Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings

Let E be either a strictly convex and reflexive Banach spaces with a uniformly Gateaux differentiable norm or a reflexive Banach spaces with a weakly sequentially continuous duality mapping, and K be a nonempty closed convex subset of E. For a family of finite many nonexpansive mappings {T-1} (l = 1, 2,..., N) and fixed contractive mapping f: K -> K, define iteratively a sequence (x(n)) as follows: x(n+1) = lambda(n+1) integral (x(n)) + (1 - lambda n(+1)) T(n+1)x(n), n >= 0, where T-n = T-n modN. We proved that {x(n)} converges strongly to p epsilon F = boolean AND(N)(n=1) F(T-n), as n -> infinity, where p is the unique solution in F to the following variational inequality: ((I - f)p, j(p - u)) <= 0 for all a epsilon F(T). The main results presented in this paper generalized, extended and improved the corresponding results of Bauschke [The approximation of fixed points of nonexpansive mappings in Hilbert space, J. Math. Anal Appl. 202 (1996) 150159], Halpen [Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957-961], Shioji and Takahashi [Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997) 3641-3645], Wittmann [Approximation of fixed points of nonexpansive mappings, Arch. Math. 59 (1992) 486-491], O' Hara et al. [Iterative approaches to fineding nearest common fixed point of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417-1426], Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279-291], Jung.[Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509-520], Zhou et al. [Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput. 173 (1) (2006) 196-212] and others. (c) 2005 Elsevier Inc. All rights reserved.