# 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:

- 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}

}

DOI: 10.1063/1.4915658

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