Abstract
In this paper, we propose two strongly convergent parallel hybrid iterative methods for finding a common element of the set of fixed points of a family of asymptotically quasi ϕ-nonexpansive mappings, the set of solutions of variational inequalities and the set of solutions of equilibrium problems in uniformly smooth and 2-uniformly convex Banach spaces. A numerical experiment is given to verify the efficiency of the proposed parallel algorithms.
Similar content being viewed by others
References
Alber, Y.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartosator, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15–50. Dekker, New York (1996)
Alber, Y., Ryazantseva, I.: Nonlinear Ill-posed Problems of Monotone Type. Spinger, Netherlands (2006)
Anh, P.K., Buong, N., Hieu, D.V.: Parallel methods for regularizing systems of equations involving accretive operators. Appl. Anal. 93, 2136–2157 (2014)
Anh, P.K., Chung, C.V.: Parallel hybrid methods for a finite family of relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 35, 649–664 (2014)
Anh, P.K., Hieu, D.V.: Parallel and sequential hybrid methods for a finite family of asymptotically quasi ϕ-nonexpansive mappings. J. Appl. Math. Comput. (2014). doi:10.1007/s12190-014-0801-6
Chang, S.-S., Kim, J.K., Wang, X.R.: Modified block iterative algorithm for solving convex feasibility problems in Banach spaces. J. Inequal. Appl. 2010, 869684 (2010)
Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht (1990)
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)
Kang, J., Su, Y., Zhang, X.: Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications. Nonlinear Anal. Hybrid Syst. 4, 755–765 (2010)
Kim, T.H., Lee, H.J.: Strong convergence of modified iteration processes for relatively nonexpansive mappings in Banach Spaces. Kyungpook Math. J. 48, 685–703 (2008)
Matsushita, S., Takahashi, W.: Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 1, 37–47 (2004)
Qin, X., Kang, S.M., Cho, Y.J.: Convergence theorems for inverse strongly monotone mappings and quasi ϕ-nonexpansive mappings. Bull. Korean Math. Soc. 46, 885–894 (2009)
Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)
Saewan, S., Kumam, P.: The hybrid block iterative algorithm for solving the system of equilibrium problems and variational inequality problems. Springer Plus 2012, 8 (2012)
Su, Y., Li, M., Zhang, H.: New monotone hybrid algorithm for hemi-relatively nonexpansive mappings and maximal monotone operators. Appl. Math. Comput. 217, 5458–5465 (2011)
Su, Y.F., Wang, Z.M., Xu, H.K.: Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings. Nonlinear Functional Analysis 71, 5616–5628 (2009)
Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point in Hilbert space. J. Math. Anal. Appl. 331, 506–515 (2007)
Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)
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)
Wang, Z., Kang, M.K., Cho, Y.J.: Convergence theorems based on the shrinking projection method for hemi-relatively nonexpansive mappings, variational inequalities and equilibrium problems. Banach. J. Math. Anal. 6, 11–34 (2012)
Zegeye, H., Shahzad, N.: Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings. Nonlinear Anal. 70, 2707–2716 (2009)
Zhang, C., Li, J., Liu, B.: Strong convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Comput. Math. Appl. 61, 262–276 (2011)
Acknowledgments
The research of the first author was partially supported by Vietnam Institute for Advanced Study in Mathematics (VIASM) and Vietnam National Foundation for Science and Technology Development (NAFOSTED).
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Nguyen Khoa Son’s 65th birthday.
Rights and permissions
About this article
Cite this article
Anh, P.K., Van Hieu, D. Parallel Hybrid Iterative Methods for Variational Inequalities, Equilibrium Problems, and Common Fixed Point Problems. Vietnam J. Math. 44, 351–374 (2016). https://doi.org/10.1007/s10013-015-0129-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10013-015-0129-z
Keywords
- Asymptotically quasi ϕ-nonexpansive mapping
- Variational inequality
- Equilibrium problem
- Hybrid method
- Parallel computation