Mark A. Krasnosel’skii

Mark A. Krasnosel’skii


Tuesday, April 27, 1920

Passed away: 

Thursday, February 13, 1997

Mark Krasnosel’skii, in full Mark Alexandrovich Krasnosel’skii, was an outstanding mathematician, a founder of the modern approach to nonlinear analysis, and a talented and unusually caring teacher who rose several generations of highly professional and gifted researchers.

He was born in Starokonstantinov, Ukraine. After leaving school, Krasnosel’skii entered the Department of Physics and Mathematics at Kyiv University. He graduated in 1942 in Kazakhstan, where the University was evacuated with the beginning of the Great Patriotic War. After graduation, Krasnosel’skii served in the Soviet Army for 4 years: he taught at the Ryazan Artillery School, evacuated in the War years to the Talgar town of the Almaty Region. In 1946, Krasnosel’skii was demobilized with the rank of lieutenant; in August of the same year, he moved to Kyiv. He worked for a few months as a descriptive geometry teacher at the Kyiv Automobile and Road Institute and then as Junior Researcher at the Institute of Mathematics, the Ukrainian Academy of Sciences.

During the first postwar years in Kyiv, Krasnosel’skii found himself in a center of the turbulent scientific life of the USSR. At the Institute of Mathematics, he listened to lectures and participated in seminars of outstanding researchers, namely, N.N. Bogolyubov, A.N. Kolmogorov, M.G. Krein, B.V. Gnedenko, M.A. Lavrent’ev, A.Yu. Ishlinsky, N.V. Efimov, A.G. Kurosh, V.E. Loshkarev, and many others.

In 1948, Krasnosel’skii defended his candidate’s dissertation on the expansion theory of Hermitian operators; in 1950, his doctoral dissertation on topological nonlinear analysis methods.

In 1953, he moved to Voronezh and headed for 15 years the Department of Functional Analysis at Voronezh University (first, within the Faculty of Physics and Mathematics; later, within the Faculty of Mathematics and Mechanics). In Voronezh, he organized a seminar on nonlinear analysis, which became popular far beyond the city. Krasnosel’skii’s scientific activities in Voronezh were very fruitful. His research interests were constantly expanding and covered many branches of modern mathematics. Krasnosel’skii discovered several lines of studies that subsequently formed the foundations of modern nonlinear analysis. He taught basic and special courses at the Department as well as led seminars.

In 1968, Krasnosel’skii moved from Voronezh to Moscow and joined the Institute of Automation and Remote Control (IARC), the USSR Academy of Sciences. At IARC, he organized the Laboratory of Mathematical Analysis Methods for Complex Systems and invited his favorite Voronezh students: A.V. Pokrovskii, N.A. Bobylev, and others. The range of R&D activities at IARC soon determined many application-oriented studies of Krasnosel’skii and his team: the Laboratory began to investigate the nonlinear problems of control theory, mathematical models of hysteresis, and many others. At the Institute, he also delivered cycles of lectures on functional analysis and modern nonlinear analysis methods. During the lectures, the small conference hall of IARC was crowded.

In the mid-1970s, Krasnosel’skii proposed an extensive program for studying systems with hysteresis and involved 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, and others). The program was connected with introducing special mathematical operations to formalize various purely phenomenological hysteresis models in plasticity theory, magnetism, etc. Program implementation required solving several unusual mathematical problems as follows: 

  • Vibrationally stable equations were singled out and studied. 
  • The extraction of individual trajectories corresponding to particular Wiener processes from the solutions of stochastic differential equations was analyzed.
  • The role of the Frobenius conditions for stochastic equations was investigated, etc. 

The resulting theory covered almost all classical hysteresis models. It allowed reducing phenomenological models of hysteresis (structural, magnetic, plastic, etc.) into applied mathematical models.

In 1990, Krasnosel’skii moved to the Institute for Information Transmission Problems (IITP), the USSR Academy of Sciences. Nikolai A. Bobylev, one of his favorite students, became Head of the Laboratory of Mathematical Analysis Methods for Complex Systems at IARC. At IITP, Krasnosel’skii’s world-class results in abstract mathematics were also interspersed with applied research (the dynamics of systems with hysteresis, the theory of desynchronized pulse systems, systems with incomplete corrections, etc.).

For more than half a century of his R&D activities, Krasnosel’skii wrote over 300 scientific papers and 14 monographs.

