Abstract

We consider a convex combination of certain Volterra operators defined on the two-dimensional simplex depending on the parameter θ, 0 ≤ θ ≤ 1, and study their trajectory behaviours. We divide the set of the parameter θ into two subsets such that on the first (resp. second) of them the corresponding Volterra operator is a regular (resp. non-ergodic) transformation.

Author information

Contact details are reproduced from the original publication and may be historical.

Uygun Jamilov

V. I. Romanovskiy Institute of Mathematics, and: Akfa University, and: National University of Uzbekistan, Tashkent, Uzbekistan

uygun.jamilov@mathinst.uz

Keywords

  • Quadratic stochastic operator
  • Volterra operator
  • cubic stochastic operator

Mathematics Subject Classification

37N25; 92D10

References

  1. S. Bernstein: Solution of a mathematical problem connected with the theory of heredity, Ann. Math. Stat. 13/1 (1942) 53–61.https://doi.org/10.1214/aoms/1177731642Google Scholar
  2. R. L. Devaney: An Introduction to Chaotic Dynamical Systems, reprint of the second 1989 edition, Studies in Nonlinearity, Westview Press, Boulder (2003).Google Scholar
  3. N. N. Ganikhodjaev, R. N. Ganikhodjaev, U. U. Jamilov (Zhamilov): Quadratic stochastic operators and zero-sum game dynamics, Ergodic Theory Dyn. Syst. 35/5 (2015) 1443–1473.https://doi.org/10.1017/etds.2013.109Google Scholar
  4. N. N. Ganikhodzhaev, D. V. Zanin: On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplex, Russ. Math. Surv. 59/3 (2004) 571–572.https://doi.org/10.1070/RM2004v059n03ABEH000744Google Scholar
  5. R. N. Ganikhodzhaev: Quadratic stochastic operators, Lyapunov functions and tournaments, Sb. Math. 76/2 (1993) 489–506.https://doi.org/10.1070/SM1993v076n02ABEH003423Google Scholar
  6. R. N. Ganikhodzhaev: Map of fixed points and Lyapunov functions for one class of discrete dynamical systems, Math. Notes 56/5 (1994) 1125–1131.https://doi.org/10.1007/BF02274660Google Scholar
  7. R. Ganikhodzhaev, F. Mukhamedov, U. Rozikov: Quadratic stochastic operators and processes: results and open problems, Inf. Dim. Analysis Quant. Prob. Rel. Topics 14/2 (2011) 279–335.https://doi.org/10.1142/S0219025711004365Google Scholar
  8. A. J. Homburg, U. U. Jamilov, M. Scheutzow: Asymptotics for a class of iterated random cubic operators, Nonlinearity 32 (2019) 3646–3660.https://doi.org/10.1088/1361-6544/ab1f24Google Scholar
  9. U. U. Jamilov: On a family of strictly non-Volterra quadratic stochastic operators, J. Phys. Conf. Ser. 697 (2016), art. no. 012013.https://doi.org/10.1088/1742-6596/697/1/012013Google Scholar
  10. U. U. Jamilov, A. Yu. Khamraev, M. Ladra: On a Volterra cubic stochastic operator, Bull. Math. Biol. 80/2 (2018) 319–334.https://doi.org/10.1007/s11538-017-0376-0Google Scholar
  11. U. U. Jamilov, M. Ladra: Non-ergodicity of uniform quadratic stochastic operators, Qual. Theory Dyn. Syst. 15/1 (2016) 257–271.https://doi.org/10.1007/s12346-015-0145-0Google Scholar
  12. U. U. Jamilov, M. Ladra: On identically distributed non-Volterra cubic stochastic operator, J. Appl. Nonlin. Dyn. 6/1 (2017) 79–90.https://doi.org/10.5890/JAND.2017.03.006Google Scholar
  13. U. U. Jamilov, M. Ladra, On a family of non-Volterra quadratic operators acting on a simplex, Qual. Theo. Dyn. Sys. 19/3 (2020), art. no. 95.https://doi.org/10.1007/s12346-020-00433-xGoogle Scholar
  14. U. U. Jamilov, A. Reinfelds: A family of Volterra cubic stochastic operators, J. Convex Analysis 28/1 (2021) 19–30.https://doi.org/10.68381/jca28003Google Scholar
  15. U. U. Jamilov, M. Scheutzow, M. Wilke-Berenguer: On the random dynamics of Volterra quadratic operators, Ergodic Theory Dyn. Syst. 37/1 (2017) 228–243.https://doi.org/10.1017/etds.2015.30Google Scholar
  16. H. Kesten: Quadratic transformations: A model for population growth I, Adv. Appl. Probab. 2 (1970) 1–82.https://doi.org/10.2307/3518344Google Scholar
  17. Y. I. Lyubich: Mathematical Structures in Population Genetics, Biomathematics 22, Springer, Berlin (1992).https://doi.org/10.1007/978-3-642-76211-6Google Scholar
  18. F. M. Mukhamedov, U. U. Jamilov, A. T. Pirnapasov: On non-ergodic uniform Lotka-Volterra operators, Math. Notes. 105 (2019) 258–264.https://doi.org/10.1134/S0001434619010280Google Scholar
  19. F. M. Mukhamedov, C. H. Pah, A. Rosli: On non-ergodic Volterra cubic stochastic operators, Qual. Theory Dyn. Syst. 18 (2019) 1225–1235.https://doi.org/10.1007/s12346-019-00334-8Google Scholar
  20. U. A. Rozikov, A. Yu. Khamraev: On cubic operators defined on finite-dimensional simplices, Ukrainian Math. J. 56/10 (2004) 1699–1711.https://doi.org/10.1007/s11253-005-0145-3Google Scholar
  21. U. A. Rozikov, A. Yu. Khamraev: On construction and a class of non-Volterra cubic stochastic operators, Nonlinear Dyn. Syst. Theory 14/1 (2014) 92–100.Google Scholar
  22. U. A. Rozikov, U. Zhamilov: F -quadratic stochastic operators, Math. Notes 83/3-4 (2008) 554–559.https://doi.org/10.1134/S0001434608030280Google Scholar
  23. U. A. Rozikov, U. U. Zhamilov: Volterra quadratic stochastic operators of a two-sex population, Ukrainian Math. J. 63/7 (2011) 1136–1153.https://doi.org/10.1007/s11253-011-0568-yGoogle Scholar
  24. S. M. Ulam: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics 8, Interscience Publishers, New York (1960).Google Scholar
  25. M. I. Zakharevich: On the behaviour of trajectories and the ergodic hypothesis for quadratic mappings of a simplex, Russ. Math. Surveys 33/6 (1978) 265–266.https://doi.org/10.1070/RM1978v033n06ABEH003890Google Scholar
  26. U. U. Zhamilov, U. A. Rozikov: On the dynamics of strictly non-Volterra quadratic stochastic operators on the two-dimensional simplex, Sb. Math. 200/9 (2009) 1339–1351.https://doi.org/10.1070/SM2009v200n09ABEH004039Google Scholar

Suggested citation

U. Jamilov. “On the Convex Combination of Volterra Operators.” Journal of Convex Analysis 29 (2022), No. 4, 1065–1082. https://doi.org/10.68381/jca29060

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.