On the existence of fixed points for Lipschitzian semigroups in Banach spaces

Let C be a nonempty bounded subset of a p-uniformly convex Banach space X, and T = {T(t) : t is an element of S} be a Lipschitzian semigroup on C with lim(n --> infinity) inf(t is an element of). ///T(t)/// < rootN(p), where N-p is normal structure coefficient of X. Suppose also there exists a nonempty bounded closed convex subset E of C with the following properties: (P-1)x is an element of E implies omega (w) (x) subset of E; (P-2)T is asymptotically regular on E. The authors prove that there exists a x is an element of E such that T(s)x = x for all s is an element of S. Further, under the similar condition, the existence of fixed points of Lipschitzian semigroups in a uniformly convex Banach space is discussed.