Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 752657, 19 pages
Bifurcation Results for a Class of Perturbed Fredholm Maps
1Dipartimento di Matematica Applicata, Università degli Studi di Firenze, Via S. Marta 3, 50139 Firenze, Italy
2Dipartimento di Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy
Received 3 March 2008; Revised 18 July 2008; Accepted 27 July 2008
Academic Editor: Fabio Zanolin
Copyright © 2008 Pierluigi Benevieri and Alessandro Calamai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We prove a global bifurcation result for an equation of the type , where is a linear Fredholm operator
of index zero between Banach spaces, and, given an open subset of , are and continuous, respectively. Under suitable
conditions, we prove the existence of an unbounded connected set of nontrivial
solutions of the above equation, that is, solutions with , whose closure
contains a trivial solution . The proof is based on a degree theory for
a special class of noncompact perturbations of Fredholm maps of index zero,
called -Fredholm maps, which has been recently developed by the authors in collaboration
with M. Furi.