# American Institute of Mathematical Sciences

December  2020, 40(12): 6837-6844. doi: 10.3934/dcds.2020135

## Gaussian iterative algorithm and integrated automorphism equation for random means

 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

Received  August 2019 Revised  November 2019 Published  December 2020 Early access  February 2020

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.
Citation: Justyna Jarczyk, Witold Jarczyk. Gaussian iterative algorithm and integrated automorphism equation for random means. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6837-6844. doi: 10.3934/dcds.2020135
##### References:
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##### References:
 [1] K. Baron and W. Jarczyk, Random-valued functions and iterative functional equations, Aequationes Math., 67 (2004), 140-153.  doi: 10.1007/s00010-003-2717-3.  Google Scholar [2] K. Baron and R. Kapica, A uniqueness-type problem for linear iterative equations, Analysis (Munich), 29 (2009), 95-101.   Google Scholar [3] K. Baron and M. Kuczma, Iteration of random-valued functions on the unit interval, Colloq. Math., 37 (1977), 263-269.  doi: 10.4064/cm-37-2-263-269.  Google Scholar [4] K. Baron and J. Morawiec, Lipschitzian solutions to linear iterative equations revisited, Aequationes Math., 91 (2017), 161-167.  doi: 10.1007/s00010-016-0455-6.  Google Scholar [5] L. V. Bogachev, G. Derfel and S. A. Molchanov, Analysis of the archetypal functional equation in the non-critical case, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications, 10th AIMS Conference. Suppl., 2015 (2015), 132–141. doi: 10.3934/proc.2015.0132.  Google Scholar [6] L. V. Bogachev, G. Derfel and S. A. Molchanov, On bounded continuous solutions of the archetypal equation with rescalling, Proc. A., 471 (2015), 20150351, 19 pp. doi: 10.1098/rspa.2015.0351.  Google Scholar [7] G. Choquet and J. Deny, Sur l'équation de convolution $\mu = \mu \ast s$, C. R. Acad. Sci. Paris, 250 (1960), 799-801.   Google Scholar [8] G. A. Derfel, A probabilistic method for studying a class of functional-differential equations, Ukrainian Math. J., 41 (1990), 1137-1141.  doi: 10.1007/BF01057249.  Google Scholar [9] Ph. Diamond, A stochastic functional equation, Aequationes Math., 15 (1977), 225-233.  doi: 10.1007/BF01835652.  Google Scholar [10] C. F. Gauss, Werke, Göttingen, 1876. Google Scholar [11] C. F. Gauss and H. Geppert, Bestimmung der Anziehung eines elliptischen Ringes: Nachlass zur Teorie des arithmetisch-geometrischen Mittels und der Modulfunktion, Engelmann, Leipzig, 1927. Google Scholar [12] J. Jarczyk, Parametrized means and limit properties of their Gaussian iterations, Appl. Math. Comput., 261 (2015), 81-89.  doi: 10.1016/j.amc.2015.03.085.  Google Scholar [13] J. Jarczyk and W. Jarczyk, Invariance of means, Aequationes Math., 92 (2018), 801-872.  doi: 10.1007/s00010-018-0564-5.  Google Scholar [14] R. Kapica and J. Morawiec, Inhomogeneous refinement equations with random affine maps, J. Difference Equ. Appl., 21 (2015), 1200–1211. doi: 10.1080/10236198.2015.1065823.  Google Scholar [15] J. L. Lagrange, Sur une nouvelle Méthode de Calcul Intégrale pour différentielles affectées d'un radical carre, Mem. Acad. R. Sci. Turin II, 2 (1784/1785), 252-312.   Google Scholar [16] J. Matkowski, Invariant and complementary quasi-arithmetic means, Aequationes Math., 57 (1999), 87-107.  doi: 10.1007/s000100050072.  Google Scholar [17] J. Matkowski, Iterations of mean-type mappings and invariant means, European Conference on Iteration Theory (Muszyna-Z{\l}ockie, 1998), Ann. Math. Sil., 13 (1999), 211-226.   Google Scholar [18] C. Radhakrishna Rao and D. N. Shanbhag, Choquet-Deny Type Functional Equations with Applications to Stochastic Models, John Wiley & Sons, Chichester, 1994.  Google Scholar [19] M. Sudzik, On a functional equation related to a problem of G. Derfel, Aequationes Math., 93 (2019), 137-148.  doi: 10.1007/s00010-018-0600-5.  Google Scholar
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