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 Gâteaux 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 {Tl} (l = 1, 2, … , N) and fixed contractive mapping f : K → K, define iteratively a sequence {xn} as follows:xn+1=λn+1f(xn)+(1-λn+1)Tn+1xn,n⩾0,whereTn = Tn modN. We proved that {xn} converges strongly to p∈F=⋂n=1NF(Tn), as n → ∞, where p is the unique solution in F to the following variational inequality:〈(I-f)p,j(p-u)〉⩽0forallu∈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) 150-159], 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.