The outstanding mathematician Mark Aleksandrovich Krasnosel'skii passed away on February 13, 1997.

M.A.Krasnosel'skii was born on April 27, 1920 in the Ukranian town of Starokonstantinov. His father Aleksandr Yakovlevich worked in the capacity of civil engineer of the ``Azovrybtrest Co.'' His mother Fanni Moiseevna taught Russian in high school. The Krasnosel'skiis had two sons. The elder, Iosif became a well-known metallurgist; he was one of the founders and leaders of the Moscow Factory of Special Alloys. The cadet, Mark chose the career of mathematician.

In 1932 the Krasnosel'skiis moved to Berdyansk. In 1938 Mark Aleksandrovich finished high school and entered the Department of Physics and Mathematics of the Kiev University. In connection with the beginning of the war, the Kiev University was moved in 1941 to Kazakhstan where it was renamed the United Ukrainian University. M.A.Krasnosel'skii graduated from this University in 1942 and joined the Soviet Army where during four year he was an instructor at the Ryazan Artillery School which during the war was evacuated to the town of Talgar in the Alma-Ata region. In 1946 Mark Aleksandrovich was separated as a lieutenant and in Augist moved to Kiev where he first taught descriptive geometry at the Kiev Highway Institute and then became a junior scientist at the Institute of Mathematics of the Ukrainian Academy of Sciences.

In the post-war Kiev, M.A.Krasnosel'skii gets into the mainstream of scientific life. He attends lectures and participates in seminars of outstanding scientists such as N.N.Bogolyubov, A.N.Kolmogorov, M.G.Krein, B.V.Gnedenko, M.A.Lavrentiev, A.Yu.Ishlinskii, N.V.Efimov, A.G.Kurosh, V.E.Loshkarev, et al.

M.A.Krasnosel'skii defended in 1948 the candidate thesis on to the theory of extension of Hermitean operators and in 1950 the doctoral thesis on topological methods of nonlinear analysis. In 1953 M.A.Krasnosel'skii moved to Voronezh where during fifteen years he headed the Chair of Functional Analysis of the Department of Physics and Mathematics (later, Mathematics and Mechanics) of the Voronezh University where his seminar on nonlinear analysis --- known well beyond Voronezh --- was started. The Voronezh period of the scientific activity of M.A.Krasnosel'skii was extremely fruitful. The focus of his scientific interests was continually increasing and included many chapters of the current mathematics. M.A.Krasnosel'skii was the initiator of a number of paths of research which underlie the modern nonlinear analysis. At the Faculty, he read main and special courses and conducted seminars.

In 1968, M.A.Krasnosel'skii moved from Voronezh to Moscow to join the Institute of Automation and Remote Control (later the Institute of Control Sciences) of the USSR Academy of Sciences where he headed the laboratory of ``Mathematical methods for analysis of complex systems.'' The specificity of the Institute of Control Sciences was reflected in the application-oriented nature of some areas of research of M.A.Krasnosel'skii during the Moscow period such as the control theory, mathmatical models of hysteresis, etc.

In 1990 M.A.Krasnosel'skii joins the Institute of Information Transmission Problems of the USSR Academy of Sciences where, as at the Institute of Control Sciences, his firstrate results in the abstract mathematical areas alternated with application-oriented studies such as the dynamics of systems with hysteresis, desychronised pulse systems, systems with incomplete corrections, etc.

During his more than semicentennial scientific activity, M.A.Krasnosel'skii wrote over three hundred papers and fourteen monographs. The main areas of his studies are outlined below.

In connection with the classical problems of von Neumann, M.A.Krasnosel'skii constructed the general theory of extensions of Hermitean operators with nondense domain of definition. He proved theorems about the invariance of defective numbers of non-Hermitean operators, established an unexpected connection of the invariance problem with the Lyusternik--Shirel'man theory of categories, and in collaboration with M.G.Krein proposed new attributes of invariance of the Noetherian index.

Mark Aleksandrovich was the author of one of the first works on the functional properties of fractional powers of self-conjugate operators. Later, in collaboration with P.E.Sobolevskii he extended the theorems obtained to the non-self-conjugate operators in non-Hilbertian spaces. The method of fractional powers is widely used today to study various boundary problems of the mathematical physics, hydrodynamic problems, etc.

The M.G.Krein theory of cones and positive operators was developed by M.A.Krasnosel'skii in new directions in collaboration with P.P.Zabreiko, E.A.Lifshits, Yu.V.Pokornii, A.V.Sobolev, V.Ya.Stetsenko, et al. Here, new classes of operators with leading simple eigenvalues were extracted, spectral interspaces estimated, some geometrical problems were solved, etc. As it turned out later, the identified classes encompass the operators of many problems of mathematical physics such as the problem of neutron multiplication.

The method of fractional powers of operators and the Krasnosel'skii theorem about the interpolation of the property of complete continuity provided the starting point for a number of studies of the theory of interpolation.

