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

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• Collection: Scientific Abstracts
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• Original Classification: S
• Document Page Count: 100
• Document Creation Date: January 3, 2017
• Document Release Date: July 31, 2000
• Sequence Number: 13
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

KWNOSELISKIY~ M. A.
USSR/MathamtIce - Operatorsj Vector 21 Jul 51
"Problen Concerning the Points of Bifurcations"
M, A. Krasnosellskiy,, Inst, of Mathj, Aced Sai USSR
Mok Ak Nauk SSSR- Vol IXM~ No 3., PP 389-392
The nunber L (in eq F r- IAF., -where A is a con-
tinnous operator operating in a real Banach space
E and satisfying the condition A9 z Q-(Q is a zero
of 9); F is an eigenvector of A) is called a point
of bifurcation of operator A if for anv pos nonzero
e there exist in the interval (L-e.,LQ eigenvalues
of the operator A to which coxx spond eigeiltilactionz
with norm as =all an desired. Submitted by Aced
A. N. Kolwogorov 16 my 51.,

USSR/Matau=tics. sonlinear P=c- 1 Dec 51
tionals
"Theory of Orlicz's Space," M. A. Krasmosellskiy,
Ya. B. Putitskiy
"DoX Ak Nauk SSSR" Vol LXXXI, No 4, pp 497-500
Application of general methods and of theorems on
nonlinear functional analysis to study of concrete
classes of nonlinear operators is effected with
aid of '.rarious concrete Banach spaces. Study of
many operAors with essentially nonpolynoviinal
nonlinearities, cannot be condbcted with aid of
spaces ordinarily appiied (C, Lp). Authors
2o2T67
USSRtNathematics - Nonlinear Func- 1 Dec 51
tionals (Contd)
propose certain assumptions relating to the theory
of L6 spaces (Orlicz spa6es) for the case where
the function (D (u) does not satisfy the .6 2-cou-
dition. (Cf. W. Orlicz, Bull international de 11-
Acad Pol, ser A, Cracovie, 1932.) SubmItted by
Acad A. N. KolmogoroV.25 Sep 51.
n2r67

AS Wrisf--L-1 -SK I Y, M.A.
USSR/Mathematics - Nonlinear Iteration
Jan-Mar 52
"Iterative Process with Minimum Residual.," M. A. Krasnosellskiy and S. 0. Kreyn
Ukraln Mat Zhur, Vol 4,, No 1. Pp 104-105
(Report given at 23 Oct 51 session of Sci Council of Inst of Math, Acad Sci Uk SSR'O)
Problem of approx soln of system of eqs Bx-b (B positive definite square n-matrix,
b known n-vector, x desired n-vector) can be solved by finding approx soln as close
J, ==
as possible to exact soln or kojesidual-vector4N-b as sm-11 as possible. Authors
propose method of max decrease of residual., namely nonlinear iteirativa process
xn+l'xn- 6n-(-RAn,An)/(B4,,BAn), converging as rapidly as geometrical progression
r-(M-m)/(M+m) Km greatest and least characteristics of B). Compare this process
with ordinary iterative process and 'steepest descent' developed by L.Ve Kantorovioh,
Also studied nonlinear operator Kf-f-(Bf f)Bf/(Bf Bf).

R Aso O.S r,
I
Krasnosel'skil, M. A. On the e
stimation ofthe number of
critical points of, functionals. , Uspehi Matem. Nauk
(N.S.) 7, no. 2(48), 157-164 (1952). -~ (Russian), 0
Let there be given on a connected manifold K a continu-
ous involut,io,n A without invariant potints,'Le., a continuous
operator on K to K for which AxOx and A*x-ax. A set of
the first kind on K is a compact set 'none of whose com.
ponents'con'tains both'x- and Ax. A semi of kind n on K is a
set each of Whose compact parts may be divided into n sets
of the first kind and for which some compact part may not
be divided into (n - 1) sets of the first kind. This concept is
closely related to that of category. The author illustrates its
use to obtain a theorem
of Lyusternik on the number of cog
'
critical points of an~eveft nonmgatiViq wealft continuous
functional on-the sphere S of Hilbert space, and related
results, without reference to the'Piro0erties of projective
space. He generalizes this concept'to manifolds on which
there are given periodic transformations of period r. 6
a Monica, Calif.).
J. 41[..DaYsskin,(SanL

