Uniformly normal structure and uniformly Lipschitzian semigroups

Assume that XX is a real Banach space with uniformly normal structure and CC is a nonempty closed convex subset of XX. We show that a κκ-uniformly Lipschitzian semigroup of nonlinear self-mappings of CC admits a common fixed point if the semigroup has a bounded orbit and if κκ is appropriately greater than one. This result applies, in particular, to the framework of uniformly convex Banach spaces.