Theory of Translation Closedness for Time Scales: With Applications in Translation Functions and Dynamic Equations [1st ed. 2020] 3030386430, 9783030386436

Table of contents :
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

Citation preview

Developments in Mathematics

Chao Wang Ravi P. Agarwal Donal O’ Regan Rathinasamy Sakthivel

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

Developments in Mathematics Volume 62

Series Editors Krishnaswami Alladi, Department of Mathematics, University of Florida, Gainesville, FL, USA Pham Huu Tiep, Department of Mathematics, Rutgers University, Piscataway, NJ, USA Loring W. Tu, Department of Mathematics, Tufts University, Medford, MA, USA

The Developments in Mathematics (DEVM) book series is devoted to publishing well-written monographs within the broad spectrum of pure and applied mathematics. Ideally, each book should be self-contained and fairly comprehensive in treating a particular subject. Topics in the forefront of mathematical research that present new results and/or a unique and engaging approach with a potential relationship to other fields are most welcome. High quality edited volumes conveying current state-of-the-art research will occasionally also be considered for publication. The DEVM series appeals to a variety of audiences including researchers, postdocs, and advanced graduate students.

More information about this series at http://www.springer.com/series/5834

Chao Wang • Ravi P. Agarwal • Donal O’Regan Rathinasamy Sakthivel

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

Chao Wang Department of Mathematics Yunnan University Kunming, Yunnan, China

Ravi P. Agarwal Department of Mathematics Texas A&M University–Kingsville Kingsville, TX, USA

Donal O’Regan School of Mathematics Statistics and Applied Mathematics National University of Ireland Galway, Ireland

Rathinasamy Sakthivel Department of Applied Mathematics Bharathiar University Coimbatore, Tamil Nadu, India

ISSN 1389-2177 ISSN 2197-795X (electronic) Developments in Mathematics ISBN 978-3-030-38643-6 ISBN 978-3-030-38644-3 (eBook) https://doi.org/10.1007/978-3-030-38644-3 Mathematics Subject Classification (2020): 34N05, 43A60, 42A75, 93A30 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

We dedicate this book to our family members: Chao Wang dedicates the book to his son Xingbo Wang and wishes him to grow up healthily and happily; Ravi P. Agarwal dedicates the book to his wife Sadhna Agarwal; Donal O’Regan dedicates the book to his wife Alice and his children Aoife, Lorna, Daniel, and Niamh; Sakthivel Rathinasamy dedicates the book to his parents Rathinasamy and Lakshmi, his wife Priyadharshini, and his son Pranav.

Preface

The theory of time scales was initiated by S. Hilger in his PhD thesis [141] in 1988 in order to unify continuous and discrete analysis. This new and exciting type of mathematics is more general and versatile than the traditional theories of differential and difference equations as it can, under one framework, mathematically describe continuous-discrete hybrid processes and hence is the optimal way forward for accurate mathematical modeling in physics, chemical technology, population dynamics, biotechnology and economics, neural networks, and social sciences. In fact, the progressive field of dynamic equations on time scales contains links and extends the classical theory of differential and difference equations. For instance, if T = Z, we have a result for difference equations, if T = R, we obtain a result for differential equations. This theory represents a powerful tool for applications to economics, population models, quantum physics among others. Not only does the new theory of the so-called dynamic equations unify the theories of differential equations and difference equations but also extends these classical cases to cases “in between,” e.g., to the so-called q-difference equations when T = q N0 := {q t : t ∈ N0 for q > 1} ∪ {0} or T = q Z := q Z ∪ {0} (which has important applications in quantum theory) and can be applied on different types of time scales like T = hN, T = N2 , and T = Tn the space of the harmonic numbers. Therefore, dealing with problems of differential equations on time scales becomes very important and meaningful in the research field of dynamic systems. Since a time scale is an arbitrary nonempty closed subset of real numbers, its irregular distribution on the real line leads to many difficulties in studying functions on time scales, especially in investigating functions defined by the translations of arguments such as periodic functions, almost periodic functions, and almost automorphic functions, etc. The classes of functions defined by the translations of arguments are referred to as Translation Functions. The concept of almost periodic functions was first proposed by H. Bohr, and such a type of real-valued functions is approximately periodic and can be to within any desired level of precision if we endow the function with suitably long, well-distributed “almost-periods” [59]. This concept was generalized by V. Stepanov, H. Weyl, and A.S. Besicovitch, among vii

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others [57]. Moreover, J.V. Neumann also introduced and studied the notion of almost periodic functions on locally compact abelian groups [60, 189]. Almost periodicity is a property of dynamical systems that seems to be repeated in their paths through phase space, but not precisely. For example, consider a planetary system, with planets in orbits moving with periods that are not commensurable (i.e., with a period vector that is disproportional to a vector of integers). A theorem of Kronecker from Diophantine approximation can be applied to demonstrate that any particular configuration that occurs once will recur to within any specified accuracy: if we wait long enough we can observe the planets all return to within a second of arc to the positions they once were in. In 1955–1962, S. Bochner observed in various contexts that a certain property enjoyed by the almost periodic functions on the group G can be applied in obtaining more concise and logical proofs of certain theorems in terms of these functions; S. Bochner called his property “almost automorphy” since it first arouse in work on differential geometry (see [61–63]). Based on well-known almost periodic and almost automorphic functions proposed by Bohr and Bochner, many new generalized concepts were introduced and studied by several researchers on the real line. However, these theories do not work on time scales since the classical concepts of almost periodic and almost automorphic functions purely depend on the translation of functions. For example, the classical definition of almost periodic functions on the real line is given as follows: For any bounded complex function f and ε > 0, we define T (f, ε) = {τ : |f (t + τ ) − f (t)| < ε for all t}, T (f, ε) is called the ε-translation set of f . We say f is Bohr almost periodic if for any ε > 0, T (f, ε) is relatively dense. However, the above definition will not be true on time scales. In fact, for an arbitrary time scale T, there may be no fixed τ ∈ R such that t + τ ∈ T for all t ∈ T. This problem is so complex that it will change the classical concept of relatively dense set on the real line, the convergence of function sequences, the completeness of function spaces, and an almost periodicity of the variable limit integrals, etc. Therefore, it is extremely insufficient just to assume t +τ := τ (t) ∈ T for all t ∈ T when we consider almost periodic problems. In fact, this general assumption also has some other serious deficiencies, for instance, (1) it is to the disadvantage of the analysis of numerical computation including the simulation of almost periodic functions since the construction of the subset of R where τ is from is unknown; (2) the relatively dense property in the sense of time scales cannot be considered under this general and abstract assumption; (3) there is no translation closedness for the vast majority of time scales, much less the translation invariance, that is, there is no τ ∈ R (i.e., there is no subset of R where τ is from) such that t + τ ∈ T for all t ∈ T, which leads to the assumption τ (t) ∈ T meaningless (we have provided several representative examples in Chap. 2). For the same reasons, the study of almost automorphic problems on time scales is also a difficult task. To overcome these difficulties, it is important to study the classification of time scales under translations and the translation closedness of time scales. Depending on the reference system of the real line, we find that an arbitrary time scale with a bounded graininess function μ may possess a well local complete closedness which is more general than translation invariance, it provides an essential condition to

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consider almost periodic problems, almost automorphic problems, and other related generalized problems on arbitrary time scales. For our accurate discussion in the book, we say a time scale is a translation time scale if a mathematical problem arising and solving must be based on a translation of the time scale. In this monograph, we establish a theory of classification and translation closedness of time scales and based on this we develop a theory of translation functions on time scales which contains (piecewise) almost periodic functions, (piecewise) almost automorphic functions, and their related generalized functions (e.g., pseudo almost periodic functions, weighted pseudo almost automorphic functions, etc.). Under 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, impulsive integro-differential equations. Also, the book provides several applications of dynamic equations on mathematical models which cover neural networks, Nicholson’s blowflies model, Lasota–Wazewska model, Keynesian Cross model, those realistic dynamical models with more complex hybrid domain are considered under different types of translation time scales. This monograph is organized in 9 chapters: In Chap. 1, we present the preliminaries and basic concepts of calculus and measure theory on time scales. In Chap. 2, we classify time scales by translation and develop a theory of translation closedness for time scales. We introduce the concepts of complete-closed translation time scales (CCTS for short), almost-complete closed translation time scales (ACCTS for short), and changing-periodic time scales. As the particular cases, the properties of the translation and almost translation invariance of periodic and almost periodic time scales are investigated. Moreover, the concept of time scale spaces is introduced and the embedding theorems of time scales are established. Based on it, almost automorphic time scales are introduced and studied. For changing-periodic time scales, some basic theorems such as the Decomposition Theorem of Time Scales and the Periodic Coverage Theorem of Time Scales are proposed and proved. In addition, we initiate the methods of delay classification analysis of delay dynamic equations on translation time scales. In Chap. 3, a theory of almost periodic functions and their generalizations such as pseudo almost periodic functions and weighted pseudo almost periodic functions is established on CCTS and changing-periodic time scales. Also, the Bohr-Transform and Mean-Value of almost periodic functions, Π -semigroup and moving-operators are proposed and discussed, which are the effective tools of investigating almost periodic and almost automorphic solutions of dynamic equations on CCTS and changing-periodic time scales. In Chap. 4, a notion of piecewise almost periodic functions and the corresponding generalizations are introduced and studied on different types of time scales. Moreover, the concepts of double-almost periodic functions and weighted piecewise pseudo double-almost periodic functions are introduced and discussed on ACCTS. In Chap. 5, we develop a theory of generalized almost automorphic functions on translation time scales. The Bochner and Bohr almost automorphic functions

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on semigroups induced by CCTS are proposed and studied. As their generalizations, the weighted (piecewise) pseudo almost periodic functions on CCTS are investigated. Moreover, a notion of local pseudo almost periodic functions on changing-periodic time scales is introduced and some basic properties are obtained. In Chap. 6, we mainly discuss nonlinear dynamic equations on translation time scales. The almost periodic generalized solutions for dynamic equations on CCTS are investigated. Under Π -semigroups on time scales, the weighted pseudo almost periodic solutions for nonlinear abstract dynamic equations are studied. In addition, based on the Clh space on changing-periodic time scales, the existence of local periodic solutions for functional dynamic equations with infinite delay is established through Krasnosel’ski˘ı’s Fixed Point Theorem. In Chap. 7, we discuss some related problems of impulsive dynamic equations on translation time scales. The Cauchy matrix and Liouville’s formula on time scales for impulsive dynamic equations are derived and their almost periodicity is analyzed. Based on it, the almost periodic solutions of impulsive delay dynamic equations are investigated and several applications are provided. In addition, εequivalent impulsive functional dynamic equations are proposed and studied. Moreover, the existence and exponential stability of weighted piecewise pseudo double-almost periodic mild solutions of impulsive evolution equations are discussed on ACCTS. In Chap. 8, on different types of translation time scales, the almost automorphic problems of different types of dynamic equations on time scales including several representative classes of Δ and ∇-dynamic equations are discussed. We mainly discuss the related problems on CCTS, changing-periodic time scales, and semigroups induced by CCTS. In Chap. 9, we focus on analyzing dynamical system models on translation time scales. The exponential dichotomies of some representative types of dynamic equations on time scales are discussed and some new mean-value criteria for exponential dichotomy are given and proved and applied to analyze the almost periodic problems of several real dynamic systems models. Moreover, the matrix measure on time scales is introduced to analyze a class of impulsive Lasota– Wazewska model on ACCTS and the existence and exponential stability of almost periodic solutions of the model are obtained. Finally, a class of double-almost periodic high-order Hopfield neural networks is proposed and some sufficient conditions for the existence and ψ-exponential stability of double-almost periodic solutions with slight vibration in time variables are established. This is a monograph devoted to developing a theory of translation time scales and applications to translation functions and dynamic equations. The study of translation closedness of time scales will not only contribute to studying translation functions such as periodic functions, almost periodic functions, and almost automorphic functions and their generalizations but also will contribute to analyzing the delays in delay dynamic equations on arbitrary time scales. These related topics on dynamic equations have become a major research field in pure and applied mathematics. In particular, this book will cover related results in the discrete and continuous cases. Moreover, some new notions of time scales are introduced and discussed in

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detail including complete-closed translation time scales (CCTS), almost-complete closed time scales (ACCTS), and changing-periodic time scales which are efficient and original in solving related problems of dynamic equations and dynamic system models with properties of translation functions. The book is written at a graduate level, and is intended for university libraries, graduate students, and researchers working in the field of general dynamic equations on time scales and it will stimulate further research into time scale theory. We acknowledge with gratitude the support of National Natural Science Foundation of China (11961077,11601470), IRTSTYN and Joint Key Project of Yunnan Provincial Science and Technology Department of Yunnan University (No. 2018FY001(-014)). Kunming, China Kingsville, TX, USA Galway, Ireland Coimbatore, India

Chao Wang Ravi P. Agarwal Donal O’Regan Rathinasamy Sakthivel

Contents

1

2

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 8 15 19 23 28 28 34 37 41 45 51 51 56 79 80 82 86 93 106 110 115 131

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3

4

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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 . . . . . . . . . . . . . .

149 161 162

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 . . . . . .

169 169 194 204 213 222 224 234

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Almost Automorphic Functions and Generalizations on Translation Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Almost Automorphic Functions on CCTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Almost Automorphic Functions on Semigroups Induced by 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 Functions on Sub-CCTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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239 239 252 265 283 283 290 290 298 303 304 310 323 324 326 328

Nonlinear Dynamic Equations on Translation Time Scales . . . . . . . . . . . . . 337 6.1 Almost Periodic Generalized Solutions for Dynamic Equations on CCTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 6.1.1 Almost Periodic Solutions for Delay Dynamic Equations . . . . 345

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6.1.2

Pseudo Almost Periodic Solutions for Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Weighted Pseudo Almost Periodic Solutions Under Π -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’ski˘ı’s Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Positive Local-Periodic Solutions for FDEID . . . . . . . . . . . . . . . . . 7

8

9

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 ε-Equivalent Impulsive Functional Dynamic Equations on ACCTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Double-Almost Periodic ε-Equivalent Impulsive FDE . . . . . . . 7.4.2 Weighted Piecewise Pseudo Double-Almost Periodic Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Almost Automorphic Dynamic Equations on Translation Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . 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 Exponential Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

347 352 363 364 370 376 389 389 399 412 437 440 458 477 477 481 493 500 505 505 506 513 519 529 531 535

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Contents

9.3 Double-Almost Periodic Analysis of High-Order Hopfield Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 9.3.1 Existence of Double-Almost Periodic Solutions with Slight Vibration in Time Variables . . . . . . . . . . . . . . . . . . . . . . . 546 9.3.2 ψ-Exponential Stability of Double-Almost Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

Chapter 1

Preliminaries and Basic Knowledge on Time Scales

1.1 Some Basic Results of Δ-Calculus on Time Scales The discrete and continuous dynamical analysis are very important for studying the dynamical behavior in the real world, many mathematicians focused on these two topics and established many classical results in this field (see Agarwal [1], Agarwal et al. [7, 12], Agarwal and O’Regan [10], Adamec [33], Aulbach [19], Aulbach and Hilger [20], Ahlbrandt et al. [24], Ahlbrandt and Peterson [23], Ahlbrandt [27, 28], Akın et al. [40], DaCunha and Davis [92], Lakshmikantham and Leela [176]). Moreover, the hybrid differential equations were also investigated and the theoretical results were established (see Dhage and Lakshmikantham [94]). Due to the complex dynamical behavior of the discrete dynamic equations and its extensive use, the study of difference equations always is a hot topic (see Ferguson and Lim [133], Kelley and Peterson [164], Kratz [165], Lakshmikantham and Trigiante [172]). In 1988, Hilger initiated the time scale calculus to unify both discrete and continuous situations (see [141–144]), since then many researchers focused their attention on this topic and many results to unify discrete and continuous analysis were published (see Agarwal and Bohner [2], Agarwal et al. [3–6, 13, 14], Agarwal and O’Regan [8, 9, 11], Aulbach and Hilger [21], Aulbach and PÖtzsche [22], Ahlbrandt et al. [25, 26], Ahlbrandt and Morian [29], Ahlbrandt and Ridenhour [30]), Akın [39], Akın et al. [41], Jackson [151], Mozyrska et al. [186], Poulsen and Wintz [198], Saker [202], Seiffertt [203], Sun and Li [204], Fra˘nková [253]). In this section, we introduce some necessary knowledge of time scales to make the book self-contained. For the notions used below we refer the reader to the books [64, 65] which summarize and organize much of time scale calculus. Throughout this section, we denote the time scale by the symbol T. A time scale T is a closed subset of R. It follows that the jump operators σ,  : T → T are defined by σ (t) = inf{s ∈ T : s > t} and (t) = sup{s ∈ T : s < t} with a stipulation that inf ∅ = sup T (i.e., σ (t) = t if T has a maximum t) and sup ∅ = inf T (i.e., ρ(t) = t if T has a minimum t), where ∅ denotes the empty set. © Springer Nature Switzerland AG 2020 C. Wang et al., Theory of Translation Closedness for Time Scales, Developments in Mathematics 62, https://doi.org/10.1007/978-3-030-38644-3_1

1

2

1 Preliminaries and Basic Knowledge on Time Scales

If σ (t) > t, we say t is right scattered, while if ρ(t) < t we say t is left-scattered. Points that are right-scattered and left-scattered at the same time are called isolated. Also, if t < sup T and σ (t) = t, then t is called right-dense, and t > inf T and ρ(t) = t, then t is called left-dense. Points that are right dense and left-dense at the same time are called dense. The mapping ν : T → [0, ∞) such that ν(t) = t − ρ(t) is called the backward graininess function, the mapping μ : T → [0, ∞) such that μ(t) = σ (t) − t is called the forward graininess function. Note that both σ (t) and ρ(t) are in T when t ∈ T, this is because T is a closed subset of R. Define     T\ ρ(sup(T)), sup T ∩ T if sup T < ∞, κ T = T if sup T = ∞.    Likewise, Tκ is defined as the set Tκ = T\ inf T, σ (inf T) ∩ T if | inf T| < ∞ and Tκ = T if inf T = −∞. If f : T → R is a function, then the function  f σ , f ρ : T → R is defined by f σ (t) = f σ (t) and f ρ (t) = f ρ(t) for all t ∈ T, respectively, i.e., f σ = f ◦ σ and f ρ = f ◦ ρ. Throughout the book, for the intervals on time scales, we make the assumption that a and b are the points in T. For a ≤ b, we will denote the time scale interval [a, b]T = {t ∈ T : a ≤ t ≤ b}. Open intervals and half-open intervals etc. are defined accordingly. Note that [a, b]κT = [a, b]T if b is left-dense and [a, b]κT = [a, b)T = [a, ρ(b)]T if b is left-scattered. Similarly, ([a, b]T )κ = [a, b]T if a is right-dense and ([a, b]T )κ = (a, b]T = [σ (a), b]T if a is right-scattered.

1.1.1 One-Sided Δ-Derivative In this subsection, we introduce the concept of one-sided Δ-derivative and some basic results. Definition 1.1 For y : T → R and t ∈ T, we define the right-side Δ-derivative of Δ (t), to be the number (if it exists) with the property that for a given ε > 0, y(t), y+ there exists a right-side neighborhood U of t (i.e., U = [t, t + δ)T for some δ > 0) such that [y(σ (t)) − y(s)] − y Δ (t)[σ (t) − s] < ε|σ (t) − s| +

for all s ∈ U. That is, the limit Δ y+ (t)

exists.

  f σ (t) − f (s) = lim σ (t) − s s→t +

(1.1)

1.1 Some Basic Results of Δ-Calculus on Time Scales

3

Definition 1.2 For y : T → R and t ∈ T, we define the left-side Δ-derivative of Δ (t), to be the number (if it exists) with the property that for a given ε > 0, y(t), y− there exists a left-side neighborhood U of t (i.e., U = (t − δ, t]T for some δ > 0) such that [y(σ (t)) − y(s)] − y Δ (t)[σ (t) − s] < ε|σ (t) − s| −

for all s ∈ U. That is, the limit Δ y− (t) = lim

s→t −

  f σ (t) − f (s) σ (t) − s

exists.

  Remark 1.1 In Definition 1.1, if | inf T| < ∞ and t ∈ inf T, σ (inf T) T , then y has the right-side Δ-derivative at t. Similarly, in Definition 1.2, if sup T < ∞ and  t ∈ ρ(sup T), sup T T , then y has the left-side Δ-derivative at t. Definition 1.3 For y : T → R and t ∈ Tκ , we define the Δ-derivative of y(t), y Δ (t), to be the number (if it exists) with the property that for a given ε > 0, there exists a neighborhood U of t (i.e., U = (t − δ, t + δ)T for some δ > 0) such that [y(σ (t)) − y(s)] − y Δ (t)[σ (t) − s] < ε|σ (t) − s| for all s ∈ U. That is, the limit   f σ (t) − f (s) y (t) = lim s→t σ (t) − s Δ

exists. From Definitions 1.1 and 1.2, one will easily obtain the following theorems. Theorem 1.1 Let y : T → R and t ∈ Tκ . Then y is Δ-differentiable if and only if Δ = yΔ. y+ − Remark 1.2 From Theorem 1.1, one will observe that let f : T → R and t ∈ T be an arbitrary right scattered point, then f is right continuous at t because f is right-side differentiable at t with (1.1). Theorem 1.2 Assume that f : T → R is a function and let t ∈ T. Then we have the following: (i) If f is right-side differentiable at t, then f is right-side continuous at t. (ii) If t is right scattered, then f is right-side differentiable at t with

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1 Preliminaries and Basic Knowledge on Time Scales

f+Δ (t)

  f σ (t) − f (t) = . σ (t) − t

(iii) If t is right dense, then f is right-side differentiable at t if and only if the limit lim

s→t +

f (t) − f (s) t −s

exists as a finite number. In this case, f+Δ (t) = lim

s→t +

f (t) − f (s) . t −s

(iv) If f is right-side differentiable at t, then   f σ (t) = f (t) + μ(t)f+Δ (t). Proof

 (i) Suppose f is right-side differentiable at t. Let ε ∈ (0, 1) and ε∗ = ε 1 + −1 |f+Δ (t)| + 2μ(t) . Then ε∗ ∈ (0, 1). By Definition 1.1, there exists a right side neighborhood U of t such that [f (σ (t)) − f (s)] − f Δ (t)[σ (t) − s] < ε|σ (t) − s| for all s ∈ U. + Therefore, for all s ∈ [t, t + ε∗ ) ∩ U , we obtain   |f (t) − f (s)| = f (σ (t)) − f (s) − f+Δ (t)[σ (t) − s]   − f (σ (t)) − f (t) − μ(t)f+Δ (t) + (t − s)f+Δ (t) ≤ ε∗ |σ (t) − s| + ε∗ μ(t) + |t − s||f+Δ (t)|   ≤ ε∗ μ(t) + |t − s| + μ(t) + |f+Δ (t)|   < ε∗ 1 + |f+Δ (t)| + 2μ(t) = ε.

Hence we have f is right-side continuous at t. (ii) If t is a right-scattered point, then f is right-side continuous at t, so lim

s→t +

f (σ (t)) − f (t) f (σ (t)) − f (t) f (σ (t)) − f (s) = = σ (t) − s σ (t) − t μ(t)

exists. Hence for given ε > 0, there is a right-side neighborhood U of t such that

1.1 Some Basic Results of Δ-Calculus on Time Scales

5

f (σ (t)) − f (s) f (σ (t)) − f (t) < ε for all s ∈ U. − σ (t) − s μ(t) It follows that [f (σ (t)) − f (s)] − f (σ (t)) − f (t) [σ (t) − s] < ε|σ (t) − s| for all s ∈ U. μ(t) Hence, we obtain f+Δ (t)

  f σ (t) − f (t) . = σ (t) − t

(iii) Assume f is right-side differentiable at t and t is right dense. Let ε > 0 be given. Since f is right-side differentiable at t, there is a right-side neighborhood U of t such that [f (σ (t)) − f (s)] − f Δ (t)[σ (t) − s] ≤ ε|σ (t) − s| +

for all s ∈ U . Since σ (t) = t we obtain that [f (t) − f (s)] − f Δ (t)[t − s] ≤ ε|t − s| + for all s ∈ U . It follows that f (t) − f (s) Δ − f+ (t) ≤ ε t −s for all s ∈ U, s = t. Therefore we get f+Δ (t) = lim

s→t +

f (t) − f (s) . t −s

On the contrary, if the limit lim

s→t +

f (t) − f (s) := f+Δ (t) t −s

exists as a finite number, then for any ε > 0, there exists a right-side neighborhood U of t such that f (t) − f (s) Δ − f+ (t) ≤ ε t −s for all s ∈ U, s = t. Since t is right-dense, we have σ (t) = t, that is,

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1 Preliminaries and Basic Knowledge on Time Scales

f (σ (t)) − f (s) Δ − f+ (t) ≤ ε, σ (t) − s so we obtain [f (σ (t)) − f (s)] − f Δ (t)[σ (t) − s] ≤ ε|σ (t) − s| +

for all s ∈ U . (iv) If σ (t) = t, then μ(t) = 0 and we have that f (σ (t)) = f (t) = f (t) + μ(t)f+Δ (t). On the other hand, if σ (t) > t, then by (ii) f (σ (t)) = f (t) + μ(t)

f (σ (t)) − f (t) = f (t) + μ(t)f+Δ (t). μ(t)

This completes the proof. 

According to Theorem 1.2, the following theorem is immediate. Theorem 1.3 Assume that f : T → R is a function and let t ∈ T. Then we have the following: (i) If f is left-side differentiable at t, then f is left-side continuous at t. (ii) If t is right scattered and f is left-side continuous at t, then f is left-side differentiable at t with f−Δ (t)

  f σ (t) − f (t) . = σ (t) − t

(iii) If t is right dense, then f is left-side differentiable at t if and only if the limit lim

s→t −

f (t) − f (s) t −s

exists as a finite number. In this case, f−Δ (t) = lim

s→t −

f (t) − f (s) . t −s

(iv) If f is left-side differentiable at t, then   f σ (t) = f (t) + μ(t)f−Δ (t). According to (ii) from Theorems 1.2 and 1.3, it immediately follows that

1.1 Some Basic Results of Δ-Calculus on Time Scales

7

Theorem 1.4 Let f : T → R and t be an isolated point in T. Then f has two one-sided Δ-derivatives and f−Δ (t) = f+Δ (t) on T , i.e., f is Δ-differentiable at t. From Theorem 1.4, we can also obtain Corollary 1.1 Let f : T → R and t be an isolated point in T. Then f is continuous at t. Remark 1.3 Note that if t is just a right-scattered point in T and f : T → R, then f may be not Δ-differentiable at t. Example 1.1 Let T = [1, 2] ∪ [3, 4] and ⎧ 2 ⎪ ⎪ ⎨t + 1, t ∈ [1, 2), f (t) = 3, t = 2, ⎪ ⎪ ⎩t, t ∈ [3, 4]. Then we have f+Δ (2) = lim

s→2+

f−Δ (2) = lim

s→2−

f (σ (2)) − f (2) f (σ (2)) − f (s) = = 0, σ (2) − s σ (2) − 2 f (σ (2)) − 5 f (σ (2)) − f (s) = = −2, σ (2) − s σ (2) − 2

so we have f+Δ (2) = f−Δ (2), i.e., f is not Δ-differentiable at t. Theorem 1.5 Assume that f : T → R is a function and let t ∈ the following:



Tκ .

Then we have

(i) If f is differentiable at t, then f is continuous at t. (ii) If f is continuous at t and t is right scattered, then f is differentiable at t with   f σ (t) − f (t) f (t) = . σ (t) − t Δ

(iii) If t is right dense, then f is differentiable at t if and only if the limit lim

s→t

f (t) − f (s) t −s

exists as a finite number. In this case, f Δ (t) = lim s→t

f (t) − f (s) . t −s

8

1 Preliminaries and Basic Knowledge on Time Scales

(iv) If f is differentiable at t, then   f σ (t) = f (t) + μ(t)f Δ (t). Remark 1.4 Notice that for T = [a, b], Definition 1.10 from the literature [64] will include the one-sided derivatives of f at a and b, where [a, b]κ = [a, b], i.e,   f Δ (a) = f+ (a) and f Δ (b) = f− (b). By Theorem 1.16 (i) of the literature [64], one will obtain that f is continuous at a and b since they are Δ-differentiable. However, f is right continuous at a and left-continuous at b, respectively. In fact,   from Sect. 1.1.1, we know that f+ (a) and f− (b) belong to one-sided Δ-derivative   of f , i.e., f+ (a) = f+Δ (a) and f− (b) = f−Δ (b), which implies that f is right continuous at a and left-continuous at b.

1.1.2 Δ-Calculus In this subsection, we will introduce some basic knowledge of Δ-calculus and measure on time scales. Definition 1.4 ([64, 65]) A function f : T → R is called regulated provided its right-sided limits exist (finite) at all right-dense points in T and its left-sided limits exist (finite) at all left-dense points in T. Definition 1.5 ([64, 65]) The function f : T → R is called rd-continuous provided that it is continuous at each right-dense point and has a left-sided limit at left dense points. The set of rd-continuous functions f : T → R will be denoted in this book by Crd (T) = Crd (T, R). The set of functions f : T → R that are Δ-differentiable 1 (T) = C 1 (T, R). and whose derivative is rd-continuous is denoted by Crd rd Definition 1.6 ([64, 65]) Assume f : T → R is a function and let t ∈ Tκ . Then we define f Δ (t) to be the number (provided it exists) with the property that given any ε > 0, there exists a neighborhood U of t (i.e., U = (t − δ, t + δ) ∩ T for some δ > 0) such that |f (σ (t)) − f (s) − f Δ (t)[σ (t) − s]| ≤ ε|σ (t) − s| for all s ∈ U , we call f Δ (t) the delta (or Hilger) derivative of f at t. A function F : T → R is called an antiderivative of f : T → R provided F Δ (t) = f (t) holds for all t ∈ Tκ , and we define the Cauchy delta integral of f by  a

t

f (s)Δs = F (t) − F (a) for all t, a ∈ T.

1.1 Some Basic Results of Δ-Calculus on Time Scales

9

Theorem 1.6 ([64, 65]) Assume f, g : T → R are differentiable at t ∈ Tκ . Then: (i) The sum f + g : T → R are differentiable at t with (f + g)Δ (t) = f Δ (t) + g Δ (t). (ii) For any constant α, αf : T → R is differentiable at t with (αf )Δ = αf Δ (t). (iii) The product f g : T → R is differentiable at t with     (f g)Δ (t) = f Δ (t)g(t) + f σ (t) g Δ (t) = f (t)g Δ (t) + f Δ (t)g σ (t) .   (iv) If f (t)f σ (t) = 0, then

1 f

is differentiable at t with

 Δ 1 f Δ (t)  . (t) = − f f (t)f σ (t)   (v) If g(t)g σ (t) = 0, then

f g

is differentiable at t and

 Δ f f Δ (t)g(t) − f (t)g Δ (t)   (t) = . g g(t)g σ (t) Theorem 1.7 ([64, 65]) If a, b, c ∈ T, α, β ∈ R, and f, g ∈ Crd , then b b b  (i) a αf (t) + βg(t) Δt = α a f (t)Δt + β a g(t)Δt; b a (ii) a f (t)Δt = − b f (t)Δt; c b c (iii) a f (t)Δt = a f (t)Δt + b f (t)Δt; b b (iv) a f (t)Δt ≤ a |f (t)|Δt. Definition 1.7 ([64, 65]) For h > 0 we define the Hilger complex numbers, the Hilger real axis, the Hilger alternating axis, and the Hilger imaginary circle as   1 Ch := z ∈ C : z = − , h   1 Rh := z ∈ Ch : z ∈ R and z > − , h   1 Ah := z ∈ Ch : z ∈ R and z < − , h

10

1 Preliminaries and Basic Knowledge on Time Scales

 Ih := z ∈ Ch

 1 1 , : z + = h h

respectively. For h = 0, let C0 := C, R0 := R, I0 = iR, and A0 := ∅. Definition 1.8 ([64, 65]) Let h > 0 and z ∈ Ch . We define the Hilger real part of z by Reh (z) :=

|zh + 1| − 1 h

and the Hilger imaginary part of z by Arg(zh + 1) , h

I mh (z) :=

where Arg(z) denotes the principle argument of z (i.e., −π < Arg(z) ≤ π ). Note that Reh (z) and I mh (z) satisfy −

1 π π < Reh (z) < ∞ and − < I mh (z) ≤ , h h h

respectively. In particular, Reh (z) ∈ Rh . Definition 1.9 ([64, 65]) Let − πh < ω ≤ number˚ ιω by ˚ ιω =

π h . We define the Hilger purely imaginary

eiωh − 1 . h

For z ∈ Ch ,˚ ιI mh (z) ∈ Ih . Theorem 1.8 ([64, 65]) If the “circle plus” addition ⊕ is defined by z ⊕ ω := z + ω + zωh, then (Ch , ⊕) is an Abelian group. For z ∈ Ch , we have z = Reh (z) ⊕ ˚ ιI mh (z). Definition 1.10 ([64, 65]) The “circle minus” substraction  on Ch is defined by −ω z  ω := z ⊕ (ω), where ω := 1+ωh . For h > 0, let Zh be the strip   π π , Zh := z ∈ C : − < I m(z) ≤ h h and for h = 0, let Z0 := C. Definition 1.11 ([64, 65]) For h > 0, the cylinder transformation ξh : Ch → Zh by

1.1 Some Basic Results of Δ-Calculus on Time Scales

ξh (z) =

11

1 Log(1 + zh), h

where Log is the principal logarithm function. For h = 0, we define ξ0 (z) = z for all z ∈ C. We define addition on Zh by z + ω := z + ω

  2π i for z, ω ∈ Zh . mod h

(1.2)

Theorem 1.9 ([64, 65]) The inverse transformation of the cylinder transformation ξh when h > 0 is given by ξh−1 (z) =

1 zh (e − 1) h

for z ∈ Zh . For h = 0, ξ0−1 (z) = z. Theorem 1.10 ([64, 65]) The cylinder transformation ξh is a group homomorphism from (Ch , ⊕) onto (Zh , +), where the addition + on Zh is defined by (1.2). Definition 1.12 ([64, 65]) A function p : T → R is called μ-regressive provided 1+μ(t)p(t) = 0 for all t ∈ Tκ . The set of all regressive and rd-continuous functions p : T → R will be denoted by R = R(T) = R(T, R). We define the set R + = R + (T, R) = {p ∈ R : 1 + μ(t)p(t) > 0, ∀ t ∈ T}. The set of all regressive functions on a time scale T forms an Abelian group under the addition ⊕ defined by p ⊕ q := p + q + μpq. Definition 1.13 ([64, 65]) If r is a μ-regressive function, then the generalized exponential function er is defined by 



t

er (t, s) = exp

ξμ(τ ) (r(τ ))Δτ s

for all s, t ∈ T, where the μ-cylinder transformation is as in ξh (z) :=

1 Log(1 + zh). h

Theorem 1.11 ([64, 65]) Assume that p, q : T → R are two μ-regressive functions. Then (i) (ii) (iii) (iv) (v)

e0 (t, s) ≡ 1 and ep (t, t) ≡ 1; ep (σ (t), s) = (1 + μ(t)p(t))ep (t, s); 1 ep (t, s) = ep (s,t) = ep (s, t); ep (t, s)ep (s, r) = ep (t, r); (ep (t, s))Δ = (p)(t)ep (t, s);

12

1 Preliminaries and Basic Knowledge on Time Scales

The following two theorems are obvious. For more inequalities on time scales, one may consult [13, 14]. Theorem 1.12 For ϕ ≥ 0 with −ϕ ∈ R + , the following inequalities hold  exp



t

− s

  t ϕ(u)Δu ≤ e−ϕ (t, s) ≤ 1 + ϕ(u)Δu for all t ≥ s. s

If ϕ is rd-continuous and nonnegative, then  1−

t

 ϕ(u)Δu ≤ eϕ (t, s) ≤ exp

s

t

 ϕ(u)Δu for all t ≥ s.

s

Theorem 1.13 If λ ∈ R + and λ(r) < 0 for all r ∈ [s, t)T , then 0 < eλ (t, s) < 1. Theorem 1.14 ([64, 65]) If p ∈ R and a, b, c ∈ T, then  Δ  σ ep (c, ·) = −p ep (c, ·) and 

b

  p(t)ep c, σ (t) Δt = ep (c, a) − ep (c, b).

a

Lemma 1.1 ([64, 65]) If z ∈ C is regressive and t0 ∈ T, then σ ez (t, t0 ) =

(z)(t) ez (t, t0 ) =− ez (t, t0 ), 1 + μ(t)z z

σ (t, t ) = e (σ (t), t ). where ez 0 z 0

Theorem 1.15 ([64, 65]) Suppose A and B are differentiable n × n-matrix-valued functions. Then (i) (ii) (iii) (iv) (v)

(A + B)Δ = AΔ + B Δ ; (αA)Δ = αAΔ if α is constant; (AB)Δ = AΔ B σ + AB Δ = Aσ B Δ + AΔ B; (A−1 )Δ = −(Aσ )−1 AΔ A−1 = −A−1 AΔ (Aσ )−1 if AAσ is invertible; (AB −1 )Δ = (AΔ − AB −1 B Δ )(B σ )−1 = (AΔ − (AB −1 )σ B Δ )B −1 if BB σ is invertible.

Definition 1.14 ([64, 65]) An n × n-matrix-valued function A on a time scale T is called regressive (with respect to T) provided I + μ(t)A(t) is invertible for all t ∈ Tκ , and the class of all such regressive and rd-continuous functions is denoted by

1.1 Some Basic Results of Δ-Calculus on Time Scales

13

R = R(T) = R(T, Rn×n ). Definition 1.15 ([64, 65]) Assume A and B are regressive n × n-matrix-valued functions on T. Then we define A ⊕ B by (A ⊕ B)(t) = A(t) + B(t) + μ(t)A(t)B(t) for all t ∈ Tκ , and we define A by (A)(t) = −[I + μ(t)A(t)]−1 A(t) for all t ∈ Tκ . Theorem 1.16 ([64, 65]) If A is regressive on T, then (A)(t) = −A(t)[I + μ(t)A(t)]−1 for all t ∈ Tκ . Theorem 1.17 ([64, 65], Putzer Algorithm) Let A ∈ R be a constant n × nmatrix. Suppose t0 ∈ T. If λ1 , λ2 , . . . , λn are the eigenvalues of A, then eA =

n−1 

ri+1 (t)Pi ,

i=1

where r(t) := (r1 (t), r2 (t), . . . , rn (t))T is the solution of the IVP ⎛

rΔ

⎞ ⎛ ⎞ 0 1 .. ⎟ ⎜0⎟ .⎟ ⎟ ⎜ ⎟ .. ⎟ r, r(t ) = ⎜0⎟ , ⎟ ⎜ ⎟ 0 .⎟ ⎜.⎟ ⎟ ⎝ .. ⎠ 0⎠ 0 0 . . . 0 1 λn

λ1 0 0 . . . ⎜ ⎜ 1 λ2 0 . . . ⎜ ⎜ = ⎜ 0 1 λ3 . . . ⎜ ⎜. . . . ⎝ .. . . . . . .

and the P-matrices P0 , P1 , . . . , Pn are recursively defined by P0 = I and Pk+1 = (A − λk+1 I )Pk , for 0 ≤ k ≤ n − 1. Definition 1.16 ([64, 65]) Let A be an m × n-matrix-valued function on T. We say that A is rd-continuous on T if each entry of A is rd-continuous on T, and the class of all such rd-continuous m × n-matrix-valued functions on T is denoted by Crd = Crd (T) = Crd (T, Rm×n ). We say that A is differentiable on T provided each entry of A is differentiable on T, and in this case we put

14

1 Preliminaries and Basic Knowledge on Time Scales

AΔ = (aijΔ )1≤i≤m,1≤j ≤n , where A = (aij )1≤i≤m,1≤j ≤n . Theorem 1.18 ([64, 65]) If A is differentiable at t ∈ Tκ , then Aσ (t) = A(t) + μ(t)AΔ (t). Definition 1.17 ([64, 65]) If the matrix-valued functions A and B are regressive on T, then we define A  B by (A  B)(t) = (A ⊕ (B))(t) for all t ∈ Tκ . If A is a matrix, then we let A∗ denote its conjugate transpose. Theorem 1.19 ([64, 65]) If A, B ∈ R are matrix-valued functions on T, then (i) (ii) (iii) (iv) (v) (vi)

e0 (t, s) ≡ I and eA (t, t) ≡ I ; eA (σ (t), s) = (I + μ(t)A(t))eA (t, s); −1 ∗ eA (t, s) = eA ∗ (t, s); −1 ∗ eA (t, s) = eA (s, t) = eA ∗ (s, t); eA (t, s)eA (s, r) = eA (t, r); eA (t, s)eB (t, s) = eA⊕B (t, s) if eA (t, s) and B(t) commute.

Consider the nonhomogeneous equation y Δ = A(t)y + f (t), where f : T → Rn is a vector-valued function. Theorem 1.20 ([64, 65]) Let A ∈ R be an n × n-matrix-valued function on T and suppose that f : T → Rn is rd-continuous. Let t0 ∈ T and y0 ∈ Rn . Then the initial value problem y Δ = A(t)y + f (t), y(t0 ) = y0 has a unique solution y : T → Rn . Moreover, this solution is given by  y(t) = eA (t, t0 )y0 +

t

eA (t, σ (τ ))f (τ )Δτ. t0

Theorem 1.21 ([64, 65]) Let A ∈ R be an n × n-matrix-valued function on T and suppose that f : T → Rn is rd-continuous. Let t0 ∈ T and x0 ∈ Rn . Then the initial value problem x Δ = −A∗ (t)x σ + f (t), x(t0 ) = x0 has a unique solution x : T → Rn . Moreover, this solution is given by

1.1 Some Basic Results of Δ-Calculus on Time Scales

 x(t) = eA∗ (t, t0 )x0 +

15

t t0

eA∗ (t, τ )f (τ )Δτ.

Lemma 1.2 ([64, 65]) A n × n-matrix-valued function A is regressive if and only if the scalar-valued function trA + μ det A is regressive, where trA denotes the trace of the matrix A, i.e., the sum of the diagonal elements of A. Theorem 1.22 ([64, 65], Liouville’s Formula) Let A ∈ R be a 2 × 2-matrixvalued function and assume that X is a solution of XΔ = A(t)X. Then X satisfies Liouville’s formula det X(t) = etrA+μ det A (t, t0 ) det X(t0 ), for t ∈ T. Theorem 1.23 ([64, 65]) Let a ∈ Tκ , b ∈ T and assume f : T × Tκ → R is continuous at (t, t), where t ∈ Tκ with t > a. Also assume that f Δ (t, ·) is rdcontinuous on [a, σ (t)]. Suppose that for each ε > 0 there exists a neighborhood U of t, independent of τ ∈ [a, σ (t)], such that |f (σ (t), τ ) − f (s, τ ) − f Δ (t, τ )(σ (t) − s)| ≤ ε|σ (t) − s| for all s ∈ U, where f Δ denotes the derivative of f with respect to the first variable. Then t t (i) g(t) := a f (t, τ )Δτ implies g Δ (t) = a f Δ (t, τ )Δτ + f (σ (t), t); b b (ii) h(t) := t f (t, τ )Δτ implies hΔ (t) = t f Δ (t, τ )Δτ − f (σ (t), t).

1.1.3 Lebesgue Δ-Measure and Δ-Measurable Function In the following, we will introduce some basic knowledge of measure theory on time scales, for more details, we refer the readers to the literatures [72, 95, 135–137]. Let F denote the class of all bounded left closed and right open intervals of T of the form [a, b)T . If a = b, then the interval reduces to the empty set. On the class + of F of semi-closed intervals, a set function m  R ∪ {0} is defined which  :F→ assigns each interval [a, b)T its length, i.e., m [a, b)T = b − a. The function m is countably additive measure on F. In fact,   1. m[a, b) = b − a ≥0 sinceb ≥ a;  T ∞ ∞ 2. m [a i=1 i , bi )T = i=1 m [ai , bi )T for pairwise disjoint intervals [ai , bi )T . 3. m(∅) = m [a, a)T = 0 holds for any a ∈ T. Definition 1.18 ([95]) Let E be any subset of T. If there exists at least one finite or countable system of intervals I ∈ F, j = 1, 2, . . . such that E ⊂ j j Ij , then  m∗ (E) = inf j m(Ij ), is called the outer measure of E, where the infimum is taken over all coverings of E by a finite or countable system of intervals Ij ∈ F.

16

1 Preliminaries and Basic Knowledge on Time Scales

Definition 1.19 ([95]) A set E ⊂ T is called m∗ -measurable or Δ-measurable if for each interval I ⊂ F, the following holds m∗ (I ) = m∗ (I ∩ E) + m∗ (I ∩ E c ), where E c = T\E. We denote the family of all m∗ -measurable subsets of T by M(m∗ ), then we have the following theorem. Theorem 1.24 ([95]) M(m∗ ) is a σ -algebra and m∗ restricted to M(m∗ ) is a countably additive measure. Definition 1.20 ([95]) m∗ restricted to M(m∗ ) is called the Δ-measure on the time scale T, denote it by μΔ . Theorem 1.25 ([95]) If {En } is an increasing sequence of sets in T, then μΔ

 ∞

 = lim μΔ (En );

En

n→∞

i=1

if {En } is a decreasing sequence of sets in T, then μΔ

 ∞

 En

= lim μΔ (En );

i=1

n→∞

that is, the Δ-measure on T is continuous. Lemma 1.3 ([72, 136]) The single-point set {t0 } ⊂ Tκ is Δ-measurable and its Δ-measure is given by μΔ ({t0 }) = σ (t0 ) − t0 . Theorem 1.26 ([72, 136]) The set of all right-scattered points of T is at most countable, that is, there are {ti }i∈I , I ⊂ N such that T\Dr = Sr = {ti }i∈I , where Dr denotes the all right-dense points and Sr denotes the all right-scattered points of T. Remark1.5 By Theorem 1.26, we will express the extension of a set E ⊆ T as E = i∈IE [ai , bi ], where ai , bi ∈ E is the right-scattered points in T and IE denotes the indices set of right-scattered points of E. Lemma 1.4 ([72, 136]) If a, b ∈ T and a ≤ b, then μΔ ([a, b)T ) = b − a, μΔ ((a, b)T ) = b − σ (a).

1.1 Some Basic Results of Δ-Calculus on Time Scales

17

If a, b ∈ Tκ and a ≤ b, then μΔ ((a, b]T ) = σ (b) − σ (a), μΔ ([a, b]T ) = σ (b) − a. Now, we introduce a relation between Lebesgue measure and Lebesgue Δmeasure. Let λ∗ be the Lebesgue outer measure on R, λ the usual Lebesgue measure and m∗ the outer measure on T. Theorem 1.27 ([72, 136]) If E ⊂ Tκ , then the following properties are satisfied: (i) m∗ (E) ≥ λ∗ (E); (ii) if E does not include right-scattered points, then m∗ (E) = λ∗ (E); (iii) The sets Dr and Sr = T\Dr are Lebesgue measurable and λ(Sr ) = 0 but μΔ (E ∩ Sr ) =

  σ (ti ) − ti , i∈IE

where IE indicates  the indices  set for all right-scattered points in E;  (iv) m∗ (E) = i∈IE σ (ti ) − ti + λ∗ (E); (v) m∗ (E) = λ∗ (E) if and only if E has no right-scattered points. Theorem 1.28 ([72]) Let E ⊂ T. Then E is Lebesgue Δ-measurable if and only if it is Lebesgue measurable. In such a case, for E ⊂ Tκ the following is true:    (i) μΔ (E) = i∈IE σ (ti ) − ti ; (ii) λ(E) = μΔ (E) if and only if E has no right-scattered point. Now, we express the extension of a set E ⊂ T as E˜ = E ∪



 ti , σ (ti ) ,

(1.3)

i∈IE

where IE denotes the indices set of right-scattered points of E. Remark 1.6 By Theorem 1.28, one can obtain ˜ where E ⊂ Tκ . μΔ (E) = λ(E), Theorem 1.29 ([72, 95]) If a set E is Lebesgue measurable, then E ∩ T is Δmeasurable. Next, we will introduce the Δ-measurable function on time scales. Definition 1.21 ([72, 95]) We say that f : T → R is Δ-measurable if for every α ∈ R, the set     f −1 (−∞, α] := t ∈ T : f (t) < α is Δ-measurable.

18

1 Preliminaries and Basic Knowledge on Time Scales

Theorem 1.30 ([72, 95]) Let f : E → R and E be a Δ-measurable set. Then for each α ∈ R, the following statements are equivalent: : f (t) < α} is Δ-measurable; : f (t) ≥ α} is Δ-measurable; : f (t) ≤ α} is Δ-measurable; : f (t) > α} is Δ-measurable; : f (t) > α} is Δ-measurable.  Remark 1.7 Let S = ni=1 αi χAi , where

(i) (ii) (iii) (iv) (v)

{t {t {t {t {t

∈E ∈E ∈E ∈E ∈E

 χAi =

1, t ∈ Ai , 0, t ∈ Ai ,

then Ai ⊂ T is Δ-measurable set for each i if and only if S is Δ-measurable function. Definition 1.22 ([72, 95]) A proposition C (t) is called Δ-almost everywhere, shortly, Δ-a.e., if   μΔ {t ∈ T : C (t) is false} = 0. Theorem 1.31 ([72, 95]) Let f be Δ-measurable on E ⊂ T and f = g Δ-a.e. Then g is Δ-measurable. Theorem 1.32 ([72, 95]) Let f, g, fn , gn : E → R be Δ-measurable functions, where E ⊂ T, n ∈ N. Then αf (α ∈ R), f ± g, f g, fg (g = 0), supn fn , infn fn , limn fn , limn fn are Δ-measurable. Theorem 1.33 ([72, 95]) Let {fn (t)} be a sequence of Δ-measurable functions on E ⊂ T. If limn→∞ fn (t) = f (t), then f (t) is Δ-measurable. Theorem 1.34 ([72, 95]) f is Δ-measurable function if and only if for any α1 < α2 , α1 , α2 ∈ R, the set {t ∈ T : α1 < f (t) < α2 } is Δ-measurable. Theorem 1.35 ([72, 95]) If f : T → R is rd-continuous, then f is Δ-measurable. Let f : E → R (where E ⊆ T) and its extension f˜ : E˜ → R be  f (t), if t ∈ E, f˜(t) =   f (ti ), if t ∈ ti , σ (ti ) ,

(1.4)

for some i ∈ IE , {ti }i∈IE represents the set of all right-scattered points in E and    E˜ = E ∪ i∈IE ti , σ (ti ) . Theorem 1.36 ([72, 95]) f is Δ measurable function if and only if f˜ is Lebesgue measurable function.

1.1 Some Basic Results of Δ-Calculus on Time Scales

19

1.1.4 Riemann Δ-Integral, Lebesgue Δ-Integral, and Some Important Convergence Theorems In what follows, we introduce the Riemann Δ-Integral , the Lebesgue Δ-integral and Fatou’s Lemma, the Monotone Convergence Theorem, and the Lebesgue Dominated Convergence Theorem on time scales. First, we introduce some basic definitions and results of the Riemann Δ-Integral. Let [a, b)T be a half closed interval in T, a partition of [a, b)T is any ordered subset P = {t0 , t1 , . . . , tn } ⊂ [a, b]T , where a = t0 < t1 < . . . < tn = b. Let f : [a, b)T → R and M = sup{f (t) : t ∈ [a, b)T }, m = inf{f (t) : t ∈ [a, b)T }, Mi = sup{f (t) : t ∈ [ti−1 , ti )T }, mi = inf{f (t) : t ∈ [ti−1 , ti )T }. The upper Darboux Δ-sum U (f, P) and the lower Darboux Δ-sum L(f, P) of the function f are defined by U (f, P) =

n 

Mi (ti − ti−1 ), L(f, P) =

i=1

n 

mi (ti − ti−1 )

i=1

with respect to the partition P. The upper Darboux integral U (f ) from a to b is defined by U (f ) = inf{U (f, P)}, and the lower Darboux integral L(f ) from a to b is defined by L(f ) = sup{U (f, P)}. Definition 1.23 ([135, 136]) f is called the Darboux Δ-integral from a to b if b U (f ) = L(f ) and this integral is denoted by a f (t)Δt. Lemma 1.5 ([135, 136]) For any δ > 0, there exists at least one partition P : a = t0 < t1 < . . . < tn = b of [a, b)T such that for each i either ti − ti−1 ≤ δ or ti − ti−1 > δ and ρ(ti ) = ti−1 , where ρ denotes the backward jump operator on T. Theorem 1.37 ([135, 136]) A bounded function f : [a, b)T → R is Darboux Δintegrable if and only if for each ε > 0 there exists δ > 0 such that U (f, P) − L(f, P) < ε for all partitions P ∈ P, P denotes the set of all partitions that posses the property indicated in Lemma 1.5. Definition 1.24 ([135, 136]) Let f : [a, b)T → R is bounded and P : a = t0 < t1 < . . . < tn = b be a partition of [a, b)T . In each interval [ti−1 , ti )T , i = 1, 2, . . . , n, we choose an arbitrary point ξi ∈ [ti−1 , ti )T and form the sum

20

1 Preliminaries and Basic Knowledge on Time Scales

SR =

n 

f (ξi )(ti − ti−1 ),

i=1

and SR is called a Riemann Δ-sum of f corresponding to the partition P. We say f is Riemann Δ-integrable from a to b if there exists a number I with the following property: for any ε > 0, there exists δ > 0 (i.e., there exists a partition Pδ ∈ P) such that |SR − I | < ε and it is independent of the way of choosing ξi ∈ [ti−1 , ti )T . The number I is called the Riemann Δ-integral of f from a to b. Theorem 1.38 ([135, 136]) A bounded function f on [a, b)T is Riemann Δintegrable if and only if it is (Darboux) Δ-integrable and the two values of the integrals are equal. Next, we introduce some basic definitions and theorems of the Lebesgue Δintegral. Definition 1.25 ([135, 136]) Let E ⊂ T be a Δ-measurable set and S : T →  [0, +∞) be a Δ-measurable simple function with S = ni=1 αi χAi , Ai = {t ∈ T : S(t) = αi }. The Lebesgue Δ-integral of S on E is defined as  S(t)Δt = E

n 

αi μΔ (Ai ∩ E).

i=1

set and S : T → [0, +∞) be Theorem 1.39 ([72]) Let E ⊂ Tκ be a Δ-measurable  a Δ-measurable simple function with S = ni=1 αi χAi and S˜ be the extension of S     given by S˜ = ni=1 αi χA˜ i , where A˜ i = Ai ∪ i∈IA ti , σ (ti ) . Then S˜ is Lebesgue i measurable and Lebesgue integrable with 

 S(t)Δt = E

˜ S(t)dt,

E˜

where E˜ is the extension of E as given in (1.3). Definition 1.26 ([72]) Let E ⊂ T measurable set and f : T → [0, +∞) be a Δ-measurable function. The Lebesgue Δ-integral of f on E is defined as 

 f (t)Δt = sup

E

S(t)Δt, E

where the supremum is taken over all Δ-measurable nonnegative simple functions S such that S ≤ f . In the following, for simplicity, the Δ-integral means the Lebesgue Δ-integral.

1.1 Some Basic Results of Δ-Calculus on Time Scales

21

Theorem 1.40 ([95, 196] Beppo Levi’s Lemma) Assume that a sequence {fn }n∈N of Δ-integrable functions on a Δ-measurable set E satisfies fn (t) ≤ fn+1 (t) Δa.e. for all n ∈ N and limn→∞ E fn (s)Δs < ∞. Then there exists a Δ-integrable  function f such that limn→∞ fn = f and limn→∞ E fn (s)Δs = E f (s)Δs. Theorem 1.41 ([95, 196] Fatou’s Lemma) Let {fn (t)}n∈N be a sequence of Δintegrable  functions defined on a Δ-measurable set E such that fn (t) ≥ 0 Δ-a.e. and lim E fn (s)Δs < ∞. Then lim fn (t) defines a Δ-integrable function on the Δ-measurable set E and   lim fn (s)Δs ≤ lim fn (s)Δs. E

E

Theorem 1.42 ([95, 196] Monotone Convergence Theorem) Let {fn }n∈N be an increasing sequence of nonnegative Δ-measurable functions defined on a Δmeasure set E and let f (t) = limn→∞ fn (t) Δ-a.e. Then 

 lim fn (t)Δt = lim

E n→∞

n→∞ E

fn (t)Δt.

Theorem 1.43 ([95, 196] Lebesgue Dominated Convergence Theorem) Let g be Δ-integrable function over E and let {fn } be sequence of Δ-measurable functions such that |fn (t)| ≤ g(t) on E and for almost all t in E we have f (t) = limn→∞ fn (t). Then 

 f (t)Δt = lim

n→∞ E

E

fn (t)Δt.

The next lemma exhibits the relation between the classical Lebesgue integral and Lebesgue Δ-integral of nonnegative Δ-measurable functions. Lemma 1.6 ([72]) Let E ⊂ Tκ be a Δ-measurable set and f : T → [0, +∞) be a Δ-measurable function and f˜ be the extension of f as (1.4). Then 

 f (s)Δs =

E

where E˜ = E ∪

 i∈IE

E˜

f˜(s)ds,

  ti , σ (ti ) .

Definition 1.27 ([72, 95]) Let E ⊂ T be a measurable set and f : T → R be a Δ– function. We say f is Δ-integrable on E if at least one of the elements measurable  + (t)Δt or − (t)Δt is finite, where f + and f − are the positive and negative f f E E part of f respectively. In this case, we define the Lebesgue Δ-integral of f on E as

22

1 Preliminaries and Basic Knowledge on Time Scales





f + (t)Δt −

f (t)Δt = E



E

f − (t)Δt. E

Remark 1.8 In Definition 1.27, the functions f + , f − : T → [0, +∞) is defined by  +

f (t) =

f (t), f (t) ≥ 0, 0, f (t) < 0,

 −

f (t) =

−f (t), f (t) ≤ 0, 0,

f (t) > 0.

Theorem 1.44 ([72, 95]) Let f, g : E → R be a Δ-integrable functions and E ⊂ T be a Δ-measurable set and α, β ∈ R. Then (i) af  and f + g is Δ-integrable  on E;  (ii) E (αf (t) + βg(t))Δt = α E f (t)Δt  + β E g(t)Δt;  (iii) If f (t) ≤ g(t) for each t ∈ E, then E f (t)Δt ≤ E g(t)Δt. Remark 1.9 Note that the classical Lebesgue integral has the same value on I = [a, b], [a, b), (a, b] and (a, b). However, the Lebesgue Δ-integral with respect to μΔ for each interval is different from each other. One can easily check that   f (t)Δt = f (t)Δt + f (b)μ(b);  

[a,b]T

[a,b)T

[a,b)T

f (t)Δt = (a,b]T



f (t)Δt = f (a)μ(a) +

f (t)Δt; (a,b)T

 [a,b)T

f (t)Δt − f (a)μ(a) + f (b)μ(b).

Theorem 1.45 ([72]) Let E ⊂ Tκ be a Δ-measurable set and f : T → R be Δmeasurable function. If f˜ be the extension of f as given in (1.4), then f is Lebesgue Δ-integrable on E if and only if f˜ is Lebesgue integrable on E˜ and   f (t)Δt = f˜(t)dt. E˜

E

Theorem 1.46 ([72]) Let E ⊂ Tκ be a Δ-measurable set. If f : T → R is Δintegrable on E, then    f (t)Δt = f (s)ds + f (ti )μ(ti ). E

E˜

i∈IE˜

Theorem 1.47 ([72, 95], Approximation of Δ-Integral) Let f be an rdcontinuous function on an interval  I ⊂ T and {In } be a sequence of intervals (where In ⊂ I for each n) such that ∞ n=1 In = {a}. Then

1.1 Some Basic Results of Δ-Calculus on Time Scales

1 n→∞ μΔ (In )

23

 f (t)Δt = f (a).

lim

In

From Theorem 1.47, it is easy to see that Corollary 1.2 ([72]) Let f be rd-continuous and bounded function on [a, b)T and F is antiderivative of f . Then  [a,b)T

f (t)Δt = F (b) − F (a).

The following theorem give a relationship between the Lebesgue Δ-integral and the Riemann Δ-integral. Theorem 1.48 ([135, 136]) Let [a, b)T be a half closed bounded interval in T and f : [a, b)T → R be bounded. If f is Riemann Δ-integrable from a to b, then f is b  Lebesgue Δ-integrable on [a, b)T and a f (t)Δt = [a,b)T f (t)Δt, where the left hand side of the equation is the Riemann Δ-integral and the right hand side of the equation is the Lebesgue Δ-integral. Theorem 1.49 ([135, 136]) Let [a, b)T be a half closed bounded interval in T and f : [a, b)T → R be bounded. Then f is Riemann Δ-integrable from a to b if and only if the set of all right dense points of [a, b)T at which f is discontinuous is a set of Δ-measure zero.

1.1.5 Henstock–Kurzweil Δ-Integral In what follows, we will introduce the Henstock–Kurzweil Δ-Integral which will be used in the book. For more details, one may consult the literature[196]. Definition 1.28 ([196]) We say δ = (δL , δR ) is a Δ-gauge for [a, b]T provided δL (t) > 0 on (a, b]T , δR (t) > 0 on [a, b)T , δL (a)(t) ≥ 0, δR (b) ≥ 0, and δR (t) ≥ μ(t) for all t ∈ [a, b)T . Definition 1.29 ([196]) A partition P for [a, b]T is a division of [a, b]T denoted by P = {a = t0 ≤ ξ1 ≤ t1 ≤ . . . tn−1 ≤ ξn ≤ tn = b} with ti > ti−1 for 1 ≤ i ≤ n and ti , ξi ∈ T. We call the points ξi tag points and the points ti end points. Sometimes we denote such a partition by P = {[u, v]; ξ }; where [u, v] denotes a typical interval in P and ξ is the associated tag point in [u, v].

24

1 Preliminaries and Basic Knowledge on Time Scales

Definition 1.30 ([196]) If δ is a Δ-gauge for [a, b]T , then we say a partition P is a δ-fine if ξi − δL (ξi ) ≤ ti−1 < ti ≤ ξi + δR (ξi ) for 1 ≤ i ≤ n. Now we can define the Henstock–Kurzweil Δ-integral. Definition 1.31 ([196]) We say that f : [a, b]T → R, is Henstock–Kurzweil delta b integrable on [a, b]T with value I = H K a f (t)Δt, provided given any ε > 0, there exists a Δ-gauge, δ, for [a, b]T such that n  I − f (ξ )(t − t ) i i i−1 < ε i=1

for all δ-fine partitions P of [a, b]T . Theorem 1.50 ([196]) If δ is a Δ-gauge for [a, b]T , then there is a δ-fine partition P for [a, b]T .  Example 1.2 Assume [a, b)T contains a countable infinite subset ∞ i=1 {ri } with σ (ri ) = ri . Define f : [a, b]T → R by  f (t) =

1,

t = ri ,

0,

t = ri .

For any given ε > 0, we define a Δ-gauge, δ, on [a, b]T by δL (ri )= δR (ri ) = ε/2i+2 , i ≥ 1, δR (t) = max{1, μ(t)} and δL (t) = 1 for t ∈ [a, b]T \ ∞ i=1 {ri }. Let P be a δ-fine partition of [a, b]T , then n ∞ ∞      ε ≤ δ f (ξ )(t − t ) (r ) + δ (r ) = < ε. i i i−1 L i R i i+1 2 i=1

i=1

i=1

Hence, even though in many cases f is not Δ-integrable on [a, b]T , we have f is Henstock–Kurzweil Δ-integrable on [a, b]T and  HK

b

f (t)Δt = 0.

a

Theorem 1.51 ([196]) Assume f : [a, b]T → R. If f is HK-Δ-integrable on b [a, b]T , then the value of the integral H K a f (t)Δt does not depend on f (b). b On the other hand, if c ∈ [a, b)T and c is right-scattered, then H K a f (t)Δt does depend on the value f (c)μ(c).

1.1 Some Basic Results of Δ-Calculus on Time Scales

25

We give the notation Nμ := {zj ∈ [a, b)T : μ(zj ) > 0} and note that Nμ is a countable set. Theorem 1.52 ([196]) Assume that F : [a, b]T → R is continuous, f : [a, b]T → R, and there is a set D with Nμ ⊂ D ⊂ [a, b]κT such that F Δ (t) = f (t) for t ∈ D and [a, b]T \D is countable. Then f is HK-Δ-integrable on [a, b]T with 

b

HK

f (t)Δt = F (b) − F (a).

a

Example 1.3 Let T = {t = 1/n : n ∈ N} ∪ {0} and define f : T → R by  f (t) =

(−1)n n, t = 1/n, L,

t = 0,

where L is any constant. Note that f is not Δ-differentiable on [0, 1]T . However, it can be shown that if F : T → R is defined by

F (t) =

⎧ ⎪ ⎪ ⎨0, 

t = 1, n (−1)k+1 k=2 k−1 ,

⎪ ⎪ ⎩− ln 2,

t = 1/n, t = 0,

then F Δ (t) = f (t) for t ∈ (0, 1)T and F is continuous on [0, 1]T . It follows by Theorem 1.52 with D = (0, 1)T , that f is HK-Δ-integrable on [0, 1]T and 

1

HK

f (t)Δt = F (1) − F (0) = ln 2.

0

1 However, it can be shown that H K 0 |f (t)|Δt does not exist (i.e., f is not absolutely HK-Δ-integrable on [0, 1]T . 

Theorem 1.53 ([196]) If f : T → R is regulated and a, b ∈ T, then f is HK-Δintegrable on [a, b]T and 

b

HK a

Moreover, if G(t) :=

t a

 f (t)Δt =

b

f (t)Δt. a

f (s)Δs, then GΔ (t) = f (t) except for a countable set.

Remark 1.10 Let δ 1 , δ be Δ-gauges for [a, b]T such that 0 < δL1 (t) ≤ δL (t) for t ∈ (a, b]T and 0 < δR1 (t) ≤ δR (t) for t ∈ [a, b)T (write δ 1 ≤ δ and we say δ 1 is finer than δ). If P1 is a δ 1 -finer partition of [a, b]T , then δ 1 is a δ-fine partition of [a, b]T ).

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1 Preliminaries and Basic Knowledge on Time Scales

Remark 1.11 If c ∈ [a, b]T and P is a δ-fine partition, then there is a δ-fine partition with c as a tag point. Remark 1.12 If c ∈ [a, b]T and ti−1 ≤ c ≤ ti is a tag point in a δ-fine partition P, then 

P = {t0 = a ≤ ξ1 ≤ . . . ≤ ti−1 ≤ c ≤ ti ≤ . . . ≤ tn = b}, where c is an end point and a tag point for the two intervals [ti−1 , c]T and [c, ti ]T is a δ-fine partition and the Riemann sum corresponding to these two partitions is the same. This follows from the simple fact that f (c)[ti − ti−1 ] = f (c)[ti − c] + f (c)[c − ti−1 ]. Remark 1.13 Let a < c < b be points in T, then we may choose the gauge δ so that δR (t), δL (t) ≤ |t − c| for t < c, t > c respectively. Then, if P is a δ-fine partition of [a, b]T , then ξi0 = c for some i0 . If ti0 −1 < ξi0 , then we may add yi0 to the partition so that {a = t0 ≤ ξ1 ≤ t1 ≤ . . . ≤ ti0 −1 ≤ ξi0 = yi0 ≤ . . . ≤ b} so that {a = t0 ≤ ξ1 ≤ t1 ≤ . . . ≤ ti0 −1 ≤ ξi0 = yi0 = c} and {c = yi0 ≤ ξ0 ≤ . . . ≤ tn = b} are δ-fine partition of [a, c]T and [c, b]T , respectively. Theorem 1.54 ([196]) Let f : [a, b]T → R. Then f is HK-Δ-integrable on [a, b]T if and only if f is HK-Δ-integrable on [a, c]T and [c, b]T . Moreover, in this case 

b

HK a



c

f (t)Δt = H K

 f (t)Δt + H K

a

b

f (t)Δt. c

Also if f, g : [a, b]T → R are HK-Δ-integrable on [a, b]T , then αf + gβ is HK-Δintegrable on [a, b]T and 

b

HK a

    αf (t) + βg(t) Δt = α H K

b a

   f (t)Δt + β H K

b

 f (t)Δt .

a

Definition 1.32 ([196]) We say a subset S of a time scale T has Δ-measure zero provided S contains no right scattered points and S has Lebesgue measure zero.

1.1 Some Basic Results of Δ-Calculus on Time Scales

27

We say a property A holds Δ-almost everywhere (Δ-a.e.) on T provided there is a subset S of T such that the property A holds for all t ∈ T\S and S has Δ-measure zero. Theorem 1.55 ([196]) If f and g are HK-Δ-integrable on [a, b]T and f (t) ≤ g(t) Δ-a.e. on [a, b)T , then 



b

b

f (t)Δt ≤ H K

HK a

g(t)Δt. a

Theorem 1.56 ([196]) Assume f is HK-Δ-integrable on [a, b]T . Then given any ε > 0 there is a Δ-gauge, δ, of [a, b]T such that  n  H K i=1

ti ti−1

f (t)Δt − f (ξi )(ti − ti−1 ) < ε

for all δ-fine partitions P of [a, b]T . Theorem 1.57 ([196] Monotone Convergence Theorem) let fk , f : [a, b]T → R and assume that (i) (ii) (iii) (iv)

fk is HK-Δ-integrable, k ∈ N; fk → f Δ-a.e. on [a, b)T ; fk ≤ fk+1 Δ-a.e. on [a, b)T , k ∈ N; b limk→∞ H K a fk (t)Δt = I .

Then f is HK-Δ-integrable on [a, b]T and 

b

I = HK

f (t)Δt. a

Theorem 1.58 ([196] Dominated Convergence Theorem) Assume that (i) fk → f Δ-a.e. on [a, b]T ; (ii) g ≤ fk ≤ h Δ-a.e. on [a, b]T ; (iii) fk , g, h are HK-Δ-integrable on [a, b]T . Then f is HK-Δ-integrable on [a, b]T and  lim H K

k→∞

a

b

 fk (t)Δt = H K

b

f (t)Δt. a

Definition 1.33 ([196]) We say f : [a, b]T → R is absolutely HK-Δ-integrable on [a, b]T provided f and |f | are HK-Δ-integrable on [a, b]T .

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1 Preliminaries and Basic Knowledge on Time Scales

Theorem 1.59 ([196]) The function f is Lebesgue Δ-integrable on [a, b]T if and only if f is absolutely HK-Δ-integrable on [a, b]T . Theorem 1.60 ([196]) The function F is absolutely continuous on [a, b]T if and only if F Δ (t) = f (t) Δ-a.e. on [a, b]T is absolutely HK-Δ-integrable function f on [a, b]T . Moreover, 

b

HK

 |F (t)|Δt = Δ

a

[a,b]T

|F Δ (t)|dt +



|F σ (ti ) − F (ti )|.

ti ∈Nμ

1.2 Some Basic Results of ∇-Calculus on Time Scales The corresponding theory for ∇-calculus was also studied extensively. From the relation between Δ and ∇-calculus theory, one can also obtain the symmetric knowledge of ∇-calculus and related theorems, which can be summarized as follows.

1.2.1 One-Sided ∇-Derivative In this subsection, we introduce the concept of one-sided ∇-derivative and some basic results. Definition 1.34 For y : T → R and t ∈ T, we define the right-side ∇-derivative of ∇ (t), to be the number (if it exists) with the property that for a given ε > 0, y(t), y+ there exists a right-side neighborhood U of t (i.e., U = [t, t + δ)T for some δ > 0) such that ∇ (t)[ρ(t) − s] < ε|ρ(t) − s| [y(ρ(t)) − y(s)] − y+ for all s ∈ U. That is, the limit ∇ y+ (t)

  f ρ(t) − f (s) = lim ρ(t) − s s→t +

(1.5)

exists. Definition 1.35 For y : T → R and t ∈ T, we define the left-side ∇-derivative of ∇ (t), to be the number (if it exists) with the property that for a given ε > 0, y(t), y− there exists a left-side neighborhood U of t (i.e., U = (t − δ, t]T for some δ > 0) such that

1.2 Some Basic Results of ∇-Calculus on Time Scales

29

∇ (t)[ρ(t) − s] < ε|ρ(t) − s| [y(ρ(t)) − y(s)] − y− for all s ∈ U. That is, the limit ∇ y− (t)

  f ρ(t) − f (s) = lim ρ(t) − s s→t −

exists.

  Remark 1.14 In Definition 1.34, if | inf T| < ∞ and t ∈ inf T, σ (inf T) T , then y has the right-side ∇-derivative at t. Similarly, in Definition 1.35, if sup T < ∞ and  t ∈ ρ(sup T), sup T T , then y has the left-side ∇-derivative at t. Definition 1.36 For y : T → R and t ∈ Tκ , we define the ∇-derivative of y(t), y ∇ (t), to be the number (if it exists) with the property that for a given ε > 0, there exists a neighborhood U of t (i.e., U = (t − δ, t + δ)T for some δ > 0) such that [y(ρ(t)) − y(s)] − y ∇ (t)[ρ(t) − s] < ε|ρ(t) − s| for all s ∈ U. That is, the limit   f ρ(t) − f (s) y (t) = lim s→t ρ(t) − s ∇

exists. From Definitions 1.34 and 1.35, one will easily obtain the following theorems. Theorem 1.61 Let y : T → R and t ∈ Tκ . Then y is ∇-differentiable if and only if ∇ = y∇ . y+ − Remark 1.15 From Theorem 1.34, one will observe that let f : T → R and t ∈ T be an arbitrary left scattered point, then f is left continuous at t because f is leftside differentiable at t with (1.5). Theorem 1.62 Assume that f : T → R is a function and let t ∈ T. Then we have the following: (i) If f is left-side differentiable at t, then f is left-side continuous at t. (ii) If t is left scattered, then f is left-side differentiable at t with f−∇ (t)

  f ρ(t) − f (t) . = ρ(t) − t

(iii) If t is left-dense, then f is left-side differentiable at t if and only if the limit

30

1 Preliminaries and Basic Knowledge on Time Scales

lim

s→t −

f (t) − f (s) t −s

exists as a finite number. In this case, f−∇ (t) = lim

s→t −

f (t) − f (s) . t −s

(iv) If f is left-side differentiable at t, then   f ρ(t) = f (t) − ν(t)f−∇ (t). Proof

 (i) Suppose f is left-side differentiable at t. Let ε ∈ (0, 1) and ε∗ = ε 1 + −1 |f−∇ (t)| + 2ν(t) . Then ε∗ ∈ (0, 1). By Definition 1.1, there exists a left-side neighborhood U of t such that [f (ρ(t)) − f (s)] − f−∇ (t)[ρ(t) − s] < ε|ρ(t) − s| for all s ∈ U. Therefore, for all s ∈ (t − ε∗ , t] ∩ U , we obtain   |f (t) − f (s)| = f (ρ(t)) − f (s) − f−∇ (t)[ρ(t) − s]   − f (ρ(t)) − f (t) + ν(t)f−∇ (t) + (t − s)f−∇ (t) ≤ ε∗ |ρ(t) − s| + ε∗ ν(t) + |t − s||f−∇ (t)|   ≤ ε∗ ν(t) + |t − s| + ν(t) + |f−∇ (t)|   < ε∗ 1 + |f−∇ (t)| + 2ν(t) = ε.

Hence we have f is left-side continuous at t. (ii) If t is a left-scattered point, then f is left-side continuous at t, so lim

s→t −

f (ρ(t)) − f (t) f (t) − f (ρ(t)) f (ρ(t)) − f (s) = = ρ(t) − s ρ(t) − t ν(t)

exists. Hence for given ε > 0, there is a left-side neighborhood U of t such that f (ρ(t)) − f (s) f (ρ(t)) − f (t) < ε for all s ∈ U. + ρ(t) − s ν(t) It follows that

1.2 Some Basic Results of ∇-Calculus on Time Scales

31

[f (ρ(t)) − f (s)] + f (ρ(t)) − f (t) [ρ(t) − s] < ε|ρ(t) − s| for all s ∈ U. ν(t) Hence, we obtain f−∇ (t)

  f ρ(t) − f (t) = . ρ(t) − t

(iii) Assume f is left-side differentiable at t and t is left-dense. Let ε > 0 be given. Since f is left-side differentiable at t, there is a left-side neighborhood U of t such that [f (ρ(t)) − f (s)] − f ∇ (t)[ρ(t) − s] ≤ ε|ρ(t) − s| −

for all s ∈ U . Since ρ(t) = t we obtain that [f (t) − f (s)] − f Δ (t)[t − s] ≤ ε|t − s| − for all s ∈ U . It follows that f (t) − f (s) ∇ − f− (t) ≤ ε t −s for all s ∈ U, s = t. Therefore we get f−∇ (t) = lim

s→t −

f (t) − f (s) . t −s

On the contrary, if the limit lim

s→t −

f (t) − f (s) := f−∇ (t) t −s

exists as a finite number, then for any ε > 0, there exists a left-side neighborhood U of t such that f (t) − f (s) ∇ − f− (t) ≤ ε t −s for all s ∈ U, s = t. Since t is left-dense, we have ρ(t) = t, that is, f (ρ(t)) − f (s) ∇ − f− (t) ≤ ε, ρ(t) − s so we obtain

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1 Preliminaries and Basic Knowledge on Time Scales

[f (ρ(t)) − f (s)] − f ∇ (t)[ρ(t) − s] ≤ ε|ρ(t) − s| −

for all s ∈ U . (iv) If ρ(t) = t, then ν(t) = 0 and we have that f (ρ(t)) = f (t) = f (t) − ν(t)f−Δ (t). On the other hand, if ρ(t) > t, then by (ii) f (ρ(t)) = f (t) + ν(t)

f (ρ(t)) − f (t) = f (t) − ν(t)f−∇ (t). ν(t)

This completes the proof. 

According to Theorem 1.62, the following theorem is immediate. Theorem 1.63 Assume that f : T → R is a function and let t ∈ T. Then we have the following: (i) If f is right-side differentiable at t, then f is right-side continuous at t. (ii) If t is left-scattered and f is right-side continuous at t, then f is right-side differentiable at t with f+∇ (t)

  f ρ(t) − f (t) . = ρ(t) − t

(iii) If t is left-dense, then f is right-side differentiable at t if and only if the limit lim

s→t +

f (t) − f (s) t −s

exists as a finite number. In this case, f+∇ (t) = lim

s→t +

f (t) − f (s) . t −s

(iv) If f is right-side differentiable at t, then   f ρ(t) = f (t) − ν(t)f+∇ (t). According to (ii) from Theorems 1.62 and 1.63, it immediately follows that Theorem 1.64 Let f : T → R and t be an isolated point in T. Then f has two one-sided ∇-derivatives and f−∇ (t) = f+∇ (t) on T , i.e., f is ∇-differentiable at t. From Theorem 1.64, we can also obtain

1.2 Some Basic Results of ∇-Calculus on Time Scales

33

Corollary 1.3 Let f : T → R and t be an isolated point in T. Then f is continuous at t. Remark 1.16 Note that if t is just a left-scattered point in T and f : T → R, then f may be not ∇-differentiable at t. Example 1.4 Let T = [1, 2] ∪ [3, 4] and

f (t) =

⎧ ⎪ ⎪t 2 + 1, t ∈ [1, 2], ⎨ t = 3,

5, ⎪ ⎪ ⎩t,

t ∈ (3, 4].

Then we have f−∇ (3) = lim

s→3−

f+∇ (3) = lim

s→3+

f (ρ(3)) − f (3) f (ρ(3)) − f (s) = = 0, ρ(3) − s ρ(3) − 3 f (ρ(3)) − 3 f (ρ(3)) − f (s) = = −2, ρ(3) − s ρ(3) − 3

so we have f+∇ (3) = f−∇ (3), i.e., f is not ∇-differentiable at t.



Theorem 1.65 Assume that f : T → R is a function and let t ∈ Tκ . Then we have the following: (i) If f is ∇-differentiable at t, then f is continuous at t. (ii) If f is continuous at t and t is left-scattered, then f is differentiable at t with f ∇ (t) =

  f ρ(t) − f (t) . ρ(t) − t

(iii) If t is left-dense, then f is ∇-differentiable at t if and only if the limit lim

s→t

f (t) − f (s) t −s

exists as a finite number. In this case, f ∇ (t) = lim s→t

f (t) − f (s) . t −s

(iv) If f is ∇-differentiable at t, then   f ρ(t) = f (t) − ν(t)f ∇ (t).

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1 Preliminaries and Basic Knowledge on Time Scales

1.2.2 ∇-Calculus In the sequel, we will introduce some basic knowledge of ∇-calculus. Definition 1.37 ([64, 65]) The function f : T → R is called ld-continuous provided that it is continuous at each left-dense point and has a right-sided limit at right-dense points. The set of ld-continuous functions f : T → R will be denoted in this book by Cld (T) = Cld (T, R). The set of functions f : T → R that are 1 (T) = ∇-differentiable and whose derivative is ld-continuous is denoted by Cld 1 Cld (T, R). Definition 1.38 ([64, 65]) The function f : T → R is called ld-continuous provided that it is continuous at each left-dense point and has a right-sided limit at each point, write f ∈ Cld (T) = Cld (T, R). Let t ∈ Tκ . Then we define f ∇ (t) to be the number (provided it exists) with the property that given any ε > 0, there exists a neighborhood U of t (i.e., U = (t − δ, t + δ) ∩ T for some δ > 0) such that |f (ρ(t)) − f (s) − f ∇ (t)[ρ(t) − s]| ≤ ε|ρ(t) − s| for all s ∈ U , we call f ∇ (t) the nabla derivative of f at t. A function F : T → R is called an antiderivative of f : T → R provided F ∇ (t) = f (t) holds for all t ∈ Tκ , and we define the Cauchy nabla integral of f by 

t

f (s)∇s = F (t) − F (a) for all t, a ∈ T.

a

Definition 1.39 ([64, 65]) A function p : T → R is called ν- regressive provided 1−ν(t)p(t) = 0 for all t ∈ Tk . The set of all regressive and ld-continuous functions p : T → R will be denoted by Rν = Rν (T) = Rν (T, R). We define the set Rν+ = Rν+ (T, R) = {p ∈ Rν : 1 − ν(t)p(t) > 0, ∀ t ∈ T}. We define circle plus addition by p ⊕ν q = p(t) + q(t) − ν(t)p(t)q(t) for all t ∈ Tκ . Theorem 1.66 ([64, 65]) The set (Rν , ⊕ν ) is an Abelian group. Definition 1.40 ([64, 65]) For p ∈ Rν , define circle minus by ν p = −

p . 1 − νp

Definition 1.41 ([64, 65]) If r is a regressive function, then the generalized exponential function eˆr is defined by 

t

eˆr (t, s) = exp s

ξˆν(τ ) (r(τ ))∇τ



1.2 Some Basic Results of ∇-Calculus on Time Scales

35

for all s, t ∈ T, where the ν-cylinder transformation is as in 1 ξˆh (z) := − Log(1 − zh). h Lemma 1.7 ([64, 65]) Assume that p, q : T → R are two ν-regressive functions. Then (i) (ii) (iii) (iv) (v)

eˆ0 (t, s) ≡ 1 and eˆp (t, t) ≡ 1; eˆp ((t), s) = (1 − ν(t)p(t))eˆp (t, s); 1 eˆp (t, s) = eˆ (s,t) = eν p (s, t); p eˆp (t, s)eˆp (s, r) = eˆp (t, r); (eˆν p (t, s))∇ = (ν p)(t)eˆν p (t, s);

Theorem 1.67 ([64, 65]) For ϕ ≥ 0 with −ϕ ∈ Rν+ , the following inequalities hold    t  t ϕ(u)∇u ≤ eˆ−ϕ (t, s) ≤ 1 + ϕ(u)∇u for all t ≥ s. exp − s

s

If ϕ is ld-continuous and nonnegative, then 

t

1−

 ϕ(u)∇u ≤ eˆϕ (t, s) ≤ exp

s

t

 ϕ(u)∇u for all t ≥ s.

s

Lemma 1.8 ([64, 65]) If λ ∈ Rν+ and λ(r) < 0 for all r ∈ (s, t]T , then 0 < eˆλ (t, s) < 1. Theorem 1.68 If p ∈ Rν and a, b, c ∈ T, then  ∇  ρ eˆp (c, ·) = −p eˆp (c, ·) and 

b

  p(t)eˆp c, ρ(t) ∇t = eˆp (c, a) − eˆp (c, b).

a

Proof Note     p(t)eˆp c, ρ(t) = p(t)eˆν p ρ(t), c   = p(t) 1 − ν(t)(ν p)(t) eˆν p (t, c) ! ν(t)p(t) = p(t) 1 + eˆν p (t, c) 1 − ν(t)p(t)

36

1 Preliminaries and Basic Knowledge on Time Scales

= p(t)

1 eˆ p (t, c) = −(ν p)(t)eˆν p (t, c) 1 − ν(t)p(t) ν

∇ (t, c) = −[eˆp (c, t)]∇ . = −eˆ νp

Thus, we have 

b

  p(t)eˆp c, ρ(t) ∇t = −

a



b

[eˆp (c, ·)]∇ (t)∇t = eˆp (c, a) − eˆp (c, b),

a



which completes the proof.

Theorem 1.69 ([64, 65]) Assume f, g : T → R are differentiable at t ∈ Tκ . Then: (i) The sum f + g : T → R are differentiable at t with (f + g)∇ (t) = f ∇ (t) + g ∇ (t). (ii) For any constant α, αf : T → R is differentiable at t with (αf )∇ = αf ∇ (t). (iii) The product f g : T → R is differentiable at t with     (f g)∇ (t) = f ∇ (t)g(t) + f ρ(t) g ∇ (t) = f (t)g ∇ (t) + f ∇ (t)g ρ(t) .   (iv) If f (t)f ρ(t) = 0, then

1 f

is differentiable at t with

 ∇ 1 f ∇ (t)  . (t) = − f f (t)f ρ(t)   (v) If g(t)g ρ(t) = 0, then

f g

is differentiable at t and

 ∇ f f ∇ (t)g(t) − f (t)g ∇ (t)   (t) = . g g(t)g ρ(t) Theorem 1.70 ([64, 65]) If a, b, c ∈ T, α, β ∈ R, and f, g ∈ Cld , then b b b  (i) a αf (t) + βg(t) ∇t = α a f (t)∇t + β a g(t)∇t; b a (ii) a f (t)∇t = − b f (t)∇t; c b c (iii) a f (t)∇t = a f (t)∇t + b f (t)∇t; b b (iv) a f (t)∇t ≤ a |f (t)|∇t.

1.2 Some Basic Results of ∇-Calculus on Time Scales

37

Lemma 1.9 If z ∈ C is regressive and t0 ∈ T, then ρ

eˆν z (t, t0 ) =

eˆν z (t, t0 ) (ν z)(t) =− eˆν z (t, t0 ), 1 − ν(t)z z

ρ

where eˆν z (t, t0 ) = eˆν z (ρ(t), t0 ). Proof From calculation, we have   ρ eˆν z (t, t0 ) = 1 − ν(t)(ν z)(t) eˆν z (t, t0 )   ν(t)z = 1+ eˆν z (t, t0 ) 1 − ν(t)z =

(ν z)(t) 1 eˆν z (t, t0 ) = − eˆν z (t, t0 ). 1 − ν(t)z z 

This proves the results.

1.2.3 Lebesgue ∇-Measure and ∇-Measurable Function In the following, we will introduce some basic knowledge of ∇-measure theory on time scales. Let G denote the class of all bounded right closed and left open intervals of T of the form (a, b]T . If a = b, then the interval reduces to the empty set. On the class + of G of semi-closed intervals, a set function m  R ∪ {0} is defined which ˆ : G → ˆ is assigns each interval (a, b]T its length, i.e., m ˆ (a, b]T = b − a. The function m countably additive measure on G. In fact,   1. m ˆ (a, ∞b]T = b − a ≥0∞sinceb ≥ a;  2. m ˆ ˆ (ai , bi ]T for pairwise disjoint intervals (ai , bi ]T . i=1 (ai , bi ]T = i=1 m 3. m(∅) ˆ =m ˆ (a, a]T = 0 holds for any a ∈ T. Definition 1.42 ([72, 95]) Let E be any subset of T. If there exists at least  one finite or countable system of intervals I ∈ G, j = 1, 2, . . . such that E ⊂ j j Ij , then  m ˆ ∗ (E) = inf j m(I ˆ j ), is called the outer measure of E, where the infimum is taken over all coverings of E by a finite or countable system of intervals Ij ∈ G. Definition 1.43 ([72, 95]) A set E ⊂ T is called m ˆ ∗ -measurable or ∇-measurable if for each interval I ⊂ G, the following holds ˆ ∗ (I ∩ E) + m ˆ ∗ (I ∩ E c ), m ˆ ∗ (I ) = m where E c = T\E.

38

1 Preliminaries and Basic Knowledge on Time Scales

ˆ m We denote the family of all m ˆ ∗ -measurable subsets of T by M( ˆ ∗ ), then we have the following theorem. ˆ m ˆ m ˆ ∗ restricted to M( ˆ ∗ ) is a Theorem 1.71 ([72, 95]) M( ˆ ∗ ) is a σ -algebra and m countably additive measure. ˆ m Definition 1.44 ([72, 95]) m ˆ ∗ restricted to M( ˆ ∗ ) is called the ∇-measure on the time scale T, denote it by μ∇ . Theorem 1.72 ([72, 95]) If {En } is an increasing sequence of sets in T, then μ∇

 ∞

 = lim μ∇ (En );

En

n→∞

i=1

if {En } is a decreasing sequence of sets in T, then μ∇

 ∞

 En

= lim μ∇ (En );

i=1

n→∞

that is, the ∇-measure on T is continuous. Lemma 1.10 ([72, 95]) The single-point set {t0 } ⊂ Tκ is ∇-measurable and its ∇-measure is given by μ∇ ({t0 }) = t0 − ρ(t0 ). Theorem 1.73 ([72, 95]) The set of all left-scattered points of T is at most countable, that is, there are {ti }i∈I , I ⊂ N such that T\Dl = Sl = {ti }i∈I , where Dl denotes the all left-dense points and Sl denotes the all left-scattered points of T. Remark  1.17 By Theorem 1.73, we will express the extension of a set E ⊆ T as E = i∈IE [ai , bi ], where ai , bi ∈ E is the left-scattered points in T and IE denotes the indices set of left-scattered points of E. Lemma 1.11 ([72, 95]) If a, b ∈ T and a ≤ b, then μ∇ ((a, b]T ) = b − a, μ∇ ((a, b)T ) = ρ(b) − a. If a, b ∈ Tκ and a ≤ b, then μ∇ ([a, b)T ) = ρ(b) − ρ(a), μ∇ ([a, b]T ) = b − ρ(a). Now, we introduce a relation between Lebesgue measure and Lebesgue ∇measure. Let λˆ ∗ be the Lebesgue outer measure on R, λˆ the usual Lebesgue measure and m ˆ ∗ the outer measure on T.

1.2 Some Basic Results of ∇-Calculus on Time Scales

39

Theorem 1.74 ([72, 95]) If E ⊂ Tκ , then the following properties are satisfied: (i) m ˆ ∗ (E) ≥ λˆ ∗ (E); (ii) if E does not include left-scattered points, then m ˆ ∗ (E) = λˆ ∗ (E); ˆ l ) = 0 but (iii) The sets Dl and Sl = T\Dl are Lebesgue measurable and λ(S μ∇ (E ∩ Sl ) =

  ti − ρ(ti ) , i∈IE

where IE indicates  the indices  set for all left-scattered points in E;  (iv) m ˆ ∗ (E) = i∈IE ti − ρ(ti ) + λˆ ∗ (E); (v) m ˆ ∗ (E) = λˆ ∗ (E) if and only if E has no left-scattered points. Theorem 1.75 ([72, 95]) Let E ⊂ T. Then E is Lebesgue ∇-measurable if and only if it is Lebesgue measurable. In such a case, for E ⊂ Tκ the following is true:    (i) μ∇ (E) = i∈IE ti − ρ(ti ) ; (ii) λˆ (E) = μ∇ (E) if and only if E has no left-scattered point. Now, we express the extension of a set E ⊂ T as E˜ = E ∪



 ρ(ti ), ti ,

(1.6)

i∈IE

where IE denotes the indices set of left-scattered points of E. Remark 1.18 By Theorem 1.75, one can obtain ˜ where E ⊂ Tκ . μ∇ (E) = λˆ (E), Theorem 1.76 ([72, 95]) If a set E is Lebesgue measurable, then E ∩ T is ∇measurable. Next, we will introduce some basic knowledge of ∇-measurable function on time scales. Definition 1.45 ([72, 95]) We say that f : T → R is ∇-measurable if for every α ∈ R, the set     f −1 (−∞, α] := t ∈ T : f (t) < α is ∇-measurable. Theorem 1.77 ([72, 95]) Let f : E → R and E be a ∇-measurable set. Then for each α ∈ R, the following statements are equivalent: (i) {t ∈ E : f (t) < α} is ∇-measurable;

40

1 Preliminaries and Basic Knowledge on Time Scales

: f (t) ≥ α} is ∇-measurable; : f (t) ≤ α} is ∇-measurable; : f (t) > α} is ∇-measurable; : f (t) > α} is ∇-measurable.  Remark 1.19 Let S = ni=1 αi χAi , where (ii) (iii) (iv) (v)

{t {t {t {t

∈E ∈E ∈E ∈E

 χAi =

1, t ∈ Ai , 0, t ∈ Ai ,

then Ai ⊂ T is ∇-measurable set for each i if and only if S is ∇-measurable function. Definition 1.46 ([72, 95]) A proposition C (t) is called ∇-almost everywhere, shortly, ∇-a.e., if   μ∇ {t ∈ T : C (t) is false} = 0. Theorem 1.78 ([72, 95]) Let f be ∇-measurable on E ⊂ T and f = g ∇-a.e. Then g is ∇-measurable. Theorem 1.79 ([72, 95]) Let f, g, fn , gn : E → R be ∇-measurable functions, where E ⊂ T, n ∈ N. Then αf (α ∈ R), f ± g, f g, fg (g = 0), supn fn , infn fn , limn fn , limn fn are ∇-measurable. Theorem 1.80 ([72, 95]) Let {fn (t)} be a sequence of ∇-measurable functions on E ⊂ T. If limn→∞ fn (t) = f (t), then f (t) is ∇-measurable. Theorem 1.81 ([72, 95]) f is ∇-measurable function if and only if for any α1 < α2 , α1 , α2 ∈ R, the set {t ∈ T : α1 < f (t) < α2 } is ∇-measurable. Theorem 1.82 ([72, 95]) If f : T → R is ld-continuous, then f is ∇-measurable. Let f : E → R (where E ⊆ T) and its extension f˜ : E˜ → R be  f (t), if t ∈ E, ˜ f (t) =   f (ti ), if t ∈ ρ(ti ), ti ,

(1.7)

for some i ∈ IE , {ti }i∈IE represents the set of all left-scattered points in E and    E˜ = E ∪ i∈IE ρ(ti ), ti . Theorem 1.83 ([72, 95]) f is ∇-measurable function if and only if f˜ is Lebesgue measurable function.

1.2 Some Basic Results of ∇-Calculus on Time Scales

41

1.2.4 Riemann ∇-Integral, Lebesgue ∇-Integral, and Some Important Convergence Theorems In what follows, we introduce the Riemann ∇-Integral, the Lebesgue ∇-integral and Fatou’s Lemma, the Monotone Convergence Theorem, and the Lebesgue Dominated Convergence Theorem on time scales. First, we introduce some basic definitions and results of the Riemann ∇-Integral. Let (a, b]T be a half closed interval in T, a partition of (a, b]T is any ordered subset Pˆ = {t0 , t1 , . . . , tn } ⊂ [a, b]T , where a = t0 < t1 < . . . < tn = b. Let f : (a, b]T → R and ˆ = inf{f (t) : t ∈ (a, b]T }, Mˆ = sup{f (t) : t ∈ (a, b]T }, m ˆ i = inf{f (t) : t ∈ (ti−1 , ti ]T }. Mˆ i = sup{f (t) : t ∈ (ti−1 , ti ]T }, m ˆ and the lower Darboux ∇-sum L(f, ˆ of the ˆ The upper Darboux ∇-sum Uˆ (f, P) P) function f are defined by ˆ = Uˆ (f, P)

n 

ˆ = ˆ P) Mˆ i (ti − ti−1 ), L(f,

i=1

n 

m ˆ i (ti − ti−1 )

i=1

ˆ The upper Darboux integral Uˆ (f ) from a to b is with respect to the partition P. ˆ and the lower Darboux integral L(f ˆ ˆ ˆ ) from a to defined by U (f ) = inf{U (f, P)}, ˆ ˆ ˆ b is defined by L(f ) = sup{U (f, P)}. Definition 1.47 ([135, 136]) f is called the Darboux ∇-integral from a to b if  ˆ ) and this integral is denoted by b f (t)∇t. Uˆ (f ) = L(f a Lemma 1.12 ([135, 136]) For any δ > 0, there exists at least one partition Pˆ : a = t0 < t1 < . . . < tn = b of (a, b]T such that for each i either ti − ti−1 ≤ δ or ti − ti−1 > δ and σ (ti−1 ) = ti , where σ denotes the forward jump operator on T. Theorem 1.84 ([135, 136]) A bounded function f : (a, b]T → R is Darboux ∇integrable if and only if for each ε > 0 there exists δ > 0 such that ˆ − L(f, ˆ 0, there exists δ > 0 (i.e., there exists a partition Pˆ δ ∈ P) such that |SˆR − Iˆ| < ε and it is independent of the way of choosing ξi ∈ (ti−1 , ti ]T . The number Iˆ is called the Riemann ∇-integral of f from a to b. Theorem 1.85 ([135, 136]) A bounded function f on (a, b]T is Riemann ∇integrable if and only if it is (Darboux) ∇-integrable and the two values of the integrals are equal. Next, we introduce some basic definitions and theorems of the Lebesgue ∇integral. Definition 1.49 ([135, 136]) Let E ⊂ T be a ∇-measurable set and Sˆ : T → n ˆ [0, +∞) be a ∇-measurable simple function with S = i=1 αi χAi , Ai = {t ∈ T : ˆ = αi }. The Lebesgue ∇-integral of Sˆ on E is defined as S(t) 

ˆ S(t)∇t = E

n 

αi μ∇ (Ai ∩ E).

i=1

Theorem 1.86 ([135, 136]) Let E ⊂ Tκ be a ∇-measurable set and Sˆ : T →  [0, +∞) be a ∇-measurable simple function with Sˆ = ni=1 αi χAi and S˜ be the     extension of Sˆ given by S˜ = ni=1 αi χA˜ i , where A˜ i = Ai ∪ i∈IA ρ(ti ), ti . Then i S˜ is Lebesgue measurable and Lebesgue integrable with  E

ˆ S(t)∇t =



˜ S(t)dt,

E˜

where E˜ is the extension of E as given in (1.6). Definition 1.50 ([135, 136]) Let E ⊂ T measurable set and f : T → [0, +∞) be a ∇-measurable function. The Lebesgue ∇-integral of f on E is defined as 

 f (t)∇t = sup

E

ˆ S(t)∇t,

E

where the supremum is taken over all ∇-measurable nonnegative simple functions Sˆ such that Sˆ ≤ f . In the following, for simplicity, the ∇-integral means the Lebesgue ∇-integral. Theorem 1.87 ([72, 95], Beppo Levi’s Lemma) Assume that a sequence {fn }n∈N of ∇-integrable functions on a ∇-measurable set E satisfies fn (t) ≤ fn+1 (t) ∇-

1.2 Some Basic Results of ∇-Calculus on Time Scales

43

 a.e. for all n ∈ N and limn→∞ E fn (s)∇s < ∞. Then there exists   a ∇-integrable function f such that limn→∞ fn = f and limn→∞ E fn (s)∇s = E f (s)∇s. Theorem 1.88 ([72, 95], Fatou’s Lemma) Let {fn (t)}n∈N be a sequence of ∇integrable  functions defined on a ∇-measurable set E such that fn (t) ≥ 0 ∇-a.e. and lim E fn (s)∇s < ∞. Then lim fn (t) defines a ∇-integrable function on the ∇-measurable set E and   lim fn (s)∇s ≤ lim fn (s)∇s. E

E

Theorem 1.89 ([72, 95], Monotone Convergence Theorem) Let {fn }n∈N be an increasing sequence of nonnegative ∇-measurable functions defined on a ∇measure set E and let f (t) = limn→∞ fn (t) ∇-a.e. Then 

 lim fn (t)∇t = lim

E n→∞

n→∞ E

fn (t)∇t.

Theorem 1.90 ([72, 95], Lebesgue Dominated Convergence Theorem) Let g be ∇-integrable function over E and let {fn } be sequence of ∇-measurable functions such that |fn (t)| ≤ g(t) on E and for almost all t in E we have f (t) = limn→∞ fn (t). Then 

 f (t)∇t = lim

n→∞ E

E

fn (t)∇t.

The next lemma exhibits a relation between the classical Lebesgue integral and the Lebesgue ∇-integral of nonnegative ∇-measurable functions. Lemma 1.13 ([72, 95]) Let E ⊂ Tκ be a ∇-measurable set and f : T → [0, +∞) be a ∇-measurable function and f˜ be the extension of f as (1.7). Then 

 f (s)∇s = E

where E˜ = E ∪

 i∈IE

E˜

f˜(s)ds,

  ρ(ti ), ti .

Definition 1.51 ([72, 95]) Let E ⊂ T be a measurable set and f : T → R be a ∇– measurable function. We say f is ∇-integrable on E if at least one of the elements   + (t)∇t or − (t)∇t is finite, where f + and f − are the positive and negative f f E E part of f respectively. In this case, we define the Lebesgue ∇-integral of f on E as 

 f (t)∇t = E

E

f + (t)∇t −



f − (t)∇t. E

44

1 Preliminaries and Basic Knowledge on Time Scales

Theorem 1.91 ([72, 95]) Let f, g : E → R be a ∇-integrable functions and E ⊂ T be a ∇-measurable set and α, β ∈ R. Then (i) af  and f + g is ∇-integrable  on E;  (ii) E (αf (t) + βg(t))∇t = α E f (t)∇t   + β E g(t)∇t; (iii) If f (t) ≤ g(t) for each t ∈ E, then E f (t)∇t ≤ E g(t)∇t. Remark 1.20 Note that the classical Lebesgue integral has the same value on I = [a, b], [a, b), (a, b] and (a, b). However, the Lebesgue ∇-integral with respect to μ∇ for each interval is different from each other. One can easily check that 

 [a,b]T

f (t)∇t =

f (t)∇t + f (a)ν(a); (a,b]T



 f (t)∇t = f (b)ν(b) + (a,b]T

f (t)∇t; (a,b)T



 [a,b)T

f (t)∇t =

f (t)∇t + f (a)ν(a) − f (b)ν(b). (a,b]T

Theorem 1.92 ([72, 95]) Let E ⊂ Tκ be a ∇-measurable set and f : T → R be ∇-measurable function. If f˜ be the extension of f as given in (1.7), then f is Lebesgue ∇-integrable on E if and only if f˜ is Lebesgue integrable on E˜ and 

 f (t)∇t =

E

E˜

f˜(t)dt.

Theorem 1.93 ([72, 95]) Let E ⊂ Tκ be a ∇-measurable set. If f : T → R is ∇-integrable on E, then 

 f (t)∇t = E

E˜

f (s)ds +



f (ti )ν(ti ).

i∈IE˜

Theorem 1.94 ([72, 95], Approximation of ∇-Integral) Let f be an ldcontinuous function on an interval  I ⊂ T and {In } be a sequence of intervals (where In ⊂ I for each n) such that ∞ n=1 In = {a}. Then 1 lim n→∞ μ∇ (In )

 f (t)∇t = f (a). In

From Theorem 1.94, it is easy to see that Corollary 1.4 ([72, 95]) Let f be ld-continuous and bounded function on (a, b]T and F is antiderivative of f . Then

1.2 Some Basic Results of ∇-Calculus on Time Scales

45

 f (t)∇t = F (b) − F (a). (a,b]T

The following theorem give a relationship between the Lebesgue ∇-integral and the Riemann ∇-integral. Theorem 1.95 ([72, 95]) Let (a, b]T be a half closed bounded interval in T and f : (a, b]T → R be bounded. If f is Riemann ∇-integrable from a to b, then f is b  Lebesgue ∇-integrable on (a, b]T and a f (t)∇t = (a,b]T f (t)∇t, where the left hand side of the equation is the Riemann ∇-integral and the right hand side of the equation is the Lebesgue ∇-integral. Theorem 1.96 ([72, 95]) Let (a, b]T be a half closed bounded interval in T and f : (a, b]T → R be bounded. Then f is Riemann ∇-integrable from a to b if and only if the set of all left dense points of (a, b]T at which f is discontinuous is a set of ∇-measure zero.

1.2.5 Henstock–Kurzweil ∇-Integral In what follows, well introduce the Henstock–Kurzweil ∇-Integral. Definition 1.52 ([196]) We say γ = (γL , γR ) is a ∇-gauge for [a, b]T provided γL (t) > 0 on (a, b]T , γR (t) > 0 on [a, b)T , γL (a) ≥ 0, γR (b) ≥ 0, and γL (t) ≥ ν(t) for all t ∈ (a, b]T . By Definition 1.29, we have the following definition: Definition 1.53 ([196]) If γ is a ∇-gauge for [a, b]T , then we say the partition P is a γ -fine if ξi − γL (ξi ) ≤ ti−1 < ti ≤ ξi + γR (ξi ) for 1 ≤ i ≤ n. Now we can define the Henstock–Kurzweil ∇-integral. Definition 1.54 ([196]) We say that f : [a, b]T → R is Henstock–Kurzweil ∇b integrable on [a, b]T with value I = H K a f (t)∇t, provided given any ε > 0, there exists a ∇-gauge, γ , for [a, b]T such that n  I − f (ξi )(ti − ti−1 ) < ε i=1

for all γ -fine partitions P of [a, b]T .

46

1 Preliminaries and Basic Knowledge on Time Scales

Theorem 1.97 ([196]) If γ is a ∇-gauge for [a, b]T , then there is a γ -fine partition P for [a, b]T .  Example 1.5 Assume (a, b]T contains a countable infinite subset ∞ i=1 {ri } with ρ(ri ) = ri . Define f : [a, b]T → R by  f (t) =

1,

t = ri ,

0,

t = ri .

For any given ε > 0, we define a ∇-gauge, δ, on [a, b]T by γL (ri )= γR (ri ) = ε/2i+2 , i ≥ 1, δL (t) = max{1, ν(t)} and δR (t) = 1 for t ∈ [a, b]T \ ∞ i=1 {ri }. Let P be a γ -fine partition of [a, b]T , then   ∞ ∞ n    ε ≤ γ f (ξ )(t − t ) (r ) + γ (r ) = < ε. i i i−1 L i R i i+1 2 i=1

i=1

i=1

Hence, even though in many cases f is not ∇-integrable on [a, b]T , we have f is Henstock–Kurzweil ∇-integrable on [a, b]T and 

b

HK

f (t)∇t = 0.

a

Theorem 1.98 ([196]) Assume f : [a, b]T → R. If f is HK-∇-integrable on b [a, b]T . Then the value of the integral H K a f (t)∇t does not depend on f (a). b On the other hand, if c ∈ (a, b]T and c is left-scattered. Then H K a f (t)∇t does depend on the value f (c)ν(c). We give the notation Nν := {zj ∈ (a, b]T : ν(zj ) > 0} and note that Nν is a countable set. Theorem 1.99 ([196]) Assume that F : [a, b]T → R is continuous, f : [a, b]T → R, and there is a set D with Nν ⊂ D ⊂ ([a, b]T )κ such that F ∇ (t) = f (t) for t ∈ D and [a, b]T \D is countable. Then f is HK-∇-integrable on [a, b]T with 

b

HK

f (t)∇t = F (b) − F (a).

a

Example 1.6 Let T = {t = 1/n2 : n ∈ N} ∪ {0} and define f : T → R by f (t) =

 (−1)n+1 n2 , t = 1/n2 , L,

t = 0,

where L is any constant. Note that f is not ∇-differentiable on [0, 1]T . However, it can be shown that if F : T → R is defined by

1.2 Some Basic Results of ∇-Calculus on Time Scales

F (t) =

⎧ ⎪ ⎪ ⎨0, 

47

t = 1, n (−1)k−1 (2k−1) , k=2 k2

⎪ ⎪ ⎩− ln 3,

t = 1/n2 , t = 0,

then F ∇ (t) = f (t) for t ∈ (0, 1)T and F is continuous on [0, 1]T . It follows by Theorem 1.99 with D = (0, 1)T that f is HK-∇-integrable on [0, 1]T and 

1

HK

f (t)∇t = F (1) − F (0) = ln 3.

0

1 However, it can be shown that H K 0 |f (t)|∇t does not exist (i.e., f is not absolutely HK-∇-integrable on [0, 1]T . 

Theorem 1.100 ([196]) If f : T → R is regulated and a, b ∈ T, then f is HK-∇integrable on [a, b]T and 

b

HK a

Moreover, if G(t) :=

t a

 f (t)∇t =

b

f (t)∇t. a

f (s)∇s, then G∇ (t) = f (t) except for a countable set.

Remark 1.21 Let γ 1 , γ be ∇-gauges for [a, b]T such that 0 < γR1 (t) ≤ γR (t) for t ∈ [a, b)T and 0 < γL1 (t) ≤ γL (t) for t ∈ (a, b]T (write γ 1 ≤ γ and we say γ 1 is finer than γ ). If P1 is a γ 1 -finer partition of [a, b]T , then γ 1 is a γ -fine partition of [a, b]T ). Remark 1.22 If c ∈ [a, b]T and P is a γ -fine partition, then there is a γ -fine partition with c as a tag point. Remark 1.23 If c ∈ [a, b]T and ti−1 ≤ c ≤ ti is a tag point in a γ -fine partition P, then 

P = {t0 = a ≤ ξ1 ≤ . . . ≤ ti−1 ≤ c ≤ ti ≤ . . . ≤ tn = b}, where c is an end point and a tag point for the two intervals [ti−1 , c]T and [c, ti ]T is a γ -fine partition and the Riemann sum corresponding to these two partitions is the same. This follows from the simple fact that f (c)[ti − ti−1 ] = f (c)[ti − c] + f (c)[c − ti−1 ]. Remark 1.24 Let a < c < b be points in T, then we may choose the gauge γ so that γL (t), γR (t) ≤ |t − c| for t < c, t > c respectively. Then, if P is a γ -fine partition of [a, b]T , then ξi0 = c for some i0 . If ti0 −1 < ξi0 , then we may add yi0 to the partition so that {a = t0 ≤ ξ1 ≤ t1 ≤ . . . ≤ ti0 −1 ≤ ξi0 = yi0 ≤ . . . ≤ b}

48

1 Preliminaries and Basic Knowledge on Time Scales

so that {a = t0 ≤ ξ1 ≤ t1 ≤ . . . ≤ ti0 −1 ≤ ξi0 = yi0 = c} and {c = yi0 ≤ ξ0 ≤ . . . ≤ tn = b} are γ -fine partition of [a, c]T and [c, b]T , respectively. Theorem 1.101 ([196]) Let f : [a, b]T → R. Then f is HK-∇-integrable on [a, b]T if and only if f is HK-∇-integrable on [a, c]T and [c, b]T . Moreover, in this case 

b

HK

 f (t)∇t = H K

a

c



b

f (t)∇t + H K

a

f (t)∇t. c

Also if f, g : [a, b]T → R are HK-∇-integrable on [a, b]T , then αf + gβ is HK-∇integrable on [a, b]T and 

b

HK

    αf (t) + βg(t) ∇t = α H K

a

b

   f (t)∇t + β H K

a

b

 f (t)∇t .

a

Definition 1.55 ([196]) We say a subset S of a time scale T has ∇-measure zero provided S contains no left-scattered points and S has Lebesgue measure zero. We say a property A holds ∇-almost everywhere (∇-a.e.) on T provided there is a subset S of T such that the property A holds for all t ∈ T\S and S has ∇-measure zero. Theorem 1.102 ([196]) If f and g are HK-∇-integrable on [a, b]T and f (t) ≤ g(t) ∇-a.e. on (a, b]T , then 

b

HK



b

f (t)∇t ≤ H K

a

g(t)∇t. a

Theorem 1.103 ([196]) Assume f is HK-∇-integrable on [a, b]T . Then given any ε > 0 there is a ∇-gauge, γ , of [a, b]T such that  n  H K i=1

ti

ti−1

f (t)∇t − f (ξi )(ti − ti−1 ) < ε

for all γ -fine partitions P of [a, b]T .

1.2 Some Basic Results of ∇-Calculus on Time Scales

49

Theorem 1.104 ([196], Monotone Convergence Theorem) Let fk , f : [a, b]T → R and assume that (i) (ii) (iii) (iv)

fk is HK-∇-integrable, k ∈ N; fk → f ∇-a.e. on (a, b]T ; fk ≤ fk+1 ∇-a.e. on (a, b]T , k ∈ N; b limk→∞ H K a fk (t)∇t = I .

Then f is HK-∇-integrable on [a, b]T and  I = HK

b

f (t)∇t. a

Theorem 1.105 ([196], Dominated Convergence Theorem) Assume that (i) fk → f ∇-a.e. on [a, b]T ; (ii) g ≤ fk ≤ h ∇-a.e. on [a, b]T ; (iii) fk , g, h are HK-∇-integrable on [a, b]T . Then f is HK-∇-integrable on [a, b]T and 

b

lim H K

k→∞



b

fk (t)∇t = H K

a

f (t)∇t. a

Definition 1.56 ([196]) We say f : [a, b]T → R is absolutely HK-∇-integrable on [a, b]T provided f and |f | are HK-∇-integrable on [a, b]T . Theorem 1.106 ([196]) The function f is Lebesgue ∇-integrable on [a, b]T if and only if f is absolutely HK-∇-integrable on [a, b]T . Theorem 1.107 ([196]) The function F is absolutely continuous on [a, b]T if and only if F ∇ (t) = f (t) ∇-a.e. on [a, b]T is absolutely HK-∇-integrable function f on [a, b]T . Moreover, 

b

HK a

|F ∇ (t)|∇t =

 [a,b]T

|F ∇ (t)|dt +



|F ρ (ti ) − F (ti )|.

ti ∈Nν

This Chapter introduces some basic knowledge on time scales including the Δ and ∇-calculus and measure theory. Concerning some basic results of boundary value problems, the existence and stability analysis and other qualitative theory of dynamic equations on time scales and some related applications on this topic, we refer to the literatures Atici and Guseinov [48], Atici et al. [49, 51], Atici and Biles [50], Tisdell and Zaidi [207], Bohner and Peterson [64, 65], Bohner and Lutz [68], Bohner and Martynyuk [69], Davis et al. [93], Guseinov and Kaymakcalan [135], Guseinov [136], Guseinov and Bohner [137], Hilscher and Zeidan [150]. Some important problems such as Lyapunov stability theory, the iterative method and Sturm-Liouville eigenvalue problems (see Kratz and Peyerimhoff [166, 167],

50

1 Preliminaries and Basic Knowledge on Time Scales

Kratz [168, 169]) were investigated on time scales and measure chains (see Kaymakçalan [155–159], Kaymakçalan et al. [160–162], Kaymakçalan and Leela 2020k7, Kaymakçalan and Rangarajan [163]. Moreover, the monotone flows and fixed points of dynamic equations and hybrid systems were also extended to time scales by Lakshmikantham and Kaymakçalan [170], Lakshmikantham and Sivasundaram [171], Lakshmikantham and Vatsala [173], Lakshmikantham [174], Lakshmikantham et al. [177] and the results of exponential dichotomies, topological decoupling, linearization and perturbation on measure chains and time scales were also obtained (see Pötzsche [199, 200]). These topics could become a future area of research.

Chapter 2

A Classification of Closedness of Time Scales Under Translations

2.1 Periodic Time Scales and Translations Invariance In 2014, Wang and Agarwal proposed the concept of almost periodic time scales and some of their basic properties were investigated (see [212, 215]). Based on it, some further results of the structure analysis of time scales were established including the decomposition and approximation theorems of time scales (see [16, 216, 217, 222, 225, 231]), etc., and the translation closedness of time scales was proposed to study for the first time [15, 223]. To arrive at the accurate qualitative analysis conclusion of dynamic equations on time scales, the most important thing is to clearly analyze the structure of time scales first. In this section, the translations invariance of time scales and periodic time scales will be introduced and investigated. Definition 2.1 ([153]) A time scale T is said to be periodic if there exists a P > 0 such that t ± P ∈ T for all t ∈ T. If T = R, the smallest positive P is called the period of the time scale. Example 2.1 The following time scales are periodic. (i) (ii) (iii) (iv) (v)

T = hZ, where h > 0, has period P = h. T = {t = k − q m : k ∈ Z, m ∈ N0 }, where 0 < q < 1, has period P = 1. T = R has P ∈ R\{0}.  an arbitrary period  ∞ (2i − 1)h, 2ih , h > 0, has period P = 2h. i=−∞   If T = k∈Z k(a + b), k(a + b) + b , where a = −b, then  σ (t) =

t,

if

t + a,

t∈

∞

if t

k=0 [k(a + b), k(a + b) + b),  ∈ ∞ k=0 {k(a + b) + b}

and

© Springer Nature Switzerland AG 2020 C. Wang et al., Theory of Translation Closedness for Time Scales, Developments in Mathematics 62, https://doi.org/10.1007/978-3-030-38644-3_2

51

52

2 A Classification of Closedness of Time Scales Under Translations

 μ(t) =

0,

if t ∈

a,

if t ∈

∞

k=0 [k(a

+ b), k(a + b) + b),

k=0 {k(a

+ b) + b},

∞

where T has period P = a + b. Obviously, if b = 0, a = 1, then T = Z. If b = 1, a = 0, then T = R. (vi) Let 0 < a < π2 and consider the time scale Ta,cos a =

∞    k(a + cos a), k(a + cos a) + a k=−∞

then  σ (t) =

if t ∈

t,

t + cos a, if t ∈



∞

k=−∞ k(a + cos a), k(a + cos a) + a ∞ k=−∞ {k(a + cos a) + a}

 ,

and  μ(t) =

0,

if t ∈

cos a, if t ∈

∞



k=−∞ k(a + cos a), k(a + cos a) + a ∞ k=−∞ {k(a + cos a) + a},



,

where T has period a + cos a. (vii) Let π2 < a < π and consider the time scale Ta,sin a =

∞    k(a + sin a), k(a + sin a) + a k=−∞

then  σ (t) =

if t ∈

t,

t + sin a, if t ∈



∞

k=−∞ k(a + sin a), k(a ∞ k=−∞ {k(a + sin a) + a}

 + sin a) + a ,

and  μ(t) =

0,

if t ∈

sin a, if t ∈

∞



k=−∞ k(a + sin a), k(a + sin a) + a ∞ k=−∞ {k(a + sin a) + a},

where T has a period a + sin a.



,



According to Definition 2.1, one can introduce the following equivalent definition of periodic time scales.

2.1 Periodic Time Scales and Translations Invariance

53

Definition 2.2 ([15, 212]) A time scale T is called a periodic time scale if     Π := τ ∈ R : t ± τ ∈ T, ∀t ∈ T ∈ {0}, ∅ . The set Π is called the periods set of T. Remark 2.1 It follows from Definition 2.2 that Π is also a closed subset of R, so Π is a time scale. Remark 2.2 Notice that the set Π in Definition 2.2 has the following equivalent forms:   Π := τ ∈ R\{0} : t ± τ ∈ T, ∀t ∈ T = ∅, or   Π := τ ∈ R : t ± τ ∈ T, ∀t ∈ T = {0}. From Example 2.1, one can obtain the set Π for each periodic time scale. Example 2.2 T = hZ, where h > 0, Π = {nh, n ∈ Z}. T = {t = k − q m : k ∈ Z, m ∈ N0 }, where 0 < q < 1, Π = {n, n ∈ Z}. T = R, Π  = {τ, τ ∈ R}. ∞ i=−∞  (2i − 1)h, 2ih , h > 0, Π = {2hn, n ∈ Z}. T = k∈Z  k(a + b), k(a + b) + b , where a = −b, Π = {(a + b)n, n ∈ Z}. π Ta,cos a = ∞ k=−∞ k(a + cos a), k(a + cos a) + a , where 0 < a < 2 , Π = {n(a +cos a), n ∈ Z}.  π (vii) Ta,sin a = ∞ k=−∞ k(a + sin a), k(a + sin a) + a , where 2 < a < π , Π = {n(a + sin a), n ∈ Z}. 

(i) (ii) (iii) (iv) (v) (vi)

Note that (ii) from Example 2.2, one can easily observe that T ∩ Π = ∅, which implies that T may be completely separated from the periodicity set Π . In the following, we will also give another example that also belongs to this case. Example 2.3 Consider the following time scale with a, b > 0: ∞    (2k + 1)(a + b), (2k + 1)(a + b) + a ,

Ta,b =

k=−∞

then  σ (t) =

t, t + a + 2b,

if t ∈

∞

if t ∈



m=1 (2k + 1)(a + b), (2k + 1)(a  ∞  m=1 (2k + 1)(a + b) + a

 + b) + a ,

54

2 A Classification of Closedness of Time Scales Under Translations

and μ(t) =

 0,

if t ∈

a + 2b,

∞

if t ∈

k=1 [k(a

∞

+ b), k(a + b) + a),

k=1 {k(a

+ b) + a}.

Note that we can consider Π = {2n(a + b), n ∈ Z} in this example. We obtain that 2na + (2n − 1)b < 2n(a + b) < (2n + 1)(a + b), where a, b > 0, n ∈ Z. Hence, we obtain 2n(a + b) ∈ T for all n ∈ Z, that is T ∩ Π = ∅. Let a = b = 1, we obtain the time scale T1,1 =

∞   4k + 2, 4k + 3], k=−∞

it follows that 4n ∈ P1,1 for n ∈ Z. Since Π = {4n, n ∈ Z}, we have T ∩ Π = ∅. 

