Multiple bifurcation from "simple" eigenvalues

Consider an operator equation G(u, lambda,) = 0 where lambda is a real parameter. Suppose 0 is a "simple" eigenvalue of the Frechet derivative G(u) at (u(0),lambda(0)). We give a hierarchy of conditions which completely determines the solution structure of the operator equation. It will be shown that multiple bifurcation as well. as simple bifurcation can occur. This extends the standard bifurcation theory from a simple eigenvalue in which only one branch bifurcates. We also discuss limit point bifurcations. Applications to semilinear elliptic equations and the homotopy method for the matrix eigenvalue problem are also given. (C) 1999 Elsevier Science Inc. All rights reserved.