Abstract
This paper uses a hybrid algorithm to find a common element of the set of solutions to a generalized mixed equilibrium problem, the set of solutions to variational inequality problems, and the set of common fixed points for a finite family of quasi-ϕ-nonexpansive mappings in a uniformly smooth and strictly convex Banach space. As applications, we utilize our results to study the optimization problem. It shows that our results improve and extend the corresponding results announced by many others recently.
Similar content being viewed by others
References
Ceng, Lu-Chuan and Yao, Jen-Chih. A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 214, 186–201 (2008)
Browder, F. E. Existence and approximation of solutions of nonlinear variational inequalities. Proc. Natl. Acad. Sci. USA 56(4), 1080–1086 (1966)
Takahashi, W. and Zembayashi, K. Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70(1), 45–57 (2008)
Takahashi, S. and Takahashi, W. Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331(1), 506–515 (2007)
Qin, X. L., Shang, M., and Su, Y. A general iterative method for equilibrium problem and fixed point problems in Hilbert spaces. Nonlinear Anal. 69(11), 3897–3909 (2008)
Cioranescu, I. Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht (1990)
Alber, Y. I. Metric and generalized projection operators in Banach spaces: properties and applications. Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (ed. Kartosator, A. G.), Marcel Dekker, New York, 15–50 (1996)
Kamimura, S. and Takahashi, W. Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13(3), 938–945 (2002)
Matsushita, S. and Takahashi, W. Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 67(6), 37–47 (2004)
Nilsrakoo, W. and Saejung, S. Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings. Fixed Point Theory Appl. 2008, Article ID 312454, 19 pages (2008) DOI: 10.1155/2008/312454
Blum, E. and Oettli, W. From optimization and variational inequalities to equilibrium problems. The Mathematics Student 63(1–4), 123–145 (1994)
Xu, H. K. Inequalities in Banach spaces with applications. Nonlinear Anal. 16(12), 1127–1138 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Contributed by Shi-sheng ZHANG
Project supported by the Natural Science Foundation of Yibin University (No. 2009Z003)
Rights and permissions
About this article
Cite this article
Zhang, Ss. Generalized mixed equilibrium problem in Banach spaces. Appl. Math. Mech.-Engl. Ed. 30, 1105–1112 (2009). https://doi.org/10.1007/s10483-009-0904-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-009-0904-6
Key words
- small generalized mixed equilibrium problem
- variational inequality
- quasi-ϕ-nonexpansive mapping
- maximal monotone operator
- monotone mapping