Fixed points and iteration of a nonexpansive mapping in a Banach space

Abstract

The following result is shown. If T T is a nonexpansive mapping from a closed convex subset D D of a Banach space into a compact subset of D D and x 1 {x_1} is any point in D D , then the sequence { x n } \{ {x_n}\} defined by x n + 1 = 2 − 1 ( x n + T x n ) {x_{n + 1}} = {2^{ - 1}}({x_n} + T{x_n}) converges to a fixed point of T T . As a matter of fact, a theorem which includes this result is proved. Furthermore, a similar result is obtained under certain restrictions which do not imply the assumption on the compactness of T T .