Strong convergence theorems for nonexpansive semigroup in Banach spaces

Let K be a nonempty closed convex subset of a reflexive and strictly convex Banach space E with a uniformly Gateaux differentiable norm, and F = {T (t): t > 0) a nonexpansive self-mappings semigroup of K, and f : K -> K a fixed contractive mapping. The strongly convergent theorems of the following implicit and explicit viscosity iterative schemes {x(n)} are proved. x(n) = alpha(n)f(x(n)) + (1 - alpha(n))T(t(n))x(n), x(n+1) = alpha(n)f(x(n)) + (1 - alpha(n))T(t(n))x(n). And the cluster point of {x(n)} is the unique solution to some co-variational inequality. (c) 2007 Elsevier Inc. All rights reserved.