Iterative approximation of fixed points of nonexpansive mappings

Let K be a nonempty closed convex subset of a real Banach space E which has a uniformly Gateaux differentiable norm and T : K → K be a nonexpansive mapping with F(T) := { x element of K : Tx = x} ≠ 0 . For a fixed δ element of (0, 1), define S : K → K by Sx := (1- δ)x+ δ Tx , for all x element of K. Assume that { zt} converges strongly to a fixed point z of T as t → 0, where zt is the unique element of K which satisfies zt = tu + (1 - t)Tzt for arbitrary u element of K. Let {αn} be a real sequence in (0, 1) which satisfies the following conditions: C1 : lim αn = 0; C2 : Σαn = ∞. For arbitrary x0 element of K, let the sequence { xn} be defined iteratively by xn+1 = αnu + (1 - αn)Sxn. Then, {xn} converges strongly to a fixed point of T. (author)