On three-dimensional mixing geometric quadratic stochastic operators

• Main Authors: Khaled, Ftameh, Pah, Chin Hee
• Format: Article
• Language: English; English
• Published: Horizon Research Publishing 2021
• Subjects: QA273 Probabilities
• Online Access: http://irep.iium.edu.my/89772/7/89772_On%20three-dimensional%20mixing%20geometric%20quadratic%20stochastic%20operators.pdf; http://irep.iium.edu.my/89772/13/89772_On%20three-dimensional%20mixing%20geometric%20quadratic%20stochastic%20operators_SCOPUS.pdf; http://irep.iium.edu.my/89772/; https://www.hrpub.org/journals/article_info.php?aid=10709
• Institution: Universiti Islam Antarabangsa Malaysia

It is widely recognized that the theory of quadratic stochastic operator frequently arises due to its enormous contribution as a source of analysis for the investigation of dynamical properties and modeling in diverse domains. In this paper, we are motivated to construct a class of quadratic stochastic operators called mixing quadratic stochastic operators generated by geometric distribution on infinite state space X . We also study regularity of such operators by investigating of the limit behavior for each case of the parameter. Some of non-regular cases proved for a new definition of mixing operators by using the shifting definition, where the new parameters satisfy the shifted conditions. A mixing quadratic stochastic operator was established on 3-partitions of the state space X and considered for a special case of the parameter ε . We found that the mixing quadratic stochastic operator is a regular transformation for 1 /4< ε <1 /2 and is a non-regular for ε <1/4 . Also, the trajectories converge to one of the fixed points. Stability and instability of the fixed points were investigated by finding of the eigenvalues of Jacobian matrix at these fixed points. We approximate the parameter ε by the parameter 6r , where we established the regularity of the quadratic stochastic operators for some inequalities that satisfy 6r . We conclude this paper by comparing with previous studies where we found some of such quadratic stochastic operators will be non-regular.