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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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