ASYMPTOTIC-BEHAVIOR OF ALMOST-ORBITS OF NONLINEAR SEMIGROUPS OF NON-LIPSCHITZIAN MAPPINGS IN HILBERT-SPACES

Let C be a nonempty closed convex subset of a Hilbert space H, F = {T(t): t > 0} be a continuous nonlinear asymptotically nonexpansive semigroup acting on C with a nonempty fixed point set F (F) , and u: [0, infinity) --> C be an almost-orbit of F. Then {u(t)} almost converges weakly to a fixed point of F, i.e., there exists an element y in F(F) such that weak-lim 1/t integral-t/0 u(r + h)dr = y uniformly for h greater-than-or-equal-to 0. This implies that {u(t)} converges weakly to a fixed point of F if and only if {u(t + h) - u(t)} converges weakly to zero as t tends to infinity for each h greater-than-or-equal-to 0.