Skip to main content
SpringerLink
Log in

The split common fixed point problem for infinite families of total quasi-asymptotically nonexpansive operators

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The main purpose in this paper is to introduce an original technique, namely, a specific way of choosing the indexes of the involved operators, by which a new iterative algorithm is developed for solving the split common fixed point problem for infinite families of total quasi-asymptotically nonexpansive operators and some strong and weak convergence theorems are established for this class of nonlinear operators in the framework of infinite-dimensional Hilbert spaces. The result is more applicable than those of other authors with related interest.

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Lopez, G., Martin, V., Xu, H.K.: Iterative algorithms for the muitiple-sets split feasibility problem. In: Censor, Y., Jiang, M., Wang, G. (eds.) Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and inverse Problems, pp. 243–279. Medical Physics Publishing, Madison (2009)

    Google Scholar 

  8. Dang, Y., Gao, Y.: The strong convergecne of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 27(015007) (2011)

  9. 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 

  10. Moudafi, A.: The split common fixed point problem for demi-contractive mappings. Inverse Probl. 26(055007), 6 (2010)

    MathSciNet  Google Scholar 

  11. Maruster, S., Popirlan, C.: On the Mann-type iteration and convex feasibility problem. J. Comput. Appl. Math. 212, 390–396 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Wang, F., Xu, H.K.: Approximation curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequalities Appl. (2010). doi:10.1155/2010/102085

  13. Xu, H.K.: Iterative methods for split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26(105018), 17 (2010)

    Google Scholar 

  14. 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 

  15. Xu, H.K.: Iterative algorithms for nonlinear operator. J. London Math. Soc. 66, 240–256 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, X.R., Chang, S., Wang, L., Zhao, Y.: Split feasibility problems for total quasi-asymptotically nonexpansive mappings. Fixed Point Theory Appl. 151, 11 (2012)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  18. Deng, W.Q., Bai, P.: An implicit iteration process for common fixed points of two infinite families of asymptotically nonexpansive mappings in banach spaces. J. Appl. Math. Article ID 602582, 6 (2013)

    Google Scholar 

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

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Statistics and Mathematics, Yunnan University of Finance and Economics Kunming, Yunnan, 650221, People’s Republic of China

    Wei-Qi Deng

Authors
  1. Wei-Qi Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei-Qi Deng.

Additional information

This work is part of the project High-Definition Tomography and it is supported by Grant No. ERC-2011-ADG 20110209 from the European Research Council.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Deng, WQ. The split common fixed point problem for infinite families of total quasi-asymptotically nonexpansive operators. Numer Algor 67, 243–256 (2014). https://doi.org/10.1007/s11075-013-9785-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-013-9785-9

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation