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.
Global continuation in euclidean spaces of the perturbed unit eigenvectors corresponding to a simple eigenvalue / Benevieri, P.; Calamai, A.; Furi, M.; Pera, M. P.. - In: TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS. - ISSN 1230-3429. - STAMPA. - 55:1(2020), pp. 169-184. [10.12775/TMNA.2019.093]
Global continuation in euclidean spaces of the perturbed unit eigenvectors corresponding to a simple eigenvalue
Calamai A.
;
2020-01-01
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.