×

An extragradient method for mixed equilibrium problems and fixed point problems. (English) Zbl 1203.47086

The authors investigate the problem of approximating a common element of the set of fixed points of a demicontractive mapping as well as the set of solutions of a mixed equilibrium problem. An extragradient method for solving the mixed equilibrium and fixed point problems are employed and some strong convergence results are established for the iterative algorithm involved.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
49J20 Existence theories for optimal control problems involving partial differential equations
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Blum, E; Oettli, W, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63, 123-145, (1994) · Zbl 0888.49007
[2] Zeng, L-C; Wu, S-Y; Yao, J-C, Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems, Taiwanese Journal of Mathematics, 10, 1497-1514, (2006) · Zbl 1121.49005
[3] Chadli, O; Wong, NC; Yao, J-C, Equilibrium problems with applications to eigenvalue problems, Journal of Optimization Theory and Applications, 117, 245-266, (2003) · Zbl 1141.49306
[4] Chadli, O; Schaible, S; Yao, J-C, Regularized equilibrium problems with application to noncoercive hemivariational inequalities, Journal of Optimization Theory and Applications, 121, 571-596, (2004) · Zbl 1107.91067
[5] Konnov, IV; Schaible, S; Yao, J-C, Combined relaxation method for mixed equilibrium problems, Journal of Optimization Theory and Applications, 126, 309-322, (2005) · Zbl 1110.49028
[6] Combettes, PL; Hirstoaga, SA, Equilibrium programming in Hilbert spaces, Journal of Nonlinear and Convex Analysis, 6, 117-136, (2005) · Zbl 1109.90079
[7] Flåm, SD; Antipin, AS, Equilibrium programming using proximal-like algorithms, Mathematical Programming, 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, Journal of Mathematical Analysis and Applications, 331, 506-515, (2007) · Zbl 1122.47056
[9] Chadli, O; Konnov, IV; Yao, J-C, Descent methods for equilibrium problems in a Banach space, Computers and Mathematics with Applications, 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, Applied Mathematics and Mechanics, 27, 1157-1164, (2006) · Zbl 1199.49010
[11] 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
[12] 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
[13] Tada, A; Takahashi, W; Takahashi, W (ed.); Tanaka, T (ed.), Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, 609-617, (2007), Yokohama, Japan · Zbl 1122.47055
[14] Noor, MA, Fundamentals of equilibrium problems, Mathematical Inequalities & Applications, 9, 529-566, (2006) · Zbl 1099.91072
[15] Yao, Y; Noor, MA; Liou, Y-C, On iterative methods for equilibrium problems, Nonlinear Analysis, 70, 497-509, (2009) · Zbl 1165.49035
[16] Zeng, 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
[17] Yao, Y; Noor, MA; Zainab, S; Liou, Y-C, Mixed equilibrium problems and optimization problems, Journal of Mathematical Analysis and Applications, 354, 319-329, (2009) · Zbl 1160.49013
[18] Yao, Y; Zhou, H; Liou, Y-C, Weak and strong convergence theorems for an asymptotically [inlineequation not available: see fulltext.]-strict pseudocontraction and a mixed equilibrium problem, Journal of the Korean Mathematical Society, 46, 561-576, (2009) · Zbl 1189.49020
[19] Mainge, P-E, Regularized and inertial algorithms for common fixed points of nonlinear operators, Journal of Mathematical Analysis and Applications, 344, 876-887, (2008) · Zbl 1146.47042
[20] Yao, Y; Liou, Y-C; Yao, J-C, An iterative algorithm for approximating convex minimization problem, Applied Mathematics and Computation, 188, 648-656, (2007) · Zbl 1121.65071
[21] Marino, G; Colao, V; Muglia, L; Yao, Y, Krasnoselski-Mann iteration for hierarchical fixed points and equilibrium problem, Bulletin of the Australian Mathematical Society, 79, 187-200, (2009) · Zbl 1165.47050
[22] Mainge, P-E; Moudafi, A, Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems, Journal of Nonlinear and Convex Analysis, 9, 283-294, (2008) · Zbl 1189.90120
[23] Moudafi, A, Weak convergence theorems for nonexpansive mappings and equilibrium problems, Journal of Nonlinear and Convex Analysis, 9, 37-43, (2008) · Zbl 1167.47049
[24] 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
[25] Marino, G; Xu, H-K, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, Journal of Mathematical Analysis and Applications, 329, 336-346, (2007) · Zbl 1116.47053
[26] Zeng, L-C; Wong, N-C; Yao, J-C, Strong convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type, Taiwanese Journal of Mathematics, 10, 837-849, (2006) · Zbl 1159.47054
[27] Zhou, H, Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 68, 2977-2983, (2008) · Zbl 1145.47055
[28] Konnov, IV; Hadjisavvas, N (ed.); Komlosi, S (ed.); Schaible, S (ed.), Generalized monotone equilibrium problems and variational inequalities, (2005), New York, NY, USA · Zbl 1100.49009
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.