Spectral Theory and Nonlinear Functional Analysis

by Lopez-Gomez; Julian

• ISBN13: 9781584882497
• ISBN10: 1584882492
• Edition: 1st
• Format: Nonspecific Binding
• Copyright: 2001-03-28
• Publisher: Chapman & Hall/

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