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 Gâteaux 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 {xn} are proved. xn = αnf (xn) +(1− αn)T (tn)xn, xn+1 = αnf (xn)+ (1−αn)T (tn)xn. And the cluster point of {xn} is the unique solution to some co-variational inequality. © 2007 Elsevier Inc. All rights reserved.