October  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  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 - A, 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.  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

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, 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

[1]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[2]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[3]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[4]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[5]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[6]

Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167

[7]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[8]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[9]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[10]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[11]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[12]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

[13]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[14]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[15]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[16]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

[17]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[18]

Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 201-215. doi: 10.3934/dcds.2020263

[19]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[20]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (83)
  • HTML views (285)
  • Cited by (0)

Other articles
by authors

[Back to Top]