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 ∈ K : Tx = x} 6= ∅. An iterative sequence {xn} is constructed for which ||xn − Txn|| → 0 as n → ∞. If, in addition, K is assumed to be bounded, this conclusion still holds without the requirement that F (T ) 6= ∅. 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-mappings, then the sequence {xn} converges strongly to a fixed point of T . Our iteration method is of independent interest.