On infinite-dimensional differential equations

Differential equations ·x(t) = f(x(t), t) are exhibited in a general infinite-dimensional Banach space, failing each of the following in turn. (i) The set S_t of solution values x(t) from a given point x(0) is compact. (ii) S_t is connected. (iii) Any point on the boundary ?S_t of S_t can be reached by a solution x with x(s) ??S_s, 0 ? s ? t.