where it is understood that/°(x)=x. Our main purpose here is to prove fixed-point theorems for nonexpansive mappings / for which the diameters of the sets 0(fn(x)) satisfy a condition introduced below, a condition which is suggested by a consideration of the Banach Contraction Principle. For such mappings /, compactness of M is seen to imply that every sequence of iterates {/"(x)} of x converges to a fixed-point of/ (which is not necessarily unique) while if M is a weakly compact, closed, and convex subset of a Banach space, then the existence of a fixed-point for/ is established. In the final section we show how the results of this paper lead in an indirect way to a generalization of Theorem 3 of [l].
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