Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Compactness of bounded trajectories of dynamical systems in infinite dimensional spaces

G. F. Webba1

a1 Mathematics Department, Vanderbilt University, Nashville, Tennessee 37235, U.S.A.


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) = S1(t)+S2(t) for t≧0, where (1) S0308210500016930_inline1 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 xX 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.

(Received September 13 1978)

(Revised January 19 1979)

(Online publication March 19 1979)


Supported in part by National Science Foundation Grant NSF 75-06332 A01.