Open Access

# On uniqueness of fixed points of quadratic stochastic operators on a 2D simplex

### Abstract

The Perron-Frobenius theorem states that a linear stochastic operator associated with a positive square stochastic matrix has a unique fixed point in the simplex and it is strongly ergodic to that fixed point. However, in general, the similar result for quadratic stochastic operators associated with positive cubic stochastic matrices does not hold true. Namely, it may have more than one fixed point in the simplex. Moreover, the uniqueness of fixed points does not imply the strong ergodicity of quadratic stochastic operators. In this paper, for some classes of positive cubic stochastic matrices, we provide a uniqueness criterion for fixed points of quadratic stochastic operators acting on a 2D simplex. Some supporting examples are also presented.

Key words: Quadratic stochastic operator, fixed point, ergodicity, contraction.

### Article Information

 Title On uniqueness of fixed points of quadratic stochastic operators on a 2D simplex Source Methods Funct. Anal. Topology, Vol. 24 (2018), no. 3, 255-264 MathSciNet MR3860805 Milestones Received 24/05/2017; Revised 10/01/2018 Copyright The Author(s) 2018 (CC BY-SA)

### Authors Information

M. Saburov
Faculty of Science, International Islamic University Malaysia, 25200 Kuantan, Pahang, Malaysia

N. A. Yusof
Faculty of Science, International Islamic University Malaysia, 25200 Kuantan, Pahang, Malaysia

### Citation Example

M. Saburov and N. A. Yusof, On uniqueness of fixed points of quadratic stochastic operators on a 2D simplex, Methods Funct. Anal. Topology 24 (2018), no. 3, 255-264.

### BibTex

@article {MFAT1085,
AUTHOR = {M. Saburov and N. A. Yusof},
TITLE = {On uniqueness of fixed points of quadratic stochastic operators on a 2D simplex},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {24},
YEAR = {2018},
NUMBER = {3},
PAGES = {255-264},
ISSN = {1029-3531},
MRNUMBER = {MR3860805},
URL = {http://mfat.imath.kiev.ua/article/?id=1085},
}

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