Atypical Bifurcation Without Compactness

Abstract

We prove a global bifurcation result for an abstract equation of the type $Lx+\lambda h(\lambda ,x)=0$, where $L:E\to F$ is a linear Fredholm operator of index zero between Banach spaces and $h:\mathbb{R}\times E\to F$ is a $C^1$ (not necessarily compact) map. We assume that $L$ is not invertible and, under suitable conditions, we prove the existence of an unbounded connected set $\Sigma$ of nontrivial solutions of the above equation (i.e. solutions $(\lambda ,x)$ with $\lambda \ne 0$) such that the closure of $\Sigma$ contains a trivial solution $(0,\bar{x})$. This result extends previous ones in which the compactness of $h$ was required. The proof is based on a degree theory for Fredholm maps of index zero developed by the first two authors.