Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps

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 is an element of K:Tx=x} not equal empty set. An iterative sequence {x(n)} is constructed for which parallel tox(n)-Tx(n)parallel to-->0 as n-->infinity. If, in addition, K is assumed to be bounded, this conclusion still holds without the requirement that F(T)not equalempty set. Moreover, if, in addition, E has a uniformly Gateaux differentiable norm and is such that every closed bounded convex subset of K has the fixed point property for nonexpansive self-mappings, then the sequence {x(n)} converges strongly to a fixed point of T. Our iteration method is of independent interest.