Positive eigenvalue-eigenvector of nonlinear positive mappings

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 {T^kx/ ||T^kx||}(\forall x\in 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.

Keywords

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

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg