On trajectory convergence of dissipative flows in Banach spaces

LetX be a convex compact in a real Banach spaceE. An actionU(t) (t0) of the semigroup ₊ onX is called dissipative if allU(t) are nonexpanding: U(t)x ₁–U(t)x ₂x ₁–x ₂. Let the spaceE be strongly normed. We prove that all trajectoriestU(t)x of the dissipative flowU(t) are converging fort if there are no two-dimensional Euclidean subspaces in the spaceE. In every two dimensional non-Euclidean spaceE (not necessarily strongly normed) all trajectories of the flow under consideration are converging.