Almost Periodic Functions and Their Applications: A Survey of Results and Perspectives

Abstract

The main aim of this survey article is to present several known results about vector-valued almost periodic functions and their applications. We separately consider almost periodic functions depending on one real variable and almost periodic functions depending on two or more real variables. We address several open problems and possibilities for further investigations of almost periodic functions, quoting more than two hundred references about the subject under our consideration.

Lyapunov Approach for Almost Periodicity in Impulsive Gene Regulatory Networks of Fractional Order with Time-Varying Delays

“…The almost periodicity concepts defined in Definitions 5-7 are adopted from [38,39]. Similar notions are applied in [35,[40][41][42][43][44][45]. We will refer the reader to [49,50] for more details and definitions of almost periodic sequences and functions.…”

“…The practical meaning of our conclusions is that when the translation rates, basal rates, and the connecting parameters in the considered fractional impulsive GRN are variable (not constants) but bounded, and the magnitudes of the impulsive perturbations satisfy conditions of Theorem 2, then the model is capable to generate a unique globally Mittag-Leffler stable almost periodic process. Indeed, the concept of almost periodicity has deep historical roots [36] and has important applications in applied mathematical models [35,[39][40][41][42]47,48]. Indeed, it is an unrealistic assumption that the behavior of the states in the GRNs are not affected by periodical environmental factors.…”

“…One of these properties is the almost periodicity of the states. On the importance of studying of almost periodic properties and processes we will direct readers to [35][36][37][38][39]. This is the main reason of the existing of numerous results on almost periodicity of different classes of applied problems [40][41][42], including fractional neural networks [43][44][45].…”

Fractional-Order Impulsive Delayed Reaction-Diffusion Gene Regulatory Networks: Almost Periodic Solutions

“…The concept of almost periodic functions was defined by several mathematicians in equivalent manner. The most common definitions are these of Bohr, Bohner, Stepanov and Weyl [66,67]. The notion has been applied to differential equations and becomes an essential topic in the qualitative analysis of their solutions [68].…”

Impulsive Fractional Differential Inclusions and Almost Periodic Waves

“…In fact, due to the importance of the notion of almost periodicity, the qualitative theory for almost periodic functions and solutions related to integer-order systems is very well developed. See, for example, [15][16][17][18][19] and the bibliography therein. It is worth noting that the theory of almost periodicity is relatively well developed for fractionalorder systems [20][21][22].…”