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Key moments of the mutual influence of the Polish and Soviet schools of nonlinear functional analysis in the 1920s--1950s

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Bogatov E. M.

Antiquitates Mathematicae

2017

tom 11

Artykuł zawiera kluczowe aspekty historii rozwoju metod jakościowych nieliniowej analizy funkcjonalnej (twierdzenia o punkcie stałym, teoria stopnia odwzorowania, teoria przestrzeni Orlicza), w~ramach których najbardziej uwidoczniły się wzajemne oddziaływania polsko-radzieckie. Do połowy XX wieku w dziedzinach tych dokonał się postęp za sprawą przedstawicieli lwowskiej szkoły matematycznej, warszawskiej szkoły matematycznej, moskiewskiej szkoły matematycznej i odeskiej szkoły matematycznej. Analiza ich rezultatów stanowi główną część niniejszej pracy. Szczególną uwagę zwracamy na badania matematyka radzieckiego M. A. Krasnosel'skiego, który korzystał z wyników J. Schaudera i K. Borsuka oraz z przestrzeni Orlicza przy rozwiązywaniu szerokiej klasy problemów z jakościowej teorii nieliniowych równań całkowych.

The paper includes key aspects of the history of development of qualitative methods of non-linear functional analysis (fixed point theorems, theory of mapping degree, theory of Orlicz spaces), within the framework of which Polish-Soviet interaction was most clearly manifested. In the period up to the mid-twentieth century these areas were advanced by the representatives of Lvov Mathematical School, Warsaw Mathematical School, Moscow Mathematical School and Odessa Mathematical School. Examination of their results is a significant part of this work. Much attention is paid to the investigations of the Soviet mathematician M.A. Krasnosel'skii, who used the achievements of J. Schauder and K. Borsuk as well as Orlicz spaces for solving a wide class of problems in qualitative theory of nonlinear integral equations.

On the history of variational methods of non-linear equations investigations and the contribution of Soviet scientists (1920s-1950s).

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Bogatov E. M.

Antiquitates Mathematicae

2020

Variational methods for solving nonlinear equations (differential, integral, etc.) are perhaps the most common methods at the present time. However, the history of their origin and development, both in the USSR and in the whole world, has not been studied enough. The author attempts to fill this gap, limiting himself mainly to 1920's-1950's, starting with Hilbert’s works on justification of Dirichlet principle and with Poincaré’s work on the study of closed geodesics on convex surfaces. The article includes an analysis of the results of Soviet mathematicians - L.А. Lyusternik, L.G. Shnirelman, S.L. Sobolev, M.A. Krasnoselskii, M.M. Vainberg et al., obtained both in the field of solvability and direct methods, and in the field of qualitative analysis. Their achievements are gauged in the context of the development of this topic in the works of their foreign colleagues - L. Tonelli, W. Ritz, L. Lichtenstein, G.D. Birkhoff, M. Morse, A. Hammerstein, M. Golomb, etc. Along the way, the question of the origin of Sobolev spaces and the mutual influence of functional analysis and variational methods for solving operator equations is investigated. In conclusion, the author gives an example of applying the qualitative theory of variational methods to the problems of nonlinear mechanics that was realized by the Soviet scientists I.I. Vorovich.

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2 Antiquitates Mathematicae

2 Bogatov E. M.

1 2020

1 2017

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Key moments of the mutual influence of the Polish and Soviet schools of nonlinear functional analysis in the 1920s--1950s

On the history of variational methods of non-linear equations investigations and the contribution of Soviet scientists (1920s-1950s).