Fixed points of asymptotically regular nonexpansive mappings on nonconvex sets

Abstract

It is shown that if $X$ is a Banach space and $C$ is a union of finitely many nonempty, pairwise disjoint, closed, and connected subsets $\left\{C_i:1\le i\le n\right\}$ of $X$, and each $C_i$ has the fixed-point property (FPP) for asymptotically regular nonexpansive mappings, then any asymptotically regular nonexpansive self-mapping of $C$ has a fixed point. We also generalize the Goebel-Schöneberg theorem to some Banach spaces with Opial's property.