Fixed point theorems for strongly inward non-self mappings defined on non-convex domains

A common fixed point theorem is proved for a pair of strongly inward, non-expansive mappings defined on a non-empty weakly compact, not necessarily star shaped, subset of a uniformly convex Banach space. This result subsumes a classical fixed point theorem, due to Browder, Ghöde and Kirk, for non-expansive mappings on convex domains, and a fixed point theorem, due to Veeramani, for non-expansive mappings on non-convex domains.