Description

• Title:
  Global continuation in Euclidean spaces of the perturbed unit eigenvectors corresponding to a simple eigenvalue
• Creator:
  Benevieri, Pierluigi;
  Calamai, Alessandro;
  Furi, Massimo;
  Pera, Maria Patrizia
• Description:
  In the Euclidean space $\mathbb R^k$, we consider the perturbed eigenvalue problem $Lx + \varepsilon N(x) = \lambda x$, $\|x\| = 1$, where $\varepsilon,\lambda$ are real parameters, $L$ is a linear endomorphism of $\mathbb R^k$, and $N\colon S^{k-1} \to \mathbb R^k$ is a continuous map defined on the unit sphere of $\mathbb R^k$. We prove a global continuation result for the \emph{solutions} $(x,\varepsilon,\lambda)$ of this problem. Namely, under the assumption that $x_* \in S^{k-1}$ is one of the two unit eigenvectors of $L$ corresponding to a simple eigenvalue $\lambda_* \in \R$, we show that, in the set of all the solutions, the connected component containing $(x_*,0,\lambda_*)$ is either unbounded or meets a solution $(x^*,0,\lambda^*)$ having $x^* \not= x_*$. 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.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2020 Topological Methods in Nonlinear Analysis
• Date:
  2020.03.07
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 55, No 1 (March 2020); 169 - 184;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.093/25033 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2019.093/25034
• Subject:
  Eigenvalues;
  eigenvectors;
  nonlinear spectral theory;
  transversality;
  intersection number;
  47J10;
  47A75;
  55M25
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.093
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other