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

摘要

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 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∈SX 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.