Spectral Theory and Nonlinear Functional Analysis

Spectral Theory and Nonlinear Functional Analysis

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 The results presented cover a thirty-year period and include recent, unpublished findings of the author and his coworkers.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

Product details

  • Paperback | 280 pages
  • 158 x 238.3 x 15.2mm | 399.17g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 2 black & white halftones
  • 1584882492
  • 9781584882497

Review quote

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.show more

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 INDEXshow more