, Volume 53, Issue 3, pp 584–603 | Cite as

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

  • Vahid DarvishEmail author
Theoretical Article


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.


Banach space Bregman mapping Fixed point System of generalized mixed equilibrium System of mixed variational inequality 



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


  1. 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. 2.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–145 (1994)Google Scholar
  3. 3.
    Aslam Noor, M., Oettli, W.: On general nonlinear complementarity problems and quasi equilibria. Matematiche (Catania) 49, 313–331 (1994)Google Scholar
  4. 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)CrossRefGoogle Scholar
  5. 5.
    Bonnans, J.F., Shapiro, A.: Perturbation analysis of optimization problem. Springer, NewYork (2000)CrossRefGoogle Scholar
  6. 6.
    Browder, F.E.: Existence and approximation of solutions of nonlinear variational inequalities. Proc. Natl. Acad. Sci. USA 56, 1080–1086 (1966)CrossRefGoogle Scholar
  7. 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. 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)CrossRefGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. 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)CrossRefGoogle Scholar
  11. 11.
    Censor, Y., Lent, A.: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 34, 321–353 (1981)CrossRefGoogle Scholar
  12. 12.
    Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)Google Scholar
  13. 13.
    Darvish, V.: Strong convergence theorem for generalized mixed equilibrium problems and Bregman nonexpansive mapping in Banach spaces, Accepted in Math. MoravicaGoogle Scholar
  14. 14.
    Flam, S.D., Antipin, A.S.: Equilibrium progamming using proximal-link algolithms. Math. Program 78, 29–41 (1997)CrossRefGoogle Scholar
  15. 15.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Grundlehren der mathematischen Wissenschaften. In: Convex Analysis and Minimization Algorithms II. 306, Springer-Verlag (1993)Google Scholar
  16. 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. 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. 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)CrossRefGoogle Scholar
  19. 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)CrossRefGoogle Scholar
  20. 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. 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)Google Scholar
  22. 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. 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. 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)CrossRefGoogle Scholar
  25. 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)CrossRefGoogle Scholar
  26. 26.
    Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl 118, 417–428 (2003)CrossRefGoogle Scholar
  27. 27.
    Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003)CrossRefGoogle Scholar
  28. 28.
    Zegeye, H.: Convergence theorems for Bregman strongly nonexpansive mappings in reflexive Banach spaces. Filomat 7, 1525–1536 (2014)CrossRefGoogle Scholar

Copyright information

© Operational Research Society of India 2015

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceAmirkabir University of TechnologyTehranIran

Personalised recommendations