Strong convergence of approximants to fixed points of Lipschitzian pseudocontractive maps

Let E be a real q-uniformly smooth Banach space which is also uniformly convex (for example, L-p or l(p) spaces, 1 < p < infinity) and K be a nonempty closed convex and bounded subset of E with phi not equal int (K). Let T K --> K be a Lipschitzian pseudocontractive mapping such that for z is an element of int (K), parallel toz - Tzparallel to < parallel tox Txparallel to, for all x is an element of partial derivative(K). Then for z(0) is an element of K arbitrary, the iteration process {z(n)} defined by z(n+1) := (1 - mu(n+1))z + mu(n) + (1)y(n); y(n) := (1 - alpha(n))z(n) + alpha(n)Tz(n) converges strongly to a fixed point of T, provided that {mu(n)} and {alpha(n)} satisfy certain conditions. Moreover, if T is strictly pseudocontractive with a nonempty fixed-point set, then it is proved that the Mann type iteration scheme converges strongly to a fixed point of T. (C) 2002 Elsevier Science Ltd. All rights reserved.