Almost periodic and asymptotically almost periodic functions: part I

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. MSC: 42A75

Interval Analysis and Calculus for Interval-Valued Functions of a Single Variable—Part II: Extremal Points, Convexity, Periodicity

“…Several additional properties and applications can be possibly established, in particular with reference to specific problems and questions such as, e.g., the study of almost-periodicity for interval-valued functions in constructive mathematics (see [34][35][36]) or the solution of interval differential equations (IDE), including the extension of ordinary and partial differential equations to the interval-valued case using gH-derivatives (examples of reflected and advanced differential equations of special type are described, e.g., in [37]).…”