On trajectory convergence of dissipative flows in Banach spaces (Q913218)

Let K be a convex compact set in a strongly normed real Banach space X. Consider a nonexpanding nonlinear semigroup U(t): \({\mathbb{R}}^+\to K\) \(\| U(t)x-U(t)y\| \leq \| x-y\|,\quad t\in {\mathbb{R}}^+,\quad x,y\in K.\) It is important to know when the trajectories \(t\to U(t)x,\) \(x\in K\), are convergent for \(t\to \infty\). For instance, the rotations \(U_ n(t)z=e^{int}z,\) in the two-dimensional Euclidean space \(X={\mathbb{C}}\) with \({\mathbb{K}}=\{z\in {\mathbb{C}}:\) \(| z| \leq 1\}\), obviously have non-convergent trajectories. It turns out that this is essentially the only counterexample: The main theorem in the paper says that if X has not any two-dimensional Euclidean subspaces, then all trajectories of U(t) are convergent. The author shows also that if X is a two-dimensional non-Euclidean space, not necessarily strongly normed, but U(t) is generated by a vector field F(x), \(x\in K\) (i.e. U(t) is a smooth flow), then again all its trajectories are convergent.