ON TRAJECTORY CONVERGENCE OF DISSIPATIVE FLOWS IN BANACH-SPACES

Let X be a convex compact in a real Banach space E. An action U(t) (t≥0) of the semigroup ℝ+ on X is called dissipative if all U(t) are nonexpanding: ∥U(t)x1-U(t)x2∥≤∥x1-x2∥. Let the space E be strongly normed. We prove that all trajectories t→U(t)x of the dissipative flow U(t) are converging for t→∞ if there are no two-dimensional Euclidean subspaces in the space E. In every two dimensional non-Euclidean space E (not necessarily strongly normed) all trajectories of the flow under consideration are converging. © 1990 Birkhäuser Verlag.