Singularities of Fredholm maps with one-dimensional kernels, II: Local behaviour and pointwise conditions

Abstract

In analogy to what happens in finite dimensions we state the Normal Form Theorem for k-singularities, introduced in the previous paper of the series. By means of that we study the local behaviour near a singularity i.e. we deduce local results of existence and multiplicity of solutions for the equation F(x) = y where F is a 0-Fredholm map and x belongs to a suitable neighbourhood of a singular point x_(o), once x_(o) is identified as one of the three kinds of singularities defined in the first paper. To this end we also start to seek alternative strategies for the determination of the type of a given singularity according to our classification. Here we give a pointwise approach for lower-order singularities which is coherent with the Thom-Boardman classification in finite dimensions. We conclude by applying the pointwise condition to a differential problem where, under suitable hypotheses, we determine a swallow's tail singularity (3-singularity).