Skip to main content
SpringerLink
Log in

Solving Common Fixed Point Problems with a Countable Family of Operators

  • Original Article
  • Published:
Vietnam Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we study common fixed point problems. The first one is a general common fixed point problem in a metric space with a countable family of operators, while the second problem is a convex feasibility problem with infinitely many constraints solved by the subgradient projection algorithm. We show that our algorithms converge to a solution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)

    Article  MathSciNet  Google Scholar 

  2. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Butnariu, D., Censor, Y., Reich, S.: Iterative averaging of entropic projections for solving stochastic convex feasibility problems. Comput. Optim. Appl. 8, 21–39 (1997)

    Article  MathSciNet  Google Scholar 

  5. Butnariu, D., Davidi, R., Herman, G.T., Kazantsev, I.G.: Stable convergence behavior under summable perturbations of a class of projection methods for convex feasibility and optimization problems. IEEE J. Sel. Top. Signal Process. 1, 540–547 (2007)

    Article  ADS  Google Scholar 

  6. Butnariu, D., Reich, S., Zaslavski, A.J.: Convergence to fixed points of inexact orbits of Bregman-monotone and of nonexpansive operators in Banach spaces. Proceedings of Fixed Point Theory and its Applications, Mexico, pp. 11–32. Yokahama Publishers (2006)

  7. Censor, Y., Davidi, R., Herman, G.T.: Perturbation resilience and superiorization of iterative algorithms. Inverse Probl. 26, 065008 (2010)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  8. Censor, Y., Davidi, R., Herman, G.T., Schulte, R.W., Tetruashvili, L.: Projected subgradient minimization versus superiorization. J. Optim. Theory Appl. 160, 730–747 (2014)

    Article  MathSciNet  Google Scholar 

  9. Censor, Y., Elfving, T., Herman, G.T.: Averaging strings of sequential iterations for convex feasibility problems. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, pp. 101–113. North-Holland, Amsterdam (2001)

    Chapter  Google Scholar 

  10. Censor, Y., Reem, D.: Zero-convex functions, perturbation resilience, and subgradient projections for feasibility-seeking methods. Math. Program. 152, 339–380 (2015)

    Article  MathSciNet  Google Scholar 

  11. Censor, Y., Zaknoonr, M.: Algorithms and convergence results of projection methods for inconsistent feasibility problems: a review. Pure Appl. Funct. Anal. 3, 565–586 (2018)

    MathSciNet  Google Scholar 

  12. Censor, Y., Zaslavski, A.J.: Convergence and perturbation resilience of dynamic string-averaging projection methods. Comput. Optim. Appl. 54, 65–76 (2013)

    Article  MathSciNet  Google Scholar 

  13. Censor, Y., Zaslavski, A.J.: Strict Fejér monotonicity by superiorization of feasibility-seeking projection methods. J. Optim. Theory Appl. 165, 172–187 (2015)

    Article  MathSciNet  Google Scholar 

  14. Djafari-Rouhani, B., Kazmi, K.R., Moradi, S., Ali, R., Khan, S.A.: Solving the split equality hierarchical fixed point problem. Fixed Point Theory 23, 351–369 (2022)

    Article  MathSciNet  Google Scholar 

  15. Gibali, A.: A new split inverse problem and an application to least intensity feasible solutions. Pure Appl. Funct. Anal. 2, 243–258 (2017)

    MathSciNet  Google Scholar 

  16. Gibali, A., Reich, S., Zalas, R.: Outer approximation methods for solving variational inequalities in Hilbert space. Optimization 66, 417–437 (2017)

    Article  MathSciNet  Google Scholar 

  17. Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  18. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York and Basel (1984)

    Google Scholar 

  19. Gurin, L.G., Poljak, B.T., Raik, E.V.: Projection methods for finding a common point of convex sets. Zhurn. Vycisl. Mat i Mat. Fiz. 7, 1211–1228 (1967)

    MathSciNet  Google Scholar 

  20. Karapinar, E., Mitrović, Z.D., Öztürk, A., Radenović, S.: On a theorem of Ćirić in b-metric spaces. Rend. Circ. Mat. Palermo 70, 217–225 (2021)

    Article  MathSciNet  Google Scholar 

  21. Kassab, W., Ţurcanu, T.: Numerical reckoning fixed points of \((\rho E)\)-type mappings in modular vector spaces. Mathematics 7, 390 (2019)

    Article  Google Scholar 

  22. Khamsi, M.A., Kozlowski, W.M.: Fixed Point Theory in Modular Function Spaces. Birkhäser/Springer, Cham (2015)

    Book  Google Scholar 

  23. Khamsi, M.A., Kozlowski, W.M., Reich, S.: Fixed point theory in modular function spaces. Nonlinear Anal. 14, 935–953 (1990)

    Article  MathSciNet  Google Scholar 

  24. Kirk, W.A.: Contraction mappings and extensions. In: Kirk, W.A., Sims, B. (eds.) Handbook of Metric Fixed Point Theory, pp. 1–34. Kluwer, Dordrecht (2001)

