Skip to main content
Log in

A fixed point method for solving a split feasibility problem in Hilbert spaces

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, a fixed method is introduced and investigated for solving a split feasibility problem. A strong convergence theorem of solutions is established in the framework of infinite dimensional Hilbert spaces. As an application, a split equality problem is also investigated.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  3. Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Censor, Y., Motova, A., Segal, A.: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl. 327, 1244–1256 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Tang, J., Chang, S.S., Dong, J.: Split equality fixed point problems for two quasi-asymptotically pseudocontractive mappings. J. Nonlinear Funct. Anal. 2017, Article ID 26 (2017)

    Google Scholar 

  7. Xu, H.K.: A variable Krasonsel’skiǐ–Mann algorithm and multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Cho, S.Y.: Strong convergence analysis of a hybrid algorithm for nonlinear operators in a Banach space. J. Appl. Anal. Comput. 8, 19–31 (2018)

  10. Chang, S.S., Wang, L., Zhao, Y.: On a class of split equality fixed point problems in Hilbert spaces. J. Nonlinear Var. Anal. 1, 201–212 (2017)

    MATH  Google Scholar 

  11. Moudafi, A.: Alternating CQ-algorithm for convex feasibility and split fixed-point problems. J. Nonlinear Convex Anal. 15, 809–818 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Gibali, A., Liu, L.-W., Tang, Y.-C.: Note on the modified relaxation CQ algorithm for the split feasibility problem. Optim. Lett. https://doi.org/10.1007/s11590-017-1148-3

  13. Cho, S.Y.: Generalized mixed equilibrium and fixed point problems in a Banach space. J. Nonlinear Sci. Appl. 9, 1083–1092 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Thianwan, S.: Strong convergence theorems by hybrid methods for a finite family of nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal. 3, 605–614 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  16. Schu, J.: Weak, strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Austral. Math. Soc. 43, 153–159 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–409 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Suzuki, T.: A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 135, 99–106 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75, 287–292 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cho, S.Y., Li, W., Kang, S.M.: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013, Article ID 199 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Moudafi, A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Qin, X.: Iterative algorithms with errors for zero points of m-accretive operators. Fixed Point Theory Appl. 2013, Article ID 148 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Qin, X., Yao, J.C.: Projection splitting algorithms for nonself operators. J. Nonlinear Convex Anal. 18, 925–935 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Opial, Z.: Weak convergence of the sequence of successive approximations of nonexpansive mappings. Bull. Am. Math. Soc. 73, 595–597 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  25. Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Suzuki, T.: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 103–123 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Moudafi, A., Al-Shemas, E.: Simultaneouss iterative methods for split equality problem. Trans. Math. Program. Appl. 1, 1–11 (2013)

    Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No.11401152.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xiaolong Qin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qin, X., Wang, L. A fixed point method for solving a split feasibility problem in Hilbert spaces. RACSAM 113, 315–325 (2019). https://doi.org/10.1007/s13398-017-0476-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-017-0476-6

Keywords

Mathematics Subject Classification

Navigation