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Title: On Volterra quadratic stochastic operators with continual state space

Abstract

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) = ∫{sub X}∫{sub X}P(x,y,A)dλ(x)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{sub n→∞} V{sup 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.

Authors:
;  [1]
  1. Department of Computational and Theoretical Sciences, Faculty of Science, International Islamic University, Jalan Sultan Ahmad Shah, Bandar Indera Mahkota, 25200 Kuantan, Pahang (Malaysia)
Publication Date:
OSTI Identifier:
22391652
Resource Type:
Journal Article
Journal Name:
AIP Conference Proceedings
Additional Journal Information:
Journal Volume: 1660; Journal Issue: 1; Conference: ICoMEIA 2014: International Conference on Mathematics, Engineering and Industrial Applications 2014, Penang (Malaysia), 28-30 May 2014; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0094-243X
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; ALGEBRA; FUNCTIONS; MATHEMATICAL OPERATORS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; PROBABILITY; SPACE; STOCHASTIC PROCESSES; TRANSFORMATIONS

Citation Formats

Ganikhodjaev, Nasir, and Hamzah, Nur Zatul Akmar. On Volterra quadratic stochastic operators with continual state space. United States: N. p., 2015. Web. doi:10.1063/1.4915658.
Ganikhodjaev, Nasir, & Hamzah, Nur Zatul Akmar. On Volterra quadratic stochastic operators with continual state space. United States. doi:10.1063/1.4915658.
Ganikhodjaev, Nasir, and Hamzah, Nur Zatul Akmar. Fri . "On Volterra quadratic stochastic operators with continual state space". United States. doi:10.1063/1.4915658.
@article{osti_22391652,
title = {On Volterra quadratic stochastic operators with continual state space},
author = {Ganikhodjaev, Nasir and Hamzah, Nur Zatul Akmar},
abstractNote = {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) = ∫{sub X}∫{sub X}P(x,y,A)dλ(x)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{sub n→∞} V{sup 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.},
doi = {10.1063/1.4915658},
journal = {AIP Conference Proceedings},
issn = {0094-243X},
number = 1,
volume = 1660,
place = {United States},
year = {2015},
month = {5}
}