×

Mixed equilibrium problems and optimization problems. (English) Zbl 1160.49013

Summary: We introduce and analyze a new hybrid iterative algorithm for finding a common element of the set of solutions of mixed equilibrium problems and the set of fixed points of an infinite family of nonexpansive mappings. Furthermore, we prove some strong convergence theorems for the hybrid iterative algorithm under some mild conditions. We also discuss some special cases. Results obtained in this paper improve the previously known results in this area.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
47H10 Fixed-point theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. Student, 63, 123-145 (1994) · Zbl 0888.49007
[2] Aslam Noor, M.; Oettli, W., On general nonlinear complementarity problems and quasi equilibria, Matematiche (Catania), 49, 313-331 (1994) · Zbl 0839.90124
[3] 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
[4] 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
[5] 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
[6] Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6, 117-136 (2005) · Zbl 1109.90079
[7] Flam, S. D.; Antipin, A. S., Equilibrium programming using proximal-like algorithms, Math. Program., 78, 29-41 (1997) · Zbl 0890.90150
[8] 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
[9] 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
[10] 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
[11] Y. Yao, Y.C. Liou, J.C. Yao, Convergence theorem for equilibrium problems and fixed point problems, Fixed Point Theory (2008), in press; Y. Yao, Y.C. Liou, J.C. Yao, Convergence theorem for equilibrium problems and fixed point problems, Fixed Point Theory (2008), in press · Zbl 1153.47058
[12] 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
[13] Tada, A.; Takahashi, W., Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, (Takahashi, W.; Tanaka, T., Nonlinear Analysis and Convex Analysis (2007), Yokohama Publishers: Yokohama Publishers Yokohama), 609-617 · Zbl 1122.47055
[14] Yao, Y.; Liou, Y. C.; Yao, J. C., Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings, Fixed Point Theory Appl., 2007 (2007), Article ID 64363, 12 pp · Zbl 1153.54024
[15] Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming using proximal-like algorithms, Math. Program., 78, 29-41 (1997) · Zbl 0890.90150
[16] Ceng, L. C.; Yao, J. C., A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math., 214, 186-201 (2008) · Zbl 1143.65049
[17] Moudafi, A., Viscosity approximation methods for fixed-point problems, J. Math. Anal. Appl., 241, 46-55 (2000) · Zbl 0957.47039
[18] Bauschke, H. H.; Borwein, J. M., On projection algorithms for solving convex feasibility problems, SIAM Rev., 38, 367-426 (1996) · Zbl 0865.47039
[19] Combettes, P. L., Hilbertian convex feasibility problem: Convergence of projection methods, Appl. Math. Optim., 35, 311-330 (1997) · Zbl 0872.90069
[20] Deutsch, F.; Yamada, I., Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings, Numer. Funct. Anal. Optim., 19, 33-56 (1998) · Zbl 0913.47048
[21] Xu, H. K., An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116, 659-678 (2003) · Zbl 1043.90063
[22] 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
[23] Shimoji, K.; Takahashi, W., Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math., 5, 387-404 (2001) · Zbl 0993.47037
[24] Xu, H. K., Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298, 279-291 (2004) · Zbl 1061.47060
[25] Aslam Noor, M., Generalized mixed quasi-equilibrium problems with trifunction, Appl. Math. Lett., 18, 695-700 (2005) · Zbl 1067.49006
[26] Aslam Noor, M., On a class of nonconvex equilibrium problems, Appl. Math. Comput., 157, 653-666 (2004) · Zbl 1070.65053
[27] Aslam Noor, M., Multivalued general equilibrium problems, J. Math. Anal. Appl., 283, 140-149 (2003) · Zbl 1039.49009
[28] Aslam Noor, M.; Rassias, Themistocles M., On nonconvex equilibrium problems, J. Math. Anal. Appl., 312, 289-299 (2005) · Zbl 1087.49009
[29] Aslam Noor, M.; Rassias, Themistocles M., On general hemiequilibrium problems, J. Math. Anal. Appl., 324, 1417-1428 (2006) · Zbl 1102.49008
[30] Yao, Y.; Aslam Noor, M.; Liou, Y. C., On iterative methods for equilibrium problems, Nonlinear Anal., 70, 497-509 (2009) · Zbl 1165.49035
[31] Aslam Noor, M.; Inayat, K.; Gupta, V., On equilibrium-like problems, Appl. Anal., 86, 807-818 (2007) · Zbl 1129.49016
[32] Aslam Noor, M.; Yao, Y.; Liou, Y. C., Extragradient method for equilibrium problems and variational inequalities, Albanian J. Math., 2, 125-138 (2008) · Zbl 1205.47063
[33] Aslam Noor, M., Some iterative algorithms for extended general variational inequalities, Albanian J. Math., 2, 265-275 (2008) · Zbl 1158.49014
[34] Bnouhachem, A.; Aslam Noor, M.; Hao, Z., Some new extragradient iterative methods for variational inequalities, Nonlinear Anal., 70, 1321-1329 (2009) · Zbl 1171.47050
[35] Aslam Noor, M., Implicit Wiener-Hopf equations and quasi variational inequalities, Albanian J. Math., 2, 15-25 (2008)
[36] Aslam Noor, M., General variational inequalities, Appl. Math. Lett., 1, 119-121 (1988) · Zbl 0655.49005
[37] Aslam Noor, M.; Inayat Noor, K.; Rassias, Th. M., Some aspects of variational inequalities, J. Comput. Appl. Math., 47, 285-312 (1993) · Zbl 0788.65074
[38] Aslam Noor, M., New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251, 217-229 (2000) · Zbl 0964.49007
[39] Aslam Noor, M., Some recent developments in general variational inequalities, Appl. Math. Comput., 152, 199-277 (2004) · Zbl 1134.49304
[40] Astern Noor, M.; Inayat Noor, K., On equilibrium problems, Appl. Math. E-Notes, 4, 125-132 (2004) · Zbl 1064.49009
[41] Aslam Noor, M., On a class of general variational inequalities, J. Adv. Math. Stud., 1, 31-42 (2008)
[42] Aslam Noor, M., Extended general variational inequalities, Appl. Math. Lett., 22, 182-186 (2009) · Zbl 1163.49303
[43] Aslam Noor, M.; Inayat Noor, K.; Yaqoob, H., On general mixed variational inequalities, Acta Appl. Math. (2008)
[44] Aslam Noor, M., Differentiable nonconvex functions and general variational inequalities, Appl. Math. Comput., 199, 623-630 (2008) · Zbl 1147.65047
[45] Aslam Noor, M., Projection iterative methods for extended general variational inequalities, J. Appl. Math. Comput. (2009) · Zbl 1213.49017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.