On Volterra Quadratic Stochastic Operators With Continual State Space

摘要

Let (X,F) be a measurable space, and S(X, F) be the set of all probability measures on (X,F) where X is a state space and F is σ-algebraon X . We consider a nonlinear transformation (quadratic stochastic operator) defined by (Vλ)(A)=∫∫P(x,y,A)dλ(y),Where P(x,y,A) is regarded as a function of two variables x and y with fixed A ∈ F. A quadratic stochastic operator V is called a regular, if for any initial measure the strong limit lim V~n (λ) is exists. In this paper, we construct a family of quadratic stochastic operators defined on the segment X = [0,1] with Borel σ-algebra F on X, prove their regularity and show that the limit measure is a Dirac measure.