Skip to main content
SpringerLink
Log in

Weak and strong convergence theorems for mixed equilibrium problems in Banach spaces

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problems and the set of fixed points for a \(\phi \)-nonexpansive mapping in Banach spaces by using sunny generalized nonexpansive retraction in Banach spaces. Moreover, we also apply our result for finding a zero point of maximal monotone operators. Finally, we give a numerical example to illustrate our main theorem.

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

Fig. 1

Similar content being viewed by others

References

  1. Agarwal, R.P., Cho, Y.J., Qin, X.: Generalized projection algorithms for nonlinear operators. Numer. Funct. Anal. Optim. 28, 1197–1215 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alber, Ya.I., Reich, S.: An iterative method for solving a class of nonlinear operator equations in Banach spaces. Pan Am. Math. J. 4, 39–54 (1994)

    Google Scholar 

  3. Alber, Ya.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)

  4. Ball, K., Carlen, E.A., Lieb, E.H.: Sharp uniform convexity and smoothness inequalities for trace norm. Invent. Math. 26, 137–150 (1994)

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  6. Chen, J.W., Cho, Y.J., Wan, Z.: Shrinking projection algorithms for equilibrium problems with a bifunction defined on the dual space of a Banach space. Fixed Point Theory Appl. 2011, 91 (2011)

    Article  MathSciNet  Google Scholar 

  7. Cho, Y.J., Argyros, I.K., Petrot, N.: Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems. Comput. Math. Appl. 50, 2292–2301 (2010)

    Article  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  9. Cho, Y.J., Kang, J.I., Qin, X.: Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems. Nonlinear Anal. 71, 4203–4214 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cho, Y.J., Petrot, N.: An optimization problem related to generalized equilibrium and fixed point problems with applications. Fixed Point Theory 11, 237–250 (2010)

    MATH  MathSciNet  Google Scholar 

  11. Cho, Y.J., Petrot, N.: On the system of nonlinear mixed implicit equilibrium problems in Hilbert spaces. J. Inequal. Appl. 2010, Article ID 437976 (2010). doi:10.1155/2010/437976

  12. Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht (1990)

    Book  MATH  Google Scholar 

  13. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  14. Gilbert, R.P., Panagiotopoulos, P.D., Pardalos, P.M.: From Convexity to Nonconvexity. Kluwer, Dordrecht (2001)

  15. Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium Problems and Variational Models. Kluwer, Dordrecht (2001)

    Google Scholar 

  16. Giannessi, F., Pardalos, P.M., Rapcsak, T.: New Trends in Equilibrium Systems. Kluwer, Dordrecht (2001)

    Google Scholar 

  17. Honda, T., Takahashi, W., Yao, J.C.: Nonexpansive retractions onto closed convex cones in Banach spaces. Taiwan. J. Math. 14, 1023–1046 (2010)

    MATH  MathSciNet  Google Scholar 

  18. Ibaraki, T., Takahashi, W.: A new projection and convergence theorems for the projections in Banach spaces. J. Approx. Theory 149, 1–14 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ibaraki, T., Takahashi, W.: Generalized nonexpansive mappings and a proximal-type algorithm in Banach spaces. Nonlinear Anal. Optim. 513, 169–188 (2010)

    Article  MathSciNet  Google Scholar 

  20. Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938–945 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Qin, X., Chang, S.S., Cho, Y.J.: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. Real World Appl. 11, 2963–2972 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Qin, X., Cho, Y.J., Kang, S.M.: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 225, 20–30 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

  24. Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  25. Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)

    MATH  Google Scholar 

  26. Takahashi, W.: Convex Analysis and Approximation Fixed Points. Yokohama Publishers, Yokohama (2009)

    Google Scholar 

  27. Takahashi, Y., Hashimoto, K., Kato, M.: On sharp uniform convexity, smoothness and strong type, cotype inequalities. J. Nonlinear Convex Anal. 3, 267–281 (2002)

    MATH  MathSciNet  Google Scholar 

  28. Takahashi, W., Zembayashi, K.: A strong convergence theorem for the equilibrium problem with a bifunction defined on the dual space of a Banach space. In: Proceeding of the 8th International Conference on Fixed Point Theory and Its Applications, pp. 197–209. Yokohama Publishers, Yokohama (2008)

  29. Takahashi, W., Zembayashi, K.: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70, 45–57 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  30. Tan, K.K., Xu, H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301–308 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  31. Wang, S., Cho, Y.J., Qin, X.: A new iterative method for solving equilibrium problems and fixed point problems for infinite family of nonexpansive mappings. Fixed Point Theory Appl. 2010, Article ID 165098 (2010)

  32. Xu, H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  33. Yao, Y., Cho, Y.J., Chen, R.: An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems. Nonlinear Anal. 71, 3363–3373 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  34. Yao, Y., Cho, Y.J., Liou, Y.: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur. J. Oper. Res. 212, 242–250 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  35. Yao, Y., Cho, Y.J., Liou, Y.: Iterative algorithms for variational inclusions, mixed equilibrium problems and fixed point problems approach to optimization problems. Cent. Eur. J. Math. 9, 640–656 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zhou, H., Gao, G., Tan, B.: Convergence theorems of a modified hybrid algorithm for a family of quasi-\(\phi \)-asymptotically nonexpansive mappings. J. Appl. Math. Comput. 32, 453–464 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to express their thanks to the reviewer for helpful suggestions and comments for the improvement of this paper. This research was supported by Thaksin University Research fund and Y.J. Cho was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number 2011-0021821).

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Science, Thaksin University (TSU), Pa Phayom, Phatthalung, 93110, Thailand

    Siwaporn Saewan

  2. Department of Mathematics Education and RINS, Gyeongsang National University, Chinju, 660-701, Korea

    Yeol Je Cho

  3. Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok, 10140, Thailand

    Poom Kumam

Authors
  1. Siwaporn Saewan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yeol Je Cho
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Poom Kumam
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Siwaporn Saewan or Yeol Je Cho.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Saewan, S., Cho, Y.J. & Kumam, P. Weak and strong convergence theorems for mixed equilibrium problems in Banach spaces. Optim Lett 8, 501–518 (2014). https://doi.org/10.1007/s11590-012-0593-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-012-0593-2

Keywords

Search

Navigation