Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

The split feasibility problem with multiple output sets in Hilbert spaces

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

We study the split feasibility problem with multiple output sets in Hilbert spaces. In order to solve this problem, we propose two new algorithms. We establish a weak convergence theorem for the first one and a strong convergence theorem for the second.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Butnariu, D., Resmerita, E.: Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstr. Appl. Anal. 2006, 1–39 (2006)

    Article  MathSciNet  Google Scholar 

  2. Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)

    Article  MathSciNet  Google Scholar 

  3. Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)

    Article  MathSciNet  Google Scholar 

  4. Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Censor, Y., Elfving, T.: A multi projection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)

    Article  MathSciNet  Google Scholar 

  6. Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its application. Inverse Probl. 21, 2071–2084 (2005)

    Article  MathSciNet  Google Scholar 

  7. Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problems. Numer. Algorithms 59, 301–323 (2012)

    Article  MathSciNet  Google Scholar 

  8. Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587–600 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Dadashi, V.: Shrinking projection algorithms for the split common null point problem. Bull. Aust. Math. Soc. 99, 299–306 (2017)

    Article  MathSciNet  Google Scholar 

  10. Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

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

    MATH  Google Scholar 

  12. Landweber, L.: An iterative formula for Fredholm integral equations of the first kind. Am. J. Math. 73, 615–624 (1951)

    Article  MathSciNet  Google Scholar 

  13. Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16, 899–912 (2008)

    Article  MathSciNet  Google Scholar 

  14. 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  MATH  Google Scholar 

  15. Moudafi, A.: The split common fixed point problem for demicontractive mappings. Inverse Probl. 26, 055007 (2010)

    Article  MathSciNet  Google Scholar 

  16. Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

    Article  MathSciNet  Google Scholar 

  17. Reich, S., Tuyen, T.M.: Iterative methods for solving the generalized split common null point problem in Hilbert spaces. Optimization (2019). https://doi.org/10.1080/02331934.2019.1655562

    Article  MATH  Google Scholar 

  18. Takahashi, S., Takahashi, W.: The split common null point problem and the shrinking projection method in Banach spaces. Optimization 65, 281–287 (2016)

    Article  MathSciNet  Google Scholar 

  19. Takahashi, W.: The split feasibility problem and the shrinking projection method in Banach spaces. J. Nonlinear Convex Anal. 16, 1449–1459 (2015)

    MathSciNet  MATH  Google Scholar 

  20. Takahashi, W.: The split common null point problem in Banach spaces. Arch. Math. 104, 357–365 (2015)

    Article  MathSciNet  Google Scholar 

  21. Tuyen, T.M.: A strong convergence theorem for the split common null point problem in Banach spaces. Appl. Math. Optim. 79, 207–227 (2019)

    Article  MathSciNet  Google Scholar 

  22. Tuyen, T.M., Ha, N.S., Thuy, N.T.T.: A shrinking projection method for solving the split common null point problem in Banach spaces. Numer. Algorithms 81, 813–832 (2019)

    Article  MathSciNet  Google Scholar 

  23. Tuyen, T.M., Ha, N.S.: A strong convergence theorem for solving the split feasibility and fixed point problems in Banach spaces. J. Fixed Point Theory Appl. 20, 140 (2018)

    Article  MathSciNet  Google Scholar 

  24. Wang, F., Xu, H.-K.: Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal. 74, 4105–4111 (2011)

    Article  MathSciNet  Google Scholar 

  25. Xu, H.-K.: Strong convergence of an iterative method for nonexpansive and accretive operators. J. Math. Anal. Appl. 314, 631–643 (2006)

    Article  MathSciNet  Google Scholar 

  26. Xu, H.-K.: A variable Krasnosel’skii–Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006)

    Article  Google Scholar 

  27. Xu, H.-K.: Iterative methods for the split feasibility problem in infinite dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010)

    Article  MathSciNet  Google Scholar 

  28. Xu, H.-K.: Averaged mappings and the gradient-projection algorithm. J. Optim. Theory Appl. 150, 360–378 (2011)

    Article  MathSciNet  Google Scholar 

  29. Yang, Q.: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Probl. 20, 1261–1266 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author was partially supported by the Israel Science Foundation (Grant 820/17), by the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund. The second author was supported by the Science and Technology Fund of Thai Nguyen University of Sciences. All the authors are very grateful to an anonymous referee for providing them with useful comments and helpful suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, The Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Simeon Reich

  2. Department of Mathematics and Informatics, Thai Nguyen University of Sciences, Thai Nguyen, Vietnam

    Minh Tuyen Truong

  3. Thai Nguyen University of Agriculture and Forestry, Thai Nguyen, Vietnam

    Thi Ngoc Ha Mai

Authors
  1. Simeon Reich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Minh Tuyen Truong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thi Ngoc Ha Mai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minh Tuyen Truong.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Reich, S., Truong, M.T. & Mai, T.N.H. The split feasibility problem with multiple output sets in Hilbert spaces. Optim Lett 14, 2335–2353 (2020). https://doi.org/10.1007/s11590-020-01555-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-020-01555-6

Keywords

Search

Navigation