On ω-limit sets of ordinary differential equations in Banach spaces

Let X be an infinite-dimensional real Banach space. We classify ω-limit sets of autonomous ordinary differential equations x′=f(x), x(0)=x0, where f:X→X is Lipschitz, as being of three types I–III. We denote by SX the class of all sets in X which are ω-limit sets of a solution to (1), for some Lipschitz vector field f and some initial condition x0∈X. We say that S∈SX is of type I if there exists a Lipschitz function f and a solution x such that S=Ω(x) and . We say that S∈SX is of type II if it has non-empty interior. We say that S∈SX is of type III if it has empty interior and for every solution x (of Eq. (1) where f is Lipschitz) such that S=Ω(x) it holds . Our main results are the following: S is a type I set in SX if and only if S is a closed and separable subset of the topological boundary of an open and connected set U⊂X. Suppose that there exists an open separable and connected set U⊂X such that , then S is a type II set in SX. Every separable Banach space with a Schauder basis contains a type III set. Moreover, in all these results we show that in addition f may be chosen Ck-smooth whenever the underlying Banach space is Ck-smooth.