Piecewise pseudo-almost periodic solution for impulsive non-autonomous high-order Hopfield neural networks with variable delays

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

“…In [ 15 ], some sufficient conditions were obtained for the existence and exponential stability of piecewise mean-square almost periodic solutions of the impulsive stochastic Nicholson’s blowflies model on translation time scales. In [ 16 , 17 , 18 , 19 , 20 ], the authors firstly introduced the concept of piecewise almost periodic and almost automorphic functions on time scales with periodicity and applied them to analyze the almost periodic solutions to neural networks and biological dynamic models.…”

Almost periodic fractional fuzzy dynamic equations on timescales: A survey

“…Furthermore, the authors studied a new impulsive Lasota‐Wazewska model with patch structure and forced perturbed terms on almost periodic timescales in [58] and introduced the new concepts of piecewise almost periodic functions and solved a general type of delay neural networks with impulsive effects in [59]. In [60], the authors established some completely new sufficient conditions of the existence and exponential stability of piecewise pseudo‐almost periodic solutions for impulsive non‐autonomous high‐order Hopfield neural networks with variable coefficients and delays by using some fixed point theorems in Banach space and the inequality technique. In 2017, Wang and Agarwal (see [61]) proposed two new concepts of mean‐square almost periodic stochastic process and provided some sufficient conditions to guarantee the existence of mean‐square almost periodic solution for a new type of neutral impulsive stochastic Lasota‐Wazewska model involving $$$ q $$$‐difference model on timescales.…”

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.…”