Iterative methods for common fixed points for a countable family of nonexpansive mappings in uniformly convex spaces

Let E be a uniformly convex real Banach space with a uniformly Gateaux differentiable norm. Let K be a closed, convex and nonempty subset of E. Let {T-i}(i=1)(infinity) be a family of nonexpansive self-mappings of K. For arbitrary fixed delta is an element of (0, 1), define a family of nonexpansive maps {S-i}(n=1)(infinity) by S-i := (1 - delta)l +delta T-i where l is the identity map of K. Let F := boolean AND(infinity)(i=1) F(T-i) not equal phi. It is proved that an iterative sequencce {x(n)} defined by x(0) is an element of K, x(n+1) = alpha(n)u +Sigma(i=1)(i >= 1) sigma(i,tn)S(i)x(n), n >= 0, converges strongly to a common fixed point of the family {T-i}(i=1)(infinity), where {alpha(n)} and {sigma(i,tn)} are sequences in (0, 1) satisfying appropriate conditions, in each of the following cases: (a) E = l(p), 1 < p < infinity; and (b) at least one of the T-i's is demicompact. (C) 2009 Published by Elsevier Ltd