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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.

47J05Equations involving nonlinear operators (general)
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
Full Text: DOI
[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