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 + εN(x) = λx, ||x|| = 1, where ε, λ are real parameters, L is a linear endomorphism of Rk, and N: Sk−1 → Rk is a continuous map defined on the unit sphere of Rk . We prove a global continuation result for the solutions (x, ε, λ) of this problem. Namely, under the assumption that x_* is one of the two unit eigenvectors of L corresponding to a simple eigenvalue λ_*, 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* ≠ 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.