# zbMATH — the first resource for mathematics

A new hybrid algorithm for variational inclusions, generalized equilibrium problems, and a finite family of quasi-nonexpansive mappings. (English) Zbl 1186.47060
Summary: We propose a new iterative scheme for finding common elements of the set of fixed points of a finite family of quasi-nonexpansive mappings, the set of solutions of variational inclusions, and the set of solutions of generalized equilibrium problems. Some strong convergence results are derived by using the concept of $$W$$-mappings for a finite family of quasi-nonexpansive mappings under suitable conditions in Hilbert spaces.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47J22 Variational and other types of inclusions
Full Text:
##### References:
 [1] Mann, WR, Mean value methods in iteration, Proceedings of the American Mathematical Society, 4, 506-510, (1953) · Zbl 0050.11603 [2] Genel, A; Lindenstrauss, J, An example concerning fixed points, Israel Journal of Mathematics, 22, 81-86, (1975) · Zbl 0314.47031 [3] Reich, S, Weak convergence theorems for nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications, 67, 274-276, (1979) · Zbl 0423.47026 [4] Nakajo, K; Takahashi, W, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, Journal of Mathematical Analysis and Applications, 279, 372-379, (2003) · Zbl 1035.47048 [5] Bigi, G; Castellani, M; Kassay, G, A dual view of equilibrium problems, Journal of Mathematical Analysis and Applications, 342, 17-26, (2008) · Zbl 1155.90021 [6] Flores-Bazán, F, Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case, SIAM Journal on Optimization, 11, 675-690, (2000) · Zbl 1002.49013 [7] Peng, J-W; Liou, Y-C; Yao, J-C, An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions, No. 2009, 21, (2009) · Zbl 1163.91463 [8] 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 [9] Blum, E; Oettli, W, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63, 123-145, (1994) · Zbl 0888.49007 [10] Combettes, PL; Hirstoaga, SA, Equilibrium programming in Hilbert spaces, Journal of Nonlinear and Convex Analysis, 6, 117-136, (2005) · Zbl 1109.90079 [11] Flåm, SD; Antipin, AS, Equilibrium programming using proximal-like algorithms, Mathematical Programming, 78, 29-41, (1997) · Zbl 0890.90150 [12] Iusem, AN; Sosa, W, Iterative algorithms for equilibrium problems, Optimization, 52, 301-316, (2003) · Zbl 1176.90640 [13] Tada, A; Takahashi, W, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, Journal of Optimization Theory and Applications, 133, 359-370, (2007) · Zbl 1147.47052 [14] Bŕezis, H, Operateur maximaux monotones, No. 5, (1973), Amsterdam, The Netherlands [15] Lemaire, B, Which fixed point does the iteration method select?, No. 452, 154-167, (1997), Berlin, Germany · Zbl 0882.65042 [16] Agarwal, RP; Cho, YJ; Huang, N-J, Sensitivity analysis for strongly nonlinear quasi-variational inclusions, Applied Mathematics Letters, 13, 19-24, (2000) · Zbl 0960.47035 [17] Ceng, L-C; Ansari, QH; Yao, J-C, On relaxed viscosity iterative methods for variational inequalities in Banach spaces, Journal of Computational and Applied Mathematics, 230, 813-822, (2009) · Zbl 1178.65074 [18] Ceng, L-C; Ansari, QH; Yao, J-C, Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces, Numerical Functional Analysis and Optimization, 29, 987-1033, (2008) · Zbl 1163.49002 [19] Ceng, LC; Chen, GY; Huang, XX; Yao, J-C, Existence theorems for generalized vector variational inequalities with pseudomonotonicity and their applications, Taiwanese Journal of Mathematics, 12, 151-172, (2008) · Zbl 1148.49004 [20] Ceng, L-C; Lee, C; Yao, J-C, Strong weak convergence theorems of implicit hybrid steepest-descent methods for variational inequalities, Taiwanese Journal of Mathematics, 12, 227-244, (2008) · Zbl 1148.49005 [21] 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 [22] Ceng, L-C; Xu, H-K; Yao, J-C, A hybrid steepest-descent method for variational inequalities in Hilbert spaces, Applicable Analysis, 87, 575-589, (2008) · Zbl 1158.47050 [23] Zeng, LC; Schaible, S; Yao, J-C, Hybrid steepest descent methods for zeros of nonlinear operators with applications to variational inequalities, Journal of Optimization Theory and Applications, 