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Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. (English) Zbl 1147.49007
Summary: In this paper, we introduce and study a relaxed extragradient method for finding solutions of a general system of variational inequalities with inverse-strongly monotone mappings in a real Hilbert space. First, this system of variational inequalities is proven to be equivalent to a fixed point problem of nonexpansive mapping. Second, by using the demi-closedness principle for nonexpansive mappings, we prove that under quite mild conditions the iterative sequence defined by the relaxed extragradient method converges strongly to a solution of this system of variational inequalities. In addition, utilizing this result, we provide some applications of the considered problem not just giving a pure extension of existing mathematical problems.

49J40Variational methods including variational inequalities
47H05Monotone operators (with respect to duality) and generalizations
47H10Fixed point theorems for nonlinear operators on topological linear spaces
[1]Browder FE, Petryshyn WV (1967) Construction of fixed points of nonlinear mappings in Hilbert Spaces. J Math Anal Appl 20: 197–228 · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[2]Goebel K, Kirk WA (1990) Topics on metric fixed-point theory. Cambridge University Press, Cambridge
[3]Korpelevich GM (1976) An extragradient method for finding saddle points and for other problems. Ekon Mate Metody 12: 747–756
[4]Liu F, Nashed MZ (1998) Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Value Analy 6: 313–344 · Zbl 0924.49009 · doi:10.1023/A:1008643727926
[5]Nadezhkina N, Takahashi W (2006) Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J Optim Theory Appl 128: 191–201 · Zbl 1130.90055 · doi:10.1007/s10957-005-7564-z
[6]Osilike MO, Igbokwe DI (2000) Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Comput Math Appl 40: 559–567 · Zbl 0958.47030 · doi:10.1016/S0898-1221(00)00179-6
[7]Suzuki T (2005) Strong convergence of krasnoselskii and mann’s type sequences for one-parameter nonexpansive semigroups without bochner integrals. J Math Anal Appl 305: 227–239 · Zbl 1068.47085 · doi:10.1016/j.jmaa.2004.11.017
[8]Takahashi W, Toyoda M (2003) Weak convergence theorems for nonexpansive mappings and monotone mappings. J Optim Theory Appl 118: 417–428 · Zbl 1055.47052 · doi:10.1023/A:1025407607560
[9]Verma RU (1999) On a new system of nonlinear variational inequalities and associated iterative algorithms. Math Sci Res, Hot-Line 3(8): 65–68
[10]Verma RU (2001) Iterative algorithms and a new system of nonlinear quasivariational inequalities. Adv Nonlinear Var Inequal 4(1): 117–124
[11]Xu HK (2004) Viscosity approximation methods for nonexpansive mappings. J Math Anal Appl 298: 279–291 · Zbl 1061.47060 · doi:10.1016/j.jmaa.2004.04.059
[12]Yao JC (1994) Variational inequalities and generalized monotone operators. Math Opera Res 19: 691–705 · Zbl 0813.49010 · doi:10.1287/moor.19.3.691
[13]Yao JC, Chadli O (2005) Pseudomonotone complementarity problems and variational inequalities. In: Crouzeix JP, Haddjissas N, Schaible S (eds) Handbook of generalized convexity and monotonicity. pp 501–558
[14]Yao Y, Yao JC (2007) On modified iterative method for nonexpansive mappings and monotone mappings. Appl Math Comput 186: 1551–1558 · Zbl 1121.65064 · doi:10.1016/j.amc.2006.08.062
[15]Zeng LC, Yao JC (2006) Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan J Math 10: 1293–1303
[16]Zeng LC, Schaible S, Yao JC (2005) Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities. J Optim Theory Appl 124: 725–738 · Zbl 1067.49007 · doi:10.1007/s10957-004-1182-z
[17]Zeng LC, Wong NC, Yao JC (2006) Strong convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type. Taiwan J Math 10(4): 837–849