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