Skip to main content

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


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.

This is a preview of subscription content, access via your institution.

Availability of data and material

Not applicable.

Code Availability

Not applicable.


  1. Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248–264 (2001)

    MathSciNet  Article  Google Scholar 

  2. Berinde, V.: Iterative Approximation of Fixed Points, 2nd edn. Springer, Berlin, Heidelberg, Germany (2007)

    MATH  Google Scholar 

  3. Blizard, W.: Multiset theory. Notre Dame Journal of Formal Logic 30, 36–66 (1989)

    MathSciNet  MATH  Google Scholar 

  4. Borg, P.: Cross-intersecting non-empty uniform subfamilies of hereditary families. Eur. J. Comb. 78, 256–267 (2019)

    MathSciNet  Article  Google Scholar 

  5. Browder, F.E.: Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z 100, 201–225 (1967)

    MathSciNet  Article  Google Scholar 

  6. Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and enlargements of monotone operators. Springer Science+Business media LLC (2008)

  7. Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Springer, Berlin (2012)

    MATH  Google Scholar 

  8. Cegielski, A., Censor, Y.: Opial-type theorems and the common fixed point problem. In: Bauschke, H.H., Burachik, R.S., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp 155–183. Springer, New York, NY, USA (2011)

  9. Censor, Y., Segal, A.: On the string averaging method for sparse common fixed points problems. Int. Trans. Oper. Res. 16, 481–494 (2009)

    MathSciNet  Article  Google Scholar 

  10. Censor, Y., Zenios, S.A.: Parallel Optimization: Theory, Algorithms, and Applications. Oxford University Press, New York, NY, USA (1997)

    MATH  Google Scholar 

  11. Combettes, P.L.: Quasi-Fejérian analysis of some optimization algorithms. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp 115–152. Elsevier, Amsterdam (2001)

  12. Crombez, G.: A geometrical look at iterative methods for operators with fixed points. Numer. Funct. Anal. Optim. 26, 157–175 (2005)

    MathSciNet  Article  Google Scholar 

  13. Dantzig, G.B., Folkman, J., Shapiro, N.: On the continuity of the minimum set of a continuous function. J. Math. Anal. Appl. 17, 519–548 (1967)

    MathSciNet  Article  Google Scholar 

  14. Lent, A., Censor, Y.: The primal-dual algorithm as a constraint-set-manipulation device. Math. Program. 50, 343–357 (1991)

    MathSciNet  Article  Google Scholar 

  15. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin, Heidelberg, Germany (1998)

    Book  Google Scholar 

  16. Salinetti, G., Wets, R.J.-B.: On the convergence of sequences of convex sets in finite dimensions. SIAM Review 21, 18–33 (1979). Addendum: SIAM Review 22, 86, (1980)

    MathSciNet  Article  Google Scholar 

  17. Zaslavski, A.J.: Approximate Solutions of Common Fixed-Point Problems. Springer International Publishing, Switzerland (2016)

    Book  Google Scholar 

Download references


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.

Author information

Authors and Affiliations



Not applicable.

Corresponding author

Correspondence to Yair Censor.

Ethics declarations

Conflict of Interests

The authors have no conflicts of interest to declare that are relevant to the content of this article.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Censor, Y., Levy, E. Limits of Eventual Families of Sets with Application to Algorithms for the Common Fixed Point Problem. Set-Valued Var. Anal 30, 1077–1088 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


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

Mathematics Subject Classification (2010)

  • 49J53
  • 47H10