In fact, from Definitions 2.1 and 2.2, we obtain that periodic time scales have a very nice invariance under translations, i.e., they all will coincide with themselves under translations of their periods. A new concept of periodic time scales in shift operators was introduced by Adívar et al. (see [31, 32]) and the periodic solutions for dynamic equations in shift operators were studied. However, the new concept cannot include the Definition 2.2 when T ∩ Π = ∅. Remark 2.3 Under Definitions 2.1 and 2.2, notice that if T is periodic under translations, then sup T = +∞ and inf T = −∞. For convenience, for τ ∈ R, let Tτ = {t + τ : t ∈ T}. We can obtain that if we choose nonzero real number τ ∈ Π , then T = Tτ if and only if T is invariant under translations. In fact, Definition 2.2 has the following equivalent forms: Definition 2.3 ([15, 212]) A time scale T is called a periodic time scale (or a translation invariant time scale) if   Π := {τ ∈ R : T ∩ Tτ = T} ∈ {0}, ∅ . Definition 2.4 ([15]) A time scale T is called a periodic time scale (or a translation invariant time scale) if   Π := {τ ∈ R : Tτ ∪ T−τ ⊂ T} ∈ {0}, ∅ . Remark 2.4 From Definition 2.3, one will obtain that Tτ ⊂ T. Furthermore, we also have T−τ ∩ T = T−τ which implies that T−τ ⊂ T. Hence, one obtains Tτ ∪ T−τ ⊂ T, then we have Definition 2.4. Conversely, it follows from Definition 2.4

2.1 Periodic Time Scales and Translations Invariance

55

that t ± τ ∈ T for any t ∈ T and τ ∈ Π , then we have Definition 2.2. Therefore, Definitions 2.2, 2.3 and 2.4 are equivalent. Theorem 2.1 If T is an invariant time scale under translations, then the graininess function μ : T → R+ is a periodic function with the same periods set Π . Proof Assume that T is invariant under translations, by Definition 2.3, we obtain T = Tτ . If t is a right dense point in T, then t + τ is also the right dense point in Tτ , so t + τ is the right dense point in T. Hence, μ(t + τ ) − μ(t) = σ (t + τ ) − σ (t) − τ = t + τ − t − τ = 0, i.e., μ(t + τ ) = μ(t). If t is a right scattered point in T, then t + τ is also the right scattered point in Tτ , so t + τ is the right scattered point in T. Without loss of generality, we assume that τ ∈ Π and τ > 0. It follows from σ (t) > t that σ (t) + τ > t + τ , so we obtain σ (t) + τ ≥ σ (t + τ ) > t + τ.

(2.1)

From (2.1), we have σ (t) − t = μ(t) ≥ σ (t + τ ) − (t + τ ) = μ(t + τ ) > 0.

(2.2)

For contradiction, if we assume that, for (2.2), μ(t) > μ(t + τ ), we can obtain μ is an decreasing function for all the right scattered points in T, which will lead to a contradiction because T is an invariant time scale under translations. Hence, μ(t + τ ) = μ(t). Therefore, μ is periodic. This completes the proof. 

According to Theorem 2.1, the below corollary follows immediately: Corollary 2.1 If T is a periodic time scale with a period τ , then μ(t + τ ) = μ(t) and σ (t + τ ) = σ (t) + τ . Further, we can derive a theorem to guarantee that for a time scale T, there exists at least an invariant time scale under translations T0∗ ⊂ T. Theorem 2.2 Let Π := {τ ∈ R : T±τ = ∅} = {0}, where T ∩ T±τ := T±τ 0 .   0  ±τ T ∈ {0}, ∅ , then T is an invariant time scale under If T0∗ := 0∗ τ ∈Π 0 translations. Proof We construct the following family of sets: C =



T±τ 0

 :A⊂Π .

τ ∈A

Obviously, T0∗ = ∅ implies that T0∗ is the minimal element in the family of sets C . 0 For contradiction, assume that there exists a τ0 ∈ Π \{0} such that T0∗ ∩ T±τ 0∗ ∈ C . ±τ0 We will obtain a contradiction if T0∗ ∩ T0∗ ⊂ T0∗ , because T0∗ is the minimal

56

2 A Classification of Closedness of Time Scales Under Translations

±τ0 0 element in C . Therefore, T0∗ ∩ T±τ 0∗ = ∅ implies that T0∗ ∩ T0∗ = T0∗ . Hence, 

T0∗ is an invariant time scale under translations. This completes the proof.

Example 2.4 Let +∞ 

2k −

k=−∞

! 1 1 , , 2k + 1 + k3 k2 + 1

then one can obtain ±2 ∈ Π = {τ ∈ R : T±τ = ∅} = {0}, it follows from 0  [2k, 2k + 1]. 

Theorem 2.2 that T0∗ = +∞ k=−∞ Remark 2.5 Note that for some particular time scales which do not satisfy the conditions of Theorem 2.2, the existence of an invariant time scale may not be guaranteed. For example, T = {−2k, 2k + 1 :k ∈ Z+ }, let Ξ1 := {2n : n ∈ Z− } τ and Ξ2 := {2n : n ∈ Z+ }, then T10∗ = + 1 : k ∈ Z+ } τ ∈Ξ1 T0 = {2k    τ − and T20∗ = τ ∈Ξ2 T0 = {2k : k ∈ Z }. Thus, T0∗ ∈ {0}, ∅ and there is no invariant time scale in T such that sup T0∗ = +∞, inf T0∗ = −∞. In fact, according Remark 2.3, an invariant time scale under translations (i.e., periodic time scale) means sup T = +∞ and inf T = −∞.

2.2 EA-Computation Method of Hausdorff Distance Between Translation Time Scales In this section, we will establish a computation method of Hausdorff distance between a time scale and its translation (we call it the endpoints approximation method or EA-method), and provide a explicit formula to calculate the distance ˆ d(T, Tx ) = fˆ(x), where T is a time scale with the bounded graininess function μ and Tx := {t + x : ∀t ∈ T}. Moreover, we propose a method to construct a distance ˆ d(T, Tx ) = f (x) which is continuous and linear, satisfying d(T, Tx ) → d(T, Tx ) ˆ (i.e., f (x) → f (x)) as some δ → 0 (see Wang et al.[213]). The positive vector δ will be illustrated later. For this, we give the following notations and definitions. In what follows, we emphasize that the time scales distance is measured by the Hausdorff distance. Definition 2.5 ([212]) Let X and Y be two non-empty subsets of a metric space (M, d). We define their Hausdorff distance d(X, Y ) by  d(X, Y ) = max

 ˜ ˜ sup inf d(x, y), sup inf d(x, y) ,

x∈X y∈Y

y∈Y x∈X

˜ ·) denotes the distance between two points (see Fig. 2.1). where d(·,

(2.3)

2.2 EA-Computation Method of Hausdorff Distance Between Translation. . .

57

Fig. 2.1 Components of the calculation of the Hausdorff distance between the green line X and the blue line Y

Hence, from Definition 2.5, if we assume that X = T1 and Y = T2 , this wellknown definition turns into   ˜ ˜ d(T1 , T2 ) = max sup inf d(t, s), sup inf d(t, s) . t∈T1 s∈T2

s∈T2 t∈T1

Let τ be a number and set the time scales: T :=

+∞ 

[αi , βi ],

+∞ 

Tτ := T+τ = {t +τ : ∀t ∈ T} :=

i=−∞

[αiτ , βiτ ].

(2.4)

i=−∞

Define the distance between two time scales, T and Tτ by  d(T, T ) = max τ

sup |αi − αiτ |, sup |βi i∈Z i∈Z

− βiτ |

 ,

(2.5)

where     αiτ := α ∈ Tτ : inf |αi − α| and βiτ := β ∈ Tτ : inf |βi − β| .

(2.6)

Note if we let X = T and Y = Tτ in Definition 2.5, then (2.3) immediately turns into (2.5) and we can calculate the distance between T and Tτ by the distance of their intervals, i.e, the formula (2.5) (see Fig. 2.2). In the sequel, we introduce some notations which will be used in our theorems later. Let Rs be the set of all right scattered points of T and Rsx be the set of all right scattered points of Tx ,   Rsx := tr,j ∈ Tx : tr,j < tr,j +1 , j ∈ Z and Ls be the set of all left scattered points of T and Lxs be the set of all left scattered points of Tx ,

58

2 A Classification of Closedness of Time Scales Under Translations

Fig. 2.2 The Hausdorff distance between the interval [αi , βi ] and the interval [αiτ , βiτ ]

ai

bi b i – b it a i – a it a it

b it

  Lxs := tl,j ∈ Tx : tl,j < tl,j +1 , j ∈ Z . For convenience, the closed continuous interval of the time scale T is denoted by [tl,j , tr,j ] for each j ∈ Z. Furthermore, μ∗ = sup μ(t),

μ∗ = inf μ(t), t∈Rs

t∈Rs

ν∗ = inf ν(t), t∈Ls

ν ∗ = sup ν(t). t∈Ls

Throughout the section, we assume that (A1 ) sup T = +∞ and inf T = −∞. Moreover, μ∗ > 0. (A2 ) T is a time scale with the bounded graininess function μ, i.e., there is a positive constant μ∗ such that μ ≤ μ∗ for all t ∈ Rs . In what follows, we will initiate an EA-computation method (i.e., endpoints approximation method) to calculate the Hausdorff distance between a time scale and its translations. ˆ Let d(T, Tx ) = fˆ(x) be a Hausdorff distance function and (A1 ) − (A2 ) hold. Let   (1) μ0∗ = inf tr,j − tr,j −1 : tr,j −1 < tr,j , tr,j , tr,j −1 ∈ Rs , j ∈Z

  (1) ν0∗ = inf tl,j − tl,j −1 : tl,j −1 < tl,j , tl,j , tl,j −1 ∈ Ls . j ∈Z

(1)

Now, for a sufficiently small number ε0 > 0 and μ0∗ , according to the definition (1) of infimum of the set, there exists j0 ∈ Z such that tr,j0 − tr,j0 −1 < μ0∗ + ε0 , i.e., (1) |tr,j0 − (tr,j0 −1 + μ0∗ )| < ε0 . Let U (tr,j0 , ε0 ) := {t ∈ R : |t − tr,j0 | < ε0 }, then it is (1)

μ

(1)

μ

easy to see tr,j0 −1 + μ0∗ := tˆr,j0 ∈ Rs 0∗ . Since tˆr,j0 ∈ Rs 0∗ ∩ U (tr,j0 , ε0 ), one can (1)    μ (1) obtain Rs 0∗ ∩ j U (tr,j , ε0 ) = ∅. Similarly, for ν0∗ , there also exists j0 ∈ Z (1)   ν such that tˆl,j  ∈ U (tl,j  , ε0 ) and Ls 0∗ ∩ j U (tl,j , ε0 ) = ∅. (1)

0

0

2.2 EA-Computation Method of Hausdorff Distance Between Translation. . .

59

Let   IR := x ∈ R : tr,j ∈ Rs , tˆr,j ∈ Rsx , tˆr,j0 ∈ U (tr,j0 , ε0 ) ,   IL := x ∈ R : tl,j ∈ Ls , tˆl,j ∈ Lxs , tˆl,j  ∈ U (tl,j  , ε0 ) , 0

0

  μ∗1 = sup |tr,j − tr,j −1 | : tr,j , tr,j −1 ∈ Rs , j ∈Z

  ν1∗ = sup |tl,j − tl,j −1 | : tl,j , tl,j −1 ∈ Ls , η1∗ = max{μ∗1 , ν1∗ }. j ∈Z

(1)

(1)

(1)

μ

(1)

(I) If η0∗ = μ0∗ ∈ IR , then let Rs 0∗ be the set of all right scattered points of Tμ0∗ and (1)  (1)  μ (1) (1) (1) (1) (1) (1) μ1∗ = inf |tr,j − tˆr,j | : tr,j ∈ Rs , tˆr,j ∈ Rs 0∗ , tˆr,j ∈ U (tr,j , ε0 ) ,

j

ν (1)

(1) 1∗,μ0∗

(1)  (1)  μ (1) (1) (1) (1) (1) = inf |tl,j − tˆl,j | : tr,j ∈ Rs , tˆr,j ∈ Rs 0∗ , tˆr,j ∈ U (tr,j , ε0 ) ,

j

(1)∗

μ1

(1)  (1) μ  (1) (1) (1) = sup |tr,j − tˆr,j | : tr,j ∈ Rs , tˆr,j ∈ Rs 0∗ .

j

(1)∗ (1) 1,μ0∗

ν

(1)  (1) μ  (1) (1) (1) = sup |tl,j − tˆl,j | : tr,j ∈ Rs , tˆr,j ∈ Rs 0∗

j

 (1)∗  (1)∗ (1)∗  = max μ1 , ν (1) . (1) , η1 1∗,μ0∗ 1,μ0∗ (1)∗ (1)∗ max{μ2 , ν 1 (1) }, where 2, k=0 ηk∗

 (1) (1) (1) and η1∗ = max μ1∗ , ν IR , then

(1)∗ η2

=

(1)∗

μ2

if

1

(1) k=0 ηk∗

1 (1)  (1) η  (1) (1) (1) = sup |tl,j − tˆl,j | : tr,j ∈ Rs , tˆr,j ∈ Rs k=0 k∗ ; j

(1)∗

(1)∗

(1) k=0 ηk∗

j

∈ IL , then η2

ν2

1

1 (1)  (1) η  (1) (1) (1) = sup |tr,j − tˆr,j | : tr,j ∈ Rs , tˆr,j ∈ Rs k=0 k∗ ,

(1)∗  (1) 2, 1k=0 ηk∗

ν

If

(1)∗

= max{ν2

(1)∗  (1) }, 2, 1k=0 ηk∗

,μ

where

1 (1)  (1) η  (1) (1) (1) = sup |tl,j − tˆl,j | : tl,j ∈ Ls , tˆl,j ∈ Ls k=0 k∗ , j

∈

60

2 A Classification of Closedness of Time Scales Under Translations

μ(1)∗ 1

(1) k=0 ηk∗

2,

1 (1)  (1) η  (1) (1) (1) = sup |tr,j − tˆr,j | : tl,j ∈ Ls , tˆl,j ∈ Ls k=0 k∗ . j

(1)

ν

(1)

(1)

Otherwise (i.e., η0∗ = ν0∗ ∈ IL ), let Ls 0∗ be the set of all left scattered points (1)

of Tν0∗ and (1)  (1)  ν (1) (1) (1) (1) (1) (1) ν1∗ = inf |tl,j − tˆl,j | : tl,j ∈ Ls , tˆl,j ∈ Ls 0∗ , tˆl,j ∈ U (tl,j , ε0 ) ,

j

(1) (1) 1∗,ν0∗

μ

(1)  (1)  ν (1) (1) (1) (1) (1) = inf |tr,j − tˆr,j | : tl,j ∈ Ls , tˆl,j ∈ Ls 0∗ , tˆl,j ∈ U (tl,j , ε0 ) ,

j

(1)∗

ν1

(1)  (1) ν  (1) (1) (1) = sup |tl,j − tˆl,j | : tl,j ∈ Ls , tˆl,j ∈ Ls 0∗ ,

j

(1)  (1) ν  (1) (1) (1) = sup |tr,j − tˆr,j | : tl,j ∈ Ls , tˆl,j ∈ Ls 0∗

(1)∗ (1) 1,ν0∗

μ

j

 (1) (1) (1) and η1∗ = max ν1∗ ,μ (1)∗

then η2

(1)∗

= max{ν2 (1)∗

ν2

(1) k=0 ηk∗

(1)∗  (1) }, 2, 1k=0 ηk∗

,μ

1,ν0∗

where

j

(1)∗  (1) 2, 1k=0 ηk∗

1

 (1)∗    (1) , η1 = max ν1(1)∗ , μ(1)∗(1) . If 1k=0 ηk∗ ∈ IL ,

1 (1)  (1) η  (1) (1) (1) = sup |tl,j − tˆl,j | : tl,j ∈ Ls , tˆl,j ∈ Ls k=0 k∗ ,

μ

if

(1) 1∗,ν0∗

1 (1)  (1) η  (1) (1) (1) = sup |tr,j − tˆr,j | : tl,j ∈ Ls , tˆl,j ∈ Ls k=0 k∗ ; j

(1)

(1)

(1) (1) ), 1∗,ν0∗

∈ IR (i.e., η1∗ ∈ IR or η1∗ = μ

η2(1)∗ = max{μ2(1)∗ , ν (1)∗ 1 2,

then

(1) k=0 ηk∗

},

where (1)∗

μ2

1 (1)  (1) η  (1) (1) (1) = sup |tr,j − tˆr,j | : tr,j ∈ Rs , tˆr,j ∈ Rs k=0 k∗ ,

(1)∗  (1) 2, 1k=0 ηk∗

ν Then

j

1 (1)  (1) η  (1) (1) (1) = sup |tl,j − tˆl,j | : tr,j ∈ Rs , tˆr,j ∈ Rs k=0 k∗ . j

2.2 EA-Computation Method of Hausdorff Distance Between Translation. . .

61

⎧ (1) η ⎪ ⎪ x if 0 ≤ x < 0∗ ⎪ 2 , ⎪ ⎪ (1) (1) (1)  η0∗ ⎪ η  η ⎪ ∗ ⎪ max 2 , η1 − 0∗ if x = 0∗ ⎪ 2 2 , ⎪ ⎪ (1) ⎪ η (1) ⎪−x + η∗ if 0∗ < x < η < η∗ , ⎪ ⎪ 1 1 2 ⎪  (1)∗  0∗ ⎪ ⎨ ∗ − η(1) if x = η(1) , max η , η 1 1 0∗ 0∗ fˆ(x) = (1) η (1) (1)∗ (1) (1) ⎪ ⎪x − η + η if η0∗ < x < η0∗ + 1∗ ⎪ 0∗ 1 2 , ⎪ ⎪ (1) (1)  ⎪  (1)∗ η1∗ η ⎪ (1) (1) ⎪ if x = η0∗ + max η1 + 2 , η1∗ − η0∗ − 1∗ ⎪ 2 ⎪ ⎪ (1) ⎪ η ⎪ (1) (1) (1) ∗ ⎪ −x + η1∗ if η0∗ + 1∗ ⎪ 2 < x < η0∗ + η1∗ < η1 , ⎪ ⎪  (1)∗ ∗  ⎩ (1) (1) (1) (1) max η2 , η1 − η0∗ − η1∗ if x = η0∗ + η1∗ .

(1)

η1∗ 2

,

(II) For each 1 ≤ i (1) ≤ i1 , i ∈ N, let (1)

I1

 (1) (1) (1) = i (1) ∈ Z+ : η(i (1) −1)∗ = μ(i (1) −1)∗ or μ (1) (i

 (1) (1) (1) = i (1) ∈ Z+ : η(i (1) −1)∗ = ν(i (1) −1)∗ or ν (1)

(1)

I2

(i

(1) −1)∗,θ (1) (i −2)∗

(1) −1)∗,θ (1) (i −2)∗

 ∈ IR , ∈ IL



(1)

I1(1)

and T

(1) (i (1) −2)∗

θ

∩

I2(1)

θ (1) Ls (i −2)∗

= ∅. Let

be the set of all left scattered points of

, where 

(1)

θ(i (1) −2)∗ =

(1)

ηk∗ + (1)

k≤i (1) −3,k∈I1 (1)



(1)

(1)

ηk∗ + η(i (1) −2)∗ , (1)

k≤i (1) −3,k∈I2

(1)

If θ(i (1) −2)∗ ∈ IR , we put θ(−1)∗ := 0, then (1) (1) (i (1) −1)∗,θ (1)

(1) θ (1)  (1)  (1) (1) (1) (1) (1) = inf |tl,j − tˆl,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i −2)∗ , tˆr,j ∈ U (tr,j , ε0 ) ,

ν

(i

−2)∗

j

(1) θ (1)  (1)  (1) (1) (1) (1) (1) (i −2)∗ ˆ ˆ ˆr,j |t μ(1) = inf − t | : t ∈ R , t ∈ R , t ∈ U (t , ε ) , s 0 s (1) r,j r,j r,j r,j r,j (i −1)∗

j

(1)∗ (1) (i (1) −1),θ (1)

(1) θ (1)  (1)  (1) (1) (1) = sup |tl,j − tˆl,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i −2)∗ ,

ν

(i

(1)∗ μ(i (1) −1)

then

j

−2)∗

= sup j



(1) |tr,j

θ

(1) (1) −2)∗

(1) (1) (1) − tˆr,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i

 ,

62

2 A Classification of Closedness of Time Scales Under Translations

 (1) (1) (1) η(i (1) −1)∗ = max μ(i (1) −1)∗ , ν (1) (i

 −1)∗,θ

(1) (i (1) −2)∗



 (1)∗ (1)∗ (1)∗ η(i (1) −1) = max μ(i (1) −1) , ν (1) (i

(1) −1),θ (1) (i −2)∗

,

,

if θ(i(1)(1) −1)∗ ∈ IR , then (1) θ (1)  (1)  (1)∗ (1) (1) (1) μi (1) = sup |tr,j − tˆr,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i −1)∗ ,

j

(1)∗ (1) i ,θ (1)

ν (1)

(i

(1) θ (1)  (1)  (1) (1) (1) = sup |tl,j − tˆl,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i −1)∗

j

−1)∗

 (1)∗ (1)∗ (1)∗ and ηi (1) = max μi (1) , ν (1) i

 (1) (i (1) −1)∗

,θ

(1)

;

if θ(i (1) −1)∗ ∈ IL , then (1)∗ νi (1)

(1)∗ (1) i ,θ (1)

μ (1)

(i

= sup j



(1) |tl,j

θ

(1) (1) −1)∗

(1) (1) (1) − tˆl,j | : tl,j ∈ Ls , tˆl,j ∈ Ls (i



,

(1) θ (1)  (1)  (1) (1) (1) = sup |tr,j − tˆr,j | : tl,j ∈ Ls , tˆl,j ∈ Ls (i −1)∗

j

−1)∗

 (1)∗ (1)∗ and ηi(1)∗ (1) = max νi (1) , μ (1) i

(1)

(1) ,θ (1) (i −1)∗

 .

(1)

Otherwise, θ(i (1) −2)∗ ∈ IL , also put θ(−1)∗ := 0, then μ

(1) (1) (i (1) −1)∗,θ (1) (i

(1) θ (1)  (1)  (1) (1) (1) (1) (1) = inf |tr,j − tˆr,j | : tl,j ∈ Ls , tˆl,j ∈ Ls (i −2)∗ , tˆl,j ∈ U (tl,j , ε0 ) , −2)∗

j

(1) θ (1)  (1)  (1) (1) (1) (1) (1) ν(i(1)(1) −1)∗ = inf |tl,j − tˆl,j | : tl,j ∈ Ls , tˆl,j ∈ Ls (i −2)∗ , tˆl,j ∈ U (tl,j , ε0 ) ,

j

μ(1)∗ (1) (i

(1) −1),θ (1) (i −2)∗

(1) θ (1)  (1)  (1) (1) (1) = sup |tr,j − tˆr,j | : tl,j ∈ Ls , tˆl,j ∈ Ls (i −2)∗ ,

j

(1) θ (1)  (1)  (1) (1) (1) (i −2)∗ ˆ ˆ ν(i(1)∗ |t , = sup − t | : t ∈ L , t ∈ L s l,j s (1) −1) l,j l,j l,j

j

then

2.2 EA-Computation Method of Hausdorff Distance Between Translation. . .

 (1) (1) (1) η(i (1) −1)∗ = max ν(i (1) −1)∗ , μ (1) (i

 (1)∗ (1)∗ (1)∗ η(i (1) −1) = max ν(i (1) −1) , μ (1) (i

 −1)∗,θ

(1) (i (1) −2)∗

 (1) −1),θ (1) (i −2)∗

63

,

,

if θ(i(1)(1) −1)∗ ∈ IL , then (1) θ (1)  (1)  (1)∗ (1) (1) (1) νi (1) = sup |tl,j − tˆl,j | : tl,j ∈ Ls , tˆl,j ∈ Ls (i −1)∗ ,

j

(1)∗ (1) i ,θ (1)

μ (1)

(i

(1) θ (1)  (1)  (1) (1) (1) = sup |tr,j − tˆr,j | : tl,j ∈ Ls , tˆl,j ∈ Ls (i −1)∗

j

−1)∗

 (1)∗ (1)∗ (1)∗ and ηi (1) = max νi (1) , μ (1)

(1) ,θ (1) (i −1)∗

i

if

(1) θ(i (1) −1)∗

 ;

∈ IR , then (1)∗ μi (1)

(1)∗ (1) i ,θ (1)

ν (1)

(i

= sup



j

(1) |tr,j

θ

(1) (1) −1)∗

(1) (1) (1) − tˆr,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i

 ,

(1) θ (1)  (1)  (1) (1) (1) = sup |tl,j − tˆl,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i −1)∗

j

−1)∗

 (1)∗ (1)∗ (1)∗ and ηi (1) = max μi (1) , ν (1) i

 (1) (i (1) −1)∗

,θ

. We put

−1

(1) k=0 ηk∗

(1)∗

= 0 and η0

= 0, then

fˆ(x) = ⎧ (1) i (1) −2 (1) i (1) −2 (1) i (1) −2 (1) η(i (1) −1)∗ ⎪ (1)∗ ⎪ x − η + η if η < x < η + , ⎪ (1) k=0 k=0 k=0 k∗ k∗ k∗ 2 ⎪ i −1 ⎪ (1) (1) (1) ⎪ ⎪ η η η (1)     (1) (1) (1) (1) ⎪ (1)∗ i −2 (1) (i −1)∗ (i −1)∗ ∗ − i −2 η(1) − (i −1)∗ if x = ⎪ + η , η η + , max ⎪ (1) k=0 k=0 ⎪ k∗ k∗ 2 2 2 i −1 1 ⎪ ⎨ (1) η   (1) (1) (1) (1) (1) −x + η1∗ if ik=0−2 ηk∗ + (i 2−1)∗ < x < ik=0−1 ηk∗ < η1∗ , ⎪     ⎪ (1) (1) (1)∗ i −1 (1) i −1 (1) ⎪ ⎪ max ηi (1) , η1∗ − k=0 ηk∗ if x = k=0 ηk∗ , ⎪ ⎪ ⎪ i (1) −1 (1) i1 i (1) −1 (1) ⎪ (1)∗ (1) ⎪ ηk∗ ≤ η1∗ , ⎪x − k=0 ηk∗ + ηi (1) if k=0 ηk∗ < x < k=0 ⎪ ⎪   1 i1 ⎩ (1) (1)∗ (2)∗ (1) η + ηi (1) , ηi (1) −1 , if x = ik=0 ηk∗ . max k=i (1) k∗

Here

i1

(1) k=0 ηk∗

≤ η1∗ and

i1 +1 k=0

ηk∗ > η1∗ .

(III) For i1 < i (2) ≤ i2 , i ∈ N, let

(1)

64

2 A Classification of Closedness of Time Scales Under Translations (2)

I1

 (2) (2) (2) = i (2) ∈ Z+ : η(i (2) −1)∗ = μ(i (2) −1)∗ or μ (2)

−1)∗,θ

(i

 (2) (2) (2) = i (2) ∈ Z+ : η(i (2) −1)∗ = ν(i (2) −1)∗ or ν (2)

(2)

I2

(i

(2) (i (2) −2)∗

(2) −1)∗,θ (2) (i −2)∗

 ∈ IR , ∈ IL



(2)

I1(2)

and T

(2) (i (2) −2)∗

θ

∩

I2(2)

θ (2) Ls (i −2)∗

= ∅. Let

be the set of all left scattered points of

, where

(2)

θ(i (2) −2)∗ =

i1 



(1)

ηk∗ +

k=0



(2)

ηk∗ + (2)

k≤i (2) −3,k∈I1

(2)

(2)

ηk∗ + η(i (2) −2)∗ , (2)

k≤i (2) −3,k∈I2

(2)

If θ(i (2) −2)∗ ∈ IR , then (2) (2) (i (2) −1)∗,θ (2)

(2) θ (2)  (2)  (2) (2) (2) (1) (1) = inf |tl,j − tˆl,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i −2)∗ , tˆr,j ∈ U (tr,j , ε0 ) ,

ν

(i

−2)∗

j

(2) θ (2)  (2) (2)  (2) (2) (2) (1) (1) μ(i (2) −1)∗ = inf |tr,j − tˆr,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i −2)∗ , tˆr,j ∈ U (tr,j , ε0 ) ,

j

(2)∗ (2) (i (2) −1),θ (2)

(2) θ (2)  (2)  (2) (2) (2) = sup |tl,j − tˆl,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i −2)∗ ,

ν

(i

j

−2)∗

(2) θ (2)  (2)  (2)∗ (2) (2) (2) μ(i (2) −1) = sup |tr,j − tˆr,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i −2)∗ ,

j

then  (2) (2) (2) η(i (2) −1)∗ = max μ(i (2) −1)∗ , ν (2) (i

 (2)∗ (2)∗ η(i(2)∗ (2) −1) = max μ(i (2) −1) , ν (2) (i

(2) −1)∗,θ (2) (i −2)∗

−1),θ

 (2) (i (2) −2)∗

 ,

,

(2)

if θ(i (2) −1)∗ ∈ IR , then (2) θ (2)  (2)  (2)∗ (2) (2) (2) μi (2) = sup |tr,j − tˆr,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i −1)∗ ,

j

2.2 EA-Computation Method of Hausdorff Distance Between Translation. . .

ν (2)∗ (2) i

(2) (i (2) −1)∗

,θ

(2) θ (2)  (2)  (2) (2) (2) = sup |tl,j − tˆl,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i −1)∗

j



 (2)∗ (2)∗ (2)∗ and ηi (2) = max μi (2) , ν (2)

(2) ,θ (2) (i −1)∗

i

if θ(i(2)(2) −1)∗ ∈ IL , then νi(2)∗ (2)

(2)∗ (2) i ,θ (2)

μ (2)

(i

65

= sup



j

(2) |tl,j

;

θ

(2) (2) −1)∗

(2) (2) (2) − tˆl,j | : tl,j ∈ Ls , tˆl,j ∈ Ls (i



,

(2) θ (2)  (2)  (2) (2) (2) = sup |tr,j − tˆr,j | : tl,j ∈ Ls , tˆl,j ∈ Ls (i −1)∗

j

−1)∗

 (2)∗ (2)∗ (2)∗ and ηi (2) = max νi (2) , μ (2) i

 (2) (i (2) −1)∗

,θ

.

Otherwise, θ(i(2)(2) −2)∗ ∈ IL , then μ

(2) (2) (i (2) −1)∗,θ (2) (i

(2) ν(i (2) −1)∗

(2) θ (2)  (2)  (2) (2) (2) (2) (2) = inf |tr,j − tˆr,j | : tl,j ∈ Ls , tˆl,j ∈ Ls (i −2)∗ , tˆl,j ∈ U (tl,j , ε0 ) ,

j

−2)∗

= inf



j

(2) |tl,j

(2)∗ μ (2) (2) (i −1),θ (2) (i

θ

(2) (2) −2)∗

(2) (2) (2) − tˆl,j | : tl,j ∈ Ls , tˆl,j ∈ Ls (i

= sup j

−2)∗



(2) |tr,j

 (2) (2) , tˆl,j ∈ U (tl,j , ε0 ) , θ

(2) (2) −2)∗

(2) (2) (2) − tˆr,j | : tl,j ∈ Ls , tˆl,j ∈ Ls (i

(2) θ (2)  (2)  (2)∗ (2) (1) (2) ν(i (2) −1) = sup |tl,j − tˆl,j | : tl,j ∈ Ls , tˆl,j ∈ Ls (i −2)∗ ,

j

then  (2) (2) (2) η(i (2) −1)∗ = max ν(i (2) −1)∗ , μ (2) (i

 (2)∗ (2)∗ (2)∗ η(i (2) −1) = max ν(i (2) −1) , μ (2) (i

(2)

if θ(i (2) −1)∗ ∈ IL , then

 −1)∗,θ

(2) (i (2) −2)∗

 (2) −1),θ (2) (i −2)∗

,

,



,

66

2 A Classification of Closedness of Time Scales Under Translations (2) θ (2)  (2)  (2) (2) (2) (i −1)∗ ˆ ˆ |t , νi(2)∗ = sup − t | : t ∈ L , t ∈ L s l,j s (2) l,j l,j l,j

j

(2)∗ (2) i ,θ (2)

μ (2)

(i

(2) θ (2)  (2)  (2) (2) (2) = sup |tr,j − tˆr,j | : tl,j ∈ Ls , tˆl,j ∈ Ls (i −1)∗

j

−1)∗

 (2)∗ (2)∗ (2)∗ and ηi (2) = max νi (2) , μ (2) i,θ

if

(2) θ(i (2) −1)∗

 ;

(i (2) −1)∗

∈ IR , then (2) θ (2)  (2)  (2)∗ (2) (2) (2) μi (2) = sup |tr,j − tˆr,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i −1)∗ ,

j

(2)∗ ν (2) (2) i ,θ (2) (i

= sup j

−1)∗

 (2)∗ (2)∗ (2)∗ and ηi (2) = max μi (2) , ν (2) i



(2) |tl,j

θ

(2) (2) −1)∗

(2) (2) (2) − tˆl,j | : tr,j ∈ Rs , tˆr,j ∈ Rs (i

(2) ,θ (2) (i −1)∗





.

Moreover, if η1∗ = max{μ∗1 , ν1∗ } = μ∗1 ∈ IR , then η2∗ = max{μ∗2 , ν2∗ }, where  (2) η∗  (2) (2) (2) μ∗2 = sup |tr,j − tˆr,j | : tr,j ∈ Rs , tˆr,j ∈ Rs 1 , j

 (2) η∗  (2) (2) (2) ν2∗ = sup |tl,j − tl,j | : tr,j ∈ Rs , tˆr,j ∈ Rs 1 ; j

if η1∗ = max{μ∗1 , ν1∗ } = ν1∗ ∈ IL , then η2∗ = max{μ∗2 , ν2∗ }, where  (2) η∗  (2) (2) (2) μ∗2 = sup |tr,j − tˆr,j | : tl,j ∈ Ls , tˆl,j ∈ Ls 1 , j

 (2) η∗  (2) (2) (2) ν2∗ = sup |tl,j − tl,j | : tl,j ∈ Ls , tˆl,j ∈ Ls 1 . j

Then

2.2 EA-Computation Method of Hausdorff Distance Between Translation. . .

67

fˆ(x) = ⎧ 1 (1) (2)∗ ⎪ x − ik=0 ηk∗ + ηi (2) −1 ⎪ ⎪ ⎪ (2) ⎪ ⎪ i1 i1 i (2) −2 (2) η(i (2) −1)∗ ⎪ (1) (1) ⎪ if η < x < η + η + , ⎪ k=i1 +1 k∗ k=0 k∗ k=0 k∗ 2 ⎪ ⎪ (2) ⎪ η ⎪  (2) i (2) −2 (2) 2 i1 ⎪ (2)∗ (1) ⎪ ηk∗ ⎪max (i 2−1)∗ + ηi (2) −1 + k=i1 +1 ηk∗ , k=1 ηk∗ − k=0 ⎪ ⎪ ⎪ ⎪ i (2) −2 (2) η(i(2)(2) −1)∗  ⎪ ⎪ ⎪ − k=i1 +1 ηk∗ − ⎪ 2 ⎪ (2) ⎪ ⎪ i1 i (2) −2 (2) η(i (2) −1)∗ ⎪ (1) ⎪ ⎪ if x = η + η + , ⎨ k=i1 +1 k∗ 2 2 k=0 ∗ k∗ −x + k=1 ηk ⎪ ⎪ (2) ⎪ i1 i (2) −2 (2) η(i (2) −1)∗ ⎪ (1) ⎪ ⎪ if η + η +

(IV) Put i0 = 0. For in0 −1 < i (n0 ) ≤ in0 , i ∈ N and n0 ≥ 1, let (n0 )

I1

 (n ) = i (n0 ) ∈ Z+ : η (n0 0 ) (i

−1)∗

 I2(n0 ) = i (n0 ) ∈ Z+ : η(n(n0 )0 ) (i

(n0 )

(n0 )

and I1

∩ I2

T

, where

(n ) θ (n0 ) (i 0 −2)∗

(n0 ) (i (n0 ) −2)∗

θ

=

= ν (n(n0 )0 ) (i

(n0 ) (n0 ) −2)∗

= ∅. Let Ls (i

i1  k=0

+

(i (n0 ) −2)∗

−1)∗

θ

i2 

(1)

ηk∗ +

(n ) 0 −1)∗,θ 0 (i (n0 ) −2)∗

(i

∈ IL





(n −1)

ηk∗0

k=in0 −2 +1 (n )

(n0 )

or ν (n(n0 ) )

in0 −1 (2)

ηk∗ + . . . + ηk∗0 +

k≤i (n0 ) −3,k∈I1

−1)∗

be the set of all left scattered points of

k=i1 +1



 ∈ IR ,

(n0 ) (n ) or μ (n0 ) (n ) (i (n0 ) −1)∗ (i 0 −1)∗,θ 0

=μ



(n )

ηk∗0 + η (n0 )

k≤i (n0 ) −3,k∈I2

(n0 ) , (i (n0 ) −2)∗

68

2 A Classification of Closedness of Time Scales Under Translations (n0 ) (i (n0 ) −2)∗

If θ

ν

∈ IR , then

(n0 ) (n0 ) (i (n0 ) −2)∗

(i (n0 ) −1)∗,θ

= inf



j

(n ) |tl,j0

(n ) − tˆl,j0 |

(n ) tr,j0

:

∈

(n ) Rs , tˆr,j0

(n0 ) (i (n0 ) −2)∗

θ

∈ Rs

 (n ) (n ) , tˆr,j0 ∈ U (tr,j0 , ε0 ) ,

(n )

μ

ν

(n0 ) (i (n0 ) −1)∗

θ (n0 )  (n )  0 (n ) (n ) (n ) (n ) (n ) = inf |tr,j0 − tˆr,j0 | : tr,j0 ∈ Rs , tˆr,j0 ∈ Rs (i −2)∗ , tˆr,j0 ∈ U (tr,j0 , ε0 ) , j

(n0 )∗

(n ) (i (n0 ) −1),θ (n0 ) (i 0 −2)∗

= sup



j

(n ) |tl,j0

(n ) − tˆl,j0 |

:

(n ) tr,j0

∈

(n0 ) (i (n0 ) −2)∗

θ

(n ) Rs , tˆr,j0

∈ Rs (n )

(n0 )∗ (i (n0 ) −1)

μ

θ (n0 )  (n )  0 (n ) (n ) (n ) = sup |tr,j0 − tˆr,j0 | : tr,j0 ∈ Rs , tˆr,j0 ∈ Rs (i −2)∗ , j

then η

η (n0 ) (i (n0 ) −1)∗

if θ

(i

(n0 )∗ (i (n0 ) −1)

= sup



j

(n0 )∗

ν (n

 (n )∗ = max μ (n0 0 )

,ν

−1)

(n ) − tˆr,j0 |

(n ) tr,j0



(n0 )

(n ) (i (n0 ) −1)∗,θ (n0 ) (i 0 −2)∗

,ν

(i

(n ) 0 ) ,θ 0 (i (n0 ) −1)∗

(n ) |tr,j0

= sup j



(n ) |tl,j0

 (n )∗ (n )∗ (n )∗ and η (n00 ) = max μ (n00 ) , ν (n0 ) i

if

−1)∗

(n0 )∗

(n ) (i (n0 ) −1),θ (n0 ) (i 0 −2)∗

,

 ,

∈ IR , then

(n )∗ μ (n00 ) i

i

 (n ) = max μ (n0 0 )

(n0 ) (i (n0 ) −1)∗

i

(n ) θ (n00 ) −1)∗ (i

i

0

:

(n ) − tˆl,j0 |

(n ) ,θ (n0 ) (i 0 −1)∗

:

∈

(n ) Rs , tˆr,j0

(n ) tr,j0

∈

∈ Rs

(n ) Rs , tˆr,j0

(n0 ) (i (n0 ) −1)∗

θ



,

(n0 ) (i (n0 ) −1)∗

θ

∈ Rs

 ;

∈ IL , then

(n )∗ ν (n00 ) i

= sup j



(n ) |tl,j0

θ

(n0 ) (n0 ) −1)∗

(n ) (n ) (n ) − tˆl,j0 | : tl,j0 ∈ Ls , tˆl,j0 ∈ Ls (i



,





,

2.2 EA-Computation Method of Hausdorff Distance Between Translation. . .

(n0 )∗

= sup

μ (n

(n ) 0 ) ,θ 0 (i (n0 ) −1)∗

i



j

(n0 ) |tr,j

(n0 ) − tˆr,j |



 (n )∗ (n )∗ (n )∗ and η (n00 ) = max ν (n00 ) , μ (n0 ) i

i

i

(n0 ) (i (n0 ) −2)∗

Otherwise, θ

0

(n ) ,θ (n0 ) (i 0 −1)∗

:

(n0 ) tl,j

∈

(n0 ) Ls , tˆl,j

69 (n0 ) (i (n0 ) −1)∗

θ

∈ Ls



.

∈ IL , then

(n0 )

μ

(n0 ) (i (n0 ) −2)∗

(i (n0 ) −1)∗,θ

= inf



j

(n ) |tr,j0

(n ) − tˆr,j0 |

:

(n ) tl,j0

∈

(n ) Ls , tˆl,j0

(n0 ) (i (n0 ) −2)∗

θ

∈ Ls

 (n ) (n ) , tˆl,j0 ∈ U (tl,j0 , ε0 ) ,

(n )

(n0 ) (i (n0 ) −1)∗

ν

θ (n0 )  (n )  0 (n ) (n ) (n ) (n ) (n ) = inf |tl,j0 − tˆl,j0 | : tl,j0 ∈ Ls , tˆl,j0 ∈ Ls (i −2)∗ , tˆl,j0 ∈ U (tl,j0 , ε0 ) , j

(n0 )∗

μ

= sup

(n0 ) (i (n0 ) −2)∗

(i (n0 ) −1),θ

ν (n(n0 )∗ (i 0 ) −1)

j

= sup



j



(n ) |tr,j0

(n0 ) |tl,j

θ

(n0 ) (n0 ) −2)∗

(n ) (n ) (n ) − tˆr,j0 | : tl,j0 ∈ Ls , tˆl,j0 ∈ Ls (i

(n0 ) − tˆl,j |

:

(n0 ) tl,j

∈

(n0 ) Ls , tˆl,j

(n0 ) (i (n0 ) −2)∗

θ

∈ Ls



,

then η(n(n0 )0 )

(n0 ) (i (n0 ) −1)∗

if θ

(i

−1)∗

η

(n0 )∗ (i (n0 ) −1)

 = max ν (n(n0 )0 ) (i

−1)∗

 (n )∗ = max ν (n0 0 ) (i

, μ(n(n0 ) ) (i

0

 (n0 ) (i (n0 ) −2)∗

−1)∗,θ

(n0 )∗

−1)

,μ

(n0 ) (i (n0 ) −2)∗

(i (n0 ) −1),θ

,

 ,

∈ IL , then (n )

θ (n0 )  (n0 )  (n0 ) (n0 ) (n0 ) (i 0 −1)∗ ˆ ˆ |t , ν (n(n00)∗ = sup − t | : t ∈ L , t ∈ L s l,j s ) l,j l,j l,j i

j

(n )

(n )∗

μ (n0 ) i

0

(n0 ) (i (n0 ) −1)∗

,θ

θ (n0 )  (n )  0 (n ) (n ) (n ) = sup |tr,j0 − tˆr,j0 | : tl,j0 ∈ Ls , tˆl,j0 ∈ Ls (i −1)∗ j

 (n )∗ (n )∗ (n )∗ and η (n00 ) = max ν (n00 ) , μ (n0 ) i

i

i

(n ) 0 ,θ 0 (i (n0 ) −1)∗



;



,

70

2 A Classification of Closedness of Time Scales Under Translations (n0 ) (i (n0 ) −1)∗

if θ

∈ IR , then

μ(n(n00)∗ i )

= sup j

(n0 )∗

ν (n i

(n ) 0 ) ,θ 0 (i (n0 ) −1)∗



(n0 ) |tr,j

= sup j



(n0 ) − tˆr,j |

(n ) |tl,j0

 (n )∗ (n )∗ (n )∗ and η (n00 ) = max μ (n00 ) , ν (n0 ) i

i

i

:

(n0 ) tr,j

(n ) − tˆl,j0 |

(n ) 0 ,θ 0 (i (n0 ) −1)∗

:

∈

(n0 ) Rs , tˆr,j

(n ) tr,j0

∈

∈ Rs

(n ) Rs , tˆr,j0

(n0 ) (i (n0 ) −1)∗

θ

,

(n0 ) (i (n0 ) −1)∗

θ

∈ Rs





 .

Moreover, if ηn∗0 −1 = max{μ∗n0 −1 , νn∗0 −1 } = μ∗n0 −1 ∈ IR , then ηn∗0 = max{μ∗n0 , νn∗0 }, where ηn∗ −1   (n ) (n ) (n ) (n ) μ∗n0 = sup |tr,j0 − tˆr,j0 | : tr,j0 ∈ Rs , tˆr,j0 ∈ Rs 0 , j

ηn∗ −1   (n ) (n ) (n ) (n ) νn∗0 = sup |tl,j0 − tl,j0 | : tr,j0 ∈ Rs , tˆr,j0 ∈ Rs 0 ; j

if ηn∗0 −1 = max{μ∗n0 −1 , νn∗0 −1 } = νn∗0 −1 ∈ IL , then ηn∗0 = max{μ∗n0 , νn∗0 }, where ηn∗ −1   (n0 ) (n0 ) (n0 ) (n0 ) μ∗n0 = sup |tr,j − tˆr,j | : tl,j ∈ Ls , tˆl,j ∈ Ls 0 , j

ηn∗ −1   (n ) (n ) (n ) (n ) νn∗0 = sup |tl,j0 − tl,j0 | : tl,j0 ∈ Ls , tˆl,j0 ∈ Ls 0 . j

Then for in0 −2 < i (n0 −1) ≤ in0 −1 , one has ⎧ 1 i (n0 −1) −1 (n0 −1) (1) (n −1)∗ ⎪ x − ik=1 ηk∗ − . . . − k=i η + η (n00 −1) ⎪ ⎪ n0 −2 +1 k∗ i ⎪ ⎪  in0 −2 i (n0 −1) −1 (n0 −1) ⎪ ⎪ if i1 η(1) + . . . + k=i η(n0 −2) + k=i η nk=1 ηk . Now, let n = n0 + 1, and 0 0 0 repeat the computation process of (2.7)-(2.10), for in0 < i ≤ in0 +1 , we can obtain that ⎧ n 0 ∗ n0 n0 ⎪ x − Λn(n0 0 ) + η(n(n00)∗ ⎪ ) if Λi (n0 ) −1 < x < Λin ≤ k=1 ηk , ⎪ i −1 i 0 ⎪ ⎪   (n0 +1)∗ in0 ⎪ (n ) (n )∗ n 0 0 0 ⎪ η + η (n0 ) if x = Λin , max η (n0 ) , ⎪ ⎪ −1 k=i (n0 ) k∗ i i 0 ⎪ ⎪ (n0 +1) ⎪ η (n ⎪ ⎪ (n0 +1)∗ n0 n0 n0 +1 (i 0 +1) −1)∗ ⎪ ⎪ x − Λ + η if Λ < x < Λ + , ⎪ in0 in0 2 i (n0 +1) −2 i (n0 +1) −1 ⎪ ⎪ (n0 +1) ⎪ ⎪ η ⎪ ⎨max  (i (n0 +1) −1)∗ + i (n0 +1) −2 η(n0 +1) + η(n0 +1)∗ , k=in0 +1 k∗ 2 i (n0 +1) −1 ˆ f (x) = (n0 +1) (n0 +1) ⎪ η (n η (n   ⎪ +1) n +1 n +1 n +1 0 0 +1) −1)∗ ⎪ (i −1)∗ (i 0 0 0 ∗ ⎪ , if x = Λ (n0 +1) + ⎪ k=1 ηk − Λi (n0 +1) −2 − 2 2 ⎪ i −2 ⎪ (n0 +1) ⎪ ⎪ η n0 +1 ∗ n0 +1 ∗ ⎪ n0 +1 ⎪ (i (n0 +1) −1)∗ ⎪ < x < Λn(n0 +1 < k=1 ηk , ⎪ 2 ⎪−x + k=1 ηkif Λi (n0 +1) −2 + i 0 +1) −1 ⎪  ⎪ (n0 +1)∗ n0 +1 ∗ n0 +1 n0 +1 ⎪ ⎪ if x = Λ (n0 +1) , max η (n0 +1) , k=1 ηk − Λ (n0 +1) ⎪ i i −1 −1 i ⎪ ⎪  ⎪ (n0 +1)∗ n0 +1 ⎩x − Λn(n0 +1 + η (n +1) if Λ (n +1) < x < Λn0 +1 ≤ n0 +1 η∗ , +1) i

where Λinn0 +1 ≤ +1

0

−1

n0 +1 k=1

0

i

i

0

0

−1

in0 +1

(n +1)

ηk∗ and Λinn0 +1 + η(in0 +1 +1)∗ > +1 0

k=1

n0 +1 k=1

0

k

ηk∗ . Therefore, the

ˆ Hausdorff distance d(T, Tx ) = fˆ(x) is a linear broken line function expressed as (2.11)–(2.13). In addition, according to (i) − (vii) in Theorem 2.3, for each n0 ∈ Z+ and in0 −1 < i (n0 ) ≤ in0 , one can easily obtain that [Λn(n0 −1 0 −1) i



Λnin0 −1 0 −1

−1

+ δ1 , Λn(n0 0 ) i −2

+ δ1 , Λinn0 −1 − δ2 ] ⊆ [Λn(n0 −1 0 −1) −1

+

i

0

η

(n0 ) (i (n0 ) −1)∗

2

− δ2



−1

, Λinn0 −1 ], −1

 , Λn(n0 0 ) ⊆ Λinn0 −1 −1 0

i

0

−2

+

η

(n0 ) (i (n0 ) −1)∗ 

2

,

76

2 A Classification of Closedness of Time Scales Under Translations

 n0 Λ (n0 ) i

−2

+

η

(n0 ) (i (n0 ) −1)∗

2

+ δ1 , Λn(n0 0 ) i −1

[Λn(n0 0 ) i

−1

− δ2



 ⊆ Λn(n0 0 ) i

+ δ1 , Λnin0 − δ2 ] ⊆ [Λn(n0 0 ) i

0

−2

−1

+

η

(n0 ) (i (n0 ) −1)∗

2

, Λn(n0 0 ) i

−1

 ,

, Λnin0 ], 0

thus, for each x ∈ I, one can obtain f (x) = fˆ(x) and f (x) is continuous on R. This completes the proof.

 Let m(·) be the Lebesgue measure function. Then one can obtain the following corollary. Corollary 2.2 For any real vector sequence {δk } with δk → 0 as k → ∞, where δk = (δ1k , δ2k ), the function sequence {fk (x)} is convergent to f (x) in measure, i.e., for any given ε > 0, it follows that   lim m x ∈ R : |fk (x) − f (x)| > ε} = 0,

k→∞

where fk (x) = ⎧ (n −1)∗ n −1 n −1 n −1 ⎪ x − Λ (n0 0 −1) + η (n00 −1) if x ∈ [Λ (n0 0 −1) + δ1k , Λin0 −1 − δ2k ], ⎪ ⎪ i −1 −1 i i 0 ⎪ ⎪ i (n0 ) ,δk n0 −1 n0 −1 n0 −1 δk ⎪ k k k k ⎪ η(n(n00)∗ ⎪ ) −1 + δ1 + ki (n0 −1) (x − Λin −1 − δ1 ) if x ∈ [Λin −1 − δ2 , Λin −1 + δ1 ] := Iin −1 , ⎪ i 0 0 0 0 ⎪ ⎪ (n0 ) ⎪ η (n ⎪   n0 −1 ⎪ n0 −1 (n0 )∗ n0 (i 0 ) −1)∗ k k ⎪ x − Λin −1 + η (n0 ) if x ∈ Λin −1 + δ1 , Λ (n0 ) + − δ2 , ⎪ ⎪ 2 i −1 i −2 0 0 ⎪ ⎪ (n0 ) (n0 ) ⎪ η η (n ⎪   (n )  ⎪ (i (n0 ) −1)∗ (i 0 ) −1)∗ k + k i 0 ,δk x − Λn0 ⎨ n0 η∗ − Λn0 − − δ − − δ1k 1 k=1 k 1 2 2 i (n0 ) −2 i (n0 ) −2 (n0 ) (n0 ) η (n η (n ⎪ ⎪ n0 (i 0 ) −1)∗ (i 0 ) −1)∗ k , Λn0 ⎪ ⎪ + − δ + + δ1k ] := I(1) , if x ∈ [Λ ⎪ 2 2 2 i (n0 ) ,δk i (n0 ) −2 i (n0 ) −2 ⎪ ⎪ (n0 ) ⎪ ⎪ η (n )   n0 n0 ∗ ⎪ (i 0 −1)∗ ⎪ ⎪ + δ1k , Λn(n0 0 ) − δ2k , ⎪ 2 ⎪−x + k=1 ηk if x ∈ Λi (n0 ) −2 + i −1 ⎪ (n0 ) ⎪ (n )∗ (2) ⎪ ⎪ η (n00 ) + δ1k + k2i ,δk (x − Λn(n0 0 ) − δ1k ) if x ∈ [Λn(n0 0 ) − δ2k , Λn(n0 0 ) + δ1k ] := I (n0 ) , ⎪ i i i −1 i −1 i −1 ,δk ⎪ ⎪ ⎪ (n )∗ n n n ⎩x − Λ 0 + η 0 if x ∈ [Λ 0 + δ k , Λ 0 − δ k ], i (n0 ) −1

i (n0 ) −1

i (n0 )

1

in0

2

where (n0 ) ,δ k k i(n0 −1) i

=

(n )∗ −1 i

η (n00 )

−

in0 −1

k=i (n0 −1)

(n −1)

ηk∗0

δ1k + δ2k

(n −1)∗ i

− η (n00 −1) + δ1k + δ2k

,

i (n0 ) −2 (n0 ) (n0 ) (n0 )∗ n0 k k ∗ k=1 ηk − Λi (n0 ) −2 − η(i (n0 ) −1)∗ − k=in0 −1 +1 ηk∗ − ηi (n0 ) −1 − δ1 + δ2

n 0 (n0 ) k1i ,δk

=

δ1k + δ2k

,

2.2 EA-Computation Method of Hausdorff Distance Between Translation. . .

(n0 ) k2i ,δk

=

(n )∗ i

η (n00 ) −

n0

n0 ∗ k=1 ηk + Λi (n0 ) −1 δ1k + δ2k

+ δ1k − δ2k

77

.

Proof In fact, note that 



lim m

k→∞



 δk Iin

in0 −1 0 by letting T = R and T = hZ, respectively. Note that if T is a periodic time scale, then T = Π indicates that T is closed to additive operation, i.e., t1 + t2 ∈ T for t1 , t2 ∈ T. Remark 2.10 From Definition 2.8, one will easily observe that there exists differences between R and T on the definitions of relatively dense set. Let Π1 := {τ ∈ R : T ∩ Tτ = ∅} = {0}, where Tτ := T + τ = {t + τ : ∀t ∈ T}. Now we introduce the definition of almost periodic time scales (i.e., almost invariant time scales under translations). Definition 2.11 ([212, 215]) We say T is an almost periodic time scale if for any given ε > 0, there exists a constant l(ε) > 0 such that each interval of length l(ε) contains a τ (ε) ∈ R such that d(T, Tτ ) < ε, i.e., for any ε > 0, the following set E{T, ε} = {τ ∈ R : d(Tτ , T) ≤ ε}

2.3 Time Scale Spaces and Completeness

81

is relatively dense in Π1 . Here τ is called the ε-translation number of T and l(ε) is called the inclusion length of E{T, ε}, E{T, ε} is called the ε-translation numbers set of T, and for simplicity, we use the notation E{T, ε} := Πε . Remark 2.11 From Definition 2.3.1, we obtain that if T is an almost periodic time scale, then sup T = +∞ and inf T = −∞. Remark 2.12 Definition 2.2 is a particular case of Definition 2.3.1. It is easy to observe that all the periodic time scales under Definition 2.2 are almost periodic time scales under Definition 2.3.1. In fact, if by letting ε = 0 in Definition 2.3.1, then we obtain the concept of periodic time scales which is equivalent to Definition 2.2. Lemma 2.1 If T is an almost periodic time scale under Definition 2.3.1, then ∀ε > 0, there exists τ ∈ Π1 such that d(T, Tτ ) ≤ ε. Proof We argue by contradiction. Assume that there exists ε0 > 0, for all τ ∈ R, one has d(T, Tτ ) > ε0 , then T is not an almost periodic time scale. Hence, ∀ε > 0, there exists τ ∈ R such that d(T, Tτ ) ≤ ε. Let Σε := {τ ∈ R : d(T, Tτ ) ≤ ε}, assume that for all τ ∈ Σε , one has τ ∈ Π1 , i.e., T ∩ Tτ = ∅. It is obvious that  τ ε∈R Σε is closed since Σε is a closed set. Now, we claim that inf d(T, T ) > 0. τ ∈Σε

In fact, for contradiction, assume that inf d(T, Tτ ) = 0. Then, for any ε > 0, there exists τε ∈ Σε such that

τ ∈Σε

d(T, Tτε ) < ε + inf d(T, Tτ ) = ε, τ ∈Σε

(2.16)

 for (2.16), let ε → 0, then τε → τ0 ∈ ε∈R Σε with T ∩ Tτ0 = ∅, and we get d(T, Tτ0 ) ≤ 0, this is a contradiction. Hence, there exists τ ∈ Σε such that τ ∈ Π1 . This completes the proof. 

Remark 2.13 According to Lemma 2.1, if T is an almost periodic time scale and for any ε > 0, there exists τ ∈ R such that d(T, Tτ ) ≤ ε, then it is rational to give tacit consent to T ∩ Tτ = ∅ for τ ∈ Π1 . According to Lemma 2.1, one will easily obtain the following equivalent definition of almost periodic time scales. Definition 2.12 ([212, 215]) We say T is an almost periodic time scale if for any given ε > 0, there exists a constant l(ε) > 0 such that each interval of length l(ε) contains a τ (ε) ∈ Π1 such that d(T, Tτ ) ≤ ε, i.e., for any ε > 0, the following set E{T, ε} = {τ ∈ Π1 : d(Tτ , T) ≤ ε}

82

2 A Classification of Closedness of Time Scales Under Translations

is relatively dense in Π1 . Here τ is called the ε-translation number of T and l(ε) is called the inclusion length of E{T, ε}, E{T, ε} is called the ε-translation numbers set of T, and for simplicity, we use the notation E{T, ε} := Πε . Remark 2.14 In Definition 2.3.1, the condition “relatively dense in Π1 ” cannot be replaced by “relatively dense in R”, or mistakes will occur. In fact, when we consider almost periodic functions on time scales, it is very important to clarify the set in which the ε-translation number set E{ε, T} is relatively dense. For example, for T = Z, if we do not clarify the condition “relatively dense in Π1 ”, then it may be “relatively dense in R”, if so, for ε = 14 , the set E{ 14 , T} = { 14 + n : n ∈ Z} is relatively dense in R but T ∩ Tτ = ∅ for all τ ∈ E{ 14 , T}, which will make it impossible to introduce almost periodic functions on T ∩ Tτ .

2.3.2 Embedding of Time Scales In this section, we introduce some isometrically isomorphic mappings to establish some embedding theorems of time scales. Now, for any x ∈ R, denote the set {t + x : ∀t ∈ T} by Tx . Let f (x) = d(T, Tx ) f be the continuous Hausdorff distance function with δ-accuracy and TBμ be the set of all the translations of the time scale T with a bounded graininess function μ : T → R+ . Note that f is a bounded non-negative continuous linear broken line function. Denote the closure set of the range of the distance function f by f (R) and f (0) = 0. First, for x ∈ R, we construct a mapping j1 :

f

TBμ

→

Tx

→

f (R), j1 (Tx ) = f (x),

and one can easily see that j1 is a continuous bijective mapping. For f (R), we introduce a distance   d¯ f (x), f (y) = f (y − x). In fact, from f (x) = d(T, Tx ), we have d(Tx , Ty ) = d(T, Ty−x ) = f (y − x) ≤ d(Tx , Tz ) + d(Tz , Ty ) = d(T, Tz−x ) + d(T, Ty−z ) = f (z − x) + f (y − z). Hence, d¯ is a distance.

  f Theorem 2.4 (TBμ , d) is isometrically isomorphic to the space f (R), d¯ under the mapping j1 .

2.3 Time Scale Spaces and Completeness

83

Proof From the construction of the mapping j1 , one has   d(Tx , Ty ) = d(T, Ty−x ) = f (y − x) = d¯ f (x), f (y) . f

Hence, j1 is an isometrically isomorphic mapping from TBμ to f (R). This completes the proof. 

