Advertisement

Journal of Global Optimization

, Volume 57, Issue 4, pp 1299–1318 | Cite as

Strong convergence for maximal monotone operators, relatively quasi-nonexpansive mappings, variational inequalities and equilibrium problems

  • Siwaporn Saewan
  • Poom Kumam
  • Yeol Je Cho
Article

Abstract

In this paper, we introduce a new hybrid iterative scheme for finding a common element of the set of zeroes of a maximal monotone operator, the set of fixed points of a relatively quasi-nonexpansive mapping, the sets of solutions of an equilibrium problem and the variational inequality problem in Banach spaces. As applications, we apply our results to obtain strong convergence theorems for a maximal monotone operator and quasi-nonexpansive mappings in Hilbert spaces and we consider a problem of finding a minimizer of a convex function.

Keywords

Hybrid projection method Inverse-strongly monotone operator  Variational inequality Equilibrium problem Relatively quasi-nonexpansive mapping Maximal monotone operator 

Mathematics Subject Classification (2000)

47H05 47H09 47H10 

Notes

Acknowledgments

First, 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 and the third author 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: 2012-0008170).

References

  1. 1.
    Alber, Y.I., Reich, S.: An iterative method for solving a class of nonlinear operator equations in Banach spaces. PanAm. Math. J. 4, 39–54 (1994)Google Scholar
  2. 2.
    Alber, Y.I.: Metric and generalized projection operators in Banach spaces: Properties and applications. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15–50. Marcel Dekker, New York (1996)Google Scholar
  3. 3.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)Google Scholar
  4. 4.
    Butnariu, D., Reich, S., Zaslavski, A.J.: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. J. Appl. Anal. 7, 151–174 (2001)CrossRefGoogle Scholar
  5. 5.
    Butnariu, D., Reich, S., Zaslavski, A.J.: Weak convergence of orbits of nonlinear operators in reflexive Banach spaces. Numer. Funct. Anal. Optim. 24, 489–508 (2003)CrossRefGoogle Scholar
  6. 6.
    Chang, S.S.: On Chidumes open questions and approximate solutions of multivalued strongly accretive mapping in Banach spaces. J. Math. Anal. Appl. 216, 94–111 (1997)CrossRefGoogle Scholar
  7. 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. 60, 2292–2301 (2010)CrossRefGoogle Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Cho, Y.J., Petrot, N.: On the system of nonlinear mixed implicit equilibrium problems in Hilbert spaces. J. Inequal. Appl. Vol. 2010, Article ID 437976, pp. 12Google Scholar
  10. 10.
    Cholamjiak, W., Suantai, S.: A hybrid methods for a countable family of multi-valued maps, equilibrium problems and variational inequality problems. Discret. Dyn. Nat. Soc., Vol. 2010 (2010), Article ID 349158, pp. 14Google Scholar
  11. 11.
    Censor, Y., Reich, S.: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 37, 323–339 (1996)CrossRefGoogle Scholar
  12. 12.
    Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht (1990)CrossRefGoogle Scholar
  13. 13.
    Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)Google Scholar
  14. 14.
    Iiduka, H., Takahashi, W.: Weak convergence of a projection algorithm for variational inequalities in a Banach space. J. Math. Anal. Appl. 339, 668–679 (2008)CrossRefGoogle Scholar
  15. 15.
    Inoue, G., Takahashi, W., Zembayashi, K.: Strong convergence theorems by hybrid methods for maximal monotone operator and relatively nonexpansive mappings in Banach spaces. J. Convex Anal. 16, 791–806 (2009)Google Scholar
  16. 16.
    Jaiboon, C., Kumam, P.: A general iterative method for solving equilibrium problems, variational inequality problems and fixed point problems of an infinite family of nonexpansive mappings. J. Appl. Math. Comput. 34, 407–439 (2010)CrossRefGoogle Scholar
  17. 17.
    Katchang, P., Kumam, P.: A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space. J. Appl. Math. Comput. 32, 19–38 (2010)CrossRefGoogle Scholar
  18. 18.
    Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938–945 (2002)CrossRefGoogle Scholar
  19. 19.
    Klin-eam, C., Suantai, S., Takahashi, W.: Strong convergence of generalized projection algorithms for nonlinear operator. Abstr. Appl. Anal. Vol. 2009 (2009), Article ID 649831, pp. 18Google Scholar
