Skip to main content
SpringerLink
Log in

A Strong Convergence Theorem for the Split Common Null Point Problem in Banach Spaces

  • Published:
Applied Mathematics & Optimization Submit manuscript

Abstract

In this paper, we study the split common null point problem. Then, using the hybrid projection method and the metric resolvent of monotone operators, we prove a strong convergence theorem for an iterative method for finding a solution of this problem in Banach spaces.

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

Fig. 1

Similar content being viewed by others

References

  1. Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-Type Mappings with Applications. Springer, New York (2009)

    MATH  Google Scholar 

  2. Alber, Y.I.: Metric and generalized projections in Banach spaces: properties and applications. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Marcel Dekker, New York (1996)

    Google Scholar 

  3. Alber, Y.I., Reich, S.: An iterative method for solving a class of nonlinear operator equations in Banach spaces. Panam. Math. J. 4, 39–54 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Editura Acad. R. S. R, Bucuresti (1976)

    Book  MATH  Google Scholar 

  5. Browder, F.E.: Nonlinear maximal monotone operators in Banach spaces. Math. Ann. 175, 89–113 (1968)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 18, 103–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

  9. Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8(2–4), 221–239 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Cioranescu, I.: Geometry of Banach Spaces. Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht (1990)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  16. Hojo, M., Takahashi, W.: A strong convegence theorem by shrinking projection method for the split common null point problem in Banach spaces. Numer. Funct. Anal. Optim. 37, 541–553 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM. J. Optim. 13, 938–945 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Reich, S.: On the asymptotic behavior of nonlinear semigroups and the range of accretive operators. J. Math. Anal. Appl. 79, 113–126 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. Reich, S.: Book review: Geometry of banach spaces, duality mappings and nonlinear problems. Bull. Am. Math. Soc. 26, 367–370 (1992)

    Article  MathSciNet  Google Scholar 

  22. Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  23. Schopfer, F., Schuster, T., Louis, A.K.: An iterative regularization method for the solution of the split feasibility problem in Banach spaces. Inverse Probl. 24(5), 055008 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)

    MATH  Google Scholar 

  25. Takahashi, W.: Convex Analysis and Approximation of Fixed Points. Yokohama Publishers, Yokohama (2000)

    MATH  Google Scholar 

  26. Takahashi, W.: The split feasibility problem in Banach spaces. J. Nonlinear Convex Anal. 15, 1349–1355 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Takahashi, W.: The split feasibility problem and the shrinking projection method in Banach spaces. J. Nonlinear Convex Anal. 16(7), 1449–1459 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Takahashi, S., Takahashi, W.: The split common null point problem and the shrinking projection method in Banach spaces. Optimization 65(2), 281–287 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Takahashi, W.: The split common null point problem for generalized resolvents in two Banach spaces. Numer. Algorithms (2016). doi:10.1007/s11075-016-0230-8

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author thanks the referees and the editor for their valuable comments and suggestions which improved the presentation of this manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics and Informatics, Thainguyen University of Sciences, Thainguyen, Vietnam

    Truong Minh Tuyen

Authors
  1. Truong Minh Tuyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Truong Minh Tuyen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tuyen, T.M. A Strong Convergence Theorem for the Split Common Null Point Problem in Banach Spaces. Appl Math Optim 79, 207–227 (2019). https://doi.org/10.1007/s00245-017-9427-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-017-9427-z

Keywords

Mathematics Subject Classification

Search

Navigation