# 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 $H$ be a real Hilbert space and denote by $S$ its unit sphere. Consider the nonlinear eigenvalue problem $Lx+ϵN(x)=λx$, where $ϵ,λ∈R$, $L:H→H$ is a bounded self-adjoint (linear) operator with nontrivial kernel and closed image, and $N:H→H$ is a (possibly) nonlinear perturbation term. A unit eigenvector $xˉ∈S∩KerL$ of $L$ (corresponding to the eigenvalue $λ=0$) is said to be *persistent* if it is close to solutions $x∈S$ of the above equation for small values of the parameters $ϵ=0$ and $λ$. 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 $λ=0$ has odd multiplicity, then the sphere $S∩KerL$ contains at least one persistent eigenvector. We provide examples in which our results apply, as well as examples showing that, if the dimension of $KerL$ 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