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 language | English |
---|---|
Pages (from-to) | 181-199 |
Number of pages | 19 |
Journal | Frontiers of Mathematics in China |
Volume | 9 |
Issue number | 1 |
DOIs | |
Publication status | Published - 1 Feb 2014 |
Keywords
- Edelstein contraction
- eigenvalue-eigenvector
- homogeneous mapping
- Nonnegative tensor
- strongly increasing
ASJC Scopus subject areas
- Mathematics (miscellaneous)