Zeitschrift für Analysis und ihre Anwendungen 24 (2005), No. 1, 137--147
Copyright Heldermann Verlag 2005
Atypical Bifurcation Without Compactness
Pierluigi Benevieri
Dip. di Matematica Applicata, Università di Firenze, Via S. Marta 3, 50139 Firenze, Italy
pierluigi.benevieri@unifi.it
Massimo Furi
Dip. di Matematica Applicata, Università di Firenze, Via S. Marta 3, 50139 Firenze, Italy
massimo.furi@unifi.it
Mario Martelli
Dept. of Mathematics, Claremont McKenna College, Claremont, CA 91711, U.S.A.
mpatrizia.pera@unifi.it
M. Patrizia Pera
Dip. di Matematica Applicata, Università di Firenze, Via S. Marta 3, 50139 Firenze, Italy
mario.martelli@claremontmckenna.edu
[Abstract-pdf]
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.
MSC: 47J15; 47H11, 34B15
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