Advertisement

ANNALI DELL'UNIVERSITA' DI FERRARA

, Volume 57, Issue 1, pp 89–108 | Cite as

A new hybrid algorithm for a system of equilibrium problems and variational inclusion

  • Thanyarat Jitpeera
  • Poom Kumam
Article

Abstract

The purpose of this paper is to consider a shrinking projection method of finding the common element of the set of common fixed points for a finite family of a ξ-strict pseudo-contraction, the set of solutions of a systems of equilibrium problems and the set of solutions of variational inclusions. Then, we prove strong convergence theorems of the iterative sequence generated by the shrinking projection method under some suitable conditions in a real Hilbert space. Our results improve and extend recent results announced by Peng, Wang, Shyu and Yao (J Inequal Appl, 2008:15, Article ID 720371, 2008), Takahashi, Takeuchi and Kubota (J Math Anal Appl 341:276–286, 2008), Takahashi and Takahashi (Nonlinear Anal 69:1025–1033, 2008) and many others.

Keywords

Generalized mixed equilibrium problem Fixed point Variational inequality Nonexpansive mapping Inverse-strongly monotone mapping 

Mathematics Subject Classification (2000)

47H09 47H05 49J05 49J25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Acedoa G.L., Xu H.K.: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 67, 2258–2271 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Brézis, H.: Opérateur maximaux monotones. In: Mathematics Studies, vol. 5. North-Holland, Amsterdam (1973)Google Scholar
  4. 4.
    Browder F.E., Petryshym W.V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20, 197–228 (1963)CrossRefGoogle Scholar
  5. 5.
    Colao, V., Acedob, G.L., Marinoa, G.: An implicit method for finding common solution of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings. Nonlinear Anal. vol. 71 (2009)Google Scholar
  6. 6.
    Cianciaruso, F., Marino, G., Muglia, L., Yao, Y.: A Hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem. Fixed Point Theory Appl. 2010, 19, Article ID 383740 (2010)Google Scholar
  7. 7.
    Combettes P.L., Hirstaoga S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chadli O., Wong N.C., Yao J.C.: Equilibrium problems with applications to eigenvalue problems. J. Optim. Theory Appl. 117(2), 245–266 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Chadli O., Schaible S., Yao J.C.: Regularized equilibrium problems with application to noncoercive hemivariational inequalities. J. Optim. Theory Appl. 121(3), 571–596 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cholamjiak, P., Suantai, S.: A New Hybrid Algorithm for Variational Inclusions, Generalized Equilibrium Problems, and a Finite Family of Quasi-Nonexpansive Mappings. Fixed Point Theory Appl. 2009, 20, Article ID 350979 (2009)Google Scholar
  11. 11.
    Goebel K., Kirk W.A.: Topics on Metric Fixed-Point Theory. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  12. 12.
    Jaiboon, C., Kumam, P.: Strong Convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities. J. Inequal. Appl. 2010, 43, Article ID 728028 (2010)Google Scholar
  13. 13.
    Jaiboon, C., Kumam, P.: A system of generalized mixed equilibrium problems and fixed point problems for pseudocontractive mappings in Hilbert spaces. Fixed Point Theory Appl. 2010, Article ID 361512 (2010)Google Scholar
  14. 14.
    Jaiboon C., Kumam P.: A general iterative method for addressing mixed equilibrium problems and optimization problems. Nonlinear Anal. Theory Methods Appl. 73(5), 1180–1202 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Jaiboon C., Chantarangsi W., Kumam P.: A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings. Nonlinear Anal. Hybrid Syst. 4, 199–215 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Jaiboon, C., Kumam, W., Kumam, P., Singta, A.: A shrinking projection method for generalized mixed equilibrium problems, variational inclusion problems and a finite family of quasi-nonexpansive mappings. J. Inequal. Appl. 2010, Article ID 458247 (2010)Google Scholar
  17. 17.
    Kangtunyakarn A., Suantai S.: A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mapping. Nonlinear Anal. 71, 4448–4460 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Katchang, P., Kumam, P.: A general iterative method of fixed points for mixed equilibrium problems and variational inclusion problems. J. Inequal. Appl. 2010, 25, Article ID 370197 (2010)Google Scholar
  19. 19.
    Kumam P., Jaiboon C.: A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems. Nonlinear Anal. Hybrid Syst. 3, 510–530 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Konnov I.V., Schaible S., Yao J.C.: Combined relaxation method for mixed equilibrium problems. J. Optim. Theory Appl. 126(2), 309–322 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Lemaire, B.: Which fixed point does the iteration method select.... In: Recent Advances in Optimization (Trier 1996). Lecture Notes in Economics and Mathematical Systems, vol. 452, pp. 154–167 (1997)Google Scholar
  22. 22.
    Moudafi, A., Théra, M.: Proximal and dynamical approaches to equilibrium problem. In: Lecture Notes in Econcmics and Mathematical Systems, vol. 477. Springer, Berlin, pp. 187–201 (1999)Google Scholar
  23. 23.
    Opial Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 595–597 (1967)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Peng, J.W., Wang, Y., Shyu, D.S., Yao, J.-C.: Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems. J. Inequal. Appl. 2008, 15, Article ID 720371 (2008)Google Scholar
  25. 25.
    Shehu, Y.: Fixed point solutions of variational inequality and generalized equilibrium problems with applications. Ann. Univ. Ferrara. doi: 10.1007/s11565-010-0102-4
  26. 26.
    Takahashi W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)zbMATHGoogle Scholar
  27. 27.
    Takahashi S., Takahashi W.: Strong convergence theorems for a generalized equilibrium problem and a nonexpansive mappings in a Hilbert space. Nonlinear Anal. Ser. A Theory Methods Appl. 69, 1025–1033 (2008)zbMATHCrossRefGoogle Scholar
  28. 28.
    Takahashi W., Takeuchi Y., Kubota R.: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276–286 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Yao, Y., Liou, Y.C., Yao, J.C.: A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems. Fixed Point Theory Appl. 2008, 15, Article ID 417089 (2008)Google Scholar
  30. 30.
    Yao, Y., Liou, Y.C., Wu, Y.J.: An Extragradient method for mixed equilibrium problems and fixed point problems. Fixed Point Theory Appl. 2009, 15, Article ID 632819 (2009)Google Scholar
  31. 31.
    Zeng L.C., Wu S.Y., Yao J.C.: Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems. Taiwanese J. Math. 10(6), 1497–1514 (2006)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Zhang S.S., Lee J.H.W., Chan C.K.: Algorithms of common solutions to quasi variational inclusion and fixed point problems. Appl. Math. Mech. 29(5), 571–581 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKing Mongkut’s University of Technology Thonburi, KMUTTBangkokThailand
  2. 2.Centre of Excellence in Mathematics, CHEBangkokThailand

Personalised recommendations