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Let K be a nonempty closed convex subset of a real Banach space E and T be a Lipschitz pseudocontractive self-map of K with F(T) := {x E K : Tx = x} ¬= O. An iterative sequence {x n } is constructed for which ∥x n - Tx n ∥ → 0 as n → oc. If, in addition, K is assumed to be bounded, this conclusion still holds without the requirement that F(T) ¬= O. Moreover, if, in addition, E has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of K has the fixed point property for nonexpansive self-trappings, then the sequence {x n } converges strongly to a fixed point of T. Our iteration method is of independent interest.

Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps