Normalized eigenvectors of a perturbed linear operator via general bifurcation

Let X be a real Banach space, A:X → X a bounded linear operator, and B:X → X a (possibly nonlinear) continuous operator. Assume that λ = 0 is an eigenvalue of A and consider the family of perturbed operators A + εB, where ε is a real parameter. Denote by S the unit sphere of X and let SA = S ∩ KerA be the set of unit 0-eigenvectors of A. We say that a vector x0 ∈ SA is a bifurcation point for the unit eigenvectors of A + εB if any neighborhood of (0, 0, x0) ∈ × × X contains a triple (ε, λ, x) with ε = 0 and x a unit λ-eigenvector of A + εB, i.e. x ∈ S and (A + εB)x = λx. We give necessary as well as sufficient conditions for a unit 0-eigenvector of A to be a bifurcation point for the unit eigenvectors of A + εB. These conditions turn out to be particularly meaningful when the perturbing operator B is linear. Moreover, since our sufficient condition is trivially satisfied when KerA is one-dimensional, we extend a result of the first author, under the additional assumption that B is of class C2