During many years, M.A.Krasnosel'skii studied together with Ya.B.Rutitskii, P.P.Zabreiko, et al. the properties of integral operators in various functional spaces. He proposed various attributes of continuity and complete continuity, differentiability over the entire space and at individual points, concavity, convexity, etc., of integral and superposition operators, and in collaboration with with A.V.Pokrovskii, established the unexpected properties of discontinuous integral operators. The system of theorems makes up a harmonic theory that found application in studying different integral equations.

Mark Aleksandrovich established that the Oplicz spaces offer a convenient tool for studying the integral equations with strong (of the exponential type) nonlinearities. In this connection, M.A.Krasnosel'skii and Ya.B.Rutitskii radically reconstructed the theory of Oplicz spaces, they extracted and studied special classes of convex functions, and established special properties of the operators in these spaces.

The topological methods of nonlinear analysis developed by Birkhoff, Kellogg, Leray, Schauder, Tikhonov, Nemytskii, et al. were oriented mostly to proving the existence theorems. In the works of M.A.Krasnosel'skii, the topological methods became a universal key to most different qualitative questions such as estimates of the number of solutions, structure of the set of solutions and conditions for its connectivity, convergence of approximate Galerkin-type methods, branching and bifurcation of solution of nonlinear problems, etc. The results obtained by Mark Aleksandrovich in this domain are well known and find wide application.

Together with P.P.Zabreiko, I.A.Bakhtin, V.V.Strygin, E.M.Mukhammadiev, E.A.Lifshits, et al., M.A.Krasnosel'skii proposed new general principles for solvability of nonlinear equations such as the principles of unilateral estimates, principle of cone stretching and compression, pronciple of the drop, first theorems about the fixed points of monotone operators, union of the Schauder principle and that of compressed maps (which underlies the rapidly developing theory of compressing operators), principle of partial conversion, etc. Proceeding from the well-known work of P.S.Urysov on integral equations, M.A.Krasnosel'skii and his students developed the theory of equations with concave operators that abounds in numerous fine facts. This theory was extensively used in the studies of various boundary problems, oscillation theory, stability studies, market models, analysis of processes in nuclear reactors, etc.

Mark Aleksandrovich put much effort to develop methods for efficient calculation of various topological characteristics of maps in infinite-dimensional spaces. Together with P.P.Zabreiko, E.A.Lifshits, V.V.Strygin, N.A.Bobylev, et al., he proposed new theorems about periodic maps and special coverings of spheres, an algorithm to calculate the index of special point in degenerate cases, principles of relatedness and invariance of rotation (relating the characteristics of various equations generated by the same problem), etc. The ability of the general methods of nonlinear analysis to calculate the topological characteristics makes them an efficient analytical tool for many particular problems.

For variational problems, M.A.Krasnosel'skii formulated the problem of stability of their solutions to a wide range of perturbations and solved it for some situations. To this end, he introduced, investigated, and for some cases calculated a new characteristic --- the genus of set. The apparatus developed is used by many researchers.

Recently, M.A.Krasnosel'skii together with N.A.Bobylev, A.M.Dement'eva, V.M.Krasnosel'skii, and E.M.Mukhamadiev proposed a general method for studying degenerate extremals. As applied, for example, to the analysis of the degenerate extremal in the classical problem of Euler, the method requires that the first non-zero number in some diagram be found. The method has found various applications. M.A.Krasnosel'skii was the author of a series of theorems on the applicability of variational schemes in the general nonlinear analysis.

M.A.Krasnosel'skii proposed qualitative methods for investigating the critical and bifurcational values of parameters which make use of very limited information about the equation under study. These methods gained wide popularity and extensive application in hydrodynamics, for studying the forms of losing stability of elastic systems, in the problems of autooscillations, etc. To analyze the bifurcational values of parameters, for example, in many cases it suffices to know only the properties of equations linearized in zero or infinity. The qualitative methods enable one to detect the families of nontrivial solutions, study the range of nonlinear problems, etc.

To analyze the branching of the solutions of general nonlinear operator equations, M.A.Krasnosel'skii (in collaboration with P.P.Zabreiko) developed the method of simple solutions, proposed their asymptotic representations, and established natural relationships between the methods of Lyapunov, Schmidt, and Nekrasov.

The method of parameter functionalization that was proposed by Mark Aleksandrovich to get rid of the parameters in nonlinear problems must be noted in its own right. It found application in various problem; in particular, it allowed one to obtain the far-reaching generalizations of the well-known Hopf theorem about the origination of autooscillatory modes from the equilibrium state.

M.A.Krasnosel'skii established various attirbutes of uniqueness, nonlocal continuability, stability, correctness, dissipativeness, solvability, etc., of the Cauchy and various boundary problems. In collaboration with S.G.Krein and P.E.Sobolevskii, he obtained the first general theorems on solvability of nonlinear differential equations with unlimited operators in functional spaces. Mark Aleksandrovich together with A.I.Perov, V.V.Strygin, N.A.Bobylev, and E.M.Mukhamadiev proposed and developed the method of direction potentials for studying periodic oscillations and constrained modes in various nonlinear systems.

