×

Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method. (English) Zbl 1197.49008

Summary: In this paper, we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real Hilbert space. Furthermore, we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters.

MSC:

49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47H10 Fixed-point theorems
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
65J15 Numerical solutions to equations with nonlinear operators
65K15 Numerical methods for variational inequalities and related problems
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Browder, F. E.; Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20, 197-228 (1967) · Zbl 0153.45701
[2] Liu, F.; Nashed, M. Z., Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal., 6, 313-344 (1998) · Zbl 0924.49009
[3] Yao, J. C., Variational inequalities with generalized monotone operators, Math. Oper. Res., 19, 691-705 (1994) · Zbl 0813.49010
[4] Zeng, L. C.; Schaible, S.; Yao, J. C., Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities, J. Optim. Theory Appl., 124, 725-738 (2005) · Zbl 1067.49007
[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] Noor, M. Aslam, Some developments in general variational inequalities, Appl. Math. Comput., 191-277 (2004) · Zbl 1134.49304
[7] Censor, Y.; Iusem, A. N.; Zenios, S. A., An interior point method with Bregman functions for the variational inequality problem with paramonotone operators, Math. Program., 81, 373-400 (1998) · Zbl 0919.90123
[8] Takahashi, W.; Toyoda, M., Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118, 417-428 (2003) · Zbl 1055.47052
[9] 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
[10] 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
[11] Korpelevich, G. M., An extragradient method for finding saddle points and for other problems, Ekonom. i Mat. Metody, 12, 747-756 (1976) · Zbl 0342.90044
[12] Yao, Y.; Yao, J. C., On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput., 186, 1551-1558 (2007) · Zbl 1121.65064
[13] Verma, R. U., On a new system of nonlinear variational inequalities and associated iterative algorithms, Math. Sci. Res. Hot-Line, 3, 65-68 (1999) · Zbl 0970.49011
[14] Verma, R. U., Iterative algorithms and a new system of nonlinear quasivariational inequalities, Adv. Nonlinear Var. Inequal., 4, 117-124 (2001) · Zbl 1014.47050
[15] Ceng, L. C.; Wang, C.; 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
[16] Acedo, G. L.; Xu, H. K., Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 67, 2258-2271 (2007) · Zbl 1133.47050
[17] 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
[18] Xu, H. K., Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.