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Journal of Convex Analysis 28 (2021), No. 1, 019--030
Copyright Heldermann Verlag 2021

A Family of Volterra Cubic Stochastic Operators

Uygun Jamilov
V. I. Romanovskiy Institute of Mathematics, Academy of Sciences, 100170 Tashkent, Uzbekistan

Andrejs Reinfelds
Institute of Mathematics and Computer Sciences, University of Latvia, Riga, Latvia
and: Faculty of Physics, Mathematics and Optometry, University of Latvia, Riga, Latvia

We consider a convex combination of Volterra cubic stochastic operators defined on a two-dimensional simplex depending on the parameter θ and study their trajectory behaviours. We show that at the values θ = 0.5 the trajectories change their orientation. Moreover, for θ small than 0.5 any Volterra cubic stochastic operator has the property being regular and it is non-ergodic while θ is greater than 0.5.

Keywords: Quadratic stochastic operator, Volterra operator, cubic stochastic operator.

MSC: 37N25; 92D10.

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