Gaussian iterative algorithm and integrated automorphism equation for random means

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December 2020, 40(12): 6837-6844. 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 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.

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 and Continuous Dynamical Systems, 2020, 40 (12) : 6837-6844. 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.
[2]	K. Baron and R. Kapica, A uniqueness-type problem for linear iterative equations, Analysis (Munich), 29 (2009), 95-101.
[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.
[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.
[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.
[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.
[7]	G. Choquet and J. Deny, Sur l'équation de convolution $\mu = \mu \ast s$, C. R. Acad. Sci. Paris, 250 (1960), 799-801.
[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.
[9]	Ph. Diamond, A stochastic functional equation, Aequationes Math., 15 (1977), 225-233. doi: 10.1007/BF01835652.
[10]	C. F. Gauss, Werke, Göttingen, 1876.
[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.
[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.
[13]	J. Jarczyk and W. Jarczyk, Invariance of means, Aequationes Math., 92 (2018), 801-872. doi: 10.1007/s00010-018-0564-5.
[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.
[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.
[16]	J. Matkowski, Invariant and complementary quasi-arithmetic means, Aequationes Math., 57 (1999), 87-107. doi: 10.1007/s000100050072.
[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.
[18]	C. Radhakrishna Rao and D. N. Shanbhag, Choquet-Deny Type Functional Equations with Applications to Stochastic Models, John Wiley & Sons, Chichester, 1994.
[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.

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.
[2]	K. Baron and R. Kapica, A uniqueness-type problem for linear iterative equations, Analysis (Munich), 29 (2009), 95-101.
[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.
[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.
[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.
[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.
[7]	G. Choquet and J. Deny, Sur l'équation de convolution $\mu = \mu \ast s$, C. R. Acad. Sci. Paris, 250 (1960), 799-801.
[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.
[9]	Ph. Diamond, A stochastic functional equation, Aequationes Math., 15 (1977), 225-233. doi: 10.1007/BF01835652.
[10]	C. F. Gauss, Werke, Göttingen, 1876.
[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.
[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.
[13]	J. Jarczyk and W. Jarczyk, Invariance of means, Aequationes Math., 92 (2018), 801-872. doi: 10.1007/s00010-018-0564-5.
[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.
[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.
[16]	J. Matkowski, Invariant and complementary quasi-arithmetic means, Aequationes Math., 57 (1999), 87-107. doi: 10.1007/s000100050072.
[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.
[18]	C. Radhakrishna Rao and D. N. Shanbhag, Choquet-Deny Type Functional Equations with Applications to Stochastic Models, John Wiley & Sons, Chichester, 1994.
[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.

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