Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 752657, 19 pages
Research Article

Bifurcation Results for a Class of Perturbed Fredholm Maps

Pierluigi Benevieri1 and Alessandro Calamai2

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 Lx+λ(h(x)+k(x))=0, where L:EF is a linear Fredholm operator of index zero between Banach spaces, and, given an open subset Ω of E, h,k:Ω×[0,+)F are C1 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 (x,λ) with λ0, whose closure contains a trivial solution (x¯,0). The proof is based on a degree theory for a special class of noncompact perturbations of Fredholm maps of index zero, called α-Fredholm maps, which has been recently developed by the authors in collaboration with M. Furi.