Differential equations in Banach spaces: Five examples (Q1071161)

The set of all solutions of an initial-value problem for an ordinary differential equation with a continuous right-hand side is connected under a condition that is always satisfied in the finite-dimensional case. We show with a two-dimensional example that ''connected'' cannot be replaced by 'arcwise connected'. In the remainder of the paper we show that results that are valid in the finite-dimensional case need not be true in the infinite-dimensional case: A nonextendable solution must not ''go from boundary to boundary'' and the set of all omega-limit points may be empty or not compact or not connected.