On the history of the fixed point method and the contribution of the soviet mathematicians (1920s--1950s.)

E. M. Bogatov

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 30-55.

Résumé

Goal. The aim of the paper is studying of Russian mathematicians contribution (V.V. Nemytskii, A.N. Tikhonov, A.A. Markov, M.G. Krein, V.L. Shmul'yan, etc.) to the development of the fixed point method for the period from the beginning 1920's until the late 1950's. Method. The reseach is based on an analysis of the original works of the listed scientists in the context of the worldwide development of nonlinear functional analysis against the backdrop of the achievements of American (J. Birkhoff, O. Kellogg), Polish (S. Banach, S. Mazur, J. Schauder, K. Borsuk, ), Italian (R. Cacciopolli), French (J. Leray) and German (E. Rothe) mathematicians. Result. The contribution of the Soviet scientists in the field of fixed point method is comparable with that of the rest of the world mathematical community in the period under review. This is confirmed both by the number of proved fixed-point theorems and by their quality. Due to the efforts of the Soviet mathematician M.A. Krasnosel'skii from the mid-1950's a fixed point method became a general method for solving a wide class of problems of a qualitative nature for a nonlinear operators analysis (until this time, the method under discussion was considered only as a tool for proving of the solvability of nonlinear integral or differential equations and their systems abstract analogues). Discussion. An analysis of the achievements at the area of the fixed-point method in the global context has shown that the development of nonlinear functional analysis (as, indeed, of any other section of mathematics) is a supranational process that is carried out by the efforts of mathematicians from different countries. This process goes beyond any scientific school, no matter how large it may be.

Détail

Keywords: history of nonlinear functional analysis, fixed point method, Schauder theorem, Tikhonov-Schauder theorem, Markov-Kakutani theorem, Leray-Schauder mapping degree, Krasnosel'skii theorem, Krein-Shmuljan theorem, nonlinear integral equations, topological analysis methods.

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