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Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. (English) Zbl 1170.47049

Summary: We introduce two iterative sequences for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space. Then we study the strong and weak convergence of the sequences.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
49J40 Variational inequalities
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[1] Alber, Y. I., Metric and generalized projection operators in Banach spaces: Properties and applications, (Kartosatos, A. G., Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (1996), Marcel Dekker: Marcel Dekker New York), 15-50 · Zbl 0883.47083
[2] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. Student, 63, 123-145 (1994) · Zbl 0888.49007
[3] Cioranescu, I., Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems (1990), Kluwer: Kluwer Dordrecht · Zbl 0712.47043
[4] Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6, 117-136 (2005) · Zbl 1109.90079
[5] Kamimura, S.; Takahashi, W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim., 13, 938-945 (2002) · Zbl 1101.90083
[6] F. Kohsaka, W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive type mappings in Banach spaces (in press); F. Kohsaka, W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive type mappings in Banach spaces (in press) · Zbl 1168.47047
[7] Matsushita, S.; Takahashi, W., Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2004, 37-47 (2004) · Zbl 1088.47054
[8] Matsushita, S.; Takahashi, W., A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory, 134, 257-266 (2005) · Zbl 1071.47063
[9] Moudafi, A., Second-order differential proximal methods for equilibrium problems, J. Inequal. Pure Appl. Math., 4 (2003), (art. 18) · Zbl 1175.90413
[10] Reich, S., A weak convergence theorem for the alternating method with Bregman distance, (Kartosatos, A. G., Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (1996), Marcel Dekker: Marcel Dekker New York), 313-318 · Zbl 0943.47040
[11] Tada, A.; Takahashi, W., Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, (Takahashi, W.; Tanaka, T., Nonlinear Analysis and Convex Analysis (2007), Yokohama Publishers: Yokohama Publishers Yokohama), 609-617 · Zbl 1122.47055
[12] Tada, A.; Takahashi, W., Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, J. Optim. Theory Appl., 133, 359-370 (2007) · Zbl 1147.47052
[13] Takahashi, S.; Takahashi, W., Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331, 506-515 (2007) · Zbl 1122.47056
[14] Takahashi, W., Nonlinear Functional Analysis (2000), Yokohama Publishers: Yokohama Publishers Yokohama · Zbl 0997.47002
[15] Takahashi, W., Convex Analysis and Approximation of Fixed Points (2000), Yokohama Publishers: Yokohama Publishers Yokohama, (in Japanese) · Zbl 1089.49500
[16] Xu, H. K., Inequalities in Banach spaces with applications, Nonlinear Anal., 16, 1127-1138 (1991) · Zbl 0757.46033
[17] Zălinescu, C., On uniformly convex functions, J. Math. Anal. Appl., 95, 344-374 (1983) · Zbl 0519.49010
[18] Zălinescu, C., Convex Analysis in General Vector Spaces (2002), World Scientific: World Scientific River Edge, NJ, USA · Zbl 1023.46003
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