Fixed points of Lipschitzian semigroups in Banach spaces

We prove the following theorem: Let p > 1 and let E be a real p-uniformly convex Banach space, and C a nonempty bounded closed convex subset of E. If T ={T-s : C --> C: s is an element of G = [0, infinity)} is a Lipschitzian semigroup such that [GRAPHICS] where c > 0 is some constant, then there exists x is an element of C such that T(s)x = x for all s is an element of G.