Implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings

Authors:
L. C. Ceng;D. R. Sahu;J. C. Yao
Affiliations:
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China;Department of Mathematics, Banaras Hindu University, Varanasi-221005, India;Department of Applied Mathematics, National Sun Yat-sen University, 804 Kaohsiung, Taiwan
Venue:
Journal of Computational and Applied Mathematics
Year:
2010

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Abstract

In this paper, we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. These implicit iterative algorithms are based on two well-known methods: extragradient and approximate proximal methods. We obtain some weak convergence theorems for these implicit iterative algorithms. Based on these theorems, we also construct some implicit iterative processes for finding a common fixed point of two mappings, such that one of these two mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive.