Weighted piecewise pseudo double-almost periodic solution for impulsive evolution equations

Abstract

In this paper, based on the concept and properties of almost-complete closedness time scales (ACCTS), we investigate the existence of weighted pseudo double-almost periodic mild solutions for non-autonomous impulsive evolution equations. We also consider the exponential stability of weighted pseudo double-almost periodic solutions. Finally, we conclude our paper by providing several illustrative applications to different types of dynamic equations and mathematical models. These applications justify the practical usefulness of the established theoretical results.

Discontinuous generalized double-almost-periodic functions on almost-complete-closed time scales

“…Moreover, the first results of some biological dynamic models, economic dynamic models, and neural networks with double-almost-periodicity were established in Sect. 4 of [33] and also in [34].…”

“…Throughout this paper, by using the same notations and definitions in Sect. 3 in [33], we will establish some fundamental results of WPDAP and WPDAP 0 .…”

“…Remark 3.2 The applications of weighted piecewise double-almost-periodic functions on impulsive evolution equations were investigated in the literature [33]. Moreover, the first results of some biological dynamic models, economic dynamic models, and neural networks with double-almost-periodicity were established in Sect.…”

“…Many natural phenomena are in such a time variable structure, for instance, the time intervals of a round for a celestial body motion, the time intervals of recurrence of a tidal flood, etc. This phenomenon will cause a time approximation when almost-periodic problems are considered (see [32][33][34]). Hence, it is necessary to introduce and study functions on "almostcomplete-closed" time scales.…”

“…Traditionally, researchers analyze functions on complete-closed time scales, which provides no difficulties in introducing and studying functions, especially for some important and basic functions like almostperiodic functions, almost-automorphic functions, and so on, because such functions are defined by their translations on the domain and their domain is at least a complete-closed time scale. However, in the Introduction part of this paper and recent works[32][33][34], ACCTS are a class of necessary and important time variable forms that exist. Obviously, if we let ε 1 → 0 in Definition 3.7, then one can easily obtain the convergence form for piecewise functions on CCTS.In view of Definition 3.7, we now introduce the second definition of piecewise continuous double-almost-periodic functions on ACCTS: Let T be an almost-complete-closed time scale.…”