BIFURCATION FROM SIMPLE EIGENVALUES,
Abstract
Let G be a mapping of a subset of a Banach space W into a Banach space Y. Let C be a curve in W such that G(C) = (0). A general version of the main problem of bifurcation theory may be stated: Given p epsilon C, determine the structure of G sup (-1) (0) in some neighborhood of p. In this work simple conditions are given under which there is a neighborhood N sub p of p such that (G sup (-1))(0) intersection (N sub p) is topologically (or diffeomorphically) equivalent to the subset (-1,1)x(0) intersection (0)x(-1,1) of the plane, and the first order behavior of G on (G sup (-1))(0) intersection (N sub p) as well as the set itself is studied. The results obtained give a new unity to that part of bifurcation theory commonly called 'bifurcation from a simple eigenvalue' as well as extend its applicability. A broad spectrum of examples is offered, including some generalizations of known results concerning non-linear eigenvalue problems for ordinary and partial differential equations. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Sep 01, 1970
- Accession Number
- AD0712438
Entities
People
- Michael G. Crandall
- Paul H. Rabinowitz
Organizations
- University of California, Los Angeles