可换Lipschitz半群的拟轨道的渐近动态

Asymptotic Behavior of Almost-orbits of Lipschitzian Semigroups

设C是实Hilbert空间H的闭凸集,G是含有单位元的可换半群。如果■={S(t):t∈G}是C上的Lipschitz映射所成半群,用k_t表示S(t)的Lipschitz常数,有lim supk_1≤1,那么■的拟轨道{u(t):t∈G}弱收敛于C中某一点的充分必要条件是对任意,h∈G,u(t+h)-u(t)弱收敛于0。

Let C be a nonempty closed convex subset of a real Hilbert Space H. Let G be a commutative semigroup with identity and let Y={S(t): t∈G} be a commutative semigroup of lipschitzian mappings S(t) on C with 1 msup k_t≤1, whine k_t is the lipschitzian constant of S(t). We show that an almos.-orbit {u(t): t∈G} of Y converges weakly to some y∈C if and only if u(t+l_t)-u(t) converges weakly to 0 for all h∈G.