His students include great mathematicians, e.g., N.A. Bobylev, P.P. Zabreiko, E.A. Lifshits, and A.V. Pokrovskii, to name a few. Over 30 of them became Dr. Sci. (Phys.—Math.). Krasnosel’skii never grudged his time for talented people. Occasionally, he turned the lessons in mathematics into round-the-clock activities. For this purpose, Krasnosel’skii and his students shared a common hobby: fishing. Gathering his favorite students to go fishing, he had endless conversations with them about mathematics between baiting and catching.

Krasnosel’skii’s main books are as follows:

  1. Analiz ustoichivosti rassinkhronizovannykh diskretnykh sistem (Stability Analysis of Desynchronized Systems), Moscow: Nauka, 1992. — 408 p. (coauthors E.A. Asarin, V.S. Kozyakin, and N.A. Kuznetsov);
  2. Positive Linear Systems: The Method of Positive Operators, Sigma Series in Applied Mathematics, vol. 5, Berlin: Hilderman Verlag, 1990. — 354 p. (coauthors Je.A. Lifshits and A.V. Sobolev);
  3. Systems with Hysteresis, Berlin—Heidelberg—New York—London—Paris—Tokyo: Springer-Verlag, 1989. — 410 p. (coauthor A.V. Pokrovskii);
  4. Pozitivnye lineinye sistemy: metod polozhitel’nykh operatorov (Positive Linear Systems: The Method of Positive Operators), Moscow: Nauka, 1985. — 256 p. (coauthors E.A. Lifshits and A.V. Sobolev);
  5. Geometrical Methods of Nonlinear Analysis, A Series of Comprehensive Studies in Mathematics, vol. 263, Berlin—Heidelberg—New York—Tokyo: Springer-Verlag, 1984. — 409 p. (coauthor P.P. Zabreiko);
  6. Sistemy s gisterezisom (Systems with Hysteresis), Moscow: Nauka, 1983. — 272 p. (coauthor A.V. Pokrovskii);
  7. Integral Operators in Spaces of Summable Functions, Leyden: Noordhoff, 1976. — 520 p. (coauthors P.P. Zabreiko, E.I. Pustylnik, and P.E. Sobolevskii);
  8. Geometricheskie metody nelineinogo analiza (Geometrical Methods of Nonlinear Analysis), Moscow: Nauka, 1975. — 512 p. (coauthor P.P. Zabreiko);
  9. Integral Equations, Leyden: Noordhoff, 1975. — 443 p. (coauthors P.P. Zabreiko, A.I. Koshelev, S.G. Mikhlin, L.S. Rakovshchik, and V.Ya. Stetsenko);
  10. Nonlinear Almost Periodic Oscillations, New York: J. Wiley, 1973. — 366 p. (coauthors V.S. Burd and Ju.S. Kolesov);
  11. N"aherungsverfahren zur L"ozung von Operatorgleichungen, Berlin: Academie-Verlag, 1973. — 324 p. (coauthors G.M. Wainikko, P.P. Sabreiko, J.B. Rutizki, and W.J. Stezenko);
  12. Rownania calkowe, Warszawa: Panstwowe Wydawnictwo Naukowe, 1972. — 456 p. (coauthors A.I. Koszelew, S.G. Michlin, L.S. Rakowszczik, W.J. Stiecenko, and P.P. Zabrejko);
  13. Functional Analysis, Groningen: Wolters-Noordhoff, 1972. — 379 p. (with many coauthors);
  14. Approximated Solutions of Operator Equations, Groningen: Walters—Noordhoff, 1972. — 484 p. (coauthors G.M. Vainikko, P.P. Zabreiko, Ja.B. Rutitcki, and V.Ja. Stecenko);
  15. Nelineinye pochti periodicheskie kolebaniya (Nonlinear Almost Periodic Oscillations), Moscow: Nauka, 1970. — 352 p. (coauthors V.Sh. Burd and Yu.S. Kolesov);
  16. Priblizhennoe reshenie operatornykh uravnenii (Approximate Solution of Operator Equations), Moscow: Nauka, 1969. — 456 p. (coauthors G.M. Vainikko, P.P. Zabreiko, Ya.B. Rutitskii, and V.Ya. Stetsenko);
  17. Integral’nye uravneniya (Integral Equations), Moscow: Nauka, 1968. — 448 p. (coauthors P.P. Zabreiko, A.I. Koshelev, S.G. Mikhlin, L.S. Rakovshchik, and V.Ya. Stetsenko);
  18. The Operator of Translation along the Trajectories of Differential Equations, Translation of Mathematical Monographs, vol. 19, Providence: American Mathematical Society, 1968. — 294 p.;
  19. Analiza funkcionalna, Warszawa: Panstwowe Wydawnictwo Naukowe, 1967. — 419 p. (with many coauthors);
  20. Operator sdviga po traektoriyam differentsial’nykh uravnenii (The Operator of Translation along the Trajectories of Differential Equations), Moscow: Nauka, 1966. — 331 p.;
  21. Integral’nye operatory v prostranstvakh summiruemykh funktsii (Integral Operators in Spaces of Summable Functions), Moscow: Nauka, 1966. — 500 p. (coauthors P.P. Zabreiko, E.I. Pustyl’nik, and P.E. Sobolevskii);
  22. Plane Vector Fields, New York: Academic Press, 1966. — 242 p. (coauthors A.I. Perov, A.I. Povolotskii, and P.P. Zabreiko);
  23. Vektorfelder in der Ebene, Berlin: Akademie-Verlag, 1966. — 197 p. (coauthors A.I. Perow, A.I. Powolozki, and P.P. Sabrejko);
  24. Topological Methods in the Theory of Nonlinear Integral Equations, Oxford—London—New York—Paris: Pergamon Press, 1964. — 395 p.;
  25. Positive Solutions of Operator Equations, Groningen: P. Noordhoff, 1964. — 381 p.
  26. Funktsional’nyi analiz (Functional Analysis), Krein, S.G., Ed., 1st ed., Moscow: Nauka, 1964. — 424 p.; 2nd ed., Moscow: Nauka, 1972. — 544 p. (with many coauthors);
  27. Vektornye polya na ploskosti (Plain Vector Fields), Moscow: Fizmatgiz, 1963. — 248 p. (coauthors A.I. Perov, A.I. Povolotskii, and P.P. Zabreiko);
  28. Convex Functions and Orlicz Spaces, Groningen: P. Noordhoff, 1961. — 249 p. (coauthor Ja.B. Rutickii);
  29. Vypuklye funktsii i prostranstva Orlicha (Convex Functions and Orlicz Spaces), Moscow: Fizmatgiz, 1958. — 271 p. (coauthor Ya.B. Rutitskii);
  30. Topologicheskie metody v teorii nelineinykh integral'nykh uravnenii (Topological Methods in the Theory of Nonlinear Integral Equations), Moscow: Gostekhizdat, 1956. — 392 p.

