Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 752657, 19 pages
doi:10.1155/2008/752657

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

Abstract

We prove a global bifurcation result for an equation of the type Lx+λ(h(x)+k(x))=0, where L:E  →  F 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 λ≠0, 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.