Gaussian iterative algorithm and integrated automorphism equation for random means

Justyna Jarczyk; Witold Jarczyk

doi:10.3934/dcds.2020135

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2020, Volume 40, Issue 12: 6837-6844. Doi: 10.3934/dcds.2020135

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Gaussian iterative algorithm and integrated automorphism equation for random means

Justyna Jarczyk^1, and
Witold Jarczyk^2, ,

1.
Institute of Mathematics, University of Zielona Góra, Szafrana 4a, PL-65-516 Zielona Góra, Poland
2.
Institute of Mathematics and Informatics, The John Paul II Catholic University of Lublin, Konstantynów 1h, PL-20-708 Lublin, Poland

^* Corresponding author: Witold Jarczyk
^* Corresponding author: Witold Jarczyk

Received: July 31, 2019

Revised: October 31, 2019

Early access: February 2020

Published: August 2020

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Abstract

Gauss-type iterates for random means are considered and their limit behaviour is studied. Among others the invariance of the limit with respect to the given random mean-type mapping $ {\bf{M}} $ is established under some relatively weak assumptions. The algorithm is applied to prove the existence and uniqueness of solutions $ \varphi $ of the equation

$ \varphi({\bf x}) = \int_{\Omega}\varphi\left({\bf{M}}({\bf x},\omega)\right)dP(\omega) $

in the class of (deterministic) means in $ p $ variables.

Keywords:

Gauss-type iterates,
random mean,
convergence of successive approximations,
invariance,
integrated automorphism equation,
existence and uniqueness of solutions.

Mathematics Subject Classification: Primary: 26A18, 37E05, 39B12; Secondary: 26E60, 60F17.

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References

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