arXiv2019
We consider the nonlinear eigenvalue problem $Lx + \varepsilon N(x) = λCx$, $\|x\|=1$, where $\varepsilon,λ$ are real parameters, $L, C\colon G \to H$ are bounded linear operators between separable real Hilbert spaces, and $N\colon S \to H$ is a continuous map defined on the unit sphere of $G$. We prove a global persistence result regarding the set $Σ$ of the solutions $(x,\varepsilon,λ) \in S \times \mathbb R\times \mathbb R$ of this problem. Namely, if the operators $N$ and $C$ are compact, under suitable assumptions on a solution $p_*=(x_*,0,λ_*)$ of the unperturbed problem, we prove that the connected component of $Σ$ containing $p_*$ is either unbounded or meets a triple $p^*=(x^*,0,λ^*)$ with $p^* \not= p_*$. When $C$ is the identity and $G=H$ is finite dimensional, the assumptions on $(x_*,0,λ_*)$ mean that $x_*$ is an eigenvector of $L$ whose corresponding eigenvalue $λ_*$ is simple. Therefore, we extend a previous result obtained by the authors in the finite dimensional setting. Our work is inspired by a paper of R. Chiappinelli concerning the local persistence property of the unit eigenvectors of perturbed self-adjoint operators in a real Hilbert space.