-4
4f

MR/Mathemtics - Iteration Process, Jul/An 52
Approximation
'*Note on the Distribution of Errors During the
Solution of a System of Linear Equations by an
Iteration Process," M. A. Krasnosellskiy, S. G.
&eyn
"Uspekh Matamat Nauk" Vol VII, No 4- (50), PP 157-
3-61
The purpose of the present note is to refute the
bypothesis that the most probable errors are al-
vays considerably leas then the max errors. As It
turns out, the max errors.are the most probable
ors. Considers the recurrent~formula
Axm + b, where A is a matrix.
C.
22

iA ri o Y
)

iopile, A. U, An extromal theatein for a hypeiellipsoid
and Its application to the solution of &,system of linear
-A9
algebraic equations. T_i:WyLS6m. Velctor-Te4z". aa
I
lizu 9, 183-197 (1952). #kussjaii)

Let A, be it symnictrk
c positive definiteaffinor [-matrix)
Lot -

er values ard J1
a
t
whose lar
est tnd least pro
nd *ff
Ut 0
.
.
p
g
de by the radius vector IF
be thean
o
at the ini P of
lc ma

R

M
aticial oviews'
th6 ,
g
p
theby rellipsoid rAr=;I.with the "principal normal plane"
PC
a
m at P-Le., with the subspace spanned by thevPctors Ar,

Vol. 14 No. 11
A2r,..-,Atr. Let Tj(x) be the
bykv
Polynomi
,

Deo here
va TA (1) -1 - Theorvni,:.Mways Un -11 i w

'H=erio*1 and Gf1phic (41+nj) (H -m)-~ Ile proof, -which, is ex ressed in
p

Methods terms of multivectors, is long.
If.K41--_.", The theorem is used to estimate.the convergeme OVA
propos d class of gradient methods of solving a linear syfiteni
ai~a.wLA:re~' let
Ax-a: Let Xo=O. 4e-r- Given xi arid
,
-x, be so'choscri in the kpdimensional
xj+j sfiace,
13, tPanned by a;, Act,. A'-'a, that jAx4j-a Js
minimized. It Is shown that iaq, 19 jadsin oi. where Is
j,i
I the angle made by al with the plane A Hi.
41~
lk

or kimt the pthor'ii method is jhe '_'I-process".de-
'
scril -i also byWrasnosel
skif andLKreln [Mat. Sbornik
qq,
N.-S. A(73), 31S these Re 14' vj
-334 (1952); V.
Iplica-
The author states that the I-proces requiresonemult

i

ed
a
tion o A by a N tep, whereas the rel
per s
'
'
"

4,
e reviewer
s two.
rocess
of Kan torovi~ and others take
[Th
p
-n And nn such difference.1
,
The reviewer suspects the theogen coud
be provW
Af
- briefly and clegantly following thc~Birman paper cited b
Y
~tlie author LUspehi Alatem, Nauk (N,S.) 5, no..3(37),
j152-155 (1950); these Hev. 12, 32, 14, 412
G. 1". FOrSY1h41 (Los Angeles. C

s Ft
All, ~4,
ZY j
tz
Mjg L

EL.SK
itmeal skiltj1d JL, and "kreln, G- Att'Iteratia
le9quo
ess in ;'resi als. Mat. S=1
11-ttlia.-Intloal F"3T.VST, 3-IS-334 (1952). ' (RussianY
Vol 14 I[q. I
S Loet B he a real positive definite matri3r. The authors
juli W A4"S~s 1963 Introduce the "a-proce2ses," a family of Rradient methodx
Virn,viartl And G~%VlMfAal 1%thodss for solving a system of linear equations Dx-b, depending
o.a a real parameter a. Let xe.be arbitrary. For each;
k-O,-J. a sequence. jx&Gj converging to 0-3-'b is
Aefined by letting *&+I -xe-cAo, vibere AA*-Bxe-k,:
vul w1we e. - (BoAe.- A.G)l(BO41A.4, A.,e). (In to and A
but Be is the nth power of D.)
a Is a xupersc6pt,
For real -(,let Is& the"y-length"of a, be (a, B Tx)1. Then the:
zi
dr-prtxess selects ee+l aptong all= a t
w Mos Lt-, ski M. A.
21 Feb 53
USSIt/Mathematics Nonlinear Integral
Equations.
"New Theorems on Existence of Solutions of Nonlinear
Integral Equations," M. A. Krasnoselskiyo Voronezh
State U
DAN SSSR,,Vo1 88, No 60 PP 949-952
Analyzes conditions discussed by A. Hamm tein
(Act& Math. 54, 1929) for solvability of Hammer-
stein's eqs. Considers functionals in a Hilbert
space only. Presented by Acad A. N. Kolmogorov.
29 Oct 52.
2587102

