On Lebesgue Quadratic Stochastic Operators with exponential measure generated by 3-partition

Quadratic stochastic operator (QSO) is a continuously expanding topic in nonlinear operator theory due to its immense applications in various disciplines. Inspired by the notion of infinite state space as there is limited literature on the study of QSO defined on such state space, in this paper, we consider a class of QSO on continuous state space, called Lebesgue QSO with exponential measure generated by three measurable partitions with three parameters. We define two different cases of three parameters which represent by a reducible QSO. We demonstrate that such a reducible QSO can be reduced to a one-dimensional setting. Then, we analyze the dynamics of such operators by employing the Jacobian matrix method and show that the operators may have either an attracting fixed point to indicate the existence of a strong limit or a non-attracting fixed point to suggest the presence of a cycle of second-order. Corresponding to the existence of a strong limit of the sequence of the reduced QSO, such operator is a regular transformation. Meanwhile, such operator is a nonregular transformation when a second-order cycle exists