Skip to main content
SpringerLink
Log in

An iterative method for solving split monotone variational inclusion and fixed point problems

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

Abstract

The purpose of this article is to propose an iterative algorithm for finding an approximate solution of a split monotone variational inclusion problem for monotone operators which is also a solution of a fixed point problem for strictly pseudocontractive maps in real Hilbert spaces. Using our iterative algorithm, we state and prove a strong convergence theorem for approximating a common solution of split monotone variational inclusion problem and fixed point problem for strictly pseudocontractive maps in the framework of real Hilbert spaces. Our result complements and extends some related results in literature.

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. López-Acedo, G., Xu, H.-K.: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 67(7), 2258–2271 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baillon, J.-B., Bruck, R.E., Reich, S.: On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houston J. Math. 4, 1–9 (1978)

    MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  4. Brézis, H.: Opérateurs maximaux monotones, in mathematics studies, vol. 5. North-Holland, Amsterdam, The Netherlands (1973)

    Google Scholar 

  5. Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20, 197–228 (1967)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Byrne, C., Censor, Y., Gibali, A., Reich, S.: Weak and strong convergence of algorithms for the split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Combettes, P.L.: The convex feasibility problem in image recovery. Adv. Imaging Electron Phys. 95, 155–453 (1996)

    Article  Google Scholar 

  13. Crombez, G.: A hierarchical presentation of operators with fixed points on Hilbert spaces. Numer. Funct. Anal. Optim. 27, 259–277 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Crombez, G.: A geometrical look at iterative methods for operators with fixed points. Numer. Funct. Anal. Optim. 26, 157–175 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kazmi, K.R., Rizvi, S.H.: An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping. Optim Lett. doi:10.1007/s11590-013-0629-2

  16. Lemaire, B.: Which fixed point does the iteration method select? In: Recent advances in optimization, vol. 452, pp. 154–157. Springer, Berlin, Germany (1997)

  17. Lin, L.-J., Chen, Y.-D., Chuang, C.-S.: Solutions for a variational inclusion problem with applications to multiple sets split feasibility problems. Fixed Point Theory Appl. 2013, 333 (2013). doi:10.1186/1687-1812-2013-333

    Article  MathSciNet  MATH  Google Scholar 

  18. Lin, L.-J.: Systems of variational inclusion problems and differential inclusion problems with applications. J. Global Optim. 44(4), 579–591 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, Y.: Strong convergence theorems for variational inclusion problems and strict pseudocontractions in Hilbert spaces. Nonlinear Funct. Anal. Appl. 17, 421–432 (2012)

    MATH  Google Scholar 

  20. Lopez, G., Martin-Marquez, V., Xu, H.K.: Iterative algorithms for the multi-sets 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 (2010)

  21. Marino, G., Xu, H.-K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329(1), 336–346 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Masad, E., Reich, S.: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8(3), 367–371 (2007)

    MathSciNet  MATH  Google Scholar 

  23. Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Moudafi, A.: The split common fixed point problem for demicontractive mappings. Inverse Probl. 26, 055007 (2010). (6pp)

    Article  MathSciNet  MATH  Google Scholar 

  25. Peng, J.W., Wang, Y., Shyu, D.S., Yao, J.-C.: Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems. J Inequalities Appl 2008, 15 (2008) (Article ID 720371)

  26. Qu, B., Xiu, N.: A note on the CQ algorithm for the split feasibility problem. Inverse Prob. 21(5), 1655–1665 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Reich, S.: Constructive techniques for accretive and monotone operators, in Applied Nonlinear Analysis, pp. 335–345. Academic Press, New York (1979)

    Google Scholar 

  28. 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 

  29. Xu, H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 2, 240–256 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. Xu, H.-K.: A variable Krasnoselskii-Mann algorithm and the multiple-set split feasibility problem. Inverse Prob. 22(6), 2021–2034 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Prob. 20(4), 1261–1266 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yang, Q., Zhao, J.: Generalized KM theorems and their applications. Inverse Prob. 22(3), 833–844 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yao, Y., Jigang, W., Liou, Y.-C.: Regularized methods for the split feasibility problem. Abstr. Appl. Anal. 2012, 13 (2012) (Article ID 140679)

  34. Zhou, H.Y.: Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 69(2), 456–462 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We gratefully acknowledge the constructive comments and suggestions of the referees which improved the presentation of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Nigeria, Nsukka, Nigeria

    Yekini Shehu

  2. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

    Ferdinard U. Ogbuisi

Authors
  1. Yekini Shehu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ferdinard U. Ogbuisi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yekini Shehu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shehu, Y., Ogbuisi, F.U. An iterative method for solving split monotone variational inclusion and fixed point problems. RACSAM 110, 503–518 (2016). https://doi.org/10.1007/s13398-015-0245-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-015-0245-3

Keywords

Mathematics Subject Classification

Search

Navigation