Annals of Functional Analysis

On divergence of any order Cesàro mean of Lotka--Volterra operators

Mansoor Saburov

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Based on some numerical calculations, S.M. Ulam has conjectured that the ergodic theorem holds true for any quadratic stochastic operator acting on the finite dimensional simplex. However, M.I. Zakharevich showed that Ulam's conjecture is false in general. Later, N.N. Ganikhodjaev and D.V. Zanin have generalized Zakharevich's example in the class of quadratic stochastic Volterra operators acting on 2D simplex. In this paper, we provide a class of Lotka--Volterra operators for which any order Cesàro mean diverges. This class of Lotka--Volterra operators encompasses all previously presented operators in this context.

Article information

Ann. Funct. Anal., Volume 6, Number 4 (2015), 247-254.

First available in Project Euclid: 1 July 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47H25: Nonlinear ergodic theorems [See also 28Dxx, 37Axx, 47A35]
Secondary: 47J35: Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25]

Lotka--Volterra operator ergodic theorem Ces\`{a}ro mean


Saburov, Mansoor. On divergence of any order Cesàro mean of Lotka--Volterra operators. Ann. Funct. Anal. 6 (2015), no. 4, 247--254. doi:10.15352/afa/06-4-247.

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  • G.M. Fikhtengolts, A course of differential and integral calculus Vol. II, Moscow, 2001.
  • N.N. Ganikhodzhaev and D.V. Zanin, On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplex, Russian Math Survey 59 (2004), no. 3. (357), 161–162.
  • R.N. Ganikhodzhaev, F.M. Mukhamedov and U.A. Rozikov, Quadratic stochastic operators: Results and open problems, Inf. Dim. Anal. Qua. Prob. Rel. Top. 14 (2011), no 2, 279–335.
  • G.H. Hardy, Divergent Series, Oxford University Press, Great Britain, 1949.
  • F. Mukhamedov and M. Saburov, On homotopy of Volterrian quadratic stochastic operators, Appl. Math Info. Sci. 4 (2010), no. 1, 47–62.
  • M. Saburov, On ergodic theorem for quadratic stochastic operators, Doklady of Uzbek Academy Sci. 6 (2007), 8–11.
  • S.M. Ulam, A collection of mathematical problems, New-York, London, 1960.
  • M.I. Zakharevich, On the behaviour of trajectories and the ergodic hypothesis for quadratic mappings of a simplex, Russian Math Survey 33 (1978), no. 6, 207–208.