Compactness of bounded trajectories of dynamical systems in infinite dimensional spaces

SynopsisThe following theorem is proved: Let S(t), t≧0 be a dynamical system in an infinite dimensional Banach space X such that S(t) = S1(t)+S2(t) for t≧0, where (1) uniformly in bounded sets of x in X, and (2) S2(t) is compact for t sufficiently large. Then, if the orbit {S(t)x: t ≧0} of x ∈ X is bounded in X, it is precompact in X. Applications are made to an age dependent population model, a non-linear functional differential equation on an infinite interval, and a non-linear Volterra integrodifferential equation.