homogeneous ramdoot fields 783
3. PAz*cv processes w1th continuous time 785
4. Limit theorems 789
5. Distr;butlonn of sums of Independent and weakly
dependent summands and InfinItely dia-
trl.%KXtl*Ca 791

. 6976h
14'L4500 16,1/600 9/155/59/000/02/005/036
AUTHORS% Keasnosellakiy, M.A., Mamedoy, Ta.D.
TI-T13i Remarkon the Appfro-&-t-ion of Differential
in the Question of the Correctness of the Cauchy
Differential Equations4n Banaoh Spaces
and Integral Inequalities
Problem for Ordinary
PERIODICAL: Nauohnyye doklady vyashey shkoly. Piaiko-matematicheakiye nauki,
1959P No. 2, pp. 32-37
TEXT: The authors abow that, with the aid of well known theorems on
differential-and integral inequalities (especially the lemma of Chaplygin)t
one can-estimate-in a,very simple-way the variations affected on the
solution of-the-integro-differatttial equations, if the right sides or the
initial conditions are subject to small perturbations.
Ye.k. Barbashin, R.I. Vishik, L.A. Lyusternik, M.G. Kreyn, A.I. Percy, and,
P.Ye. Sobolevskiy are mentioned.
There are 12 referencest 10 Soviet, 1 American and 1 German.
ASSOCIATIONs Voronezhakiy gosudaretyennyy universitet (Voronezh state
University)
SUBMITTEDi February 20, 1959
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32503
S/044/6-1/000/oil/026/049
C111/C444
AUTHORS; Krasnosellskiy, M. A., Rutitskiy, Ya,. B,,, Sultanov. R..
-------------------------------
TITLE: On a non-linear operator, operatinU in 3paces of kibstra-
functions
PERIODICAL: Referativnyy zhurnal, Matematika, no. 11, 1961, 71,,
abstract 11B397.(Izv. AN. Azerb. SSR. Ser. fiz-m~item
i tekhn. n., 1959, no. 3, 15-21)
TEXT: Investigated are certain properties of tile nper~itoz
fu(t) = f(t,u(t)) N
which transforms a subset of a certain Banach spaco B into
Banach 8pace B,. One assumes that the abstract function f(t,u) With
values in B 1 is strongly measurable for every fixed u C B., and that
the operator f(t,u) is strongly continuous with respect to u for
almost all t6_.CL;Xj is a bounded closed set of the finite
Euclidian space. In the article it is proved that the theorems c-n the
continuity and boundedness of the operator f which former1y have be;~n
roved for the spaces Lp, Lp
ard 1/3 (U) of vector function3, for Orllie.~.~, opu,;Ps
E

32503
S/044/6I/OG-0/O,-'/O26/O4-)
On a non-linear operator, operating . . . C111/C444
etc., hold for broad classes of abstract function,,zpaces Phe
of a I'M - space" is introduced as follows; Let B be the It!
of all measurable abstract functions u(t) with vk~-Iues in the _H~ina~,,,
space B- let B be made a complete Banach space b~~ aid of a rer,_'jin
norm 1) liv . The space ~ is called 11,je- space". if the follow_zn~_7, ..n~ma-
B
tions are satisfied:
u o if and only if u (t) = 0 almost ever-;--here 4/
1.) There -'s B
I/
2.) B contains all functions taking a constant value on A3
V
3 ) B contains together with the abstract function u( T ) a I] +',I ri,~ r;ns
U~t) WE (t), ?~(t) being the characteristic function of the measurabl-:-
s e t E C_ the r e 11 u zes 1) - -:!n- 1) u
13
4.) Out of the condition ~ujj 0 there followi3
B
un(t) converge to 0 with respect to the norm.
The authors investigate certain properties of the _Jntrodu,~el -X
under certain additional conditions and prove til,,e
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8/044/61/000/011/026/049
on a non-linear operator, operating . . . C111/C444
YO wo
operator f, transf"Eming a subset of B into B , where B is the Oct. 0'
those functions Of B which have absolutely continuous norms, It is
V
said that an abstract function u(t) e B possesses an absolutely
continuous norm, if 0 u zeE 11 - -4 0 for mes E ---% 0. Purther on it -is
B V
proved that the operator f is bounded in every sphere T 3C_ B At the
end of the paper the case mes IL = oo is considered.
