Fixed point properties and proximinality in Banach spaces

In this paper we prove the existence of a fixed point for several classes of mappings (mappings admitting a center, nonexpansive mappings, asymptotically nonexpansive mappings) defined on the closed convex subsets of a Banach space satisfying some proximinality conditions. In particular, we derive a sufficient condition, more general than weak star compactness, such that if CC is a bounded closed convex subset of ℓ1ℓ1 satisfying this condition, then every nonexpansive mapping T:C→CT:C→C has a fixed point.