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

Abstract

As an effective tool to unify discrete and continuous analysis, time scale calculus have been widely applied to study dynamic systems in both theoretical and practical aspects. In addition to such a classical role of unification, the dynamic equations on time scales have their own unique features which the difference and differential equations do not possess and these advantages have been highlighted in describing some complicated dynamical behavior in the hybrid time process. In this review article, we conduct a survey of abstract analysis and applied dynamic equations on hybrid time scales, some recent main results and the related developments on hybrid time scales will be reported and the future research related to this research field is discussed. The results presented in this article can be extended and generalized to study both pure mathematical analysis and real applications such as mathematical physics, biological dynamical models and neural networks, etc.

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]).…”