Compactness characterizations via Fixed Point Theorems

P.K. Lin and Y. Sternfeld characterized norm-compactness in Banach spaces via a fixed-point theorem in the following way:

Let $ C $ be a closed convex subset of a Banach space. Then $ C $ is norm-compact if and only if there is some $ L>1 $ such that every $ L $- Lipschitzian mapping $ T: C \to C $ has a fixed point (a mapping  is said to be $ L $-Lipschitzian if $ \Vert Tx-Ty \Vert \le L \Vert x-y \Vert $ for every $ x,y $ in its domain).

 In this talk we will study how this theorem can be extended, replacing norm-compactness by weak compactness and considering more restricted families of mappings, such as  uniform Lipschitzian mappings, nonexpansive mappings and so on.