Generalized Lyapunov--Schmidt reduction for parametrized equations at near singular points

. The Lyapunov-Schmidt reduction is generalized to the case of imperfect singularities. The results presented neither need very precise information about the location of the (near) singularities nor a precise knowledge of (near) null spaces. Keywords: Lyapunov-Schmidt reduction, unfolding, imperfect singularities 1991 MSC Classification: 58C15 1 Introduction Let X; Y be Banach spaces, let F : \Theta D ` R p \Theta X ! Y be an k 1 times continuously differentiable nonlinear mapping such that the partial derivative F u (; u) of F with respect to u is a Fredholm operator of index zero. We want to study the solution manifold of F (; u) = 0 in a neighbourhood of an approximate zero ( 0 ; u 0 ) 2 int( \Theta D) of F . In practice, one often is not interested in the manifold itself, but in finitely many key parameters ¯ i (; u). Thus we assume that we are also given a vector valued function ¯ : \Theta D ! R m defining k times continuously differentiable parameter functionals ¯ i..