# zbMATH — the first resource for mathematics

A hybrid extragradient viscosity approximation method for solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. (English) Zbl 1186.47065
From the summary: We introduce a new hybrid extragradient viscosity approximation method for finding the common element of the set of equilibrium problems, the set of solutions of fixed points of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality problem for a $$\beta$$-inverse-strongly monotone mapping in Hilbert spaces. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of a variational inequality, which is the optimality condition for a minimization problem.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
Full Text:
##### References:
 [1] Takahashi W: Nonlinear Functional Analysis, Fixed Point Theory and Its Application. Yokohama, Yokohama, Japan; 2000:iv+276. · Zbl 0997.47002 [2] Ceng, L-C; Yao, J-C, Relaxed viscosity approximation methods for fixed point problems and variational inequality problems, Nonlinear Analysis: Theory, Methods & Applications, 69, 3299-3309, (2008) · Zbl 1163.47052 [3] Ceng, L-C; Cubiotti, P; Yao, J-C, An implicit iterative scheme for monotone variational inequalities and fixed point problems, Nonlinear Analysis: Theory, Methods & Applications, 69, 2445-2457, (2008) · Zbl 1170.47040 [4] Ceng, L-C; Yao, J-C, An extragradient-like approximation method for variational inequality problems and fixed point problems, Applied Mathematics and Computation, 190, 205-215, (2007) · Zbl 1124.65056 [5] Ceng, L-C; Petruşel, A; Yao, J-C, Weak convergence theorem by a modified extragradient method for nonexpansive mappings and monotone mappings, Fixed Point Theory, 9, 73-87, (2008) · Zbl 1223.47072 [6] Shang, M; Su, Y; Qin, X, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, No. 2007, 6, (2007) [7] Browder, FE; Petryshyn, WV, Construction of fixed points of nonlinear mappings in Hilbert space, Journal of Mathematical Analysis and Applications, 20, 197-228, (1967) · Zbl 0153.45701 [8] Liu, F; Nashed, MZ, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Analysis, 6, 313-344, (1998) · Zbl 0924.49009 [9] Rockafellar, RT, On the maximality of sums of nonlinear monotone operators, Transactions of the American Mathematical Society, 149, 75-88, (1970) · Zbl 0222.47017 [10] Ceng, L-C; Al-Homidan, S; Ansari, QH; Yao, J-C, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, Journal of Computational and Applied Mathematics, 223, 967-974, (2009) · Zbl 1167.47307 [11] Ceng, L-C; Yao, J-C, Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings, Applied Mathematics and Computation, 198, 729-741, (2008) · Zbl 1151.65058 [12] Ceng, L-C; Schaible, S; Yao, JC, Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings, Journal of Optimization Theory and Applications, 139, 403-418, (2008) · Zbl 1163.47051 [13] Ceng, L-C; Yao, J-C, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, Journal of Computational and Applied Mathematics, 214, 186-201, (2008) · Zbl 1143.65049 [14] Kumam P, P, Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space, Turkish Journal of Mathematics, 33, 85-98, (2009) · Zbl 1223.47083 [15] Peng, J-W; Yao, J-C, A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems, Taiwanese Journal of Mathematics, 12, 1401-1432, (2008) · Zbl 1185.47079 [16] Peng, J-W; Yao, J-C, A modified CQ method for equilibrium problems, fixed points and variational inequality, Fixed Point Theory, 9, 515-531, (2008) · Zbl 1172.47051 [17] Flåm, SD; Antipin, AS, Equilibrium programming using proximal-like algorithms, Mathematical Programming, 78, 29-41, (1997) · Zbl 0890.90150 [18] Korpelevič, GM, An extragradient method for finding saddle points and for other problems, Èkonomika i Matematicheskie Metody, 12, 747-756, (1976) · Zbl 0342.90044 [19] Takahashi, W; Toyoda, M, Weak convergence theorems for nonexpansive mappings and monotone mappings, Journal of Optimization Theory and Applications, 118, 417-428, (2003) · Zbl 1055.47052 [20] Iiduka, H; Takahashi, W, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Analysis: Theory, Methods & Applications, 61, 341-350, (2005) · Zbl 1093.47058 [21] Deutsch, F; Yamada, I, Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings, Numerical Functional Analysis and Optimization, 19, 33-56, (1998) · Zbl 0913.47048 [22] Xu, H-K, Iterative algorithms for nonlinear operators, Journal of the London Mathematical Society, 66, 240-256, (2002) · Zbl 1013.47032 [23] Xu, H-K, An iterative approach to quadratic optimization, Journal of Optimization Theory and Applications, 116, 659-678, (2003) · Zbl 1043.90063 [24] Yamada, I; Butnariu, D (ed.); Censor, Y (ed.); Reich, S (ed.), The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, No. 8, 473-504, (2001), Amsterdam, The Netherlands · Zbl 1013.49005 [25] Marino, G; Xu, H-K, A general iterative method for nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications, 318, 43-52, (2006) · Zbl 1095.47038 [26] Plubtieng, S; Punpaeng, R, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Journal of Mathematical Analysis and Applications, 336, 455-469, (2007) · Zbl 1127.47053 [27] Wangkeeree, R, An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings, No. 2008, 17, (2008) · Zbl 1170.47051 [28] Colao, V; Marino, G; Xu, H-K, An iterative method for finding common solutions of equilibrium and fixed point problems, Journal of Mathematical Analysis and Applications, 344, 340-352, (2008) · Zbl 1141.47040 [29] Opial, Z, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bulletin of the American Mathematical Society, 73, 591-597, (1967) · Zbl 0179.19902 [30] Blum, E; Oettli, W, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63, 123-145, (1994) · Zbl 0888.49007 [31] Shimoji, K; Takahashi, W, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese Journal of Mathematics, 5, 387-404, (2001) · Zbl 0993.47037 [32] Yao, Y; Liou, Y-C; Yao, J-C, Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings, No. 2007, 12, (2007) · Zbl 1153.54024 [33] Suzuki, T, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, Journal of Mathematical Analysis and Applications, 305, 227-239, (2005) · Zbl 1068.47085 [34] Xu, H-K, Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical Analysis and Applications, 298, 279-291, (2004) · Zbl 1061.47060 [35] Qin, X; Shang, M; Su, Y, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Analysis: Theory, Methods & Applications, 69, 3897-3909, (2008) · Zbl 1170.47044
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.