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

TL;DR

This paper extends the theory of Fredholm maps with one-dimensional kernels by establishing a normal form theorem, analyzing local solution behavior near singularities, and applying pointwise conditions to classify singularities, including a swallow's tail.

Contribution

It introduces a normal form theorem for k-singularities in infinite dimensions and develops a pointwise classification method aligned with finite-dimensional Thom-Boardman theory.

Findings

Established local existence and multiplicity results near singularities.

Developed a pointwise approach for classifying lower-order singularities.

Applied the theory to identify a swallow's tail singularity in a differential problem.

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).