Nonlinear Spectral Theory

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.

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.

"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