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

0352543 - MÚ 2011 RIV US eng J - Journal Article
Hájek, Petr Pavel - Vivi, P.
On omega-limit sets of ordinary differential equations in Banach spaces.
Journal of Mathematical Analysis and Applications. Roč. 371, č. 2 (2010), s. 793-812. ISSN 0022-247X. E-ISSN 1096-0813
R&D Projects: GA AV ČR IAA100190801
Institutional research plan: CEZ:AV0Z10190503
Keywords : omega-limit set * ODE in Banach space
Subject RIV: BA - General Mathematics
Impact factor: 1.174, year: 2010
http://www.sciencedirect.com/science/article/pii/S0022247X10004798

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.
Permanent Link: http://hdl.handle.net/11104/0192032