A FIXED POINT THEOREM FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

Let K be a subset of a Banach space X. A mapping F.K-+KI& said to be asymptotically nonexpansive if there exists a sequence {ki} of real numbers with £?-+1 as /'-►co such that WF'x—F'yW^kiWx—yW, yE K. It is proved that if AT is a non- empty, closed, convex, and bounded subset of a uniformly convex Banach space, and if F-.K-+K is asymptotically nonexpansive, then F has a fixed point. This result generalizes a fixed point theorem for nonexpansive mappings proved independently by F. E. Browder, D. Gohde, and W. A. Kirk.