MEAN VALUE ITERATION OF NONEXPANSIVE MAPPINGS IN A BANACH SPACE

This paper applies a certain method of iteration, of the mean value type introduced by W. R. Mann, to obtain two theorems on the approximation of a fixed point of a mapping of a Banach space into itself which is nonexpansive (i.e., a mapping which satisfies ∥ Tx − Ty ∥ ≦ ∥ x − y ∥for each x and y). Thefirst theorem obtains convergence of the iterates to a fixed point of a nonexpansive mapping which maps a compact convexsubset of a rotund Banach space into itself. The second theorem obtains convergence to a fixed point provided that the Banach space is uniformly convex and the iterating transformation is nonexpansive, maps a closed bounded convex subsetof the space into itself, and satisfies a certain restriction on the distance between any point and its image. © 1969 by Pacific Journal of Mathematics.