Remarks on some fixed point theorems

A compact Hausdorff pseudo-topology is introduced on every closed convex bounded subset of a uniformly convex Banach space and is used to prove a previous theorem of the author. In [7], we used a transfinite induction method which depends on the structure of the real line to prove the following fixed point theorem for multivalued nonexpansive mappings: THEOREM 1. Let K be a closed convex bounded nonempty subset of a uniformly convex Banach space and let T: K ' ( (K) be a nonexpansive mapping, where ( (K) denotes the family of nonempty compact (not necessarily convex) subsets of K, equipped with the Hausdorff metric. Then T has a fixed point, i.e. there exists x E K such that x E Tx. The properties of real numbers we used are the order property and the separability, or more explicitly, that a decreasing nonnegative transfinite sequence indexed by ordinals less than the uncountable ordinal Q must be eventually constant. Recently, Caristi and Kirk [1], [2], [5] have proven the following theorem and have given several interesting applications: THEOREM 2 [1], [2]. Let X be a complete metric space, and let g: X -* X be a self-map of X. Suppose that there exists a lower semicontinuous nonnegative real-valued mapping ( such that for all x in X, (1) d (x, g(x)) < ((x) g-(x)). Then g has a fixed point. Chi Song Wong [8] has given a simple proof of the Caristi-Kirk theorem by using the transfinite induction method mentioned above. On the other hand, we define a pseudo-compact-Hausdorff-topology on any closed convex bounded subset of a uniformly convex Banach space and give Theorem 1 a simpler and more conceptual proof. It is our feeling that the existence of such a compact Hausdorff pseudo-topology may well serve to explain the similarity between uniform convexity and compactness in some aspects of geometric fixed point theory. Presented to the Society, January 20, 1975 under the the title A remark on a fixed point theorem; received by the editors September 5, 1975. AMS (MOS) subject classifications (1970). Primary 46A05. Copyright ? 1977, American Mathematical Society