Annals of Functional Analysis

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

Mansoor Saburov

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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


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References

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