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

Abstract

Let X be an infinite-dimensional real Banach space. We classify ω-limit sets of autonomous ordinary differential equations x ′ = f ( x ) , x ( 0 ) = x 0 , where f : X → X is Lipschitz, as being of three types I–III. We denote by S X 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 x 0 ∈ X . We say that S ∈ S X is of type I if there exists a Lipschitz function f and a solution x such that S = Ω ( x ) and { x ( t ) : t ⩾ 0 } ∩ S = ∅ . We say that S ∈ S X is of type II if it has non-empty interior. We say that S ∈ S X 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 { x ( t ) : t ⩾ 0 } ⊂ S . Our main results are the following: S is a type I set in S X 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 S = U ¯ , then S is a type II set in S X . 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 C k -smooth whenever the underlying Banach space is C k -smooth.