On volterra quadratic stochastic operators with continual state space - Ganikhodjaev, Nasir; Hamzah, Nur Zatul Akmar (American Institute of Physics, 2015)
Let FX ),( be a measurable space, and FXS ),( be the set of all probability measures on FX ),( where X is a state space and F is V - algebraon X . We consider a nonlinear transformation (quadratic stochastic operator) defined by ³³ X X ( O)( OO ydxdAyxPAV )()(),,() , where AyxP ),,( is regarded as a function of two variables x and y with fixed FA . A quadratic stochastic operator V is called a regular, if for any initial measure the strong limit lim O)( nnV fo is exists. In this paper, we construct a family of quadratic stochastic operators defined on the segment X > @1,0 with Borel V - algebra F on X, prove their regularity and show that the limit measure is a Dirac measure.