Now, let TBμ be the set of all time scales with bounded graininess functions. Denote DBC0 (R) the set of all Hausdorff distance functions of time scales with δ-accuracy from the space TBμ , and note f (0) = 0 for all f ∈ DBC0 (R). Let ZBC(R) be the set of all bounded continuous linear broken line functions on R.   Theorem 2.5 ZBC(R),  ·  is a Banach space equipped with the norm  ·  = supt∈T | · |. Proof Let {fk (x)} ⊂ ZBC(R) be a Cauchy sequence, and then there exists a constant N > 0 such that n, m > N implies |fn (x) − fm (x)| < ε, ∀x ∈ R.

(2.17)

Let the extreme point sets of the bounded continuous linear broken line functions y = fn (x) and y = fm (x) be En = {xn1 , xn2 , . . . , xnk } and Em = {xm1 , xm2 , . . . , xmk }, respectively. Now, let the set s(En ∪ Em ) = {smn1 , smn2 , . . . , smnk , . . . : smnk < smnk+1 , smnk , smnk+1 ∈ En ∪ Em , k ∈ Z+ }. Then we claim that the Cauchy sequence is convergent. In fact, it follows from (2.17) that |fn (smni ) − fm (smni )| < ε, ∀i ∈ Z+ , which indicates that {fk (smni )} is a Cauchy real number sequence for each i ∈ Z+ . Hence, for any ε > 0, there exists constant cmni such that |fn (smni ) − cmni | < ε, ∀i ∈ Z+ .   (i)   (i) Denote afn := smni , fn (smni ) , bfn := smni+1 , fn (smni+1 ) . Thus,  fn (smni+1 ) − fn (smni ) (x − smni ) + fn (smni ), fn (x) = fn(i) (x) : fn(i) (x) = smni+1 − smni  + smni ≤ x ≤ smni+1 i ∈ Z ,

84

2 A Classification of Closedness of Time Scales Under Translations

which implies the bounded continuous linear broken line function y = fn (x) is    uniquely determined by the set M = af(i)n , bf(i)n , i ∈ Z+ . Next, for each i ∈ Z+ , we construct a bounded continuous linear broken line function  cmni+1 − cmni f (x) = f (i) (x) : f (i) (x) = (x − smni ) + cmni , smni+1 − smni  + smni ≤ x ≤ smni+1 i ∈ Z , then, for each i ∈ Z+ and note that smni ≤ x ≤ smni+1 , we will obtain |fn(i) (x) − f (i) (x)|

    fn (smni+1 ) − cmni+1 − fn (smni ) − cmni = smni+1 − smni × (x − smni ) + fn (smni ) − cmni ≤ |fn (smni+1 ) − cmni+1 | + |fn (smni ) − cmni | +|fn (smni ) − cmni | < ε + ε + ε = 3ε.

Hence, one can obtain that |fn (x) − f (x)| ≤ sup |fn(i) (x) − f (i) (x)| ≤ 3ε, i∈Z+

which shows that fn (x) → f (x) as n → ∞ and f (x) is also a bounded continuous linear broken line function, i.e., f ∈ ZBC(R). This completes the proof. 

  Theorem 2.6 DBC0 (R),  ·  is a Banach space equipped with the norm  ·  = supt∈T | · |. Proof According to Theorem 2.3, one can obtain that DBC0 (R) ⊂ ZBC(R) is a closed set. Thus, it follows from Theorem 2.5 that DBC0 (R),  ·  is also a Banach space. This completes the proof. 

Next, we construct a mapping j2 :

TBμ

→

DBC0 (R),

T

→

j2 (T) = f,

and one can easily see that j2 is a surjective mapping, but j2 is not an injective mapping. In fact, note that d(Ty , Tx+y ) = d(T, Tx ) = f (x) for all y ∈ R, so the preimage of f is not unique. Definition 2.13 ([213]) We say T1 and T2 are f -equivalent if f1 (x) = f2 (x) = f (x) for all x ∈ R, i.e., d(T1 , Tx1 ) = d(T2 , Tx2 ) for all x ∈ R, and denote it by

2.3 Time Scale Spaces and Completeness

85

T1 ∼f T2 . We say T1 is not equivalent to T2 if f1 (x) ≡ f2 (x) for some x ∈ R, and denote it by T1 ∼ T2 . Remark 2.15 Notice that if T1 ∼f T2 and T2 ∼f T3 , then T1 ∼f T3 . Definition 2.14 ([213]) The set of all f -equivalent elements is called the f f equivalent class of TBμ , and we denote it by TBμ . Now, we construct a f -equivalent class set T"Bμ of TBμ as follows:  f  T"Bμ = TBμ : ∀f ∈ DBC0 (R) and introduce a distance  f f  d˜ TBμ1 , TBμ2 = sup |f1 (x) − f2 (x)| = f1 − f2 .

(2.18)

x∈R

Next, we introduce a mapping: j3 :

T"Bμ

→ DBC0 (R),

f

→

TBμ

f,

and j3 is a bijective mapping. From (2.18), one obtain the following theorem immediately. ˜ is isometrically isomorphic to the space (DBC0 (R),  · ) Theorem 2.7 (T"Bμ , d) under the mapping j3 . f

f

Remark 2.16 If T1 ∈ TBμ1 , T2 ∈ TBμ2 and f1 ≡ f2 for some x ∈ R, then     j1−1 f1 (0) = T1 ∼ T2 = j1−1 f2 (0) . In fact, from Definition 2.13, one can obtain f1 (x) = f2 (x) = f (x) for all x ∈ R if and only if T1 ∼f T2 . f

For convenience, we introduce the definitions of generating element of TBμ .   f Definition 2.15 ([213]) We say j1−1 f (0) is the generating element of TBμ , and denote it by Tf .   f Remark 2.17 Notice that TBμ = {Txf : x ∈ R} = {t + x : ∀t ∈ Tf } : x ∈ R . Theorem 2.8

 f  f TBμ , d is a complete metric space, where TBμ denotes the closure

f

of TBμ .

  Proof Since f (R), d¯ is a complete metric space, from Theorem 2.4, it follows   f 

that TBμ , d is a complete metric space. This completes the proof.

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2 A Classification of Closedness of Time Scales Under Translations

Remark 2.18 According to Theorem 2.8, all Cauchy sequences in the space f

(TBμ , d) are convergent. f

g

g

Theorem 2.9 For any T1 ∈ TBμ and T2 ∈ TBμ , if f ≡ g, then Tx1 ∈ TBμ and f

Tx2 ∈ TBμ for any x ∈ R. f

Proof We argue by contradiction. If T1 ∈ TBμ and there exists x1 ∈ R such that g

x +y

y

Tx11 ∈ TBμ , i.e., d(Tx11 , T11 ) = g(y), we have d(T1 , T1 ) = g(y) for all y ∈ R, g which implies that T1 ∈ TBμ . Hence, d(T1 , Tx1 ) = f (x) = g(x), i.e., f (x) ≡ g(x) for all x ∈ R, and this is a contradiction. This completes the proof. 

f

Remark 2.19 Note if T1 ∈ TBμ and f ≡ g for all x ∈ R, there is no translation of g T1 in TBμ . f

Theorem 2.10 For a time scale T ∈ TBμ , if its translations time scale sequence {Tαn } is a Cauchy sequence, then there is a time scale T0 such that Tαn → T0 ∈ f

TBμ as n → ∞. Proof From Remark 2.18, the result follows immediately.



2.3.3 Approximation Time Scale Spaces Induced by Functions In this section, based on Sect. 2.3.2, we introduce two types of approximation time scale spaces by adopting the real function approximation on R. For convenience, let APd (R) and AAd (R) be the sets of all almost periodic distance functions and almost automorphic distance functions, respectively. Denote the sets of all almost periodic functions and almost automorphic functions on R by AP (R) and AA(R), respectively. It is easy to see that AP d (R) and AAd (R) are  closed subsets  in AP (R) and AA(R), respectively. Hence, APd (R),  ·  and AAd (R),  ·  are Banach spaces. AP , d) ˜ ⊂ (T"Bμ , d) ˜ is an almost periodic Definition 2.16 ([213]) We say (T"Bμ AP ,f AP ∈ T" , then f ∈ APd (R). equivalent class time scale space if for any T Bμ

Bμ

 AP ,f   f  Definition 2.17 ([213]) We say TBμ , d ⊂ TBμ , d is an almost periodic time scale space affiliated to the function f if f is almost periodic at the origin under the ¯ Here we say T ∈ T AP ,f is an almost periodic time scale. distance d. Bμ Remark 2.20 In Definition 2.17, f is almost periodic at the origin under the distance d¯ implies that for any ε > 0, there is at least a τ (ε) in the interval with the inclusion length of l such that d¯ f (0 + τ ), f (0) = f (τ ) < ε, that is, for any real sequence {τn }, there is a subsequence {τnk } and c ∈ f (R) such that

2.3 Time Scale Spaces and Completeness

87

  d¯ f (τnk ), c → 0 as k → ∞. Therefore, the almost periodic time scales introduced in Definition 2.17 completely fulfills Definition 2.12. Theorem 2.11 f is almost periodic at the origin under the distance d¯ if and only if f ∈ APd (R). Proof This result is clear since |f (x + τ ) − f (x)| = |d(T, Tx ) − d(T, Tx+τ )| ≤ d(Tx , Tx+τ ) = d(T, Tτ )   = d¯ f (0), f (τ ) = |f (0 + τ ) − f (0)| ≤ sup |f (x + τ ) − f (x)|, x∈R

so one can obtain   d¯ f (0), f (0 + τ ) = sup |f (x + τ ) − f (x)|. x∈R



This completes the proof. AP ,f

is an almost Theorem 2.12 For a time scale T and any x ∈ R, Tx ∈ TBμ periodic time scale if and only if f is almost periodic at the origin under the distance ¯ d. AP ,f

Proof If Tx ∈ TBμ is an almost periodic time scale, then for any ε > 0, there is at least a τ (ε) in the interval with the inclusion length of l such that   d(Tx , Tx+τ ) = d¯ f (x), f (x + τ )

  = d(T, Tτ ) = f (τ ) = d¯ f (0 + τ ), f (0) < ε,

¯ which implies that f is almost periodic at the origin under d. ¯ from Definition 2.17 Conversely, if f is almost periodic at the origin under d, and Remark 2.20,   d¯ f (0 + τ ), f (0) = f (τ ) = d(T, Tτ ) = d(Tx , Tx+τ ) < ε, AP ,f

which shows that Tx ∈ TBμ proof.

is an almost periodic time scale. This completes the 

  AP , d) ˜ and T AP ,f , d are complete metric spaces. Theorem 2.13 (T"Bμ Bμ

AP if Proof From Definition 2.16 and Theorem 2.7, one can obtain T AP ,f ∈ T"Bμ   AP , d) ˜ and only if f ∈ APd (R). Since APd (R),  ·  is a Banach space, then (T"Bμ is a complete metric space. Moreover, from Theorem 2.4 and the completeness of

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2 A Classification of Closedness of Time Scales Under Translations

   AP ,f  the space f (R), d¯ , it follows that TBμ , d is a complete metric space. This completes the proof. 

AP ,f

Theorem 2.14 For any T ∈ TBμ

and real sequence {αn }, there exists g ∈ H (f ) AP ,g

and a subsequence {αnk } ⊂ {αn } such that Tαnk → T0 ∈ TBμ H (f ) is the hull of f .

as k → ∞, where

Proof Since f ∈ APd (R), then for any real sequence {αn }, there exists a subsequence {αnk } such that limk→∞ f (x + αnk ) = g(x) uniformly for x ∈ R. Hence, we have     n = j1−1 g(x) = Txg . lim j1 f (x + αn ) = lim Tx+α f

n→∞

n→∞

  Let Tg = j1−1 g(0) = T0 , and we obtain the desired result. This completes the proof.

 AP , d) ˜ is an almost periodic equivalent class time scale space, Theorem 2.15 If (T"Bμ  AP ,f  AP ,f AP then for any TBμ ∈ T"Bμ , one has TBμ , d is an almost periodic time scale space. AP ,f AP , one obtains that f ∈ Proof From Definition 2.16, for any TBμ ∈ T"Bμ APd (R). Hence, for any ε > 0, there is at least a τ (ε) in the interval with the inclusion length of l such that

  ε > sup |f (x + τ ) − f (x)| = |f (0 + τ ) − f (0)| = f (τ ) = d¯ f (τ ), f (0) , t∈R

 AP ,f  i.e., f is almost periodic at the origin. Therefore, TBμ , d is an almost periodic time scale space. This completes the proof. 

 AP ,f  Corollary 2.4 For any f ∈ DBC0 (R), if f ∈ APd (R), then TBμ , d is an almost periodic time scale space. Proof If f is an almost periodic function on R, then for any ε > 0, there is at least a τ (ε) in the interval with the inclusion length of l such that   ε > sup |f (x + τ ) − f (x)| > |f (0 + τ ) − f (0)| = f (τ ) = d¯ f (0 + τ ), f (0) , x∈R

¯ according which implies that f is almost periodic at the origin under the distance d,  AP ,f  to Theorem 2.12, and this result is clear. Hence, TBμ , d is an almost periodic time scale space. This completes the proof. 

 AP ,f  Corollary 2.5 TBμ , d is an almost periodic time scale space if and only if f ∈ APd (R).

2.3 Time Scale Spaces and Completeness

89

 AP ,f  Proof If TBμ , d is an almost periodic time scale space, then f is almost ¯ and we have for any ε > 0, there is at least a τ (ε) in periodic at the origin under d, the interval with the inclusion length of l such that     ε > d¯ f (0 + τ ), f (0) = f (τ ) = d¯ f (x + τ ), f (x) = d(Tx , Tx+τ ) ≥ |d(T, Tx ) − d(T, Tx+τ )| = |f (x + τ ) − f (x)|, which implies that f ∈ APd (R). From Corollary 2.4, we obtain the desired result. 

 f  Remark 2.21 Note that TBμ , d is isometrically isomorphic to the space   f (R), d¯ under the mapping j1 but it is not isometrically isomorphic to the   space f (R), | · | , where | · | is an absolute value norm. In what follows, according to some embedding theorems above, we introduce the ˜ almost automorphy of time scales in the f -equivalent class space (T"Bμ , d). AA , d) ˜ ⊂ (T"Bμ , d) ˜ is an almost automorphic Definition 2.18 ([213]) We say (T"Bμ AA,f ∈ T"AA , then f ∈ AAd (R). equivalent class time scale space if for any T Bμ

Bμ

 AA,f   f  Definition 2.19 ([213]) We say TBμ , d ⊂ TBμ , d is an almost automorphic time scale space affiliated to the function f if f is an almost automorphic function on R, i.e, for any real sequence {αn }, there exists a subsequence {αnk } such that lim f (x + αnk ) = g(x) and

k→∞

lim g(x − αnk ) = f (x)

k→∞

or lim

lim f (x + αnk1 − αnk2 ) = f (x)

k2 →∞ k1 →∞ AA,f

is called an almost automorphic time scale.

AA,f

is an almost automorphic time scale, then for any

for each x ∈ R, and T ∈ TBμ Theorem 2.16 If T1 ∈ TBμ

x+αnk

real sequence {αn }, there exists a subsequence {αnk } such that T1 AA,g TBμ as k → ∞ for each x ∈ R. AA,f

Proof Since T1 ∈ TBμ

is an almost automorphic time scale, then x+αnk

lim j1 (T1

k→∞

) = lim f (x + αnk ) = g(x).

From the continuity of j1 , we have

k→∞

→ Tx2 ∈

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2 A Classification of Closedness of Time Scales Under Translations x+αnk

lim T1

k→∞

  AA,g = j1−1 g(x) = Tx2 ∈ TBμ

for each x ∈ R. This completes the proof.   AA , d) ˜ and T AA,f , d are complete metric spaces. Theorem 2.17 (T"Bμ Bμ



AA,f AA if Proof From Theorem 2.7 and Definition 2.18, one can obtain TBμ ∈ T"Bμ   AA , d) ˜ and only if f ∈ AAd (R). Since AAd (R),  ·  is a Banach space, then (T"Bμ is a complete metric space. Moreover, from Theorem 2.4, from the completeness of    AA,f  the space f (R), d¯ , it follows that TBμ , d is a complete metric space. This completes the proof. 

AA , d) ˜ is an almost automorphic equivalent class time scale Theorem 2.18 If (T"Bμ   AA,f AA , one has T AA,f , d is an almost automorphic space, then for any TBμ ∈ T"Bμ Bμ time scale space. AA,f AA , one obtains that f ∈ Proof From Definition 2.18, for any TBμ ∈ T"Bμ  AA,f  AAd (R). Therefore, TBμ , d is an almost automorphic time scale space. This completes the proof. 

The following theorem is obvious. AP ,f

Theorem 2.19 If T ∈ TBμ

AA,f

, then T ∈ TBμ

.

Proof If f ∈ APd (R), then f ∈ AAd (R). According to Definition 2.19, T ∈ AA,f TBμ . This completes the proof.

 In the following, we will provide a convenient criterion to guarantee the almost periodicity and almost automorphy of time scales. Denote all the right scattered points of T by Rs = {ti ∈ T : i ∈ Z} and Rs induces a set μ(Rs ) = {μ(ti ), i ∈ Z}. Similarly, denote all the right scattered points of Tx by Rsx = {ti ∈ Tx : i ∈ Z} and Rsx induces a set μx (Rsx ) = {μx (ti ), i ∈ Z}. Theorem 2.20 For a time scale T with a bounded graininess function μ, if μ(Rs ) forms an almost periodic sequence, i.e., for any ε > 0, there exists a positive integer p(ε) such that any set consisting of p(ε) consecutive integers contains at least one integer l for which |μ(ti+l ) − μ(ti )| < ε for all i ∈ Z, then T is an almost periodic time scale; if μ(Rs ) forms an almost automorphic sequence, i.e., for every sequence of integer {kl }l∈Z there exists a subsequence {kn }n∈Z , such that lim lim μ(ti+kn −km ) = μ(ti ) for each i ∈ Z,

n→∞ m→∞

then T is an almost automorphic time scale.

2.3 Time Scale Spaces and Completeness

91

Proof For a time scale T and its translation Tτ , by using the relationship between d(T, Tτ ) and the graininess function μ (i.e., Fig. 2.4), one can obtain d(T, Tτ ) = sup |μτ (t + τ ) − μ(t)| = f (τ ) = sup |μ(t + τ ) − μ(t)|,

(2.19)

t∈Rs

t∈T

where t + τ ∈ T is right scattered point, μτ : Tτ → R+ and f (x) = d(T, Tx ). Then one can select a τ such that ti+l = ti + τ since μ(Rs ) is an almost periodic sequence, i.e., for any ε > 0, there is at least a τ (ε) in the interval with the inclusion length of l such that d(T, Tτ ) = sup |μτ (t + τ ) − μ(t)| = f (τ ) = sup |μ(ti+l ) − μ(ti )| < ε, t∈T

i∈Z

¯ which indicates that f is almost periodic at the origin under the distance d, according to Theorem 2.12, T is an almost periodic time scale. Moreover, from (2.19), since μ(Rs ) forms an almost automorphic sequence, for any real sequence sl , there exists a subsequence {sn } such that lim lim |f (x + sn − sm ) − f (x)| ≤ lim lim d(Tx+sn −sm , T) − d(Tx , T) m→∞ n→∞ m→∞ n→∞

≤ lim lim d(Tx+sn −sm , Tx ) m→∞ n→∞ = lim lim sup μx (t + sn − sm ) − μx (t) , m→∞ n→∞ t∈Tx

(2.20)

where t + sn − sm ∈ Tx is right scattered point, μx : Tx → R+ and f (x) = d(T, Tx ). Since μ(Rs ) forms an almost automorphic sequence, then μ(Rsx ) also forms an almost automorphic sequence for each x ∈ R, i.e., for every sequence of integer {kl }l∈Z there exists a subsequence {kn }n∈Z , such that lim lim μx (ti+kn −km ) = μx (ti ) for each i ∈ Z,

n→∞ m→∞

so one can select a subsequence {kn } such that ti+kn −km = ti + sn − sm since μ(Rsx ) is an almost automorphic sequence. Hence, for each x ∈ R, by (2.20), we have lim lim |f (x + sn − sm ) − f (x)| ≤ lim lim sup μx (t + sn − sm ) − μx (t) m→∞ n→∞

m→∞ n→∞ t∈Tx

= lim lim sup μx (t + sn − sm ) − μx (t) m→∞ n→∞ t∈R x s

= lim lim sup μx (ti+kn −km ) − μx (ti ) = 0. m→∞ n→∞ i∈Z

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2 A Classification of Closedness of Time Scales Under Translations

Therefore, we obtain lim lim |f (x + sn − sm ) − f (x)| = 0

m→∞ n→∞

for each x ∈ R, i.e., f is an almost automorphic function. According to Definition 2.19, T is an almost automorphic time scale. This completes the proof. 

Example 2.6 Let a > 1, and consider the following time scale: Pa,| sin t+sin √5t| =

∞ 

[pm , a + pm ],

m=1

where pm = (m − 1)a +

m−1 

sin

 ka + | sin a + sin

k=1

+ sin

√

√

5a| + | sin(2a + | sin a + sin

5(2a + | sin a + sin

√ 5a|)|

√ 5a|)  √ √ + sin 5((k − 1)a + | sin a + sin 5a|)| # $% & + . . . + | sin((k − 1)a + | sin a + sin

k terms

+ sin

√

 √ 5 ka + | sin a + sin 2a|

√ +| sin(2a + | sin a + sin 5a|) √ √ + sin 5(2a + | sin a + sin 5a|)|

√ 5a|)  √ √ + sin 5((k − 1)a + | sin a + sin 5a|)| . # $% & + . . . + | sin((k − 1)a + | sin a + sin

k terms

Then, it follows that σ (t) =

 t,

if t ∈ ∪∞ m=1 [pm , a + pm ), √ t + | sin t + sin 5t|, if t ∈ ∪∞ m=1 {a + pm }

and  μ(t) =

if t ∈ ∪∞ m=1 [pm , a + pm ), √ | sin t + sin 5t|, if t ∈ ∪∞ m=1 {a + pm }.

0,

√

5a|)

2.3 Time Scale Spaces and Completeness

93

Clearly, μ(Rs ) forms an almost periodic sequence, and then T is an almost periodic time scale. 

Example 2.7 Let a > 1, similar to the construction of pm in Example 2.6, consider the time scale Pa,cos

⎧ ⎨t, where σ (t) = ⎩t + cos

1 √ |2+sin t+sin 5t|

=

∞ 

[pm , a + pm ],

m=1

if t ∈ ∪∞ m=1 [pm , a + pm ),

1 √ , |2+sin t+sin 5t|

if t ∈ ∪∞ m=1 {a + pm }

and μ(t) =

⎧ ⎨0,

if t ∈ ∪∞ m=1 [pm , a + pm ),

⎩cos

1 √ , |2+sin t+sin 5t|

if t ∈ ∪∞ m=1 {a + pm }.

One can obtain that μ(Rs ) forms an almost automorphic sequence, and then T is an almost automorphic time scale. 

2.3.4 The Properties of Almost Periodic Time Scales In this section, we will provide some properties of almost periodic time scales mainly including some approximation properties in the sense of the Hausdorff distance. For convenience, note that d(T, Tx ) = f (x) denotes the continuous Hausdorff distance function with δ-accuracy. Lemma 2.2 Let T be an almost periodic time scale. If τ ∈ Πε , then there exists ˆ τ + δ) ˆ Π1 ⊂ Π ε . some sufficiently small number δˆ > 0 such that (τ − δ, Proof Let f (x) = d(T, Tx ), x ∈ R. Since T is an almost periodic time scale, for any ε > 0, there exists a constant l(ε) > 0 such that each interval of length l(ε) contains a τ ∈ Π1 such that F (τ ) = d(T, Tτ ) − ε = f (τ ) − ε < 0, where F (x) = f (x)−ε. Note that F is a continuous function on R, then there exists ˆ τ + δ), ˆ one has F (x) < 0. Therefore, for all a δˆ > 0 such that for all x ∈ (τ − δ, ˆ ˆ x ∈ (τ − δ, τ + δ)Π1 , we have x ∈ Πε . This completes the proof. 

According to Definition 2.3.1, we obtain the following useful lemmas: Lemma 2.3 Let T be an almost periodic time scale. If τ1 , τ2 ∈ Πε , then there exists τ1 + τ2 ] such that ξ0 ∈ Πε , where ξ0 ∈ (τ1 ,

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2 A Classification of Closedness of Time Scales Under Translations



(τ1 , τ1 + τ2 ], τ1 < τ1 + τ2 , (τ1 , τ1 + τ2 ] = [τ1 + τ2 , τ1 ), τ1 > τ1 + τ2 . Proof From the condition of the lemma, we obtain d(T, Tτ1 +τ2 ) < d(T, Tτ1 ) + d(Tτ1 , Tτ1 +τ2 ) ≤ 2ε. Case I If d(T, Tτ1 +τ2 ) ≤ ε, then ξ0 = τ1 + τ2 , so we get the desired result. Case II Let 2ε ≥ d(T, Tτ1 +τ2 ) > ε. Note (2.5), let τ1 + τ2 ], f (x) = d(T, Tx ), x ∈ [τ1 , and note f (x) is continuous on R. Now, let F (x) = f (x) − ε, and we obtain that F (τ1 ) = f (τ1 ) − ε < 0, F (τ1 + τ2 ) = f (τ1 + τ2 ) − ε > 0, so, there exists some ξ ∈ (τ1 , τ1 + τ2 ) such that F (ξ ) = 0, i.e., f (ξ ) = ε. According to Lemma 2.2, we obtain that there exists some sufficiently small number ˆ ξ + δ) ˆ Π1 ⊂ Πε . Note that (ξ − δ, ˆ ξ + δ) ˆ Π1 ⊂ (τ1 , δˆ > 0 such that (ξ − δ, τ1 + τ2 ]. This completes the proof. 

Remark 2.22 From Lemma 2.3, for almost periodic time scales, one will obtain that Πε is an infinite number set. Lemma 2.4 For any ε > 0 and τ1 ∈ Πε , if t0 ∈ T ∩ Tτ1 , then there exists τ2 ∈ Πε \{τ1 } such that t0 ∈ T ∩ Tτ2 . Proof From Lemma 2.3, there exists ξ ∈ Πε such that d(T, Tξ ) < ε. Case I If for t0 ∈ T ∩ Tτ1 , we have t0 ∈ T ∩ Tξ , then τ2 = ξ ∈ Πε . Case II If for t0 ∈ T ∩ Tτ1 , we have t0 ∈ T ∩ Tξ , then t0 ∈ Tξ . ξ

Case (i) Assume that there exists some i0 ∈ Z such that βi0 > t0 > βi0 . Then we denote γ0 = d(T, Tξ ). According to (2.6), we have ξ +γ0

αi

  := α ∈ Tξ +γ0 : inf |αi − α| ,

ξ +γ0

βi

  := β ∈ Tξ +γ0 : inf |βi − β| . ξ

Note that for sufficiently small ε > 0, we are able to guarantee that αi , αi and ξ βi , βi keep the same sign, respectively. Hence, we have αi − (α ξ + γ0 ) = |αi − α ξ | − γ0 = |αi − α ξ +γ0 |, i i i

(2.21)

2.3 Time Scale Spaces and Completeness

95

βi − (β ξ + γ0 ) = |βi − β ξ | − γ0 = |βi − β ξ +γ0 |. i i i

(2.22)

From (2.21) and (2.22), we obtain that d(T, T

ξ +γ0

 ) = max 

ξ +γ sup |αi − αi 0 |, sup |βi i∈Z i∈Z

ξ +γ − βi 0 |



ξ ξ = max sup |αi − αi | − γ0 , sup |βi − βi | − γ0 i∈Z



i∈Z

≤ d(T, T ) ≤ ε. ξ

Hence, we have ξ + γ0 ∈ Πε and t0 ∈ T ∩ Tξ +γ0 . ξ

Case (ii) If there exists some i0 ∈ Z such that αi0 > t0 > αi0 . Then we denote γ0 = d(T, Tξ ). According to (2.6), we have ξ −γ0

αi

  := α ∈ Tξ −γ0 : inf |αi − α| ,

ξ −γ0

βi

  := β ∈ Tξ −γ0 : inf |βi − β| . ξ

ξ

Note that for sufficiently small ε > 0, we can guarantee that αi , αi and βi , βi keep the same sign, respectively. Hence, we obtain αi − (α ξ − γ0 ) = γ0 − |αi − α ξ | = |αi − α ξ −γ0 |, i i i

(2.23)

βi − (β ξ − γ0 ) = γ0 − |βi − β ξ | = |βi − β ξ −γ0 |. i i i

(2.24)

It follows from (2.23) and (2.24) that   ξ −γ ξ −γ d(T, Tξ −γ0 ) = max sup |αi − αi 0 |, sup |βi − βi 0 | i∈Z

i∈Z

 ξ ξ = max sup γ0 − |αi − αi | , sup γ0 − |βi − βi | 

i∈Z

i∈Z

≤ d(T, Tξ ) ≤ ε. Thus, we obtain ξ − γ0 ∈ Πε and t0 ∈ T ∩ Tξ −γ0 . This completes the proof. Remark 2.23 From Lemma 2.4, one  obtains that if τ1 ∈ Πε and t0 ∈ T the set τ ∈ Πε \{τ1 } : t0 ∈ T ∩ Tτ is an infinite number set.

∩ Tτ1 ,



then

In the sequel, we will establish some convergence theorems of almost periodic time scales under translations. Theorem 2.21 Let T be an almost periodic time scale. Then for any given real   n sequence α ⊂ Π1 , there exists a subsequence α ⊂ α such that {Tα } converges to

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2 A Classification of Closedness of Time Scales Under Translations

a time scale T0 , i.e., for any given ε > 0, there exists N0 > 0 such that n > N0 implies d(Tαn , T0 ) < ε. Moreover, T0 is an almost periodic time scale. Proof For any given ε > 0, let l = l( 4ε ) be an inclusion length of E{T, 4ε }. For any       given subsequence α = {αn }, we will obatin αn = τn + γn , where τn ∈ E{T, 4ε }  and 0 ≤ γn ≤ l, n = 1, 2, . . . . Therefore, there exists a subsequence γ = {γn } ⊂   γ = {γn } such that γn → s as n → ∞ and 0 ≤ s ≤ l. ˆ Also, there exists δ(ε) > 0 such that |t1 − t2 | < δˆ implies d(Tt1 , Tt2 )
2|α1 | in which there is no ε0 -translation number of T. Next, taking α2 =     1 2 (a1 + b1 ), obviously, α2 − α1 ∈ (a1 , b1 ), so α2 − α1 ∈ E{T, ε0 }; then there   exists an interval (a2 , b2 ) with b2 − a2 > 2(|α1 | + |α2 |) such that there is no ε0  translation number of T in this interval. Next, selecting α3 = 12 (a2 + b2 ), obviously,     α3 − α2 , α3 − α1 ∈ E{T, ε0 }. One can continue to repeat this process again and     again to find α4 , α5 , . . . , such that αi − αj ∈ E{T, ε0 }, i > j. Hence, for any i = j, i, j = 1, 2, . . . , without loss of generality, let i > j, we have 







d(Tαi , Tαj ) = d(Tαi −αj , T) ≥ ε0 . 

Therefore, for the sequence {αn }, there is no convergent subsequence of the time 

scale sequence {Tαn }, which is a contradiction. Hence, T is an almost periodic time scale. This completes the proof. 

From Theorems 2.21 and 2.22, we are able to obtain the following equivalent definition of almost periodic time scales. 

Definition 2.20 ([213]) Let T be a time scale. If for any given sequence α , there  exists a subsequence α ⊂ α such that {Tαn } converges to a time scale T0 , then T is called an almost periodic time scale. From the definition of the limit for a sequence of sets, one can obtain the following theorem immediately according to Definition 2.20. Theorem 2.23 The time scale T is an almost periodic time scale if and only if for   any given sequence α , there exists a subsequence α ⊂ α such that

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2 A Classification of Closedness of Time Scales Under Translations

lim Tαn :=

n→∞

∞ ∞  

Tαk =

j =1 k=j

∞ ∞   j =1 k=j

Tαk := lim Tαn n→∞

exists. Corollary 2.6 The time scale T is a periodic time scale with period ω if and only   if for any given sequence α , there exists a subsequence α ⊂ α such that lim Tαn = Tα0 ,

n→∞

where α0 is some constant. 

Proof In fact, if T has period ω, then from the given sequence {αn } we may select a sequence {αn } such that {αn (mod ω)} converges to α0 . Then limn→∞ Tαn = Tα0 . This completes the proof. 

In the following, we will establish some relationship between an almost periodic time scale and its graininess function μ. For this, we need a lemma as follows. Lemma 2.5 If T is an almost periodic time scale, then for any ε > 0 and a right dense point t ∈ T, there exists a constant l(ε) > 0 such that there exists a τ (ε) ∈ Π1 in each interval with length l(ε) and there exists a sequence {tn } ⊆ T ∩ Tτ such that limn→∞ tn = t, tn > t, n ∈ N. Proof Since T is an almost periodic time scale, then for any ε > 0, there exists a constant l(ε) > 0 and there exists a τ (ε) ∈ Π1 in each interval with length l(ε) such that d(T, Tτ ) ≤ ε. Because t ∈ T is a right dense point in T, then there exists a sequence {tn } ⊆ T and tn > t, n ∈ N such that limn→∞ tn = t. We argue by contradiction. If there exists some N > 0 such that {tn }n≥N ⊂ Tτ , then t ∈ Tτ , which implies that there exists some t ∗ ∈ Tτ such that d(T, Tτ ) ≥ |t − t ∗ | > 0, this is a contradiction. Hence, we have {tn } ⊆ Tτ . Therefore, there exists a sequence 

{tn } ⊆ T ∩ Tτ and limn→∞ tn = t, tn > t, n ∈ N. This completes the proof. Theorem 2.24 If T is an almost periodic time scale, then for any ε > 0, there exists a constant l(ε) > 0 such that each interval of length l(ε) contains a τ (ε) ∈ E{ε, μ} such that |μ(t + τ ) − μ(t)| < ε,

for all t ∈ T ∩ T−τ .

(2.25)

Proof First, if T be an almost periodic time scale, we claim that: for any τ ∈ Πε , if t is a right scattered point in T, then t + τ is the right scattered point in T; if t is a right dense point in T, then t + τ is the right dense point in T. In fact, if t ∈ T ∩ T−τ is a right scattered point in T, then t + τ is the right scattered point in T. We argue by contradiction. Assume that t + τ ∈ T is a right dense point in T, then by Lemma 2.5, there exists a sequence {tn } ⊆ (T ∩ Tτ ) and tn ≥ t +τ, n ∈ N such that limn→∞ tn = t +τ . Hence, we have limn→∞ tn −τ = t,

2.3 Time Scale Spaces and Completeness

99

Fig. 2.4 The relationship between d(T, Tτ ) and the graininess function μ

h1

h2

T

t Tt

t+t mt (t+t)

h3 m (t)

i.e., {tn − τ } ⊆ (T ∩ T−τ ) ⊆ T. So t is a right dense point in T, which is a contradiction. Thus, t + τ is the right scattered point in T. On the other hand, if t ∈ T∩T−τ is a right dense point in T, then t +τ is the right dense point in T. In fact, by Lemma 2.5, there exists a sequence {tn } ⊆ T ∩ T−τ and tn ≥ t, n ∈ N such that limn→∞ tn = t. Then we have limn→∞ tn + τ = t + τ and {tn + τ } ⊆ T ∩ Tτ ⊂ T. Thus, t + τ is the right dense point in T. Hence, t ∈ T ∩ T−τ implies that t, t + τ ∈ T. When t is a right dense point, then t + τ is also the right dense point, so we have μ(t) = μ(t + τ ) = 0. When t is a right scattered point, then t + τ is also the right scattered point and |μ(t) − μ(t + τ )| ≤ d(T, Tτ ) < ε. 

This completes the proof.

Remark 2.24 From Fig. 2.4, let T be an almost periodic time scale and μτ : Tτ → R+ be the graininess function on Tτ , then we obtain that   |μ(t) − μτ (t + τ )| ≤ max |μτ (t + τ ) − h3 |, |μ(t) − h3 | = max{h2 , h1 } ≤ d(T, Tτ ) < ε for all t ∈ T.

(2.26)

Particularly, if t ∈ (T ∩ T−τ ) ⊂ T, then t + τ ∈ T and μτ (t + τ ) = μ(t + τ ). Hence, from (2.26), we can also obtain (2.25). In fact, if t + τ ∈ T, i.e., t ∈ T ∩ T−τ , then we get μτ (t + τ ) = μ(t). Remark 2.25 The inequality (2.25) can also be written as |σ (t + τ ) − σ (t) − τ | < ε

for all t ∈ T ∩ T−τ ,

which indicates that if T is τ -periodic, then σ (t + τ ) = σ (t) + τ .

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2 A Classification of Closedness of Time Scales Under Translations

Now, from Definition 2.3.1 and the Hausdorff distance (2.5), we can give the third concept of almost periodic time scales by the graininess function μ as follows: Definition 2.21 ([212, 213]) Let μ : T → R+ , μ(t) = σ (t) − t and μτ : Tτ → R+ . We say T is an almost periodic time scale if for any ε > 0, the set Π ∗ = {τ ∈ Π1 : |μτ (t + τ ) − μ(t)| < ε, ∀t ∈ T}

(2.27)

is relatively dense in Π1 . Remark 2.26 Note that Definition 2.21 implies the following: (1) If t ∈ (T ∩ T−τ ) ⊂ T, then t + τ ∈ T and μτ (t + τ ) = μ(t + τ ). Hence, from (2.27), we obtain (2.25). Furthermore, if t ∈ T ∩ T−τ is a right dense point in T, it follows from (2.27) that t + τ ∈ T is also a right dense point. Similarly, if t ∈ T ∩ T−τ is a right scattered point in T, by (2.27), then t + τ ∈ T is also a right scattered point. In fact, if t ∈ T ∩ T−τ is a right dense point in T, then μ(t) = 0. From (2.27), we know that μ(t + τ ) = 0, so t + τ ∈ T is a right dense point. Similarly, if t ∈ T ∩ T−τ is a right scattered point, then μ(t) > 0. From (2.27), we have μ(t +τ ) = |μ(t)−μ(t)+μ(t +τ )| ≥ |μ(t)|−|μ(t +τ )−μ(t)| ≥ μ(t)−ε > 0. Hence, t + τ is a right scattered point. (2) From Fig. 2.4 and (2.27), we obtain that d(T, Tτ ) ≤ sup |μτ (t + τ ) − μ(t)| < ε, t∈T

which indicates that T is an almost periodic time scale. (3) From Fig. 2.4 and Remark 2.24, Tτ is a τ -translation of T. Let μτ : Tτ → R+ be the graininess function of Tτ , then we obtain that  μτ (t + τ ) =

μ(t),

t + τ ∈ T,

μ(t + τ ), t + τ ∈ T.

(2.28)

Thus, from (2.28), we can simplify Definition 2.21 as follows: Definition 2.22 ([212]) Let μ : T → R+ be a graininess function of T. We say T is an almost periodic time scale if for any ε > 0, the set Π ∗ = {τ ∈ Π1 : |μ(t + τ ) − μ(t)| < ε, ∀t ∈ T ∩ T−τ } is relatively dense in Π1 .

2.3 Time Scale Spaces and Completeness

101

We are now in the position to prove an interesting theorem related to almost periodic time scales. This result plays an important role in maintaining some properties and formulas of almost periodic functions on periodic time scales. First, by adopting the notations (2.4), we obtain the following lemma. Lemma 2.6 For any given ε > 0 and any αi∗0 ∈ N (αi , ε), there is a C(τ ) such that αi∗0 + C(τ ) ∈ N(αi+1 , ε), where N (α, ε) denotes the ε-neighbourhood of α, τ ∈ Π1 is a 2ε -translation number and τ − αiτ |. C(τ ) = sup |αi+1 i

Proof In fact, we have τ |αi + C(τ ) − αi+1 | = αi + sup |αi+1 − αiτ | − αi+1 i τ = sup |αi+1 − αiτ | − (αi+1 − αi ) i

≤

τ |αi+1

− αiτ − αi+1 + αi |

τ < |αi+1 − αi+1 | + |αiτ − αi |
0, let T be an almost periodic time scale and τ be a ε ˜ ˜ 2 -translation number. Then, there exists a τ -periodic time scale T with T = (T ∩ Tτ ) ∪ Z and μΔ (Z) < ε. Proof Similar to the proof of Lemma 2.6, we can also obtain that for any given ε > 0 and any βi∗0 ∈ N(βi , ε), there also exists the same C(τ ) such that βi∗0 + C(τ ) ∈ N(βi+1 , ε), so one will obtain the following τ -periodic time scale T˜ =

+∞  i0 =−∞

[αi∗0 , βi∗0 ].

Further, let T ∩ Tτ :=

+∞ 

[αi∗τ , βi∗τ ],

i=−∞

we can obtain that αi∗τ ∈ N(αi , ε) and βi∗τ ∈ N (βi , ε), then we choose αi∗0 < αi∗τ and αi∗0 ∈ N(αi , ε), and βi∗0 > βi∗τ and βi∗0 ∈ N(βi , ε), it follows that

2.3 Time Scale Spaces and Completeness

103

(T ∩ Tτ ) ⊂ T˜ ⊂ (T ∪ Tτ ). Because T is an almost periodic time scale, for any ε > 0, we have   μΔ (T ∪ Tτ )\(T ∩ Tτ ) < ε,   ˜ ˜ which implies that μΔ T\(T ∩ Tτ ) < ε. Finally, let Z = T\(T ∩ Tτ ), then we τ have T˜ = (T ∩ T ) ∪ Z. This completes the proof.

 Definition 2.23 ([216]) We say that {T˜ τk } is a τk -periodic time scale sequence if T˜ τk is a τk -periodic time scale for each k ∈ Z+ . Next, using Definition 2.23, we shall prove the following theorem: Theorem 2.26 (Approximation Theorem for Almost Periodic Time Scales) Let T be an almost periodic time scale. Then there exists a τk -periodic time scale sequence {T˜ τk } such that T = lim T˜ τk , where T˜ τk is a τk -periodic time scale, k→∞

where     1 1 1 , T , where E , T is the -translation number set of T. τk ∈ E 2k 2k 2k Proof In view of Definition 2.3.1, for each number k > 0, there exists a constant 1 1 l( 2k ) > 0 such that each interval of length l( 2k ) contains a τk such that d(T, Tτk )
0, 

 1 τk ∈ E ,T . 2k Hence, T = lim T˜ τk . This completes the proof.



k→∞

An immediate consequence of Theorem 2.26 is the following corollary: Corollary 2.7 If T is an almost periodic time scale then, there exists a τk -periodic time scale sequence {T˜ τk } with T˜ τk ⊂ T˜ τk+1 or T˜ τk ⊃ T˜ τk+1 such that T = lim T˜ τk . k→∞

Theorem 2.27 Let {T˜ τk } be a convergent τk -periodic time scale sequence, where supk∈Z+ |τk | < +∞. Then, T = limk→∞ T˜ τk is almost periodic.

2.3 Time Scale Spaces and Completeness

105

Proof For any ε > 0, there exists some k0 > 0 such that k > k0 implies d(T, T˜ τk )
0, there exists a constant l(ε) := supk∈Z+ |τk | such that each interval of length l(ε) contains a τ (ε) = n0 τk , n0 ∈ Z+ , k > k0 such that d(T, Tτ (ε) ) < ε. This completes the proof. 

Combining Theorems 2.26 and 2.27, we get the following important corollary: Corollary 2.8 A time scale T is almost periodic if and only if T is the limit set of a τk -periodic time scale sequence {T˜ τk }. From Corollary 2.8, we immediately have the following interesting equivalent definition of almost periodic time scales: Definition 2.24 ([216]) We say that T is an almost periodic time scale if there exists a τk -periodic time scale sequence {T˜ τk } such that T = limk→∞ T˜ τk holds. Remark 2.27 Some results obtained for almost periodic functions on periodic time scales can now be developed to almost periodic time scales by using Corollary 2.8, because an almost periodic time scale is the limit set of some τk -periodic time scale sequence. Theorem 2.28 Let T be an almost periodic time scale. Then for any ε > 0 the ε-translation number set E{ε, T} forms a time scale. Proof Consider a sequence {τn } ⊂ E{ε, T} satisfying τn → τ as n → ∞. For each n ∈ N, we have d(T, Tτn ) < ε and note that d(T, Tx ) = f (x) is a continuous function on R, then we obtain d(T, Tτ ) = f (τ ) = lim f (τn ) = lim d(T, Tτn ) ≤ ε, n→∞

n→∞

which indicates that τ ∈ E{ε, T}. This completes the proof.



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2 A Classification of Closedness of Time Scales Under Translations

2.4 Complete-Closed Translations Time Scales (CCTS) From the concept of periodic time scales (i.e., the invariant time scales under translations), all the time scales demonstrated in the example below are not periodic but they have the complete closedness under translations attached with translation direction. Example 2.8 T = hZ+ , where h > 0, Π = {nh, n ∈ Z+ }. T = {t = k − q m : k ∈ Z+ , m ∈ N0 }, where 0 < q < 1, Π = {n, n ∈ Z+ }. T = R+ , Π = {τ, τ ∈ R+ }.  ∞ + i=0(2i −1)h, 2ih , h > 0, Π =  {2hn, n ∈ Z }. T = k∈Z− k(a+b), k(a+b)+b , where a = −b, Π = {(a+b)n, n ∈ Z− }.    Ta,cos a = 0k=−∞ k(a + cos a), k(a + cos a) + a , where 0 < a < π2 , Π = {n(a +cos a), n ∈ Z− }.  (vii) Ta,sin a = 0k=−∞ k(a + sin a), k(a + sin a) + a , where π2 < a < π , Π = {n(a + sin a), n ∈ Z− }. 

(i) (ii) (iii) (iv) (v) (vi)

According to the concept of periodic time scales in the sense of Definition 2.2, one can see that (i)–(vii) from Example 2.8 are not periodic time scales since that it follows from (i)–(iv) that inf T < +∞. Also, it is obvious that sup T < +∞ from (v)–(vii). In fact, we take (iv) for example, for any t ∈ T, we have t + 2h ∈ T, but there exists t0 = −h ∈ T such that t0 − 2h = −3h ∈ T. Nevertheless, all the time scales from Example 2.8 have a very nice closedness in translation for time variables if the time scale is only translated towards the assigned direction, for example, in (iv), because for any t ∈ T, we have t + 2h ∈ T. In this section, we will introduce the concept of complete-closed time scales and investigate some of their basic properties. For τ ∈ R, let Tτ = {t + τ : ∀t ∈ T} and Tτ = T ∩ Tτ . Definition 2.25 ([212]) We introduce three types of intersection for a translation time scale as follows: (1) We say T is a positive-direction intersection time scale if for any p > 0, there exists a number P > p and P ∈ Π1 , we call TP the positive-direction intersection for T. (2) We say T is a negative-direction intersection time scale if for any q < 0, there exists a number Q < q and Q ∈ Π1 , we call TQ the negative-direction intersection for T. (3) We say T is a bi-direction intersection time scale if for any p > 0 and q < 0, there exists two numbers P > p, Q < q and P , Q ∈ Π1 , we call TP ∪ TQ the bi-direction intersection for T. (4) We say T is an oriented-direction intersection time scale if T is a positivedirection intersection time scale or a negative-direction intersection time scale.

2.4 Complete-Closed Translations Time Scales (CCTS)

107

Remark 2.28 From Definition 2.25, we obtain that if sup T = +∞, inf T = −∞, then T is a bi-direction intersection time scale since for any τ ∈ R, we have Tτ = {−∞, +∞}. If inf T = −∞, then T is a negative-direction intersection time scale since for any τ ∈ R, we have Tτ = {−∞}. If sup T = +∞, then T is a positivedirection intersection time scale since for any τ ∈ R, we have Tτ = {+∞}. Note that T defined in Definition 2.2 is a particular bi-direction intersection time scale according to (3) in the Definition 2.25. In fact, for any τ ∈ Π , we have T±τ = T. Next, we introduce a concept of complete-closed time scales attached with translation direction which are more general than Definition 2.2. Definition 2.26 ([212]) We say T is a complete-closed time scale (CCTS) if   Π0 := {τ ∈ R : Tτ ⊆ T} ∈ {0}, ∅ . We say Π0 is the complete closedness translation number set of CCTS. Furthermore, we can describe it in detail as follows: (a) if for any p > 0, there exists a number P > p and P ∈ Π0 , we say T is a positive-direction CCTS or a positive-direction periodic time scale; (b) if for any q < 0, there exists a number Q < q and Q ∈ Π0 , we say T is a negative-direction CCTS or a negative-direction periodic time scale; (c) if ±τ ∈ Π0 , we say T is a bi-direction CCTS or a bi-direction periodic time scale; (d) we say T is a periodic time scale if T is a bi-direction complete-closed time scale; (e) we say T is an oriented-direction CCTS or an oriented-direction periodic time scale if T is a positive-direction CCTS or a negative-direction CCTS. Remark 2.29 From Definition 2.26, we obtain that a bi-direction CCTS is an oriented-direction CCTS. Furthermore, one can easily observe that Definition 2.2 is just a concept of bi-direction CCTS, it is just a particular case of Definition 2.26. Under Definition 2.26, (i)–(iv) from Example 2.8 are positive-direction CCTS, (v)– (vii) are negative-direction CCTS. Example 2.9 Consider the following oriented-direction CCTS: T1 =

+∞ 

[k(a + b), k(a + b) + a], a, b ≥ 0, a + b > 0,

k=0

T2 =

+∞ 

[k(a + b), k(a + b) + a], a, b ≤ 0, a + b < 0.

k=0

We can obtain that T1 is a positive-direction CCTS and T2 is a negative-direction CCTS with the translation number a + b, but they are not invariant time scales

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2 A Classification of Closedness of Time Scales Under Translations

under translations (i.e., periodic time scales) in the sense of Definition 2.2 because inf T1 = sup T2 = 0. 

Remark 2.30 From Definition 2.26, if T is a complete-closed time scale, i.e., there exists some τ = 0 such that Tτ ⊆ T (i.e., (Tτ )−τ ⊆ T−τ ), then one has T ⊆ T−τ , i.e., T ∩ T−τ = T, and vice versa. According to Remark 2.30, one can obtain the following equivalent definition of CCTS immediately. Definition 2.27 ([212]) We say T is called a complete-closed time scale (CCTS) if   Π˜ 0 := {τ ∈ R : T−τ ∩ T = T} ∈ {0}, ∅ .

(2.29)

We say Π˜ 0 is the translation number set of CCTS. In the following, we provide a lemma to guarantee that one can abstract a complete-closed time scale from an arbitrary time scale T.   " ⊆ Π1 and Π " ∈ {0}, ∅ be closed with respect to additive Lemma 2.7  Let Π  operation. If τ ∈Π " Tτ = ∅, then " Tτ is a complete-closed time scale. τ ∈Π   " . Let Proof We consider the following family of sets C = ∩τ ∈A Tτ : A ⊂ Π ∩τ ∈Π " Tτ := T0 . Obviously, T0 = ∅ implies that T0 is the minimal element in the " we obtain family of sets C , and for any τ0 ∈ Π,    τ0 T0 ∩ Tτ00 = ∩τ ∈Π " Tτ ∩ ∩τ ∈Π " Tτ     τ0 τ +τ0 = ∩τ ∈Π ) " Tτ ∩ ∩τ ∈Π " (T ∩ T     τ +τ0 = ∩τ ∈Π ∩ Tτ0 . " Tτ ∩ ∩τ ∈Π "T

(2.30)

", +) is closed with respect to additive operation, then Π " + τ0 := {τ + τ0 : Since (Π τ +τ τ τ 0 ˜ " ∀τ ∈ Π } ⊆ Π. Hence, we obtain ∩τ ∈Π = ∩τ ∈Π "T "+τ0 T ⊇ ∩τ ∈Π "T ⊃ τ 0 ∩τ ∈Π " Tτ . Obviously, one can also observe that T ⊃ Tτ0 ⊃ ∩τ ∈Π " Tτ , so it follows from (2.30) that T0 ∩ Tτ00 = ∩τ ∈Π " Tτ = T0 , which implies that ∩τ ∈Π " Tτ is a complete-closed time scale according to (2.29) from Definition 2.27. This completes the proof. 

From Lemma 2.7, we obtain the following corollary directly. " ⊆ Π1 and the pair (Π, " +) be an Abelian group. If Corollary 2.9 Let Π   " Tτ = ∅, then " Tτ is a bi-direction CCTS. τ ∈Π τ ∈Π Example 2.10 Let    2 3 n + T = − 2k, 2k + 1, k ∈ N , , ,n ∈ Z . 3 4 n+1 

2.4 Complete-Closed Translations Time Scales (CCTS)

109

For this time scale, we obtain Π ∗ = {Π˜ 1 , Π˜ 2 , Π˜ 2 , Π˜ 4 } ⊂ Π1 , where Π˜ 1 = {2k, k ∈ Z},   3 n 2 Π˜ 2 = − , − , − , n ∈ Z+ , 3 4 n+1 Π˜ 3 = {2k, k ∈ Z+ }, Π˜ 4 = {2k, k ∈ Z− }. We calculate that     ∩τ ∈Π˜ 1 Tτ = ∩τ ∈Π˜ 3 Tτ ∩ ∩τ ∈Π˜ 4 Tτ = {−2k, k ∈ N} ∩ {2k + 1, k ∈ N} = ∅. and Π˜ 2 is not closed with respect to additive operation. Hence, Π˜ 1 , Π˜ 2 does not satisfy Lemma 2.7. Further, T1∗ := ∩τ ∈Π˜ 3 Tτ = {−2k, k ∈ N} = ∅, T2∗ := ∩τ ∈Π˜ 4 Tτ = {2k + 1, k ∈ N} = ∅, and Π˜ 3 and Π˜ 4 are closed with respect to additive operation, and according to Lemma 2.7, we obtain ∩τ ∈Π˜ 3 Tτ and ∩τ ∈Π˜ 4 Tτ are CCTS. In fact, through calculation, for any τ3 ∈ Π˜ 4 and τ4 ∈ Π˜ 3 , one can easily obtain that Tτ1∗3 ⊆ T1∗ and 

Tτ2∗4 ⊆ T2∗ . Example 2.11 Let    T = ∪+∞ 2n + 1 + [2k, 2k + 1] k=−∞  2n +

 n − ,n ∈ Z . 1−n

n , n ∈ Z+ n+1

For this time scale, we obtain Π ∗ = {Π˜ 1 , Π˜ 2 , Π˜ 3 } ⊂ Π1 , where Π˜ 1 = {2k, k ∈ Z},  ˜ Π2 = − 2n − 1 −

 n + ,n ∈ Z , n+1   n Π˜ 3 = − 2n − , n ∈ Z− . 1−n

We calculate that



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2 A Classification of Closedness of Time Scales Under Translations



Tτ =

τ ∈Π˜ 1

+∞ 

[2k, 2k + 1] = ∅,

k=−∞

and Π˜ 1 is an Abelian group. According to Corollary 2.9, ∩τ ∈Π˜ 1 Tτ is a bi-direction CCTS. In fact, through calculation, we see that ∩τ ∈Π˜ 1 Tτ is actually a bi-direction CCTS. 

2.5 Almost Complete-Closed Time Scales Under Translations (ACCTS) In this section, we introduce the concept of almost complete-closed time scales which is more general than the concept of almost periodic time scales (i.e., Definition 2.3.1). Let τ be a number and Aετ be a subset of R, A denotes the closure of the set A, and we set the time scales:  [αi , βi ], T := i∈I

T := T + τ = {t + τ : ∀t ∈ T} := τ



 [αiτ , βiτ ], i∈I

[α˜ iτ , β˜iτ ]

:= T\Aετ ,

i∈I

and define the distance between two time scales, T\Aετ and Tτ by  d(T\Aετ , Tτ )

= max

sup |α˜ iτ i∈I

− αiτ |,

 τ τ ˜ sup |βi − βi | , i∈I

where I is an infinite index set. We introduce the following notations:   αiτ := α ∈ Tτ : inf |αi − α| ,   βiτ := β ∈ Tτ : inf |βi − β| and   α˜ iτ := α ∈ T\Aετ : inf |αiτ − α| ,   β˜iτ := β ∈ T\Aετ : inf |βiτ − β| .

2.5 Almost Complete-Closed Time Scales Under Translations (ACCTS)

111

In what follows, we will give three equivalent definitions of ACCTS. Let Π1 := {τ ∈ R : Tτ = ∅} = {0}, where Tτ = T ∩ Tτ , Tτ := T + τ = {t + τ : ∀t ∈ T}. Definition 2.28 ([214]) We say T is an almost complete-closed time scale (ACCTS) if for any given ε1 > 0, there exist a constant l(ε1 ) > 0 such that each interval of length l(ε1 ) contains a τ (ε1 ) and sets Aετ1 such that d(T\Aετ1 , Tτ ) < ε1 i.e., for any ε1 > 0, the following set E{T, ε1 } = {τ ∈ Π1 : d(T\Aετ1 , Tτ ) < ε1 } := Πε1 is relatively dense in Π1 . Here, τ is called the ε1 -translation number of T, l(ε1 ) is called the inclusion length of E{T, ε1 }, and E{T, ε1 } the ε1 -translation set of T, Aετ1 1 is called the ε1 -improper set of T, RT (τ, ε1 ) := T ∩ (∪τ ∈Πε1 T−τ \Aε−τ ) the ε1 -main ε1 ε1 −τ ε1 region of T, where A−τ = (Aτ ) := {a − τ : a ∈ Aτ }. Furthermore, we can describe it in detail as follows: (a) if for any p > 0, there exists a number P > p and τ ∈ E{T, ε1 } ∩ (P , +∞), then we say T is a positive-direction ACCTS; (b) if for any q < 0, there exists a number Q < q and τ ∈ E{T, ε1 } ∩ (−∞, Q), then we say T is a negative-direction ACCTS; (c) for exist numbers Q < q, P > p and ±τ ∈ E{T, ε1 }∩  any p > 0, q < 0, there  (−∞, Q) ∪ (P , +∞) , then we say T is a bi-direction ACCTS; (d) we say T is an oriented-direction ACCTS if T is a positive-direction ACCTS or a negative-direction ACCTS. 1 Remark 2.31 Since ±τ ∈ E{T, ε1 } in Definition 2.12, then there exists Aε±τ = ε1 ±τ ±τ ±τ T\T such that d(T\A±τ , T ) ≤ d(T, T ) < ε1 , which implies that almost periodic time scales are bi-direction ACCTS with inf T = −∞, sup T = +∞.

Remark 2.32 Definition 2.28 includes Definition 2.26. In fact, if let ε1 → 0 in Definition 2.28, then there exists A0τ = T\Tτ such that d(T\A0τ , Tτ ) = 0, which implies that Tτ ⊆ T. In fact, one can observe that if T = Tτ , then the 0improper set A0τ = ∅, i.e, T is a bi-direction CCTS; if Tτ ⊂ T, then A0τ = T\Tτ = ∅, i.e., T is an oriented-direction CCTS. Next, we provide some sufficient and necessary conditions to guarantee that a time scale is bi-direction CCTS or ACCTS.

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2 A Classification of Closedness of Time Scales Under Translations

Lemma 2.8 A time scale is a bi-direction CCTS if and only if there exists a 0improper set A0τ such that A0τ = ∅. Proof If T is a bi-direction CCTS, then there exists a set Π˜ 0 = {τ ∈ R : Tτ ∪T−τ ⊆ T} = {0}, so Tτ ⊆ T. For any t ∈ T, we have t − τ ∈ T−τ ⊆ T, thus, t ∈ Tτ , so we obtain T ⊆ Tτ . Hence, T = Tτ . From Definition 2.28, we can take the 0-improper set A0τ = T\Tτ = ∅. From Definition 2.28, if the 0-improper set of T is empty, i.e., A0τ = ∅, then d(T, Tτ ) = 0 with τ = 0, which implies that T = Tτ , so for any t ∈ T, we have t + τ ∈ Tτ = T, i.e., t ∈ T−τ , thus, T ⊆ T−τ . Furthermore, for any t ∈ T−τ , we have t + τ ∈ T = Tτ , so t ∈ T, i.e., T−τ ⊆ T. Hence, T = T−τ , i.e., T is a bi-direction CCTS. This completes the proof. 

Lemma 2.9 A time scale is an almost periodic time scale if and only if there exists ε-improper set Aετ such that μΔ (Aετ ) < ε. Proof Assume that there exists the ε-improper set of T such that μΔ (Aετ ) < ε, and ε = T\Tτ , then it follows from d(T\Aε , Tτ ) < ε that d(T, Tτ ) = we can τ τ  take A  ε max d(T\Aτ , Tτ ), μΔ (Aετ ) < ε, i.e., T is an almost periodic time scale. If T is an almost periodic time scale, for any τ ∈ Πε , let Aετ = T\Tτ , and we can obtain μΔ (Aετ ) ≤ d(T, Tτ ) < ε. This completes the proof. 

