Mark Krasnosel'skii was born in the town of Starokostiantyniv in Ukraine on the 27 April 1920 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 Mark 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. He graduated in 1942, in the middle of the war, served four years in the Soviet Army, became Candidate in Science in 1948, with a dissertation on self-adjoint extensions of operators with nondense domains, before getting the title of Doctor in Science in 1950, with a thesis on investigations in Nonlinear Functional analysis.
Mark Krasnosel'skii's has authored or co-authored some three hundred papers and fourteen monographs. Nonlinear techniques are roughly classified into analytical, topological and variational methods. Mark Krasnosel'skii has contributed to all three aspects in a significant way, as well as to their application to many types of integral, differential and functional equations coming from mechanics, engineering, and control theory. 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 ofpartial 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 desire to always 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.
Selected works
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, Translation of Mathematical Monographs, 19, 294p.
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, Grundlehren Der Mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, 263, 409p.
