# Zeitschrift für Analysis und ihre Anwendungen

Volume 36, Issue 1, 2017, pp. 99–128
DOI: 10.4171/ZAA/1581

Published online: 2017-01-06

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

Pierluigi Benevieri[1], Alessandro Calamai[2], Massimo Furi[3] and Maria Patrizia Pera[4]

(1) Università degli Studi di Firenze, Italy
(2) Università Politecnica delle Marche, Ancona, Italy
(3) Università degli Studi di Firenze, Italy
(4) Università degli Studi di Firenze, Italy

Let $H$ be a real Hilbert space and denote by $S$ its unit sphere. Consider the nonlinear eigenvalue problem $Lx + \epsilon N(x) = \lambda x$, where $\epsilon, \lambda \in \mathbb R$, $L : H \to H$ is a bounded self-adjoint (linear) operator with nontrivial kernel and closed image, and $N : H \to H$ is a (possibly) nonlinear perturbation term. A unit eigenvector $\bar x \in S \cap \mathrm {Ker} L$ of $L$ (corresponding to the eigenvalue $\lambda = 0$) is said to be persistent if it is close to solutions $x \in S$ of the above equation for small values of the parameters $\epsilon \neq 0$ and $\lambda$. 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 $N$ is Lipschitz continuous and the eigenvalue $\lambda = 0$ has odd multiplicity, then the sphere $S\cap \mathrm {Ker} L$ contains at least one persistent eigenvector. We provide examples in which our results apply, as well as examples showing that, if the dimension of $\mathrm {Ker} L$ is even, then the persistence phenomenon may not occur.

Keywords: Fredholm operators, nonlinear spectral theory, eigenvalues, eigenvectors, degree theory

Benevieri Pierluigi, Calamai Alessandro, Furi Massimo, Pera Maria Patrizia: On the Persistence of the Eigenvalues of a Perturbed Fredholm Operator of Index Zero under Nonsmooth Perturbations. Z. Anal. Anwend. 36 (2017), 99-128. doi: 10.4171/ZAA/1581