A FIXED POINT THEOREM FOR ASYMPTOTICALLY
Nonexpansive Mappings, K. Goebel, Fx-F'y ktWx-yW · 2005
Let K be a subset of a Banach space X. mapping F.K-+KI& said to be if there exists a sequence {ki} of real numbers with £?-+1 as /'-►co such that WF'x—F'yW^kiWx—yW, x, yE K. It is proved that if AT is a nonempty, closed, convex, and bounded subset of a uniformly convex Banach space, and if F-.K-+K is nonexpansive, then F has a fixed point. This result generalizes a fixed point theorem for mappings proved independently by F. E. Browder, D. Gohde, and W. A. Kirk. In 1965, F. E. Browder [1] and D. Gohde [4] independently proved that every self-mapping of a closed convex and bounded subset of a uniformly convex Banach space has a fixed point. This result was also obtained by W. A. Kirk [5], under assumptions slightly weaker in a technical sense, and another proof, more geometric and elementary in nature, has recently been given by K. Goebel [3]. Our purpose here is to extend Browder's result to a more general class of transformations which we shall call asymptotically nonexpansive mappings. Banach space X is called uniformly convex (Clarkson [2]) if for each £>0 there is a «5(e)>0 such that if ||x|| = ||_y|| = l then ||(x+_y)/2|| /. Furthermore, the function -(0, 1] may be assumed to be increasing. Definition. Let A be a subset of a Banach space X. transformation F.K-^-K is said to be if for arbitrary x, y £ K, \\Fx-Fy < \\x-y\\. Received by the editors November 15, 1971. AMS 1969 subject classifications. Primary 4785.