Approximation and convergence rate of nonlinear eigenvalues: Lipschitz perturbations of a bounded self-adjoint operator

Abstract

We consider nonlinear eigenvalue problems of the form (⁎)Tx+ϵB(x)=λx, where T is a self-adjoint bounded linear operator acting in a real Hilbert space H, and B:H→H is a (possibly) nonlinear continuous perturbation term. Assuming that λ0 is an isolated eigenvalue of finite multiplicity of T, we ask if for ϵ≠0 and small there are “eigenvalues” of (⁎) near λ0, that is, numbers λϵ for which (⁎) is satisfied by some normalized “eigenvector” xϵ of T+ϵB. In this paper we recall some recent results giving an affirmative answer to this question, and for these cases we prove – assuming in addition Lipschitz continuity on B – upper and lower bounds for the perturbed eigenvalues λϵ which are determined by those for the nonlinear Rayleigh quotient 〈B(v),v〉/〈v,v〉 with v in the eigenspace Ker(T−λ0I). This yields in particular information on the rate of convergence of λϵ to λ0 as ϵ→0. Applications are given in the sequence space l2, and in the Sobolev space H01 to deal with some nonlinearly perturbed ordinary or partial differential equations.