turkmath.org

It is a well-known fact that the asymptotic behavior of certain physical systems constitutes an integral part of physics and mathematics. The solutions of equations are typically expressed by one-parameter operator semigroups, and therefore, it is essential to know their long-term behavior. One needs to have effective methods and techniques for the examination of their asymptotic behavior. Great attention has been attracted to the study of the connection between uniform ergodicities and ergodic coefficients of semigroups defined on the classical function spaces. However, in those investigations, the limiting operator was taken as a rank-one projection. If one wants to consider the limiting operators as some projections then the standard ergodic coefficient is no longer effective. Consequently, a generalized Dobrushin ergodicity coefficient $δ_P(T)$  for Markov operators (acting on an abstract state space) concerning a projection P has been introduced.

The main goal of the talk is to explore stability and perturbation bounds for positive $C_0$-semigroups defined on abstract state spaces using a generalized Dobrushin ergodicity coefficient. It is worth noting that we also establish that the uniform and weak stabilities of time average Markov operators through the generalized ergodicity coefficient shed new light on this subject.