Skip to main content
SpringerLink
Log in

Strong convergence theorem for a system of generalized mixed equilibrium problems and finite family of Bregman nonexpansive mappings in Banach spaces

  • Theoretical Article
  • Published:
OPSEARCH Aims and scope Submit manuscript

Abstract

In this paper we introduce a general iterative algorithm for finding the common element of the set of common fixed points of a finite family of Bregman nonexpansive mappings and the set of solutions of systems of generalized mixed equilibrium problems. As an application, we find a solution for system of mixed variational inequality.

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. Alber, Y.I.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, A.G. (ed.): Theory and Applications of Nonlinear Operator of Accretive and Monotone Type, pp 15–50. Marcel Dekker, New York (1996)

    Google Scholar 

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

    Google Scholar 

  3. Aslam Noor, M., Oettli, W.: On general nonlinear complementarity problems and quasi equilibria. Matematiche (Catania) 49, 313–331 (1994)

    Google Scholar 

  4. Bauschke, H.H., Borwein, J.M., Combettes, P.L.: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Commun. Contemp. Math. 3, 615–647 (2001)

    Article  Google Scholar 

  5. Bonnans, J.F., Shapiro, A.: Perturbation analysis of optimization problem. Springer, NewYork (2000)

    Book  Google Scholar 

  6. Browder, F.E.: Existence and approximation of solutions of nonlinear variational inequalities. Proc. Natl. Acad. Sci. USA 56, 1080–1086 (1966)

    Article  Google Scholar 

  7. Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3, 459–470 (1977)

    Google Scholar 

  8. Butnariu, D., Resmerita, E.: Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstr. Appl. Anal. Art. ID 84919, 1–39 (2006)

    Article  Google Scholar 

  9. Butnariu, D., Iusem, A.N.: Totally convex functions for fixed points computation and infinite dimensional optimization. Applied optimization, vol. 40. Kluwer Academic, Dordrecht (2000)

    Book  Google Scholar 

  10. Ceng, L.C., Yao, J.C.: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 214, 186–201 (2008)

    Article  Google Scholar 

  11. Censor, Y., Lent, A.: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 34, 321–353 (1981)

    Article  Google Scholar 

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

    Google Scholar 

  13. Darvish, V.: Strong convergence theorem for generalized mixed equilibrium problems and Bregman nonexpansive mapping in Banach spaces, Accepted in Math. Moravica

  14. Flam, S.D., Antipin, A.S.: Equilibrium progamming using proximal-link algolithms. Math. Program 78, 29–41 (1997)

    Article  Google Scholar 

  15. Hiriart-Urruty, J.B., Lemaréchal, C.: Grundlehren der mathematischen Wissenschaften. In: Convex Analysis and Minimization Algorithms II. 306, Springer-Verlag (1993)

  16. Kohsaka, F., Takahashi, W.: Proximal point algorithms with Bregman functions in Banach spaces. J. Nonlinear Convex Anal. 6, 505–523 (2005)

    Google Scholar 

  17. Kumam, W., Witthayarat, U., Kumam, P., Suantaie, S., Wattanawitoon, K.: Convergence theorem for equilibrium problem and Bregman strongly nonexpansive mappings in Banach spaces, Optimization: A journal of mathematical programming and operations research (2015). doi:10.1080/02331934.2015.1020942

  18. Martn-Marquez, V., Reich, S., Sabach, S.: Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete Contin. Dyn. Syst. Ser. S. 6, 1043–1063 (2013)

    Article  Google Scholar 

  19. Peng, J.W., Yao, J.C.: Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems. Math. Comp. Model. 49, 1816–1828 (2009)

    Article  Google Scholar 

  20. Phelps, R.P.: Convex Functions, Monotone Operators, and Differentiability, second ed.. In: Lecture Notes in Mathematics, p 1364. Springer Verlag, Berlin (1993)

    Google Scholar 

  21. Reich, S.: A weak convergence theorem for the alternating method with Bregman distances. In: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Marcel Dekker, pp. 313–318. New York (1996)

  22. Reich, S., Sabach, S.: A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces. J. Nonlinear Convex Anal. 10, 471–485 (2009)

    Google Scholar 

  23. Reich, S., Sabach, S.: Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Optim. Its Appl. 49, 301–316 (2011)

    Google Scholar 

  24. Reich, S., Sabach, S.: Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim. 31, 22–44 (2010)

    Article  Google Scholar 

  25. Suantai, S., Cho, Y.J., Cholamjiak, P.: Halperns iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces. Comput. Math. Appl. 64, 489–499 (2012)

    Article  Google Scholar 

  26. Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl 118, 417–428 (2003)

    Article  Google Scholar 

  27. Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003)

    Article  Google Scholar 

  28. Zegeye, H.: Convergence theorems for Bregman strongly nonexpansive mappings in reflexive Banach spaces. Filomat 7, 1525–1536 (2014)

    Article  Google Scholar 

Download references

Acknowledgements

I would like to thank anonymous referees for a thorough and detailed report with many helpful comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Amirkabir University of Technology, Hafez Ave., P.O. Box 15875-4413, Tehran, Iran

    Vahid Darvish

Authors
  1. Vahid Darvish
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vahid Darvish.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Darvish, V. Strong convergence theorem for a system of generalized mixed equilibrium problems and finite family of Bregman nonexpansive mappings in Banach spaces. OPSEARCH 53, 584–603 (2016). https://doi.org/10.1007/s12597-015-0245-2

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12597-015-0245-2

Keywords

Search

Navigation