Weak Convergence for Reversible Lipschitzian Semigroups

The weak convergence for a class of Lipschitzian semigroups is discussed. It is proved that in a Hilbert space the weak asymptotic regularity implies the weak convergence of the semigroup; while in a uniformly convex Banach space with a Frechet differentiable norm, the asymptotic regularity implies the weak convergence of the semigroup. The existence of a retract for the semigroup is also discussed.