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Frontiers of Mathematics in China

, Volume 9, Issue 1, pp 181–199 | Cite as

Positive eigenvalue-eigenvector of nonlinear positive mappings

  • Yisheng SongEmail author
  • Liqun Qi
Research Article

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 k x/‖T k x‖} (∀ xP +) 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 

MSC

47A52 47J10 47H09 15A48 47H07 

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Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHong KongChina
  2. 2.College of Mathematics and Information ScienceHenan Normal UniversityXinxiangChina

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