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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Stochastic fixed points and nonlinear Perron–Frobenius theorem
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by E. Babaei, I. V. Evstigneev and S. A. Pirogov PDF
Proc. Amer. Math. Soc. 146 (2018), 4315-4330 Request permission

Abstract:

We provide conditions for the existence of measurable solutions to the equation $\xi (T\omega )=f(\omega ,\xi (\omega ))$, where $T:\Omega \rightarrow \Omega$ is an automorphism of the probability space $\Omega$ and $f(\omega ,\cdot )$ is a strictly nonexpansive mapping. We use results of this kind to establish a stochastic nonlinear analogue of the Perron–Frobenius theorem on eigenvalues and eigenvectors of a positive matrix. We consider a random mapping $D(\omega )$ of a random closed cone $K(\omega )$ in a finite-dimensional linear space into the cone $K(T\omega )$. Under the assumptions of monotonicity and homogeneity of $D(\omega )$, we prove the existence of scalar and vector measurable functions $\alpha (\omega )>0$ and $x(\omega )\in K(\omega )$ satisfying the equation $\alpha (\omega )x(T\omega )=D(\omega )x(\omega )$ almost surely.
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Additional Information
  • E. Babaei
  • Affiliation: Department of Economics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
  • MR Author ID: 1040683
  • Email: esmaeil.babaeikhezerloo@postgrad.manchester.ac.uk
  • I. V. Evstigneev
  • Affiliation: Department of Economics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
  • MR Author ID: 210292
  • Email: igor.evstigneev@manchester.ac.uk
  • S. A. Pirogov
  • Affiliation: Department of Mechanics and Mathematics, Moscow State University, ul. Lenĭnskiye Gory, 1, Moscow, Russia, 119234 –and– the Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetny 19-1, Moscow, 127051, Russia
  • MR Author ID: 231708
  • Email: s.a.pirogov@bk.ru
  • Received by editor(s): January 25, 2017
  • Received by editor(s) in revised form: December 29, 2017
  • Published electronically: June 13, 2018
  • Communicated by: Yingfei Yi
  • © Copyright 2018 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 146 (2018), 4315-4330
  • MSC (2010): Primary 37H10, 37H15; Secondary 37H05, 37H99
  • DOI: https://doi.org/10.1090/proc/14075
  • MathSciNet review: 3834661