On firmly nonexpansive mappings
Ryszard Smarzewski · Proceedings of the American Mathematical Society · 1991
It is shown that any $\lambda$-firmly, $0 < \lambda < 1$ , nonexpansive mapping $T:C \to C$ has a fixed point in $C$ whenever $C$ is a finite union of nonempty, bounded, closed convex subsets of a uniformly convex Banach space.