Bifurcation results for a class of perturbed Fredholm maps

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 CM1 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 (x̄, 0). The proof is based on a degree theory fora 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.