KRASNOSELISKIY, M.A.
USSR/ Mathematics - Nonlinear Integrals 1 April 1953
"Differentiability of Nonlinear Integral Operators in Orlicz Spaces", M.A.
Krasnosellskiy and Ya. B. Rutitskiy
DAN SSSR) Vol 89, No 4, pp 601-604
Investigate the operator Hf(x)=-fGK(x,y) F Cy.,f(y)3 dy., where G is a compact set of
n-dimensional spaces, and show that this 6perator with extensive classes of kernels
K(x,y) and nonlinear functions F(x,u) can be studied by nears of OrUC2 Spaces (see
A. Zygmund, Trigonometric Series, 1939)- State that the general principles of func-
tional analysis permit one to investigate the eq f =I Hf (H is a nonlinear operator)
but finer theorems (namely, on bifurcation points, stability of solutions, eigen-
functions, etc.) are successful in establishing when It is a differential operator.
One author cites earlier work (Ya. B. Rutitskiy, Dopovidi Akad Nauk RSH, No 3, 1952).
PreBented by Acad A.N. Kolmogorov, 2 Feb 53
256T99

MPS~70SELISKIYO M. A.
Yiri; tional I'lethod
I Jul 53
I'Varit-tional Methods in the Problem of Bifureat-;,)n Pol.,Ac, A. Kracnosellskiy Vnd
A. I. Fovoloiskiy
D.AJI SSSR, Vol 91, No 1, Tr
p 19-22
Generalize resulte of itive at igat ions of nonlinear operatcra A that operate in -Ilanach e-one
E and traxisform zero 0 of this space to zero 0; namely, operators A of' tlhe I*orm JG, where
J ie a certain unitF-ry operation coincidin.-I -aith unit I in one iiar,~.riautl subspace :)f
linear o-Derator B a!,d equal to -1 on the orthogoaal complement, G (0e--9) is a &radieat
o-oerator of a weakly continuous functioaal defined la Hilbert space H and poeeess~.,s at
Point 15 q Frpchet derivative of B (thia derivative a linear celf-adjoint positive-
defiaite ooerator). Presented by Aced A. 11. Kol-,vioe;orov 2.2 A~pr 53.
266T79

U~81q/Vathematics Nonlinear Integrals 11 Sep 53
"The Structure of a Certain Operator," 14. Vaynberg
DAN SSSR, Vol 92, No 2, pp 213-216
Considers the roblem of whether a given op;--rator h generated by a real
function f(u,x~ depends upon the structural propef-ties of f(u,x), where
f(u,x) is defined for all real u a-d for all x in the meavurable set B of
Euclidean space s of dimensions by the equality hu f(u(x),x). Notes that
h was studied earlier by V. V. Nemytskiy (Matem Sbor. 41, 438 (1934)), by
the author in 1949, and by M. A. Krasnosellskiy (Ukrain Matem Zhurn. 2, No
3, 1951). Completes the investigation of the 'continuity of h for an ex-
tensive class of functional spaces, and rhows. that the necessary and
sufficient criterion of continuity. Presented by Acad S. L. Sobolev 13
Jul 53.
269T74

XUBHOMISKIY, N.A.; UDYZHUSKIT. L-A-
~ "~ -,
. T;, ~. ~-
Gonditions for total continuity of P.G.Urymohn's operator valid
In the apace LP. Trudy Roak.mt.ob-va 3:307-320 154. MAL 7:7)
(Operators (Hathomtics)) (Spaces. Generalized)

KUSIIOSZLISIaY, M.A.; IADYZHENSKIT, L.A.
Structure of the spectrum of positive heterogeneous opsr~Ltors.
Trudy Roek.zat.ob-va 3-.)21-346 154. (KLRL 7:7)
(Operators (Mathematics) (Topology)

KRhSNGSEL'SK%Y, Mj~.
USSR/Mathematics - Nonlinear analysis FD_u62
Card 1/1 Pub. 118-3/30
Author Krasnosellskiy, M. A.
Title problems of nonlinear analysis
Periodical : Usp. mat. nauk, 9, No 3(61), 57-125, Jul-Sep 1954
Abstract : The author presents a survey article in which considers a number of
problems In the theory of nonlinear equations, differential or integral,
such as are found in mathematical physics and technology. He treats in
particular the transition to operator equations (e.g. choice of space,
operators of P. S. Uryson and Hammerstein, operator of Lyapunov, applica-
tion of Orlicz spaces, differential operators, potential operators); the
existence and uniqueness of solutions (e.g. choice of method of study,
-method of successive approximations, principle of the fixed point, the
lerey-Schauder method, index of solution, variational method of proving
existence theorems, approximate solution by Galerkin method); eigen-func-
tions of nonlinear operators (e.g. existence of eigen-vectors, problem
of points of bifurcation, spectral studyieigenfunctions of positive oper-
ators). The author thanks V. L Sobolev, Fifty-five references; e.g.
P.S. Aleksandrov, M. M. Vaynberg, F. R. Gantmakher, M. G. Kreyn, L. A.
Iadyzhenskiy, Ya. B. Rutitskiy, A. L Povolotskiy, A. I. Nekrasov, etc.
InBtitution
Submitted

