Abstract
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 \({F(S)\cap\Omega_{A}}\). 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.
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References
Antipin A.S.: Methods for solving variational inequalities with related constraints. Comput. Math. Math. Phys. 40, 1239–1254 (2000)
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)
Browder F.E.: Existence of periodic solutions for nonlinear equations of evolution. Proc. Nat. Acad. Sc. USA 55, 1100–1103 (1965)
Browder F.E., Petryshyn W.V.: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20, 197–228 (1967)
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)
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)
Geobel K., Kirk W.A.: Topics on Metric Fixed-point Theory. Cambridge University Press, Cambridge, England (1990)
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)
Hu S., Papageorgiou N.S.: Handbook of multivalued analysis, vol. I: theory. Kluwer Academic Publishers, Dordrecht (1997)
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)
Korpelevich G.M.: The extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976)
Liu F., Nashed M.Z.: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Anal. 6, 313–344 (1998)
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)
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)
Nakajo K., Takahashi W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)
Opial Z.: Weak convergence of the sequence of successive approximations for nonlinear mappings. Bull. Amer. Math. Soc. 73, 591–597 (1967)
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)
Takahashi W., Toyoda M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)
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)
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Nicolas Hadjisavvas was supported by grant no. 227-\({\varepsilon}\) of the Greek General Secretariat of Research and Technology. Ngai-Ching Wong was supported partially by a grant of Taiwan NSC 96-2115-M-110-004-MY3.
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Ceng, LC., Hadjisavvas, N. & Wong, NC. Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J Glob Optim 46, 635–646 (2010). https://doi.org/10.1007/s10898-009-9454-7
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DOI: https://doi.org/10.1007/s10898-009-9454-7
Keywords
- Hybrid extragradient-like approximation method
- Variational inequality
- Fixed point
- Monotone mapping
- Nonexpansive mapping
- Demiclosedness principle