Synopses & Reviews
Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence.
This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences--for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue. Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory.
Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.
Synopsis
Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence.
This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences--for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue. Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory.
Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.
Synopsis
Includes bibliographical references (p. [161]-165) and index.
Synopsis
Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence.
This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences--for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue. Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory.
Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.
About the Author
Boris Buffoni holds a Swiss National Science Foundation Professorship in Mathematics at the Swiss Federal Institute of Technology-Lausanne. John Toland is Professor of Mathematical Sciences at the University of Bath and a Senior Research Fellow of the UK's Engineering and Physical Sciences Research Council
Table of Contents
Preface ix
Chapter 1. Introduction 1
1.1 Example: Bending an Elastic Rod I 2
1.2 Principle of Linearization 5
1.3 Global Theory 6
1.4 Layout 7
PART 1. LINEAR AND NONLINEAR FUNCTIONAL ANALYSIS 9
Chapter 2. Linear Functional Analysis 11
2.1 Preliminaries and Notation 11
2.2 Subspaces 13
2.3 Dual Spaces 14
2.4 Linear Operators 15
2.5 Neumann Series 16
2.6 Projections and Subspaces 17
2.7 Compact and Fredholm Operators 18
2.8 Notes on Sources 20
Chapter 3. Calculus in Banach Spaces 21
3.1 Fréchet Differentiation 21
3.2 Higher Derivatives 27
3.3 Taylor's Theorem 31
3.4 Gradient Operators 32
3.5 Inverse and Implicit Function Theorems 35
3.6 Perturbation of a Simple Eigenvalue 38
3.7 Notes on Sources 40
Chapter 4. Multilinear and Analytic Operators 41
4.1 Bounded Multilinear Operators 41
4.2 Fa