We present an abstract framework for asymptotic analysis of convergence based on the notions of eventual families of sets that we define. A family of subsets of a given set is called here an “eventual family” if it is upper hereditary with respect to inclusion. We define accumulation points of eventual families in a Hausdorff topological space and define the “image family” of an eventual family. Focusing on eventual families in the set of the integers enables us to talk about sequences of points. We expand our work to the notion of a “multiset” which is a modification of the concept of a set that allows for multiple instances of its elements and enable the development of “multifamilies” which are either “increasing” or “decreasing”. The abstract structure created here is motivated by, and feeds back to, our look at the convergence analysis of an iterative process for asymptotically finding a common fixed point of a family of operators.
Bibliographical noteFunding Information:
The work of Yair Censor is supported by the Israel Science Foundation and the Natural Science Foundation China, ISF-NSFC joint research program Grant No. 2874/19.
© 2022, The Author(s), under exclusive licence to Springer Nature B.V.
- Common fixed-points
- Eventual families
- Firmly nonexpansive operators
- Hausdorff topological space
- Set convergence
ASJC Scopus subject areas
- Statistics and Probability
- Numerical Analysis
- Geometry and Topology
- Applied Mathematics