Analytic Theory of Global Bifurcation : : An Introduction / / John Toland, Boris Buffoni.

• Superior document: Title is part of eBook package: De Gruyter Princeton Series in Applied Mathematics eBook-Package
• VerfasserIn: Buffoni, Boris,; Toland, John,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2016]; ©2003
• Year of Publication: 2016
• Language: English
• Series: Princeton Series in Applied Mathematics ; 55
• Physical Description: 1 online resource (184 p.) :; 5 line illus.

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.

Table of Contents

Frontmatter -- Contents -- Preface -- Chapter 1. Introduction -- Part 1. Linear and Nonlinear Functional Analysis -- Chapter 2. Linear Functional Analysis -- Chapter 3. Calculus in Banach Spaces -- Chapter 4. Multilinear and Analytic Operators -- Part 2. Analytic Varieties -- Chapter 5. Analytic Functions on Fn -- Chapter 6. Polynomials -- Chapter 7. Analytic Varieties -- Part 3. Bifurcation Theory -- Chapter 8. Local Bifurcation Theory -- Chapter 9. Global Bifurcation Theory -- Part IV. Stokes Waves -- Chapter 10. Steady Periodic Water Waves -- Chapter 11. Global Existence of Stokes Waves -- Bibliography -- Index