This monograph establishes a theory of classification and translation closedness of time scales, a topic that was first studied by S. Hilger in 1988 to unify continuous and discrete analysis. The authors develop a theory of translation function on time scales that contains (piecewise) almost periodic functions, (piecewise) almost automorphic functions and their related generalization functions (e.g., pseudo almost periodic functions, weighted pseudo almost automorphic functions, and more). Against the background of dynamic equations, these function theories on time scales are applied to study the dynamical behavior of solutions for various types of dynamic equations on hybrid domains, including evolution equations, discontinuous equations and impulsive integro-differential equations.
The theory presented allows many useful applications, such as in the Nicholson`s blowfiles model; the Lasota-Wazewska model; the Keynesian-Cross model; in those realistic dynamical models with a more complex hibrid domain, considered under different types of translation closedness of time scales; and in dynamic equations on mathematical models which cover neural networks. This book provides readers with the theoretical background necessary for accurate mathematical modeling in physics, chemical technology, population dynamics, biotechnology and economics, neural networks, and social sciences.

Autorentext

Chao Wang is a Professor and PhD in Mathematics at Yunnan University in China. Dr. Wang has authored the book "Theory of Translation Closedness for Time Scales" (978-3-030-38643-6), published by Springer. His research focuses on the fields of nonlinear dynamic systems, control theory, fuzzy dynamic equations, fractional differential equations, bifurcation theory, nonlinear analysis, and numerical modeling. Ravi P. Agarwal is a Professor at the Texas A&M University-Kingsville, USA. He completed his PhD at the Indian Institute of Technology, Madras, India, in 1973. Dr. Agarwal has published 1700 research articles in several different fields and authored or co-authored 50 books, including "Theory of Translation Closedness for Time Scales" (978-3-030-38643-6), published by Springer.

Inhalt
Preface.- Preliminaries and Basic Knowledge on Time Scales.- A Classification of Closedness of Time Scales under Translations.- Almost Periodic Functions and Generalizations on Complete-Closed Time Scales.- Piecewise Almost Periodic Functions and Generalizations on Translation Time Scales.- Almost Automorphic Functions and Generalizations on Translation Time Scales.- Nonlinear Dynamic Equations on Translation Time Scales.- Impulsive Dynamic Equations on Translation Time Scales.- Almost Automorphic Dynamic Equations on Translation Time Scales.- Analysis of Dynamical System Models on Translation Time Scales.- Index.