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Solvability of nonlinear equations and boundary value problems. (English) Zbl 0453.47035
These lecture notes formed the basis of a course “Nonlinear Problems” given by the author at the Department of Mathematical Analysis of Charles University in Prague during the school year 1976–77. The subject of interest is the study of the solvability of nonlinear operator equations. The text is not intended as a textbook. Auxiliary facts about linear differential equations, function spaces, the properties of the Leray-Schauder degree as well as the main results about the surjectivity of monotone coercive operators are used without proofs. The exposition proceeds so that before general results are presented, the method is illustrated by applying it to the Dirichlet problem for a simple ordinary differential equation of the second order. Where possible, the general results are applied to boundary value problems for nonlinear ordinary differential equations as well as for nonlinear partial differential equations. They are also used to prove the existence of periodic solutions of nonlinear ordinary differential equations and periodic solutions of nonlinear partial differential equations both of parabolic and hyperbolic types.
The main goal of these notes is to give a survey of the results concerning the solvability of noncoercive nonlinear equations. This topic has been very popular recently and many related papers have appeared during the last ten years. There is given an almost complete list of references up to 1978. The book is divided into twelve parts.
The parts I–V deal with some auxiliary facts about boundary value problems for differential equations and some generalizations of the results of Landesman-Lazer type (nonlinear perturbations of linear noninvertible operators). In parts VI and VII the method of a priori estimates and the method of truncated equations are explained. The variational approach to potential equations is treated in part VIII. The abstract results of Kazdan-Warner type and the method of Peter Hess are given in part IX. In part X, the author studies the periodic problem and boundary value problem for equations with rapid nonlinearities. Part XI is devoted to the equations with jumping nonlinearities. The last part XII (epilogue) deals with periodic solutions of both nonlinear heat equation and nonlinear telegraph equation. There are presented some results concerning multiple solutions of nonlinear equations. Finally there is an extensive bibliography containing 354 titles.
Each part of the book contains open problems unsolved up to the year 1978. Many mathematicians are working on the topics of this lecture note and some of the open problems have been solved up to now. These open problems show the way of further investigation and so Fučík’s book is a significant contribution to the development of nonlinear functional analysis.

MSC:
47H05 Monotone operators and generalizations
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34L99 Ordinary differential operators
34G20 Nonlinear differential equations in abstract spaces
35D30 Weak solutions to PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35K05 Heat equation
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35L70 Second-order nonlinear hyperbolic equations
35L99 Hyperbolic equations and hyperbolic systems
47E05 General theory of ordinary differential operators
47F05 General theory of partial differential operators
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H10 Fixed-point theorems
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
47J05 Equations involving nonlinear operators (general)
47H99 Nonlinear operators and their properties