Analytic Theory of Global Bifurcation

Description

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. Preface ix Chapter 1. Introduction 1 Part 1. Linear and Nonlinear Functional Analysis Chapter 2. Linear Functional Analysis 11 Chapter 3. Calculus in Banach Spaces 21 Chapter 4. Multilinear and Analytic Operators 41 Part 2. Analytic Varieties Chapter 5. Analytic Functions on Fn 61 Chapter 6. Polynomials 70 Chapter 7. Analytic Varieties 78 Part 3. Bifurcation Theory Chapter 8. Local Bifurcation Theory 103 Chapter 9. Global Bifurcation Theory 114 Part IV. Stokes Waves Chapter 10. Steady Periodic Water Waves 127 Chapter 11. Global Existence of Stokes Waves 152 Bibliography 161 Index 167