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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:
47J05 Equations involving nonlinear operators (general)
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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[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., (), 187-201
[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
[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
[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
[10] Takahashi, W., Nonlinear functional analysis, (2000), Yokohama Publishers Yokohama
[11] Takahashi, W., Convex analysis and approximation of fixed points, (2000), Yokohama Publishers Yokohama, (in Japanese)
[12] Takahashi, W., Introduction to nonlinear and convex analysis, (2005), Yokohama Publishers Yokohama, (in Japanese)
[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
[14] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. math., 58, 486-491, (1992) · Zbl 0797.47036
[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
[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 1153.47058
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