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

Let X be an infinite-dimensional real Banach space. We classify omega-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 omega-limit sets of a solution to (1), for some Lipschitz vector field f and some initial condition x(0) is an element of X. We say that S is an element of S-x is of type I if there exists a Lipschitz function integral and a solution x such that S = Omega(x) and {x(t): t >= 0} boolean AND S = empty set. We say that S is an element of S-x is of type II if it has nonempty interior. We say that S is an element of 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 = Omega(x) it holds {x(t). t >= 0} subset of 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 subset of X. Suppose that there exists an open separable and connected set U subset of c X such that S = (U) over bar. 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 integral may be chosen C-k-smooth whenever the underlying Banach space is C-k-smooth (C) 2010 Elsevier Inc. All rights reserved.