×

Strong convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities. (English) Zbl 1187.47048

Summary: We introduce a new iterative scheme for finding the common element of the set of solutions of generalized equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of variational inequality problems for a relaxed \((u,v)\)-cocoercive and \(\xi \)-Lipschitz continuous mapping in a real Hilbert space. Then, we prove the strong convergence to a common element of the above three sets under some suitable conditions. Our result can be considered as an improvement and refinement of previously known results.

MSC:

47J25 Iterative procedures involving nonlinear operators
49J40 Variational inequalities
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Stampacchia, G, Formes bilinéaires coercitives sur LES ensembles convexes, Comptes Rendus Academy of Sciences, 258, 4413-4416, (1964) · Zbl 0124.06401
[2] Gabay, D; Fortin, M (ed.); Glowinski, R (ed.), Applications of the method of multipliers to variational inequalities, 299-331, (1983), The Netherlands
[3] Noor, MA; Oettli, W, On general nonlinear complementarity problems and quasi-equilibria, Le Matematiche, 49, 313-331, (1994) · Zbl 0839.90124
[4] Blum, E; Oettli, W, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63, 123-145, (1994) · Zbl 0888.49007
[5] Flåm, SD; Antipin, AS, Equilibrium programming using proximal-like algorithms, Mathematical Programming, 78, 29-41, (1997) · Zbl 0890.90150
[6] Qin, X; Shang, M; Su, Y, Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems, Mathematical and Computer Modelling, 48, 1033-1046, (2008) · Zbl 1187.65058
[7] 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
[8] Takahashi W: Nonlinear Functional Analysis, Fixed Point Theory and Its Applications. Yokohama, Yokohama, Japan; 2000:iv+276. · Zbl 0997.47002
[9] Verma, RU, Generalized system for relaxed cocoercive variational inequalities and projection methods, Journal of Optimization Theory and Applications, 121, 203-210, (2004) · Zbl 1056.49017
[10] Verma, RU, General convergence analysis for two-step projection methods and applications to variational problems, Applied Mathematics Letters, 18, 1286-1292, (2005) · Zbl 1099.47054
[11] Rockafellar, RT, On the maximality of sums of nonlinear monotone operators, Transactions of the American Mathematical Society, 149, 75-88, (1970) · Zbl 0222.47017
[12] 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
[13] Moudafi, A; Théra, M, Proximal and dynamical approaches to equilibrium problems, No. 477, 187-201, (1999), Berlin, Germany · Zbl 0944.65080
[14] Qin, X; Cho, YJ; Kang, SM, Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications, Nonlinear Analysis: Theory, Methods & Applications, 72, 99-112, (2010) · Zbl 1225.47106
[15] Cho, YJ; Qin, X; Kang, SM, Some results for equilibrium problems and fixed point problems in Hilbert spaces, Journal of Computational Analysis and Applications, 11, 294-316, (2009) · Zbl 1223.47075
[16] Cho, YJ; Qin, X; Kang, JI, Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems, Nonlinear Analysis: Theory, Methods & Applications, 71, 4203-4214, (2009) · Zbl 1219.47105
[17] Hu, CS; Cai, G, Viscosity approximation schemes for fixed point problems and equilibrium problems and variational inequality problems, Nonlinear Analysis: Theory, Methods & Applications, 72, 1792-1808, (2010) · Zbl 1226.47072
[18] Huang, N-J; Lan, H-Y; Teo, KL, On the existence and convergence of approximate solutions for equilibrium problems in Banach spaces, No. 2007, 14, (2007)
[19] Jaiboon C, Kumam P: Strong convergence theorems for solving equilibrium problems and fixed point problems of -strict pseudo-contraction mappings by two hybrid projection methods. Journal of Computational and Applied Mathematics. In press · Zbl 1191.65065
[20] Jaiboon, C; Kumam, P, A hybrid extragradient viscosity approximation method for solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings, No. 2009, 32, (2009) · Zbl 1186.47065
[21] 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
[22] 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
[23] Liu, Q-Y; Zeng, W-Y; Huang, N-J, An iterative method for generalized equilibrium problems, fixed point problems and variational inequality problems, No. 2009, 20, (2009) · Zbl 1186.47068
[24] Moudafi, A, Weak convergence theorems for nonexpansive mappings and equilibrium problems, Journal of Nonlinear and Convex Analysis, 9, 37-43, (2008) · Zbl 1167.47049
[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, No. 2008, 15, (2008) · Zbl 1161.65050
[26] 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
[27] 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
[28] Zeng, W-Y; Huang, N-J; Zhao, C-W, Viscosity approximation methods for generalized mixed equilibrium problems and fixed points of a sequence of nonexpansive mappings, No. 2008, 15, (2008) · Zbl 1219.47136
[29] Combettes, PL; Hirstoaga, SA, Equilibrium programming in Hilbert spaces, Journal of Nonlinear and Convex Analysis, 6, 117-136, (2005) · Zbl 1109.90079
[30] 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
[31] Shang, M; Su, Y; Qin, X, Strong convergence theorem for nonexpansive mappings and relaxed cocoercive mappings, International Journal of Applied Mathematics and Mechanics, 3, 24-34, (2007)
[32] 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
[33] 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
[34] Xu, H-K, Iterative algorithms for nonlinear operators, Journal of the London Mathematical Society, 66, 240-256, (2002) · Zbl 1013.47032
[35] Xu, HK, An iterative approach to quadratic optimization, Journal of Optimization Theory and Applications, 116, 659-678, (2003) · Zbl 1043.90063
[36] 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
[37] Chang, S-S; Lee, HWJ; Chan, CK, A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization, Nonlinear Analysis: Theory, Methods & Applications, 70, 3307-3319, (2009) · Zbl 1198.47082
[38] Yao, Y; Noor, MA; Liou, Y-C, On iterative methods for equilibrium problems, Nonlinear Analysis: Theory, Methods & Applications, 70, 497-509, (2009) · Zbl 1165.49035
[39] Cho, YJ; Qin, X, Generalized systems for relaxed cocoercive variational inequalities and projection methods in Hilbert spaces, Mathematical Inequalities & Applications, 12, 365-375, (2009) · Zbl 1160.47051
[40] 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
[41] 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
[42] 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
[43] Xu, H-K, Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical Analysis and Applications, 298, 279-291, (2004) · Zbl 1061.47060
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.