Convergences of nonexpansive iteration processes in Banach spaces ✩

Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm and S be a mapping of the form S = α0I + α1T1 + α2T2 +· · ·+αkTk, where αi 0, α0 > 0, k i=0 αi = 1 and Ti :E →E (i = 1, 2, . . . , k) is a nonexpansive mapping. For an arbitrary x0 ∈ E, let {xn} be a sequence in E defined by an iteration xn+1 = Sxn, n = 0, 1, 2, . . . .We establish a dual weak almost convergence result of {xn} in a reflexive Banach space with a uniformly Gâteaux differentiable norm. As a consequence of the result, a weak convergence result of {xn} is also given.  2002 Elsevier Science (USA). All rights reserved.