Positive eigenvalue-eigenvector of nonlinear positive mappings |
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Authors: | Yisheng Song Liqun Qi |
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Institution: | 1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China; 2. College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China |
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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/ ||Tkx||}(?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. |
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Keywords: | Nonnegative tensor Edelstein contraction strongly increasing homogeneous mapping eigenvalue-eigenvector |
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