## Generalized mixed equilibrium problem in Banach spaces.(English)Zbl 1178.47051

From the summary: 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-$$\varphi$$-nonexpansive mappings in a uniformly smooth and strictly convex Banach space. As applications, we utilize our results to study an optimization problem.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H05 Monotone operators and generalizations 47H10 Fixed-point theorems 47J20 Variational and other types of inequalities involving nonlinear operators (general)
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### References:

 [1] 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) · Zbl 1143.65049 [2] Browder, F. E. Existence and approximation of solutions of nonlinear variational inequalities. Proc. Natl. Acad. Sci. USA 56(4), 1080–1086 (1966) · Zbl 0148.13502 [3] 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) · Zbl 1170.47049 [4] 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) · Zbl 1122.47056 [5] 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) · Zbl 1170.47044 [6] Cioranescu, I. Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht (1990) · Zbl 0712.47043 [7] 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) · Zbl 0883.47083 [8] Kamimura, S. and Takahashi, W. Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13(3), 938–945 (2002) · Zbl 1101.90083 [9] 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) · Zbl 1088.47054 [10] 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 · Zbl 1203.47061 [11] Blum, E. and Oettli, W. From optimization and variational inequalities to equilibrium problems. The Mathematics Student 63(1–4), 123–145 (1994) · Zbl 0888.49007 [12] Xu, H. K. Inequalities in Banach spaces with applications. Nonlinear Anal. 16(12), 1127–1138 (1991) · Zbl 0757.46033
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