[Abstracter's note: Complete translation.]
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16(1) 05256
AUTHORS; Mrasnosellskiy,M.A., and Ladyzhenskiy,L.A. SOV/140-59-5-12/25
TITLE: On the Extent of the Notion u.-Concave Operator
PERIODICAL: Izvestiya vysshikh uchebnykh zavedeniy. Katematika, 1959,
Nr 5, pp 112-121 (USSR)
ABSTRACT: The authors consider
(1) Af(x) -JG[xyT(y)]dy
F
An operator A in the Banach space E which is partially ordered
with the aid of a cone K, is called u 0-concave if it is positive
and monotone and if there exists a positive element u 0 so that:
1 ) For every LPEK(I tP0 / 0) there exist 04-, A, so that
(2) ofuo
Theorem 2 t Every regular cone is normal.
Let u be a fixed 'element of K different from zero. Let Eu denote the set of
such x6 E that f*or certain-a - a(x) it holds i - a u,--x4a u . The smallest
a for which this inequation is satisfied is called u-norm of a and is denoted
with IIXIIU
Theorem 3 t Tn nrder that K is normal it is necessary and sufficient that an
M >0 exists so that for every y 6K it holds i
(4) x 11~~M 11 y 11 - 11 x 11 y (XeEy)
The positive functional fl, defined on K is called strongly increasing if for
rx)
all hn G_ K (n - 1, 2.... ) from hn 0>0 (n - 1129..) it follows
lim f(h 1+ --- + h n) - 00
n-*.oo
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86382
Regalar and Perfectly Regular Cones S/020J60/135/002/002/036
C111/C222
Theorem 4 : Let a functional-strongly increasing on K be bounded on the
intersection of the cone with each sphere of E. Then K is perfectly regular.
Theorem 5 1 It on K a monotone strongly increasing functional can be defined
then K is perfectly regular.
A linear functional f(x) is called uniformly positive if it holds
(5) f (x);?, a 11 X 11 (X E10
where a> 0 . It is said that K admits a costing if there exists a cons-K so
that every x6K different from zero is an inner element of K I and furthermore
it lies in KI with a spherical neighborhood.of radius bjjxjj, where b does
not depend on x .
Thelorem-6 t In order that K admits a coating it is necessary and sufficient
that-on--K a uniformly positive linear functional can be defined.
Theorem 7 1 Every cone admitting a coating is perfectly regular.
Theorem 8 On K let be defined a linear completely continuous (on K) operator
A, where
(6) Ax 11 _-~ a 11 x 1! (x r-- K)
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86382
Regular and Perfectly Regular Cones S/020/60/135/002/002/036
CIII/C222
Then K admits the coating.
Theo-rem 9 relates to linear operators with respect to which a certain cone
is invariant.
The author mentions M.G. Kreyn# V.Ya. Stetsenko, D.P. Millman and I.A.
Bakhtin.
There are 4 Soviet references.
ASSOCIATIONt Voronezhekiy gosudarstvennyy universitet (Voronezh State
University)
PRESENTED: June 15, 1960, by P.S. Aleksandrov, Academician
SUBMITTEDt June 11, 1960
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86027
S/02 6o/135/003/005/039
C1 11%222
AUTHOR: Krasnosellskiy, M.A.
TITLEi -nafT-onaryPoinFB--o-? -Cone - Compressing or Cone - Extending
Operators
PERIODICALs Doklady Akademil nauk SSSR, ig6o, V01,135, N0~3, PP-527-530
TEXT: -Let K be a cone in the real Banach space E. Let xO so that I 1U,
(1) Ai-.3~ x (x C K, 1,; xR) .
then A has at least one fixed point in K.
Furthermore let AG - 0, where 0 is the zero point of E.