v:r

ZRASHOSELI SKIY, M.A.; RUTITSKIY, Ta.13.
......
Linear functional@ in Orlicz spees. Dokl. AN sso-R 97 no.4:581-584
Ag '54. (MLRA 70)
1. Predstavleno akademikom F.A.Alakoandrovym,
(Functional analysis) (Spaces, Generalized)

KWKOSSLISKIY' N.A.
Two remarks on the wthod of sequential approximations, Usp.
mat,usuk. 10 uooltl23-127 155 WMA 8:6)
(Approximate computation)(Topology)

A~ con, mo
Uat!
P4,MdeocC 0-.

lit
c'

IMASHOMISKIY, H,A., (Voronezh)
Stability of critical values of even functionals on a share.
Mat.abor.37 n0-2:301-322 S-0 155. (KIRA 9:1)
(Functional analysis) (Topolog7)

. . . . . . . . . . .

-':T,P lla~l
pace %V'1 V.U

7, 1
wgi,
~jk.'7. 70.
41 lz
17
P'g

~Xi

Kr~ As No SF_t~' 5 K '~~ N) . R -
SUBJECT U$SR/MATHEYATICS/IntograL equationn CARD 1/.5 P(; - I t' 6
AUTHOR 13ACI[TIS I.A., WUSNOSV'1,SKk;r M.A..
TITLE To the problem on the *Longitudinal fluxure of u bi'am of variabte
flexural r1gidity.
PERIODICLL Doklady Akod. Nauk. 621-624 0955)
reviewed 7/1956
The author uses the method of the nQn-linear functional analy3i~~ for the in-
-vestigation of the longitudinal flexure of a chin blam of vavia.Jb flexurai
rigidity which is fastened by a hinge. Ot-Le catil of the beam cat, r'-C)v.: in the
horizontal plane. The corresponding differartial eqaation 'be
P Y T,
dB2
with the boundary conditions
(2) Y(O) - Y(1) - 0
(P is the charge, ?(a) the fleximil rigidity. o the lengtb of the Currod
beam, y the corresponding deviation from the equilibrium pogit'Lon). By
d2
-v- . -1(s) the solution of this equation can be reduced to tha letarm.Linati*or,
de2

Doklady Akad. N44 105, 621-624 (1955) GARD 213 PC,
of 49(s) of the integral eqi.tur-ivii ~,,Ker,-A
0(ort) T(t)dtl i-
and the determingtion of Y(s) 0-~
y(s)
f or
G(s,t )
f --~r ,
The operator B la Cb;183Aered on the spricre Ty,~O (C the Spaca of the fuartiaca
being continuous on [0,I] ) of tadlu5 ! '/29 It 19 ,complete op- T atd differeati4i,le
according to Prechat, whore its FreOiotls derJvatlve in the 79ro Doint of
the'space is the operator DT(e) Ef Oe--y(B)-,6-i/2, 1han
B [ t til \F(a) (04t4,-I). if (0 X-,
P2(s))? thozi there exists an Oe- 3uch thet B 'Pi (9) -B ~P2 (s) >L6~9) 8 (1 -8)
The charge P 0 is called cri ti cal if tn~t7 arb.1trary El J:~, 0 there exists a

Doklady Akad. Nauk j25X 621-624 (1955) CARD 3/3 PG - 166
solution of (1)-(2) being different from zero, which satisfies the inequation
I Y (814 E if at the same timejP-P.je_S . The critical forces of the con-
sidered problem agree with the eigenvalues P k of the boundary value problem
ey ~ P S(S)y Y(O) 0.
ds2
The investigation of the qiiqstion, when, (l)-(2) admits small solutions, yields
the theorems For critical charges Pk (k-1-,2,...') the equation f(s) - FB ~(s)
has no small solutions being different from zero. To every Pk there corrvaponds
an interval A _(Pitp~+42) such that for P Cr A the equation f(8) - PB P(3)
k k k
has solutions being different from 2ero, which 'for P ----)-P,, tend to zero
together with their second derivatives. The proofs of the theorems and lemmas
are sketched.
INSTITUTIM Public University Voronei.

