Nonlinear Spectral Theory
Nonlinear spectral theory is a relatively new field of mathematics, which is far from being complete, and many fundamental questions still remain open. The main focus of this book is therefore formulated by the authors as the following question: How should we define a spectrum for nonlinear operators in such a way that it preserves useful properties of the linear case but admits applications to a possibly large variety of nonlinear problems? Contrary to the linear case, the spectrum of a nonlinear operator contains practically no information on this operator. The authors convince the reader that it is not the intrinsic structure of the spectrum itself, which leads to interesting applications, but its property of being a useful tool for solving nonlinear equations. The book is an excellent presentation of the “state-of-the-art” of contemporary nonlinear spectral theory as well as a glimpse of the diversity of directions in which current research is moving.
The whole text consists of 12 chapters. The authors recall basic facts on the spectrum of a bounded linear operator in the first chapter. In Chapter 2, some numerical characteristics providing quantitative descriptions of certain mapping properties of nonlinear operators are studied. The classical Kuratowski measure of noncompactness plays a key role here. Chapter 3 is devoted to general invertibility results. In particular, conditions that guarantee that the local invertibility of a nonlinear operator implies its global invertibility are of interest. The Rhodins and the Neuberger spectra are studied in Chapter 4. In Chapter 5, the authors study a spectrum for Lipschitz continuous operators, first proposed by Kachurovskij in 1969, and a spectrum for linearly bounded operators, introduced recently by Dörfner. Chapter 6 discusses the spectrum for certain special continuous operators introduced by Furi, Martelli and Vignoli in 1978, and its modification introduced recently by Appell, Giorgieri and Väth. The Feng spectrum is discussed in detail in Chapter 7. Chapter 8 is devoted to the study of “local spectrum” due to Väth, which in the literature is called “phantom”. In Chapter 9, the authors investigate a modification of the Feng spectrum proposed by Feng and Webb and another spectrum introduced by Singhof-Weyer and Weberand Infante-Webb. Chapter 10 is devoted to the study of nonlinear eigenvalue problems. The authors concentrate on the notion of a “nonlinear eigenvalue”, nonlinear analogue of the Krein-Rutman theorem, connected eigenvalues, etc. Chapter 11 contains a description on how numerical ranges may be used to localize the spectrum of a nonlinear operator on the real line or in the complex plane. Selected applications are presented in the last Chapter.
The exposition of nonlinear spectral theory in this book is self-contained. All major statements are proved; each definition and notion is carefully illustrated by examples. To understand this text does not require any special knowledge and only modest background of nonlinear analysis and operator theory is required. The book is addressed not only to mathematicians working in analysis but also to non-specialists wanting to understand the development of spectral theory for nonlinear operators in the last 30 years. The bibliography is rather exhaustive and so this text will certainly serve as an excellent reference book for many years. I am convinced that at least one copy of this book should be in any mathematical library.