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 QSO study defined on such a state space, we consider a QSO class on continuous state space in this work. It is known as Lebesgue QSO, which is an exponential measure generated by three measurable partitions with three parameters. We specify two distinct cases of three parameters, which are represented by reducible QSOs. We demonstrate that such a reducible QSO can be reduced to a one-dimensional simplex. Consequently, we analyse the dynamics of such operators by employing the first derivative 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 second-order cycle. Corresponding to a strong limit of the sequence of the reduced QSO, such an operator is regular. Meanwhile, such an operator is a nonregular transformation when a second-order cycle exists.