THE STRUCTURE OF THE ZERO SETS OF NONLINEAR MAPPINGS NEAR GENERIC MULTIPARAMETER EIGENVALUES

Let X and Y be real Banach spaces, let f: R(p) x X --> Y be a C-l mapp ing, l greater than or equal to 2, and let L(lambda) = Df(lambda, 0), for lambda is an element of R(p). Suppose also that f(lambda, 0) = 0, for all lambda, and at some point lambda(0), dim N(L(lambda(0)) = n > 0, codim R(L(lambda(0))) = m > 0. We examine the structure of the zero set off in a neighbourhood of (lambda(0), 0), and give conditions und er which this zero set is diffeomorphic to the zero set of the lower o rder terms in the Taylor expansion of f near (lambda(0), 0). These res ults tell us about the structure of the set of non-trivial solutions b ifurcating from zero when there are several parameters and the dimensi on of the null space of the linearization is greater than 1. Finally, we consider the genericity of the conditions used. (C) 1995 Academic P ress, Inc.