Almost periodicity

Summary

This chapter investigates almost periodic (in short a.p.) solutions of the strongly wellposed (ACP ₂) in a Banach space E, such solutions corresponding to “almost standing waves” in applications.

In Section 7.1, we look at the Cauchy problem for the incomplete second order equation u″(t) = Au(t) (t ∈ R). It is known that in this case the first propagator C(t) is a cosine operator function, and the second propagator S(t) a sine operator function. We characterize (Theorem 1.2) a.p. cosine (or sine) operator functions in terms of the spectral properties of A. In the case when E is a Hilbert space, we clarify the relation between a.p. cosine operator functions and sine operator functions and show their structures in terms of the mean value P _λ (see Theorem 1.3 and the statement above it). For general Banach spaces, we show (Theorem 1.7) that, if S(t) is a.p., so are all the solutions of the incomplete (ACP ₂).