On the Lebesgue nonlinear transformations

TL;DR

This paper introduces and analyzes Lebesgue quadratic stochastic operators, demonstrating their regularity and dynamic properties on Lebesgue measures within the interval [0,1], advancing understanding of nonlinear measure transformations.

Contribution

It introduces a new class of quadratic stochastic operators on Lebesgue measures and proves their regularity, contributing to the theory of nonlinear measure transformations.

Findings

Proves the regularity of Lebesgue quadratic stochastic operators.

Analyzes the dynamics of these operators on [0,1].

Establishes foundational properties of nonlinear measure transformations.

Abstract

In this paper, we introduce a quadratic stochastic operators on the set of all probability measures of a measurable space. We study the dynamics of the Lebesgue quadratic stochastic operator on the set of all Lebesgue measures of the set [0,1]. Namely, we prove the regularity of the Lebesgue quadratic stochastic operators