Global continuation in Euclidean spaces of the perturbed unit eigenvectors corresponding to a simple eigenvalue - Benevieri, Pierluigi; Calamai, Alessandro; Furi, Massimo; Pera, Maria Patrizia (Topological Methods in Nonlinear Analysis, 2020)
In the Euclidean space $\mathbb R^k$, we consider the perturbed eigenvalue problem $Lx + ε N(x) = λ x$, $\|x\| = 1$, where $ε,λ$ are real parameters, $L$ is a linear endomorphism of $\mathbb R^k$, and $N\colon S^{k-1} → \mathbb R^k$ is a continuous map defined on the unit sphere of $\mathbb R^k$. We prove a global continuation result for the solutions $(x,ε,λ)$ of this problem. Namely, under the assumption that $x_* \in S^{k-1}$ is one of the two unit eigenvectors of $L$ corresponding to a simple eigenvalue $λ_* \in \R$, we show that, in the set of all the solutions, the connected component containing $(x_*,0,λ_*)$ is either unbounded or meets a solution $(x^*,0,λ^*)$ having $x^* ot= x_*$. Our result is inspired by a paper of R.\ Chiappinelli concerning the local persistence property of eigenvalues and eigenvectors of a perturbed self-adjoint operator in a real Hilbert space.