  20. 20.
    Kohsaka, F., Takahashi, W.: Existence and approximation of fixed points of firmly nonexpansivetype mappings in Banach spaces. SIAM J. Optim. 19, 824–835 (2008)CrossRefGoogle Scholar
  21. 21.
    Kumam, P.: A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping. J. Appl. Math. Comput. 29, 263–280 (2009)CrossRefGoogle Scholar
  22. 22.
    Li, J.: On the existence of solutions of variational inequalities in Banach spaces. J. Math. Anal. Appl. 295, 115–126 (2004)CrossRefGoogle Scholar
  23. 23.
    Liu, Y.: Strong convergence theorems for variational inequalities and relatively weak nonexpansive mappings. J. Glob. Optim. 46, 319–329 (2010)CrossRefGoogle Scholar
  24. 24.
    Martinet, B.: Regularization d’ in\(\acute{e}\)quations variationelles par approximations successives. Revue Francaise d’informatique et de Recherche operationelle 4, 154–159 (1970)Google Scholar
  25. 25.
    Matsushita, S., Takahashi, W.: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. J. Approx. Theory 134, 257–266 (2005)CrossRefGoogle Scholar
  26. 26.
    Matsushita, S., Takahashi, W.: Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2004, 37–47 (2004)CrossRefGoogle Scholar
  27. 27.
    Moudafi, A.: Second-order differential proximal methods for equilibrium problems. J. Inequal. Pure Appl. Math. 4, Article 18 (2003)Google Scholar
  28. 28.
    Nilsrakoo, W., Saejung, S.: Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings. Fixed Point Theory Appl. 2008, Article ID 312454, pp. 19 (2008)Google Scholar
  29. 29.
    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)CrossRefGoogle Scholar
  30. 30.
    Qin, X., Cho, Y.J., Kang, S.M.: Approximating zeros of monotone operators by proximal point algorithms. J. Glob. Optim. 46, 75–87 (2010)CrossRefGoogle Scholar
  31. 31.
    Qin, X., Cho, S.Y., Kang, S.M.: Strong convergence of shrinking projection methods for quasi-\(\phi \)-nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 234, 750–760 (2010)CrossRefGoogle Scholar
  32. 32.
    Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)CrossRefGoogle Scholar
  33. 33.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)CrossRefGoogle Scholar
  34. 34.
    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)Google Scholar
  35. 35.
    Saewan, S., Kumam P., Wattanawitoon, K.: Convergence theorem based on a new hybrid projection method for finding a common solution of generalized equilibrium and variational inequality problems in Banach spaces. Abstr. Appl. Anal., vol. 2010, Article ID 734126, pp. 26Google Scholar
  36. 36.
    Su, Y., Wang, D., Shang, M.: Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings. Fixed Point Theory Appl. 2008, Article ID 284613, pp. 8 (2008)Google Scholar
  37. 37.
    Su, Y., Shang, M., Qin, X.: A general iterative scheme for nonexpansive mappings and inverse-strongly monotone mappings. J. Appl. Math. Comput. 28, 283–294 (2008)CrossRefGoogle Scholar
  38. 38.
    Takahashi, W.: Nonlinear Functional Analysis: Fixed Point Theory and Its Application. Yokohama-Publishers, Yokohama (2000)Google Scholar
  39. 39.
    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)CrossRefGoogle Scholar
  40. 40.
    Takahashi, W., Zembayashi, K.: Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings. Fixed Point Theory Appl. 2008, Article ID 528476, pp. 11 (2008)Google Scholar
  41. 41.
    Xu, H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)CrossRefGoogle Scholar
  42. 42.
    Zegeye, H., Shahzad, N.: Strong convergence for monotone mappings and relatively weak nonexpansive mappings. Nonlinear Anal. 70, 2707–2716 (2009)CrossRefGoogle Scholar
  43. 43.
    Zegeye, H., Shahzad, N.: A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems. Nonlinear Anal. 74, 263–272 (2011)CrossRefGoogle Scholar
  44. 44.
    Zhou, H., Gao, X.: An iterative method of fixed points for closed and quasi-strict pseudo-contractions in Banach spaces. J. Appl. Math. Comput. 33, 227–237 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Faculty of ScienceThaksin University (TSU)Phat Tha LungThailand
  2. 2.Department of Mathematics, Faculty of ScienceKing Mongkut’s University of Technology Thonburi (KMUTT)BangkokThailand
  3. 3.Department of Mathematics Education and RINSGyeongsang National UniversityChinjuRepublic of Korea

Personalised recommendations