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M.A.Krasnosel'skii

#### (27.04.1920 --- 13.02.1997)

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Mark Mark Alexandrovich Krasnosel'skii, renowned for his farreaching
and innovative work on nonlinear functional analysis and its applications,
died while working at his desk in Moscow on the 13th of February
1997, just 6 weeks short of his 77th birthday. He had spent parts of
1995 and 1996 as a Humboldt-Preis-Tr"ager at the Universit"at Regensburg
and the Weierstrass--Institut in Berlin. His ideas and results have had
a profound impact on mathematics and its applications throughout the international
mathematical community and in Germany in particular.

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Mark Krasnosel'skii was born on the 27th of April 1920 in the town
of Starokonstantinov in the Ukraine where his father worked as a
construction engineer and his mother taught in an elementary school. In
1932 the Krasnosel'skii family moved to Berdyansk and in 1938 entered the
physico-mathematical faculty of Kiev University, which was evacuated
at the beginning of World War II to Kazakhstan where it became
known as the Joint Ukrainian University. After graduating in 1942
Mark Krasnosel'skii spent the next four years as an artillery officier
in the Soviet Army and in August 1946 he returned to Kiev to commence
his graduate studies while holding a junior position in the Mathematical
Institute of the Ukranian Academy of Sciences. In 1948 Krasnosel'skii defended
his *Kandidat of Science* (equivalent to Ph.D.) dissertation
on extension theory for Hermite operators and in 1950 he completed his
*Doctor of Science* (similar to Habilitation) dissertation on topological
methods in nonlinear analysis. In 1953 he was appointed to a Chair in Functional
Analysis at the Voronezh University and during the 15 years that he stayed
there his seminar on nonlinear analysis became well-known far
beyond Voronezh. In 1968 Krasnosel'skii moved from Voronezh to Moscow where
he worked at the USSR Academy of Sciences, first at what is now called
the Institute of Control Sciences and then from 1990 until his death at
the Institute for Information Transmissions Problems.

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In over a half century of scientific activity Mark Krasnosel'skii
wrote more than 380 scientific articles and 14 monographs.
He was an indefatable hard worker, an outstanding teacher,
an energetic scientific organisor and an inspiration to all those
who came into contact with him. Throughout his scientific career he
worked and published extensively with many other mathematicians and scientists
in Russia and further a field. In addition, 33 of his students
completed the Doctor of Science degree. His scientific interests
were extensive and covered many aspects of modern mathematics.
In particular, he was responsible for opening up important new mathematical
directions, the development of which created the foundations of modern
nonlinear analysis.

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Mark Krasnosel'skii was the first to investigate the functional analytical
properties of fractional powers of operators, at first for self-adjoint
operators and then for more general situations. His theorem on the
interpolation of complete continuity of such fractional power operators
has been a basic tool in the theory of partial differential equations.
Of comparable importance in applications is his extensive collection of
works on the theory of positive operators, in particular results in which
spectral gaps were estimated. His work on integral operators and superposition
operators has also found many theoretical and practical applications. A
major reason for this was his ceaseless desire always to find readily
verifiable conditions and estimates for whatever functional properties
were under consideration. This is perhaps best seen in his work on topological
methods in nonlinear analysis which he developed into a
universal method for finding answers to such qualitative problems
such as evaluating the number of solutions, describing the structure
of a solution set and conditions for the connectedness of this set,
convergence of Galerkin type approximations, the bifurcation of solutions
in nonlinear systems, and so on. Krasnosel'skii also presented many new
general principles on solvability of a large variety of nonlinear equations,
including one-sided estimates, cone stretching and contractions,
fixed-point theorems for monotone operators and a combination of the Schauder
fixed point and contraction mapping theorems that was the genesis
of condensing operators. He suggested a new general method for investigating
degenerate extremals in variational problems and developed qualitative
methods for studying critical and bifurcation parameter values based on
restricted information of nonlinear equations such as the properties
of equations linearized at zero or at infinity, which have been very useful
in determining the existence of bounded or periodic solutions. After
he moved to Moscow he turned his attention increasingly to discontinuous
processes and operators, in connection firstly with nonlinear
control systems and then with a mathematically rigorous formulation of
hysteresis which encompasses most classical models of hysteresis and is
now standard. He also became actively involved with the analysis of desynchronized
systems and the justification of the harmonic balance method commonly used
by engineers. This summary of Mark Krasnosel'skii's mathematical
contributions is but a brief and selective sketch. A more thorough appreciation
is warranted and will undoubtedly appear in the future.

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Mark Krasnosel'skii was of noble personality, straightforward and uncompromising
in matters of principle but always open minded and ready to help
everyone. He was a Big Man in both body and mind. He was a superb
mathematician.

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P.E. Kloeden

Weierstrass-Institut, Berlin

with assistance from

N.A.Bobylev, V.S.Kozyakin, A.M.Krasnosel'skii,

A.V.Pokrovskii and B.I.Sadovsky.