ON ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS

Abstract

Let K be a nonempty closed convex subset of a real Banach space E, T : K -> K a uniformly continuous asymptotically pseudocontractive mapping having T(K) bounded with sequence {k_n}_{n>=0} \subset [1, \infty), \lim_{n -> \infty}k_n =1 such that p \in F(T) = {x \in K : Tx = x}. Let {\alpha_n}_{n>=0} \subset [1, \infty] be such that \sum_{n>=0}\alpha_n^2 = \infty and \lim_{n->\infty}\alpha_n =0. For arbitrary x_0 \in K let {x_n}_{n>=0} be iteratively defined by

x_{n+1} = (1-\alpha_n) x_n + \alpha_n nT^n x_n, n>=0.

Then {x_n}_{n>=0} converges strongly to p \in F(T).