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Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. (English) Zbl 1198.47081
Summary: The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points F(S) of a nonexpansive mapping S and the set of solutions Ω A of the variational inequality for a monotone, Lipschitz continuous mapping A. We introduce a hybrid extragradient-like approximation method which is based on the well-known extragradient method and a hybrid (or outer approximation) method. The method produces three sequences which are shown to converge strongly to the same common element of F(S)Ω A . As applications, the method provides an algorithm for finding the common fixed point of a nonexpansive mapping and a pseudocontractive mapping, or a common zero of a monotone Lipschitz continuous mapping and a maximal monotone mapping.
MSC:
47J25Iterative procedures (nonlinear operator equations)
47J20Inequalities involving nonlinear operators
47H09Mappings defined by “shrinking” properties
References:
[1]Antipin A.S.: Methods for solving variational inequalities with related constraints. Comput. Math. Math. Phys. 40, 1239–1254 (2000)
[2]Antipin A.S., Vasiliev F.P.: Regularized prediction method for solving variational inequalities with an inexactly given set. Comput. Math. Math. Phys. 44, 750–758 (2004)
[3]Browder F.E.: Existence of periodic solutions for nonlinear equations of evolution. Proc. Nat. Acad. Sc. USA 55, 1100–1103 (1965) · Zbl 0135.17601 · doi:10.1073/pnas.53.5.1100
[4]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 · doi:10.1016/0022-247X(67)90085-6
[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 · doi:10.1016/j.amc.2007.01.021
[6]Ceng L.C., Yao J.C.: On the convergence analysis of inexact hybrid extragradient proximal point algorithms for maximal monotone operators. J. Comput. Appl. Math. 217, 326–338 (2007) · Zbl 1144.47048 · doi:10.1016/j.cam.2007.02.010
[7]Geobel K., Kirk W.A.: Topics on Metric Fixed-point Theory. Cambridge University Press, Cambridge, England (1990)
[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, 362–374 (2004) · Zbl 1068.65087 · doi:10.1016/j.jmaa.2004.04.068
[9]Hu S., Papageorgiou N.S.: Handbook of multivalued analysis, vol. I: theory. Kluwer Academic Publishers, Dordrecht (1997)
[10]Iiduka H., Takahashi W.: Strong convergence theorem by a hybrid method for nonlinear mappings of nonexpansive and monotone type and applications. Adv. Nonlinear Var. Inequal. 9, 1–10 (2006)
[11]Korpelevich G.M.: The extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976)
[12]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 · doi:10.1023/A:1008643727926
[13]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 · doi:10.1007/s10957-005-7564-z
[14]Nadezhkina N., Takahashi W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230–1241 (2006) · Zbl 1143.47047 · doi:10.1137/050624315
[15]Nakajo K., Takahashi W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003) · Zbl 1035.47048 · doi:10.1016/S0022-247X(02)00458-4
[16]Opial Z.: Weak convergence of the sequence of successive approximations for nonlinear mappings. Bull. Amer. Math. Soc. 73, 591–597 (1967) · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[17]Solodov M.V., Svaiter B.F.: An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions. Math. Oper. Res. 25, 214–230 (2000) · Zbl 0980.90097 · doi:10.1287/moor.25.2.214.12222
[18]Takahashi W., Toyoda M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003) · Zbl 1055.47052 · doi:10.1023/A:1025407607560
[19]Zeng L.C., Yao J.C.: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan J. Math. 10, 1293–1303 (2006)