Convergence theorems by hybrid projection methods for Lipschitz-continuous monotone mappings and a countable family of nonexpansive mappings
In this paper, we introduce two iterative schemes for finding a common element of the set of a common fixed points of a countable family of nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping in a Hilbert space by using the hybrid projection methods in the mathematical programming. Then we prove strong convergence theorems by the hybrid projection methods for a monotone, Lipschitz-continuous mapping and a countable family of nonexpansive mappings. Moreover, we apply our result to the problem for finding a common fixed point of two mappings, such that one of these mappings is nonexpansive and the other is taken from the more general class of Lipschitz pseudocontractive mappings. Our results extend and improve the results of Nadezhkina and Takahashi [SIAM J. Optim. 16 (2006), 1230–1241], Zeng and Yao [Taiwanese J. Math. 10 (2006), 1293–1303] and many authors.