Many of these books are presented in the Institute’s database:Красносельский+М.А.

The list of journal papers by Krasnosel’skii is available at:

His publications can be found at:

Articles about M.A. Krasnosel’skii

1. N. A. Bobylev, E. A. Gorin, A. Yu. Ishlinskii, S. P. Novikov, V. M. Tikhomirov, Mark Aleksandrovich Krasnosel’skii (Obituary)Uspekhi Mat. Nauk, 53:1(319) (1998), 199—201; Russian Math. Surveys53:1 (1998), 195—198.
2. E. A. Asarin, I. A. Bakhtin, N. A. Bobylev, V. A. Bondarenko, V. Sh. Burd, E. A. Gorin, S. V. Emelyanov, P. P. Zabreiko, L. A. Ivanov, V. S. Kozyakin, A. M. Krasnosel’skii, N. A. Kuznetsov, A. B. Kurzhanskii, A. Yu. Levin, E. M. Mukhamadiev, A. I. Perov, Yu. V. Pokornyi, A. V. Pokrovskii, D. I. Rachinskii, V. V. Strygin, Ya. Z. Tsypkin, V. V. Chernorutski, Memory of M. A. Krasnosel’skiiAvtomat. i Telemekh., 1998, no. 2, 179—184.
3. M. A. Krasnosel’skii (Obituary)Avtomat. i Telemekh., 1997, no. 3, 237.
4. N. N. Bogolyubov, N. A. Bobylev, P. P. Zabreiko, A. Yu. Ishlinskii, V. P. Maslov, Yu. A. Mitropol’skii, S. P. Novikov, Mark Alexandrovich Krasnosel’skii (on His Seventieth Birthday)Uspekhi Mat. Nauk45:2(272) (1990), 225—227; Russian Math. Surveys45:2 (1990), 231—234.
5. N. N. Bogolyubov, A. Yu. Ishlinskii, L. V. Kantorovich, B. N. Sadovskii, S. L. Sobolev, V. A. Trapeznikov, N. A. Bobylev, Mark Aleksandrovich Krasnosel’skii (on His Sixtieth Birthday)Uspekhi Mat. Nauk, 36:2(218) (1981), 215—220; Russian Math. Surveys36:2 (1981), 205—212.

Also, see the Wikipedia page devoted to Krasnosel’skii:

Scopus Author ID: 7003664706