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 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 R^k, and NS^k-1 → R^k is a continuous map defined on the unit sphere of R^k. We prove a global continuation result for the solutions (x,ε,λ) of this problem. Namely, under the assumption that x_* S^k-1 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^* 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.