FIXED-POINT THEOREMS FOR CERTAIN CLASSES
Lawrence Peter Belluce, W. A. Kirk · 2016
where it is understood that f0(x) = x. Our main purpose here is to prove fixed-point theorems for nonexpansive mappings f for which the diameters of the sets O(fP(x)) satisfy a condition introduced below, a condition which is suggested by a consideration of the Banach Contraction Principle. For such mappings f, compactness of M is seen to imply that every sequence of iterates {fn(x) } of x converges to a fixed-point of f (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 f is established. In the final section we show how the results of this paper lead in an indirect way to a