KIP s Xi o Er.:_ C!~A-
Call Nr: AF 2208825
~d All cal Congress
up ~9~~r uol -union Mathemati (Cont. ) Moscow,
ur
Y 711,
!p
Sect. Rpts., Izdatel'stvo AN SSSH, Moscow, 1956, 237 pp.
__Krasn On the Investigation
9ffa1.Lak,jZj'P M. A. (Voronezh).
of Bifurcation 61-nl'S of Non-linear Equation. 204-205
Kreyn, S. G. (Voronezh). Mathematical Problems in
the Theory of Motion of Solid Bodies With Fluid-
filled Cavities. 205
Kupradze, V, D. (Tbilisi). On Some New Research at
the University of Tbilisi in the Mathematical Theory
of Elasticity. 205
Mikhaylov, G. K. (Moscow). Precise Solution of a
Problem on Stabilized Motion of Ground Water in Vertical
Plane With Free Surface and Feeding Zone. 205-206
Mention iB made of Polubarinova-Kochina, P. Ya.
Movchan, A. A. (Moscow). Linear Oscillations of a Plate
Moving In Gas at High Velocity. 2o6
Card 68/80

K P\ ns)\j or-C s r\ i y, m - n ,
SUBJECT USSR/MLTHEMLTICS/Functional analysis CARD 1/1 PG - 544
AUTHOR KRASNOSELISKIJ M.A.
TITLE Topological methods in the theory of non-linear integral
equations. (Modern problems of mathematics).
PERIODICAL Moscow: State Publication for technioal-theoretical literature
392 p. (1956)
reviewed 1/1957
In the present book the author c6mpiles moat of the researches on the non-
linear analysis in the Banach spaces, researches combined essentially with
the method of Leray-Schauder. The book contains six chapters. In the first
chapter the author studies the integral operators to which the abstract
methods in the following chapters are applied. The second chapter contains
the notions and the fundamental theorems: The rotation of vector fields (in
the sense of the author being equivalent to the topological degree of Leray-
Schauder), the theorems of Brouwer, Hopf, Leray-Schauder, Kusternik-gnirell-
man-Borsuk.
The notions and theorems of tha combinatoric topology used in this theory are
deduced partially. Then (Chapter III and IV) these methods are applied to
more concrete problemss the existence of solutions and imper values, ramification
points, non-linear 9 ectral analysis (the author defines a resolvent for
non-linear operators5, asymptotically linear operators, Liapunov theorems.

SOV/1 24- 57-4- 3911
T rans lation I rom: Ref e rativnyy zhurnal. Meklianika, 19 5 7, Nr 4, p I I (USSR)
AUTHOR: Krasnoset'skiy, M. A.
TITLE: On the Investigation of Points of Forking of Nonlinear EqUations (0b
issledovanii tochek bifurkatsii nelineynykii uravneniy)
PERIODICAL: Tr. 3-go Vses. matem. s"yezda. Vol 1. Moscow, AN SSSR, 1956,
pp 204-205
ABSTRACT: Bibliographic entry
Card 1/1

-7'
Elf,
;r't,
It
- N

A ~ N 0 S K %Y' M - f) -
SUBJECT. USSR/,MATHEMATICS/Theory of approximations CARD 1/1 PG - 429
AUTHOR KRASKOSELISKIJ M.A.
TITLE On some approximative methods for the determination of the
eigenvalues and eigenvectors of a positive definite matrix.
PERIODICAL Uspechi mat. Nauk 11, 3, 151-158 (1956)
reviewed 12/1956
The author proposes some methods for the approximative computation of the
eigenvalues and eigenvectors of a positive definite, quadratic, symmetric
matrix of n-th order. The matrix is considered as an operator in the En.
The proposed methods are analytic analogues to the well-known methods for
the construction of point sequences x k (k_O,1,2,...) on the ellipsoid
(Ax,x) - 1 which converge to the endpoint of one of the semiaxes. Compare
Kantorovitf (Uspechi mat~,Ilauk , 6, 69-185 (1948)), KostarYuk (Doklady Akad.
Nauk 18-2- 531-534 (1954) Lanczoe (journ. of Research of the Nat.Bureau of
Standards AL 200 (1950 etc.-

KRASNOSICLI SKIT 013y, V. I.
The Voronezh Seminar on functional analysis. Usp.mat.nauk 11 no.5:
249-250 S-0 156. (MM 10:2)
(Voronezh--Functional analysis)

SUBJECT USSR/MATHEMLTICS/Functional analysis CARD 1/1 PG - 410
AUTHOR KRASVOSEWSKIJ M.A.
TITLE On a boundary value problem.
PERIODICAL Izvestija Akad. Nauk 20, 241-252 (1956)
reviewed 1211956
For the non-linear boundary value problem
Y11 . f(XOYFYI)
Y(O) - y0r) - 0
the author gives new conditions for the existence of the solution. If the
boundary value problem possesses a trivial solution, the author gives
conditions for the existenoe of a second, non-vanishing solution. Methods
of the functional analysis are used.

