It is shown that every asymptotically regular or λ-firmly nonexpansive mapping T:CC has a fixed point whenever C is a finite union of nonempty weakly compact convex subsets of a Banach space X which is uniformly convex in every direction. Furthermore, if {Ti}iI is any compatible family of strongly nonexpansive self-mappings on such a C and the graphs of Ti, i I, have a nonempty intersection, then Ti, iI, have a common fixed point in C.