Lemma 2.10 A time scale is an oriented-direction ACCTS if and only if there exists a ε- improper set Aετ such that μΔ (Aετ ) > 0. Proof If a time scale is ACCTS, from Definition 2.28, we can obtain that μΔ (Aετ ) ≥ 0. By Lemma 2.8, one can take A0τ = ∅ such that μΔ (A0τ ) = 0 if and only if T is bi-direction CCTS. Hence, there exists a ε- improper set Aετ such that μΔ (Aετ ) > 0 if a time scale is ACCTS. If there exists the ε-improper set Aετ such that μΔ (Aετ ) > 0, according to Definition 2.28, T is an oriented-direction ACCTS. This completes the proof. 

Remark 2.33 According to Definition 2.28, one can obtain that for any τ ∈ Πε , it follows from d(T\Aετ , Tτ ) < ε that d(T−τ \Aε−τ , T) < ε. Let μτ : Tτ → R+ be the graininess function of Tτ and Aτ be a set, satisfying  μτ (t + τ ) =

μ(t),

t + τ ∈ T\Aτ ,

μ(t + τ ), t + τ ∈ T\Aτ .

(2.31)

Then, from (2.31), we can simplify Definition 2.28 as follows: Definition 2.29 ([214]) Let μ : T → R, μ(t) = σ (t) − t. We say T is an almost complete-closed time scale if for any ε > 0, there exists the set A−τ such that the set "T (τ )} Π ∗∗ = {τ ∈ Π1 : |μ(t + τ ) − μ(t)| < ε, ∀t ∈ R

2.5 Almost Complete-Closed Time Scales Under Translations (ACCTS)

113

Changing-periodic time scales

Time scales with bounded graininess function

ACCTS

Almost periodic time scales Where A

CCTS

Periodic time scales B denotes A includes B

Fig. 2.5 A new inclusion relation of time scales

"T (τ ) := T ∩ (∪τ ∈Π T−τ \A−τ ), A−τ := {a − τ : is relatively dense in Π1 , where R a ∈ Aτ }. Remark 2.34 Since Π ∗∗ is relatively dense in Π1 in Definition 2.29, one can observe that the graininess function μ is bounded. In fact, we can provide a new inclusion relation of time scales (see Fig. 2.5), which indicates that ACCTS is the most general type of independent variables with almost periodicity, the concept of changing-periodic time scales will be introduced in Sect. 2.6. We can also introduce an equivalent definition that depend on sequential convergence. Definition 2.30 ([214]) If a time scale T fulfills the following conditions: 



(1) for any given sequence α ⊂ (0, +∞)Π1 , there exists a subsequence α ⊂ α n and a sequence {A−αn } such that {T−α \A−αn } converges to a time scale T0 , we say T is a positive-direction ACCTS;   (2) for any given sequence α ⊂ (−∞, 0)Π1 , there exists a subsequence α ⊂ α n −α and a sequence {A−αn } such that {T \A−αn } converges to a time scale T0 , we say T is a negative-direction ACCTS;   (3) for any given sequence α ⊂ (−∞, +∞)Π1 , there exists a subsequence α ⊂ α n −α and a sequence {A−αn } such that {T \A−αn } converges to a time scale T0 , we say T is a bi-direction ACCTS; (4) if T is a positive-direction ACCTS or a negative-direction ACCTS, we say T is an oriented-direction ACCTS.

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2 A Classification of Closedness of Time Scales Under Translations

Example 2.12 Let a > 1 and consider the following time scale ∞ 

Pa,| sin √3t+sin √7t| =

[pm , a + pm ],

m=1

where pm = (m − 1)a +

m−1 

√ sin 3

k=1

 ka + | sin

√

3a + sin

√ 7a|

√ √ √ +| sin 3(2a + | sin 3a + sin 7a|) √ √ √ + sin 7(2a + | sin 3a + sin 7a|)| √ √ √ + . . . + | sin 3((k − 1)a + | sin 3a + sin 7a|)  √ √ √ + sin 7((k − 1)a + | sin 3a + sin 7a|)| # $% & 

+ sin

k terms

√ √ √ 7 ka + | sin 3a + sin 7a|

√ √ √ +| sin 3(2a + | sin 3a + sin 7a|) √ √ √ + sin 7(2a + | sin 3a + sin 7a|)| √ √ √ + . . . + | sin 3((k − 1)a + | sin 3a + sin 7a|)  √ √ √ + sin 7((k − 1)a + | sin 3a + sin 7a|)| . # $% & k terms

Then we have σ (t) =

 t,

if t ∈ ∪∞ m=1 [pm , a + pm ), √ √ t + | sin 3t + sin 7t|, if t ∈ ∪∞ m=1 {a + pm }

and  μ(t) =

if t ∈ ∪∞ m=1 [pm , a + pm ), √ √ | sin 3t + sin 7t|, if t ∈ ∪∞ m=1 {a + pm }.

0,

This time scale satisfies item (a) from Definition 2.28, thus, it is a positive-direction ACCTS. 

Remark 2.35 Since the Definition 2.12 is a particular case of Definition 2.28, if T is a bi-direction ACCTS (i.e., an almost periodic time scale), then RT (τ, ε) = T ∩ (∪TΠε ), where Πε = E{T, ε}, TΠε = T−τ : −τ ∈ E{T, ε} .

2.6 Changing-Periodic Time Scales

115

2.6 Changing-Periodic Time Scales In this section, we address the periodic coverage phenomenon on arbitrary unbounded time scales and initiate a new idea, namely, we introduce the concept of changing-periodic time scales. We discuss some properties of this new concept and give several examples. We establish a basic decomposition theorem of time scales which provides bridges between a CCTS and an arbitrary time scale with a bounded graininess function μ. The concept changing-periodic time scales will help in understanding and removing the serious deficiency which arises in the study of classical functions on arbitrary time scales. The following is the famous Zorn’s Lemma which will play an important role in establishing and proving the main results. Lemma 2.11 ([245], Zorn’s Lemma) Suppose (P , ) is a partially ordered set. A subset T is totally ordered if for any s, t in T we have s  t or t  s. Such a set T has an upper bound u in P if t  u for all t in T . Suppose a non-empty partially ordered set P has the property that every non-empty chain has an upper bound in P . Then the set P contains at least one maximal element. In order to introduce the concept of changing-periodic time scales precisely and concisely, we need the following definitions. Definition 2.31 ([16, 212, 217]) We say a time scale is an infinite time scale if one of the following conditions is satisfied: (i) sup T = +∞ and inf T = −∞; (ii) sup T = +∞ or inf T = −∞. Remark 2.36 Under Definition 2.31, it is obvious that an infinite time scale indicates it has at least an infinite boundary. For a simple example take T =  +∞ k=1 [2k, 2k + 1]. Definition 2.32 ([16, 212, 217]) Let T be a time scale, we say T is a zero-periodic time scale if and only if there exists no nonzero real number ω such that t + ω ∈ T for all t ∈ T. Remark 2.37 By Definition 2.32, it follows that a finite union of the closed intervals can be regarded as a zero-periodic time scale. The single point set {a} can also be regarded as a closed interval, since {a} = [a, a]. For convenience, T0 denote the zero-periodic time scale. Remark 2.38 Definition 2.32 indicates that for a zero-periodic time scale, t +ω ∈ T for all t ∈ T if and only if ω = 0. Hence, we one can easily see that a finite union of the closed intervals  is a zero-periodic time scale. For example, T = [0, 1] ∪ [2, 3] ∪ [4, 5] and T = 100 k=0 [2k, 2k + 1], etc. Definition 2.33 ([16, 212, 217]) A timescale sequence {Ti }i∈Z+ is called well connected sequence if and only if for i = j , one has Ti ∩ Tj = {tijk }k∈Z , where

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2 A Classification of Closedness of Time Scales Under Translations

{tijk } is the countable points set or an empty set, and tijk is called the connected point between Ti and Tj for each k ∈ Z, the set {tijk } is called the connected points set of this well connected sequence. Remark 2.39 Clearly the connected points set of a well connected sequence can be  an empty set. Hence, it follows that if ∞ T i=1 i = ∅, then {Ti }i∈Z+ is well connected. Remark 2.40 Under Definition 2.33, we will give some examples to help the reader to understand Definition 2.33. We provide a time scale sequence {Ti }i∈Z+ . (a) Let T1 =

∞ 

[4k, 4k + 1],

k=0

T2 =

∞ 

[4k + 2, 4k + 3],

k=0

 Ti =

4i − 1 2

 for i ≥ 3.

Then Ti ∩ Tj = ∅ if i = j . According to Definition 2.33, one has {tijk }k∈Z = ∅, then such a timescale sequence is a well connected sequence. (b) Let T1 =

∞ 

[2k, 2k + 1],

k=0

 ∞    4k − 1 , 2 k=1   4i − 1 1 Ti = + for i ≥ 3. 2 3

T2 = {1, 2}

k } We can obtain T1 ∩T2 = {t12 k∈Z = {1, 2} is a countable set. Further, T1 ∩Ti = ∅ and T2 ∩ Ti = ∅ for i ≥ 3, Ti ∩ Tj = ∅ for i, j ≥ 3, i = j , that is, k k {t1i }k∈Z = {t2i }k∈Z = {tijk }k∈Z = ∅ for i ≥ 3, i = j.

According to Definition 2.33, such a timescale sequence is a well connected sequence.  (c) Let T1 = ∞ k=0 [k(1 + a), k(1 + a) + 1], where a > 0 and

∞  1 < a. n2 n=1

2.6 Changing-Periodic Time Scales

117

For i ≥ 2, let Ti =

∞ 

k(1 + a) + 1 +

k=0

One can obtain that  T1 ∩ Tj =

! i−1 i   1 1 . , k(1 + a) + 1 + i2 i2 i =2 0 i =2 0 0

k } {t12 k∈Z = k } {t1j k∈Z

0

∞

k=0 {k(1 + a) + 1},

i = 1, j = 2,

= ∅, i = 1, j = 2,

for i = 1, j = 2, we have ⎧ ⎪ {t k } = Ti ∩ Tj ⎪ ⎨ ij k∈Z  i  = ∞ Ti ∩ T j = k=0 k(1 + a) + 1 + i0 =2 ⎪ ⎪ ⎩ k {tij }k∈Z = ∅, j = i + 1.

1 i02



, j = i + 1,

According to Definition 2.33, such a timescale sequence is a well connected sequence. (d) If T1 =

∞ 

[2k, 2k + 1]

k=0

and Ti =

∞  k=0

1 1 [2k + , 2k + 1 + ], i i

then for any i = j , Ti ∩ Tj is an uncountable set, such a timescale sequence is not a well connected sequence. We now introduce a new concept of time scales called “changing-periodic time scales”. Definition 2.34 ([16, 212, 217]) Let T be an infinite time scale. We say T is a changing-periodic or a piecewise-periodic time scale if the following conditions are fulfilled:  ∞    (a) T = Tr and {Ti }i∈Z+ is a well connected timescale sequence, Ti i=1

where Tr =

k  i=1

[αi , βi ] and k is some finite number, and [αi , βi ] are closed

intervals for i = 1, 2, . . . , k or Tr = ∅;

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2 A Classification of Closedness of Time Scales Under Translations

(b) S subsets of R with 0 ∈ Si for each i ∈ Z+ and Λ =   iis∞ a nonempty R0 , where R0 = {0} or R0 = ∅; i=1 Si (c) for all t ∈ Ti and all ω ∈ Si , we have t + ω ∈ Ti , i.e., Ti is an ω-periodic time scale; (d) for i = j , for all t ∈ Ti \{tijk } and all ω ∈ Sj , we have t + ω ∈ T, where {tijk } is the connected points set of the timescale sequence {Ti }i∈Z+ ; (e) R0 = {0} if and only if Tr is a zero-periodic time scale and R0 = ∅ if and only if Tr = ∅; and the set Λ is called a changing-periods set of T, Ti is called the periodic subtimescale of T and Si is called the periods subset of T or the periods set of Ti , Tr is called the remain time scale of T and R0 the remain periods set of T. Remark 2.41 From Definition 2.34, it follows that if T is a changing-periodic time scale, then the remain time scale Tr is a finite union of some closed intervals or Tr = ∅, i.e., R0 = {0} or R0 = ∅. Remark 2.42 From condition (c) in Definition 2.34, for all connected points tijk , i = j , we have tijk ∈ Ti and tijk ∈ Tj , for all ω ∈ Si and ω ∈ Sj , we have tijk + ω ∈ Ti ⊂ T and tijk + ω ∈ Tj ⊂ T. Now we introduce the following related concept. Definition 2.35 ([16, 212, 217]) A changing-periodic time scale is called complete if and only if its remain time scale is an empty set, i.e, Tr = ∅; similarly, Tr = ∅ if and only if T is non-complete. We now prove the following proposition. Proposition 2.1 All periodic time scales are particular changing-periodic time scales and complete.  Proof Let T = Tj , S =  Sj ,  j ∈ Z+ . Since T = T  ∅, by 2.34,    Definition ∞ we can take ∞ T = T ∅ and S = S ∅ , where S i =j i =j i=1 i i=1 i is the periods set of T, then Tr = ∅ and R0 = ∅. Now by Definition 2.34 and Definition 2.35, we obtain the desired result. This completes the proof. 

Remark 2.43 If T is a constant-periodic time scale, i.e., T is an ω-periodic time scale, where ω is a constant, then Λ = Π = {nω : n ∈ Z} := S. We also note that although Π can be written as Π = ∞ n=1 {nω}, we cannot regard Si as {iω} for all i ∈ Z since for any i = j , t ∈ Ti , we have t + j ω ∈ Ti , which contradicts the condition (d) in Definition 2.34. Therefore, Si0 = S, Tr = ∅ for some i0 ∈ Z+ and for i = i0 , Si = ∅. Hence, all constant-periodic time scales are particular changingperiodic time scales (see Fig. 2.6). From Definition 2.34, the following properties of changing-periodic time scales are immediate.

2.6 Changing-Periodic Time Scales

119

Fig. 2.6 T and S come in pair when T is periodic

Theorem 2.29 If T be a changing-periodic time scales, then the following hold:  (1) If t ∈ T, then there exists some i ∈ Z+ such that t ∈ Ti Tr . Furthermore, if T is complete, then t ∈ Ti . (2) If t ∈ Ti , ω ∈Si ,  then t + ω ∈ Ti ⊂ T. (3) If i = j , then Ti Tj \{tijk } = ∅, Si Sj = ∅. (4) If t ∈ Ti \{tijk }, t + ω ∈ T for i ∈ Z+ , then ω ∈ Si . (5) If ω ∈ Si , t + ω ∈ T for i ∈ Z+ , then t ∈ Ti . Proof From Definition 2.34, the conclusions (1) and (2) are obvious. Next, we will prove (3)–(5).   To prove (3), if there exists some t ∈ Ti ∩ Tj \{tijk }, i = j , then t + α ∈ Ti , t ∈ Tj , ∀α ∈ Si and t + β ∈ Tj , t ∈ Ti , ∀β ∈ Sj , but this contradicts condition (d) in Definition 2.34. Similarly, if there exists some τ ∈ Si ∩ Sj , i = j , then t + τ ∈ Ti , τ ∈ Sj , ∀t ∈ Ti and t + τ ∈ Tj , τ ∈ Si , ∀t ∈ Tj , which also contradicts condition (d) in Definition 2.34. To prove (4), we assume that ω ∈ Si . Case (i) ω ∈ R0 , then there exists Sj such that ω ∈ Sj , i = j , but t + ω ∈ T for t ∈ Ti \{tijk }, which contradicts condition (d) in Definition 2.34. Case (ii) ω ∈ R0 , then from t + ω ∈ T, we have t ∈ Tr , but this contradicts that t ∈ Ti . To prove (5), we assume that t ∈ Ti .

120

2 A Classification of Closedness of Time Scales Under Translations

Case (i) t ∈ Tr , then there exists Tj such that t ∈ Tj , i = j and t is obviously not a connective point between Ti and Tj , then t + ω ∈ T for ω ∈ Si , which contradicts condition (d) in Definition 2.34. Case (ii) t ∈ Tr , from t + ω ∈ T, we have ω ∈ R0 , but this contradicts that ω ∈ Si . This completes the proof. 

In view of the characteristics of changing-periodic  time scales (Theorem 2.29), we can introduce an index function τt : T → Z+ {0} such that for t ∈ T, t ∈ Tτt holds. This function plays a very important role in introducing well defined functions on time scales. Formally, we have the following definition of τt . Definition 2.36 ([16, 212, 217]) Let T be a changing-periodic time scale, then the function τ  τ: T → Z+ {0}  ∞

 Ti

→

i, where t ∈ Ti , i ∈ Z+

Tr

→

0, where t ∈ Tr

→

τt

i=1

t

is called an index function for T, where the corresponding periods set of Tτt is denoted as Sτt . In what follows we shall call Sτt as the adaption set generated by t, and all the elements in Sτt will be called the adaption factors for t. Remark 2.44 From Definition 2.36, if τt is an index function for T, then it immediately follows that (i) for t ∈ T, we have t ∈ Tτt ; (ii) for each i ∈ Z+ , t1 , t2 ∈ Ti if and only if τt1 = τt2 = i. Furthermore, for t1 , t2 ∈ Tr = ∅ if and only if τt1 = τt2 = 0, i.e., Tr = T0 , where T0 is a zero-periodic time scale (see Fig. 2.7). Obviously, if Tr = ∅, then τ : T → Z+ . Remark 2.45 Let T be a changing-periodic time scale, for all t ∈ T and all ω ∈ Sτt , we have t + ω ∈ Tτt ⊂ T. It is also obvious that if Si is an adaption set generated by some given t0 ∈ Ti , then Sτt0 is also the adaption set for all t ∈ Ti since τt0 = τt = i for all t ∈ Ti . Now we can prove the following proposition: Proposition 2.2 T is an ω-periodic time scale if and only if we can obtain its index function τt ≡ z for all t ∈ T, where z denotes some positive integer.

2.6 Changing-Periodic Time Scales Fig. 2.7 The index function τt : T → Z+ {0} with Tr = ∅

121

T

Z + U{0}

tt

T1

1

T2

2

° °

t1 , t2

° °

i

Ti ° °

Tr

° °

° °

° °

O

Proof If T is an ω-periodic time scale, then there is S = {nω : n ∈ Z} such that for all t ∈ T and all τ˜ ∈ S, we have t + τ˜ ∈ T. Thus, by Proposition 2.1 and Remark 2.44, there exists some positive integer z such that T = Tz and S = Sz , i.e., we can choose the index function τt ≡ z for all t ∈ T. If τt ≡ z for all t ∈ T, then there exists an adaption set Sτt = Sz for all t ∈ T such that for any ω ∈ Sτt , we have t + ω ∈ Tτt = T. Hence, T is ω-periodic. This completes the proof. 

Remark 2.46 All changing-periodic time scales can be equipped with the corresponding index functions such that for all t ∈ T and all ω ∈ Sτt , t + ω ∈ Tτt ⊂ T. We are now in the position to prove the following important theorem which classifies the time scales with bounded graininess function μ as changing-periodic time scales. For this, we give the following lemma. Lemma 2.12 Let T be an infinite time scale with a bounded graininess function μ : T → R+ . If there exists τ0 such that T ∩ Tτ0 is a maximum element in {T ∩ Tτ : τ ∈ R}, then T ∩ Tτ0 is an infinite time scale with a bounded graininess function μT∩Tτ0 , where μT∩Tτ0 : T ∩ Tτ0 → R+ is the graininess function for the time scale T ∩ Tτ0 . Proof First, for T is an infinite time scale, we obtain T ∩ Tτ0 is also an infinite time scale. Moreover, since T ∩ Tτ0 has a maximum element in {T ∩ Tτ : τ ∈ R}, then we claim that   T ∩ Tτ0 ∈ ∅, {∞} .

(2.32)

In fact, for contradiction, assume that T ∩ Tτ0 = {∞}, then for any τ ∈ R, we have T ∩ Tτ = {∞}, which implies that T = {a} ∪ {∞}, where {a} is a single point set, this is a contradiction because T has a bounded graininess function μ.

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2 A Classification of Closedness of Time Scales Under Translations

Next, we will show that μT∩Tτ0 is bounded. For contradiction. Assume that μT∩Tτ0 is unbounded, then for any Mn > 0, there exists right scattered point tn ⊂ T ∩ Tτ0 such that μT∩Tτ0 (tn ) > Mn , n = 1, 2, . . .. Without loss of generality, we assume that M1 < M 2 < . . . < M n < . . . , then there exist t1 , t2 , . . . , tn , . . . such that μT∩Tτ0 (tn ) > Mn → +∞ as n → ∞. Let T˜ 1 =

∞    tn , σT∩Tτ0 (tn ) ,

(2.33)

n=1

by (2.32) and (2.33), we obtain   T˜ 1 ∩ T ∩ Tτ0 = {∞},

(2.34)

from (2.34), for any n ∈ Z+ , we have   tn , σT∩Tτ0 (tn ) T ∩ Tτ0 = ∅, τ0 + ∗ which will lead to μτ0 (tn∗ ) → +∞ as n → +∞,  whereτμτ0 : T → R and tn is τ ∗ 0 0 ˜ ˜ the right scattered point in T satisfying tn = tn ∈ T : inf |tn − tn | , so μτ0 is unbounded. On the other hand, since T is an infinite time scale with a bounded graininess function μ : T → R+ and the time scale Tτ0 is a translation of the time scale T, then Tτ0 is also an infinite time scale with a bounded graininess function μτ0 : Tτ0 → R+ , this is a contradiction. This completes the proof. 

Theorem 2.30 If T is an infinite time scale and the graininess function μ : T → R+ is bounded, then T is a changing-periodic time scale. Proof Without loss of generality, we assume that sup T = +∞ and inf T = −∞. From Remark 2.28, T is an oriented-direction intersection time scale. For any p > 0, q < 0, we choose some n1 , n2 such that ¯ p < n1 μ,

q > −n2 μ, ¯

where μ¯ = supt∈T μ(t) and ¯ n2 μ} ¯ max{n1 μ,

μ¯ + L,

2.6 Changing-Periodic Time Scales

123

where L > 0 is the length of a closed finite continuous interval with the largest length in T. We denote the set   ¯ q] ∪ [p, n1 μ] ¯ . I = T ∩ Tτ : τ ∈ [−n2 μ, Clearly, I forms a semi-ordered set with respect to the inclusion relation and I is closed. Denote I∗ any subset of I and is totally ordered. We consider two cases. Case (1)   I∗ = Tτn ∈ I : Tτn ⊃ Tτn+1 , n ∈ N , where Tτn := T ∩ Tτn , then Tτ1 ∈ I∗ ⊂ I and Tτ1 is an upper bound of I∗ in I. Case (2)   I∗ = Tτn ∈ I : Tτn ⊂ Tτn+1 , n ∈ N , then lim (T ∩ Tτn ) =

n→∞

∞ 

(T ∩ Tτn ) := T ∩ Tτ∞

n=1

is an upper bound of I∗ . Because I is closed, then T ∩ Tτ∞ ∈ I. According to Zorn’s lemma, there exists some τ0 ∈ [−n2 μ, ¯ q] ∪ [p, n1 μ] ¯ such that T ∩ Tτ0 is the maximum element in I. Note that since μ is bounded, according to Lemma 2.12, T ∩ Tτ0 is an infinite time scale with a bounded graininess function and sup(T ∩ Tτ0 ) = +∞ and inf(T ∩ Tτ0 ) = −∞. Now we show that T is a changing-periodic time scales. We divide the proof into three steps. Step I We can select a time scale T10 such that T10 ⊂ T is the largest periodic subtimescale in T. For this, we make a continuous translation of T to find a number τ1 such that T ∩ Tτ1 := T1 is the maximum. Next, consider a translation of T1 again to find a number τ2 such that T1 ∩ Tτ12 := T2 is the maximum. Continue this process n n times to find a number τn such that Tn−1 ∩ Tτn−1 := Tn is the maximum. This process leads to a decreasing sequence of timescale sets: T ⊃ T1 ⊃ T2 ⊃ . . . ⊃ Tn ⊃ . . . . Hence, it follows that lim Tn =

n→∞

∞  n=1

Tn = T10 .

(2.35)

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2 A Classification of Closedness of Time Scales Under Translations 1

This shows that there exists some τ01 such that (T10 )τ0 ⊆ T10 through the translation 1

1

1

of the number τ01 and (T10 )τ0 = (T10 )τ0 ∩ T10 is maximum. In fact, if (T10 )τ0 ⊆ T10 , then there exists T1∗ 0 such that 1

1 τ0 1 1 T1∗ 0 = (T0 ) ∩ T0 ⊂ T0 .

(2.36)

Furthermore, from (2.35), we have T10

= lim Tn = n→∞

∞ 

Tn ⊆ T1∗ 0 ,

n=1

this is in contradiction with (2.36). Step II For the time scale T∗1 := T\T10 , where A denotes the closure of the set A, by replacing T with T∗1 , and repeating the Step I, we can obtain the periodic

sub-timescale T20 . For the time scale T∗2 := T\(T10 ∪ T20 ), by replacing T with T∗2 , and repeating the Step I, we can obtain the periodic sub-timescale T30 . Similarly, we can obtain T40 , . . . , Tn0 . . .. Obviously, the timescale sequence {Ti0 }i∈Z+ is well  j connected and Ti0 ∩ T0 \{tijk 0 } = ∅ for i = j , where {tijk 0 } is the connected points  0 i j T0 is set between Ti0 and T0 . If for some sufficiently large n0 , still T∗n0 = T\ ni=1 an infinite time scale, then we repeat the Step I again until the remaining timescale  i T\ ∞ i=1 T0 = ∅, or a finite union of the closed continuous intervals.

Step III Letting the set Π of T be as Π=

 ∞ i=1

Si



R0 =

 ∞

{nτ0i ,

n ∈ Z}



R0 ,

i=1

where R0 = {0} or ∅, from Steps I and II, it follows that Si ∩ Sj = ∅ if i = j .   k  ∞   i   From Steps I, II, III, we find T0 [αi , βi ] = T, where k is some i=1 i=1   ∞  i T0 = T. finite number and [αi , βi ] are closed intervals for i = 1, 2, . . . , k, or i=1

Therefore, T is a changing-periodic time scale. This completes the proof.



Remark 2.47 For any oriented-direction periodic time scale T, we can add {−∞, +∞} to the time scale such that inf T = −∞ and sup T = +∞. For example, for the time scales (i)–(iv) from Example 2.8, we can add {−∞} to each time scale such that inf T = −∞ if we need, and they will still be positive-direction periodic time scales. From the proof of Theorem 2.30, we have the following proposition:

2.6 Changing-Periodic Time Scales

125

 Proposition 2.3 If T = ∞ Z+ , then i=1 Ti , where Ti is ωi -periodic for each i ∈  i i there exists a well connected timescale sequence {T0 }i∈Z+ such that T = ∞ i=1 T0 , i i where T0 is ω0 -periodic. Furthermore, T is a complete changing-periodic time scale. Proof Since Ti is periodic for each i ∈ Z+ , the graininess function μi : Ti → R+ is bounded for each i ∈ Z+ . Thus, the graininess function μ of T is also bounded. Therefore, in view of Theorem 2.30, T is a changing-periodic time scale. Further, from the proof of Theorem 2.30, one obtains that T can be decomposed into the union of a well connected periodic timescale sequence {Ti0 }i∈Z+ and the remain time scale Tr = ∅. Now by Definitions 2.34 and 2.35, T is a complete changingperiodic time scale. This completes the proof. 

Now we shall demonstrate some complete changing-periodic time scales. Example 2.13 Let k ∈ Z, and consider the following time scale: T=

 +∞ 

3 3 1 (2k + 1), (2k + 1) + 2 2 12

k=−∞

!

√ √ √ ! 3 2 3 2 3 (2k + 1), (2k + 1) + . 2 2 5

   +∞ k=−∞

We denote +∞ 

T1 =

k=−∞

3 3 1 (2k + 1), (2k + 1) + 2 2 12

!

and T2 =

+∞  k=−∞

√ √ √ ! 3 2 3 2 3 (2k + 1), (2k + 1) + . 2 2 5

Then, by a direct calculation the set Π2 is   √  3 2n, n ∈ Z := S1 ∪ S2 . Π2 = 3n, n ∈ Z This time scale is a changing-periodic time scale according to Definition 2.34. Example 2.14 Let k ∈ Z, and consider the following time scale: T=

 +∞  k=0

3 1 3 − (2k + 1), − (2k + 1) − 2 2 12

!



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2 A Classification of Closedness of Time Scales Under Translations

√ √ √ !   +∞  3 2 3 3 2 . (2k + 1), (2k + 1) + 2 2 5 k=0

We denote T1 =

+∞  k=0

3 3 1 − (2k + 1), − (2k + 1) − 2 2 12

!

and T2 =

+∞  k=0

√ √ √ ! 3 2 3 2 3 (2k + 1), (2k + 1) + . 2 2 5

Then, by direct calculation the set Π2 is   √  3 2n, n ∈ N := S1 ∪ S2 . Π2 = − 3n, n ∈ N According to Definition 2.34, this time scale is a changing-periodic time scale in which T1 is a negative-direction periodic sub-timescale and T2 is a positivedirection periodic sub-timescale. 

Example 2.15 Consider T = {−4k, 4k + 3 : k ∈ N}. Note that T1 = {−4k : k ∈ N} and T2 = {4k + 3 : k ∈ N} are oriented-direction periodic time scales, i.e., T1 is a negative-direction periodic time scale and T2 is a positive-direction periodic time scale. Hence, T is a changing-periodic time scale. 

The above examples lead to the following immediate propositions: Proposition 2.4 Let Ti be constant-periodic time scales for all i ∈ I . Then may not be a constant-periodic time scale, where I is an index number set.

 i∈I

Ti

Proposition 2.5 Let Ti be constant-periodic time scales with ωi -period for all i ∈ I , and ωi be a natural number for each  i ∈ I . If all the numbers in the set {ωi }i∈I have a lowest common multiple ω, then i∈I Ti is a ω-periodic time scale. Example 2.16 Let k ∈ Z, and consider the time scale: T=

 +∞  k=−∞

1 1 1 2k + , 2k + + 3 3 23

   +∞ k=−∞

!   +∞ 

3 3 1 5k + , 5k + + 4 4 37

k=−∞

1 1 1 3k + , 3k + + 2 2 16

!

! ,

then one will see that Π = {2n, n ∈ Z} ∪ {3n, n ∈ Z} ∪ {5n, n ∈ Z}, and T has the constant period 30. 

2.6 Changing-Periodic Time Scales

127

Now we will provide some changing-periodic time scales with bounded graininess functions. Example 2.17 Let a > 1, t > a, t ∈ T, and consider the following time scale: Pa,e−t =

∞ 

[pm , a + pm ],

m=1

where pm = (m − 1)a +

m−1 

 exp −

 ka + exp(−a) + exp

k=1

 + exp #

   − 2a + exp(−a) + . . .

!    . − (k − 1)a + exp(−a) $% & k terms

Then, we have  σ (t) =

if t ∈ ∪∞ m=1 [pm , a + pm ),

t,

t + e−t , if t ∈ ∪∞ m=1 {a + pm }

and μ(t) =

 0,

if t ∈ ∪∞ m=1 [pm , a + pm ),

e−t , if t ∈ ∪∞ m=1 {a + pm }.

Similarly, we also have  ρ(t) =

if t ∈ ∪∞ m=1 (pm , a + pm ],

t,

t − e−t , if t ∈ ∪∞ m=1 {pm }

and  ν(t) =

0,

if t ∈ ∪∞ m=1 (pm , a + pm ],

e−t , if t ∈ ∪∞ m=1 {pm }.

Since the graininess function μ for the time scale Pa,e−t is bounded, then it is a changing-periodic time scale.

 Corollary 2.10 Almost periodic time scales are changing-periodic time scales.

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2 A Classification of Closedness of Time Scales Under Translations

Time Scales With Bounded Graininess Function

Almost Periodic Time Scales

Periodic Time Scales

ChangingPeriodic Times Scales

Fig. 2.8 The containing relationship of four classes of time scales

Proof If T is an almost periodic time scale, then μ : T → R+ is bounded. Thus, it follows from Theorem 2.30 that T is a changing-periodic time scale. This completes the proof. 

Remark 2.48 Time scales considered in Examples 2.13–2.15 and 2.17 are changing-periodic time scales rather than almost periodic time scales. Therefore, the changing-periodic time scales strictly include almost periodic time scales (see Fig. 2.8). Now we will give the following theorem, which plays an important role in establishing classical functions on changing-periodic time scales. Theorem 2.31 (Decomposition Theorem of Time Scales) Let T be an infinite time scale and the graininess function μ : T → R+ be bounded. Then T is a changing-periodic  ∞timescale, i.e., there exists a countable periodic decomposition   Tr and Ti is ω-periodic sub-timescale, ω ∈ Si , i ∈ Z+ , Ti such that T = i=1

where Ti , Si , Tr satisfy the conditions in Definition 2.34. Proof According to Theorem 2.30, we obtain that T is a changing-periodic time scale, so one can obtain the decomposition of the time scale T directly from the Definition 2.34. The proof is complete. 

From the definition of the index function (i.e., Definition 2.36), it is easy to observe that a decomposition of a time scale will be determined by its index function τ . In fact, as a consequence of Theorem 2.31, we have the following result. Theorem 2.32 (Periodic Coverage Theorem of Time Scales) Let T be an infinite time scale and the graininess function μ : T → R+ be bounded. Then T can be covered by countable periodic time scales. Proof From Theorem 2.31 and Definition 2.34, we obtain T=

 ∞ i=1

Ti



Tr ,

2.6 Changing-Periodic Time Scales

129

where Ti is periodic for each i ∈ Z+ and Tr = ∅ or Tr is a zero-periodic time scale. This completes the proof. 

In view of Theorem 2.31, Proposition 2.3 and Definition 2.34, it is possible to introduce another concept of changing-periodic time scales. Definition 2.37 ([16, 212, 217]) We say that T is a changing-periodic time scale if and only if T is a countable union of periodic time scales. + Remark 2.49 According to kRemark 2.37, for any fixed k ≥ 1, k ∈ Z , a finite unionof closed  intervals i=1 [αi , βi ] := Tr is a zero-periodic time scale. Thus, ∞   Tr can be regarded as an union of periodic time scales. Therefore, T= Ti i=1

by Proposition 2.3, it follows that Definition 2.34 is equivalent to Definition 2.37. Remark 2.50 From the above Definitions 2.34 and 2.37, we note that that Ti and Si appear in pairs for i ∈ Z+ , likewise, Tr and R0 also crop up in pair (see Fig. 2.9). Remark 2.51 For simplicity, since the remain timescale Tr can be regarded as the zero-periodic time scale, then a changing-periodic time scale can be denoted by  T= ∞ T which includes Tr . i=1 i Theorem 2.33 Let T be an infinite time scale with sup T = +∞ and inf T = −∞. If μ : T → R+ is bounded, then T contains at least an oriented-direction periodic time scale. Proof From Remark 2.28, T is a bi-direction intersection time scale. For any p > 0, q < 0, we take some n1 , n2 such that p < n1 μ, ¯ q > −n2 μ, ¯ where ¯ n2 μ} ¯ μ¯ + L, L > 0 is the length of a closed μ¯ = supt∈T μ(t) and max{n1 μ, finite interval with the largest length in T. We denote the set by   I = T ∩ Tτ : τ ∈ [−n2 μ, ¯ q] ∪ [p, n1 μ] ¯ .

¥

T1

T2

T3

S1

S2

S3

T4

S4

T5

T6

S5

S6

... = UTi i=1

Tr

¥ ... = USi R0 = {0} i=1

Fig. 2.9 Ti and Si come in pair and one-to-one for i ∈ Z+ when T is changing-periodic

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2 A Classification of Closedness of Time Scales Under Translations

Clearly, I forms a semi-ordered set with respect to the inclusion relation and I is closed. Let I∗ be any totally ordered subset of I. We consider two cases: Case (1)   I∗ = Tτn ∈ I : Tτn ⊃ Tτn+1 , n ∈ N , where Tτn := T ∩ Tτn , then Tτ1 ∈ I∗ ⊂ I and Tτ1 is an upper bound of I∗ in I. Case (2)   I∗ = Tτn ∈ I : Tτn ⊂ Tτn+1 , n ∈ N , then lim (T ∩ Tτn ) =

n→∞

∞ 

(T ∩ Tτn ) = T ∩ Tτ∞

n=1

is an upper bound of I∗ . Because I is closed, then T ∩ Tτ∞ ∈ I. According to Zorn’s lemma, there exists some τ0 ∈ [−n2 μ, ¯ q] ∪ [p, n1 μ] ¯ such that T ∩ Tτ0 is the maximum element in I. Note that since μ is bounded, according to Lemma 2.12, T ∩ Tτ0 is an infinite time scale with a bounded graininess function and sup(T ∩ Tτ0 ) = +∞ or inf(T ∩ Tτ0 ) = −∞. We can find a time scale T10 such that T10 ⊂ T is the largest periodic sub-timescale in T. For this, we make a continuous translation of T to find a number τ1 such that T ∩ Tτ1 := T1 arrive at the maximum. Next, consider a translation of T1 again to find a number τ2 such that T1 ∩ Tτ12 := T2 arrive at the maximum. Continue this n process n times to find a number τn such that Tn−1 ∩ Tτn−1 := Tn arrive at the maximum. This process leads to a decreasing sequence of the time scale sets: T ⊃ T1 ⊃ T2 ⊃ . . . ⊃ Tn ⊃ . . . . Hence, it follows that limn→∞ Tn = T10 . This shows that there exists some τ01 such 1

1

that (T10 )τ0 ⊆ T10 through the translation of this number and (T10 )τ0 is maximum (in 1

1

fact, if (T10 )τ0 ⊆ T10 , then there exists T20 such that T20 = (T10 )τ0 ∩ T10 ⊂ T10 , this is a contradiction since limn→∞ Tn = T10 ), obviously, this also implies that T10 is not a finite union of the closed intervals. Now we claim that T10 = ∅. In fact, if T10 = ∅, then there exists some sufficiently large n0 such that Tn0 = ∅, i.e., there exists some sub-timescale Tn0 −1 ⊂ T such that Tn0 −1 has no intersection with itself through the translation of the number τn0 τn and Tn0 −1 ∩ Tn00−1 is the maximum element in In0 = {Tn0 −1 ∩ Tτn0 −1 : τ = 0} = ∅,

2.7 Some Compactness Criteria on Time Scales

131

which means that Tn0 −1 is a single point set, but this is a contradiction since Tn0 −1 is an infinite time scale with a bounded graininess function according to Lemma 2.12. Therefore, T10 is a τ01 -periodic sub-timescale whose translation direction is determined by the sign of the number τ01 , that is, T10 is a completely closed sub-timescale unit in T. This completes the proof. 

Remark 2.52 Under the conditions in Theorem 2.33, we obtain that T contains at least one invariant unit under translations which is attached with a translation direction (i.e. it is an oriented-periodic time scale). Example 2.18 Consider the time scale T = {−6k, 6k+5 : k ∈ N}. One will observe that T contains a positive-direction periodic time scale T1 = {6k + 5 : k ∈ N} and a negative-direction periodic time scale T2 = {−6k : k ∈ N}, respectively. In fact, because T is a changing-periodic time scale, such a decomposition is obvious. 

2.7 Some Compactness Criteria on Time Scales For more details of the compactness criteria in the functional analysis, one may consult the literatures Conway [75], Fra˘nková [134], Guo and Lakshmikantham [138], Gordon [140], Morris and Noussair [184], Vidyasagar [211], Wilansky [245], Xiao and Liang [246]. In this section, we introduce some new definitions and establish new characterization results of compact sets in functional spaces on time scales which will play an important role in studying abstract discontinuous dynamic equations on time scales. First, let δL+∞ , δR+∞ : [T0 , +∞)T → R+ ∪ {0}. By using the same idea from the literature [196], we can extend the Δ-gauge for [a, b]T to [T0 , +∞)T . Definition 2.38 ([218]) We say δ +∞ = (δL+∞ , δR+∞ ) is a Δ-gauge for [T0 , +∞)T provided δL+∞ (t) > 0 on (T0 , +∞)T and δL+∞ (T0 ) ≥ 0, δR+∞ (t) > 0 on [T0 , +∞)T and δR+∞ (t) ≥ μ(t) for all t ∈ [T0 , +∞)T . For a Δ-gauge, δ +∞ , we always assume δL+∞ (T0 ) ≥ 0 (we will sometimes not even point this out). For T0 ∈ T and a Banach space (X,  . ), let    G [T0 , +∞)T , X := x : [T0 , +∞)T → X; lim x(s) = x(t + ) and lim x(s) = x(t − ) exist and are finite,

s→t +

s, t < +∞ and

s→t −

sup

 x(t) < +∞ .

t∈[T0 ,+∞)T

  Endow G [T0 , +∞)T , X with the norm x∞ =

sup t∈[T0 ,+∞)T

x(t).

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2 A Classification of Closedness of Time Scales Under Translations

Lemma 2.13

    G [T0 , +∞)T , X ,  · ∞ is a Banach space.

Proof Let {xn } be an arbitrary Cauchy sequence in G, i.e., for any ε > 0, there exists N such that n, m > N implies xn (t) − xm (t) < ε for all t ∈ [T0 , +∞)T .

(2.37)

Since X is a Banach space, for each t ∈ [T0 , +∞)T , {xn (t)} ⊂ X is a Cauchy sequence so xn (t) → x(t). Hence, let m → +∞ in (2.37), and we have xn (t) − x(t) < ε for all t ∈ [T0 , +∞)T . Furthermore, since {xn } ⊂ G, for any ε > 0 and t0 ∈ [T0 , +∞)T , there exists  δ +∞ > 0, t ∈ t0 − δL+∞ (t0 ), t0 T with xn (t) − xn (t0+ ) < ε. Hence, we obtain x(t) − x(t0+ ) ≤ x(t) − xn (t) + xn (t) − xn (t0+ ) +xn (t0+ ) − x(t0+ ) ≤ 3ε.

(2.38)

  Similarly, for t ∈ t0 , t0 + δR+∞ (t0 ) T , we also obtain x(t) − x(t0− ) ≤ x(t0 ) − xn (t) + xn (t) − xn (t0− ) +xn (t0− ) − x(t0− ) ≤ 3ε.

(2.39)

Therefore, from (2.38) and (2.39), we obtain x ∈ G. Hence, G is a Banach space. 

In the following, we will introduce the definition of a partition P for [T0 , +∞)T . Definition 2.39 ([218]) A partition P for [T0 , +∞)T is a division of [T0 , +∞)T denoted by   P P = T0 = t0P ≤ η1 ≤ t1P ≤ . . . ≤ tn−1 ≤ ηn ≤ tnP ≤ . . . < . . . P for i = 1, 2, . . . and t , η ∈ T. We call the points η tag points and with tiP > ti−1 i i i the points ti end points.

Definition 2.40 ([218]) If δ +∞ is a Δ-gauge for [T0 , +∞)T , then we say a partition P is δ +∞ -fine if P < tiP ≤ ηi + δR+∞ (ηi ) ηi − δL+∞ (ηi ) ≤ ti−1

for i = 1, 2, . . . .

2.7 Some Compactness Criteria on Time Scales

133

Remark 2.53 From Definition 2.40, one can observe that if a partition P is δ +∞ fine for [T0 , +∞)T , then for any closed interval [a, b]T ⊂ [T0 , +∞)T , there exist a δ-fine partition P ∗ and P ∗ ⊂ P.   Definition 2.41 ([218]) A set A ⊂ G [T0 , +∞)T , X is called uniformly equiregulated, if it has the following property: for every ε > 0 and t0 ∈ [T0 , +∞)T , there is a δ +∞ = (δL+∞ , δR+∞ ) such that 





(a) If x ∈ A, t ∈ [T0 , +∞)T and t0 −δL+∞ (t0 ) < t < t0 , then x(t0− )−x(t ) < ε.    (b) If x ∈ A, t ∈ [T0 , +∞)T and t0 < t < t0 + δR+∞ (t0 ), then x(t0+ ) − x(t ) < ε. From Definition 2.41, we obtain the following theorem.   Theorem 2.34 A set A ⊂ G [T0 , +∞)T , X is uniformly equi-regulated, if and only if, for every ε > 0, there is a δ +∞ -fine partition P: T0 = t0P < t1P < t2P < . . . < tnP < . . . < . . . such that 

x(t ) − x(t) ≤ ε,

(2.40)



for every x ∈ A and [t, t ]T ⊂ (tjP−1 , tjP )T , j = 1, 2, . . .. Proof Let ε > 0 be given and let   D = ξ ; ξ ∈ (T0 , +∞)T such that there is a partition P: T0 = t0P < t1P < . . . < tkP = ξ for which (2.40) holds with j = 1, 2, . . . , k.   (i) If A ⊂ G [T0 , +∞)T , X is uniformly equi-regulated, then there is a δR+∞ (T0 ) > 0 such that x(t) − x(T0+ )
ξ1 > T0 . Since x ∈ A, then there is a δL+∞ (ξ2 ) such that x(ξ2− ) − x(t)
0, there is a δ +∞ -fine partition P: T0 = t0P < t1P < t2P < . . . < tkP < . . . < . . . such that 

x(t) − x(t ) ≤ ε,

(2.41)



for every x ∈ A and [t, t ]T ⊂ (tjP−1 , tjP )T , j = 1, 2, . . .. is δ +∞ -fine, we have Let ηj be a tag −1 , tj )T . Since this partition  of (tj+∞  +∞ P P (tj −1 , tj )T ⊂ ηj − δL (ηj ), ηj + δR (ηj ) T . Therefore, the inequal   ity (2.41) holds, for t, t ∈ ηj − δL+∞ (ηj ), ηj + δR+∞ (ηj ) T . Taking t = ηj−    and t ∈ ηj − δL+∞ (ηj ), ηj T , then the inequality (2.41) remains true. Also,    if t = ηj+ and t ∈ ηj , ηj + δR+∞ (ηj ) T , the inequality (2.41) is fulfilled. Then, from Definition 2.41, it follows that A is uniformly equi-regulated. This completes the proof. 

  Definition 2.42 ([218]) Let A ⊂ G [T0 , +∞)T , X . We say A is uniformly Cauchy if for any ε > 0, there exist T1 ∈ (T0 , +∞)T and a δ +∞ = (δL+∞ , δR+∞ )fine partition P: t0P = T1 < t1P < t2P < . . . < tnP < . . . < . . . such that 





(a) If x ∈ A, t , t0 ∈ [T1 , +∞)T and t0 −δL+∞ (t0 ) < t < t0 , then x(t0− )−x(t ) < ε.   (b) If x ∈ A, t , t0 ∈ [T1 , +∞)T and t0 < t < t0 + δR+∞ (t0 ), then x(t0+ ) −  x(t ) < ε.

2.7 Some Compactness Criteria on Time Scales 





135 





P , t P ) , i, j = (c) If x ∈ A, t ∈ [a , b ]T ⊂ (tjP−1 , tjP )T , t ∈ [a , b ]T ⊂ (ti−1 i T 



1, 2, . . ., then x(t ) − x(t ) < ε.

Remark 2.54 From Definition 2.42, one can observe that if   A ⊂ G [T0 , +∞)T , X is uniformly Cauchy, then there exists T1 ∈ (T0 , +∞)T such that A is uniformly equi-regulated on [T1 , +∞)T .   Theorem 2.35 Assume that a set A ⊂ G [T0 , +∞)T , X is uniformly equiregulated and uniformly Cauchy, and for any t ∈ [T0 , +∞)T , there is a number βt such that, for x ∈ A, x(t) − x(t − ) ≤ βt , x(t + ) − x(t) ≤ βt , t ∈ [T0 , +∞)T .

(2.42)

Then there is a constant K > 0 such that x(t) − x(T0 ) ≤ K, for every x ∈ A and t ∈ [T0 , +∞)T . Proof Since A is uniformly Cauchy, according to Definition 2.42, there exists T1 ∈ (T0 , +∞)T such that x(t) − x(T1 ) < 1, t ∈ [T1 , +∞)T .

(2.43)

Let C be the set of all τ ∈ (T0 , T1 ]T such that there exists Kτ > 0 such that x(t) − x(T0 ) ≤ Kτ , for any x ∈ A and t ∈ [T0 , τ ]T . Since A is uniformly equi-regulated, there is a δR+∞ (T0 ) such that x(t) − x(T0+ ) ≤ 1,   for every x ∈ A and t ∈ T0 , T0 + δR+∞ (t0 ) T . This fact together with the hypothesis imply that x(t) − x(T0 ) ≤ x(t) − x(T0+ ) + x(T0+ ) − x(T0 ) ≤ 1 + βT0 := KT0 +δ +∞ ,     for every t ∈ T0 , T0 + δR (T0 ) T and x ∈ A. Hence, T0 , T0 + δR (T0 ) T ⊂ C. Denote τ0 = sup C. As a consequence of the uniformly equi-regulatedness of A, there is a δL+∞ (τ0 ) > 0 such that x(t) − x(τ0− ) ≤ 1 for x ∈ A and t ∈   τ0 − δL+∞ (τ0 ), τ0 T .

136

2 A Classification of Closedness of Time Scales Under Translations

  Let τ ∈ C ∩ τ0 − δL+∞ (τ0 ), τ0 T . Then x(t) − x(T0 )  ≤ x(t) − x(τ0− ) + x(τ0− ) − x(τ ) +x(τ ) − x(T0 ) ≤ 1 + 1 + Kτ = 2 + Kτ , for every x ∈ A and t ∈ (τ, τ0 )T . Also, x(τ0− ) − x(T0 ) ≤ x(τ0− ) − x(τ ) + x(τ ) − x(T0 ) ≤ 1 + Kτ . These inequalities and hypothesis imply that x(τ0 ) − x(T0 ) ≤ x(τ0 ) − x(τ0− ) + x(τ0− ) − x(T0 ) ≤ βτ0 + 1 + Kτ .

(2.44)

Thus τ0 ∈ C, where Kτ0 = βτ0 + 1 + Kτ . If τ0 < T1 , then since A is uniformly equi-regulated, there is a δR+∞ (τ0 ) > 0 such that   x(t) − x(τ0+ ) ≤ 1, for any x ∈ A and t ∈ τ0 , τ0 + δR (τ0 ) T , which implies x(t) − x(T0 ) ≤ x(t) − x(τ0+ ) + x(τ0+ ) − x(τ0 ) + x(τ0 ) − x(T0 ) ≤ 1 + βτ0 + Kτ0 = Kτ0 +δ +∞ (τ0 ) , R

  for t ∈ τ0 , τ0 + δR+∞ (τ0 ) T and x ∈ A. Thus τ0 + δR+∞ (τ0 ) ∈ C which contradicts the fact that τ0 = sup C. Therefore, τ0 = T1 . Hence, by (2.44), we have x(T1 ) − x(T0 ) ≤ Kτ0 . Combining with (2.43), we have x(t) − x(T0 ) ≤ x(t) − x(T1 ) + x(T0 ) − x(T1 ) ≤ 1 + Kτ0 , for t ∈ (T1 , +∞)T . Then we can get the desired result.



Now,  we give some  sufficient conditions to guarantee that A is relatively compact in G [T0 , +∞)T , X .   Theorem 2.36 Let A ⊂ G [T0 , +∞)T , X be uniformly equi-regulated and uniformly Cauchy, for every t ∈ [T0 , +∞)T , and let the set {x(t); x ∈ A} be   relatively compact in X. Then the set A is relatively compact in G [T0 , +∞)T , X .

2.7 Some Compactness Criteria on Time Scales

137

Proof Since A is uniformly Cauchy, ∀x ∈ A, by (a), (b) from Definition 2.42, for any ε > 0, there exists T1 ∈ (T0 , +∞)T , there is a δ +∞ = (δL+∞ , δR+∞ )-fine partition P1 : t0P1 = T1 < t1P1 < t2P1 < . . . < tnP1 < . . . < . . . such that     x(tjP1 + ) − x(t ) < ε for t ∈ tjP1 , tjP1 + δR+∞ (tjP1 ) T ,

(2.45)

    x(tjP1 − ) − x(t ) < ε for t ∈ tjP1 − δL+∞ (tjP1 ), tjP1 T ,

(2.46)

    x(T1+ ) − x(t ) < ε for t ∈ T1 , T1 + δR+∞ (T1 ) T

(2.47)

for each j = 0, 1, 2, . . .. From (c) in Definition 2.42, we have ' '   'x T1 + δ +∞ (T1 ) − x(t)' < ε

(2.48)

R

 1  1 for t ∈ tjP−1 + δR+∞ (tjP−1 ), tjP1 − δL+∞ (tjP1 ) T and 



x(t ) − x(t ) < ε.

(2.49)

From (2.46) and (2.48), we have ' '  'x(T1 + δ +∞ (T1 ) − x(t P1 − )' j

R

'    ' ≤ 'x T1 + δR+∞ (T1 ) − x tjP1 − δL+∞ (tjP1 ) ' '  ' +'x(tjP1 − ) − x tjP1 − δL+∞ (tjP1 ) ' < 2ε

(2.50)

for each j = 0, 1, 2, . . .. Similarly, from (2.45), (2.47) and (2.49), we also have 



x(T1+ ) − x(tjP1 + ) ≤ x(T1+ ) − x(t ) + x(tjP1 + ) − x(t ) 



+x(t ) − x(t ) < 3ε

(2.51)

for each j = 0, 1, 2, . . .. Hence, from (2.48), we obtain ' '   'x T1 + δ +∞ (T1 ) − x(t)' < 4ε, R

 1  1 t ∈ tjP−1 + δR+∞ (tjP−1 ), tjP1 − δL+∞ (tjP1 ) T . From (2.46) and (2.50), we get

(2.52)

138

2 A Classification of Closedness of Time Scales Under Translations

' '   'x T1 + δ +∞ (T1 ) − x(t)' R ' '  = x(tjP1 − ) − x(t) + 'x(T1 + δR+∞ (T1 ) − x(tjP1 − )'   < 4ε, t ∈ tjP1 − δL+∞ (tjP1 ), tjP1 T .

(2.53)

Similarly, from (2.47) and (2.51), we also obtain ' ' P1 + 'x(t ) − x(t)' = x(T1+ ) − x(t) + x(T1+ ) − x(tjP1 + ) j   < 4ε, t ∈ tjP1 , tjP1 + δR+∞ (tjP1 ) T , so we obtain   x(T1+ ) − x(t) < 4ε, t ∈ tjP1 , tjP1 + δR+∞ (tjP1 ) T .

(2.54)

Further, since A is uniformly equi-regulated on [T0 , +∞)T , given ε > 0, there is a δ +∞ -fine partition P2 : P2 = T1 , t0P2 = T0 < t1P2 < . . . < tK

such that 

x(t ) − x(t) < ε 2 for every [t, t ]T ⊂ (tjP−1 , tjP2 )T , j ∈ {1, 2, . . . , K}. Obviously, P = P1 ∪ P2 is a δ +∞ -fine partition for [T0 , +∞)T . Let {xn ; n ∈ N} be a given sequence. By assumption, the set 



   xn (tjP2 ), xn T1 + δR+∞ (T1 ) , xn (T1+ ), n ∈ N

is relatively compact in X for every j = 0, 1, 2, . . . , K. Therefore, we can find a subsequence of indexes {nk ; k ∈ N} ⊂ {n; n ∈ N} such that the set     xnk (tjP2 ), xnk T1 + δR+∞ (T1 ) , xnk (T1+ ), k ∈ N is also relatively compact in X for every j = 0, 1, . . . , K. Using this fact, we can find the elements {yj ; j = 0, 1, 2, . . . , K, K + 1, K + 2} ⊂ X such that   yj = lim xnk (tjP2 ), yK+1 = lim xnk T1 + δR+∞ (T1 ) and yK+2 = lim xnk (T1+ ). k→∞

k→∞

Therefore, there exists N ∈ N such that for every k > N, we have xnk (tjP2 ) − yj  < Let q > k, and then

ε . 4

k→∞

2.7 Some Compactness Criteria on Time Scales

xnq (tjP2 ) − yj 
k. Then t = tjP2 for some j ∈ {0, 1, . . . , K} and, in this case, we have xnk (t) − xnq (t) ≤ xnk (tjP2 ) − yj  + xnq (tjP2 ) − yj 
0, such that for any finite mutually disjoint open interval (xi , yi )T ⊂ [T0 , +∞)T (i = 1, 2, . . . , n) satisfying n  (yi − xi ) < δ i=1

implies n  f (xi ) − f (yi ) < ε. i=1



 Lemma 2.14 Let A ⊂ G0 [T0 , +∞)T , Rn be uniformly bounded and   Z = t ∈ [T0 , +∞)T : x is not Δ-differentiable at t and μΔ (Z ) = 0 for all x ∈ A, i.e., x is Δ-differentiable at [T0 , +∞)T \Z , there exists M > 0 such that |x Δ (t)| ≤ M for all x ∈ A. Then A is uniformly equiregulated, and for any ε > 0, there exists T1 > 0 such that for any x ∈ A, the following is fulfilled: 

x (s)Δs ≤ ε. Δ

[T1 ,+∞)T \Z

Proof From the condition of the theorem, since x is Δ-differentiable at [T0 , +∞)T \Z and there exists M > 0 such that |x Δ (t)| ≤ M, according to Corollary 1.68 from [64], we can obtain for all t ∈ [T0 , +∞)T \Z , x(t) satisfies the Lipschitz condition x(t1 ) − x(t2 ) ≤ M|t1 − t2 |, ∀t1 , t2 ∈ [T0 , +∞)T \Z . So for each N0 ∈ Z+ , we can take δ = N0  j =1

|tj1 − tj2 |
for every x ∈ A and [t , t ]T ⊂ (tjP−1 . . . > N1 . Hence, we have 



Pi − + −  i x(t ) − x(tjP−1 ) ≤ x(tjPi ) − x(t ) j  + 3   i +|x(t ) − x(t )| + x(t ) − x(tjP−1 ) ≤ N , 2 i    i Pi    i , tj −1 + δR+∞ (tjP−1 ) T . Thus, where t ∈ tjPi − δL+∞ (tjPi ), tjPi T , t ∈ tjP−1 from (2.58), we obtain Ni Ni +∞  +∞  +∞    Pi − 1 Ni Pi + x(t = ) − x(t ) = < +∞, j j −1 2Ni 2Ni i=1 j =1

i=1 j =1

i=1

which implies that for any ε > 0, there exists i0 > 0 such that Ni +∞   Pi − + x(t ) − x(t Pi ) ≤ ε, j

i=i0 j =1

j −1

2.7 Some Compactness Criteria on Time Scales

147 Pi0

which implies that for any ε > 0, there exists T1 ≥ t0 P :=

+∞ 

and a partition:

Pi ,

i=i0 Pi0

Pi0

i.e., t0

= t0P < t1

tNPi +1
0 such that |x Δ (t)| ≤ M for all x ∈ A. Then there exists T1 > 0 such that A is uniformly Cauchy on [T1 , +∞)T . Proof According to Lemma 2.14, for any ε > 0, there exists T1 > 0 such that for any x ∈ A, the following is fulfilled: 

x (s)Δs ≤ ε. Δ

[T1 ,+∞)T \Z

Hence, for any t1 , t2 ∈ Z , t1 , t2 > T1 , we obtain  |x(t1 ) − x(t2 )| =

t2 t1

 x (s)Δs < Δ

x (s)Δs ≤ ε. Δ

[T1 ,+∞)T \Z

148

2 A Classification of Closedness of Time Scales Under Translations



This completes the proof. In what follows, we will give the following useful corollaries:   Corollary 2.13 Let A ⊂ G0 [T0 , +∞)T , Rn be uniformly bounded and   Z = t ∈ [T0 , +∞)T : x is not Δ-differentiable at t

and μΔ (Z ) = 0 for all x ∈ A, there exists M > 0 such that |x Δ (t)| ≤ M for all x ∈ A. Then A is relatively compact in G0 . Proof According to Theorem 2.41, A is uniformly equi-regulated, uniformly Cauchy. Further, since A is uniformly bounded, so it satisfies (2.42), by Theorem 2.37, we get the desired result immediately. 

Let  BC [T0 , +∞)T := x ∈ BC([T0 , +∞)T , Rn ) and

sup

 |x(t)| < ∞ ,

t∈[T0 ,+∞)T

where BC([T0 , +∞)T , Rn ) denotes the set of all bounded continuous functions on [T0 , +∞)T . Then we can obtain the following corollary: Corollary 2.14 Let A ⊂ BC [T0 , +∞)T be uniformly bounded and for all x ∈ A, x is Δ-differentiable and there exists M > 0 such that |x Δ (t)| < M. Then A is relatively compact in BC . Proof Since x is Δ-differentiable on [T0 , +∞)T , thus, A is equi-absolutely continuous on [T0 , +∞)T . Hence, for any t1 , t2 ∈ [T0 , +∞)T , we can obtain  |x(t1 ) − x(t2 )| =

t2

t1

which means that 

+∞ t1

x Δ (s)Δs ≤ 2

sup

|x(t)| < 2M0 ,

t∈[T0 ,+∞)T

x Δ (s)Δs < 2M0 , M0 is some constant. 



