Global continuation in Euclidean spaces of the perturbed unit eigenvectors corresponding to a simple eigenvalue (Q2179277)

A nonlinear perturbed eigenvalue problem on a finite-dimensional Euclidean space is considered. The main result, Theorem 3.8, implies a Rabinowitz-type continuation result, formulated as Proposition 1.2 in the Introduction. It provides a global version of \textit{R. Chiappinelli}'s result [Nonlinear Funct. Anal. Appl. 8, No. 4, 557--579 (2003; Zbl 1064.47060)] in the finite-dimensional case. A possible sharpening of the result is left open as conjecture. The proof of the main result is interesting and detailed. Several examples are given.