Korpusov M.O., Ovchinnikov A.V., Panin A.A. Lectures In Nonlinear Functional Analysis: Synopsis Of Lectures Given At The Faculty Of Physics Of Lomonosov Moscow State University

Singapore: World Scientific Publishing Company, 2022. — 376 p. — ISBN 9811248923.This monograph is addressed to mathematicians and physicists who apply methods of nonlinear functional analysis to the study of various problems of mathematical physics. It is based on lectures given by the authors for graduate and undergraduate students specializing in mathematical physics at the Faculty of Physics of the M. V. Lomonosov Moscow State University. We introduce and discuss in detail the notions of Gateaux and Frechet derivatives and various types of continuity of non-linear operators. We describe variational methods, including M. A. Krasnoselsky’s theory of genus of a set. Galerkin’s method combined with the monotony and compactness methods, the method of lower and upper solutions, and the topological Schauder principle are presented. The monograph ends with chapters devoted to the abstract Picard theorem, which generalizes the classical theorem on the existence and uniqueness of a solution to an ordinary differential equation, and its applications to Sobolev-type equations.True PDF