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

Abstract

LetEbe a complete locally convex space (l.c.s.) andf:R→Ea continuous function; thenfis said to be almost-periodic (a.p.) if, for every neighbourhood (of the origin inE)U, there existsℓ=ℓ(U)>0such that every interval[a,a+ℓ]of the real line contains at least oneτpoint such thatf(t+τ)−f(t)∈Ufor everyt∈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.