M.A.Krasnosel'skii in collaboration with V.Sh.Burd and Yu.S.Kolesov developed basically new methods for analyzing almost-periodic nonlinear oscillations as applied to the pendulum theory, problem of self-regulation, etc. He proposed (together with S.G.Krein) a new approach to substantiating the main theorems of the method of Bogolyubov--Krylov averaging and pointed out to its new applications to the bifurcation theory and other problems. Together with A.V.Pokrovskii, he proposed an original approach to studying the absolute stability which requires only that uniqueness of some explicitly written equations be analyzed. This approach enables one to study the absolute stability of some systems with many nonlinear units. M.A.Krasnosel'skii proposed together with N.A.Bobylev new methods for studying the Lyapunov systems.

Some publications of M.A.Krasnosel'skii (in part, together with Ya.B.Rutitskii, V.A.Chechik, et al.) were devoted to the general theory of applied methods, the applicability of the Galerkin, Galerkin - Petrov, and Ritz methods to nonlinear problems, schemes of a posteriori estimates of the errors of approximate solutions, etc. Mark Aleksandrovich developed new methods for determining the estimates of the spectral radii of linear operators which are important for analyzing the rate of convergence of iterative procedures, proposed new variants of the Seidel - Nekrasov method, etc. M.A.Krasnosel'skii together with S.G.Krein developed the method of minimal mismatches of the solution of nonlinear problem which found wide application. In collaboration with I.V.Emelin and N.P.Panskikh, he developed the spurt method based on the ideas of control of variable-structure systems. Together with I.V.Emelin he studied the method of stopping iterative procedures which regularizes a wide class of incorrect problems. Together with A.V.Pokrovskii, he developed and investigated the method of shuttle iterations as applied to solving approximatly various boundary problems and studying the oscillations in variable-structure systems (in case of discontinuous operators).

In collaboration with N.A.Bobylev and N.A.Kuznetsov, M.A.Krasnosel'skii considered in detail the method of harmonic balance which is of considerable engineering importance. The range of its applicability was described, the convergence theorems proved, and the convergence rate estimated.

M.A.Krasnosel'skii developed new approaches to studying the dynamics of control systems with nonlinear units having discontinuious characteristics. A series of exquisite theorems allows one to find in the equations with discontinuous operators solutions that are correct in various perturbations, to estimate their number, to determine them with high accuracy, to take into account the inevitable small noise in systems, etc. The method developed is applicable to variable-structure systems, problem such as the M.A.Lavrentev problem of break-off flows, etc.

In the mid-1970s M.A.Krasnosel'skii suggested an extensive project for studying systems with hysteresis and invited a large group of his students (A.V.Pokrovskii, V.S.Kozyakin, P.P.Zabreiko, A.F.Kleptsyn, E.A.Lifshits, N.I.Grachev, D.I.Rachinskii, V.V.Chernorutskii, et al.) to participate in it. The project was based on special mathematical operations corresponding to different phenomenological models of hysteresis in the plasticity theory, magnetism, etc. Its realization demanded that some surprising problems be solved: vibrostable equations were identified and solved, possibility of extracting the individual trajectories of stochastic differential equations that correspond to individual Wiener processes was investigated, significance of the Frobenius conditions of full integrability was studied for the stochastic equations, etc. Virtually all classical models of hysteresis fit into the mathematical theory constructed, which enabled one to reduce the phenomenological models of (constructive, magnetic, plastic, etc.) hysteresis to usable mathematical models.

In recent years, M.A.Krasnosel'skii together with E.A.Asarin, A.F.Kleptsyn, V.S.Kozyakin, and N.A.Kuznetsov was actively engaged in the theory of desynchronized systems. He proposed methods of qualitative analysys of desynchronized systems, developed an apparatus for studying their stability, and identified engineering applications.

Understandably, a sufficiently full presentation of the scientific results obtained by M.A.Krasnosel'skii cannot be made within the framework of this obituary. The above survey is very brief and only touches upon the main areas of the scientific activity of M.A.Krasnosel'skii.

The scientific activity of M.A.Krasnosel'skii always went hand in hand with his pedagogocal work. The desire and ability of M.A.Krasnosel'skii to attract gifted young people to the research activity made themselves manifest from the beginning of his research work. The reserve of scientific enthusiasm and optimism obtained by the students of M.A.Krasnosel'skii during the years of communication and joint research inspires them for many years. Scores of his students have scinetific degrees; over thirty of them are doctors of science, professors leading their own research domains and scientific schools.

Mark Aleksandrovich actively worked until his last days and was full of energy and plans.

*
E.A.Asarin,
I.A.Bakhtin,
N.A.Bobylev,
V.A.Bondarenko,
V.Sh.Burd,
V.V.Chernorutskii,
S.V.Emel'yanov,
E.A.Gorin,
L.A.Ivanov,
V.S.Kozyakin,
A.M.Krasnosel'skii,
A.B.Kurzhanskii,
N.A.Kuznetsov,
A.Yu.Levin,
E.M.Mukhamadiev,
A.I.Perov,
Yu.V.Pokornii,
A.V.Pokrovskii,
D.I.Rachinskii,
B.I.Sadovskii,
V.V.Strygin,
Ya.Z.Tsypkin,
P.P.Zabreiko
*