KRA ~ PIOZEI~.' Z,- K 1 1. f, M4A.
SUBJECT USSR/ MATHRITATICS/Lategrai vqueLicns
AUTHOR MBITOSELOSKIJ M.A.
TITLE On the equations of A.I.Nekrasoy of
of a heavy liquid.
CARD 1/3 PG - 368
the theory of surface waves
PERIODICAL Moklady Akad. Nauk 109, 456-459 (1956)
reviewed 11/1956
Nekrasov has shown that the non-vamishing solutions of the integral equation
21r
S K(x'y)s'n2(y) -dy
0 1 +Pfs in 4f (t) dt
0
sin nx sin ny
n-1 /A n
determine the form of the waves on the surface of a heavy liquid. Here the
positive eigenvalues /A. are different in dependence of the fact if the depth
is finite or infinite. The papameter is determined by the characteristics
of the stream. The author.applies metl*hods of the functional analysis and
topological considerations in order to investigate the solutions of this
integral equation without constructing the solution. The initial point of the
investigation is the statement that the operator

Doklady Akad. Nauk 109, 456-459 (1956) CARD 2/3 PG - 368
Or
A.( (f, P) - 14 K(X.Y) ysLn tW_ dy
f siny (t)dt
0
is completely continuous on a sufficiently small sphere of functions being
continuous on [0,2'X], and admits the representation
A( f , ~A) - /AB f + C(I ./A ) + 1)(1 , IA),,
Here B is a linear integral operator which is determined by the kernel K(X.-Y),
2qr y
C( 1 K(xoy) f (Y) I f ~(t)dl dy
0 0
and X is of higher order than C( T,1A) in tf . Now from an earlier
result of~he author follows that (1) possesses smanon-vanishing solutions
for certain 1A , which lie in the neighborhood of each ~An . In order to find
these ~\-Values the author applies very interesting topological considerations
(see: Bachtin and Krasnoselslkij, Doklady Akad, Nauk 101. 4, (1955)) which
llokladY Akad. Naii 1~ -12L, 456-45Q (19rj:,
load 10 the theorems which red"ce CAR D 3113 PG 3,-r,
appli cations' The capabi the --,omDu4ation to M1~14jr
to `-111,0 7ery'u.,3ef., e applj,,,:,,j, of I t um at
and 121imber assertions thIle 1110thods ,,, ' the
Of the no On the distr ables the & *
3 1'allar ones. n-vanishing 301,1ti... ibutilon, e,i,,., 1.1tilor
f nce,
INSTITUTIONs Univercit., VOI-oney.

SUBJECT USSR/31ATMUTICS/Functional analysis CARD 1/3 PG - 711
AUTHOR MWNOSEL#SKIJ M.A., nZJN S.G., SOBOLEVSKI P.R.
TITLE On differential equations with bounded operators in Banach spaces.
PERIODICAL Doklady Akad-Nauk 111 19-22 (1956)
reviewed 4/1957
The authors consider the equation
ax A(t)x + f(t,x),
TF W
where x(t) is the sought function with a range of values in the Banaoh space 9,
A(t) and f(t,m) are operators in Z and besides A(t) is unbounded, closed and
linear for every t. A solution is sought which satisfies the initial condition
(2) X(0) - X02
where x0 belongs to the region of definition D(A) of the operator A(O). The
authors use the theory of semigroups Pad therefore it is assumed that A(t) is
the generating operator of a strongly continuous semigroup of bounded operators
T( >0) for every t. At first the linear equation
dx .Ax + f (t)
rt
is considered, where A is, independent of t. Let Q be the linear operator

