Bifurcation from simple eigenvalues

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 ϵ C, determine the structure of G−1{0} in some neighborhood of p. In this work simple conditions are given under which there is a neighborhood Np of p such that G−1{0} ∩ Np is topologically (or diffeomorphically) equivalent to the subset (−1, 1) × {0} ∪ {0} × (−1, 1) of the plane, and the first order behavior of G on G−1{0} ∩ Np as well as the set itself is studied. The results obtained help unify that part of bifurcation theory commonly called “bifurcation from a simple eigenvalue” as well as they extend its applicability. A broad spectrum of examples is offered, including some generalizations of known results concerning nonlinear eigenvalue problems for ordinary and partial differential equations.