On the Persistence of the Eigenvalues of a Perturbed Fredholm Operator of Index Zero under Nonsmooth Perturbations

  • Pierluigi Benevieri

    Università degli Studi di Firenze, Italy
  • Alessandro Calamai

    Università Politecnica delle Marche, Ancona, Italy
  • Massimo Furi

    Università degli Studi di Firenze, Italy
  • Maria Patrizia Pera

    Università degli Studi di Firenze, Italy

Abstract

Let be a real Hilbert space and denote by its unit sphere. Consider the nonlinear eigenvalue problem , where , is a bounded self-adjoint (linear) operator with nontrivial kernel and closed image, and is a (possibly) nonlinear perturbation term. A unit eigenvector of (corresponding to the eigenvalue ) is said to be persistent if it is close to solutions of the above equation for small values of the parameters and . We give an affirmative answer to a conjecture formulated by R. Chiappinelli and the last two authors in an article published in 2008. Namely, we prove that, if is Lipschitz continuous and the eigenvalue has odd multiplicity, then the sphere contains at least one persistent eigenvector. We provide examples in which our results apply, as well as examples showing that, if the dimension of is even, then the persistence phenomenon may not occur.

Cite this article

Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera, On the Persistence of the Eigenvalues of a Perturbed Fredholm Operator of Index Zero under Nonsmooth Perturbations. Z. Anal. Anwend. 36 (2017), no. 1, pp. 99–128

DOI 10.4171/ZAA/1581