Let A compress K if there exist positive R,r so that (1) and
(2) Ax,- (x CE K,!Ix!l:,
,-~ r)
is satisfied, Let A extend K if
(3) kx,-~--x (xCK,Pxkj>,R)
and
(4) Ax~~,x
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S/020/60/135/003/005/039
C111/C222
Stationary Points of Cone - Compressing or Cone - Extending Operators
Theorem 21 Let the completely continuous operator A extend or compress K.
Then A has at least one fixed point different from zero in K.
The proofs of the theorems 1 and 2 are given with topological methods,
A linear operator B which leaves K fixed, is called u 0-b3unded from below
(above) if to every x GK there exist an integer p - p(x) and a DOSitiVe
number Y,...4*(x) so that Bpx>_'.!~u 0 (Bpx < eAu0); here u 0 is an element of
K different from zero.
For an examination of the conditions of theorems 1,2 the author uses:
Theorem 3i Let the linear operator B uo-bounded from below satisfy the
condition Bu 0>( I +;_-. du 0 1 Eo '>~O. Then it holds Bx;~7-x for all x!:-K, x
Theorem 4t Let the linear operator B u 0-bounded from above satisfy the
Oondition Bu 0 so that for OZ-uo (-F>0) the relation
A(t~ X)> tAx I Atx ~ tAx (0 O, so that A(tx)> (1 +)?)tAx. The authors
t
prove the following t eoremst (1) if an operator A monotonic on
K 0 positive function 6~(r)
exi a to s o. -that A (X + Y) ~-, Ax +y 11) z0 (XI x + YEK , yEK), where
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S/19 61/002/003/001/005
Method of successive approximations B112YB203
z0 is a certain element in K differing from the zero element. (2) If the
conditions of theorem 1 are fulfilled, and if there is only one fixed
point x*, then the latter is the limit element of the successive
approximations x. = Ax n-1 (n - 1, 2p .,o), whatever element x 0 is the
Initial element of this approximation. (3) If the equation x X Ax with
the concave operator A on the cone X has a unique solution x* differing
from the zero solution, and if one of the three conditions (a, b, c) of
theorem 1 is fulfilledg then the sequence x n ' Ax n-1 converges with
reepe~ot to its normp whatever point x EK is the initial point of the
0
approximation. (4) If the equation I a Ax with the u 0-concave operator A
in the cone X has, a solution x* differing from the zero solutiong then
the sequence x. a Ax n-1 converges for all'xc~K with respect to' its uo-norm
toward x* (the u 0-norm of x is the smallest number ? for which the
inequality - 9 UO 0) differing from zero
Aw> t 0Av + F_0U0(Eo - EO(v, w, t,)> 0) follows from tov< w0. Then (1)-(2) has at least one solution which does
not vanish identically.
Theorem 6s Let f(X,,YZ) be non-decreasing in z; f(x,y,z)> 0 for z >0
and almost all ~Xpyjrrj. Let
f(.,y,-A,)-,Xro f(.,Y,,) (~X,YjGn ; 0?q_>I;!~1; 7-'ZO), (9)
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S/020/61/137/005/001/026
Non-trivial solutions... C111/C222
where Y,4 2(1-o(). Then (1)-(2) cannot have more than one non-negative
solution being not E 0.
Theorem 7t Let exist So> 0 and Mo> 0 so that
r(x,y,z,p,q)-,-a 3 z ( f x 0 y 1 16 0 4 z!L: So I -ou,< p, q < cc) (10)
r (x . y. z . p I q) >, a4z (fx, yj cr z ),' MO; - co-1p, q 4 co ) , (11)
where ~,> 2, a 39 a4 > 0. Then (1)-(2) has at least one solution beside of
the trivial one.
Theorem 8: Let exist a sequence R n --iicof so that
f(x,yp7,;ppq).4,az (S Rn 4 z O~R n) 9
where,rl> 2 and S>O is aurficiently small. Let oxist a sequence R OD
so that f(XY,Z,I),q)-4a (1+z 2)r2 (0 rc z 44)
n n
where 9*2