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9781584882497

Spectral Theory and Nonlinear Functional Analysis

by ;
  • ISBN13:

    9781584882497

  • ISBN10:

    1584882492

  • Edition: 1st
  • Format: Nonspecific Binding
  • Copyright: 2001-03-28
  • Publisher: Chapman & Hall/

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Supplemental Materials

What is included with this book?

Summary

This Research Note addresses several pivotal problems in spectral theory and nonlinear functional analysis in connection with the analysis of the structure set of zeroes of a general class of nonlinear operators. Appealing to a broad audience, it contains many important contributions to linear algebra, linear functional analysis, nonlinear function analysis, and topology. The author gives several applications of the abstract theory to reaction diffusion equations and system. The results presented cover a thirty-year period and cut across a variety of mathematical fields.

Table of Contents

Preface ix
Introduction
1(20)
General assumptions and basic concepts
2(4)
Some new results
6(2)
Historical remarks
8(13)
Bifurcation from simple eigenvalues
21(48)
Simple eigenvalues and transversality
22(5)
The theorem of M. G. Crandall and P. H. Rabinowitz
27(5)
Local bifurcation diagrams
32(2)
The exchange stability principle
34(9)
Applications
43(26)
A Simple application
43(3)
The Selkov scheme for glycolysis
46(23)
First general bifurcation results
69(18)
Lyapunov-Schmidt reductions
71(4)
The theorem of J. Ize
75(7)
The global alternative of P. H. Rabinowitz
82(1)
The theorem of D. Westreich
83(4)
The algebraic multiplicity
87(42)
Motivating the concept of transversality
89(3)
Transversal eigenvalues
92(13)
Algebraic eigenvalues
105(14)
Analytic families
119(6)
Simple degenerate eigenvalues
125(4)
Other fundamental properties of the multiplicity
129(34)
The multiplicity of R. J. Magnus
130(7)
Relations between χ and μ
137(8)
The fundamental theorem
145(3)
The classical algebraic multiplicity
148(6)
Finite dimensional characterizations
154(3)
The parity of the crossing number
157(6)
Global bifurcation theory
163(34)
Preliminaries
166(2)
Local bifurcation
168(4)
Global behavior of the bounded components
172(7)
Unilateral global bifurcation
179(11)
Unilateral bifurcation for positive operators
190(7)
Applications
197(54)
Positive solutions of semilinear elliptic problems
197(19)
Coexistence states for elliptic systems
216(15)
Examples
231(15)
Example 1
231(5)
Example 2
236(8)
Example 3
244(2)
A further application
246(5)
References 251(10)
Index 261

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