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On iterative methods for equilibrium problems. (English) Zbl 1165.49035
Summary: We introduce a hybrid iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of finitely many nonexpansive mappings. We prove that the approximate solution converges strongly to a solution of a class of variational inequalities under some mild conditions, which is the optimality condition for some minimization problem. We also give some comments on the results of {\it S. Plubtieng} and {\it R. Punpaeng} [J. Math. Anal. Appl. 336, No. 1, 455--469 (2007; Zbl 1127.47053)]. Results obtained in this paper may be viewed as an improvement and refinement of the previously known results in this area.

49M37Methods of nonlinear programming type in calculus of variations
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
58E05Abstract critical point theory
Full Text: DOI
[1] Combettes, P. L.; Hirstoaga, S. A.: Equilibrium problems in Hilbert spaces. J. nonlinear convex anal. 6, 117-136 (2005) · Zbl 1109.90079
[2] 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
[3] Marino, G.; Xu, H. K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. math. Anal. appl. 318, 43-52 (2006) · Zbl 1095.47038
[4] Plubtieng, S.; Punpaeng, R.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. math. Anal. appl. 336, 455-469 (2007) · Zbl 1127.47053
[5] Blum, E.; Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. student 63, 123-145 (1994) · Zbl 0888.49007
[6] 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
[7] Atsushiba, S.; Takahashi, W.: Strong convergence theorems for a finite family of nonexpansive mappings and applications. Indian J. Math. 41, 435-453 (1999) · Zbl 1055.47514
[8] Takahashi, W.; Shimoji, K.: Convergence theorems for nonexpansive mappings and feasibility problems. Math. comput. Modelling 32, 1463-1471 (2000) · Zbl 0971.47040
[9] Chadli, O.; Wong, N. C.; Yao, J. C.: Equilibrium problems with applications to eigenvalue problems. J. optim. Theory appl. 117, 245-266 (2003) · Zbl 1141.49306
[10] Chadli, O.; Schaible, S.; Yao, J. C.: Regularized equilibrium problems with an application to noncoercive hemivariational inequalities. J. optim. Theory appl. 121, 571-596 (2004) · Zbl 1107.91067
[11] Konnov, I. V.; Schaible, S.; Yao, J. C.: Combined relaxation method for mixed equilibrium problems. J. optim. Theory appl. 126, 309-322 (2005) · Zbl 1110.49028
[12] Combettes, P. L.; Hirstoaga, S. A.: Equilibrium programming in Hilbert spaces. J. nonlinear convex anal. 6, 117-136 (2005) · Zbl 1109.90079
[13] Chadli, O.; Konnov, I. V.; Yao, J. C.: Descent methods for equilibrium problems in a Banach space. Comput. math. Appl. 48, 609-616 (2004) · Zbl 1057.49009
[14] L.C. Zeng, S.Y. Wu, J.C. Yao, Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems, Taiwanese J. Math. 2007 (in press) · Zbl 1121.49005
[15] Ding, X. P.; Lin, Y. C.; Yao, J. C.: Predictor--corrector algorithms for solving generalized mixed implicit quasi-equilibrium problems. Appl. math. Mech. 27, 1157-1164 (2006) · Zbl 1199.49010
[16] Tada, A.; Takahashi, W.: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. Nonlinear analysis and convex analysis, 609-617 (2007) · Zbl 1122.47055
[17] Noor, M. Aslam; Oettli, W.: On general nonlinear complementarity problems and quasi equilibria. Mathematiche (Catania) 49, 313-331 (1994) · Zbl 0839.90124
[18] Noor, M. Aslam; Noor, K. Inayat: On equilibrium problems. Appl. math. E-notes 4, 125-132 (2004) · Zbl 1064.49009
[19] Noor, M. Aslam: Fundamentals of equilibrium problems. Math. inequal. Appl. 9, 529-566 (2006) · Zbl 1099.91072
[20] Noor, M. Aslam: Some classes of equilibrium problems. Nonlinear anal. Forum 12, xxx (2007)
[21] Noor, M. Aslam: Regularized mixed quasi equilibrium problems. J. appl. Math. comput. 23, 183-191 (2007) · Zbl 1111.49005
[22] Noor, M. Aslam: Predictor--corrector methods for multivalued hemiequilibrium problems. Appl. math. Comput. 181, 721-731 (2006) · Zbl 1148.65307
[23] Noor, M. Aslam: Mixed quasi equilibrium-like problems. J. appl. Math. stoch. Anal. 2006, 1-12 (2006)
[24] Noor, M. Aslam: Invex equilibrium problems. J. math. Anal. appl. 302, 463-475 (2005) · Zbl 1058.49007
[25] Noor, M. Aslam: Fundamentals of mixed quasi variational inequalities. Int. J. Pure appl. Math. 15, 137-258 (2004) · Zbl 1059.49018
[26] Noor, M. Aslam; Noor, K. Inayat; Gupta, V.: On equilibrium-like problems. Appl. anal. 86, 807-818 (2007) · Zbl 1129.49016