In this paper, we introduce two iterative algorithms based on the relaxed extragradient-like method for finding a common element of the solution set of a general system of generalized mixed equilibria and the fixed point set of a strictly pseudocontractive mapping in a real Hilbert space. We will prove the weak convergence of the iterative algorithm under some mild conditions, while the strong convergence is obtained under some more restrictive conditions.
References
F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228. 0153.45701 10.1016/0022-247X(67)90085-6F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228. 0153.45701 10.1016/0022-247X(67)90085-6
F. Liu and M. Z. Nashed, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal., 6 (1998), 313-344. MR1690160 0924.49009 10.1023/A:1008643727926F. Liu and M. Z. Nashed, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal., 6 (1998), 313-344. MR1690160 0924.49009 10.1023/A:1008643727926
J. C. Yao, Variational inequalities with generalized monotone operators, Math. Oper. Res., 19 (1994), 691-705. 0813.49010 10.1287/moor.19.3.691J. C. Yao, Variational inequalities with generalized monotone operators, Math. Oper. Res., 19 (1994), 691-705. 0813.49010 10.1287/moor.19.3.691
L. C. Zeng, S. Schaible and J. C. Yao, Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities, J. Optim. Theory Appl., 124 (2005), 725-738. 1067.49007 10.1007/s10957-004-1182-zL. C. Zeng, S. Schaible and J. C. Yao, Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities, J. Optim. Theory Appl., 124 (2005), 725-738. 1067.49007 10.1007/s10957-004-1182-z
L. C. Ceng and J. C. Yao, An extragradient-like approximation method for variational inequality problems and fixed point problems, Appl. Math. Comput., 190 (2007), 205-215. MR2335441 1124.65056 10.1016/j.amc.2007.01.021L. C. Ceng and J. C. Yao, An extragradient-like approximation method for variational inequality problems and fixed point problems, Appl. Math. Comput., 190 (2007), 205-215. MR2335441 1124.65056 10.1016/j.amc.2007.01.021
B. T. Kien, M. M. Wong, N. C. Wong and J. C. Yao, Solution existence of variational inequalities with pseudomonotone operators in the sense of Brézis, Journal of Optimization Theory and Applications, 140 (2009), 249-263. MR2472208 10.1007/s10957-008-9446-7B. T. Kien, M. M. Wong, N. C. Wong and J. C. Yao, Solution existence of variational inequalities with pseudomonotone operators in the sense of Brézis, Journal of Optimization Theory and Applications, 140 (2009), 249-263. MR2472208 10.1007/s10957-008-9446-7
Y. Censor, A. N. Iusem and S. A. Zenios, An interior point method with Bregman functions for the variational inequality problem with paramonotone operators, Math. Program., 81 (1998), 373-400. 0919.90123Y. Censor, A. N. Iusem and S. A. Zenios, An interior point method with Bregman functions for the variational inequality problem with paramonotone operators, Math. Program., 81 (1998), 373-400. 0919.90123
W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428. 1055.47052 10.1023/A:1025407607560W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428. 1055.47052 10.1023/A:1025407607560
N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201. 1130.90055 10.1007/s10957-005-7564-zN. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201. 1130.90055 10.1007/s10957-005-7564-z
L. C. Zeng and J. C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J. Math., 10 (2006), 1293-1303. MR2253379 1110.49013 10.11650/twjm/1500557303 euclid.twjm/1500557303
L. C. Zeng and J. C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J. Math., 10 (2006), 1293-1303. MR2253379 1110.49013 10.11650/twjm/1500557303 euclid.twjm/1500557303
G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekonom. i Mat. Metody, 12 (1976), 747-756. 0342.90044G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekonom. i Mat. Metody, 12 (1976), 747-756. 0342.90044
Y. Yao and J. C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput., 186 (2007), 1551-1558. 1121.65064Y. Yao and J. C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput., 186 (2007), 1551-1558. 1121.65064
R. U. Verma, On a new system of nonlinear variational inequalities and associated iterative algorithms, Math. Sci. Res. Hot-Line, 3 (1999), 65-68. 0970.49011R. U. Verma, On a new system of nonlinear variational inequalities and associated iterative algorithms, Math. Sci. Res. Hot-Line, 3 (1999), 65-68. 0970.49011
R. U. Verma, Iterative algorithms and a new system of nonlinear quasivariational inequalities, Adv. Nonlinear Var. Inequal., 4 (2001), 117-124. 1014.47050R. U. Verma, Iterative algorithms and a new system of nonlinear quasivariational inequalities, Adv. Nonlinear Var. Inequal., 4 (2001), 117-124. 1014.47050
L. C. Ceng, C. Y. Wang and J. C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res., 67 (2008), 375-390. 1147.49007 10.1007/s00186-007-0207-4L. C. Ceng, C. Y. Wang and J. C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res., 67 (2008), 375-390. 1147.49007 10.1007/s00186-007-0207-4
G. L. Acedo and H. K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 67 (2007), 2258-2271. MR2331876 1133.47050 10.1016/j.na.2006.08.036G. L. Acedo and H. K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 67 (2007), 2258-2271. MR2331876 1133.47050 10.1016/j.na.2006.08.036
T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227-239. 1068.47085 10.1016/j.jmaa.2004.11.017T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227-239. 1068.47085 10.1016/j.jmaa.2004.11.017
