This Research Note addresses several pivotal problems in spectral theory and nonlinear functional analysis in connection with the analysis of the structure of the set of zeroes of a general class of nonlinear operators. It features the construction of an optimal algebraic/analytic invariant for calculating the Leray-Schauder degree, new methods for solving nonlinear equations in Banach spaces, and general properties of components of solutions sets presented with minimal use of topological tools. The author also gives several applications of the abstract theory to reaction diffusion equations and systems.
The results presented cover a thirty-year period and include recent, unpublished findings of the author and his coworkers. Appealing to a broad audience, Spectral Theory and Nonlinear Functional Analysis contains many important contributions to linear algebra, linear and nonlinear functional analysis, and topology and opens the door for further advances.
General Assumptions and Basic Concepts
Some New Results
BIFURCATION FROM SIMPLE EIGENVALUES
Simple Eigenvalues and Transversality
The Theorem of M.G. Crandall and P.H. Rabinowitz
Local Bifurcation Diagrams
The Exchange Stability Principle
FIRST GENERAL BIFURCATION RESULTS
The theorem of J. Ize
The Global Alternative of P.H. Rabinowitz
The Theorem of D. Westreich
THE ALGEBRAIC MULTIPLICITY
Motivating the Concept of Transversality
Simple Degenerate Eigenvalues
FUNDAMENTAL PROPERTIES OF THE MULTIPLICITY
The Multiplicity of R.J. Magnus
Relations between c and m
The Fundamental Theorem
The Classical Algebraic Multiplicity
Finite Dimensional Characterizations
The Parity of the Crossing Number
GLOBAL BIFURCATION THEORY
Global Behavior of the Bounded Components
Unilateral Global Bifurcation
Unilateral Bifurcation for Positive Operators
Positive Solutions o Semilinear Elliptic Problems
Coexistence States for Elliptic Systems
A Further Application
"The book is both an excellent introduction to some novel ideas about nonlinear eigenvalue problems and an exposition of a range of earlier results scattered in different papers and expounded in book form for the first time here."
- Mathematical Reviews, Issue 2002
"This is a nice introductory text about classical functional analysis…" This book will be interesting and useful for many mathematicians, scientists, graduate and undergraduate students."