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

Abstract

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 { T l } ( l = 1, 2, … , N) and fixed contractive mapping f : K → K, define iteratively a sequence { x n } as follows: x n + 1 = λ n + 1 f ( x n ) + ( 1 - λ n + 1 ) T n + 1 x n , n ⩾ 0 , where T n = T n mod N . We proved that { x n } converges strongly to p ∈ F = ⋂ n = 1 N F ( T n ) , as n → ∞, where p is the unique solution in F to the following variational inequality: 〈 ( I - f ) p , j ( p - u ) 〉 ⩽ 0 for all u ∈ 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.