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On Regularity of Diagonally Positive Quadratic Doubly Stochastic Operators

Results in Mathematics volume 72, pages 1907–1918 (2017)Cite this article

Abstract

The classical Perron–Frobenius theorem says that a trajectory of a linear stochastic operator associated with a positive square stochastic matrix always converges to a unique fixed point. In general, an analogy of the Perron–Frobenius theorem does not hold for a quadratic stochastic operator associated with a positive cubic stochastic matrix. Namely, its trajectories may converge to different fixed points depending on initial points or may not converge at all. In this paper, we show regularity of quadratic doubly stochastic operators associated with diagonally positive cubic stochastic matrices. This is a nonlinear analogy of the Perron–Frobenius theorem for positive doubly stochastic matrices.

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Authors and Affiliations

  1. Department of Computational and Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P.O. Box 25200 , Kuantan, Pahang, Malaysia

    Mansoor Saburov

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  1. Mansoor Saburov
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Correspondence to Mansoor Saburov.

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Saburov, M. On Regularity of Diagonally Positive Quadratic Doubly Stochastic Operators. Results Math 72, 1907–1918 (2017). https://doi.org/10.1007/s00025-017-0723-3

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  • DOI: https://doi.org/10.1007/s00025-017-0723-3

Mathematics Subject Classification

  • 47H25
  • 47H60
  • 37A30

Keywords

  • Quadratic doubly stochastic operator
  • cubic stochastic matrix
  • regularity