Takahashi, Wataru; Zembayashi, Kei Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. (English) Zbl 1170.47049 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 1, 45-57 (2009). 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. Cited in 7 ReviewsCited in 218 Documents 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 Keywords:Banach space; equilibrium problem; relatively nonexpansive mapping; strong convergence; weak convergence; resolvent PDF BibTeX XML Cite \textit{W. Takahashi} and \textit{K. Zembayashi}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 1, 45--57 (2009; Zbl 1170.47049) Full Text: DOI OpenURL References: [1] Alber, Y.I., Metric and generalized projection operators in Banach spaces: properties and applications, (), 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 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) · 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, (), 313-318 · Zbl 0943.47040 [11] Tada, A.; Takahashi, W., Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, (), 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 [15] Takahashi, W., Convex analysis and approximation of fixed points, (2000), Yokohama Publishers Yokohama, (in Japanese) [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 River Edge, NJ, USA · Zbl 1023.46003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.