Skip to main content
SpringerLink
Log in

Hybrid iterative method for split monotone variational inclusion problem and hierarchical fixed point problem for a finite family of nonexpansive mappings

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

Abstract

In this paper, we propose a hybrid iterative method to approximate a common solution of split monotone variational inclusion problem and hierarchical fixed point problem for a finite family of nonexpansive mappings in real Hilbert spaces. We prove that sequences generated by the proposed hybrid iterative method converge strongly to a common solution of these problems. Further, we discuss some applications of the main result. We also discuss a numerical example to demonstrate the applicability of the iterative method. The method and results presented in this paper extend and unify the corresponding known results in this area.

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. Aoyama, K., Kimura, Y., Takahashi, W.: Maximal monotone operators and maximal monotone functions for equilibrium problems. J. Convex Anal. 15(2), 395–409 (2008)

    MathSciNet  MATH  Google Scholar 

  2. Atsushiba, S., Takahashi, W.: Strong Convergence Theorems for a Finite Family of Nonexpansive Mappings and Applications, in: B.N. Prasad birth centenary commemoration volume. Indian J. Math. 41(3), 435–453 (1999)

    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  Google Scholar 

  4. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  5. Bnouhachem, A.: Strong convergence algorithm for split equilibrium problem and hierarchical fixed point problems. Sci. World J. 2014, Article ID 390956, 12 (2014)

    Article  Google Scholar 

  6. Bnouhachem, A., Ansari, Q.H., Yao, J.C.: An iterative algorithm for hierarchical fixed point problems for a finite family of nonexpansive mappings. Fixed Point Theory Appl. 2015:111, 13 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Bnouhachem, A., Ansari, Q.H., Yao, J.C.: Stromg convergence algorithm for hierarchical fixed point problems of a finite family of nonexpansive mappings. Fixed Point Theory 17(1), 47–62 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Bre~zis, H.: Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Mathematical Studies(Amsterdam: North-Holand) 5, 759–775 (1973)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13(4), 759–775 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Cabot, A.: Proximal point algorithm controlled by a slowly vanishing term: application to hierarchical minimization. SIAM J. Optim. 15, 555–572 (2005)

    Article  MathSciNet  Google Scholar 

  13. Ceng, L.C., Petruşel, A.: Krasnoselski-mann iterations for hierarchical fixed point problems for a finite family of nonself mappings in Banach spaces. J. Optim. Theory Appl. 146, 617–639 (2010)

    Article  MathSciNet  Google Scholar 

  14. 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 

  15. Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59(2), 301–323 (2012)

    Article  MathSciNet  Google Scholar 

  16. Colao, V., Marino, G., Xu, H.K.: An iterative method for finding common solutions of equilibrium and fixed point problems. J. Math. Anal. Appl. 344, 340–352 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  18. Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)

    MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  20. Kazmi, K.R., Ali, R., Furkan, M.: Krasnoselski-mann type iterative method for hierarchical fixed point problem and split mixed equilibrium problem. Numerical Algorithms (2017), https://doi.org/10.1007/s11075-017-0316-y

    Article  MathSciNet  Google Scholar 

  21. Kazmi, K.R., Rizvi, S.H.: Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem. J. Egy. Math. Soc. 44–51, 21 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Kazmi, K.R., Rizvi, S.H.: An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping. Optim. Letters 8, 1113–1124 (2014)

    Article  MathSciNet  Google Scholar 

  23. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  24. Marino, G., Colao, V., Muglia, L., Yao, Y.: Krasnoselski-mann iteration for hierarchical fixed-points and equilibrium problem. Bull. Aust. Math. Soc. 79, 187–200 (2009)

    Article  MathSciNet  Google Scholar 

  25. Moudafi, A.: Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Probl. 23, 1635–1640 (2007)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Moudafi, A., Mainge, P.-E.: Towards viscosity approximations of hierarchical fixed-point problems. Fixed Point Theory Appl. 2006, Article ID 95453 (2006)

    Article  MathSciNet  Google Scholar 

  28. Moudafi, A., Mainge, P.-E.: Strong convergence of an iterative method for hierarchical fixed-point problems. Pacific J. Optim. 3, 529–538 (2007)

    MathSciNet  MATH  Google Scholar 

  29. Nakajo, K., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroup. J. Math. Anal. Appl. 279, 372–379 (2003)

    Article  MathSciNet  Google Scholar 

  30. Shehu, Y., Ogbuisi, F.U.: An iterative method for solving split monotone variational inclusion and fixed point problems. RACSAM 110(2), 503–518 (2016)

    Article  MathSciNet  Google Scholar 

  31. Takahashi, W.: Weak and strong convergence theorems for families of nonexpansive mappings and their applications. Ann. Univ. Mariae Curie-Sklodowska. 51, 277–292 (1997)

    MathSciNet  MATH  Google Scholar 

  32. Takahashi, W., Shimoji, K.: Convergence theorems for nonexpansive mappings and feasibility problems. Math. Comput. Modelling. 32, 1463–1471 (2000)

    Article  MathSciNet  Google Scholar 

  33. Yamada, I., Ogura, N.: Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619–655 (2004)

    Article  MathSciNet  Google Scholar 

  34. Yang, Q., Zhao, J.: Generalized KM theorem and their applications. Inverse Probl. 22, 833–844 (2006)

    Article  MathSciNet  Google Scholar 

  35. Yao, Y.: A general iterative method for a finite family of nonexpansive mappings. Nonlinear Anal. 66, 2676–2687 (2007)

    Article  MathSciNet  Google Scholar 

  36. Yao, Y., Liou, Y.C.: Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Probl. 24, 501–508 (2008)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the anonymous referees for their constructive and helpful comments and suggestions towards the improvement of the original manuscript.

Author information

Author notes

  1. Rehan Ali

    Present address: Department of Mathematics, Jamia Millia Islamia, New Delhi, 110025, India

  2. Mohd Furkan

    Present address: University Polytechnic, Aligarh Muslim University, Aligarh, 202002, India

Authors and Affiliations

  1. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    K. R. Kazmi, Rehan Ali & Mohd Furkan

Authors
  1. K. R. Kazmi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rehan Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohd Furkan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. R. Kazmi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kazmi, K.R., Ali, R. & Furkan, M. Hybrid iterative method for split monotone variational inclusion problem and hierarchical fixed point problem for a finite family of nonexpansive mappings. Numer Algor 79, 499–527 (2018). https://doi.org/10.1007/s11075-017-0448-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-017-0448-0

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation