Global continuation in Euclidean spaces of the perturbed unit eigenvectors corresponding to a simple eigenvalue

In the Euclidean space $Rk$, we consider the perturbed eigenvalue problem $Lx + e N(x) = l x$, $|x| = 1$, where $e,l$ are real parameters, $L$ is a linear endomorphism of $Rk$, and $Ncolon Ssp{k-1} o Rk$ is a continuous map defined on the unit sphere of $Rk$. We prove a sort of global continuation of the emph{solutions} $(x,e,l)$ of this problem. Namely, under the assumption that $x_* in Ssp{k-1}$ is one of the two unit eigenvectors of $L$ corresponding to a simple eigenvalue $l_* in R$, we show that, in the set of all the solutions, the connected component containing $(x_*,0,l_*)$ is either unbounded or meets a solution $(xsp*,0,lsp*)$ having $xsp* ot= x_*$. Our result is inspired by a paper of R.! Chiappinelli regarding the local persistence property of eigenvalues and eigenvectors of a perturbed self-adjoint operator in a real Hilbert space. Our results are related to some papers by Chiappinelli, in which he studied a “local” persistence property of eigenvalues and eigenvectors of self-adjoint operators in real Hilbert spaces.