141, 75-91, (2009) · Zbl 1178.47050 [24] 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 [25] Chang, SS, Set-valued variational inclusions in Banach spaces, Journal of Mathematical Analysis and Applications, 248, 438-454, (2000) · Zbl 1031.49018 [26] Cholamjiak, P, A hybrid iterative scheme for equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces, No. 2009, 18, (2009) · Zbl 1167.65379 [27] Ding, XP, Perturbed Ishikawa type iterative algorithm for generalized quasivariational inclusions, Applied Mathematics and Computation, 141, 359-373, (2003) · Zbl 1030.65071 [28] Fang, Y-P; Huang, N-J, [inlineequation not available: see fulltext.]-monotone operator and resolvent operator technique for variational inclusions, Applied Mathematics and Computation, 145, 795-803, (2003) · Zbl 1030.49008 [29] Kangtunyakarn, A; Suantai, S, A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings, Nonlinear Analysis: Theory, Methods & Applications, 71, 4448-4460, (2009) · Zbl 1167.47304 [30] Kangtunyakarn, A; Suantai, S, Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings, Nonlinear Analysis: Hybrid Systems, 3, 296-309, (2009) · Zbl 1226.47076 [31] Kumam, P, A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping, Nonlinear Analysis: Hybrid Systems, 2, 1245-1255, (2008) · Zbl 1163.49003 [32] Nilsrakoo, W; Saejung, S, Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications, Journal of Mathematical Analysis and Applications, 356, 154-167, (2009) · Zbl 1162.47051 [33] Peng, J-W; Wang, Y; Shyu, DS; Yao, J-C, Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems, No. 2008, 15, (2008) · Zbl 1161.65050 [34] 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 [35] 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 [36] Petruşel, A; Yao, J-C, An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems, Central European Journal of Mathematics, 7, 335-347, (2009) · Zbl 1195.49017 [37] Plubtieng, S; Sriprad, W, A viscosity approximation method for finding common solutions of variational inclusions, equilibrium problems, and fixed point problems in Hilbert spaces, No. 2009, 20, (2009) · Zbl 1186.47075 [38] Schaible, S; Yao, J-C; Zeng, L-C, A proximal method for pseudomonotone type variational-like inequalities, Taiwanese Journal of Mathematics, 10, 497-513, (2006) · Zbl 1116.49004 [39] Verma, RU, [inlineequation not available: see fulltext.]-monotonicity and applications to nonlinear variational inclusion problems, Journal of Applied Mathematics and Stochastic Analysis, 2004, 193-195, (2004) · Zbl 1064.49012 [40] Zeng, L-C; Guu, SM; Yao, J-C, Hybrid approximate proximal point algorithms for variational inequalities in Banach spaces, No. 2009, 17, (2009) · Zbl 1173.49009 [41] Zeng, LC; Lin, LJ; Yao, J-C, Auxiliary problem method for mixed variational-like inequalities, Taiwanese Journal of Mathematics, 10, 515-529, (2006) · Zbl 1259.49010 [42] Zeng, LC; Yao, J-C, A hybrid extragradient method for general variational inequalities, Mathematical Methods of Operations Research, 69, 141-158, (2009) · Zbl 1169.65070 [43] Zhang, S-S; Lee, JH; Chan, CK, Algorithms of common solutions to quasi variational inclusion and fixed point problems, Applied Mathematics and Mechanics, 29, 571-581, (2008) · Zbl 1196.47047 [44] 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 [45] Martinez-Yanes, C; Xu, H-K, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Analysis: Theory, Methods & Applications, 64, 2400-2411, (2006) · Zbl 1105.47060 [46] Takahashi W: Nonlinear Functional Analysis. Yokohama, Yokohama, Japan; 2000. · Zbl 0997.47002 [47] Atsushiba, S; Takahashi, W, Strong convergence theorems for a finite family of nonexpansive mappings and applications, Indian Journal of Mathematics, 41, 435-453, (1999) · Zbl 1055.47514 [48] 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 [49] Takahashi, W, Weak and strong convergence theorems for families of nonexpansive mappings and their applications, Annales Universitatis Mariae Curie-Skłodowska. Sectio A, 51, 277-292, (1997) · Zbl 1012.47029 [50] Takahashi W: Convex Analysis and Approximation of Fixed Points. Volume 2. Yokohama, Yokohama, Japan; 2000. · Zbl 1089.49500
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.