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A new hybrid algorithm for a system of mixed equilibrium problems, fixed point problems for nonexpansive semigroup, and variational inclusion problem. (English) Zbl 1216.47097
Summary: The purpose of this paper is to consider a shrinking projection method for finding the common element of the set of common fixed points for nonexpansive semigroups, the set of common fixed points for an infinite family of a $$\xi$$-strict pseudocontractions, the set of solutions of a system of mixed equilibrium problems, and the set of solutions of a variational inclusion problem. Strong convergence of the sequences generated by the proposed iterative scheme is obtained. The results presented in this paper extend and improve some well-known results in the literature.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H20 Semigroups of nonlinear operators 47J22 Variational and other types of inclusions
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##### References:
 [1] 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 [2] Combettes, PL; Hirstoaga, SA, Equilibrium programming in Hilbert spaces, Journal of Nonlinear and Convex Analysis, 6, 117-136, (2005) · Zbl 1109.90079 [3] Blum, E; Oettli, W, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63, 123-145, (1994) · Zbl 0888.49007 [4] Chadli, O; Wong, NC; Yao, JC, Equilibrium problems with applications to eigenvalue problems, Journal of Optimization Theory and Applications, 117, 245-266, (2003) · Zbl 1141.49306 [5] Chadli, O; Schaible, S; Yao, JC, Regularized equilibrium problems with application to noncoercive hemivariational inequalities, Journal of Optimization Theory and Applications, 121, 571-596, (2004) · Zbl 1107.91067 [6] Konnov, IV; Schaible, S; Yao, JC, Combined relaxation method for mixed equilibrium problems, Journal of Optimization Theory and Applications, 126, 309-322, (2005) · Zbl 1110.49028 [7] Moudafi, A; Théra, M, Proximal and dynamical approaches to equilibrium problems, No. 477, 187-201, (1999), Berlin, Germany · Zbl 0944.65080 [8] 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 [9] Jitpeera, T; Kumam, P, An extra gradient type method for a system of equilibrium problems, variational inequality problems and fixed points of finitely many nonexpansive mappings, Journal of Nonlinear Analysis and Optimization: Theory & Applications, 1, 71-91, (2010) · Zbl 1413.47112 [10] Cianciaruso, F; Marino, G; Muglia, L; Yao, Y, A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem, (2010) · Zbl 1203.47043 [11] Cholamjiak, P; Suantai, S, A new hybrid algorithm for variational inclusions, generalized equilibrium problems, and a finite family of quasi-nonexpansive mappings, (2009) · Zbl 1186.47060 [12] Jaiboon, C; Kumam, P, Strong convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities, (2010) · Zbl 1187.47048 [13] Jaiboon, C; Chantarangsi, W; Kumam, P, A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings, Nonlinear Analysis. Hybrid Systems, 4, 199-215, (2010) · Zbl 1179.49011 [14] Kumam, P; Jaiboon, C, A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems, Nonlinear Analysis. Hybrid Systems, 3, 510-530, (2009) · Zbl 1221.49010 [15] Shehu, Y, Fixed point solutions of variational inequality and generalized equilibrium problems with applications, Annali dell’Universita di Ferrara, 56, 345-368, (2010) · Zbl 1206.47085 [16] Yao, Y; Liou, Y-C; Yao, J-C, A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems, (2008) · Zbl 1203.47087 [17] Yao, Y; Liou, Y-C; Wu, Y-J, An extragradient method for mixed equilibrium problems and fixed point problems, (2009) [18] Hao, Y, On variational inclusion and common fixed point problems in Hilbert spaces with applications, Applied Mathematics and Computation, 217, 3000-3010, (2010) · Zbl 1227.65047 [19] Ansari, QH; Yao, JC, Iterative schemes for solving mixed variational-like inequalities, Journal of Optimization Theory and Applications, 108, 527-541, (2001) · Zbl 0999.49008 [20] Brézis, H, Opérateur maximaux monotones, No. 5, (1973), Amsterdam, The Netherlands [21] Lemaire, B, Which fixed point does the iteration method select?, No. 452, 154-167, (1997), Berlin, Germany · Zbl 0882.65042 [22] Takahashi, W; Takeuchi, Y; Kubota, R, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications, 341, 276-286, (2008) · Zbl 1134.47052 [23] Takahashi, S; Takahashi, W, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Analysis. Theory, Methods & Applications, 69, 1025-1033, (2008) · Zbl 1142.47350 [24] Zhang, S-S; Lee, JHW; Chan, CK, Algorithms of common solutions to quasi variational inclusion and fixed point problems, Applied Mathematics and Mechanics. English Edition, 29, 571-581, (2008) · Zbl 1196.47047 [25] 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, (2008) · Zbl 1161.65050 [26] Saeidi, S, Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings, Nonlinear Analysis. Theory, Methods & Applications, 70, 4195-4208, (2009) · Zbl 1225.47110 [27] Katchang, P; Kumam, P, A general iterative method of fixed points for mixed equilibrium problems and variational inclusion problems, (2010) · Zbl 1189.47066 [28] Kumam, W; Jaiboon, C; Kumam, P; Singta, A, A shrinking projection method for generalized mixed equilibrium problems, variational inclusion problems and a finite family of quasi-nonexpansive mappings, (2010) · Zbl 1206.47077 [29] Liu, M; Chang, SS; Zuo, P, An algorithm for finding a common solution for a system of mixed equilibrium problem, quasivariational inclusion problem, and fixed point problem of nonexpansive semigroup, (2010) · Zbl 1206.47078 [30] Jitpeera T, Kumam P: A new hybrid algorithm for a system of equilibrium problems and variational inclusion.Annali dell’Universita di Ferrara. In press · Zbl 1368.47064 [31] Acedo, GL; Xu, H-K, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Analysis. Theory, Methods & Applications, 67, 2258-2271, (2007) · Zbl 1133.47050 [32] 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 [33] Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan; 2000:iv+276. · Zbl 0997.47002 [34] 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 [35] Zhou, Haiyun, Convergence theorems of fixed points for [inlineequation not available: see fulltext.]-strict pseudo-contractions in Hilbert spaces, Nonlinear Analysis. Theory, Methods & Applications, 69, 456-462, (2008) · Zbl 1220.47139 [36] 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 [37] Chang SS: Variational Inequalities and Related Problems. Chongqing Publishing House, China; 2007. [38] Shimizu, T; Takahashi, W, Strong convergence to common fixed points of families of nonexpansive mappings, Journal of Mathematical Analysis and Applications, 211, 71-83, (1997) · Zbl 0883.47075 [39] Tan, K-K; Xu, HK, The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces, Proceedings of the American Mathematical Society, 114, 399-404, (1992) · Zbl 0781.47045
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