Atypical bifurcation without compactness

We prove a global bifurcation result for an abstract equation of the type $Lx + lambda h(lambda,x) = 0$, where $L: E o F$ is a linear Fredholm operator of index zero between Banach spaces and $h: mathbb R imes E o F$ 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 $lambda eq 0$) such that the closure of $Sigma$ contains a trivial solution $(0,ar x)$. 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.