Doklady Akad.Nauk 111,- 19-22 (1956) CARD 2/3 PG - 711
t
qx(t) -f T(t-x)x(C)dV.
0
Theorems a) Q acts and is continuous in the apace C L of functions which
satisfy the Lipschitz condition. If for I >- 0 the semigroup T( ~) in con-
tinuous with respect to the norm (condition Cn according to Hill , then Q
acts from CL to C Iand is continuous. b) if A-1 is completely continuous,
then Q as an operator from C L to 0 is completely continuous too.
Theorem: Let T(I ) satisfy the condition Cn and let f(t) be continuous and
have a strongly bounded variation. For x 04-'-.D(A) the formula
x(t) - T(t)xo + qf(t)
yields the solution of (1)-(2). dx
Let be given a homogeneous linear equation - . A(t)z and let be satisfied
dt
the condition CK) C(t) - A(t) A- A-'(t) bounded and strongly continuous in t.
dt
Theorems If cK) is satisfied, then 1) the operators A(t) have a common region

Doklady Akad.Nauk 111, 19-22 (1956) Cin 3/3 ?G - 711
a a
of definition, 2) the operators B(t,s) - I(t A-I a are continuous with respect
t s
~O~B
to the norm in t and a and 3) the derivative t s is strongly continuous
t
at
for every a in t.
it 1) and 3) are satisfied, then K) is satisfied too. This theorem and a further
one are int.'direct connection with the investigations of Kato (J.Math.Soc.Jap.
.2,. no.2, (1953)).
Then the non-linear equation (1) is treated. A generalized solution of (l)-(2)
mesna a function x(t) which satisfies the operator equation
(3) X(t) - Qf [tox(t)] + U(t'O)xo
For the proof of the theorems of existence theorems of fixed points are used.
For a sufficient smoothness of f(t,x) in some cases it can be shown that the
generalized solutions the existence of which was proved, are ordinary solutions
of (1). Some examples are considered.

/V 0 5 1
SUBJECT USSR/MATHEMATICS/Funotional analysis CARD 1/2 PG - 612
AUTHOR KRASNOSELISKI N.A.
TITLE On the application of the methods of non-linear functional analysis
to some problems on periodic solutions of equations of non-linear
mechanics.
PERIODICAL Dolklady Akad.Nauk Ill. 283-286 (1956)
reviewed 2/1957
With means of non-linear functional analysis the following questions of non-
linear mechanics are treateds Existence of periodic solutions, uniqueness of
thmm, dependence of these solutions on the parameters of the right side of an
equation etce The starting point of the considerations is the statement that
to every system of ordinary differential equations there can be associated an
equation with a completely continuous operator such that the solutions of this
equation only determine the periodic solutions of the system. By aid of theorems
on fixed points the author obtains sufficient conditions for the existence of
periodic solutions. E.g. let be given the system
(1) x i + gi(ttxl, ... txnv:'lg ... 91n) - 0
where the g iare continuous and posseas the period 21T in t. If the condition
(2) n n 2 n 2- X*
Z Xigi(tPx10"'Xn'y1 .... Yd -.4s. xi + b E lyil +0
ino i-1

Doklady Akad.Hauk 111.L 283-286 (1956) CARD 2/2 PG - 612
with 0 < )r< 21 a,b,c numbers and a 0
For what maximal class of operators A the set of bifurcation points of A
is identical with the set of characteristic values of B ? Two partial
results are givens 1) If A is the gradient of a weakly continuous functional#
then the sets are identicall 2) Every characteristic value -A0 of B with
an odd multiplicity is a bifurcation point of A.
Furthermore the estion is treated for which 'N the equation (1) has
small solutions PO.

KR.ASNOSELISKIT,.M.A.; KFXYN, B.G.; MYSHKIS, A.D.
k The broadened sessions of the Voronezh Seminar on Functional
Analysis in March 1957. Usp.mat.nauk 12 no.4:241-250 Jl-Ag 157.
(MIRA 10:10)
(Voronezh--Functional analysis)

KRASNMLISKIT, N.A.; SOBOLEY, V.I.
The decomposition of linear operators. Unp.mat.uauk 12 no.4:313-317
J1-Ag '57.!,.-. . (6perators (Mathematics)) '(KIRA 10:10):

~~RRSNOSC, C:SK\ ly, M- n -
SUBJECT UJSR/MATHEMLTICS/Functional analysis CARD 113 PO - 874
AUTHOR KRASNOSELISKIJ N.A.p KR3JN S.G., SOBOLEVSKIJ P.E.
TITLE On differential equations with unbounded operators in the
Hilbert space.
PERIODICAL Doklady Akad.Nauk jjjL 990-993 (1957)
reviewed 6/1957
Joining a paper of Kato (J.,Math.Soo.Japan, ~, 2, (1953)) the authors
investigate the equation
(1) dx + A(t)x - f(t)
dt
in the Hilbert space H. Kato constructed the solution of (1) in the Banach
space in the form
(2) X(t) - U(t,o)x 0 + Qf(t),
where the solution of the homogeneous equation has the form
XM - u(tes)xo
I
with a continuous and bounded operator U(t,s) and with the initial condition

