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Some new extragradient iterative methods for variational inequalities. (English) Zbl 1171.47050
From the summary: We suggest and analyze some new extragradient iterative methods for finding the common element of the fixed points of a nonexpansive mapping and the solution set of the variational inequality for an inverse strongly monotone mapping in a Hilbert space. We also consider the strong convergence of the proposed method under some mild conditions. Several special cases are also discussed.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J40 Variational inequalities
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[1] Auslender, A., Optimization méthodes numériques, (1976), Mason Paris · Zbl 0326.90057
[2] Bnouhachem, A.; Aslam Noor, M., Numerical comparison between prediction-correction methods for general variational inequalities, Appl. math. comput., 186, 496-505, (2007) · Zbl 1119.65056
[3] Giannessi, F.; Maugeri, A.; Pardalos, P.M., Equilibrium problems: nonsmooth optimization and variational inequality models, (2001), Kluwer Academic Press Dordrecht, Holland · Zbl 0979.00025
[4] Glowinski, R.; Lions, J.L.; Tremoliers, R., Numerical analysis of variational inequalities, (1981), North-Holland Amsterdam, Holland
[5] Goebel, K.; Kirk, W.A., Topics on metric fixed-point theory, (1990), Cambridge University Press Cambridge, England · Zbl 0708.47031
[6] Harker, P.T.; Pang, J.S., Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, Algorithms appl. math. program., 48, 161-220, (1990) · Zbl 0734.90098
[7] He, B.S.; Liao, L.Z., Improvement of some projection methods for monotone variational inequalities, J. optim. theory appl., 112, 111-128, (2002) · Zbl 1025.65036
[8] He, B.S.; Yang, Z.H.; Yuan, X.M., An approximate proximal-extragradient type method for monotone variational inequalities, J. math. anal. appl., 300, 2, 362-374, (2004) · Zbl 1068.65087
[9] Iiduka, H.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear anal., 61, 341-350, (2005) · Zbl 1093.47058
[10] Kinderlehrer, D.; Stampacchia, G., An introduction to variational inequalities and their applications, (2000), SIAM Philadelphia · Zbl 0988.49003
[11] Korpelevich, G.M., The extragradient method for finding saddle points and other problems, Matecon, 12, 747-756, (1976) · Zbl 0342.90044
[12] Lions, J.L.; Stampacchia, G., Variational inequalities, Comm. pure appl. math., 20, 493-512, (1967) · Zbl 0152.34601
[13] Marino, G.; Xu, H.K., Convergence of generalized proximal point algorithms, Comm. pure appl. anal., 3, 791-808, (2004) · Zbl 1095.90115
[14] Nadezhkina, N.; Takahashi, W., Weak convergence theorem by an extragradient method for nonexpansive and monotone mappings, J. optim. theory appl., 128, 191-201, (2006) · Zbl 1130.90055
[15] Aslam Noor, M., New extragradient-type methods for general variational inequalities, J. math. anal. appl., 277, 379-395, (2003) · Zbl 1033.49015
[16] Aslam Noor, M., Some developments in general variational inequalities, Appl. math. comput., 152, 199-277, (2004) · Zbl 1134.49304
[17] Aslam Noor, M.; Bnouhachem, A., On an iterative algorithm for general variational inequalities, Appl. math. comput., 185, 155-168, (2007) · Zbl 1119.65058
[18] Opial, Z., Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. amer. math. soc., 73, 591-597, (1967) · Zbl 0179.19902
[19] Patriksson, M., Nonlinear programming and variational inequality problems: A unified approach, (1999), Kluwer Academic Publishers Dordrecht, Holland · Zbl 0913.65058
[20] Rockafellar, R.T., On the maximality of sums nonlinear monotone operators, SIAM trans. amer. math. soc., 149, 75-88, (1970) · Zbl 0222.47017
[21] Stampacchia, G., Formes bilineaires coercitivies sur LES ensembles convexes, C. R. acad. sci. Paris, 258, 4413-4416, (1964) · Zbl 0124.06401
[22] Suzuki, T., Strong convergence of Krasnoselskii and mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. math. anal. appl., 305, 227-239, (2005) · Zbl 1068.47085
[23] Takahashi, W.; Toyoda, M., Weak convergence theorems for nonexpansive mapping, J. optim. theory appl., 118, 417-428, (2003) · Zbl 1055.47052
[24] Zeng, L.C.; Yao, J.C., Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J. math., 10, 5, 1293-1303, (2006) · Zbl 1110.49013
[25] Xu, H.K., Viscosity approximation methods for nonexpansive mappings, J. math. anal. appl., 298, 279-291, (2004) · Zbl 1061.47060
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