The structure of the zero sets of nonlinear mappings near generic multiparameter eigenvalues
Let \(X\) and \(Y\) be real Banach spaces, let \(f: \mathbb{R}^p\times X\to Y\) be a \(C^\ell\) mapping, \(\ell\geq 2\), and let \(L(\lambda)= Df (\lambda, 0)\), for \(\lambda\in \mathbb{R}^p\). Suppose also that \(f(\lambda, 0) =0\), for all \(\lambda\), and at some point \(\lambda^0\), \(\dim N(L (\lambda^0))= n>0\), \(\text{codim } R(L( \lambda^0))= m>0\). The author examines the structure of the zero set of \(f\) in a neighbourhood of \((\lambda^0, 0)\), and gives conditions under which this zero set is diffeomorphic to the zero set of the lower order terms in the Taylor expansion of \(f\) near \((\lambda^0, 0)\). These results tell something about the structure of the set of non-trivial solutions bifurcating from zero when there are several parameters and the dimension of the null space of the linearization is greater than 1. Finally, the author considers the genericity of the conditions used.
scientific article; zbMATH DE number 811227