Solvability of nonlinear equations and boundary value problems.

*(English)*Zbl 0453.47035These 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.

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.

Reviewer: Pavel Drábek (Plzeň)

##### 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 |