Theory of Translation Closedness for Time Scales: With Applications in Translation Functions and Dynamic Equations

Preface
Contents
1 Preliminaries and Basic Knowledge on Time Scales
	1.1 Some Basic Results of Δ-Calculus on Time Scales
		1.1.1 One-Sided Δ-Derivative
		1.1.2 Δ-Calculus
		1.1.3 Lebesgue Δ-Measure and Δ-Measurable Function
		1.1.4 Riemann Δ-Integral, Lebesgue Δ-Integral, and Some Important Convergence Theorems
		1.1.5 Henstock–Kurzweil Δ-Integral
	1.2 Some Basic Results of -Calculus on Time Scales
		1.2.1 One-Sided -Derivative
		1.2.2 -Calculus
		1.2.3 Lebesgue -Measure and -Measurable Function
		1.2.4 Riemann -Integral, Lebesgue -Integral, and Some Important Convergence Theorems
		1.2.5 Henstock–Kurzweil -Integral
2 A Classification of Closedness of Time Scales Under Translations
	2.1 Periodic Time Scales and Translations Invariance
	2.2 EA-Computation Method of Hausdorff Distance Between Translation Time Scales
	2.3 Time Scale Spaces and Completeness
		2.3.1 Almost Periodic Time Scales
		2.3.2 Embedding of Time Scales
		2.3.3 Approximation Time Scale Spaces Induced by Functions
		2.3.4 The Properties of Almost Periodic Time Scales
	2.4 Complete-Closed Translations Time Scales (CCTS)
	2.5 Almost Complete-Closed Time Scales Under Translations(ACCTS)
	2.6 Changing-Periodic Time Scales
	2.7 Some Compactness Criteria on Time Scales
	2.8 Analysis of General Delays on Translation Time Scales
		2.8.1 Delay Systems on Time Scales with a Monotone Interval Length
		2.8.2 Delay Systems on Periodic Time Scales
		2.8.3 Delay Systems on Almost Periodic Time Scales
3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales
	3.1 Almost Periodic Functions
	3.2 Bohr-Transform and Mean-Value
	3.3 Generalized Pseudo Almost Periodic Functions
	3.4 Π-Semigroup and Moving-Operators
	3.5 The Equivalence of Two Concepts of Relatively Dense Sets
	3.6 Abstract Weighted Pseudo Almost Periodic Functions
	3.7 Almost Periodic Functions on Changing-Periodic Time Scales
4 Piecewise Almost Periodic Functions and Generalizations on Translation Time Scales
	4.1 Piecewise Almost Periodic Functions on CCTS
	4.2 Weighted Piecewise Pseudo Almost Periodic Functions on CCTS
	4.3 Weighted Piecewise Pseudo Double-Almost Periodic Functions on ACCTS
5 Almost Automorphic Functions and Generalizations on Translation Time Scales
	5.1 Almost Automorphic Functions on CCTS
	5.2 Almost Automorphic Functions on Semigroups Inducedby CCTS
		5.2.1 Bochner Almost Automorphic Functions on Semigroups
		5.2.2 Bohr Almost Automorphic Functions on Semigroups
	5.3 Equivalence of Bochner and Bohr Almost Automorphy on Semigroup Related to Time Scales
	5.4 Weighted Pseudo Almost Automorphic Functions on CCTS
	5.5 Weighted Piecewise Pseudo Almost Automorphic Functions
	5.6 Local Almost Automorphic Functions on Changing-Periodic Time Scales
		5.6.1 Local Semigroup on Changing-Periodic Time Scales
		5.6.2 Local Almost Automorphic Functions on Translation Invariant Sub-Timescales
		5.6.3 Local Pseudo Almost Automorphic Functionson Sub-CCTS
6 Nonlinear Dynamic Equations on Translation Time Scales
	6.1 Almost Periodic Generalized Solutions for Dynamic Equations on CCTS
		6.1.1 Almost Periodic Solutions for Delay Dynamic Equations
		6.1.2 Pseudo Almost Periodic Solutions for DynamicEquations
	6.2 Weighted Pseudo Almost Periodic Solutions UnderΠ-Semigroup
	6.3 Local-Periodic Solutions on Changing-Periodic Time Scales
		6.3.1 The Clh Space on Changing-Periodic Time Scales
		6.3.2 Preliminary Results of Krasnosel\'skiĭ\'s Fixed Point Theorem
		6.3.3 Positive Local-Periodic Solutions for FDEID
7 Impulsive Dynamic Equations on Translation Time Scales
	7.1 The Cauchy Matrix and Liouville\'s Formula on Time Scales
	7.2 Piecewise Almost Periodic Solutions on CCTS
	7.3 Weighted Piecewise Pseudo Almost Periodic Solutions on CCTS
	7.4 -Equivalent Impulsive Functional Dynamic Equationson ACCTS
		7.4.1 Double-Almost Periodic -Equivalent Impulsive FDE
		7.4.2 Weighted Piecewise Pseudo Double-Almost Periodic Mild Solutions
8 Almost Automorphic Dynamic Equations on Translation TimeScales
	8.1 Weighted Pseudo Almost Automorphic Solutions on CCTS
	8.2 Abstract Almost Automorphic Impulsive -Dynamic Equations
	8.3 Semilinear Automorphic Dynamic Equations on Changing-periodic Time Scales
	8.4 Almost Automorphic Solution on Semigroups Induced by CCTS
9 Analysis of Dynamical System Models on Translation Time Scales
	9.1 Exponential Dichotomies of Impulsive Dynamic Systems with Applications
		9.1.1 Exponential Type of Bounds of Solutions for Impulsive Dynamic Equations
		9.1.2 Some New Mean-Value Criteria for ExponentialDichotomy
		9.1.3 Applications of Exponential Dichotomy on Almost Periodic Impulsive Dynamic Equations
	9.2 Almost Periodic Analysis of Impulsive Lasota-Wazewska Model on ACCTS
		9.2.1 Matrix Measure on Time Scales and Its Properties
		9.2.2 Existence and Exponential Stability of Almost Periodic Solutions of the Model on ACCTS
	9.3 Double-Almost Periodic Analysis of High-Order Hopfield Neural Networks
		9.3.1 Existence of Double-Almost Periodic Solutions with Slight Vibration in Time Variables
		9.3.2 ψ-Exponential Stability of Double-Almost Periodic Solutions
References
Index