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doi: 10.3934/dcds.2020135

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

Received  August 2019 Revised  November 2019 Published  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.
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.
Citation: Justyna Jarczyk, Witold Jarczyk. Gaussian iterative algorithm and integrated automorphism equation for random means. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020135
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

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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

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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, di {f}{f} erential equations and applications, 10th AIMS Conference. Suppl., 2015 (2015), 132–141. doi: 10.3934/proc.2015.0132.  Google Scholar

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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

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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-di {f}{f} erential 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

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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. Di {f}{f} erence 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 di {f}{f} érentielles a {f}{f} ectées 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

show all references

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, di {f}{f} erential 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-di {f}{f} erential 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. Di {f}{f} erence 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 di {f}{f} érentielles a {f}{f} ectées 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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