Thus, for any ε > 0, there exists T1 > 0, and we have t2 > t1 > T1 implies  t 2 Δ |x(t1 ) − x(t2 )| =  x (s)Δs < ε, 



t1

i.e., A is uniformly Cauchy. According to Theorem 2.37, A is relatively dense in BC . This completes the proof.



2.8 Analysis of General Delays on Translation Time Scales

149

Remark 2.58 Note that if for all x ∈ A ⊂ BC , x has uniformly bounded Δderivatives, then one can obtain that A is equi-absolutely continuous, which will lead to that A is uniformly Cauchy. Hence, the uniformly boundedness of A and the uniformly boundedness of Δ-derivatives functions of A can guarantee A is relatively compact.

2.8 Analysis of General Delays on Translation Time Scales Throughout this section we shall denote an arbitrary time scale by T=

+∞ 

[αi , βi ],

i=−∞

where [αi , βi ] is a continuous interval of R for all i ∈ Z (it is easy to see that if αi , βi are left and right-dense points for all i ∈ Z, then T = R, if αi , βi are isolated points for all i ∈ Z, then T is an isolated time scale).

2.8.1 Delay Systems on Time Scales with a Monotone Interval Length In what follows, we let hi = βi − αi , i ∈ Z. We begin with the following definition of time scales with a monotone interval length. Definition 2.44 ([216]) We say that T is a time scale with a monotone nondecreasing interval length if hi+1 ≥ hi , i ∈ Z and that T is a time scale with a monotone non-increasing interval length if hi+1 ≤ hi , i ∈ Z. If hi = hi+1 , i ∈ Z, then T is called a time scale with a constant interval length. We will construct general delays that satisfy t + τ (t) ∈ T for each of the above three classes of time scales. (1) The first class of time scales with hi+1 ≥ hi , i ∈ Z. Let a function F be defined as F :

+∞ 

[αi , βi ] −→

i=−∞

t Then, it follows that

+∞ 

{βi }

i=−∞

−→

βi ,

where t ∈ [αi , βi ].

150

2 A Classification of Closedness of Time Scales Under Translations

σ (βi ) = αi+1 , μ(βi ) = αi+1 − βi , i ∈ Z. Hence, for all t ∈ [αi , βi ), we have t + hi + μ(βi ) ∈ [σ (βi ), βi + hi + μ(βi )], i ∈ Z, and thus t + hi + μ(βi ) ∈ [σ (βi ), βi + hi+1 + μ(βi )] = [αi+1 , βi+1 ], i ∈ Z.

(2.59)

Further, for all t ∈ {βi }, i ∈ Z, we find t + μ(βi ) + hi ∈ [αi+1 , βi+1 ], i ∈ Z.

(2.60)

In view of (2.59) and (2.60), we can let τi (t) = hi + μ(βi ), t ∈ [αi , βi ], i ∈ Z, and hence it follows that t + τi (t) ∈ [αi+1 , βi+1 ] ⊂ T, i ∈ Z. Therefore, if t ∈ [αi1 , βi1 ], i1 ∈ Z, then t+

i 1 +n

τi (t) ∈ [αi1 +n+1 , βi1 +n+1 ], i1 ∈ Z, n ∈ N.

i=i1

i1 +n Further, note that τi (t) > 0 for all t ∈ [αi , βi ], i ∈ Z. We shall call i=i τi (t) 1 an advanced delay adapted for the time scales. We are now in the position to state the following lemma: Lemma 2.15 For a given time scale T with hi ≤ hi+1 , i ∈ Z, the delay on the time scale has the following general form: +∞     τ (t) = hi + μ F (t) , t ∈ [αi , βi ], i

i=−∞

where 1 +n    i hi + μ F (t) = τi (t), if F (t) ∈ {βi1 }, i1 ∈ Z, n ∈ N.

i

i=i1

2.8 Analysis of General Delays on Translation Time Scales

151

We say that τ (t) is an advanced delay adapted for the time scale if n < +∞, and the system   x Δ (t) = g t, x(t + τ (t)) ,

(2.61)

where g : T × Rn → Rn and t ∈ T is called an advanced delay system. Example 2.19 Let T=

 k∈Z

! k(k + 1) (k + 1)(k + 2) ka + b, ka + b , 2 2

where a, b > 0. Then

σ (t) =

⎧ ⎪ ⎨t,

 ka + k(k+1) 2 b, ka + k∈Z   ka + (k+1)(k+2) b if t ∈ 2

if t ∈

⎪ ⎩t + a,

(k+1)(k+2)  b , 2

k∈Z

and

μ(t) =

⎧ ⎪ ⎨0,

 ka + k(k+1) 2 b, ka + k∈Z   b . ka + (k+1)(k+2) if t ∈ 2

if

⎪ ⎩a,

t∈

(k+1)(k+2)  b , 2

k∈Z

Thus, this time scale T is with a strictly increasing interval length. Further, by Lemma 2.15, we find the delay on this time scale as: τ (t) =

i 1 +n i=i1

  (i1 + 1)(i1 + 2) b , i1 ∈ Z, n ∈ N. (ib + a), if F (t) ∈ i1 a + 2

Note that τ (t) is actually a variable delay because the delay is dependent on t. In fact, T is unsuitable for a constant delay because the general delay τ (t) is dependent on t. 

 Example 2.20 If T = [k(a + b), k(a + b) + b], where a = −b, then k∈Z

σ (t) =

⎧ ⎪ ⎨t,

if t ∈

⎪ ⎩t + a,

if t ∈



[k(a + b), k(a + b) + b),

k∈Z



k∈Z

and

{k(a + b) + b}

152

2 A Classification of Closedness of Time Scales Under Translations

μ(t) =

⎧ ⎪ ⎨0,

if t ∈

⎪ ⎩a,

if t ∈



[k(a + b), k(a + b) + b),

k∈Z



{k(a + b) + b}.

k∈Z

Thus, this time scale T is with a constant interval length. Further, by Lemma 2.15, we can obtain the delay on this time scale as: τ (t) =

i 1 +n

(b + a) = n(a + b), if F (t) ∈ {i1 (a + b) + b}, i1 ∈ Z, n ∈ N.

i=i1

Note that τ (t) is independent on t. Thus, τ (t) can be a constant delay. Now we let    Π := τ (t) : t ∈ [k(a + b), k(a + b) + b], ∀n ∈ N k∈Z

= {n(a + b) : n ∈ N, a = −b}. If we consider a function τ0 : T → Π , then τ0 (t) is a variable delay, and t ± τ0 (t) ∈ T. Therefore, this time scale is suitable for a constant as well as a variable delay. Further, because for any τ ∈ Π , we have t ± τ ∈ T for all t ∈ T and Π forms a time scale, T is also suitable for finite distributed delays, which are defined as: 

b1

x(t + θ )ΔΠ θ, where t ∈ T, θ ∈ [a1 , b1 ]Π .

(2.62)

a1

This integral is over the time scale Π . Note that Π is an infinite set; thus, T is also suitable for infinite distributed delays, which are given as: 

+∞

x(t + θ )ΔΠ θ, where t ∈ T, θ ∈ [a1 , +∞)Π .

(2.63)

a1



This integral is over the time scale Π .

Remark 2.59 It follows from the time scale considered in Example 2.20 that if b = 0, a = h, h > 0, then T = hZ. If b = 1, a = 0, then T = R. Hence, these two classical time scales satisfy Π = T. Thus, it follows that 

b1

a1



τ

x(t + θ )ΔΠ θ = 0

 x(t + θ )Δθ,

+∞ a1

 x(t + θ )ΔΠ θ =

+∞

x(t + θ )Δθ.

0

(2.64)

2.8 Analysis of General Delays on Translation Time Scales

If Π = T, then (2.64) will be invalid. From this example, it is clear that b θ )ΔΠ θ is more general than a11 x(t + θ )Δθ .

153

 b1 a1

x(t +

(2) The second class of time scales with hi+1 ≤ hi , i ∈ Z. We define a function G as follows G:

+∞ 

+∞ 

[αi , βi ] −→

i=−∞

t

{αi }

i=−∞

−→

αi ,

where t ∈ [αi , βi ].

Then, it follows that ρ(αi ) = βi−1 , ν(αi ) = αi − βi−1 , i ∈ Z. Hence, for all t ∈ (αi , βi ], we have t − hi − ν(αi ) ∈ [αi − hi − ν(αi ), βi − hi − ν(αi )], i ∈ Z, and thus t − hi − ν(αi ) ∈ [αi − hi−1 − ν(αi ), βi − hi − ν(αi )] = [αi−1 , βi−1 ], i ∈ Z. (2.65) For all t ∈ {αi }, i ∈ Z, one can easily show that t − ν(αi ) − hi ∈ [αi−1 , βi−1 ], i ∈ Z.

(2.66)

From (2.65) and (2.66), we can let τi (t) = −hi − ν(αi ), t ∈ [αi , βi ], i ∈ Z and thus it follows that t + τi (t) ∈ [αi−1 , βi−1 ] ⊂ T, i ∈ Z. Therefore, if t ∈ [αi1 , βi1 ], i1 ∈ Z, then t+

i1 

τi (t) ∈ [αi1 −n−1 , βi1 −n−1 ], i1 ∈ Z, n ∈ N.

i=i1 −n

1 Further, note that τi (t) < 0 for all t ∈ [αi , βi ], i ∈ Z. We shall call ii=i τi (t) 1 −n the backward delay adapted for the time scale. We can now state the following lemma:

154

2 A Classification of Closedness of Time Scales Under Translations

Lemma 2.16 For the time scale T with hi ≥ hi+1 , i ∈ Z, the delay of the time scale has the following general form: τ (t) = −

+∞     [αi , βi ], hi + ν G(t) , t ∈ i

i=−∞

where i1     hi + ν G(t) = τi (t), if G(t) ∈ {αi1 }, i1 ∈ Z, n ∈ N. i=i1 −n

i

We say that τ (t) is a backward delay adapted for the time scale if n < +∞, and (2.61) is called a backward delay system. Example 2.21 Let T=

 k∈Z



   ! 1 ka + 2 − k−1 b, ka + 2 − k b , 2 2 1

where a, b > 0. Then,

σ (t) =

⎧ ⎪ ⎨t, ⎪ ⎩t + a,

   1 ka + 2 − 2k−1 b, ka + 2 − k∈Z     ka + 2 − 21k b if t ∈

if t ∈



1 2k

  b ,

k∈Z

and

μ(t) =

⎧ ⎪ ⎨0, ⎪ ⎩a,

    1 ka + 2 − 2k−1 b, ka + 2 − k∈Z     ka + 2 − 21k b . if t ∈

if t ∈

1 2k

  b ,

k∈Z

Thus, this time scale T is with a monotone strictly decreasing interval length. Further, by Lemma 2.16, we can obtain the delay on this time scale as follows:       i1  b 1 τ (t) = + a , if G(t) ∈ i1 a + 2 − i b , i1 ∈ Z, n ∈ N. 2i 21 i=i1 −n

We note that τ (t) is actually a variable delay because the delay is dependent on t. Therefore, this T is unsuitable for a constant delay because the general delay τ (t) is dependent on t. 

2.8 Analysis of General Delays on Translation Time Scales

155

Example 2.22 Let a > 1, t > a, t ∈ T, and consider the following time scale Pa,| cos t| =

∞ 

[pm , a + pm ],

m=1

where pm = (m − 1)a +

m−1 

cos

k=1

 ka + | cos a| + | cos(2a + | cos a|)| + . . .  +| cos((k − 1)a + | cos a|)| . # $% & k terms

Then,  ρ(t) =

if t ∈ ∪∞ m=1 (pm , a + pm ],

t,

t − | cos t|, if t ∈ ∪∞ m=1 {pm }

and ν(t) =

 0,

if t ∈ ∪∞ m=1 (pm , a + pm ],

| cos t|, if t ∈ ∪∞ m=1 {pm }.

Thus, this time scale T is with a monotone constant interval length, and by Lemma 2.16, we can obtain the delay on this time scale as follows: τ (t) = −

i1      a + cos G(t) , if G(t) ∈ {pi1 }, i1 ∈ Z+ , n ∈ N, t > a. i=i1 −n

Note that τ (t) is strictly dependent on t, and hence τ (t) is a variable delay.



Example 2.23 Consider the time scale in Example 2.20 and follow the same steps, then we obtain the delay on this time scale as: τ (t) = −

i1 

(b + a) = −n(a + b), if G(t) ∈ {i1 (a + b)}, i1 ∈ Z, n ∈ N.

i=i1 −n

Note that τ (t) is independent on t, and hence τ (t) can be taken as a constant delay. Now we let

156

2 A Classification of Closedness of Time Scales Under Translations

   Π := τ (t) : t ∈ [k(a + b), k(a + b) + b], ∀n ∈ N k∈Z

= {−n(a + b) : n ∈ N, a = −b}. If we consider a function τ0 : T → Π , then it is clear that τ0 (t) is a variable delay and t ± τ0 (t) ∈ T. Thus, this time scale is suitable for a constant as well as variable delays. Further, because for any τ ∈ Π , we have t ± τ ∈ T for all t ∈ T and Π forms a time scale, T is also suitable for finite distributed delays, which are defined as (2.62). Note that Π is an infinite set; thus, T is also suitable for infinite distributed delays, which are given as (2.63). 

Remark 2.60 It follows from the time scale considered in Example 2.20 that if b = 0, a = h, h > 0, then T = hZ. If b = 1, a = 0, then T = R. Hence, these two classical time scales satisfy Π = T. Thus, (2.64) also holds. If Π = T, then (2.64) b will be invalid. From this example, it is clear that a11 x(t + θ )ΔΠ θ is more general b than a11 x(t + θ )Δθ . (3) The third class of time scales with hi+1 = hi , i ∈ Z. For this case, we follow the same arguments as in cases (1) and (2), and obtain the following lemma: Lemma 2.17 For the time scale T with hi = hi+1 , i ∈ Z, the delays of the time scales have the following two general forms: +∞     τ1 (t) = hi + μ F (t) , t ∈ [αi , βi ], i

i=−∞

where 1 +n    i hi + μ F (t) = τi (t), if F (t) ∈ {βi1 }, i1 ∈ Z, n ∈ N

i

i=i1

and τ2 (t) = −

+∞     hi + ν G(t) , t ∈ [αi , βi ], i

i=−∞

where i1     hi + ν G(t) = τi (t), if G(t) ∈ {αi1 }, i1 ∈ Z, n ∈ N. i

i=i1 −n

2.8 Analysis of General Delays on Translation Time Scales

157

We say that τ1 (t) is an advanced delay adapted for the time scales if n < +∞, and (2.61) is called an advanced delay system. Similarly, we say that τ2 (t) is a backward delay adapted for the time scales if n < +∞, and (2.61) is called a backward delay system. Example 2.24 Let a > 1, t > a, t ∈ T, and consider the following time scale ∞ 

Pa,cos t = where pm = (m − 1)a +

m−1 

sin

k=1

[pm , a + pm ],

m=1



ka + | sin a| + | sin(2a + | sin a|)| + . . .  +| sin((k − 1)a + | sin a|)| . # $% & k terms

Then,  ρ(t) =

if t ∈ ∪∞ m=1 (pm , a + pm ],

t,

t − | sin t|, if t ∈ ∪∞ m=1 {pm }

and ν(t) =

 0,

if t ∈ ∪∞ m=1 (pm , a + pm ],

| sin t|, if t ∈ ∪∞ m=1 {pm }.

Similarly, we obtain  σ (t) =

if t ∈ ∪∞ m=1 [pm , a + pm ),

t,

t + | sin t|, if t ∈ ∪∞ m=1 {a + pm }

and μ(t) =

 0,

if t ∈ ∪∞ m=1 [pm , a + pm ),

| sin t|, if t ∈ ∪∞ m=1 {a + pm }.

Thus, T is a time scale with a constant interval length, and by Lemma 2.17, we can find the delays on this time scale as: τ1 (t) =

i 1 +n i=i1

    a + sin F (t) , if F (t) ∈ {a + pi1 }, i1 ∈ Z+ , n ∈ N, t > a.

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2 A Classification of Closedness of Time Scales Under Translations

and τ2 (t) = −

i1      a + sin G(t) , if G(t) ∈ {pi1 }, i1 ∈ Z+ , n ∈ N, t > a. i=i1 −n

Note that τ1 (t), τ2 (t) are strictly dependent on t, and hence, τ1 (t), τ2 (t) are variable delays. 

Example 2.25 Let a > 1, t > a, t ∈ T, and consider the following time scale Pa,e−t =

∞ 

[pm , a + pm ],

m=1

where pm = (m − 1)a +

m−1 

 exp −

 ka + exp(−a) + exp

k=1

 + exp #

   − 2a + exp(−a) + . . .

!    . − (k − 1)a + exp(−a) $% & k terms

Then,  σ (t) =

if t ∈ ∪∞ m=1 [pm , a + pm ),

t,

t + e−t , if t ∈ ∪∞ m=1 {a + pm }

and μ(t) =

 0, e−t ,

if t ∈ ∪∞ m=1 [pm , a + pm ), if t ∈ ∪∞ m=1 {a + pm }.

Similarly, we obtain  ρ(t) =

if t ∈ ∪∞ m=1 (pm , a + pm ],

t,

t − e−t , if t ∈ ∪∞ m=1 {pm }

and  ν(t) =

0,

if t ∈ ∪∞ m=1 (pm , a + pm ],

e−t , if t ∈ ∪∞ m=1 {pm }.

2.8 Analysis of General Delays on Translation Time Scales

159

Thus, T is a time scale with a constant interval length, and by Lemma 2.17, we can find the delays on this time scale as: τ1 (t) =

i 1 +n

  a + e−(F (t)) , if F (t) ∈ {a + pi1 }, i1 ∈ Z+ , n ∈ N, t > a

i=i1

and τ2 (t) = −

i1    a + e−(G(t)) , if G(t) ∈ {pi1 }, i1 ∈ Z+ , n ∈ N, t > a. i=i1 −n

Note that τ1 (t), τ2 (t) are strictly dependent on t, and hence, τ1 (t), τ2 (t) are variable delays. 

Example 2.26 If we consider the time scale in Example 2.20, then we can obtain the delays on this time scale as: τ1 (t) =

i 1 +n

(b + a) = n(a + b), if F (t) ∈ {i1 (a + b) + b}, i1 ∈ Z, n ∈ N.

i=i1

and τ2 (t) = −

i1 

(b + a) = −n(a + b), if G(t) ∈ {i1 (a + b)}, i1 ∈ Z, n ∈ N.

i=i1 −n

Note that τ1 (t), τ2 (t) are independent on t, and thus, can be taken as constant delays. Denote    Π1 := τ1 (t) : t ∈ [k(a + b), k(a + b) + b], ∀n ∈ N k∈Z

and    [k(a + b), k(a + b) + b], ∀n ∈ N . Π2 := τ2 (t) : t ∈ k∈Z

If we consider τ01 : T → Π1 , then τ01 (t) is a variable delay. Thus, such a time scale is suitable for both constant and variable delays. Similarly, if we consider τ02 : T → Π2 , then τ02 (t) is a variable delay. Further, because for any τ ∈ Π := Π1 ∪ Π2 , we have t ± τ ∈ T for all t ∈ T, and Π1 , Π2 form time scales, it follows that Π is also a time scale. Thus, T is also suitable for finite distributed delays, which are described as:

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2 A Classification of Closedness of Time Scales Under Translations



b1

x(t ± θ )ΔΠ θ, where t ∈ T, θ ∈ [a1 , b1 ]Π ;

(2.67)

a1

here, the integral is over the time scale Π . We also note that Π is an infinite set, and thus T is also suitable for infinite distributed delays, which are described as: 

+∞

x(t ± θ )ΔΠ θ, where t ∈ T, θ ∈ [a1 , +∞)Π ;

(2.68)

a1

here, the integral is over the time scale Π .



In view of the above analysis, we can now state and prove the following theorem: Theorem 2.42 Let T be a time scale with a monotone interval length. If the general delay τ (t) is dependent on t, then the time scale is only suitable for variable delays, not constant delays. If the general delay τ (t) is independent on t, then the time scale is not only suitable for constant and variable delays, but also for distributed delays. Further, for the latter case, if we denote Π := {τ (t) : t ∈ T}, where τ (t) is a general delay on T, then the general distributed delay system can be written as:   x Δ (t) = g t,

b1

 x(t ± θ )ΔΠ θ , θ ∈ [a1 , b1 ]Π , t ∈ T.

(2.69)

a1

Proof From Lemmas 2.15, 2.16, and 2.17 it follows that if the general delay τ (t) is dependent on t, then there is no constant τ such that t ± τ ∈ T. Hence, in this case the time scale is only suitable for variable delays. Similarly, if the general delay τ (t) is independent of t, then the set Π is independent of t, and for any τ ∈ Π , t ± τ ∈ T, and thus we can take any constant τ ∈ Π as a constant delay, or make a function τ : T → Π into a variable delay, or take a distributed delay as (2.67). This completes the proof. 

The following result is also an immediate consequence of Lemmas 2.15, 2.16, and 2.17. Theorem 2.43 Let T be a time scale with a monotone interval length. If τ (t) denotes the general delay in Lemmas 2.15, 2.16, and 2.17, then the general delay system that belongs to the corresponding time scale categories can be written as (2.61).

2.8 Analysis of General Delays on Translation Time Scales

161

2.8.2 Delay Systems on Periodic Time Scales In the following, we present some examples of time scales and their corresponding sets Π which will be used in our discussion of time delays. Example 2.27 The following time scales are periodic:  (a) T = +∞ i=−∞ [2(i − 1)h, 2ih], h > 0. (b) T = hZ. (c) T = R. (d) T = {t = k − q n : k ∈ Z, n ∈ N0 }, where 0 < q < 1 has period p = 1. It is well known that (a), (b), (c), (d) are four representative classes of periodic time scales. For (a), one can show that Π = {2nh : n ∈ Z, h > 0}. Clearly, for all t ∈ T and τ ∈ Π , we have t ± τ ∈ T. Thus, we can find a function τ0 : T → Π , which in view of the property of the set Π satisfies t ± τ0 (t) ∈ T. This τ0 (t) can be a variable delay for this time scale. Now we will investigate (b). If T = hZ, then the set Π is Π = {nh : n ∈ Z, h > 0}. For all t ∈ T and τ ∈ Π , we have t ± τ ∈ T. We also note that T = Π . From the property of Π , it follows that T is closed with respect to addition, i.e., ∀t1 , t2 ∈ T, one has t1 + t2 ∈ T; and T is symmetric, i.e., ∀t ∈ T, −t ∈ T. Thus, we can find a function τ0 : T → T such that t ± τ0 (t) ∈ T. Hence, τ0 (t) can be a variable delay for this time scale. Next, we shall consider (c). If T = R, then Π = R. Hence, we can repeat the same discussion as for (b) and obtain the same conclusion. Finally, for (d), we have Π = {n : n ∈ Z}, and hence the conclusion is obvious. 

Example 2.28 For the time scale T considered in Example 2.20, we find the set Π as: Π = {n(a + b) : n ∈ Z}. Note that for this time scale, if b = 0, a = h, h > 0, then T = hZ. If b = 1, a = 0, then T = R. Therefore, the cases (b) and (c) in Example 2.27 can be unified by this example. Further, we can find a function τ0 : T → Π such that t ± τ0 (t) ∈ T. Hence, τ0 (t) can be a variable delay for this time scale. 

Remark 2.61 The periodic time scales are suitable for constant, variable, and distributed delays because of the property of the set Π . If T is a periodic time scale, then the following type delay systems will always be meaningful:

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2 A Classification of Closedness of Time Scales Under Translations

  x Δ (t) = g t, x(t ± τ0 (t)) , where g : T × Rn → Rn , τ0 : T → Π. Remark 2.62 For any a1 , b1 ∈ Π , one can always consider the finite distributed delay as (2.67). Because Π is an infinite set, we can also choose an infinite distributed delay, which is given as (2.68). Thus, on periodic time scales, the distributed delay systems (2.69) will always be relevant. Theorem 2.44 Let T be a periodic time scale. Then, T is suitable for constant, variable, as well as distributed delays. Moreover, the general distributed delay systems can be written as (2.69). Proof From the Definition of periodic time scales, it is clear that we can choose any τ ∈ Π as a constant delay, construct a function τ : T → Π which is a variable delay, or consider the distributed delay as (2.67). This completes the proof. 

The following result is an immediate consequence of Lemma 2.17. Theorem 2.45 Let T be a periodic time scale and τ (t) be the general delay in Lemma 2.17. Then, the general delay system that belong to the periodic time scales can be written as:   x Δ (t) = g t, x(t ± τ (t)) , where τ : T → Π .

2.8.3 Delay Systems on Almost Periodic Time Scales In what follows we shall use the following notations: En denotes Rn or Cn , D denotes an open set in En or D = En , and S denotes an arbitrary compact subset of D. On combining the idea that the functions approximate each other along with the approximation of their respective time scales (i.e., the respective domains of the functions), the concepts of almost periodic functions can be defined as: Definition 2.45 ([216]) Let T be an almost periodic time scale, i.e., T satisfies Definition 2.12. A function f ∈ Crd (T × D, En ) is called an almost periodic function in t ∈ T uniformly for x ∈ D if the ε2 -translation set of f E{ε2 , f, S} = {τ ∈ Πε1 : |f (t+τ, x)−f (t, x)| < ε2 , f or all (t, x) ∈ (T ∩ T−τ ) × S} is a relatively dense set for all ε2 > ε1 > 0 with d(T, T−τ ) < ε1 and for each compact subset S of D; that is, for any given ε2 > ε1 > 0 with d(T, T−τ ) < ε1 and each compact subset S of D, there exists a constant l(ε2 , S) > 0 such that each interval of length l(ε2 , S) contains a τ (ε2 , S) ∈ E{ε2 , f, S} such that

2.8 Analysis of General Delays on Translation Time Scales

|f (t + τ, x) − f (t, x)| < ε2 ,

163

for all (t, x) ∈ (T ∩ T−τ ) × S.

This τ is called the ε2 -translation number of f and l(ε2 , S) is called the inclusion length of E{ε2 , f, S}. Remark 2.63 The positive numbers ε1 in Definition 2.12 should be sufficiently small. Obviously, if ε1 = 0, then Definition 2.45 is equivalent to the definition of almost periodic functions on periodic time scales. It is worth noting that because ε2 > ε1 > 0, if ε1 > 0 is an arbitrarily small positive number, then ε2 can be an arbitrarily small positive number, i.e., the approximation of the translation of time scales is the prerequisite that guarantees the approximation of the translation of functions defined on time scales. Remark 2.64 By Definition 2.12, if E{T, ε1 } = {τ ∈ R : d(T−τ , T) < ε1 } then by Definition 2.45, one can easily obtain E{ε2 , f, S} ⊂ E{ε1 , T}. Particularly, if T is a periodic time scale with a period p, then there must exist n0 ∈ Z such that n0 p happens to be the ε-almost period of the function f . Remark 2.65 Let   Πε = E{T, ε}, TΠε = T−τ : −τ ∈ E{T, ε} . Then, Definition 2.45 can be stated as follows: Definition 2.46 ([212]) Let T be an almost periodic time scale. A function f ∈ Crd (T × D, En ) is called an almost periodic function in t ∈ T uniformly for x ∈ D if the ε2 -translation set of f E{ε2 , f, S} = {τ ∈ Πε1 : |f (t + τ, x) − f (t, x)| < ε2 , f or all (t, x) ∈ (T ∩ (∪−τ TΠε1 )) × S} is a relatively dense set for all ε2 > ε1 > 0 and for each compact subset S of D; that is, for any given ε2 > ε1 > 0 and each compact subset S of D, there exists a constant l(ε2 , S) > 0 such that each interval of length l(ε2 , S) contains a τ (ε2 , S) ∈ E{ε2 , f, S} such that |f (t + τ, x) − f (t, x)| < ε2 ,

for all (t, x) ∈ (T ∩ (∪−τ TΠε1 )) × S.

Here τ is called the ε2 -translation number of f and l(ε2 , S) is called the inclusion length of E{ε2 , f, S}. For convenience, we shall denote ∪−τ TΠε := ∪TΠε .

164

2 A Classification of Closedness of Time Scales Under Translations

Definition 2.47 ([216]) Assume that T is an almost periodic time scale. Let f ∈   Crd (T × D, En ); if for any given sequence α , there exists a subsequence α ⊂ α such that the limit set T0 of {T−αn } exists and Tα f (t, x) exists uniformly on T0 × S, then f (t, x) is called an almost periodic function in t uniformly for x ∈ D. Theorem 2.46 Let T be an almost periodic time scale. If g : T → Rn is almost periodic and τ : T → Πε is also almost periodic, then g t − τ (t) is an almost periodic function on T. Proof Because τ : T → Πε is almost periodic, for any fixed t ∈ T it follows that τ (t) ∈ E{T, ε}, i.e., for any ε > 0 it follows that d(T, Tτ (t) ) < ε for any fixed   t ∈ T. Moreover, for any given ε > ε > 0 there exists l(ε ) such that each interval  of length l(ε ) contains a τ ∗ ∈ Πε such that 

|τ (t + τ ∗ ) − τ (t)| < ε , t ∈ T ∩ (∪TΠε ). 

Hence, by the continuity of the function g, it follows that there exists a δ < ε such that      g t + τ ∗ − τ (t + τ ∗ ) − g t + τ ∗ − τ (t) < ε . 2

(2.70)

Further, since t − τ (t) ∈ T and t − τ (t) ∈ ∪TΠε it follows that t − τ (t) ∈ T ∩ (∪TΠε ). Because g is almost periodic, in view of Definition 2.46, we have      g t + τ ∗ − τ (t) − g t − τ (t) < ε , t ∈ T ∩ (∪TΠε ). 2

(2.71)

Now from (2.70) and (2.71), we find         g t + τ ∗ − τ (t + τ ∗ ) − g t − τ (t) = g t + τ ∗ − τ (t + τ ∗ ) − g t + τ ∗ − τ (t)     + g t + τ ∗ − τ (t) − g t − τ (t) 


1, and consider the following time scale: Pa,| sin t+sin √5t| =

∞  m=1

[pm , a + pm ],

2.8 Analysis of General Delays on Translation Time Scales

165

where pm = (m − 1)a +

m−1 

sin

 ka + | sin a + sin

k=1

+ sin

√

√

5a| + | sin(2a + | sin a + sin

5(2a + | sin a + sin

√

5a|)

√ 5a|)|

√ 5a|)  √ √ + sin 5((k − 1)a + | sin a + sin 5a|)| # $% & + . . . + | sin((k − 1)a + | sin a + sin

k terms

 √ √ + sin 5 ka + | sin a + sin 2a| √ +| sin(2a + | sin a + sin 5a|) √ √ + sin 5(2a + | sin a + sin 5a|)|

√ + . . . + | sin((k − 1)a + | sin a + sin 5a|)  √ √ + sin 5((k − 1)a + | sin a + sin 5a|)| . # $% & k terms

Then, it follows that σ (t) =

 t,

if t ∈ ∪∞ m=1 [pm , a + pm ), √ t + | sin t + sin 5t|, if t ∈ ∪∞ m=1 {a + pm }

and  μ(t) =

if t ∈ ∪∞ m=1 [pm , a + pm ), √ | sin t + sin 5t|, if t ∈ ∪∞ m=1 {a + pm }.

0,

Clearly, T is an almost periodic time scale because μ is an almost periodic function. However, there does not exist a period p such that t ± p ∈ T for all t ∈ T. From Lemma 2.17, we can obtain the general delay on this time scale as: τ1 (t) =

i 1 +n

√       a + sin F (t) + sin 5 F (t) ,

i=i1

if F (t) ∈ {a + pi1 }, i1 ∈ Z+ , n ∈ N, t > a, and

166

2 A Classification of Closedness of Time Scales Under Translations i1  √       a + sin G(t) + sin 5 G(t) ,

τ2 (t) = −

i=i1 −n

if G(t) ∈ {pi1 }, i1 ∈ Z+ , n ∈ N, t > a. We note that τ1 (t), τ2 (t) are strictly 

dependent on t. Thus, τ1 (t), τ2 (t) are variable delays. We will now discuss distributed delays on almost periodic time scales. Theorem 2.47 Let T be an almost periodic time scale and x : T → Rn be an almost periodic function. Then, the integral 

b a

x(t + θ )ΔΠε θ, θ ∈ [a, b]Πε

is almost periodic on T, where the integral is over the time scale Πε . 

Proof Since x : T → Πε is an almost periodic function, for any given ε > ε > 0,   there exists l(ε ) such that each interval of length l(ε ) contains a τ ∗ ∈ Πε and |x(t + τ ∗ ) − x(t)|
0, the δ-neighborhood U (t0 , δ) of a given point t0 ∈ T is the set of all points t ∈ T such that d(t, t0 ) < δ, i.e.,   U (t0 , δ) = t ∈ T : d(t, t0 ) < δ . From Definition 2.2 we have the following theorem. Theorem 3.1 Let T be a periodic time scale defined in Definition 2.2. If T = R, then Π = R; if T = R, then Π is an isolated periodic time scale. For convenience, let Ai = [ai , bi ]T and Ait be the sub-timescale which the argument t belongs to, obviously, it ∈ Z+ , we introduce the following definitions.  Definition 3.1 Let T and Π be time scales, where T = i∈Z+ Ai and Π is defined in Definition 2.2. Then we say Π is an adjoint set of T if there exists a bijective mapping: F :

T



A ∈ Ai , i ∈ Z

 +

→



Π

 → B ∈ τ ∈ R : t + τ ∈ T, ∀t ∈ T ,

i.e., F (A) = B. Now F is called the adjoint mapping between T and Π . Definition 3.2 Suppose the adjoint mapping F : T → Π is continuous and fulfills:   (1) for any τ ∈ Π, t0 ∈ T, F (Ait0 +τ ) = F Ait0 + τ = F (Ait0 ) + τ holds; (2) if t1 , t2 ∈ T and t1 ≤ t2 , then F (Ait1 ) ≤ F (Ait2 ). We say (T, F, Π ) a regular matched space for the time scale T under translations. Lemma 3.1 If the time scale T is periodic in the sense of Definition 2.2 and (T, F, Π ) is a regular matched space, then for any fixed point t0 ∈ T, there exists a suitable adjoint mapping Fˆ : T → Π such that Fˆ (Ait0 ) = 0. Proof Since the time scale T is periodic in the sense of Definition 2.2, then 0 is also the identity element in (Π, +). From Definition 2.2, there exists an inverse element −F (Ait0 ) ∈ Π \{0} such that   F (Ait0 ) + − F (Ait0 ) = 0, so there exists a suitable constant b ∈ Π \{0} such that 

F (Ait ) + b = Fˆ (Ait ) = 0. This completes the proof. 0

0

3.1 Almost Periodic Functions

171

Remark 3.1 From condition (2) in Definition 3.2, if F (Ait0 ) = 0 for a fixed t0 ∈ T, then it is easy to obtain that F (Ait ) ≤ 0 for t ≤ t0 and F (Ait ) ≥ 0 for t ≥ t0 . For convenience, we always assume that (T, F, Π ) is a regular matched space for a periodic time scale T under translations. By Definition 3.2, the following lemma is immediate. Lemma 3.2  Let T be periodic  and τ ∈ [0, +∞)  Π . If t0 ∈ [t,t + τ ]T , then F (Ait0 ) ∈ F (Ait ), F (Ait+τ ) Π . If F (Ait0 ) ∈ F (Ait ), F (Ait+τ ) Π , then there exists some t0∗ ∈ Ait such that t0 ∈ [t0∗ , t0∗ + τ ]T . Proof From condition (2) in Definition 3.2, we have   F (Ait0 ) ∈ F (Ait ), F (Ait+τ ) Π .   If F (Ait0 ) ∈ F (Ait ), F (Ait+τ ) Π , then we obtain min Ait ≤ t0 ≤ max Ait+τ = max Ait + τ, i.e., t0 − τ ∈ Ait . Note that for any t0∗ ∈ Ait , we have t0∗ − τ ≤ min Ait , so there is some t0∗ ∈ Ait such that t0∗ − τ ≤ t0 − τ ≤ t0∗ , i.e., t0 ∈ [t0∗ , t0∗ + τ ]T . The proof is completed.



  ∗ From Lemma 3.2, we obtain that if F (Ait0 ) ∈ F (Ait ), F (Ait+τ ) Π , then t0 is not unique. For convenience of discussion in the future, we give the following remark.   Remark 3.2 We stipulate that F (Ait0 ) ∈ F (Ait ), F (Ait+τ ) Π is written only when t0 ∈ [t, t + τ ]T .

We introduce some notations, En denotes Rn or Cn , D denotes an open set in En or D = En , S denotes an arbitrary compact subset of D. In the sequel, we will describe some types of continuity of functions on time scales, which will be important for studying the properties of almost periodic functions. Definition 3.3 (Continuous Functions) f : X → En is called continuous at t0 ∈ X ⊆ T if for any ε > 0, there exists U (t0 , δ) such that for any s ∈ U (t0 , δ), |f (s) − f (t0 )| < ε. f is called continuous on X provided that it is continuous for every t ∈ X. Definition 3.4 (Uniformly Continuous Functions) f : X → En is called uniformly continuous on X ⊆ T, if for any ε > 0, there exists δ(ε) such that for any t1 , t2 ∈ X with |t1 − t2 | < δ(ε) implies

172

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

|f (t1 ) − f (t2 )| < ε. Definition 3.5 (Regulated Functions) f : X → En is called regulated if given any ε > 0, for any right dense point t0 ∈ X ⊆ T, there exist a constant a1 and δ > 0 such that t ∈ (t0 , t0 + δ)T implies |f (t) − a1 | < ε; for any left dense point t0 ∈ X ⊆ T, there exist a constant a2 and δ > 0 such that t ∈ (t0 − δ, t0 )T implies |f (t) − a2 | < ε. Definition 3.6 (Rd-Continuous Functions) Let a be a constant. Now f : X → En is called rd-continuous if given any ε > 0, for any right dense point t0 ∈ X ⊆ T, there exists δ > 0 such that t ∈ (t0 , t0 + δ)T implies |f (t) − f (t0 )| < ε; for any left dense point t0 ∈ X ⊆ T, there exists δ > 0 such that t ∈ (t0 − δ, t0 )T implies 

|f (t) − a | < ε, 

where a is some constant. Proposition 3.1 For any right scattered point t0 ∈ X ⊆ T and a function f : X → En , there exists δ > 0 such that t ∈ [t0 , t0 + δ)T implies |f (t) − f (t0 )| < ε. Proof Since t is a right scattered point, then f is right-side continuous. The proof is complete. 

Definition 3.7 (Uniformly Rd-Continuous Functions) f : X → En is called uniformly rd-continuous if given any ε > 0, for any right dense points t1 , t2 ∈ X ⊆ T, there exists δ(ε) > 0 such that |t1 − t2 | < δ implies |f (t1 ) − f (t2 )| < ε; for any left dense point t0 ∈ X ⊆ T, there exists δ(ε) > 0 such that t ∈ (t0 − δ, t0 )T implies 

|f (t) − a | < ε,

3.1 Almost Periodic Functions

173



where a is some constant. Proposition 3.2 If f : X → En is uniformly rd-continuous. Then for any ε > 0 and any t0 ∈ T, there exists δ(ε) > 0 such that for any points t1 , t2 ∈ [t0 , t0 + δ)T , if |t1 − t2 | < δ then |f (t1 ) − f (t2 )| < ε. Proof If t0 ∈ T is a right dense point, there exists some sufficiently small δ > 0 such that all the point in [t0 , t0 + δ)T are right dense points, by Definition 3.7, for any points t1 , t2 ∈ [t0 , t0 + δ)T , if |t1 − t2 | < δ, then |f (t1 ) − f (t2 )| < ε. If t0 ∈ T is a right scattered point, then by Proposition 3.1 and Definition 3.7, we have t1 = t2 = t0 , so |f (t1 ) − f (t2 )| < ε. The proof is completed. 

Similar to the finite covering theorem in functional analysis [75], one can easily obtain that  Lemma 3.3 Let [a, b]T ⊂ T be a closed interval. If [a, b]T ⊆ (Gα ∩ T), where α∈I

I is an index set, and for every α ∈ I , Gα is an open set in R, then there exist n  α1 , α2 , . . . , αn ∈ I, such that [a, b]T ⊆ (Gαk ∩ T). k=1

We introduce some notations. The set of all rd-continuous functions is denoted by Crd and the set of all the bounded rd-continuous functions is denoted by BCrd . Lemma 3.4 Let f ∈ Crd ([a, b]T , En ). Then f is uniformly rd-continuous on [a, b]T . Proof Let x0 be an arbitrary right dense point in [a, b]T . Since f (x) is rdcontinuous at x0 , then for any ε > 0, there exists δ(ε, x0 ) > 0 such that 

|x − x0 |
0 such that each interval of length l(ε, S) contains a τ (ε, S) ∈ E{ε, f, S} such that |f (t + τ, x) − f (t, x)| < ε,

for all (t, x) ∈ T × S.

Here, τ is called the ε-translation number of f and l(ε, S) is called the inclusion length of E{ε, f, S}.

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177

Remark 3.3 In Definition 3.9, the ε-translation set E{ε, f, S} is relatively dense in Π , rather than in T since T ∩ Π may be an empty set (i.e., T ∩ Π = ∅). Example 3.1 Consider the following time scale where a, b > 0, Pa,b =

∞   (2k + 1)(a + b), (2k + 1)(a + b) + a]. k=−∞

Clearly,  σ (t) =

if t ∈

t, t + a + 2b,

∞

if t ∈



k=−∞ (2k + 1)(a + b), (2k + 1)(a   ∞ k=−∞ (2k + 1)(a + b) + a

 + b) + a ,

and  μ(t) =

0, a + 2b,

if t ∈

∞

if t ∈

k=−∞ [(2k

∞

+ 1)(a + b), (2k + 1)(a + b) + a),

k=−∞ {(2k

+ 1)(a + b) + a}.

Note that here Π = {2n(a + b), n ∈ Z}. However, 2na + (2n − 1)b < 2n(a + b) < (2n + 1)(a + b), where a, b > 0, n ∈ Z, which implies that 2n(a + b) ∈ T for all n ∈ Z, that is T ∩ Π = ∅.



Remark 3.4 In Example 3.1, let a = b = 1, we can obtain the time scale P1,1 =

∞   4k + 2, 4k + 3]. k=−∞

Clearly 4n ∈ P1,1 for n ∈ Z. Since Π = {4n, n ∈ Z}, we have T ∩ Π = ∅. 



Theorem 3.2 Let T be a periodic time scale defined in Definition 2.2, α = {αn } ⊂ Π be a number sequence and E{ε, f, S} be an ε-translation set. Then 



(i) there exists an ε-translation number sequence τ = {τn } ⊂ E{ε, f, S} such      that {αn − τn = γn } = γ with 0 ≤ γn ≤ l(ε, S), where l(ε) is the inclusion length; (ii) if there is no right scattered point in T, then there is a Cauchy sequence γ =  {γn } ⊂ γ ; (iii) if there exist right scattered points in T, then there exist subsequences α =   {αn } ⊂ α , τ = {τn } ⊂ τ and constant s ∈ Π such that αn − τn = γn ≡ s.

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3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

Proof 

(i) For any interval with length of l, there exists τn ∈ E{ε, f, S}, thus, we can   choose a proper interval with length of l such that 0 ≤ αn − τn ≤ l. According    to the definition of Π , we obtain that γn = αn − τn ∈ Π . (ii) If there is no right scattered point in T, then T = R. By (i), there exists a   subsequence γ = {γn } ⊂ γ = {γn } such that γn → s as n → ∞, where 0 ≤ s ≤ l. (iii) If there exist right scattered points in T, i.e., T = R, then by Theorem 3.1, Π is an isolated periodic time scale. Let ω > 0 be the smallest period of Π and    αn = nα  ω, τn = nτ  ω and γn = nγ  ω = (nα  − nτ  )ω, where nα  , nτ  ∈ Z, n n n n n n n nγ  ∈ Z+ . Then by (i), we have n

0 ≤ nγ  ω ≤ l(ε, S), n

which implies that there exists some subsequence γ = {γn ≡ s}, where s ∈ Π is some constant. The proof is completed. 

For convenience, we denote by A P(T) the set of all rd-continuous almost periodic functions on the time scale T and introduce some notations. Let α = {αn } and β = {βn } be two sequences. Then β ⊂ α means that β is a subsequence of α;  α + β = {αn + βn }; −α = {−αn }; and α and β are common subsequences of α    and β , respectively, which means that αn = αn(k) and βn = βn(k) for some given function n(k). We will introduce the translation operator T , Tα f (t, x) = g(t, x) means that g(t, x) = lim f (t + αn , x) and is written only when the limit exists. The mode n→+∞

of convergence, e.g. pointwise, uniform, etc., will be specified at each use of the symbol. Theorem 3.3 Let f ∈ Crd (T×D, En ) be almost periodic in t uniformly for x ∈ D. Then it is uniformly rd-continuous in t and bounded on T × S. Proof For a given ε ≤ 1 and some compact set S ⊂ D, there exists a constant l(ε, S) such that in any interval of length l(ε, S), there exists τ ∈ E{ε, f, S} such that |f (t + τ, x) − f (t, x)| < ε ≤ 1,

for all (t, x) ∈ T × S.

Because f ∈ Crd (T × D, Rn ), for any (t, x) ∈ ([t0 , t0 + l(ε, S)]T ) × S and t0 ∈ T, there exists an M > 0 such that |f (t, x)| < M. For any given t ∈ T, take τ ∈ E(ε, f, S) ∩ [F (Ait0 ) − F (Ait ), F (Ait0 ) − F (Ait ) + l(ε, S)]Π , then τ + F (Ait ) ∈ E(ε, f, S) ∩ [F (Ait0 ), F (Ait0 ) + l(ε, S)]Π ,

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179

i.e., F (Ait+τ ) ∈ E(ε, f, S) ∩ [F (Ait0 ), F (Ait0 +l )]Π . Hence, for x ∈ S, we obtain |f (t + τ, x)| < M

and

|f (t + τ, x) − f (t, x)| < 1.

Thus for all (t, x) ∈ T × S, we have |f (t, x)| < M + 1. Moreover, for any ε > 0, let l1 = l1 ( 3ε , S) be an inclusion length of E( 3ε , f, S). One can choose a proper point t0∗ ∈ T such that f (t, x) is uniformly rd-continuous on ([t0∗ , t0∗ + l1 ]T ) × S. Hence there exists a positive constant δ = δ( 3ε , S), for any right dense points t1 , t2 ∈ [t0∗ , t0∗ + l1 ]T and |t1 − t2 | < δ, |f (t1 , x) − f (t2 , x)|
0 so that |t1 − t2 | < δ, for x ∈ S, implies |f (t1 , x) − f (t2 , x)|
N implies integer N = N (ε0 , X) |g(tm , xm )| ≤ M +

ε0 as m → ∞, 3

which contradicts (3.5). Therefore, g(t, x) is rd-continuous in T uniformly for x ∈ D. Finally, for any compact set S ⊂ D and given ε > 0, one can select τ ∈ E{ε, f, S} such that for all (t, x) ∈ T × S, the following holds: |f (t + βn + τ, x) − f (t + βn , x)| < ε. Let n → +∞, for all (t, x) ∈ T × S, we have |g(t + τ, x) − g(t, x)| ≤ ε, which implies that E{ε, g, S} is relatively dense. Therefore, g(t, x) is almost periodic in t uniformly for x ∈ D. This completes the proof. 



Theorem 3.5 Let f ∈ Crd (T × D, En ). If for any sequence α ⊂ Π , there exists  α ⊂ α such that Tα f (t, x) exists uniformly on T×S, then f (t, x) is almost periodic in t uniformly for x ∈ D. Proof For contradiction, if this is not true, then there exist ε0 > 0 and S ⊂ D such that for any sufficiently large l > 0, there exists an interval with length of l so that there is no ε0 -translation numbers of f (t, x) in this interval, that is, every point in this interval does not belong to E{ε0 , f, S}.  Now, one can choose a number α1 ∈ Π and an interval (a1 , b1 ) with b1 − a1 >  2|α1 |, where a1 , b1 ∈ Π such that there is no ε0 -translation numbers of f (t, x) in    this interval. Next, one can select α2 = 12 (a1 + b1 ), obviously, α2 − α1 ∈ (a1 , b1 ),   then α2 −α1 ∈ E{ε0 , f, S}; similarly, one can find an interval (a2 , b2 ) with b2 −a2 >   2(|α1 | + |α2 |), where a2 , b2 ∈ Π such that there is no ε0 -translation numbers of

3.1 Almost Periodic Functions

183 

f (t, x) in this interval. Again, one can choose α3 = 12 (a2 + b2 ), it is easy to observe     that α3 − α2 , α3 − α1 ∈ E{ε0 , f, S}. Hence, one can repeat this process again and     again to find α4 , α5 , . . . , such that αi − αj ∈ E{ε0 , f, S}, i > j. So, for any i = j, i, j = 1, 2, . . . , without loss of generality, let i > j, for x ∈ S, we can obtain sup (t,x)∈T×S





|f (t + αi , x) − f (t + αj , x)| =

sup (t,x)∈T×S





|f (t + αi − αj , x) − f (t, x)| ≥ ε0 . 

Therefore, there is no uniformly convergent subsequence of {f (t + αn , x)} for (t, x) ∈ T×S, this is a contradiction. Thus, f (t, x) is almost periodic in t uniformly for x ∈ D. This completes the proof. 

From Theorems 3.4 and 3.5, we can obtain the following equivalent definition of uniformly almost periodic functions on periodic time scales. 

Definition 3.10 Let f ∈ Crd (T × D, En ), if for any given sequence α ⊂ Π, there  exists a subsequence α ⊂ α such that Tα f (t, x) exists uniformly on T × S, then f (t, x) is called an almost periodic function in t uniformly for x ∈ D. Theorem 3.6 If f ∈ Crd (T × D, En ) is almost periodic in t uniformly  for x ∈ D and ϕ(t) is almost periodic with {ϕ(t) : t ∈ T} ⊂ S, then f t, ϕ(t) is almost periodic in t. 



Proof For any given sequence α ⊂ Π, there exists α ⊂ α , ψ(t), g(t, x) such that Tα ϕ(t) = ψ(t) exists uniformly on T and Tα f (t, x) = g(t, x) exists uniformly on T × S, where ψ(t) is almost periodic and g(t, x) is almost periodic in t uniformly for x ∈ D. Therefore, g(t, x) is uniformly rd-continuous in T for x ∈ S. In addition, for any given ε > 0, there exists δ( 2ε ) > 0 such that for any x1 , x2 ∈ S and all t ∈ T, |x1 − x2 | < δ( 2ε ) implies |g(t, x1 ) − g(t, x2 )|
0 so that n ≥ N0 (ε) implies ε , ∀(t, x) ∈ T × S, 2   ε |ϕ(t + αn ) − ψ(t)| < δ , ∀t ∈ T, 2

|f (t + αn , x) − g(t, x)|
0 so that |t1 − t2 | < δ(ε1 , S) implies |f (t1 , x) − f (t2 , x)| < ε1 ,

∀x ∈ S,

where ε1 = 2ε and t1 , t2 ∈ [t0 , t0 + δ)T . Now, one can choose η∗ = δ( 2ε , S) = δ(ε1 , S) and L = l(ε1 , S) + 2η∗ , where l(ε1 , S) is the inclusion length of E(ε1 , f, S). For any a ∈ Π , consider an interval [a, a + L]Π , taking   τ ∈ E(f, ε1 , S) ∩ a − η∗ , a + η∗ + l(ε1 , S) Π ,

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3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

we obtain   τ − η∗ , τ + η∗ Π ⊂ [a, a + L]Π .   Hence, for all ξ ∈ τ − η∗ , τ + η∗ Π , we have |ξ − τ | ≤ η∗ . Therefore, for any (t, x) ∈ T × S, |f (t + ξ, x) − f (t, x)| ≤ |f (t + ξ, x) − f (t + τ, x)| +|f (t + τ, x) − f (t, x)| ≤ ε. Hence, we let α = τ − η∗ and η = 2η∗ , then [α, α + η]Π ⊂ E(ε, f, S). This completes the proof. 

Theorem 3.11 If f, g ∈ Crd (T × D, En ) are almost periodic in t uniformly for x ∈ D, then for any ε > 0, E(f, ε, S) ∩ E(g, ε, S) is a nonempty relatively dense set in Π . Proof For f, g are almost periodic in t uniformly for x ∈ D, they are uniformly rd-continuous   in T uniformly for x ∈ S. Hence, for any given ε > 0, one can take δi = δi 2ε , S (i = 1, 2); and l1 = l1 ( 2ε , S), l2 = l2 ( 2ε , S) as the inclusion lengths of E(f, 2ε , S), E(g, 2ε , S), respectively. From Definition 3.9 and Theorem 3.10, let η = η(ε, S) ∈ Π be the smallest positive period of the time scale Π (note that for T = R, we can choose η = η(ε, S) = min{δ1 , δ2 }), then we can choose Li = li + η (i = 1, 2), L = max(L1 , L2 ). Hence, one can select 2ε -translation numbers of f (t, x) and g(t, x): τ1 = mη and τ2 = nη, respectively, where τ1 , τ2 ∈ [a, a + L]Π , m, n are integers, and we have |τ1 − τ2 | ≤ L. Let m − n = s, then s can only be taken from a finite number set {s1 , s2 , . . . , sp }. When m − n = sj , j = 1, 2, . . . , p, denote the 2ε -translation numbers of f (t, x) j j j j and g(t, x) by τ1 , τ2 , respectively, i.e., τ1 − τ2 = sj η, j = 1, 2, . . . , p, and we j j choose T = max{|τ1 |, |τ2 |}. j

For any a ∈ Π , on the interval [a + T , a + T + L]Π , we can take 2ε -translation numbers of f (t, x) and g(t, x) as τ1 and τ2 , respectively, so there exists some integer sj such that j

j

τ1 − τ2 = sj η = τ1 − τ2 . Let j

j

τ (ε, S) = τ1 − τ1 = τ2 − τ2 ,

3.1 Almost Periodic Functions

187

then τ (ε, S) ∈ [a, a + L + 2T ]Π , and for any (t, x) ∈ T × S, we have j

j

|f (t + τ, x) − f (t, x)| ≤ |f (t + τ1 − τ1 , x) − f (t − τ1 , x)| j

+|f (t − τ1 , x) − f (t, x)| < ε and j

j

|g(t + τ, x) − g(t, x)| ≤ |g(t + τ2 − τ2 , x) − g(t − τ2 , x)| j

+|g(t − τ2 , x) − g(t, x)| < ε. Therefore, there exists at least a τ = τ (ε, S) on any interval [a, a + L + 2T ]Π with the length L+2T such that τ ∈ E(f, ε, S)∩E(g, ε, S). The proof is completed.  According to Definition 3.9, it easily follows that Theorem 3.12 If f ∈ Crd (T × D, En ) is almost periodic in t uniformly for x ∈ D, then for any α ∈ R, b ∈ Π , the functions αf (t, x), f (t + b, x) are almost periodic in t uniformly for x ∈ D. Theorem 3.13 If f, g ∈ Crd (T × D, En ) are almost periodic in t uniformly for x ∈ D, then f + g, f g are almost periodic in t uniformly for x ∈ D, if inft∈T |g(t, x)| > (t,x) 0, then fg(t,x) are almost periodic in t uniformly for x ∈ D. Proof According to Theorem 3.11, for any ε > 0, E(f, 2ε , S) ∩ E(g, 2ε , S) is a nonempty relatively dense set. It is easy to observe that if τ ∈ E(f, 2ε , S) ∩ E(g, 2ε , S), then τ ∈ E(f + g, ε, S). Hence   ε ε E(f, , S) ∩ E(g, , S) ⊂ E(f + g, ε, S). 2 2 Therefore, E(f + g, ε, S) is relatively dense in Π , which indicates that f + g is almost periodic in t uniformly for x ∈ D. On the other hand, let sup (t,x)∈T×S

|f (t, x)| = M1 ,

sup

|g(t, x)| = M2 ,

(t,x)∈T×S

and select τ ∈ E(f, 2ε , S) ∩ E(g, 2ε , S), then for all (t, x) ∈ T × S, one can obtain |f (t + τ, x)g(t + τ, x) − f (t, x)g(t, x)| ≤ |g(t + τ, x)||f (t + τ, x) − f (t, x)| +|f (t, x)||g(t + τ, x) − g(t, x)| ≤ (M1 + M2 )ε ≡ ε1 .

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3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

Therefore, τ ∈ E(fg, ε1 , S), i.e., E(fg, ε1 , S) is relatively dense in Π , which implies that f g is almost periodic in t uniformly for x ∈ D. Finally, let inf |g(t, x)| = N and take τ ∈ E(g, ε, S), then for all (t, x) ∈ (t,x)∈T×S

T × S, one has ε 1 1 g(t + τ, x) − g(t, x) = − g(t + τ, x) g(t, x) g(t + τ, x)g(t, x) < N 2 ≡ ε2 , that is, τ ∈ E( g1 , ε2 , S). Hence, g1 is almost periodic in t uniformly for x ∈ D, so fg is almost periodic in t uniformly for x ∈ D. The proof is completed.

 Theorem 3.14 If fn ∈ Crd (T × D, En ), n = 1, 2, . . . are almost periodic in t for x ∈ D, and the sequence {fn (t, x)} uniformly converges to f (t, x) on T × S, then f ∈ Crd (T × D, En ) is almost periodic in t uniformly for x ∈ D. Proof For any ε > 0, there exists sufficiently large n0 such that for all (t, x) ∈ T × S, |f (t, x) − fn0 (t, x)|
g :=

inf

(t,x)∈T×S

F (t, x),

3.1 Almost Periodic Functions

189

for any ε > 0, there exist t1 and t2 such that ε F (t1 , x) < g + , 6

ε F (t2 , x) > G − , 6

∀x ∈ S.

Let l = l(ε1 , S) be an inclusion length of E(f, ε1 , S). One can choose some n0 ∈ N such that n0 F (Ait1 ) − F (Ait2 ) := d ≥ |t1 − t2 | and let ε1 =

ε 6d .

For any t ∈ T, take

  τ ∈ E(f, ε1 , S) ∩ F (Ait ) − F (Ait1 ), F (Ait ) − F (Ait1 ) + l Π . Hence, we have   F (Ait1 ) + τ ∈ E(f, ε1 , S) ∩ F (Ait ), F (Ait ) + l Π and  F (Ait2 ) + τ ∈ E(f, ε1 , S)    F (Ait ) + F (Ait2 ) − F (Ait1 ), F (Ait ) + F (Ait2 ) − F (Ait1 ) + l Π ⊆ [F (Ait ) − L, F (Ait ) + L]Π , where L = l + d. Let si = ti + τ, (i = 1, 2), then s1 , s2 ∈ [t − L, t + L]T .

(3.6)

Moreover, for all x ∈ S, we obtain F (s2 , x) − F (s1 , x) = F (t2 , x) − F (t1 , x)  t2  − f (t, x)Δt + t1

t2 +τ

f (t, x)Δt

t1 +τ



= F (t2 , x) − F (t1 , x) +

t2

[f (t + τ, x) − f (t, x)]Δt

t1

> G−g−

ε ε − ε1 d = G − g − , 3 2

that is     ε F (s1 , x) − g + G − F (s2 , x) < . 2

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3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

Since F (s1 , x) − g ≥ 0,

G − F (s2 , x) ≥ 0,

in any interval with length L, then ε F (s1 , x) < g + , 2

ε F (s2 , x) > G − . 2

ε Now, let ε2 = 4L , we will prove that if τ ∈ E(f, ε2 , S), then τ ∈ E(F, ε, S). In fact, for any (t, x) ∈ T × S, by (3.6), one can take s1 , s2 ∈ [t, t + 2L]T such that

ε F (s1 , x) < g + , 2

ε F (s2 , x) > G − , 2

then for τ ∈ E(f, ε2 , S), we have F (t + τ, x) − F (t, x) = F (s1 + τ, x) − F (s1 , x)  s1  s1 +τ + f (t, x)Δt − f (t, x)Δt t



t+τ t1

ε [f (t + τ, x) − f (t, x)]Δt > g − (g + ) − 2 t ε > − − ε2 2L = −ε 2 and F (t + τ, x) − F (t, x) = F (s2 + τ, x) − F (s2 , x)  s2  s2 +τ + f (t, x)Δt − f (t, x)Δt t

t+τ

ε < + ε2 2L = ε. 2 That is, for τ ∈ E(f, ε2 , S), there is τ ∈ E(F, ε, S), thus, F (t, x) is almost periodic in t uniformly for x ∈ D. The proof is completed. 

Theorem 3.16 If f ∈ Crd (T × D, En ) is almost periodic in t uniformly for x ∈ D, F (·) is uniformly continuous on the value field of f (t, x), then F ◦ f ∈ Crd (T × D, En ) is almost periodic in t uniformly for x ∈ D. Proof In fact, since F is uniformly continuous on the value field of f (t, x), and f (t, x) is almost periodic in t uniformly for x ∈ D, there exists a real number sequence α = {αn } ⊆ Π such that     Tα (F ◦ f ) = lim F f (t + αn , x) = F lim f (t + αn , x) = F (Tα f ) n→+∞

n→+∞

3.1 Almost Periodic Functions

191

holds uniformly on T × S. Hence, F ◦ f is almost periodic in t uniformly for x ∈ D. The proof is completed.

