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

Abstract

In this paper, we introduce the concept of almost-complete-closed time scales (ACCTS) that allows independent variables of functions to possess almost-periodicity under translations. For this new type of time scale, a class of piecewise functions with double-almost-periodicity is proposed and studied. Based on these, concepts of weighted pseudo-double-almost-periodic functions (WPDAP) in Banach spaces and a translation-almost-closed set are introduced. Further, we prove that the function space WPDAP 0 affiliated to WPDAP is a translation-almost-closed set. Then, by introducing the concept of almost-uniform convergence for piecewise functions on ACCTS and using measure theory on time scales, some composition theorems of WPDAP and the completeness of the function space are proved. MSC: 26E70; 43A60

A Survey of Function Analysis and Applied Dynamic Equations on Hybrid Time Scales

“…Based on the results of approximation property of time scales, a new type of almost periodic functions called double-almost periodic functions was proposed and applied to study neural networks and biological dynamic models, and some new results of the existence and stability of the double-almost periodic solutions were established (see [ 26 , 27 ]). Moreover, these results were also extended to discontinuous cases and some notions of piecewise double-almost periodic functions and their generalizations were put forward and applied to study the impulsive dynamic equations and models (see [ 28 , 29 , 30 , 31 ]).…”

Almost periodic fractional fuzzy dynamic equations on timescales: A survey

“…In 2019, Wang et al (see [66]) introduced a notion of local pseudo‐almost automorphic functions on an arbitrary timescale with a bounded graininess function $$$ \mu $$$ based on the concepts of changing‐periodic timescales and obtained some properties of local pseudo‐almost automorphic functions. In addition, Wang et al (see [67]) introduced the concepts of almost‐complete‐closed timescales (ACCTS) and weighted pseudo‐double‐almost‐periodic functions (WPDAP) in Banach spaces and a translation‐almost‐closed set.…”

Fundamental solution matrix and Cauchy properties of quaternion combined impulsive matrix dynamic equation on time scales

“…However, there is no work on the combined matrix dynamic equations on time scales under quaternionic background. Moreover, the dynamic equations with impulses demonstrate their advantages in describing the dynamical behavior with a sudden change or an impact, it is significant to investigate the impulsive dynamic equations on hybrid domains (see [43][44][45][46][47][48][49]). Motivated by the above, since impulsive dynamic equations play a vital role in depicting the natural phenomena with sudden changes in the practical applications (see [11,19]), we will introduce a quaternion matrix combined-exponential function and study its properties.…”