Limits of Eventual Families of Sets with Application to Algorithms for the Common Fixed Point Problem

Yair Censor, Eliahu Levy

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)1077-1088
Number of pages12
JournalSet-Valued and Variational Analysis
Issue number3
Early online date23 Mar 2022
StateE-pub ahead of print - 23 Mar 2022

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature B.V.


  • Common fixed-points
  • Cutters
  • Eventual families
  • Firmly nonexpansive operators
  • Hausdorff topological space
  • Multifamily
  • Multiset
  • Set convergence

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Numerical Analysis
  • Geometry and Topology
  • Applied Mathematics


Dive into the research topics of 'Limits of Eventual Families of Sets with Application to Algorithms for the Common Fixed Point Problem'. Together they form a unique fingerprint.

Cite this