Nonlinear Spectral Theory

Jürgen Appell; Espedito De Pascale; Alfonso Vignoli

Published by De Gruyter 2004

Nonlinear Spectral Theory

Jürgen Appell , Espedito De Pascale and Alfonso Vignoli

Volume 10 in the series De Gruyter Series in Nonlinear Analysis and Applications

https://doi.org/10.1515/9783110199260

Overview
Contents

About this book

In view of the eminent importance of spectral theory of linear operators in many fields of mathematics and physics, it is not surprising that various attempts have been made to define and study spectra also for nonlinear operators. This book provides a comprehensive and self-contained treatment of the theory, methods, and applications of nonlinear spectral theory.

The first chapter briefly recalls the definition and properties of the spectrum and several subspectra for bounded linear operators. Then some numerical characteristics for nonlinear operators are introduced which are useful for describing those classes of operators for which there exists a spectral theory. Since spectral values are closely related to solvability results for operator equations, various conditions for the local or global invertibility of a nonlinear operator are collected in the third chapter. The following two chapters are concerned with spectra for certain classes of continuous, Lipschitz continuous, or differentiable operators. These spectra, however, simply adapt the corresponding definitions from the linear theory which somehow restricts their applicability. Other spectra which are defined in a completely different way, but seem to have useful applications, are defined and studied in the following four chapters. The remaining three chapters are more application-oriented and deal with nonlinear eigenvalue problems, numerical ranges, and selected applications to nonlinear problems.

The only prerequisite for understanding this book is a modest background in functional analysis and operator theory. It is addressed to non-specialists who want to get an idea of the development of spectral theory for nonlinear operators in the last 30 years, as well as a glimpse of the diversity of the directions in which current research is moving.

Author / Editor information

Jürgen Appell is Professor at the Mathematics Institute of the University of Würzburg, Germany.

Espedito De Pascale is Professor at the Mathematics Department of the University of Calabria, Italy.

Alfonso Vignoli is Professor at the Mathematics Department of the University of Rome, Italy.

Reviews

"This text will certainly serve as an excellent reference book for many years."
EMS Newsletter 3/2005

"The book can be considered as a systematic introduction to this area, emphasizing the diversity of directions in which current research in nonlinear spectral ist developing."
S. Cobzas in: Studia Universitatis Babes-Boyai 1/2005

"This carefully written book is a comprehensive treatment of the subject including new results several of them archieved by the author. [...] This research monograph is recommended to scientists interested in the theory of nonlinar operators and their applications."
R. Schumann in ZAA 4/2004

"One of the features of the book is that it provides many examples, varying from trivial to profound, to illustrate exactly what can and cannot happen. Each chapter ends with a valuable "Notes and remarks'' section; some of these contain some open problems. The book is well written, the presentation is self-contained, and the text may be read by anyone with a background in functional analysis who knows a little nonlinear analysis." Mathematical Reviews

Topics

Frontmatter Requires Authentication Unlicensed Licensed Download PDF	i
Contents Requires Authentication Unlicensed Licensed Download PDF	ix
Introduction Requires Authentication Unlicensed Licensed Download PDF	1
Chapter 1. Spectra of Bounded Linear Operators Requires Authentication Unlicensed Licensed Download PDF	11
Chapter 2. Some Characteristics of Nonlinear Operators Requires Authentication Unlicensed Licensed Download PDF	40
Chapter 3. Invertibility of Nonlinear Operators Requires Authentication Unlicensed Licensed Download PDF	67
Chapter 4. The Rhodius and Neuberger Spectra Requires Authentication Unlicensed Licensed Download PDF	94
Chapter 5. The Kachurovskij and Dörfner Spectra Requires Authentication Unlicensed Licensed Download PDF	109
Chapter 6. The Furi–Martelli–Vignoli Spectrum Requires Authentication Unlicensed Licensed Download PDF	130
Chapter 7. The Feng Spectrum Requires Authentication Unlicensed Licensed Download PDF	159
Chapter 8. The Väth Phantom Requires Authentication Unlicensed Licensed Download PDF	184
Chapter 9. Other Spectra Requires Authentication Unlicensed Licensed Download PDF	214
Chapter 10. Nonlinear Eigenvalue Problems Requires Authentication Unlicensed Licensed Download PDF	268
Chapter 11. Numerical Ranges of Nonlinear Operators Requires Authentication Unlicensed Licensed Download PDF	303
Chapter 12. Some Applications Requires Authentication Unlicensed Licensed Download PDF	336
Backmatter Requires Authentication Unlicensed Licensed Download PDF	375

(Deutsch)

Language: English
Publisher: De Gruyter
Audience: Research Mathematicians, in particular Analysts, and Graduate Students in Mathematics; Academic Libraries
Pages
- Front matter: 11
- Main content: 408
Illustrations
- Illustrations: 1
- Other: Num. tab.
Keywords: Spectral theory (Mathematics); Nonlinear theories

eBook

Published: August 22, 2008
ISBN: 9783110199260

Hardcover

Published: April 26, 2004
ISBN: 9783110181432