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Relaxed extragradient iterative methods for variational inequalities. (English) Zbl 1229.65109

Summary: We suggest and analyze some new relaxed extragradient iterative methods for finding a common element of the solution set of a variational inequality, the solution set of a general system of variational inequalities and the set of fixed points of a strictly pseudo-contractive mapping defined on a real Hilbert space. Strong convergence of the proposed methods under some mild conditions is established.

MSC:

65K15 Numerical methods for variational inequalities and related problems
49J40 Variational inequalities
49M25 Discrete approximations in optimal control
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[1] Bnouhachem, A.; Noor, M.A.; Hao, Z., Some new extragradient iterative methods for variational inequalities, Nonlinear anal., 70, 1321-1329, (2009) · Zbl 1171.47050
[2] Ceng, L.C.; Ansari, Q.H.; Yao, J.C., Viscosity approximation methods for generalized equilibrium problems and fixed point problems, J. global optim., 43, 487-502, (2009) · Zbl 1172.47045
[3] Ceng, L.C.; Huang, S., Modified extragradient methods for strict pseudo-contractions and monotone mappings, Taiwanese J. math., 13, 4, 1197-1211, (2009) · Zbl 1184.47034
[4] Ceng, L.C.; Wang, C.Y.; Yao, J.C., Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. methods oper. res., 67, 375-390, (2008) · Zbl 1147.49007
[5] Ceng, L.C.; Yao, J.C., An extragradient-like approximation method for variational inequality problems and fixed point problems, Appl. math. comput., 190, 205-215, (2007) · Zbl 1124.65056
[6] Ceng, L.C.; Yao, J.C., Relaxed viscosity approximation methods for fixed point problems and variational inequality problems, Nonlinear anal., 69, 3299-3309, (2008) · Zbl 1163.47052
[7] Iiduka, H.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear anal., 61, 341-350, (2005) · Zbl 1093.47058
[8] Korpelevich, G.M., An extragradient method for finding saddle points and for other problems, Ekon. mate. metody, 12, 747-756, (1976) · Zbl 0342.90044
[9] Lions, J.L.; Stampacchia, G., Variational inequalities, Commun. pure appl. math., 20, 493-512, (1967) · Zbl 0152.34601
[10] Marino, G.; Xu, H.K., Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. math. anal. appl., 329, 336-346, (2007) · Zbl 1116.47053
[11] Nadezhkina, N.; Takahashi, W., Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. optim. theory appl., 128, 191-201, (2006) · Zbl 1130.90055
[12] Nadezhkina, N.; Takahashi, W., Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. optim., 16, 4, 1230-1241, (2006) · Zbl 1143.47047
[13] 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
[14] Takahashi, W.; Toyoda, M., Weak convergence theorems for nonexpansive mappings and monotone mappings, J. optim. theory appl., 118, 417-428, (2003) · Zbl 1055.47052
[15] Verma, R.U., On a new system of nonlinear variational inequalities and associated iterative algorithms, Math. sci. res. hot-line, 3, 8, 65-68, (1999) · Zbl 0970.49011
[16] Xu, H.K.; Kim, T.H., Convergence of hybrid steepest-descent methods for variational inequalities, J. optim. theory appl., 119, 1, 185-201, (2003) · Zbl 1045.49018
[17] Yao, Y.; Liou, Y.C.; Kang, S.M., Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method, Comput. math. appl., 59, 3472-3480, (2010) · Zbl 1197.49008
[18] Zeng, L.C.; Wong, N.C.; Yao, J.C., Strong convergence theorems for strictly pseudocontractive mappings of browder – petryshyn type, Taiwanese J. math., 10, 837-849, (2006) · Zbl 1159.47054
[19] 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, 1293-1303, (2006) · Zbl 1110.49013
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