Fixed Points and Iteration of a Nonexpansive Mapping in a Banach Space

Abstract

The following result is shown. If $T$ is a nonexpansive mapping from a closed convex subset $D$ of a Banach space into a compact subset of $D$ and ${x_1}$ is any point in $D$, then the sequence $\{ {x_n}\}$ defined by ${x_{n + 1}} = {2^{ - 1}}({x_n} + T{x_n})$ converges to a fixed point of $T$. 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 $T$.