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 $h\colon \mathbb R \times E \to F$ is a $C\sp{1}$ (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 $\lambda \neq 0$) 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.