This article is cited in 2 papers

MATHEMATICS

On the history of the positive operators (1900s - 1960s) and the contribution of M. A. Krasnosel'skii

E. M. Bogatov

Stary Oskol Technological Institute of National Research University of Science and Technology "MISIS";Branch of National Research University of Science and Technology «MISIS» in Gubkin town of Belgorod Region Gubkin

Abstract: Goal. The aim of the work is studying of the contribution of foreign and domestic mathematicians, in particular M. A. Krasnoselskii, to the development of the theory of linear and nonlinear positive operators for the period from the mid-1990s until the end of the 1960s. Method. The study is based on an analysis of original works of O. Perron, G. Frobenius, R. Jentzscli, P. S. Urysolm, M. G. Krein, M. A. R.utman, M. A. Krasnoselskii and others in the context of the global process of development of functional analysis. Result. The contribution of domestic scientists in the field of positive operators was larger than that of the rest of the world mathematical community in the period under review. Soviet mathematicians M. G. Krein and his student M. A. Rutman in the 1940s created the theory of cones and linear positive operators A in space of infinite dimension. They applied this theory to the study of the solvability of equations of the form Ax = $\lambda$x. . Thanks to the efforts of another Krein student - M. A. Krasnoselskii - the theory of positive operators has become a general method for solving a wide class of problems of a qualitative nature, related to the analysis of nonlinear operator equations, since the mid 1950s (proof of new fixed point theorems and theorems about the spectrum structure of the operator A), investigation of the bifurcation value of the parameter $\mu$ in the equation x = A(x,$\mu$), substantiation of the successive approximations method for the equation Ax = $\lambda$x. in a cone of Banach space for nonlinear operator A and so on). Besdides, in the framework of the theory created by Krasnoselskii, a number of important applied problems were solved. Discussion. Analysis of developments in the field of positive operators showed that in one country (USSR) may be formed conditions for the successful creation and development of a separate scientific field. Of great importance here is the scale of the scientists who stood at the origins of this direction - M. G. Krein and M. A. Krasnoselskii.

Keywords: history of nonlinear functional analysis, Perron-Frobenius theorem, Jentzscli theorem, Krasnosel'skii cone theorem, positive operators, theory of cones, Hammerstein equation, Urysolm equation.

UDC: 51 (091)

DOI: 10.18413/2687-0959-2020-52-2-105-127