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Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. (English) Zbl 1147.47052
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.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H10Fixed point theorems for nonlinear operators on topological linear spaces
49J40Variational methods including variational inequalities
47H09Mappings defined by “shrinking” properties
91B50General equilibrium theory in economics
References:
[1]Takahashi, W.: Convex Analysis and Approximation of Fixed Points. Yokohama Publishers, Yokohama (2000)
[2]Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
[3]Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
[4]Flam, S.D., Antipin, A.S.: Equilibrium programming using proximal-like algorithms. Math. Program. 78, 29–41 (1997) · Zbl 0890.90150 · doi:10.1007/BF02614504
[5]Moudafi, A., Thera, M.: Proximal and dynamical approaches to equilibrium problems. In: Lecture Notes in Economics and Mathematical Systems, vol. 477, pp. 187–201. Springer, New York (1999)
[6]Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)
[7]Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953) · doi:10.1090/S0002-9939-1953-0054846-3
[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 · doi:10.1016/S0022-247X(02)00458-4
[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 · doi:10.1023/A:1025407607560
[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 · doi:10.1090/S0002-9904-1967-11761-0