The construction of an asymptotic center with a fixed-point property
Michael L. Edelstein · Bulletin of the American Mathematical Society · 1972
Given a bounded sequence {u n : n = 1,2,...} of points in a closed convex subset C of a uniformly convex Banach space, c m denotes the point in C with the property that among all closed balls centered at points of C and containing {u m ,u m+lt ...} the one centered at c m is of smallest radius.It is shown that the sequence {c m : m = 1,2,...} converges (strongly) to a point ceC called the asymptotic center of {u n } with respect to C. Further, for a class of mappings ƒ of C into itself, which contains all nonexpansive mappings, f(c) = c whenever an x e C exists such that ƒ "(x) = «",« = 1,2,... . AM S 1969 subject classifications.