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Zeitschrift für Analysis und ihre Anwendungen


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Volume 24, Issue 1, 2005, pp. 137–147
DOI: 10.4171/ZAA/1233

Published online: 2005-03-31

Atypical Bifurcation Without Compactness

Pierluigi Benevieri[1], Massimo Furi[2], Mario Martelli[3] and Maria Patrizia Pera[4]

(1) Universita di Firenze, Italy
(2) Università di Firenze, Italy
(3) Claremont McKenna College, United States
(4) Universita di Firenze, Italy

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.

Keywords: Oriented Fredholm maps, global bifurcation, topological degree

Benevieri Pierluigi, Furi Massimo, Martelli Mario, Pera Maria Patrizia: Atypical Bifurcation Without Compactness. Z. Anal. Anwend. 24 (2005), 137-147. doi: 10.4171/ZAA/1233