Weak topology and Browder–Kirk’s theorem on hyperspace

Let K be a weakly compact, convex subset of a Banach space X with normal structure. Browder–Kirk’s theorem states that every non-expansive mapping T which maps K into K has a fixed point in K. Suppose now that WCC(X) is the collection of all non-empty weakly compact convex subsets of X. We shall define a certain weak topology Tw on WCC(X) and have the above-mentioned result extended to the hyperspace (WCC(X);Tw). © 2007 Elsevier Inc. All rights reserved.