NORMALIZED EIGENVECTORS OF A PERTURBED LINEAR OPERATOR VIA GENERAL BIFURCATION

Abstract

AbstractLetXbe a real Banach space,A:X→Xa bounded linear operator, andB:X→Xa (possibly nonlinear) continuous operator. Assume that λ = 0 is an eigenvalue ofAand consider the family of perturbed operatorsA+ ϵB, where ϵ is a real parameter. Denote bySthe unit sphere ofXand letSA=S∩ KerAbe the set of unit 0-eigenvectors ofA. We say that a vectorx0∈SAis abifurcation pointfor the unit eigenvectors ofA+ ϵBif any neighborhood of (0,0,x0) ∈$\R$×$\R$×Xcontains a triple (ϵ, λ,x) with ϵ ≠ 0 andxa unit λ-eigenvector ofA+ ϵB, i.e.x∈Sand (A+ ϵB)x= λx.We give necessary as well as sufficient conditions for a unit 0-eigenvector ofAto be a bifurcation point for the unit eigenvectors ofA+ ϵB. These conditions turn out to be particularly meaningful when the perturbing operatorBis linear. Moreover, since our sufficient condition is trivially satisfied when KerAis one-dimensional, we extend a result of the first author, under the additional assumption thatBis of classC2.