On Ω-limit Sets of Ordinary Differential Equations in Banach Spaces

We classify ω-limit sets of autonomous ordinary differential equations x′ = f(x), x(0) = x0, where f is Lipschitz, in infinite dimensional Banach spaces as being of three types I-III. Let S ⊂ X be a Polish subset of a Banach space X. S is of type I if there exists a Lipschitz function f and a solution x such that S = Ω(x) and x ∩ S = ∅. S is of type II if it has non-empty interior and there exists a Lipschitz function f and a solution x such that S = Ω(x). S is of type III if it has empty interior and x ⊂ S for every solution x (of an equation where f is Lipschitz) such that S = Ω(x). Our main results are the following: S is a type I set if and only if there exists an open and connected set U ⊂ X such that S ⊂ ∂U . Suppose that there exists an open separable and connected set U ⊂ X such that S = U . Then S is a type II set. 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.