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

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