Positive eigenvalue-eigenvector of nonlinear positive mappings

Yisheng Song, Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

8 Citations (Scopus)

Abstract

We show that an (eventually) strongly increasing and positively homogeneous mapping T defined on a Banach space can be turned into an Edelstein contraction with respect to Hilbert's projective metric. By applying the Edelstein contraction theorem, a nonlinear version of the famous Krein-Rutman theorem is presented, and a simple iteration process {Tkx/{norm of matrix}Tkx{norm of matrix}} (∀ x ∈ P+) is given for finding a positive eigenvector with positive eigenvalue of T. In particular, the eigenvalue problem of a nonnegative tensor A can be viewed as the fixed point problem of the Edelstein contraction with respect to Hilbert's projective metric. As a result, the nonlinear Perron-Frobenius property of a nonnegative tensor A is reached easily.
Original languageEnglish
Pages (from-to)181-199
Number of pages19
JournalFrontiers of Mathematics in China
Volume9
Issue number1
DOIs
Publication statusPublished - 1 Feb 2014

Keywords

  • Edelstein contraction
  • eigenvalue-eigenvector
  • homogeneous mapping
  • Nonnegative tensor
  • strongly increasing

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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