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