1st Edition

Spectral Theory and Nonlinear Functional Analysis

By Julian Lopez-Gomez Copyright 2001
    278 Pages
    by Chapman & Hall

    278 Pages
    by Chapman & Hall

    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    The results presented cover a thirty-year period and include recent, unpublished findings of the author and his coworkers.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.

    INTRODUCTION
    General Assumptions and Basic Concepts
    Some New Results
    Historical Remarks
    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
    Applications
    FIRST GENERAL BIFURCATION RESULTS
    Lyapunov-Schmidt Reductions
    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
    Transversal Eigenvalues
    Algebraic Eigenvalues
    Analytic Families
    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
    Preliminaries
    Local Bifurcation
    Global Behavior of the Bounded Components
    Unilateral Global Bifurcation
    Unilateral Bifurcation for Positive Operators
    APPLICATIONS
    Positive Solutions o Semilinear Elliptic Problems
    Coexistence States for Elliptic Systems
    Examples
    A Further Application
    REFERENCES
    INDEX

    Biography

    Julian Lopez-Gomez

    "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."
    -Mathematical Reviews