Iterative algorithm for common fixed points of infinite family of nonexpansive mappings in banach spaces

Let c be a nonempty closed convex subset of a real uniformly smooth banach space x,{t k} k = 1 ∞ : c → c an infinite family of nonexpansive mappings with the nonempty set of common fixed points k = 1 fix (t k),and f: c → c a contraction. we introduce an explicit iterative algorithm x n + 1 = n f (x n) + (1 - n) l n x n,where l n = k = 1 n ( k / s n) t k,s n = k = 1 n k,and w k 0 with k = 1 k = 1. under certain appropriate conditions on { n },we prove that { x n } converges strongly to a common fixed point x * of { t k } k = 1,which solves the following variational inequality: x * - f (x*),j (x * - p) 0,p k = 1 fix (t k),where j is the (normalized) duality mapping of x. this algorithm is brief and needs less computational work,since it does not involve w -mapping. copyright © 2012 songnian he and jun guo.