# Atypical Bifurcation Without Compactness

### Pierluigi Benevieri

Universita di Firenze, Italy### Massimo Furi

Università di Firenze, Italy### Maria Patrizia Pera

Universita di Firenze, Italy### Mario Martelli

Claremont McKenna College, United States

## Abstract

We prove a global bifurcation result for an abstract equation of the type $Lx+λh(λ,x)=0$, where $L:E→F$ is a linear Fredholm operator of index zero between Banach spaces and $h:R×E→F$ 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 $Σ$ of nontrivial solutions of the above equation (i.e. solutions $(λ,x)$ with $λ=0$) such that the closure of $Σ$ contains a trivial solution $(0,xˉ)$. 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.

## Cite this article

Pierluigi Benevieri, Massimo Furi, Maria Patrizia Pera, Mario Martelli, Atypical Bifurcation Without Compactness. Z. Anal. Anwend. 24 (2005), no. 1, pp. 137–147

DOI 10.4171/ZAA/1233