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Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. (English) Zbl 1142.47350
Summary: We introduce an iterative method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space and then obtain that the sequence converges strongly to a common element of two sets. Using this result, we prove three new strong convergence theorems in fixed point problems, variational inequalities and equilibrium problems.

MSC:
47J05Equations involving nonlinear operators (general)
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
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References:
[1] Blum, E.; Oettli, W.: From optimization and variational inequalities to equilibrium problems, Math. stud. 63, 123-145 (1994) · Zbl 0888.49007
[2] Combettes, P. L.; Hirstoaga, A.: Equilibrium programming in Hilbert spaces, J. nonlinear convex anal. 6, 117-136 (2005) · Zbl 1109.90079
[3] Iiduka, H.; Takahashi, W.: Weak convergence theorem by Cesàro means for nonexpansive mappings and inverse-strongly monotone mappings, J. nonlinear convex anal. 7, 105-113 (2006) · Zbl 1104.47059
[4] A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems, J. Nonlinear Convex Anal. (in press) · Zbl 1167.47049
[5] Moudafi, A.; Théra, M.: Proximal and dynamical approaches to equilibrium problems, Lecture notes in economics and mathematical systems 477, 187-201 (1999) · Zbl 0944.65080
[6] Opial, Z.: Weak covergence of the sequence of successive approximations for nonexpansive mappings, Bull. amer. Math. soc. 73, 591-597 (1967) · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[7] Suzuki, T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. math. Anal. appl. 305, 227-239 (2005) · Zbl 1068.47085 · doi:10.1016/j.jmaa.2004.11.017
[8] Tada, A.; Takahashi, W.: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, J. optim. Theory appl. 133, 359-370 (2007) · Zbl 1147.47052
[9] 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 · doi:10.1016/j.jmaa.2006.08.036
[10] Takahashi, W.: Nonlinear functional analysis, (2000) · Zbl 0997.47002
[11] Takahashi, W.: Convex analysis and approximation of fixed points, (2000) · Zbl 1089.49500
[12] Takahashi, W.: Introduction to nonlinear and convex analysis, (2005)
[13] Takahashi, W.; Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings, J. optim. Theory appl. 118, 417-428 (2003) · Zbl 1055.47052 · doi:10.1023/A:1025407607560
[14] Wittmann, R.: Approximation of fixed points of nonexpansive mappings, Arch. math. 58, 486-491 (1992) · Zbl 0797.47036 · doi:10.1007/BF01190119
[15] Xu, H. K.: Another control condition in an iterative method for nonexpansive mappings, Bull. austral. Math. soc. 65, 109-113 (2002) · Zbl 1030.47036 · doi:10.1017/S0004972700020116
[16] Y. Yao, Y.C. Liou, R. Chen, Convergence theorems for fixed point problems and variational inequality problems, J. Nonlinear Convex Anal. (in press) · Zbl 1225.47119 · doi:10.1002/mana.200610817