Spectral Theory and Nonlinear Functional Analysis: 1st Edition (Paperback) book cover

Spectral Theory and Nonlinear Functional Analysis

1st Edition

By Julian Lopez-Gomez

Chapman and Hall/CRC

280 pages

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Paperback: 9781584882497
pub: 2001-03-28
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Description

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The results presented cover a thirty-year period and include recent, unpublished findings of the author and his coworkers.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

Reviews

"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

Table of Contents

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

About the Series

Chapman & Hall/CRC Research Notes in Mathematics Series

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Subject Categories

BISAC Subject Codes/Headings:
MAT007000
MATHEMATICS / Differential Equations
MAT037000
MATHEMATICS / Functional Analysis
SCI040000
SCIENCE / Mathematical Physics