Skip to main content
SpringerLink
Log in

Convergence theorems for nonspreading mappings and nonexpansive multivalued mappings and equilibrium problems

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

Abstract

In this paper, we introduce some new iterative sequences for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of nonspreading mappings and a finite family of nonexpansive multivalued mappings in Hilbert space. We establish some weak and strong convergence theorems of the sequences generated by our iterative process. The results obtained in this paper extend and improve some recent known results.

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. Li X.B., Li S.J.: Existence of solutions for generalized vector quasi-equilibrium problems. Optim. Lett. 4, 17–28 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Giannessi F., Maugeri G., Pardalos P.M.: Equilibrium problems: nonsmooth optimization and variational inequality models. Kluwer Academics Publishers, Dordrecht, Holland (2001)

    Google Scholar 

  5. Moudafi, A., Thera, M.: Proximal and dynamical approaches to equilibrium problems, In: Lecture Note in Economics and Mathematical Systems, vol. 477, pp. 187-201. Springer-Verlag, New York (1999)

  6. Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear Analysis and Variational Problems. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  7. Tada A., Takahashi W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133, 359–370 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Takahashi S., Takahashi W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–515 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ceng L.C., Al-Homidan S., Ansari Q.H., Yao J.C.: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J. Comput. Appl. Math. 223, 967–974 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang, S., Marino, G., Liou, Y.C.: Strong convergence theorems for variational inequality, equilibrium and fixed point problems with applications. J. Glob. Optim. doi:10.1007/s10898-011-9754-6

  11. Qin X., Cho S.Y., Kang S.M.: An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. J. Glob. Optim. 49, 679–693 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Shehu, Y.: A new iterative scheme for a countable family of relatively nonexpansive mappings and an equilibrium problem in Banach spaces. J. Glob. Optim. doi:10.1007/s10898-011-9775-1

  13. Qin X., Cho S.Y., Kang S.M.: Iterative algorithms for variational inequality and equilibrium problems with applications. J. Glob. Optim. 48, 423–445 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu Y.: Strong convergence theorems for variational inequalities and relatively weak nonexpansive mappings. J. Glob. Optim. 46, 319–329 (2010)

    Article  MATH  Google Scholar 

  15. Kamraksa, U., Wangkeeree, R.: Existence theorems and iterative approximation methods for generalized mixed equilibrium problems for a countable family of nonexpansive mappings. J. Glob. Optim. doi:10.1007/s10898-011-9739-5

  16. Moudafi A.: Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9, 37–43 (2008)

    MathSciNet  MATH  Google Scholar 

  17. 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  MathSciNet  MATH  Google Scholar 

  18. Chang S.S., Lee H.W.J., Kim J.K.: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 70, 3307–3319 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Song Y., Wang H.: Convergence of iterative algorithms for multivalued mappings in Banach spaces. Nonlinear Anal. 70, 1547–1556 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shahzad N., Zegeye H.: On Mann and Ishikawa iteration schemes for multivalued maps in Banach space. Nonlinear Anal. 71, 838–844 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Eslamian M., Abkar A.: One-step iterative process for a finite family of multivalued mappings. Math. Comput. Modell. 54, 105–111 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kohsaka F., Takahashi W.: Fixed point theorems for a class of nonlinear mappings relate to maximal monotone operators in Banach spaces. Arch. Math. (Basel). 91, 166–177 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kohsaka F., Takahashi W.: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM J. Optim. 19, 824–835 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Iemoto S., Takahashi W.: Approximating commom fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space. Nonlinear Anal. 71, 2082–2089 (2009)

    Article  MathSciNet  Google Scholar 

  26. Kangtunyakarn, A.: Convergence theorem of common fixed points for a family of nonspreading mappings in Hilbert space. Optim. Lett. doi:10.1007/s11590-011-0326-y

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Osilike M.O., Igbokwe D.I.: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Comput. Math. Appl. 40, 559–567 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Young Researchers Club, Babol Branch, Islamic Azad University, Babol, Iran

    Mohammad Eslamian

Authors
  1. Mohammad Eslamian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad Eslamian.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eslamian, M. Convergence theorems for nonspreading mappings and nonexpansive multivalued mappings and equilibrium problems. Optim Lett 7, 547–557 (2013). https://doi.org/10.1007/s11590-011-0438-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-011-0438-4

Keywords

Search

Navigation