Abstract

It is shown that every asymptotically regular or λ-firmly nonexpansive mapping T:C→C 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 {T i}i∈I is any compatible family of strongly nonexpansive self-mappings on such a C and the graphs of T i, i ∈I, have a nonempty intersection, then T i, i∈I, have a common fixed point in C.