Relationships between properties that imply the weak fixed point property

A Banach space, X, satisfies the weak fixed point property (w-FPP) if every nonexpansive mapping, T, on every weak compact convex nonempty subset, C, has a fixed point. Over the past 30 years or so, a number of Banach space p roperties have been shown to imply the w-FPP, such as Opial's condition, we ak normal structure, property (K), R(X)< 2, and property (M). The relevant papers are, respectively, Opial [13], Kirk [10], Sims [15], Garcia-Falset [ 4], and Garcia-Falset and Sims [5]. Section 2 gives the definitions and dis cusses these properties and Section 3 looks at the interrelationships among them.