### Journals Information

**
Mathematics and Statistics Vol. 10(4), pp. 861 - 867 DOI: 10.13189/ms.2022.100417 Reprint (PDF) (307Kb) **

## Dynamics of Nonlinear Operator Generated by Lebesgue Quadratic Stochastic Operator with Exponential Measure

**Nur Zatul Akmar Hamzah ^{1}^{,2}^{,*}, Siti Nurlaili Karim ^{1}, Mathuri Selvarajoo ^{3}, Noor Azida Sahabudin ^{4}**

^{1}Department of Computational and Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, Bandar Indera Mahkota, 25200, Kuantan, Pahang, Malaysia

^{2}Dynamical Systems and Their Applications Unit, Faculty of Science, International Islamic University Malaysia, Bandar Indera Mahkota, 25200, Kuantan, Pahang, Malaysia

^{3}Department of Mathematics, Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450, Shah Alam, Selangor, Malaysia

^{4}Faculty of Computing, College of Computing and Applied Sciences, Universiti Malaysia Pahang, 26300, Gambang, Pahang, Malaysia

**ABSTRACT**

Quadratic stochastic operator (QSO) is a branch of nonlinear operator studies initiated by Bernstein in 1924 through his presentation on population genetics. The study of QSO is still ongoing due to the incomplete understanding of the trajectory behavior of such operators given certain conditions and measures. In this paper, we intend to introduce and investigate a class of QSO named Lebesgue QSO which gets its name from the Lebesgue measure as the measure is used to define the probability measure of such QSO. The broad definition of Lebesgue QSO allows the construction of a new measure as its family of probability measure. We construct a class of Lebesgue QSO with exponential measure generated by 3-partition with three different parameters defined on continual state space . Also, we present the dynamics of such QSO by describing the fixed points and periodic points of the system of equations generated by the defined QSO using a functional analysis approach. The investigation is concluded by the regularity of the operator, where such Lebesgue QSO is either regular or nonregular depending on the parameters and defined measurable partitions. The result of this research allows us to define a new family of functions of the probability measure of Lebesgue QSO and compare their dynamics with the existing Lebesgue QSO.

**KEYWORDS**

Quadratic Stochastic Operator, Lebesgue Measure, Regularity, Partition, Periodic Points

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Nur Zatul Akmar Hamzah , Siti Nurlaili Karim , Mathuri Selvarajoo , Noor Azida Sahabudin , "Dynamics of Nonlinear Operator Generated by Lebesgue Quadratic Stochastic Operator with Exponential Measure," Mathematics and Statistics, Vol. 10, No. 4, pp. 861 - 867, 2022. DOI: 10.13189/ms.2022.100417.

(b). APA Format:

Nur Zatul Akmar Hamzah , Siti Nurlaili Karim , Mathuri Selvarajoo , Noor Azida Sahabudin (2022). Dynamics of Nonlinear Operator Generated by Lebesgue Quadratic Stochastic Operator with Exponential Measure. Mathematics and Statistics, 10(4), 861 - 867. DOI: 10.13189/ms.2022.100417.