Tada, A.; Takahashi, W. Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. (English) Zbl 1147.47052 J. Optim. Theory Appl. 133, No. 3, 359-370 (2007). The authors introduce two iterative sequences for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in a Hilbert space. Then they show that one of the sequences converges strongly and the other converges weakly. Reviewer: Jinhai Chen (Hongkong) Cited in 12 ReviewsCited in 212 Documents MSC: 47J25 Iterative procedures involving nonlinear operators 47H10 Fixed-point theorems 49J40 Variational inequalities 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 91B50 General equilibrium theory Keywords:equilibrium problems; nonexpansive mappings; firmly nonexpansive mappings; weak and strong convergence; monotonicity PDF BibTeX XML Cite \textit{A. Tada} and \textit{W. Takahashi}, J. Optim. Theory Appl. 133, No. 3, 359--370 (2007; Zbl 1147.47052) Full Text: DOI References: [1] Takahashi, W., Convex Analysis and Approximation of Fixed Points (2000), Yokohama: Yokohama Publishers, Yokohama [2] Takahashi, W., Nonlinear Functional Analysis (2000), Yokohama: Yokohama Publishers, Yokohama · Zbl 0997.47002 [3] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. Stud., 63, 123-145 (1994) · Zbl 0888.49007 [4] Flam, S. D.; Antipin, A. S., Equilibrium programming using proximal-like algorithms, Math. Program., 78, 29-41 (1997) · Zbl 0890.90150 [5] Moudafi, A.; Thera, M., Proximal and dynamical approaches to equilibrium problems, Lecture Notes in Economics and Mathematical Systems, vol. 477, 187-201 (1999), New York: Springer, New York · Zbl 0944.65080 [6] Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6, 117-136 (2005) · Zbl 1109.90079 [7] Mann, W. R., Mean value methods in iteration, Proc. Am. Math. Soc., 4, 506-510 (1953) · Zbl 0050.11603 [8] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279, 372-379 (2003) · Zbl 1035.47048 [9] Takahashi, W.; Toyoda, M., Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 113, 417-428 (2003) · Zbl 1055.47052 [10] Opial, Z., Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Am. Math. Soc., 73, 591-597 (1967) · Zbl 0179.19902 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.