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

We consider nonlinear eigenvalue problems of the form (*) Tx + epsilon B(x) = lambda x, 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 lambda(0) is an isolated eigenvalue of finite multiplicity of T, we ask if for epsilon not equal 0 and small there are "eigenvalues" of (*) near lambda(0), that is, numbers lambda(epsilon) for which (*) is satisfied by some normalized "eigenvector" x of T delta B. 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 lambda(epsilon) which are determined by those for the nonlinear Rayleigh quotient < B(nu), nu >)/<nu, nu > with nu in the eigenspace Ker(T - lambda I-0). This yields in particular information on the rate of convergence of lambda(epsilon) to lambda(0) as E -> 0. Applications are given in the sequence space l(2), and in the Sobolev space H-0(1) to deal with some nonlinearly perturbed ordinary or partial differential equations.