Skip to main content
SpringerLink
Log in

The split common fixed point problem for multivalued demicontractive mappings and its applications

  • 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 article, we consider the split common fixed point problem for two infinite families of multivalued mappings in real Hilbert spaces. We introduce an algorithm based on the viscosity method for solving the split common fixed point problem for two infinite families of multivalued demicontractive mappings. We establish a strong convergence result under some suitable conditions. As applications, we also apply our main result to the split variational inequality problem and the split common null point problem. Finally, we give the numerical example for supporting our main theorem.

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. Alsulami, S.M., Latif, A., Takahashi, W.: Strong convergence theorems by hybrid methods for split feasibility problems in Hilbert spaces. J. Nonlinear Convex Anal. 16, 2521–2538 (2015)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  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  MATH  Google Scholar 

  4. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)

    Book  MATH  Google Scholar 

  5. 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 

  6. Cegielski, A.: General method for solving the split common fixed point problem. J. Optim. Theory Appl. 165, 385–404 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Chidume, C.E., Ezeora, J.N.: Krasnoselskii-type algorithm for family of multi-valued strictly pseudo-contractive mappings. Fixed Point Theory Appl. (2014). https://doi.org/10.1186/1687-1812-2014-111

    MathSciNet  MATH  Google Scholar 

  9. Chidume, C.E., Bello, A.U., Ndambomve, P.: Strong and \(\Delta \)-convergence theorems for common fixed points of a finite family of multivalued demicontractive mappings in CAT(0) spaces. Abstr. Appl. Anal., (2014). https://doi.org/10.1155/2014/805168

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

    MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  12. 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 

  13. Eslamian, M.: General algorithms for split common fixed point problem of demicontractive mappings. Optimization 65, 443–465 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Isiogugu, F.O., Osilike, M.O.: Convergence theorems for new classes of multivalued hemicontractive-type mappings. Fixed Point Theory Appl. 93, 12 (2014). https://doi.org/10.1186/1687-1812-2014-93

    MathSciNet  MATH  Google Scholar 

  15. Iiduka, H., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal. 61, 341–350 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Latif, A., Eslamian, M.: Strong convergence and split common fixed point problem for set-valued operators. J. Nonlinear Convex Anal. 17, 967–986 (2016)

    MathSciNet  Google Scholar 

  17. Latif, A., Qin, X.: A regularization algorithm for a splitting feasibility problem in Hilbert spaces. J. Nonlinear Sci. Appl. 10, 3856–3862 (2017)

    Article  MathSciNet  Google Scholar 

  18. Latif, A., Vahidi, J., Eslamian, M.: Strong convergence for generalized multiple-set split feasibility problem. Filomat 30(2), 459–467 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Moudafi, A.: The split common fixed-point problem for demicontractive mappings. Inverse Probl. 26, 587–600 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Moudafi, A.: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal. 74, 4083–4087 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Moudafi, A.: Viscosity-type algorithms for the split common fixed-point problem. Adv. Nonlinear Var. Inequal. 16, 61–68 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Palta, J.R., Mackie, T.R.: Intensity-Modulated Radiation Therapy: The State of The Art. Medical Physics Publishing, Madison (2003)

    Google Scholar 

  24. Qu, B., Xiu, N.: A note on the \(CQ\) algorithm for the split feasibility problem. Inverse Probl. 21, 1655–1665 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Song, Y., Cho, Y.J.: Some note on Ishikawa iteration for multi-valued mappings. Bull. Korean Math. Soc. 48, 575–584 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shehu, Y., Cholamjiak, P.: Another look at the split common fixed point problem for demicontractive operators. Rev. R. Acad. Cien. Exactas Fís. Nat. Ser. A Mat. 110, 201–218 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Takahashi, W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama (2009)

    MATH  Google Scholar 

  28. Tang, Y.C., Peng, J.G., Liu, L.W.: A cyclic algorithm for the split common fixed point problem of demicontractive mappings in Hilbert spaces. Math. Model. Anal. 17, 457–466 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tang, Y.C., Peng, J.G., Liu, L.W.: A cyclic and simultaneous iterative algorithm for the multiple split common fixed point problem of demicontractive mappings. Bull. Korean Math. Soc. 51, 1527–1538 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tufa, A.R., Zegeye, H., Thuto, M.: Convergence theorems for non-self mappings in CAT(0) Spaces. Numer. Funct. Anal. Optim. 38, 705–722 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wen, M., Peng, J.G., Tang, Y.C.: A cyclic and simultaneous iterative method for solving the multiple-sets split feasibility problem. J. Optim. Theory Appl. 166, 844–860 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  35. Xu, H.K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26 (2010). https://doi.org/10.1088/0266-5611/26/10/105018/meta

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions for improving this work and the Thailand Research Fund under the project RTA 5780007 and Chiang Mai University for the financial support. The first author was supported by the Royal Golden Jubilee (RGJ) Ph.D. Scholarship.

Author information

Authors and Affiliations

  1. Ph.D Degree Program in Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Pachara Jailoka

  2. Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Suthep Suantai

Authors
  1. Pachara Jailoka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suthep Suantai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Suthep Suantai.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jailoka, P., Suantai, S. The split common fixed point problem for multivalued demicontractive mappings and its applications. RACSAM 113, 689–706 (2019). https://doi.org/10.1007/s13398-018-0496-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-018-0496-x

Keywords

Mathematics Subject Classification

Search

Navigation