Compactness of bounded trajectories of dynamical systems in infinite dimensional spaces

The following theorem is proved: Let S ( t ), t ≧0 be a dynamical system in an infinite dimensional Banach space X such that S ( t ) = S 1 ( t )+ S 2 ( t ) for t ≧0, where (1) uniformly in bounded sets of x in X , and (2) S 2 ( 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.