Annals of Functional Analysis

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

Mansoor Saburov

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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

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

Dates
First available in Project Euclid: 1 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.afa/1435764015

Digital Object Identifier
doi:10.15352/afa/06-4-247

Mathematical Reviews number (MathSciNet)
MR3365995

Zentralblatt MATH identifier
1339.47076

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.afa/1435764015


Export citation

References

  • 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.
    Mathematical Reviews (MathSciNet): MR2116542
    Digital Object Identifier: doi:10.4213/rm744
    Zentralblatt MATH: 1145.47021
  • 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.
    Mathematical Reviews (MathSciNet): MR2813492
    Zentralblatt MATH: 1242.60067
    Digital Object Identifier: doi:10.1142/S0219025711004365
  • G.H. Hardy, Divergent Series, Oxford University Press, Great Britain, 1949.
    Mathematical Reviews (MathSciNet): MR30620
  • F. Mukhamedov and M. Saburov, On homotopy of Volterrian quadratic stochastic operators, Appl. Math Info. Sci. 4 (2010), no. 1, 47–62.
    Mathematical Reviews (MathSciNet): MR2575080
  • 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.
    Mathematical Reviews (MathSciNet): MR120127
    Zentralblatt MATH: 0086.24101
  • 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.
    Mathematical Reviews (MathSciNet): MR526020

Duke University Press

Annals of Functional Analysis

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more