A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space

Teck-Cheong Lim
1974 Bulletin of the American Mathematical Society  
Let C be a nonempty weakly compact convex subset of a Banach space X, and ^(C) be the family of nonempty compact subsets of C equipped with the Hausdorff metric. Let T: C-^€(C) be a nonexpansive mapping, i.e. for each x, y e C, H(T(x) 9 T(y))£\\x-y\\ 9 where H (A, B) denotes the Hausdorff distance between A and B. A point x e C is called a fixed point of T if x e Tx. Fixed point theorems for such mappings T have been established by Mar kin [11] for Hubert spaces, by Browder [2] for spaces
more » ... weakly continuous duality mapping, and by Lami Dozo [7] for spaces satisfying OpiaPs condition. Lami Dozo's result is also generalized by Assad and Kirk [1]. By making use of Edelstein's asymptotic center [4], [5] , we are able to prove Theorem 1. Let C be a closed convex subset of a uniformly convex Banach space and let {wj be a bounded sequence in C. The asymptotic center x of {wj in (or with respect to) C is the unique point in C such that lim sup \\x -u t \\ = infjlim sup \\y -wj| :y e c}. The number r=inf{lim supjjy-wj :y e C} is called the asymptotic radius of {wj in C. Existence of the unique asymptotic center is proved by Edelstein in [5]. Results on ordinal numbers used here may be found in [13]. THEOREM 1. Let X be a uniformly convex Banach space and C be a closed convex bounded nonempty subset of X. Let T: C-^€{C) be a nonexpansive mapping from C into the family of nonempty compact subsets of AM S (MOS) subject classifications (1970). Primary 46A05.
doi:10.1090/s0002-9904-1974-13640-2 fatcat:xinqalc2prechlhe4fw7tlc6du