Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems

Authors:
Lu-Chuan Ceng;Nicolas Hadjisavvas;Ngai-Ching Wong
Affiliations:
Department of Mathematics, Shanghai Normal University, Shanghai, China 200234;Department of Product and Systems Design Engineering, University of the Aegean, Hermoupolis, Greece 84100;Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan 804
Venue:
Journal of Global Optimization
Year:
2010

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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 驴 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.