Abstract:
In the USSR, the development of the calculus of variations as a whole (L.A. Lyusternik, L.G. Shnirelman, 1929) began historically first. Somewhat later the theory of semi-ordered spaces (L.V. Kantorovich, 1935), the fixed point method (A.N. Tikhonov, 1935; A.A. Markov, 1936), the theory of cones (M.G. Krein, 1937), and the theoretical substantiation of approximate methods for solving operator equations (L.V. Kantorovich, 1939) were appeared. In the late 1940s the listed branches of nonlinear functional analysis were supplemented by the theory of Orlicz spaces (M.A. Krasnoselskii, Ya.B. Rutitskii), and in the early 1950s. – direct methods of the calculus of variations (M.M. Vainberg, M.A. Krasnoselskii) and the theory of the degree of mapping (M.A. Krasnoselskii). If we supplement this list with the theory of branching and bifurcations (M.A. Krasnoselskii - the middle of the 1950s, V.A. Trenogin - the end of the 1950s) as well as the prehistory and evolution of these sections until the early 1970s, then we get a whole picture of the half-century history of nonlinear functional analysis in the USSR. The author has set as his goal to reconstruct this picture with an emphasis on the development of qualitative methods, ideas and approaches to the study of general operator equations.
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