Doklady Akad.11auk 112, 990-993 (1957) CARD 213 PG - 874
X(B) - X0 and t
Qf(t) - j U(t,s)f(a)ds.
0
In the special case considered by the authors, about U and Q more exact
assertions can be made. Here it is assumed that 1) A(t) is selfadjoint and
(A(t)x,x)>.,(x,x), 2) for 0,6 1, A-c~(t) is differentiable, where
V( d -0(
Cv-(t) - A (t) Tt A (t) are uniformly bounded with respect to 0~ and t.
3) CI(t) is strongly continuous in t and bounded. It is shown that under
certain conditions of 1) and 3) there follows the condition 2). Furthermores
x(t) - U(t,s)x0 satisfies the homogeneous equation for all x 0 e H. For
t> s and 0.4ry, 4 2 the operators A MU(t,s) are bounded, where
AO'(t)U(t,s)1j 4M(t-s)-t(. This estimation also holds for 0(- 2 if
IIC(t)-C(E;)II (- L)t-sj~ . The estimation holds for all Ot if A is constant.
If f(t) satisfies the condition Lip t with F_ 0 a positive bounded function is obtained,
for u -;Poo either a positive bounded function or zero is obtained.
b
Let Ake-f K[s,t, Y"t)]dt + f(s). Let the equation y- A%p,
a
where f(s) is a non-negative function, have a positive solution
,P* (S) -
Then the sequence b
Card 2/3 Yn+I (S) f K [a, t, T .(t) ] dt + f (s)
a

On the Theory of Equations With Concave Operators SOV/20-123-1-3/56
converges uniformly to f*(s) for every non-negative function
qc(s), 'fc(s)*O-
Two further theorems contain refinements of these assertions
for some special cases (e.g. for special fK1,1101 -concave
operators).
There are 8 Soviet references.
ASSOCIATIONiVoronezhekiy gosudarstvennyy universilet (Voronezh State University)
PRESENTED: June 9, 1958, by P.S.Aleksandrov, Academician
SUBMITTEDs May 10, 1958
Card 3/3

AUTHORS: Krasnosellskiy,M.A. and Perov,A.T. SOV/20-123-2-6/50
TITLE: On a Principle of the Existence of Bounded, Periodic and
Periodic Solutions of a System of Ordinary Differential Equations
(0b odnom printsipe sushchestvovaniya ogranichennykh,
periodicheskikh i pochti-periodicheskikh resheniy u. sistemy
obyknovennykh differentsiallnykh uravneniy')
PERIODICALs Doklady Akademii nauk SSSR, 1956, Vol -123,Nr 2,pp 235-238 (USSR.)
ABSTRAM Given the system
N A - f(t,x),
dt
where x - ) and f - (fi~'-Yf ), f f4(t,xlg-.";Xr
(XII ... 'xn n
and the fi are continuous in -ooct'x,,-...,X < +oD . Let further
~%(X))and r(x) be two continuously differentiable funciions,
,,(-x - NW,
(f(t,x),grad-X(x)) f > 0
X
for /Ix fl R >0. Let m - min -A (x), M max -A(x). On the set T
n hxII-R llxft-R
of those zC-E for which m :C--A(x)4M, jjrjj>H, let r(x) satisfy
Card 1113

On a Principle of the Existence of Bounded, Periodic and SOV/20-123-2-6, 50
A !moot, Periodic SolutionEi of a Systam of Orliriry
Diffl~rential Equations
the 4=dition. n
(f(t~x),grad~,(x)+grad~A ~X~ 0,
L
where lim Ir +f)3
xtT, ~ .11 _~* 4-C0
Theorem- Under the given ansumptions has at least one
uniformly bounded solution an If the f are per 4,jdiC
i I
in t' then (1) has at loast nr,,j V;iriocLic solution of the same
psriod~ If -the fi are almost--ppriodic (uniformly in every spiier,-)
in t, then (1) bas at least one ajuost~-per.iodlc solut;.or..
The theorem holas in a strengtheri,~;d form if i,nstead rf A(-x* ."'~x)
it is assumed that outside of a certain sphere grad-A(x) ~ 0
and that the field of grad -~,(x) has a nonvanishing rotation on
spheres of a sufficiently large radius (see [Ref 1, 71).
The proof of the theorems in based on the theoremz On the
boundary rof a bounded domain G(En let the vector fields
Card 2/3 f(t,x), -co-ct