Strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings

Let C be a nonempty closed convex subset of a real strictly convex and reflexive Banach space E which has a uniformly Gateaux differentiable norm. Let f : C -> C be a given contractive mapping and {T-n}(n-1)(infinity) : C -> C be an infinite family of nonexpansive mappings such that the common. fixed point sets F := boolean AND(infinity)(n-1) F(T-n) not equal empty set. Let {alpha(n)} and {beta(n)} be two real sequences in [0, 1]. For given x(0) is an element of C arbitrarily, let the sequence {x(n)} be generated iteratively by x(n+1) = alpha(n)f(x(n)) + beta(n)x(n) + (1 - alpha(n) - beta(n))W(n)x(n), where W-n is the W-mapping generated by the mappings T-n; Tn-1,..., T-1 and xi(n), xi(n-1),..., xi(1). Suppose the iterative parameters {alpha(n)} and {beta(b)} satisfy the following control conditions: (C1) lim(n ->infinity) alpha(n) = 0; (C2) Sigma(infinity)(n-0) alpha(n) = infinity; (B5) lim sup(n ->infinity) beta(n) < 1. Then the sequence {x(n)} converges strongly to p is an element of F where p is the unique solution in F to the following variational inequality: <(I - f)p,j(p - x*)> <= 0 for all x* is an element of F. (C) 2008 Elsevier Inc. All rights reserved.