Commemorative plaque to Mark Krasnoselsky in Voronezh

Mark Aleksandrovich Krasnoselsky (Russian: Ма́рк Алекса́ндрович Красносе́льский; 27 April 1920, Starokostiantyniv – 13 February 1997, Moscow) was a Soviet and Russian mathematician renowned for his work on nonlinear functional analysis and its applications.

Biography

Scientific achievements

Krasnoselsky has authored or co-authored some three hundred papers and fourteen monographs. Nonlinear techniques are roughly classified into analytical, topological and variational methods. Mark Krasnoselsky 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.

Krasnoselsky 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 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.

Krasnoselsky 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

1. Krasnosel'skii, M.A. (1964), Topological Methods in the Theory of Nonlinear Integral Equations, Oxford-London-New York City-Paris: Pergamon Press, 395p.
2. Krasnosel'skii, M.A.; Rutickii, Ya.B. (1961), Convex Functions and Orlicz Spaces, Groningen: P.Noordhoff Ltd, 249p.
3. Krasnosel'skii, M.A. (1964), Positive Solutions of Operator Equations, Groningen: P.Noordhoff Ltd, pp. 381 pp
4. Krasnosel'skii, M.A.; Perov, A.I.; Povolockii, A.I.; Zabreiko, P.P. (1966), Plane Vector Fields, New York City: Academic Press, 242p.
5. Krasnosel'skii, M.A.; Gorin, E.A.; Vilenkin, N.Ya.; Kostyuchenko, A.G.; Maslov, V.P.; Mityagin, B.S.; Petunin, Yu.I.; Rutitskij, Ya.B.; Sobolev, V.I.; Stetsenko, V.Ya.; Faddeev, L.D.; Tsitlanadze, E.S. (1972), Functional Analysis, Groningen: Wolters-Noordhoff Publ., 379p.
6. Krasnosel'skii, M.A. (1968), The Operator of Translation along the Trajectories of Differential Equations, Providence: American Mathematical Society, Translation of Mathematical Monographs, 19, 294p.
7. Krasnosel'skii, M.A.; Pustylnik, E.I.; Sobolevskii, P.E.; Zabreiko, P.P. (1976), Integral Operators in Spaces of Summable Functions, Leyden: Noordhoff International Publishing, 520 p.
8. Krasnosel'skii, M.A.; Koshelev, A.I.; Mikhlin, S.G.; Rakovshchik, L.S.; Stet'senko, V.Ya.; Zabreiko, P.P. (1975), Integral Equations, Leyden: Noordhoff International Publishing, 443p.
9. Krasnosel'skii, M.A.; Rutitcki, Ja.B.; Stecenko, V.Ja.; Vainikko, G.M.; Zabreiko, P.P. (1972), Approximate Solutions of Operator Equations, Groningen: Walters - Noordhoff Publ., 484p.
10. Krasnosel'skii, M.A.; Burd, V.S.; Kolesov, Ju.S. (1973), Nonlinear Almost Periodic Oscillations, New York City: John Wiley, 366p.
11. Krasnosel'skii, M.A.; Zabreiko, P.P. (1984), Geometrical Methods of Nonlinear Analysis, Berlin-Heidelberg-New York City-Tokyo: Springer Verlag, Grundlehren Der Mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, 263, 409p.
12. Krasnosel'skii, M.A.; Pokrovskii, A.V. (1989), Systems with Hysteresis, Berlin-Heidelberg-New York City-Paris-Tokyo: Springer Verlag, 410p.
13. Krasnosel'skii, M.A.; Lifshits, Je.A.; Sobolev, A.V. (1990), Positive Linear Systems: The method of positive operators, Sigma Series in Applied Mathematics, vol. 5, Berlin: Helderman Verlag, pp. 354 pp
14. Asarin, E.A.; Kozyakin, V.S.; Krasnosel'skii, M.A.; Kuznetsov, N.A. (1992), Asynchronous system stability analysis, Moscow: Nauka, 408p., [Russian].

References

1. The article is based on official obituaries, see those by Prof. P.E. Kloeden and Prof. E.A. Asarin, et al.
2. List of selected papers
3. Book of memoires
4. Complete papers (pdf): v.1,v.2, v.3, v.4, v.5, v.6, v.7