An Example; Mann Iteration for Strictly Pseudo-contractive Mappings

We have seen (Chapter 6) that the Mann iteration method has been successfully employed in approximating fixed points (when they exist) of nonexpansive mappings. This success has not carried over to the more general class of pseudo-contractions. If K is a compact convex subset of a Hilbert space and T : K → K is Lipschitz, then, by Schauder fixed point theorem, T has a fixed point in K. All efforts to approximate such a fixed point by means of the Mann sequence when T is also assumed to be pseudo-contractive proved abortive. In 1974, Ishikawa introduced a new iteration scheme and proved the following theorem.