 Theorem 3.17 A function f ∈ Crd (T × D, En ) is almost periodic in t uniformly   for x ∈ D if and only if for every pair of sequences α , β ⊆ Π , there exist common   subsequences α ⊂ α , β ⊂ β such that Tα+β f (t, x) = Tα Tβ f (t, x).

(3.7)

Proof If f (t, x) is almost periodic in t uniformly for x ∈ D, for any two sequences     α , β ⊆ Π , there exists subsequence β ⊂ β such that Tβ  f (t, x) = g(t, x) holds uniformly on T × S and g(t, x) is almost periodic in t uniformly for x ∈ D.       Take α ⊂ α so that α , β are the common subsequences of α , β , respectively,   then there exists α ⊂ α such that Tα  g(t, x) = h(t, x) holds uniformly on T × S.       Similarly, take β ⊂ β so that β , α are the common subsequences of β , α ,   respectively, then there exist common subsequence α ⊂ α , β ⊂ β such that Tα+β f (t, x) = k(t, x) holds uniformly on T × S. According to the above, it is easy to see that Tβ f (t, x) = g(t, x),

Tα g(t, x) = h(t, x)

hold uniformly on T × S. Thus, for all ε > 0, if n is sufficiently large, then for any (t, x) ∈ T × S, we have |f (t + αn + βn , x) − k(t, x)| < |g(t, x) − f (t + βn , x)|
0 and t0 ∈ T, and subsequences α ⊂        γ , β ⊂ γ , s ⊆ Π , where α = {αn }, β = {βn }, s = {sn }, such that 







|f (t0 + sn + αn , x) − f (t0 + sn + βn , x)| ≥ ε0 > 0. 





(3.8) 

According to (3.7), there exist common subsequences α ⊂ α , s ⊂ s such that for all (t, x) ∈ T × S, we have Ts  +α  f (t, x) = Ts  Tα  f (t, x). 









(3.9) 





Taking β ⊂ β so that β , α , s are common subsequences of β , α , s , respectively, such that for all (t, x) ∈ T × S, we have Ts  +β  f (t, x) = Ts  Tβ  f (t, x).

(3.10)



Similarly, taking α ⊂ α satisfying α, β, s are common subsequences of   α , β , s , respectively, according to (3.9), for all (t, x) ∈ T × S, we have 

Ts+α f (t, x) = Ts Tα f (t, x).

(3.11)

Since Tα f (t, x) = Tβ f (t, x) = Ts f (t, x), by (3.10) and (3.11), for all (t, x) ∈ T × S, one can obtain Ts+β f (t, x) = Ts+α f (t, x), that is, for all (t, x) ∈ T × S, it follows that lim f (t + sn + βn , x) = lim f (t + sn + αn , x).

n→+∞

n→+∞

(3.12)

In (3.12), let t = t0 , then it contradicts (3.8). Therefore, f (t, x) is almost periodic in t uniformly for x ∈ D. The proof is completed. 

Definition 3.12 Let mij (t, x) ∈ Crd (T × D, E)(1, 2, . . . , n; j = 1, 2, . . . , m) be in t uniformly for x ∈ D, then the matrix function M(t, x) = almost periodic  mij (t, x) n×m is called almost periodic in t uniformly for x ∈ D.

3.1 Almost Periodic Functions

193

If one adopts the matrix norm: |M(t, x)| =

( ij

m2ij (t, x), then the definition

above can be rewritten as Definition 3.13 A matrix function M ∈ Crd (T × D, En×m ) is almost periodic in t uniformly for x ∈ D if and only if for any ε > 0, the translation set E(M, ε, S) = {τ ∈ Π : |M(t + τ, x) − M(t, x)| < ε, ∀(t, x) ∈ T × S} is a relatively dense set in Π . Theorem 3.18 Definition 3.12 is equivalent to Definition 3.13. Proof In fact, if M(t, x) is almost periodic in t uniformly for x ∈ D, by Definition 3.12, every element mij (t, x) is almost periodic in t uniformly for x  ∈ D. Thus, forε any ε > 0, there exists nonempty relatively dense set Θ = E mij (t, x), √mn , S . For any τ ∈ Θ, we have i,j

|M(t + τ, x) − M(t, x)| =



!1/2 |mij (t + τ, x) − mij (t, x)|

2

< ε.

i,j

On the other hand, if for any ε > 0, E(M, ε, S) is a relatively dense set, then for any i = 1, 2, . . . , n; j = 1, 2, . . . , m and τ ∈ E(M, ε, S), we obtain |mij (t + τ, x) − mij (t, x)| < |M(t + τ, x) − M(t, x)| < ε, ∀(t, x) ∈ T × S. Hence, every element mij (t, x) is almost periodic in t uniformly for x ∈ D, that is, M(t, x) is almost periodic in t uniformly for x ∈ D. The proof is completed. 

Definition 3.14 A rd-continuous matrix function M(t, x) is called normal if for   any sequence α ⊆ Π , there exists subsequence α ⊂ α such that Tα M(t, x) exists uniformly on T × S. Theorem 3.19 A rd-continuous matrix function M(t, x) is normal if and only if M(t, x) is almost periodic in t uniformly for x ∈ D. Proof If M(t, x) is normal, then every element mij (t, x) satisfies Definition 3.10, so M(t, x) is almost periodic in t uniformly for x ∈ D. On the other hand, if M(t, x) is almost periodic in t uniformly for x ∈ D, by   Definition 3.12, for any sequence α ⊆ Π , there exists subsequence α1 ⊂ α such that Tα1 m11 (t, x) exists uniformly on T × S. Hence, there exists α2 ⊂ α1 , such that Tα2 m12 (t, x) exists uniformly on T × S; we can repeat this step m × n times, then one can get a series of subsequences satisfying: α = {αk } = αmn ⊂ αmn−1 ⊂ . . . ⊂ α2 ⊂ α1 ⊂ α



194

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

such that Tα mij (t, x),

i = 1, 2, . . . , n,

j = 1, 2, . . . , m 

exist uniformly on T × S. Therefore, there exists subsequence α ⊂ α such that Tα M(t, x) exists uniformly on T × S, that is, M(t, x) is normal. The proof is completed. 

3.2 Bohr-Transform and Mean-Value For more details of the integral-transforms on time scales, we refer the reader to the literatures Marks et al. [188], Thomas [208]. Let f (t, x) ∈ Crd (T × D, En ) be almost periodic in t uniformly for x ∈ D and π − h < λ ≤ πh . One can introduce the Bohr-transform: 1 T →+∞ T

a(f, λ, x, t0 ) = lim



t0 +T

∗

f (t, x)e−iλ(t−t ) Δt, T ∈ Π, ∀t0 ∈ T,

t0

(3.13) √ where i = −1 and fixed t ∗ ∈ T. Obviously, for a fixed (f, λ, x, t0 ), we have a(f, λ, x, t0 ) ∈ En . Definition 3.15 Assume that t0 ∈ T and T ∈ Π . We say a(f, 0, x, t0 ) is the meanvalue of f (t, x) if 1 0 < a(f, 0, x, t0 ) = lim T →∞ T



t0 +T

f (t, x)Δt < +∞.

t0

Theorem 3.20 Let λ ∈ [− πh , πh ]. Then a(f, λ, x, t0 ) defined by (3.13) exists uniformly for x ∈ S and it is uniformly continuous on S with respect to x. Proof For any η1 ∈ Π, η1 > 0, we can introduce a sequence {iη1 } = {ηi }i∈Z+ ⊂ t +η Π , next, we will demonstrate the sequence { η1i t00 i f (t, x)Δt}i∈Z+ converges uniformly with respect to x ∈ S. For any integer m, n and x ∈ S, let t0 ∈ T, η0 = 0 and take ηm , ηn ∈ Π , we have  1 η n

t0 +ηn

t0

1 f (t, x)Δt − ηm



t0 +ηm

t0

f (t, x)Δt

 t +ηn  t0 +ηmn 0 1 ηm ≤ f (t, x)Δt − f (t, x)Δt ηm ηn t0 t0  t +ηmn  t0 +ηm 0 1 + f (t, x)Δt − ηn f (t, x)Δt η m η n t0 t0

3.2 Bohr-Transform and Mean-Value

≤

195

 t0 +ηkn m  η1  t0 +ηn f (t, x)Δt − f (t, x)Δt ηm ηn t0 t0 +η(k−1)n k=1

n   +



t0 +ηkm

t0 +η(k−1)m

k=1

t0 +ηm

f (t, x)Δt −

t0

! f (t, x)Δt .

(3.14)

We consider the following integral form: 

t0 +ηa+s

t0 +ηa

 f (t, x)Δt −

t0 +ηs

f (t, x)Δt,

(3.15)

t0

where s = n, a = (k − 1)n, k = 1, 2, . . . , m or s = m, a = (k − 1)m, k = 1, 2, . . . , n. For arbitrary a, s, we can evaluate (3.15). Let G = sup |f (t, x)|. For any ε > 0, let l = l( 4ε , S) be the inclusion length (t,x)∈T×S

of E(f, 4ε , S), by taking 

   ε τ ∈ E f, , S ∩ ηa − F (Ait0 ), ηa − F (Ait0 ) + l Π , 4 i.e.,     ε F (Ait0 ) + τ ∈ E f, , S ∩ ηa , ηa + l Π . 4 Moreover, one can also get     ε ηs + τ ∈ E f, , S ∩ ηa+s − F (Ait0 ), ηa+s − F (Ait0 ) + l Π . 4 Let L = l + |F (Ait0 )|. Then for all x ∈ S, we get  t +ηa+s  t0 +ηs 0 f (t, x)Δt − f (t, x)Δt t0 +ηa t0   t +τ +ηs  t +ηs  t +ηa+s  t +τ  0 0 0 0 = f (t, x)Δt − + + t0 +τ



t0 +ηs

≤

t0

+ ≤

t0 +ηa

|f (t + τ, x) − f (t, x)|Δt

t0



t0 +τ +ηs

t0 +ηa+s t0 +τ +ηs

 |f (t, x)|Δt +

εηs + 2LG. 4

t0 +τ t0 +ηa

|f (t, x)|Δt (3.16)

196

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

According to (3.16), we can reduce (3.14) to the following:  1 η n

t0 +ηn

t0

f (t, x)Δt −



1 ηm

t0 +ηm t0

f (t, x)Δt

  ! εηm η1 εηn < + 2LG + n + 2LG m ηm ηn 4 4   ε 2LG 1 1 = + + → 0, n, m → ∞. 2 η1 m n 

According to the Cauchy convergence criterion, the sequence 

1 ηi





t0 +ηi

f (t, x)Δt t0

i∈N

converges uniformly with respect to x ∈ S. For any sufficiently large 0 < T ∈ Π , there exist 0 < ηn ∈ Π such that 0 < ηn < T ≤ ηn+1 , so for all x ∈ S, we have 

t0 +T

 f (t, x)Δt −

t0 +ηn

t0

t0

f (t, x)Δt ≤ G(T − ηn ) ≤ Gη1 .

Therefore,  t +T  t +T  t0 +ηn  t0 +ηn 0 1 0 1 < 1 f (t, x)Δt − f (t, x)Δt f (t, x)Δt − f (t, x)Δt T T ηn t0 t0 t0 t0    t0 +ηn   1 1 1 Gη1 1 2G − |f (t, x)|Δt ≤ − + → 0, n → +∞. ηn G < + ηn T T ηn T n t0

Hence, for all x ∈ S, 1 a(f, 0, x, t0 ) = lim n→+∞ ηn



t0 +ηn

f (t, x)Δt. t0

 t +T Besides, for T1 t00 f (t, x)Δt is continuous with respect to x and S is an arbitrary compact set in En , a(f, 0, x, t0 ) is uniformly continuous on S. ∗ It is easy to observe that f (t, x)e−iλ(t−t ) is almost periodic in t uniformly ∗ for x ∈ D. Since a(f, λ, x, t0 ) = a(f (t, x)e−iλ(t−t ) , 0, x, t0 ), it is obvious that a(f, λ, x, t0 ) exists uniformly for x ∈ S and it is uniformly continuous on S. This completes the proof. 

3.2 Bohr-Transform and Mean-Value

197

Theorem 3.21 Assume that T ∈ Π and f ∈ Crd (T × D, En ) is almost periodic in t uniformly for x ∈ D. Then 1 lim T →+∞ T



α+T

  ∗ f (t, x)e−iλ(t−t ) Δt := m f (t, x), λ, x, α

α

uniformly holds for all α ∈ T and for tα ∈ T if α − tα ∈ Π , then     m f (t, x), λ, x, α = a f (t + α − tα , x)e−iλ(α−tα ) , λ, x, tα .   ∗ Proof For m(f, λ, x, α) = m f (t, x)e−iλ(t−t ) , 0, x, α , it is sufficient to demonstrate for x ∈ S, ∀α ∈ T, the following uniformly exists: 1 m(f, 0, x, α) = lim n→+∞ T



α+T

f (t, x)Δt. α

 ε Take l = l , S and t0 ∈ T, then 4 

 τ ∈E

   ε , f, S ∩ F (Aiα ) − F (Ait0 ), F (Aiα ) − F (Ait0 ) + l Π , 4

i.e.,    ε F (Ait0 ) + τ ∈ E , f, S ∩ F (Aiα ), F (Aiα ) + l Π 4 

and there exists some n0 ∈ N such that n0 |F (Ait0 ) − F (Aiα )| := l0 ≥ |t0 − α|. Now, let L = l + l0 , it is easy to get F (Ait0 ) + τ ∈ [F (Aiα ), F (Aiα ) + L]Π , so t0 + τ ∈ [α, α + L]T . Let G = sup |f (t, x)|, for x ∈ S, we obtain (t,x)∈T×S

 α+T  1 1 t0 +T f (t, x)Δt − f (t, x)Δt T T t0 α   t +τ +T  t +T  α+T  t0 +τ  0 0 1 = f (t, x)Δt − + + T t0 +τ t0 t0 +τ +T α

(3.17)

198

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales



1 ≤ T

t0 +T

 |f (t + τ, x) − f (t, x)|Δt +

t0



t0 +τ

+



1 ≤ T

t0 +τ +T

|f (t, x)|Δt

|f (t, x)|Δt

α



α+T

εT 2LG + 4 T

 =

2LG ε + , 4 T

(3.18)

so we have  1 nT

α+nT

f (t, x)Δt −

α

1 T



t0 +T t0

f (t, x)Δt

n !  t0 +T  α+kT 1  1 = f (t, x)Δt − f (t, x)Δt n T α+(k−1)T t0 k=1

≤

ε 2LG + , 4 T

(3.19)

for (3.19), let n → ∞, for x ∈ S, we have  t0 +T ε 2LG a(f, 0, x, α) − 1 , f (t, x)Δt ≤ + T t0 4 T

(3.20)

by using the triangle inequality, according to (3.18) and (3.20), and we can take 8LG T > such that ε  α+T 1 ε ≤ + 4LG < ε, f (t, x)Δt − a(f, 0, x, α) T 2 T α Hence, we can easily get (3.17) uniformly exists for α ∈ T and m(f, 0, x, α) =   a f (t, x), 0, x, α . Furthermore, if α − tα ∈ Π , then 1 T



α+T α

1 f (t, x)Δt = T



tα +T

f (t + α − tα , x)Δt.

tα

  Therefore, a f (t + α − tα , x), 0, x, tα uniformly exists for α ∈ T and     m f (t, x), 0, x, α = a f (t + α − tα , x), 0, x, tα . It is easy to see f (t, x)e−iλ(t−t we have

∗)

is almost periodic in t uniformly for x ∈ D, thus,

3.2 Bohr-Transform and Mean-Value

199

    ∗ m f (t, x), λ, x, α = m f (t, x)e−iλ(t−t ) , 0, x, α   ∗ = a f (t + α − tα , x)e−iλ(t+α−tα −t ) , 0, x, tα   = a f (t + α − tα , x)e−iλ(α−tα ) , λ, x, tα .   Hence, m f (t, x), λ, x, α uniformly exists for α ∈ T. This completes the proof.



From the proof Theorem 3.21 and (3.18), we immediately have the theorem below. Theorem 3.22 If f ∈ Crd (T × D, En ) is almost periodic in t uniformly for x ∈ D, then for any fixed α, t0 ∈ T, the following holds: 1 T →+∞ T



α+T

lim

1 T →+∞ T

f (t, x)Δt = lim

α



t0 +T

f (t, x)Δt. t0

Remark 3.5 From Theorem 3.22, one will obtain that a(f, λ, x, t0 ) and m(f, λ, x, t0 ) are independent of t0 when f is almost periodic in t uniformly for x ∈ D. For convenience, we use the notations a(f, λ, x) and m(f, λ, x) instead of a(f, λ, x, t0 ) and m(f, λ, x, t0 ). In Theorem 3.20 and Theorem 3.21, one can take λ = 0, then   a f (t, x), 0, x = lim

1 T →+∞ T



t0 +T

  f (t, x)Δt := mt f (t, x)

(3.21)

t0

and if α − tα ∈ Π , then  1 t0 +T lim f (t + α − tα , x)Δt T →+∞ T t0   := mt f (t + α − tα , x)

  a f (t + α − tα ), 0, x =

(3.22)

uniformly converge for x ∈ S and for x ∈ S, α ∈ T, respectively. Definition 3.16 We say (3.21) and (3.22) are the mean value and strong mean-value of f (t, x), respectively. Theorem 3.23 Let t0 ∈ T. Then (3.13) has the following two equivalent forms: 1 a(f, λ, x) = lim T →+∞ T =−



t0 +T

t0

˚ ιλ 1 lim ˚ ιλ T →+∞ T

f (t, x)e˚ιλ (t, t ∗ )Δt 

t0 +T

t0

σ ∗ f (t, x)e ˚ ιλ (t, t )Δt.

(3.23)

200

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

Proof Notice that e˚ιλ (t, t ∗ ) =

1 1 1 = = ∗ ∗ e˚ιλ (t, t ) e eiλh −1 (t, t ) eξ −1 (iλ) (t, t ∗ ) 

h



= exp

t

−

t0



h

  ∗ ξh ξh−1 (iλ) Δt = e−iλ(t−t ) .

Then (3.13) is equivalent to 1 T →+∞ T



t0 +T

f (t, x)e˚ιλ (t, t ∗ )Δt.

lim

t0

By Lemma 1.1, we have e˚ιλ (t, t ∗ ) = −

˚ ιλ σ e (t, t ∗ ). ˚ ιλ ˚ιλ 

Hence, (3.23) is equivalent to (3.13). The proof is complete. Lemma 3.7 Let T be an almost periodic time scale and − πh < λ ≤ then mt (e

iλ(t−t ∗ )

1 ) = lim T →+∞ T



t0 +T

π h.

If λ = 0,

∗

eiλ(t−t ) Δt = 0, ∀t0 ∈ T.

t0

Proof By Theorem 3.23, we have ∗

mt (eiλ(t−t ) ) = = =

1 T →+∞ T



t0 +T

lim

1 T →+∞ T



t0 t0 +T

lim

1 T →+∞ T



∗

t0 t0 +T

lim

t0

ιλ 1 ˚ = lim T →+∞ T ˚ ιλ



1 T →+∞ T

eiλ(t−t ) Δt = lim e˚ιλ (t, t ∗ )Δt e˚ιλ (t, t ∗ )Δt t0 +T

t0

σ ∗ e ˚ ιλ (t, t )Δt

t=t ιλ 1 1 + h˚ e˚ιλ (t ∗ , t) t=t0 +T 0 T →+∞ T ˚ ιλ ιλ −iλ(t−t ∗ ) t=t0 1 1 + h˚ e = − lim t=t0 +T T →+∞ T ˚ ιλ = − lim



t0 +T

t0

e˚ιλ (t, t ∗ )Δt

3.2 Bohr-Transform and Mean-Value

= − lim

T →+∞

201

ιλ 1 1 + h˚ ιλ T ˚

  t=t × cos λ(t − t ∗ ) − i sin λ(t − t ∗ ) t=t0 +T = 0. 0



The proof is completed.

Theorem 3.24 Let f ∈ Crd (T × be almost periodic in t uniformly for x ∈ D. For any finite set of distinct real numbers λ1 , λ2 , . . . , λN with D, En )

−

π π ≤ λ1 , λ2 , . . . , λ N ≤ h h

and any finite set of real or complex n-dimensional vectors b1 , b2 , . . . , bN , the following holds: mt (|f (t, x) −

N 

∗

bk eiλk (t−t ) |2 ) = mt (|f (t, x)|2 ) −

k=1

N 

|a(f, λk , x)|2

k=1

+

N 

|bk − a(f, λk , x)|2 .

(3.24)

k=1

Proof Note that |f (t, x)|2 = !f (t, x), f (t, x)", where ! , " denotes the usual inner product in En , is almost periodic in t for x ∈ D, so it has a mean-value and f (t, x) denotes the conjugate of f (t, x), then mt (|f (t, x) −

N 

∗

bk eiλk (t−t ) |2 )

k=1

= mt (!f (t, x) −

N 

∗

bk eiλk (t−t ) , f (t, x) −

k=1

= mt (|f (t, x)|2 ) −

N 

N 

∗

bk e−iλk (t−t ) ")

k=1

!bk , a(f, λk , x)"

k=1

−

N N N    ∗ !bk , a(f, λk , x)" + !bl , bj "mt (ei(λl −λj )(t−t ) ), l=1 j =1

k=1

by Lemma 3.7, we obtain mt (|f (t, x) −

N 

∗

bk eiλk (t−t ) |2 )

k=1

= mt (|f (t, x)|2 ) −

N  k=1

!bk , a(f, λk , x)" −

N  k=1

!bk , a(f, λk , x)" +

N  j =1

!bj , bj "

202

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

= mt (|f (t, x)|2 ) −

N 

|a(f, λk , x)|2 +

k=1

N 

|bk − a(f, λk , x)|2 .

k=1



The proof is completed.

Remark 3.6 By Theorem 3.24, one can take λ1 , λ2 , . . . , λN as Fourier exponents of f (t, x) and bk = a(f, λk , x)(k = 1, 2, . . . , N), then one can obtain the best approximation of uniformly periodic function f (t, x) on time scales. Then (3.24) can be reduced into mt (|f (t, x) −

N 

bk e

iλk (t−t ∗ ) 2

| ) = mt (|f (t, x)| ) − 2

k=1

N 

|a(f, λk , x)|2 .

k=1

From Remark 3.6, one can easily get the following corollary: Corollary 3.2 Let f ∈ Crd (T × D, En ) be almost periodic in t uniformly for x ∈ D. Then N 

|a(f, λk , x)|2 ≤ mt (|f (t, x)|2 ).

k=1

Theorem 3.25 Let f ∈ Crd (T×D, En ) be almost periodic in t uniformly for x ∈ D and λ ∈ [− πh , πh ]. Then there is a countable set of real numbers Λ ⊂ [− πh , πh ] such that a(f, λ, x) ≡ 0 on S if λ ∈ Λ. Proof For f (t, x) is uniformly almost periodic, then for all (t, x) ∈ T × S, there exists M > 0 such that |f (t, x)| ≤ M. Therefore, for any n ∈ Z+ , the real number set !   1 π π : |a(f, λ, x)| > λ∈ − , h h n +∞ 

is finite (If it is infinite, then

|a(f, λk , x)|2 >

+∞

k=1

1 k=1 n

→ +∞, this is a

contradiction with Corollary 3.2). Hence, for any fixed x ∈ S, one can obtain the set  Λ= λ∈

−

π π , h h

!

 : a(f, λ, x) = 0

is countable. Furthermore, by Corollary 3.2, one can see N 

sup |a(f, λk , x)|2 ≤ M 2 .

k=1 x∈S

3.2 Bohr-Transform and Mean-Value

203

Thus, there is a countable set Λ ⊂ [− πh , πh ] such that a(f, λ, x) ≡ 0 on S if λ ∈ Λ. The proof is completed. 

Definition 3.17 Let the set  Λ= λ∈

−

π π , h h

!

 : a(f, λ, x) ≡ 0, x ∈ S .

We say that Λ is an exponent set of f (t, x) on S, it is denoted by exp(f ), and λ ∈ Λ, a(f, λ, x) are called Fourier exponent and Fourier coefficient of f (t, x), respectively. Remark 3.7 By the above discussion, if f (t, x) is almost periodic in t uniformly for x ∈ D, then for any S ⊂ D, Λ ⊂ [− πh , πh ] is countable. If λk ∈ Λ, then its corresponding Fourier coefficient a(f, λk , x) = 0. Hence, f (t, x) can be expanded in Fourier series:  ∗ a(f, λk , x)eiλk (t−t ) , where t ∗ ∈ T, λk ∈ Λ, x ∈ S. f (t, x) ∼ k

We have though, that since Corollary 3.2 holds for any finite set of numbers in Λ, that the sum over Λ converges, i.e. the following corollary holds: Corollary 3.3 Let f ∈ Crd (T × D, En ) be almost periodic in t uniformly for x ∈ D. Then for all λk ∈ Λ, the following holds: ∞ 

|a(f, λk , x)|2 ≤ mt (|f (t, x)|2 ).

k=1

Just apply the definition of a(f, λ, x), and one can easily have the following theorem: Theorem 3.26 Let f, g ∈ Crd (T × D, En ) be almost periodic in t uniformly for x ∈ D and λ ∈ Λ. Then |a(f, λ, x) − a(g, λ, x)| ≤ f (·, x) − g(·, x). Consequently, if {fn (t, x)} converge uniformly to f (t, x) on T × S, then a(fn , λ, x) → a(f, λ, x) uniformly on S. In particular, the fourier series of fn converge formally to the fourier series of f . Theorem 3.27 If f ∈ Crd (T × D, R) is a non-negative almost periodic function in t uniformly for x ∈ D and f ≡ 0, then a(f, 0, x) > 0. 

Proof Let f (t0 , x) = M > 0 and select some δ > so that f (t, x) ≥ 2M 3 on   (t0 − δ, t0 + δ)T × S. Let l ∈ Π be an inclusion length for E{ M , f, S} and take 3 l > 2δ (In fact, one can choose 0 < τ0 ∈ Π such that nτ0 = l ∈ Π, n is some positive integer). If h ∈ Π, t0 ∈ T, we can choose

204

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

   M , f, S ∩ F (Ait0 ) + h + δ − F (Ai  ), F (Ait0 ) + h + δ − F (Ai  ) + l Π . τ ∈E t0 t0 3 

Then     F Ai  + τ ∈ F (Ait0 ) + h + δ, F (Ait0 ) + h + l + δ Π , t0





so either t0 + τ or t0 − 2δ + τ ∈ [t0 + h, t0 + h + l]T since l > 2δ. In the first case    if t0 + τ ∈ [t0 + h, t0 + h + l]T , then (t0 + τ − δ, t0 + τ )T ⊂ [t0 + h, t0 + h + l]T ,   so t ∈ (t0 − δ + τ, t0 + τ )T we have |f (t, x)| ≥ |f (t − τ, x)| − |f (t − τ, x) − f (t, x)| ≥

2M M M − = . 3 3 3

The second case can be handled similarly. In either case we have 

t0 +h+l

f (t, x)Δt >

t0 +h

since that f (t, x) ≥ get

M 3

M δ 3

on a subinterval with length δ. Now let h = (n − 1)l, so we 

t0 +nl

t0 +(n−1)l

f (t, x)Δt ≥

M δ 3

and a fortiori 1 Nl



t0 +N l

t0

f (t, x)Δt =

N  1  t0 +nl Mδ . f (t, x)Δt ≥ Nl 3l t0 +(n−1)l n=1

Hence, we obtain a(f, 0, x) ≥

Mδ >0 3l

by letting N → ∞. The proof is completed.



3.3 Generalized Pseudo Almost Periodic Functions In this section, we introduce the concept of a type of generalized pseudo almost periodic functions and obtain some of their main properties. Let A P(T)n = {g ∈ Crd (T, En ) : g is almost periodic in t },

3.3 Generalized Pseudo Almost Periodic Functions

205

A P(T × D)n = {g ∈ Crd (T × D, En ) : g is almost periodic in t uniformly for x ∈ D},  ˜ PA P0 (T)n = ϕ ∈ BCrd (T, En ) : ϕ is Δ-measurable such that 1 lim r→+∞ 2r



t0 +r

t0 −r

 |ϕ(s)|Δs = 0, where t0 ∈ T, r ∈ Π

˜ P0 (T × D)n PA  = ϕ ∈ BCrd (T × D, En ) : lim

1 r→+∞ 2r



t0 +r

t0 −r

|ϕ(s, x)|Δs = 0

 uniformly for x ∈ D, where t0 ∈ T, r ∈ Π , where BCrd (T × D, En ) denotes the set of all bounded rd-continuous functions on T × D. Definition 3.18 A function f ∈ Crd (T × D, En ) is called generalized pseudo almost periodic in t uniformly for x ∈ D if f = g + ϕ, where g ∈ A P(T × D)n ˜ P0 (T × D)n . and ϕ ∈ PA Remark 3.8 Note that g and ϕ are uniquely determined. Indeed, for any fixed t0 ∈ T and r ∈ Π since  t0 +r 1 |ϕ(s, x)|Δs = 0, N(ϕ) = lim r→+∞ 2r t −r 0 ˜ P(T × D)n , f = g1 + ϕ1 = g2 + ϕ2 one has N (g1 − g2 ) = 0, then if f ∈ PA which implies that g1 = g2 , thus, ϕ1 = ϕ2 . g and ϕ are called the almost periodic component and the ergodic perturbation, respectively, of the function f . Denote by ˜ P(T × D)n the set of generalized pseudo almost periodic functions uniformly PA for x ∈ D. ∞ Example 3.2 Let T = k=1 [2k, 2k + 1], f (t) = g(t) + ϕ(t) and F (t, x) = f (t) cos x, where g(t) = sin t + sin π t, t ∈ T, ϕ(t) = − Then one will obtain

1 , t ∈ T. tσ (t)

206

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales



1 lim r→+∞ 2r

t0 +r

t0 −r

1 |ϕ(s)|Δs = lim r→+∞ 2r



t0 +r t0 −r

1 Δs sσ (s)

1 1 t0 +r · = 0. r→+∞ 2r s t −r 0

= lim

˜ P0 (T). Therefore, f ∈ PA ˜ P(T), F ∈ PA ˜ P(T × D). Thus, ϕ ∈ PA



˜ P(T × D)n , then Theorem 3.28 If f ∈ PA 

1 lim r→+∞ 2r

t0 +r

f (s, x)Δs := M(f )

t0 −r

exists and is finite. Moreover, M(f ) = M(g). Proof Indeed 1 r→+∞ 2r



lim

t0 +r t0 −r

1 r→+∞ 2r



f (s, x)Δs = lim

t0 +r

t0 −r

1 r→+∞ 2r



+ lim

g(s, x)Δs

t0 +r

t0 −r

ϕ(s, x)Δs.

Since g ∈ A P(T × D)n then 1 r→+∞ 2r lim



t0 +r

t0 −r

g(s, x)Δs

exists and is finite by Theorem 3.20. Furthermore, one has −|ϕ(s, x)| ≤ ϕ(s, x) ≤ |ϕ(s, x)|. Then 1 − lim r→+∞ 2r



t0 +r

t0 −r

1 |ϕ(s, x)|Δs ≤ lim r→+∞ 2r 1 ≤ lim r→+∞ 2r

 

t0 +r t0 −r t0 +r t0 −r

ϕ(s, x)Δs |ϕ(s, x)|Δs.

Then 1 r→+∞ 2r lim



t0 +r t0 −r

|ϕ(s, x)|Δs = M(ϕ) = 0.

Hence, M(f ) = M(g). The proof is completed.



3.3 Generalized Pseudo Almost Periodic Functions

207

˜ P(T × D)n , then a(f, λ, x) = Theorem 3.29 If f = g + ϕ is an element of PA a(g, λ, x). ∗

Proof In fact, for each λ ∈ [− πh , πh ], the function f (s, x)e−iλ(s−t ) is also pseudo almost periodic with the almost periodic component which equals to ∗ ˜ P(T × g(s, x)e−iλ(s−t ) . So, we can define the Fourier coefficients for f ∈ PA D)n as follows 1 r→+∞ 2r lim

1 r→+∞ 2r

= lim



t0 +r

t0 −r



t0 +r

t0 −r

∗

f (s, x)e−iλ(s−t ) Δs = a(f, λ, x)   ∗ g(s, x) + ϕ(s, x) e−iλ(s−t ) Δs = a(g, λ, x). 

The proof is completed.

Remark 3.9 Pseudo almost periodic functions admit the same Fourier exponents as their almost periodic component. By Definition 3.18, one can easily show the following theorem: ˜ P(T × D)n and g is its almost periodic component, Theorem 3.30 If f ∈ PA then we have g(T × S) ⊂ f (T × S). Furthermore, if ϕ is rd-continuous and bounded, we have f  ≥ g ≥

inf

(t,x)∈T×S

|g(t, x)| ≥

inf

(t,x)∈T×S

|f (t, x)|,

where f (T × S) and g(T × S) are the value field of f and g on T × S, respectively, f (T × S) is the closure of f (T × S) and S is an arbitrary compact subset of D. Definition 3.19 A closed subset C of T is said to be an ergodic zero set of T if   μΔ ([t0 − r, t0 + r)T ) ∩ C → 0 as r → ∞, where t0 ∈ T, r ∈ Π. 2r ˜ P0 (T × D)n , the proof of the following theorem is By the definition of PA straightforward. ˜ P0 (T × D)n if and only if for ε > 0, the set Theorem 3.31 A function ϕ ∈ PA Cε = {t ∈ T : |ϕ(t, x)| ≥ ε} is an ergodic zero subset of T. Theorem 3.32 ˜ P0 (T × D) if and only if |ϕ|2 ∈ PA ˜ P0 (T × D). (i) A function ϕ ∈ PA

208

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

˜ P0 (T × D)n if and only if the norm function |Φ(·, x)| ∈ (ii) Φ ∈ PA ˜ P0 (T × D). PA Proof (i) The sufficiency follows since 1 2r



t0 +r

t0 −r

|ϕ(s, x)|Δs ≤



1 2r

=

t0 +r

t0 −r

1 2r



t0 +r

t0 −r

!1/2  |ϕ(s, x)|2 Δs

t0 −r

!1/2 |ϕ(s, x)|2 Δs

!1/2

t0 +r

1Δs

,

˜ P0 (T × D)n . then ϕ ∈ PA The necessity follows from the fact that 1 2r



t0 +r t0 −r

1 |ϕ(s, x)| Δs ≤ ϕ 2r 2



t0 +r

t0 −r

|ϕ(s, x)|Δs,

˜ P0 (T × D). because ϕ ∈ BCrd (T × D, En ), then |ϕ|2 ∈ PA (ii) By (i), Φ = (ϕ1 , ϕ2 , . . . , ϕn ) ∈ PA P0 (T × D)n if and only if ϕi ϕ¯i ∈ ˜ P0 (T × D), i = 1, 2, . . . , n. The latter is equivalent to |Φ(·, x)|2 = PA n  ˜ P0 (T × D), which, again by (i), is equivalent to |ϕi (·, x)|2 ∈ PA i=1

˜ P0 (T × D). |Φ(·, x)| ∈ PA

The proof is completed.



For H = (h1 , h2 , . . . , hn ) ∈ En , suppose that H ∈ D for all t ∈ T. Define H × ι : T → T × D by   H × ι(t) = t, h1 (t), h2 (t), . . . , hn (t) . ˜ P(T × D)n , let G = (g1 , g2 , . . . , gn ) and For f = (f1 , f2 , . . . , fn ) ∈ PA Φ = (ϕ1 , ϕ2 , . . . , ϕn ), where gi and ϕi are the almost periodic component and the ergodic perturbation of fi , respectively, i = 1, 2, . . . , n. ˜ P(T × Theorem 3.33 Let S be an arbitrary compact subset of D and f ∈ PA ˜ P(T)n and F (T) ⊂ D)n be continuous in x ∈ S uniformly for t ∈ T. If F ∈ PA ˜ P(T)n , where F (T) denotes the value field of F. D, then f ◦ (F × ι) ∈ PA Proof Let f = g + ϕ and F = G + Φ with G = (g1 , g2 , . . . , gn ) ∈ A P(T)n , as above. Note that f ◦ (F × ι) = g ◦ (F × ι) + ϕ ◦ (F × ι) = g ◦ (G × ι) + [g ◦ (F × ι) − g ◦ (G × ι) + ϕ ◦ (F × ι)].

3.3 Generalized Pseudo Almost Periodic Functions

209

It follows from Theorem 3.30 that G(T) ⊂ F (T) ⊂ D. Therefore, g ◦ (G × ι) ∈ A P(T)n . To finish the proof, we need to show the function h = g ◦ (F × ι) − g ◦ ˜ P0 (T)n . (G × ι) + ϕ ◦ (F × ι) ∈ PA ˜ P0 (T)n . First we show that g ◦ (F × ι) − g ◦ (G × ι) ∈ PA It is trivial in the case g = 0. So we assume that g = 0. Let D1 = F (T). Then the function g is uniformly continuous in D1 uniformly for t ∈ T, i.e., for ε > 0 and t ∈ T, there exists a δ > 0 such that |x1 − x2 | < δ implies |g(t, x1 ) − g(t, x2 )|
0 such that r ≥ T implies μΔ ([t0 − r, t0 + r)T ∩ Cδ ) ε < . 2r 4g

(3.27)

It follows from (3.25), (3.26) and (3.27) that 1 2r =



t0 +r t0 −r

    g s, F (s) − g s, G(s) Δs



1 2r  +

[t0 −r,t0 +r)T \Cδ

[t0 −r,t0 +r)T ∩Cδ

    g s, F (s) − g s, G(s) Δs



ε μΔ ([t0 − r, t0 + r)T ∩ Cδ ) ≤ + 2g < ε. 2 2r ˜ P0 (T)n . Therefore, g ◦ (F × ι) − g ◦ (G × ι) ∈ PA ˜ P0 (T)n . Note that f and g are Finally, we show that ϕ ◦ (F × ι) ∈ PA continuous in D1 uniformly for t ∈ T, so is ϕ. Since D1 is compact in En , then f, g, ϕ are uniformly continuous in D1 . Hence, there exist, say m, open balls Ok m  with center x k ∈ D1 , k = 1, 2, . . . , m, and radius δ(x k , ε/2) such that D1 ⊂ Ok k=1

and |ϕ(t, x) − ϕ(t, x k )| < The set

ε , x ∈ Ok , t ∈ T. 2

(3.28)

210

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

Bk = {t ∈ T : F (t) ∈ Ok } is open and T =

m 

Bk . Let Ek = Bk \

k=1

k−1  j =1

Bj , then Ek

(3.29) 

Ej = ∅ when k =

j, 1 ≤ k, j ≤ m. ˜ P0 (T)n , then there is a number T0 > 0 such that Since each ϕ(·, x (k) ) ∈ PA  t0 +r m  1 ε |ϕ(s, x (k) )|Δs < , r ≥ T0 . 2r t0 −r 2

(3.30)

k=1

It follows from (3.28), (3.29) and (3.30) that 1 2r



t0 +r

t0 −r

  ϕ s, F (s) Δs

m        1  ϕ s, F (s) − ϕ s, x (k) + |ϕ(s, x (k) | Δs ≤ 2r Ek ∩[t0 −r,t0 +r)T k=1

ε  1 + 2 2r m

≤

k=1



t0 +r t0 −r

|ϕ(s, x (k) )|Δs < ε,

˜ P0 (T)n . The proof is completed. which shows that ϕ ◦ (F × ι) ∈ PA



Define E0 (T × D)n = {f ∈ Crd (T × D, En ) : f (t, x) → 0 uniformly for x ∈ D as |t| → ∞}. E0 (T)n = {f ∈ Crd (T, En ) : f (t) → 0 as |t| → ∞}. Definition 3.20 Let A A P(T × D)n be the set of all the functions f of the form f = g + ϕ, where g ∈ A P(T × D)n and ϕ ∈ E0 (T × D)n . The members of A A P(T × D)n are called asymptotically almost periodic functions in t uniformly for x ∈ D. ˜ P0 (T × D)n and A A P(T × D)n ⊂ It is obvious that E0 (T × D)n ⊂ PA ˜ PA P(T × D)n . Corollary 3.4 Let f ∈ A A P(T × D)n be continuous in S uniformly for t ∈ T. If F ∈ A A P(T)n and F (T) ⊂ D, then f ◦ (F × ι) ∈ A A P(T)n , where S is an arbitrary compact subset of D.

3.3 Generalized Pseudo Almost Periodic Functions

211

Proof As in the proof of Theorem 3.33, we have f ◦ (F × ι) = g ◦ (F × ι) + ϕ ◦ (F × ι) = g ◦ (G × ι) + [g ◦ (F × ι) − g ◦ (G × ι) + ϕ ◦ (F × ι)], where g ◦ (G × ι) ∈ A P(T)n . By the hypothesis that Φ = F − G ∈ E0 (T)n and ϕ ∈ E0 (T × D)n , and because the uniform continuity  of g in S, it follows that g ◦ (F × ι) − g ◦ (G × ι) ∈ E0 (T)n . Moreover, since ϕ t, F (t) ≤ supx∈D ϕ(t, x), then ϕ ◦ (F × ι) ∈ E0 (T)n . The proof is completed. 

Theorem 3.34 Suppose g ∈ A P(T × D)n and for every ε > 0,   μΔ t ∈ [t0 − r, t0 + r)T : g(t, x) > −ε → 1 as r → +∞, 2r where t0 ∈ T, r ∈ Π . Then g ≥ 0 for all T × S, where S is an arbitrary compact subset of D. 

Proof Suppose the conclusion does not hold. This implies that g(t0 , x) < 0 for   some t0 ∈ T. Choose ε > 0 and ε < −g(t0 , x).    By rd-continuity, there exists δ > 0 so that t ∈ (t0 , t0 + δ)T implies g(t0 , x) < −ε. In view of the Definition 3.9, there exists l(ε, S) > 0 so that in each interval I with the length l, there exists an 2ε -almost period τ with the property that |g(t + τ, x) − g(t, x)|
0 and c ∈ Π such that ϕ(t, x) > α˜ for t ≥ t0 + c, which yields 1 r



t0 +r

t0 −r

1 |ϕ(s, x)|Δs = r ≥



t0 +c t0 −r

 |ϕ(s, x)|Δs +

t0 +r

t0 +c

! |ϕ(s, x)|Δs

1 α(r ˜ − c). r

Passing to the limit as r → ∞, we obtain 1 lim r→∞ 2r



t0 +r t0 −r

|ϕ(s, x)|Δs ≥ α, ˜

˜ P0 (T × D)n . which contradicts the fact that ϕ ∈ PA (ii) Assuming f ≥ 0, we will show that g ≥ 0. In fact, we have f = g + ϕ, with 1 r→∞ 2r lim



t0 +r t0 −r

|ϕ(s, x)|Δs = 0.

Thus, there exists {cn }n∈N ⊂ Π, cn → +∞ as n → ∞ such that g(t +cn , x) → g(t, x) for all (t, x) ∈ T × S. Furthermore, for any ε > 0 and r > 0, one can get   μΔ t ∈ [t0 − r, t0 + r)T : ϕ(t, x) > ε → 0, as r → ∞ which implies that μΔ {t ∈ [t0 − r, t0 + r)T : g(t, x) > −ε} → 1 as r → +∞, 2r where t0 ∈ T, r ∈ Π . By Theorem 3.31, one can have g(t, x) ≥ 0 for all (t, x) ∈ T × S. The proof is completed.



3.4 Π -Semigroup and Moving-Operators

213

˜ P0 (T), but lim ϕ(t) Remark 3.10 There exist functions ϕ such that ϕ is in PA |t|→∞   [2n, 2n + 1] and T = does not exist. Consider, for example, let T = n∈Z  1 [2n, 2n + n5/2 ]. Take n∈N

ϕ(t) =

√  n, t ∈ T , 

t ∈ T\T .

0,

Obviously, T is a positive periodic time scale and Π = {2n, n ∈ N}. It is clear that lim ϕ(t) does not exist, noting that {2n + 1}n∈N are right scattered points, so

|t|→∞

1 r→∞ 2r lim



t0 +r

t0 −r

1 n→∞ 2n

|ϕ(s)|Δs = lim

n  

2k+

1 k 5/2

√

kΔs +

k=−n 2k

n  

!

2k+1

1 k=−n 2k+ k 5/2

0Δs

   n ∞ 1  1 1 2 · = lim = 0. = lim 2 n→∞ 2n n→∞ 2n k k2 k=−n

k=1

3.4 Π-Semigroup and Moving-Operators This section introduces the definitions of Π -semigroup and moving-operators on complete-closed time scales under translations and we obtain some of their main properties. Theorem 3.36 If T is a periodic time scale, then the graininess function μ : T → R+ is a periodic function. Proof Assume that T is a periodic time scale, i.e., T is invariant under translation, then by Definition 2.3, we have T = Tτ . If t is a right dense point in T, then t + τ is also a right dense point in Tτ , so t + τ is a right dense point in T. Hence, μ(t + τ ) − μ(t) = σ (t + τ ) − σ (t) − τ = t + τ − t − τ = 0, i.e., μ(t + τ ) = μ(t). If t is a right scattered point in T, then t + τ is also a right scattered point in Tτ , so t + τ is a right scattered point in T. Without loss of generality, we assume that τ ∈ Π and τ > 0. It follows from σ (t) > t that σ (t) + τ > t + τ , so we have σ (t) + τ ≥ σ (t + τ ) > t + τ.

(3.31)

From (3.31), we obtain σ (t) − t = μ(t) ≥ σ (t + τ ) − (t + τ ) = μ(t + τ ) > 0.

(3.32)

214

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

If in (3.32), μ(t) > μ(t +τ ), then clearly μ is decreasing at all right scattered points in T, which leads to a contradiction because the time scale T is invariant under translations. Hence, μ(t + τ ) = μ(t). Therefore, μ is periodic. This completes the proof. 

Definition 3.21 We say the set Π in Definition 2.3 is called an invariant translation number set for T. Theorem 3.37 Let a time scale T be invariant under translations and Π be a translation number set. Then (a) (b) (c) (d)

Π is a time scale. For all τ1 , τ2 ∈ Π , we have τ1 + τ2 ∈ Π . For all τ1 , τ2 , τ3 ∈ Π , we have (τ1 + τ2 ) + τ3 = τ1 + (τ2 + τ3 ). There exists an element 0 ∈ Π , such that for all elements τ ∈ Π , the equation 0 + τ = τ + 0 = τ holds. (e) For each τ ∈ Π , there exists an element −τ ∈ Π such that τ +(−τ ) = 0, where 0 is the identity element. (f) For all τ1 , τ2 ∈ Π , we have τ1 + τ2 = τ2 + τ1 . Proof All the parts follow from the definition, so we omit the details.



Remark 3.11 Because Π is a time scale, we denote its graininess function by μΠ : Π → R+ , and its forward jump operator as σΠ (τ1 ) = inf{τ2 ∈ Π : τ2 > τ1 }. From Theorem 3.37, the following result follows immediately. Theorem 3.38 The pair (Π, +) is an Abelian group. From the proof of Proposition 4.4 in [208], the following lemma is also immediate. Lemma 3.8 For the time scale Π , Π κ has constant graininess, that is μΠ (τ ) = h, τ ∈ Π κ , for some h ≥ 0. Remark 3.12 Since Π is an Abelian group, sup Π = +∞, inf Π = −∞. For convenience, we let Π + = [0, +∞)Π . Let X be a Banach space, and Tτ : X → X be a transformation. Obviously, {Tτ : τ ∈ Π } is a set containing only one parameter. We define the multiplication as Tτ1 Tτ2 = Tτ1 +τ2 .

(3.33)

It follows that     Tτ1 Tτ2 Tτ3 = Tτ1 Tτ2 Tτ3 = Tτ1 +τ2 +τ3 , I = T0 is the identity, and T−τ is the inverse element of Tτ . From these definitions, the following theorem is clear.

3.4 Π -Semigroup and Moving-Operators

215

Theorem 3.39 {Tτ : τ ∈ Π } is an operator group with respect to the multiplication defined by (3.33). It is an Abelian group. In view of Theorem 3.39, we are now in a position to introduce some basic concepts which will be needed to define a Π -semigroup for an invariant time scale under translations. Definition 3.22 Let a time scale T be invariant under translations, and {Tτ } be a family of bounded linear operators on Banach space X. If for all τ1 , τ2 ∈ Π + the following holds: Tτ1 +τ2 = Tτ1 Tτ2 ,

(3.34)

then {Tτ : τ ∈ Π + } is called a one-parameter operator semigroup; if (3.34) holds for all τ ∈ Π , we call {Tτ : τ ∈ Π } a one-parameter operator group. Definition 3.23 Let T be an invariant time scale under translations, and {Tτ : τ ∈ Π + } be an operator group on a Banach space X, i.e., Tτ1 Tτ2 = Tτ1 +τ2 ,

τ1 , τ2 ∈ Π + ,

T0 = I.

If for every τ0 ≥ 0 and any ε > 0, there is a neighborhood U of τ0 (i.e., U = (τ0 − δ, τ0 + δ)Π + for some δ > 0) such that Tτ x − Tτ0 x < ε

for all τ ∈ U,

then we call {Tτ : τ ∈ Π + } the strong-continuous operator semigroup or the Π semigroup. Theorem 3.40 Let a time scale T be invariant under translations, and {Tτ : τ ∈ Π + } be an operator semigroup on the Banach space X. For any ε > 0 and x ∈ X there exists a neighborhood U = (τ1 − δ, τ1 + δ)Π + for some δ > 0, such that ' ' 'T|σ (τ )−τ | x − x ' ≤ ε Π 1 2

for all τ2 ∈ U,

(3.35)

then {Tτ : τ ∈ Π + } is a Π -semigroup. Proof For any L ∈ Π, L > 0, we claim that   sup Tτ  : τ ∈ [0, L]Π < +∞.

(3.36)

In fact, for any x ∈ X, we can take h ∈ Π + \{0}, c > 0 such that   sup Tτ x : τ ∈ [0, h]Π ≤ c. Now for τ ∈ [0, L]Π , let τ = kh + r, r ∈ Π , where k ≤ follows that

L h,

0 ≤ r < h. Then, it

216

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

Tτ x = Tkh Tr x ≤ Tkh c.   Hence (3.36) holds. In what follows we let M := sup Tτ  : τ ∈ [0, L]Π . For any ε > 0, there is δ, such that for τ2 ∈ (τ1 − δ, τ1 + δ)Π + , we have (i) If τ2 > τ1 , then σΠ (τ1 ) = τ1 , and we have Tτ2 x − Tτ1 x ≤ TσΠ (τ1 ) (Tτ2 −σΠ (τ1 ) − I )x + Tτ1 (TσΠ (τ1 )−τ1 − I )x ≤ 2Mε. In the above σΠ (τ1 ) = τ1 . In fact, if σΠ (τ1 ) > τ1 , then τ1 is a right scattered point, which implies that τ2 = τ1 , and this contradicts τ2 > τ1 . (ii) If τ2 ≤ τ1 , then τ2 ≤ τ1 ≤ σΠ (τ1 ), which yields 0 ≤ σΠ (τ1 ) − τ1 ≤ σΠ (τ1 ) − τ2 . Hence, we have Tτ2 x − Tτ1 x ≤ Tτ2 (I − TσΠ (τ1 )−τ2 )x + Tτ1 (TσΠ (τ1 )−τ1 − I )x ≤ 2Mε. Therefore, for τ2 ∈ (τ1 − δ, τ1 + δ)Π + , the following holds: Tτ2 x − Tτ1 x ≤ 2Mε. Hence, from Definition 3.23, {Tτ : τ ∈ Π + } is a Π -semigroup. This completes the proof. 

In the following, we introduce the definition of infinitesimal generator of a Π semigroup. Definition 3.24 Let T be an invariant time scale under translations and {Tτ : τ ∈ Π + } be a Π -semigroup on a Banach space X. Let D denote a subset of X, which has the property that for each x ∈ D there exists a y ∈ X such that for any ε > 0, there is a neighborhood U = (τ1 − δ, τ1 + δ)Π + for some δ > 0 such that ' '(T|σ

Π (τ1 )−τ2 |

' − I )x − y|σΠ (τ1 ) − τ2 |' < ε|σΠ (τ1 ) − τ2 |,

τ2 ∈ U.

(3.37)

We define A : D → X satisfying Ax = y, where y is fixed by (3.37). In what follows we call this A the infinitesimal generator of this Π -semigroup. Remark 3.13 From (3.37), it follows that ' ' ' T|σΠ (τ1 )−τ2 | − I ' ' ' ' σ (τ ) − τ x − Ax ' < ε. Π 1 2 By Lemma 3.8, μΠ (τ ) ≡ h > 0 is a constant, thus A is independent of the variable τ. (i) If T = R, then Π = R. Thus from (3.37), we have ' ' '1 ' ' lim ' (Tt − I )x − y ' ' = 0, t→0 t

3.4 Π -Semigroup and Moving-Operators

217

hence 1 A = lim (Tt − I ). t→0 t (ii) If T = hZ, then Π = hZ. Thus from (3.37), we obtain ' ' '1 ' ' (Th − I ) − y ' = 0, 'h ' hence A=

1 (Th − I ). h

Definition 3.25 ([75]) A linear operator T from one topological vector space X to another one Y is said to be densely defined if the domain of T is a dense subset of X and the range of T is contained within Y. Theorem 3.41 Let T be an invariant under translations time scale, {Tτ : τ ∈ Π + } be a Π -semigroup on Banach space X satisfying (3.35), and A be the infinitesimal generator of the Π -semigroup. Then A is a closed densely defined operator and for every x ∈ D(A), the following holds: (Tτ x)ΔΠ = A(Tτ x) = Tτ Ax,

(3.38)

that is  (Tτ x) − x =

τ



τ

ATs xΔΠ s =

0

Ts AxΔΠ s,

(3.39)

0

where D(A) denotes the domain of the operator A and ΔΠ is the differential operator over the time scale Π . Proof First we show that A is a densely defined operator. Note that for any x ∈ X, we have '  |σΠ (τ )−τ | ' 1 2 ' ' ' Tθ xΔΠ θ − |σΠ (τ1 ) − τ2 |x ' ' ' 0

' ' =' '

|σΠ (τ1 )−τ2 |

0

≤ |σΠ (τ1 ) − τ2 |

' ' (Tθ x − x)ΔΠ θ ' ' sup

0≤θ≤|σΠ (τ1 )−τ2 |

< |σΠ (τ1 ) − τ2 |ε.

Tθ x − x (3.40)

218

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

Let y =

τ 0

Tθ xΔΠ θ, then 

T|σΠ (τ1 )−τ2 | y − y =  = 

τ 0

τ +|σΠ (τ1 )−τ2 | |σΠ (τ1 )−τ2 |



Tθ xΔΠ θ 

|σΠ (τ1 )−τ2 |

Tθ xΔΠ θ −

Tθ xΔΠ θ 0

τ |σΠ (τ1 )−τ2 |

=

τ

Tθ xΔΠ θ − 0

τ +|σΠ (τ1 )−τ2 |

= 

(Tθ+|σΠ (τ1 )−τ2 | x − Tθ x)ΔΠ θ



|σΠ (τ1 )−τ2 |

Tθ (Tτ x)ΔΠ θ −

0

Tθ xΔΠ θ. 0

Since (3.40) holds for any x ∈ X, then it follows that (T|σΠ (τ1 )−τ2 | y − y) − |σΠ (τ1 ) − τ2 |(Tτ x − x) ' '  |σΠ (τ )−τ | 1 2 ' ' ' Tθ (Tτ x − x)ΔΠ θ − |σΠ (τ1 ) − τ2 |(Tτ x − x)' =' ' 0

≤ |σΠ (τ1 ) − τ2 |ε. Therefore, y ∈ D(A), so D(A) = X. Next, we will show that (3.38) and (3.39) hold. Since lim

τ2 →τ1

(T|σΠ (τ1 )−τ2 | − I )Tτ1 x Tτ1 (T|σΠ (τ1 )−τ2 | − I )x = lim = Tτ1 Ax, τ2 →τ1 |σΠ (τ1 ) − τ2 | |σΠ (τ1 ) − τ2 |

we have Tτ1 (T|σΠ (τ1 )−τ2 | − I )x − |σΠ (τ1 ) − τ2 |Tτ1 Ax ≤ Tτ1 (T|σΠ (τ1 )−τ2 | − I )x − |σΠ (τ1 ) − τ2 |Ax ≤ Tτ1 ε|σΠ (τ1 ) − τ2 |,

(3.41)

and so, Tτ x ∈ D(A). From (3.41), we also have (T|σΠ (τ1 )−τ2 | − I )x − |σΠ (τ1 ) − τ2 |Ax ≤ ε|σΠ (τ1 ) − τ2 |. (i) If τ2 > τ1 , then from (3.42) and Theorem 3.40 it follows that (TσΠ (τ1 ) − Tτ2 )x − (σΠ (τ1 ) − τ2 )Tτ1 Ax ≤ TσΠ (τ1 ) (I − Tτ2 −σΠ (τ1 ) )x − (σΠ (τ1 ) − τ2 )TσΠ (τ1 ) Ax +(σΠ (τ1 ) − τ2 )TσΠ (τ1 ) Ax − (σΠ (τ1 ) − τ2 )Tτ1 Ax

(3.42)

3.4 Π -Semigroup and Moving-Operators

219

≤ TσΠ (τ1 ) (τ2 − σΠ (τ1 ))Ax − (I − Tτ2 −σΠ (τ1 ) )x +Tτ1 (I − TσΠ (τ1 )−τ1 )Ax(τ2 − σΠ (τ1 )) ≤ Mε(τ2 − σΠ (τ1 )), where M := sup{Tτ  : τ ∈ [0, L]Π }, and L ∈ Π is any fixed positive constant. In the above it is necessary that σΠ (τ1 ) = τ1 , since if σΠ (τ1 ) > τ1 , then τ1 is right scattered point, which implies that τ2 = τ1 , and this contradictions our assumption that τ2 > τ1 . (ii) If τ2 ≤ τ1 , then it follows from τ2 ≤ τ1 ≤ σΠ (τ1 ) that 0 ≤ τ1 − τ2 ≤ σΠ (τ1 ) − τ2 . Hence, from (3.42) and Theorem 3.40, we obtain (TσΠ (τ1 ) − Tτ2 )x − (σΠ (τ1 ) − τ2 )Tτ1 Ax ≤ Tτ2 (TσΠ (τ1 )−τ2 − I )x − (σΠ (τ1 ) − τ2 )Tτ2 Ax +(σΠ (τ1 ) − τ2 )Tτ2 Ax − (σΠ (τ1 ) − τ2 )Tτ1 Ax ≤ Tτ2 (TσΠ (τ1 )−τ2 − I )x − (σΠ (τ1 ) − τ2 )Ax) +Tτ2 (I − Tτ1 −τ2 )Ax(σΠ (τ1 ) − τ2 ) ≤ Mε(σΠ (τ1 ) − τ2 ), where M := sup{Tτ  : τ ∈ [0, L]Π }, and L ∈ Π is any fixed positive constant. Therefore, (Tτ x)ΔΠ = Tτ Ax = ATτ x. Since (3.39) is the integral form of (3.38), we can conclude that (3.39) holds. Finally, we show that A is a closed operator. Let xn ∈ D(A), xn → x, Axn → y, then by (3.42), we have (T|σΠ (τ1 )−τ2 | − I )x − |σΠ (τ1 ) − τ2 |y = lim (T|σΠ (τ1 )−τ2 | − I )xn − |σΠ (τ1 ) − τ2 |Axn  n→∞

≤ ε|σΠ (τ1 ) − τ2 |. Hence, x ∈ D(A) and Ax = y, that is, A is a closed operator. This completes the proof. 

Theorem 3.42 Let T be an invariant time scale under translations and X be a Banach space. Assume that {Tτ : τ ∈ Π + } is a Π -semigroup, A is the infinitesimal generator of the Π -semigroup and D(A) = X, eA (τ1 + τ2 , 0) = eA (τ1 , 0)eA (τ2 , 0) for all τ1 , τ2 ∈ Π + . Then, Tτ = eA (τ, 0), τ ∈ Π + , where D(A) denotes the domain of A.

220

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

Proof From Theorem 3.41, we have Δ  eA (τ, 0)x Π = AeA (τ, 0)x = eA (τ, 0)Ax. Further, since eA (τ, 0) is Δ-differentiable on Π , from Definition 1.3, for any ε > 0, there is δ, such that for τ2 ∈ (τ1 − δ, τ1 + δ)Π + , it follows that ' ' '(eA (σΠ (τ1 ), 0) − eA (τ2 , 0))x − (σΠ (τ1 ) − τ2 )AeA (τ1 , 0)x ' ≤ ε|σΠ (τ1 ) − τ2 |.

