Bifurcation Results for a Class of Perturbed Fredholm Maps

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.