Uniformly rd-piecewise almost periodic functions with applications to the analysis of impulsive Δ-dynamic system on time scales

Almost periodic fractional fuzzy dynamic equations on timescales: A survey

“…Besides, almost periodic theory was developed by Bohr during 1923Bohr during -1925 (see [42,43]) and could be used to describe many natural phenomena in engineering, life sciences, information sciences, and control theory (see [44][45][46]). Since then, many papers have been devoted to solving the problems of almost periodicity on timescales (see [47][48][49][50][51]). In [47], the author introduced the dynamic equations on timescales and their applications.…”

“…In [53], Wang introduced a class of impulsive stochastic Nicholson's blowflies model with patch structure and nonlinear harvesting terms on timescales and established some sufficient conditions for the existence and exponential stability of piecewise mean-square almost periodic solutions for the model with infinite delays by using contraction mapping principal and Gronwall-Bellman inequality technique. In 2015, Wang et al established some new generalizations of invariance under translation timescales and almost periodic functions in [54] and introduced two equivalent concepts of uniformly rd-piecewise almost periodic functions on timescales and solved an impulsive non-autonomous Nicholson's blowflies system model with patch structure and multiple nonlinear harvesting terms in [51]. In addition, Wang et al put forward the concept of changing-periodic timescales.…”

Almost Anti-Periodic Discrete Oscillation of General N-Dimensional Mechanical System and Underactuated Euler-Lagrange System

“…Almost periodic oscillation is a hot research field in the study of dynamic equations (see ). In recent years, many results of the various types of dynamic equations have been established related to almost periodic background including the results of the almost periodic analysis on the stochastic dynamic equations, fuzzy dynamic equations, and the dynamic equations on hybrid domains (see [3,[12][13][14][16][17][18][19][20][22][23][24]). Based on the theory of translation closedness for time scales, the almost periodic functions and their generalizations were well defined and applied to study different types of dynamic models on time scales (see [1,3,[12][13][14][15][16][17][18][19][20]23,[25][26][27][28][29]).…”

“…Anti-periodic functions were widely used in homogenization theory for composite materials and the continualization of discrete lattices (see [31,32]). It is well known that although antiperiodic oscillation belongs to a periodic oscillation, it is able to reflect a more particularly accurate oscillation in a period through the switch of its oscillation direction, and it has been widely applied in many interdisciplines (see [4][5][6][8][9][10][11][12][13][14][17][18][19][20][21]23,[25][26][27][28][29][30]33,34]).…”

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.…”