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∈S}be a Lipschitzian semigroup on C with lim inf|||T(t)||| ＜ Np, where Np isn→∞t∈s the normal structure coefficient of X. Suppose also there exists a nonempty bounded closed convex subset E of C with the following properties: (P1)x ∈ E implies ωw(x) E; (P2)T is asymptotically regular on E. The authors prove that there exists a z ∈ E such that T(s)z = z for all s ∈ S. Further, under the similar condition, the existence of fixed points of Lipschitzian semigroups in a uniformly convex Banach space is discussed.