    Chapter  Google Scholar 

  25. Kolobov, V.I., Reich, S., Zalas, R.: Finitely convergent deterministic and stochastic iterative methods for solving convex feasibility problems. Math. Program. 194, 1163–1183 (2022)

    Article  MathSciNet  Google Scholar 

  26. Kong, T.Y., Pajoohesh, H., Herman, G.T.: String-averaging algorithms for convex feasibility with infinitely many sets. Inverse Probl. 35, 045011 (2019)

    Article  MathSciNet  ADS  Google Scholar 

  27. Kopecká, E., Reich, S.: A note on alternating projections in Hilbert space. J. Fixed Point Theory Appl. 12, 41–47 (2012)

    Article  MathSciNet  Google Scholar 

  28. Kubota, R., Takahashi, W., Takeuchi, Y.: Extensions of Browder’s demiclosedness principle and Reich’s lemma and their applications. Pure Appl. Funct. Anal. 1, 63–84 (2016)

    MathSciNet  Google Scholar 

  29. Lenzen, F., Becker, F., Lellmann, J., Petra, S., Schnörr, C.: A class of quasi-variational inequalities for adaptive image denoising and decomposition. Comput. Optim. Appl. 54, 371–398 (2013)

    Article  MathSciNet  Google Scholar 

  30. Masad, E., Reich, S.: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8, 367–371 (2007)

    MathSciNet  Google Scholar 

  31. Nilsrakoo, W., Saejung, S.: Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings. Fixed Point Theory Appl. 2008, 312454 (2008)

    Article  MathSciNet  Google Scholar 

  32. Okeke, G.A., Abbas, M., de la Sen, M.: Approximation of the fixed point of multivalued quasi-nonexpansive mappings via a faster iterative process with applications. Discrete Dyn. Nat. Soc. 2020, 8634050 (2020)

    Article  MathSciNet  Google Scholar 

  33. Okeke, G.A., Ugwuogor, C.I.: Iterative construction of the fixed point of Suzuki’s generalized nonexpansive mappings in Banach spaces. Fixed Point Theory 23, 633–652 (2022)

    Article  MathSciNet  Google Scholar 

  34. Rakotch, E.: A note on contractive mappings. Proc. Amer. Math. Soc. 13, 459–465 (1962)

    Article  MathSciNet  Google Scholar 

  35. Reem, D., De Pierro, A.R.: A new convergence analysis and perturbation resilience of some accelerated proximal forward-backward algorithms with errors. Inverse Probl. 33, 044001 (2017)

    Article  MathSciNet  ADS  Google Scholar 

  36. Reich, S., Tuyen, T.M.: Projection algorithms for solving the split feasibility problem with multiple output sets. J. Optim. Theory Appl. 190, 861–878 (2021)

    Article  MathSciNet  Google Scholar 

  37. Reich, S., Zaslavski, A.J.: Genericity in Nonlinear Analysis. Developments in Mathematics, vol. 34. Springer, New York (2014)

  38. Takahashi, W.: The split common fixed point problem and the shrinking projection method for new nonlinear mappings in two Banach spaces. Pure Appl. Funct. Anal. 2, 685–699 (2017)

    MathSciNet  Google Scholar 

  39. Takahashi, W.: A general iterative method for split common fixed point problems in Hilbert spaces and applications. Pure Appl. Funct. Anal. 3, 349–369 (2018)

    MathSciNet  Google Scholar 

  40. Tam, M.K.: Algorithms based on unions of nonexpansive maps. Optim. Lett. 12, 1019–1027 (2018)

    Article  MathSciNet  Google Scholar 

  41. Wang, S., Yu, L., Guo, B.: An implicit iterative scheme for an infinite countable family of asymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2008, 350483 (2008)

    Article  MathSciNet  Google Scholar 

  42. Wangkeeree, R.: An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings. Fixed Point Theory Appl. 2008, 134148 (2008)

    Article  MathSciNet  Google Scholar 

  43. Zaslavski, A.J.: Approximate Solutions of Common Fixed-Point Problems. Springer Optimization and Its Applications, vol. 112. Springer, Cham (2016)

  44. Zaslavski, A.J.: Algorithms for Solving Common Fixed Point Problems. Springer Optimization and Its Applications, vol. 132. Springer, Cham (2018)

  45. Zaslavski, A.J.: Solving feasibility problems with infinitely many sets. Axioms 12, 273 (2023)

    Article  Google Scholar 

Download references

Acknowledgements

The author thanks the referees for careful reading of the paper and useful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Technion–Israel Institute of Technology, Haifa, Israel

    Alexander J. Zaslavski

Authors
  1. Alexander J. Zaslavski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander J. Zaslavski.

Additional information

Dedicated to Professor Tamas Terlaky.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zaslavski, A.J. Solving Common Fixed Point Problems with a Countable Family of Operators. Vietnam J. Math. (2024). https://doi.org/10.1007/s10013-024-00681-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10013-024-00681-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation