Almost periodic and asymptotically almost periodic functions: part I

Dhaou Lassoued,Rahim Shah,Tongxing Li

doi:10.1186/s13662-018-1487-0

Abstract

In this paper, we review indispensable properties and characterizations of almost periodic functions and asymptotically almost periodic functions in Banach spaces. Special accent is put on the Stepanov generalizations of almost periodic functions and asymptotically almost periodic functions. We also recollect some basic results regarding equi-Weyl-almost periodic functions and Weyl-almost periodic functions. The class of asymptotically Weyl-almost periodic functions, introduced in this work, seems to be not considered elsewhere even in the scalar-valued case. We actually introduce eight new classes of asymptotically almost periodic functions and analyze relations between them. In order to make a picture as complete and clear as possible, several illustrating examples and counter-examples are given. It is worth noting that the topics dealt with in this paper seem to be of an intrinsic connection with the problem of existence and uniqueness of solutions of differential and difference equations, in both determinist and stochastic cases.

Highlights

The theory of almost periodic functions has gradually been increased to a comprehensive and extensive theory by the contributions of numerous mathematicians
The theory of almost periodic functions was developed in its main features by Bohr as a generalization of pure periodicity in three rather long papers [1–3], under the common title ‘Zur Theorie der fastperiodischen Funktionen’ in 1925 and 1926
Afterwards, the theory of almost periodic functions was continuously getting established by several mathematicians like Amerio and Prouse [4], Levitan [5], Besicovitch, Bochner, von Neumann, Fréchet, Pontryagin, Lusternik, Stepanov, Weyl, etc.; with respect to this matter, we cite [6–10] and the references therein

Summary

Introduction

The theory of almost periodic functions has gradually been increased to a comprehensive and extensive theory by the contributions of numerous mathematicians. By BUC(I : X) we denote the space consisting of all bounded uniformly continuous functions from I to X. Letting f ∈ AP([0, ∞) : X) and f ∈ BUC([0, ∞) : X), f ∈ AP([0, ∞) : X) To see this, it suffices to apply assertion (7) from the same theorem by noticing that the sequence defined by fn(t) := n[f (t + 1/n) – f (t)], t ≥ 0 of almost periodic functions converges uniformly to f (t) for t ≥ 0, because fn(t) – f (t) t+1/n. This important class of functions will not be considered in the sequel (for further details concerning this intriguing topic and connections between almost periodicity and Carleman spectrum of functions, one may refer to the monograph [21] and the references cited therein)

Asymptotically almost periodic functions

Asymptotically Stepanov almost periodic functions

Asymptotically Weyl almost periodic functions