Almost-periodicity in linear topological spaces and applications to abstract differential equations

Abstract

Let E be a complete locally convex space (l.c.s.) and f:R→E a continuous function; then f is said to be almost-periodic (a.p.) if, for every neighbourhood (of the origin in E) U, there exists ℓ=ℓ(U)>0 such that every interval [a,a+ℓ] of the real line contains at least one τ point such that f(t+τ)−f(t)∈U for every t∈R. We prove in this paper many useful properties of a.p. functions in l.c.s, and give Bochner's criteria in Fréchet spaces.