Bifurcation Results for a Class of Perturbed Fredholm Maps. (31st July 2008) - Benevieri, Pierluigi; Calamai, Alessandro
Abstract : We prove a global bifurcation result for an equation of the typeL x + λ ( h ( x ) + k ( x ) ) = 0, whereL : E → F is a linear Fredholm operator of index zero between Banach spaces, and, given an open subsetΩ ofE, h, k : Ω × [ 0, + ∞ ) → F areC 1 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 ) withλ ≠ 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, calledα -Fredholm maps, which has been recently developed by the authors in collaboration with M. Furi.