(3.43)

Hence, (i) if τ2 > τ1 , then it follows from (3.43) that eA (σΠ (τ1 ), 0)[I − eA (τ2 − σΠ (τ1 ), 0)x −(σΠ (τ1 ) − τ2 )eA (τ1 , σΠ (τ1 ))Ax] ≤ eA (σΠ (τ1 ), 0)[I − eA (τ2 − σΠ (τ1 ), 0)x −(σΠ (τ1 ) − τ2 )eA (τ1 , σΠ (τ1 ))Ax] ≤ Mε|σΠ (τ1 ) − τ2 |. In the above σΠ (τ1 ) = τ1 . Indeed, if σΠ (τ1 ) > τ1 , then τ1 is a right scattered point, and then τ2 = τ1 , which is a contradiction since τ2 > τ1 . (ii) If τ2 ≤ τ1 , then it follows from τ2 ≤ τ1 ≤ σΠ (τ1 ), that 0 ≤ τ1 − τ2 ≤ σΠ (τ1 ) − τ2 . Hence, from (3.43) we find eA (τ2 , 0)[(eA (σΠ (τ1 ) − τ2 , 0) − I )x − (σΠ (τ1 ) − τ2 )Ax +(σΠ (τ1 ) − τ2 )(I − eA (τ1 , τ2 ))Ax] ≤ eA (τ2 , 0)[(eA (σΠ (τ1 ) − τ2 , 0) − I )x − (σΠ (τ1 ) − τ2 )Ax] +Mε|σΠ (τ1 ) − τ2 | ≤ 2Mε|σΠ (τ1 ) − τ2 |, where M := sup{eA (τ, 0) : τ :∈ [0, L]Π }, and L ∈ Π is any fixed positive constant. From (i) and (ii), we obtain ' ' '(eA (|σΠ (τ1 ) − τ2 |, 0) − I )x − |σΠ (τ1 ) − τ2 |Ax ' ≤ 2Mε|σΠ (τ1 ) − τ2 |. Therefore, A is the infinitesimal generator of {Tτ : τ ∈ Π + }. This completes the proof.



3.4 Π -Semigroup and Moving-Operators

221

Remark 3.14 If (i) T = R, then Π = R, Tτ = eA (τ, 0) = eAτ . Clearly, it satisfies Tτ1 +τ2 = Tτ1 Tτ2 . If (ii) T = Z, then Π = Z, Tτ = eA (τ, 0) = (I + A)τ , which also satisfies Tτ1 +τ2 = Tτ1 Tτ2 . Now we will introduce a new concept that will be needed later. Definition 3.26 Let A be the infinitesimal generator of the Π -semigroup. We call e˜A (t, t0 ), t0 ∈ T the exponential function generated by A on the time scale T. We also let Tt = e˜A (t, t0 ) and call Tt the moving-operator on T. Remark 3.15 In Fig. 3.1 we give a relationship between T, Π, A, and Tt . Note that if T = Π , then the Π -semigroup will strictly include the continuous (T = R) and the discrete (T = Z) cases of the C0 -semigroup. Let X be a Banach space, and consider the following system: x Δ = Ax(t),

x(t0 ) = x0 , t0 ∈ T,

(3.44)

where A is the infinitesimal generator of a Π -semigroup satisfying all the conditions in Theorem 3.42, and x : T → X. Theorem 3.43 The fundamental solution of the system (3.44) can be expressed as x(t) = Tt x0 ,

The time scale

T

Generates

The set ∏

Generates

Generates

˜ Exponential function e A (t,t0)

∏ – Semigroup

Generates

Equilvalent to

the infinitesimal generator A of the ∏ – Semigroup

The moving operator Ft Generates

Fig. 3.1 The generation relationship of T, Π -semigroup, A, Tt

222

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

Proof From Definition 3.26, Tt = e˜A (t, t0 ), thus x Δ = (Tt x0 )Δ = ATt x(t0 ) = Ax(t). Therefore, Tt x0 is the fundamental solution of (3.44). This completes the proof.  From Theorem 3.43, the following result follows immediately. Theorem 3.44 Let A be the infinitesimal generator of the Π -semigroup, and let Tt be the moving-operator on T. Then (Tt x)Δ = A(Tt x) = Tt Ax, that is  (Tt x) − x =

t t0

 ATs xΔs =

t

Ts AxΔs.

t0

Remark 3.16 Note the Π -semigroups studied in this book are more general than the C0 -semigroups introduced in [145]. If we let T = Π , then the Π -semigroups will turn into the C0 -semigroups in [145]. If Π = T, for example, Π ∩ T = ∅ (see the Example 3.1), the results in [145] cannot be applied to study abstract dynamic equations on periodic time scales.

3.5 The Equivalence of Two Concepts of Relatively Dense Sets In this section, we will introduce two equivalent definitions of relatively dense sets on semigroups induced by complete-closed time scales under translations. Definition 3.27 Let T be a complete-closed time scale. If   Π + := [0, +∞)Π ∈ ∅, {0} , then we say (Π + , +) is a positive direction semigroup induced by the time scale T; if   Π − := (−∞, 0]Π ∈ ∅, {0} , then we say (Π − , +) is a negative direction semigroup induced by the time scale T. +∞ Remark 3.17 According to Definition 3.27, let T1 = k=0 [2k, 2k + 1], then Π + = {2n : n ∈ N}. However, Π1− = {0}, so Π1+ is a positive direction semigroup induced  by the positive direction complete-closed time scale T1 . Conversely, let T2 = 0k=−∞ [2k, 2k + 1], then Π2− = {2n : n ∈ N− }, where N− = {−n, n ∈ N}.

3.5 The Equivalence of Two Concepts of Relatively Dense Sets

223

However, Π2+ = {0}, so Π − is a negative direction semigroup induced by the negative direction complete-closed time scale T2 .  From Remark 3.17, we can observe that T = T1 ∪ T2 = +∞ k=−∞ [2k, 2k + 1] and its period set Π = Π1+ ∪ Π2− , obviously, (Π, +) forms an Abelian group and T is a periodic time scale exactly. In fact, we can obtain the following theorem. Theorem 3.45 Let T be a complete-closed time scale. Then T is a periodic time scale if and only if (Π, +) is an Abelian group. Proof If T is a periodic time scale, then by Definition 2.26 (d), for any τ1 , τ2 , τ3 ∈ Π , we obtain that (i) for all τ1 , τ2 ∈ Π , we have τ1 + τ2 ∈ Π ; (ii) for all τ1 , τ2 , τ3 ∈ Π , we have (τ1 + τ2 ) + τ3 = τ1 + (τ2 + τ3 ); (iii) there exists an element 0 ∈ Π , such that for all elements τ ∈ Π , the equation 0 + τ = τ + 0 = τ holds; (iv) for each τ ∈ Π , there exists an element −τ ∈ Π such that τ + (−τ ) = 0, where 0 is the identity element; (v) for all τ1 , τ2 ∈ Π , we have τ1 + τ2 = τ2 + τ1 . Hence, (Π, +) is an Abelian group. On the other hand, if (Π, +) is an Abelian group, then by (2.29), for any τ ∈ Π \{0}, there exists −τ ∈ Π \{0} such that T−τ ⊆ T and Tτ ⊆ T, which indicates that T−τ ∪ Tτ ⊆ T, i.e., for any t ∈ T, we have t ± τ ∈ T. Hence, T is periodic. This completes the proof. 

Now, we denote the set {1, 2, . . . , m} by Λ and introduce the following concept. Definition 3.28 A subset E of a semigroup Π + induced by time  scales is relatively dense if there exists elements s1 , s2 , . . . , sm in Π + such that i∈Λ (si + E) = Π + , where si + E = {si + e : e ∈ E}. Remark 3.18 Note that Definition 3.28 can be extended to the relatively dense set E of a group Πinduced by time scales, i.e., if there exists elements s1 , s2 , . . . , sm in Π such that i∈Λ (si + E) = Π , where si + E = {si + e : e ∈ E}. Hence, for any s1∗ , s2∗ ∈ Π , there exist some τ1 , τ2 ∈ E and si , sj such that s1∗ − s2∗ = sj − si + τ1 − τ2 since s1∗ = sj + τ1 and s2∗ = si + τ2 . Definition 3.29 ([214]) A subset E of Π + is called relatively dense if there exists a positive number L ∈ Π + such that [a, a + L]Π + ∩ E = ∅ for all a ∈ Π + . The number L is called the inclusion length. Theorem 3.46 Definition 3.28 is equivalent to Definition 3.29. Proof Let E be a relatively dense set that satisfies Definition 3.28, for contradiction, assume that E does not satisfy Definition 3.29, then there exists some a0 ∈ Π + , for any L ∈ Π + , we have [a0 , a0 + L]Π + ∩ E = ∅, which implies that for any s ∈ Π + , we have [a0 + s, a0 + L + s]Π + ∩ (s + E) = ∅, so there exists no elements s1 , s2 , . . . , sm in Π + such that i∈Λ (si +E) = Π + , this contradicts Definition 3.28.

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3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

On the other hand, let E be a relatively dense set that satisfies Definition 3.29, then there exists a positive number L ∈ Π + such that [a, a + L]Π + ∩ E = ∅ for all a ∈ Π + , thus we have  (s + E) = Π + . (3.45) s∈[0,L]Π +

According to the Finite Covering Theorem, there exist some open interval (a1 , b1 )Π + , (a2 , b2 )Π + , . . . , (am−1 , bm−1 )Π + m−1 such that [0, L]Π + ⊆ i=1 (ai , bi )Π + . Now, we can choose the suitable numbers s1 , s2 , . . . , sm−1 , sm = 0 ∈ Π + such that s1 + E ⊇ (a1 , b1 )Π + , s2 + E ⊇ (a2 , b2 )Π + , . . . , sm−1 + E ⊇ (am−1 , bm−1 )Π + . Hence, from (3.45), we obtain 

(s + E) = Π + ,

(3.46)

s∈∪m i=1 (si +E)

so (3.46) yields

m

i=1 (si

+ E) = Π + . This completes the proof.



By Theorem 3.46, it is obvious that for the Abelian group (Π, +), the following two definitions are also equivalent. Definition 3.30 A subset E of a group Π induced by time  scales is relatively dense if there exists elements s1 , s2 , . . . , sm in Π such that i∈Λ (si + E) = Π , where si + E = {si + e : e ∈ E}. Definition 3.31 A subset E of Π is called relatively dense if there exists a positive number L ∈ Π + such that [a, a + L]Π ∩ E = ∅ for all a ∈ Π . The number L is called the inclusion length.

3.6 Abstract Weighted Pseudo Almost Periodic Functions In this section, we will introduce the concept of abstract weighted pseudo almost periodic functions on time scales and provide some basic properties. For this, we need the following definitions. Definition 3.32 Let T be a complete-closed time scales under translations and X be a Banach space. A function f ∈ Crd (T×X, X) is called an almost periodic function in t ∈ T uniformly for x ∈ X if the ε-translation set of f E{ε, f, D} = {τ ∈ Π : f (t + τ, x) − f (t, x) < ε, f or all (t, x) ∈ T × D}

3.6 Abstract Weighted Pseudo Almost Periodic Functions

225

is a relatively dense set in Π for all ε > 0 and for each compact subset D of X; that is, for any given ε > 0 and each compact subset D of X, there exists a constant l(ε, D) > 0 such that each interval of length l(ε, D) contains a τ (ε, D) ∈ E{ε, f, D} such that f (t + τ, x) − f (t, x) < ε,

for all t ∈ T × D.

Here, τ is called the ε-translation number of f and l(ε, D) is called the inclusion length of E{ε, f, D}. Let X be a Banach space and T be a periodic time scale, by the Sect. 3.1, the following Theorems 3.47–3.50 can be easily obtained in the same way. Theorem 3.47 Let f ∈ Crd (T×X, X) be almost periodic in t uniformly for x ∈ X. Then it is uniformly rd-continuous and bounded on T × D; here D is any compact subset of X. Corollary 3.5 Let f ∈ Crd (T, X) be almost periodic. Then it is uniformly rdcontinuous and bounded on T. Theorem 3.48 If F ∈ C(R × X, X) is almost periodic in t uniformly for x ∈ X, then F (t, x) is also continuous on T × X and almost periodic in t uniformly for x ∈ X. Corollary 3.6 If F ∈ C(R, X) is almost periodic, then F ∈ Crd (T, X) is almost periodic on T. Theorem 3.49 If fn ∈ Crd (T × X, X), n = 1, 2, . . . are almost periodic in t for x ∈ X, and the sequence {fn (t, x)} uniformly converges to f (t, x) on T × D, then f (t, x) is almost periodic in t uniformly for x ∈ X; here D is any compact subset of X. Corollary 3.7 If fn ∈ Crd (T, X), n = 1, 2, . . . are almost periodic on T, and the sequence {fn (t)} uniformly converges to f (t) on T, then f (t) is almost periodic on T. Theorem 3.50 Let f ∈ Crd (T×X, X) be almost periodic in t uniformly for x ∈ X, and let  t f (s, x)Δs, t0 ∈ T. F (t, x) = t0

Then F : T × X → X is almost periodic in t uniformly for x ∈ X if and only if F (t, x) is bounded on T × D; here D is any compact subset of X. Theorem 3.51 If u ∈ Crd (T, X) and g(t) ˆ : T → Π are almost periodic, and E{ε, u} ∩ E{ε, g} ˆ = ∅, then u t − g(t) ˆ is almost periodic.

226

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

Proof Since u : T → X is almost periodic, for any ε > 0, there exists a τ such that '    ' ε '< . 'u t + τ − g(t) ˆ − u t − g(t) ˆ 2 Now from Theorem 3.47, we have '    ' ε '< . 'u t + τ − g(t ˆ + τ ) − u t + τ − g(t) ˆ 2 Hence, it follows that '    ' ' 'u t + τ − g(t ˆ + τ ) − u t − g(t) ˆ '    ' ' ˆ + τ ) − u t + τ − g(t) ˆ = 'u t + τ − g(t '    ' ε ' ≤ + ε = ε. ˆ − u t − g(t) ˆ +'u t + τ − g(t) 2 2 

This completes the proof. Similar to Definition 3.16, we also introduce the following concept.

Definition 3.33 Assume that X is a Banach space, f ∈ Crd (T × X, X). Then m(f ) is called mean-value of f (t, x) if 1 m(f ) = lim T →∞ T



t0 +T

f (t, x)Δt ∈ X,

t0

where t0 ∈ T, T ∈ Π. By Theorem 3.21, the following result follows immediately: Theorem 3.52 Let f ∈ Crd (T×X, X) be almost periodic in t uniformly for x ∈ X. Then m(f ) exists uniformly for x ∈ X. Remark 3.19 If f ∈ Crd (T × X, X) is almost periodic in t uniformly for x ∈ X, then f (t, x) is almost periodic in t uniformly for x ∈ X. This follows from the fact that for any ε > 0, there is a τ ∈ E{ε, f, D} such that f (t + τ, x) − f (t, x) ≤ f (t + τ, x) − f (t, x) < ε. Hence, by Theorem 3.52, we have 1 T →∞ T

m(f ) = lim



t0 +T

f (t, x)Δt

t0

exists uniformly for x ∈ X, where t0 ∈ T. Let X, Y be two Banach spaces endowed with the norms  · X and  · Y , respectively. We denote by B(X, Y) the Banach space of all bounded linear

3.6 Abstract Weighted Pseudo Almost Periodic Functions

227

operators from X to Y. This is denoted by B(X) when X = Y and the norm is denoted by  · . Now BCrd (T, X) is the space of bounded rd-continuous function from T to X equipped with the supremum norm defined by u∞ = supt∈T u(t). Let U be the collection of all functions (weights) ρˆ : T → (0, +∞) which are locally integrable over T such that ρ(t) ˆ > 0 for almost each t ∈ T. For each r > 0 and ρˆ ∈ U, we set  m(r, ρ, ˆ t0 ) =

t0 +r t0 −r

ρ(t)Δt, ˆ where t0 ∈ T, r ∈ Π.

Let   U∞ = ρˆ ∈ U : lim m(r, ρ, ˆ t0 ) = ∞ ; r→∞   ˆ >0 . UB = ρˆ ∈ U∞ : ρˆ is bounded and inf ρ(t) t∈T

Before introducing the concept of weighted pseudo almost periodic functions, we need to define the “weighted ergodic” spaces P AP0 (T, X, ρ) ˆ and P AP0 (T × X, X, ρ). ˆ Let ρˆ ∈ U∞ . We define ˆ P AP0 (T, X, ρ)  = f ∈ BCrd (T, X) : lim

1 r→∞ m(r, ρ, ˆ t0 )  where r ∈ Π, t0 ∈ T



t0 +r

t0 −r

f (t)ρ(t)Δt ˆ = 0,

and  ˆ = f ∈ BCrd (T × X, X) : P AP0 (T × X, X, ρ) 1 r→∞ m(r, ρ, ˆ t0 ) lim



t0 +r

t0 −r

f (t, x)ρ(t)Δt ˆ =0

 uniformly for x ∈ X, where r ∈ Π, t0 ∈ T .

Remark 3.20 If ρ(t) ˆ ≡ 1, t0 = 0 ∈ T, then P AP0 (T, X, ρ) ˆ and P AP0 (T × X, X, ρ) ˆ are reduced to ergodic spaces PA P0 (T, X) and PA P0 (T × X, X) respectively, which are defined as    r 1 PA P0 (T, X) = f ∈ BCrd (T, X) : lim f (t)Δt = 0 r→∞ 2r −r

228

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

and  PA P0 (T × X, X) = f ∈ BCrd (T × X, X) : 1 r→∞ 2r lim



r −r

 f (t, x)Δt = 0, uniformly for x ∈ X .

Definition 3.34 A function f ∈ BCrd (T, X) is called pseudo almost periodic, if f = g + φ where g ∈ A P(T, X) and φ ∈ PA P0 (T, X). We denote the set of all such functions by PA P(T, X). Definition 3.35 A function f ∈ BCrd (T × X, X) is called pseudo almost periodic in t uniformly for x ∈ X, if f = g + φ where g ∈ A P(T × X, X) and φ ∈ PA P0 (T × X, X). We denote the set of all such functions by PA P(T × X, X). Definition 3.36 Let ρˆ ∈ U∞ . A function f ∈ BCrd (T, X) is called weighted pseudo almost periodic, if f = g + φ where g ∈ A P(T, X) and φ ∈ P AP0 (T, X, ρ). ˆ We denote the set all such functions by W P AP (T × X, X, ρ). ˆ Definition 3.37 Let ρˆ ∈ U∞ . A function f ∈ BCrd (T × X, X) is called weighted pseudo almost periodic in t uniformly for x ∈ X, if f = g + φ where g ∈ A P(T × X, X) and φ ∈ P AP0 (T × X, X, ρ). ˆ We denote the set all such functions by W P AP (T × X, X, ρ). ˆ Theorem 3.53 Let ρˆ ∈ U∞ be fixed. Then the decomposition of a weighted pseudo almost periodic function f = g+φ where g ∈ A P(T, X) and φ ∈ P AP0 (T, X, ρ) ˆ is unique. Proof Let f ∈ W P AP (T, X, ρ), ˆ if f = g1 +φ1 = g2 +φ2 , then we have g1 −g2 = φ2 − φ1 . Hence, there exists some positive constant c such that 1 c · sup g1 (t) − g2 (t) ≤ lim r→∞ m(r, ρ, ˆ t0 ) t∈T 1 = lim r→∞ m(r, ρ, ˆ t0 ) 1 r→∞ m(r, ρ, ˆ t0 )

≤ lim

  

t0 +r t0 −r t0 +r t0 −r t0 +r t0 −r

1 r→∞ m(r, ρ, ˆ t0 )

+ lim



g1 (s) − g2 (s)ρ(s)Δs ˆ φ1 (s) − φ2 (s)ρ(s)Δs ˆ φ1 (s)ρ(s)Δs ˆ

t0 +r t0 −r

φ2 (s)ρ(s)Δs ˆ = 0,

so g1 − g2 = 0 = φ1 − φ2 , i.e., g1 = g2 and φ1 = φ2 . This completes the proof.



Theorem 3.54 If ρˆ ∈ U∞ , then (W P AP (T × X, X, ρ), ˆ  · ∞ ) is a Banach space.

3.6 Abstract Weighted Pseudo Almost Periodic Functions

229

Proof For any convergent sequence {fn } ⊂ W P AP0 (T × X, X, ρ) ˆ with fn → f uniformly for t ∈ T, we can obtain 1 r→∞ m(r, ρ, ˆ t0 ) lim

1 ≤ lim r→∞ m(r, ρ, ˆ t0 )



t0 +r

t0 −r



t0 +r

t0 −r

1 + lim r→∞ m(r, ρ, ˆ t0 )



f (s, x)ρ(s)Δs ˆ fn (s, x)ρ(s)Δs ˆ

t0 +r

t0 −r

fn (s, x) − f (s, x)ρ(s)Δs, ˆ

by letting n → ∞ we have 1 lim r→∞ m(r, ρ, ˆ t0 )



t0 +r

t0 −r

f (s, x)ρ(s)Δs ˆ = 0,

which indicates W P AP0 (T × X, X, ρ) ˆ is a closed subspace of BCrd (T × Ω, X). Therefore, W P AP0 (T × X, X, ρ) ˆ is itself a Banach  space. Then by Theorems 3.49 and 3.53, we have W P AP (T × X, X, ρ), ˆ  · ∞ is a Banach space. The proof is completed. 

By Definition 3.36 and Theorem 3.53, the following theorem is immediate. Theorem 3.55 If f ∈ W P AP (T × X, X, ρ), ˆ then f (t, x) is bounded on T × D; here D is any compact subset of X. Definition 3.38 A closed subset C of T is said to be a weighted ρ-ergodic ˆ zero set in T if  ˆ C∩[t0 −r,t0 +r)T ρ(t)Δt → 0 as r → ∞, where t0 ∈ T. m(r, ρ, ˆ t0 ) Theorem 3.56 A function φ ∈ P AP0 (T × X, X, ρ) ˆ if and only if for ε > 0, the set Cε = {t ∈ T : φ(t, x) ≥ ε} is a weighted ρ-ergodic ˆ zero subset of T. Proof (a) Necessity. For contradiction, suppose that there exists ε0 > 0 such that  lim

C∩[t0 −r,t0 +r)T

r→∞

ρ(t)Δt ˆ

m(r, ρ, ˆ t0 )

= 0.

Then there exists δ > 0 such that for every n ∈ N,  C∩[t0 −rn ,t0 +rn )T

ρ(t)Δt ˆ

m(rn , ρ, ˆ t0 )

≥ δ for some rn > n, where rn ∈ Π.

230

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

Denote   Mr,ε,t0 (φ) := t ∈ [t0 − r, t0 + r)T : φ(t, x) ≥ ε , then we get 1 m(rn , ρ, ˆ t0 ) =

1 m(rn , ρ, ˆ t0 )  ×



t0 +rn

t0 −rn

φ(s, x)ρ(s)Δs ˆ



φ(s, x)ρ(s)Δs ˆ + Mrn ,ε0 ,t0 (φ)

[t0 −r,t0 +r)T \Mrn ,ε0 ,t0 (φ)

1 m(rn , ρ, ˆ t0 )

≥

ε0 m(rn , ρ, ˆ t0 )

≥

1 m(rn , ρ, ˆ t0 )

φ(s)ρ(s)Δs ˆ



φ(s, x)ρ(s)Δs ˆ Mrn ,ε0 ,t0 (φ)



ρ(s)Δs ˆ ≥ ε0 δ, Mrn ,ε0 ,t0 (φ)

and this contradicts the assumption.  C∩([t −r,t +r)

ρ(t)Δt ˆ

0 0 T (b) Sufficiency. Assume that lim m(r,ρ,t ˆ 0) r→∞ there exists r0 > 0 such that for every r > r0 ,

 C∩[t0 −r,t0 +r)T

ρ(t)Δt ˆ

m(r, ρ, ˆ t0 )


0,

ε , M

where M := supt∈T φ(t) < ∞. Now, we have 1 m(r, ρ, ˆ t0 ) =

1 m(r, ρ, ˆ t0 )  +



t0 +r t0 −r

φ(s, x)ρ(s)Δs ˆ



φ(s, x)ρ(s)Δs ˆ Mr,ε,t0 (φ)

[t0 −r,t0 +r)T \Mr,ε,t0 (φ)

≤

M



C∩[t0 −r,t0 +r)T

 φ(s, x)ρ(s)Δs ˆ

ρ(s)Δs ˆ

m(r, ρ, ˆ t0 )  ε + ρ(s)Δs ˆ ≤ 2ε. m(r, ρ, ˆ t0 ) [t0 −r,t0 +r)T \Mr,ε,t0 (φ)

3.6 Abstract Weighted Pseudo Almost Periodic Functions

231

Therefore, 1 r→∞ m(r, ρ, ˆ t0 ) lim



t0 +r t0 −r

φ(s, x)ρ(s)Δs ˆ = 0,

that is, φ ∈ P AP0 (T × X, X, ρ). ˆ This completes the proof.



Theorem 3.57 If ρˆ ∈ U∞ , then the following hold: (i) A function φ ∈ P AP0 (T × D, X, ρ) ˆ if and only if φ(·, x)2 ∈ P AP0 (T × D, R, ρ); ˆ where D is any compact subset of X. (ii) φ ∈ P AP0 (T × D, X, ρ) ˆ if and only if the norm function φ(·, x) ∈ P AP0 (T × D, R, ρ). ˆ Proof (i) The sufficiency follows since 1 m(r, ρ, ˆ t0 )



1 ≤ m(r, ρ, ˆ t0 ) =

1 m(r, ρ, ˆ t0 )

t0 +r

t0 −r





φ(s, x)ρ(s)Δs ˆ

t0 +r t0 −r t0 +r t0 −r

!1/2  φ(s, x) ρ(s)Δs ˆ

t0 +r

2

t0 −r

!1/2 φ(s, x)2 ρ(s)Δs ˆ

!1/2 ρ(s)Δs ˆ

,

then φ ∈ P AP0 (T × D, X, ρ). ˆ The necessity follows from the fact that 1 m(r, ρ, ˆ t0 )



t0 +r

t0 −r

φ(s, x)2 ρ(s)Δs ˆ ≤ φ

1 m(r, ρ, ˆ t0 )



t0 +r

t0 −r

φ(s, x)ρ(s)Δs, ˆ

because φ ∈ P AP0 (T × D, X, ρ), ˆ then φ2 ∈ P AP0 (T × D, R, ρ). ˆ (ii) By (i), φ ∈ P AP0 (T × D, X, ρ) ˆ if and only if φ(·, x)2 ∈ P AP0 (T × D, R, ρ). ˆ The latter is equivalent to φ(·, x)2 = [φ(·, x)]2 ∈ P AP0 (T × D, R, ρ), ˆ which, again by (i), is equivalent to φ(·, x) ∈ P AP0 (T×D, R, ρ). ˆ 

The proof is completed.

Lemma 3.9 Let f be a positive direction Bochner almost periodic function on a semigroup Π + . If NT ⊂ T is finite and ε > 0 is given, the set   Cε (NT ) = τ ∈ Π + : max |f (t + τ ) − f (t)| < ε t∈NT

is relatively dense.

(3.47)

232

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

Proof For the sake of contradiction, assume that there exists a finite set NT ⊂ T and an ε > 0 such that the set Cε (NT ) of (3.47) Then there  is not relatively dense. + . Now, we will do not exist s1 , s2 , . . . , sm ∈ Π + such that m (s + C ) = Π i ε i=1 construct a sequence {τn } ⊂ Π + and {τ˜n } ⊂ Π + determined by {τn } such that τ˜n + τk ∈ Cε for k < n. For n = 1 let τ1 be an arbitrary element of Π + . Now, we select τ1 , τ2 , . . . , τn ∈  Π + such that τ˜m +τl ∈ Cε , m > l, there will exist an element τ˜ = τ˜n+1 with τ˜n+1 ∈ nj=1 (τj +Cε ). Then we also have τ˜n+1 +τk ∈ Cε for k ≤ n. From this constructed sequence we can extract a subsequence {τi∗ } = τ ∗ , τi∗ = τni and a sequence τ˜ ∗ ⊂ τ˜ depending on τ ∗ such that Tτ ∗ f (t) = g(t) and Tτ˜ ∗ g(t) = f (t) for t ∈ NT . Let τ˜j∗ = τ˜nj be selected so that maxt∈NT |g(t + τ˜j∗ )−f (t)| < ε/2, and then choose τi∗ = τni with nj > ni such that max |f (t + τi∗ + τ˜j∗ ) − f (t)| ≤ max |f (t + τi∗ + τ˜j∗ ) − g(t + τ˜j∗ )|

t∈NT

t∈NT

+ max |g(t + τ˜j∗ ) − f (t)| < ε/2 + ε/2 = ε. t∈NT

Hence τi∗ + τ˜j∗ = τni + τ˜nj ∈ Cε which is a contradiction with the construction of the sequence because nj > ni . Thus, Cε (NT ) must be relatively dense. This completes the proof. 

By Lemma 3.9 and Definition 3.30, the following lemma is obvious. Lemma 3.10 Let (Π, +) be an Abelian group and g : T → X is an almost periodic function. For fixed t0 ∈ T let Bε := {τ ∈ Π : g(t0 + τ ) − g(t0 ) < ε}. Then there exist s1 , . . . , sm ∈ Π such that m 

(si + Bε ) = Π.

i=1

Theorem 3.58 If f ∈ W P AP (T × X, X, ρ) ˆ and g is its almost periodic component, then g(T × D) ⊂ f (T × D)

(3.48)

and f ∞ ≥ g∞ ≥

inf

(t,x)∈T×D

g(t, x) ≥

inf

(t,x)∈T×D

f (t, x),

where f (T × D) denotes the value field of f on T × D, f (T × D) denotes the closure of f (T × D) and is the same as g(T × D); here D is an arbitrary compact subset of X.

3.6 Abstract Weighted Pseudo Almost Periodic Functions

233

Proof Fix f ∈ W P AP (T × X, X, ρ). ˆ There exist g ∈ A P and a bounded continuous function φ with vanishing weighted mean-value such that f = g + φ. For contradiction, if (3.48) is not true, then there exists t0 ∈ T, ε > 0 such that g(t0 , x) − f (t0 , x) ≥ 2ε, t ∈ T.

(3.49)

Let s1 , s2 , . . . , sm be as in Definition 3.28, t ∗ ∈ T and write s˜i = si − t0 (1 ≤ i ≤ m), η = max |˜si |. 1≤i≤m

For T ∈ Π with |T | > η, let   (i) Bε,T ,t ∗ = F (Ait ∗ ) − T + η − F (Ais˜i ), F (Ait ∗ ) + T − η − F (Ais˜i )  (F (Ait0 ) + Bε ), 1 ≤ i ≤ m, where Bε is as in Lemma 3.10 for the function g(t, x). Using Lemma 3.10 gives that m   (i)  F (Ais˜i ) + Bε,T ,t ∗ = [F (Ait ∗ ) − T + η, F (Ait ∗ ) + T − η)Π , i=1

that is, there exists t ∗∗ ∈ Ait ∗ such that m      ∗∗ (i)  − T + η, t ∗∗ + T − η T (t0 + Bε ). s˜i + Bε,T ,t ∗ = t i=1

Hence 

 [t ∗∗ −T +η,t ∗∗ +T −η)T

ρ(t)Δt ˆ = ≤

(i)

∪m si +Bε,T ,t ∗∗ ) i=1 (˜ m   (i)

i=1

s˜i +Bε,T ,t ∗∗

ρ(t)Δt ˆ

ρ(t)Δt ˆ



≤ m max

1≤i≤m s˜i +B (i) ∗∗ ε,T ,t

ρ(t)Δt, ˆ

so 1 1 1 1 = ≥  . m(T − η, ρ, ˆ t ∗∗ ) m ρ(t)Δt ˆ ρ(t)Δt ˆ (i) [t ∗∗ −T +η,t ∗∗ +T −η)T s˜ +B i

ε,T ,t ∗∗

(3.50)

234

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

By (3.49), for any t ∈ t0 + Bε , we have φ(t, x) = f (t, x) − g(t, x) ≥ g(t0 , x) − f (t, x) − g(t, x) − g(t0 , x) > ε. This and (3.50) together yields that 

1 m(T − η, ρ, ˆ t ∗∗ )  ≥ε·

[t ∗∗ −T +η,t ∗∗ +T −η)T

φ(t, x)ρ(t)Δt ˆ

ˆ [t ∗∗ −T +η,t ∗∗ +T −η)T ρ(t)Δt m(T − η, ρ, ˆ t ∗∗ )

 ˆ ε ε [t ∗∗ −T +η,t ∗∗ +T −η)T ρ(t)Δt  · ≥ ≥ , as T → ∞. m max1≤i≤m s˜ +B (i) m ρ(t)Δt ˆ i

ε,T ,t ∗∗

This is a contradiction. Hence, (3.48) holds. The proof is completed.



From Theorem 3.58, we can easily prove the following theorem. Theorem 3.59 If f ∈ W P AP (T × X, X, ρ) ˆ satisfy the Lipschitz condition f (t, x) − f (t, y) ≤ Lf x − y, for all x, y ∈ X, t ∈ T,   ˆ implies f ·, φ0 (·) ∈ W P AP (T, X, ρ). ˆ then φ0 ∈ W P AP (T, X, ρ)

3.7 Almost Periodic Functions on Changing-Periodic Time Scales We now propose a completely new concept of almost periodic functions on changing-periodic time scales, which not only includes the concept of almost periodic functions on periodic time scales, but also includes the concept of almost periodic functions on almost periodic time scales, and it is more general and comprehensive. For this, we need the following notations: Let α τ = {αnτ } ⊂ Sτt and β τ = {βnτ } ⊂ Sτt be two adaption factors sequences for t under the index function τ . Then β τ ⊂ α τ means that β τ is a subsequence of α τ ; α τ + β τ = {αnτ + βnτ }; −α τ = {−αnτ }; En denotes Rn or Cn , D denotes an open set in En or D = En , and S denotes an arbitrary compact subset of D. We will also need the translation operator Tα τ , Tα τ f (t, x) = g(t, x) which means that g(t, x) = lim f (t + αnτ , x) provided n→+∞

the limit exists. Definition 3.39 Let T be a changing-periodic time scale, i.e., T satisfies Definition 2.34. A function f ∈ C(T × D, En ) is called a local-almost periodic function in t ∈ T uniformly for x ∈ D if the ε-translation number set of f

3.7 Almost Periodic Functions on Changing-Periodic Time Scales

235

E{ε, f, S} = {τ˜ ∈ Sτt : |f (t + τ˜ , x) − f (t, x)| < ε, f or all (t, x) ∈ T × S} is a relatively dense set for all ε > 0 and for each compact subset S of D; that is, for any given ε > 0 and each compact subset S of D, there exists a constant l(ε, S) > 0 such that each interval of length l(ε, S) contains a τ˜ (ε, S) ∈ E{ε, f, S} such that |f (t + τ˜ , x) − f (t, x)| < ε,

for all(t, x) ∈ T × S;

here, τ˜ is called the ε-local translation number of f and l(ε, S) is called the local inclusion length of E{ε, f, S}. Remark 3.21 Since the changing-periodic time scales include periodic and almost periodic time scales, if T is a τ˜ -periodic time scale, then Tr = ∅, R0 = ∅ and Sτt = {nτ˜ : n ∈ Z}; if T is an almost periodic time scale, then μ is bounded, so T is a changing-periodic time scale. Remark 3.22 In Definition 3.39, we require E{ε, f, S} to be a relatively dense set. Thus, by the definition and the property of Sτt , we can obtain the local almost periodicity on the periodic sub-timescale Tτt from the Definition 3.39 (see Fig. 3.2). Now we give another definition which in view of Theorem 2.31 is equivalent to Definition 3.39. Definition 3.40 Assume that T is a changing-periodic time scale. Let f ∈ C(T ×  D, En ), if for any given adaption factors sequence (α τ ) ⊂ Sτt , there exists a

Fig. 3.2 The local-almost periodicity of the function f occurs on the periodic sub-timescale T2 through ε-local translation number τ0 , but the function f is without almost periodicity on the periodic sub-timescale T1

236

3 Almost Periodic Functions and Generalizations on Complete-Closed Time Scales 6 f(t) at the right scattered and left dense point

4 2 0 −2 −4 f(t) at the right dense and left dense point

−6 −8

0

5

10

15

20

25

30

Fig. 3.3 f (t) is a local-almost periodic function on the periodic sub-timescale of the set  ∞ [pm , a + pm ). However, f (t) will lose its almost periodicity at the right scattered points m=1 ∞ m=1 {a + pm } 

subsequence α τ ⊂ (α τ ) such that Tα τ f (t, x) exists uniformly on T × S, then f (t, x) is called a local-almost periodic function in t uniformly for x ∈ D. Example 3.3 Consider the changing-periodic time scale given in Example 2.12, with a = 3, and define 

√ 7t +

√ sin t + cos 5t, f (t) = √ t e−2t cos t + 5e 100 sin 13t, sin

1 7

if t ∈ if t ∈

∞

m=1 [pm , a

∞

m=1 {a

+ pm ),

+ pm },

then f (t) is a local-almost periodic function on T. It is worth noting  that f (t) is almost periodic only on a local part of this time scale. In fact, if t ∈ ∞ m=1 {a + pm }, then the function f (t) is not almost periodic on these points since f (t) becomes unbounded as t increases. Hence, f (t) is only a local-almost periodic function on the subset of ∞ m=1 [pm , a + pm ). From Fig. 3.3, and in view of Theorem 2.31, it is clear that f (t) is local-almost periodic on the periodicsub-timescale of the set ∞ [p , a + p ), except at the right scattered points ∞ m m m=1 m=1 {a + pm }, and thus by the definition of f (t), it will not be almost periodic on the whole time scale T. From the Definition 3.40, we have the following proposition: Proposition 3.3 Let T be a changing-periodic time scale. If f ∈ C(T × D, En ) is a local-almost periodic in t uniformly for x ∈ D, then f ∈ C(T0 × D, En ) is localalmost periodic in t uniformly for x ∈ D, where T0 ⊂ T is a changing-periodic time scale.

3.7 Almost Periodic Functions on Changing-Periodic Time Scales

237

Proof Let f ∈ C(T × D, En ) be uniformly local-almost periodic, then, by  Definition 3.40, for any adaption factors sequence (α τ ) ⊂ Sτt ⊂ Π , there exists a  subsequence α τ ⊂ (α τ ) such that Tα τ f (t, x) exists uniformly on T × S, where S is any compact set in D. Consequently, Tα τ f (t, x) exists uniformly on T0 × S. This completes the proof. 

Next, we have the following definition: Definition 3.41 Let f, g ∈ C(T×D, En ) be uniformly local-almost periodic and T be a changing-periodic time scale. We say f and g are synchronously local-almost periodic if f, g are almost periodic on the same periodic sub-timescales of T. From Theorem 2.31, we can deduce the following result for synchronously localalmost periodic functions. Theorem 3.60 If f, g ∈ C(T×D, En ) are two synchronously local-almost periodic functions, then for any ε > 0, the intersection of ε-local translation number sets of f  and g is a nonempty relatively dense set, i.e., E{ε, f, S} E{ε, g, S} is a nonempty relatively dense set. Proof If f, g ∈ C(T × D, En ) are two synchronously local-almost periodic functions, then by Definition 3.41, f, g are almost periodic on the same periodic sub-timescales of T. Now from Theorem 3.11, the desired conclusion follows immediately. This completes the proof. 

In what follows, we will give the concept of combinable-almost periodic functions on changing-periodic time scales by Definition 3.39. Definition 3.42 Let T be a changing-periodic time scale. If there exists an ωi0 periodic sub-timescale set {Ti0 }i0 ∈I such that the period set {ωi0 }i0 ∈I has a lowest common multiple ω and f is almost periodic on Ti0 for each i0 , where I is a combinable index number set, then f is called a combinable-almost periodic function onT or f is called an almost periodic function on the ω-periodic sub timescale i0 ∈I Ti0 . Further, if i0 ∈I Ti0 = T, then f is called the globally combinable-almost periodic function on T. Corollary 3.8 The function f is globally combinable-almost periodic on T if and only if f is an almost periodic function on the periodic time scale T.  Proof Since Ti is an ωi -periodic sub-timescale and T = i∈I Ti , then we obtain T is an ω-periodic time scale, where ω is a lowest common multiple of {ωi }i∈I and I is a combinable index number set. Hence, from Definition 3.42 the desired conclusion follows immedaitely. This completes the proof. 

Remark 3.23 From Corollary 3.8 it follows that the concept of almost periodic functions on periodic time scales is equivalent to the concept of globally combinable-almost periodic functions on changing-periodic time scales.

Chapter 4

Piecewise Almost Periodic Functions and Generalizations on Translation Time Scales

4.1 Piecewise Almost Periodic Functions on CCTS The discontinuous dynamic equations have been widely studied by many researchers and many classical results were published (see Akhmet and Turan [34, 35, 38], Akhmet and Yılmaz [36], Akhmet et al. [37], Afonso et al. [46, 47], Benchohra et al. [52, 53], Bainov and Milusheva [54], Lakshmikantham et al. [175], Samoilenko and Perestyuk [201], Tang et al. [206], Wang [232]). The discontinuous functions were also introduced on time scales and applied to study impulsive dynamic equations on time scales (see Chang and Li [89], Graef and Ouahab [139], Hatipoˇglu et al. [146], Mesquita [183], Wang et al. [218, 227, 229] and Wang and Agarwal [228, 230]). To study almost periodic problems on time scales, the discontinuous almost periodic functions were introduced by Wang and Agarwal in the literatures [233, 234] with the applications to the almost periodic biological dynamic models on time scales. In addition, a new almost periodicity called double almost periodicity was also introduced and applied to study several dynamic system models on time scales (see Wang et al. [214, 240]). In this section, we will introduce and study the piecewise almost periodic functions on complete-closed time scales. Definition 4.1 We say ϕ : T → Rn is piecewise rd-continuous with respect to a sequence {τi } ⊂ T which satisfy τi < τi+1 , i ∈ Z, if ϕ(t) is continuous on [τi , τi+1 )T and rd-continuous on T\{τi }. Furthermore, [τi , τi+1 )T , i ∈ Z, are called intervals of continuity of the function ϕ(t). Now, we give an example of this kind of rd-piecewise continuous functions on a time scale T. Example 4.1 Let a, b > 0, k ∈ Z and consider the time scale

© Springer Nature Switzerland AG 2020 C. Wang et al., Theory of Translation Closedness for Time Scales, Developments in Mathematics 62, https://doi.org/10.1007/978-3-030-38644-3_4

239

240

4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . . +∞ 

Pa,b =

[k(a + b, k(a + b) + a].

k=−∞

Then σ (t) =

 t

if t ∈ ∪+∞ k=−∞ [k(a + b), k(a + b) + a)

t + b if t ∈ ∪+∞ k=−∞ {k(a + b) + a},

and we can define a function ⎧ √ 1 2 ⎪ ⎪ ⎨cos 7t + √ 7 cos t + t ϕ(t) = sin t + sin 3t, ⎪ ⎪ ⎩(a + b) sin t

if t ∈ ∪+∞ k=−∞ [k(a + b), k(a + b) + a/2)

if t ∈ ∪+∞ k=−∞ [k(a + b) + a/2, k(a + b) + a) if t ∈ ∪+∞ k=−∞ {k(a + b) + a}.

Then ϕ(t) is rd-piecewise continuous with respect to a sequence {k(a + b) + a/2}, k ∈ Z.

 For convenience, P Crd (T, Rn ) denotes the set of all piecewise continuous functions with respect to a sequence {τi }, i ∈ Z. For any integers i and j , denote j j τi = τi+j − τi and consider the sequence {τi }, i, j ∈ Z. It is easy to verify that the j number τi , i, j ∈ Z satisfy j

j

j

j −k

k − τik , τi − τik = τi+k . τi+k − τi = τi+j

(4.1)

Definition 4.2 For any ε > 0, let Γε ⊂ Π be a set of real numbers and {τi } ⊂ T. j We say {τi }, i, j ∈ Z is equipotentially almost periodic on a periodic time scale T if for r ∈ Γε ⊂ Π , there exists at least one integer k such that |τik − r| < ε, for all i ∈ Z. In the following, we will give the definition of piecewise rd-continuous almost periodic functions with respect to the sequence {τi , }i∈Z on a periodic time scale T. Definition 4.3 Let T be a periodic time scale and assume that {τi } ⊂ T satisfying j the derived sequence {τi }, i, j ∈ Z, is equipotentially almost periodic. A function ϕ ∈ P Crd (T, Rn ) is said to be piecewise rd-continuous almost periodic (short for rd-piecewise almost periodic) if: (i) for any ε > 0, there is a positive number δ = δ(ε) such that if the points     t and t belong to the same interval of continuity and |t − t | < δ, then   ϕ(t ) − ϕ(t ) < ε; (ii) for any ε > 0, there is relative dense set Γε ⊂ Π of ε-almost periods such that if τ ∈ Γε , then ϕ(t + τ ) − ϕ(t) < ε for all t ∈ T which satisfies the condition |t − τi | > ε, i ∈ Z.

4.1 Piecewise Almost Periodic Functions on CCTS

241

Lemma 4.1 If ϕ ∈ P Crd (T, Rn ) is rd-piecewise almost periodic, then for any ε > 0, there exists a relative dense set of intervals of a fixed length γε ∈ Π , which consist of ε-almost periods of the function ϕ(t). Proof Let l be the density index of the set Γ , where Γ is the set of 2ε -almost periods of the function ϕ(t), and the number γ2ε = δ( 2ε ) be determined by using the uniform   continuity of this function, i.e. if t and t belongs to the same interval of continuity     of the function ϕ(t) and |t − t | < γ2ε , then ϕ(t ) − ϕ(t ) < 2ε . Let L = l + γε and consider an arbitrary interval [a, a + L]Π . By the definition   of an almost period, there is an 2ε -almost period r ∈ [a − γ2ε , a + γ2ε + l]Π , then [r − γ2ε , r + γ2ε ]Π ⊂ [a, a + L]Π . Let ξ be an arbitrary number from the interval [r − γ2ε , r + γ2ε ]Π . By  using the inequality |ξ − r| < δ and let t = t − r + ξ , for all t ∈ T, we have  ε |t − τi | > ε, |t − τi | > 2 and 





ϕ(t + ξ ) − ϕ(t) ≤ ϕ(t + r) − ϕ(t ) + ϕ(t ) − ϕ(t) < This completes the proof.

ε ε + = ε. 2 2 

Lemma 4.2 Let ϕ ∈ P Crd (T, Rn ) be a rd-piecewise almost periodic function with values in the set E ⊂ Rn . If F (y)is an uniformly continuous function defined on the set E, then the function F ϕ(t) is rd-piecewise almost periodic in t. Proof Using the uniform continuity of the function F , this theorem is easy to establish so we omit it here.

 Lemma 4.3 For any two rd-piecewise almost periodic functions with respect to the same sequence {τi } ⊂ T, for any ε > 0 there exists a relative dense set of their common ε-almost periods. Proof Let ϕ1 (t) and ϕ2 (t) be two rd-piecewise almost periodic functions with the same sequence {τi } ⊂ T. By Lemma 4.1, there exist numbers l1 and l2 such that every interval [a, a + l1 ]Π and [a, a + l2 ]Π contain the corresponding 4ε -almost   periods r1 and r2 which are multiples of γε (where γε is the proper positive number in Π ). If we set l = max(l1 , l2 ), then in each segment [a, a + l]Π , there exists a pair       of 4ε -almost periods r1 = n γε ∈ Π and r2 = n γε ∈ Π with integers n and n .       Since r1 − r2 = (n − n )γε = nγε and |nγε | ≤ l, we can see that nγε can take only    a finite number of values. Let these values be n1 γε , n2 γε , . . . , np γε with the pairs of p p 1 1 2 2 almost periods (r1 , r2 ), (r1 , r2 ), . . . , (r1 , r2 ) as their representatives, i.e. such that  r1s − r2s = ns γε , s = 1, . . . , p. Set maxs {|r1s |} = T . Let [a, a + l + 2T ]Π be an arbitrary segment of length l + 2T . In the segment [a + T , a + l + T ]Π , take two     ε 4 -almost periods of the functions ϕ1 (t) and ϕ2 (t), r1 = n γε and r2 = n γε , and let  r1 − r2 = ns γε = r1s − r2s . Hence we get r = r1 − r1s = r2 − r2s , r ∈ [a, a + l + 2T ]Π .

(4.2)

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4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

The set Γ ⊂ Π of all the numbers given by relation (4.2) is relatively dense and  consists of multiples of γε . Now, let us show that there is a subset Γ0 ⊂ Γ , relatively dense in Π , such that, for all r ∈ Γ0 , |t + r − τi | > 2ε if |t − τi | > ε, i ∈ Z.   Let l = l(ε) be the density index of the set Γ ⊂ Π , and l = l (ε) be the density    ε index of the set Γ ⊂ Π which are multiples of γε for 4 . Clearly, γε can be chosen    so that the inequalities l, l < +∞ hold. Set l = max(l, l ), then for any interval       [a, a + l ]Π , there exist such integers m, m , and q that mγε , m γε ∈ [a, a + l ]Π and q



|τk − mγε |
ε and t ∈ (τk +ε, τk+1 −ε)T . Then t +r ∈ (τk +r +ε, τk+1 +r −ε)T . Since |τkh − r| < 2ε for k ∈ Z, we have that r ∈ (τih − 2ε , τih + 2ε )Π , consequently, ε ε < t + r < τk+1 + τk+h+1 − τk+1 + − ε, 2 2   ε ε t + r ∈ τk+h + , τk+h+1 − , 2 2 T

τk+h − τk + τk + ε −

i.e. |t + r − τi | >

ε , i = 0, ±1, ±2, . . . . 2

Hence the set Γ0 is nonempty and relatively dense in Π . Therefore, for j = 1, 2, r ∈ Γ0 , one has 











ϕj (t + r) − ϕj (t) = ϕj (t + (m − ms )γε ) − ϕj (t) 



≤ ϕj (t + (m − ms )γε ) − ϕj (t + m γε )

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243 



+ϕj (t) − ϕj (t + m γε ) ε ε < + = ε. 2 2 

This completes the proof. Now, introduce the set   B = {tk } : tk ∈ T, tk < tk+1 , k ∈ Z, lim tk = ±∞ , k→±∞

which denotes all unbounded increasing sequences of real numbers. Let X be a Banach space, Ω be an open set in X or Ω = X, and S denotes an arbitrary compact subset of Ω. Definition 4.4 The functions f, g ∈ P Crd (T × Ω, X) are said to be ε-equivalent uniformly for x ∈ Ω or f, g possess uniform ε-equivalence for x ∈ Ω, and denote ε f ∼ g, if for all ε > 0 and for each compact subset S of Ω, the following conditions hold: (i) The points of possible discontinuity of these functions can be enumerated f g f g tk , tk , admitting a finite multiplicity by the order in T, so that |tk − tk | < ε.      (ii) There exist strictly increasing sequences of numbers {tk }, {tk }, tk < tk+1 , tk <  tk+1 , k ∈ Z, for which we have 



sup 











f (t, x) − g(t , x) < ε, |tk − tk | < ε, ∀x ∈ S, k ∈ Z.

t∈(tk ,tk+1 )T ,t ∈(tk ,tk+1 )T

Remark 4.1 One can see that if f = g in Definition 4.4, it follows from (ii) that f (t, x) is uniformly continuous in t for any x ∈ S. By d(f, g) = inf ε we denote the distance between functions f ∈ P Crd (T × Ω, X) and g ∈ P Crd (T × Ω, X), and by P Crd ϕ the set of all functions ϕ ∈ P Crd (T × Ω, X), for which d(f, ϕ) is a finite number. It is easy to verify that P Crd ϕ is a metric space. Next, we will introduce the concept of uniformly rd-piecewise almost periodic functions on time scales. Definition 4.5 (The First Definition) The function ϕ ∈ P Crd (T × Ω, X) is said to be rd-piecewise almost periodic in t uniformly for x ∈ Ω, if for any ε > 0 and for each compact subset S of Ω, the set   T¯ (ε, ϕ, S) = {τ ∈ Π : d ϕ(t + τ, x), ϕ(t, x) < ε, ∀(t, x) ∈ T × S} is relatively dense set.

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4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

Remark 4.2 By Definition 4.5, one can easily see that ϕ ∈ P Crd (T × Ω, X) is ε rd-piecewise almost periodic in t uniformly for x ∈ Ω if and only if ϕ(t + τ, x) ∼ ϕ(t, x). From Definition 4.5, the following statement follows. Definition 4.6 Let T be a periodic time scale and assume that {tk } ⊂ T satisfying j the derived sequence {tk }, k, j ∈ Z, is equipotentially almost periodic. A function ϕ ∈ P Crd (T × Ω, X) is said to be rd-piecewise almost periodic in t uniformly for x ∈ Ω if for any ε > 0 and for each compact subset S of Ω: 



(i) there is a positive number δ = δ(ε, S) such that if the points t and t belong to     the same interval of continuity and |t −t | < δ, then ϕ(t , x)−ϕ(t , x) < ε; (ii) there is relative dense set Γε ⊂ Π of ε-almost periods such that if τ ∈ Γε , then ϕ(t + τ, x) − ϕ(t, x) < ε for all (t, x) ∈ T × S which satisfies the condition |t − tk | > ε, k ∈ Z. Obviously, a rd-piecewise almost periodic function can be regarded as a special case of an uniformly rd-piecewise almost periodic function. So, in the following, we mainly discuss the basic properties of uniformly rd-piecewise almost periodic functions. The basic properties of rd-piecewise almost periodic functions can be derived directly from the corresponding ones of uniformly rd-piecewise almost periodic functions. Theorem 4.1 Let ϕ ∈ P Crd (T × Ω, X) be rd-piecewise almost periodic in t uniformly for x ∈ Ω. Then it is uniformly rd-continuous on T\B and bounded on T × S. Proof Let ϕ ∈ P Crd (T × Ω, X) be a rd-piecewise almost periodic function and rd-continuous on T\B, similar to the proof of Theorem 3.3, we can obtain ϕ is uniformly rd-continuous on T\B. Now, let l = l(1, S) be the density index of the set Γ1 , for any t0 ∈ T, M= 



max

(t,x)∈[t0 ,t0 +l]T ×S 





ϕ(t, x), ϕ(t , x) − ϕ(t , x) ≤ M1



if |t − t | ≤ 1 and t , t belong to the same interval of continuity of the function ϕ(t, x) in t. By the definition of almost periodicity, for any t ∈ T, |t − tk | > 1, k ∈ Z, there exists 1-almost period r ∈ [F (Ait0 ) − F (Ait ), F (Ait0 ) − F (Ait ) + l]Π , i.e., r + F (Ait ) ∈ [F (Ait0 ), F (Ait0 ) + l]Π such that t + r ∈ [t0 , t0 + l]T and ϕ(t + r, x) − ϕ(t, x) < 1. It follows that 

ϕ(t, x) < M + M1 + 1 for any (t, x) ∈ T × S. This completes the proof.

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245

Let T , P ∈ B and let s(T ∪ P ) : B → B be a map such that the set s(T ∪ P ) forms a strictly increasing sequence. For D ⊂ R and 0 < h ∈ Π , we introduce the  notations θh (D) = {t +h : t ∈ D}, Fh (D) = D∩{θh (D)}. Denote by φ = ϕ(t), T the element from the space P Crd (T × Ω,  sequence of real  X) × B and for every numbers{sn }, n = 1, 2, . . . with , we shall consider θ φ = ϕ(t + s , x), T + s s n n n  the sets ϕ(t + sn , x), T + sn ⊂ P Crd × B, where T + sn := T sn = {tk + sn : k ∈ Z, n = 1, 2, . . .}.   Definition 4.7 The sequence {φn },φn = ϕn (t, x),   Tn ∈ P Crd (T × Ω, X) × B is convergent to φ, φ = ϕ(t, x), T , ϕ(t, x), T ∈ P Crd (T × Ω, X) × B, if and only if for any ε > 0 and for each compact subset S of Ω, there exists n0 > 0 such that n ≥ n0 implies d(T , Tn ) < ε, ϕn (t, x) − ϕ(t, x) < ε,    uniformly for (t, x) ∈ T\Fε s(Tn ∪ T ) × S, d(·, ·) is an arbitrary distance in B. Now we will introduce the second concept of rd-piecewise almost periodic functions on time scales which is equivalent to Definition 4.5. Before this, we will prove the following two lemmas which will play an important role in the proof of the equivalence of these two definitions. For convenience, we will introduce the translation operator S, let we denote by Sα+β φ and Sα Sβ φ the limits lim θαn +βn (φ) and lim θαn ( lim θβm φ), respecn→∞ n→∞ m→∞ tively and are written only when the limits exist. Lemma 4.4 Let f ∈ P Crd (T × Ω, X) be rd-piecewise almost periodic in t  uniformly for x ∈ Ω. Then for any given sequence α ⊂ Π , there exists a  subsequence β ⊂ α and g ∈ P Crd (T × Ω, X) such that Sβ f˜(t, x) = g(t, ˜ x)   ˜ on P Crd (T × Ω, X) × B, where f = f (t, x), Tf , g˜ = holds uniformly  g(t, x), Tg , Tf , Tg ∈ B and g ∈ P Crd (T×Ω, X) is rd-piecewise almost periodic in t uniformly for x ∈ Ω. Proof For any ε > 0 and for each compact subset S of Ω, let l = l(ε/4, S) be an   inclusion length of T¯ (ε/4, f, S). For any given sequence α = {αn } ⊂ Π , we denote       αn = τn + γn , where τn ∈ T¯ (ε/4, f, S), γn ∈ Π and 0 ≤ γn ≤ l, n = 1, 2 . . . .  (In fact, for any interval with length of l, there exists τn ∈ T¯ (ε/4, f, S), thus we   can choose a proper interval with length of l such that 0 ≤ αn − τn ≤ l, from the    definition of Π , it is easy to see that γn = αn − τn ∈ Π .) Therefore, there exists a   subsequence γ = {γn } ⊂ γ = {γn } such that γn → s as n → ∞, 0 ≤ s ≤ l. Also, by Definition 4.5, there exists δ(ε, S) > 0 so that t10 , t20 belong to the same intervals of continuity of f and |t10 − t20 | < δ implies f (t10 , x) − f (t20 , x)
ε, |t20 − tk | > ε, k ∈ Z. 2 1

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4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

Since γ is a convergent sequence, there exists N = N (δ, S) so that p, m ≥ n    implies |γp − γm | < δ. Now, one can take α ⊂ α , τ ⊂ τ = {τn } such that α, τ common with γ , then for any integers p, m ≥ N , we have f (t + τp − τm , x) − f (t, x) ≤ f (t + τp − τm , x) − f (t + τp , x) +f (t + τp , x) − f (t, x) ε ε ε < + = , 4 4 2 and τ −τm

d(Tf p

τ −τm

, Tf ) ≤ d(Tf p

τ −τ

τ

τ

, Tf p ) + d(Tf p , Tf ) ≤

ε ε ε + = , 4 4 2

τ

where Tf p m and Tf p are the points of discontinuity of functions f (t + τp − τm ) and f (t + τp ), that is   ε ¯ ,f . (αp − αm ) − (γp − γm ) = τp − τm ∈ T 2 Hence we can obtain f (t + alphap , x) − f (t + αm , x) ≤

sup

f (t + αp , x) − f (t + αm , x)

(t,x)∈T×S

≤

sup

f (t + αp − αm , x) − f (t, x)

(t,x)∈T×S

≤

sup

f (t + αp − αm , x) − f (t + γp − γm , x)

(t,x)∈T×S

+

f (t + γp − γm , x) − f (t, x)

sup

(t,x)∈T×S

≤

ε ε + = ε. 2 2

Meanwhile, α

α −αm

d(Tf p , Tfαm ) = d(Tf p

α −αm

≤ d(Tf p

, Tf ) γ −γm

, Tf p

γ −γm

) + d(Tf p

, Tf ) ≤

ε ε + = ε. 2 2

(k)

Thus, we can take sequences α (k) = {αn }, k = 1, 2, . . . , and α (k+1) ⊂ α (k) ⊂ α such that, for any integers m, p and all (t, x) ∈ T × S, the following holds: (k) |f (t + αp(k) , x) − f (t + αm , x)|
N implies 



f (tm + βn , xm ) − g(tm , xm )
0, we can find an interval with length of l and there is no ε0 -translation numbers of f (t, x) in this interval, that is, every point in this interval is not in T¯ (ε, f, S).  One can take a number α1 ∈ Π and find an interval (a1 , b1 ) with b1 − a1 >  2|α1 |, where a1 , b1 ∈ Π such that there is no ε0 -translation numbers of f (t, x) in    this interval. Next, taking α2 = (1/2)(a1 + b1 ), obviously, α2 − α1 ∈ (a1 , b1 ),   so α2 − α1 ∈ T¯ (ε0 , f, S), then one can find an interval (a2 , b2 ) with b2 − a2 >   2(|α1 | + |α2 |), where a2 , b2 ∈ Π such that there is no ε0 -translation numbers of     f (t, x) in this interval. Next, taking α3 = (1/2)(a2 + b2 ), obviously, α3 − α2 , α3 −  α1 ∈ T¯ (ε0 , f, S). One can repeat these processes, again and again one can find     α4 , α5 , . . . , such that αi − αj ∈ T¯ (ε0 , f, S), i > j . Hence, for any i = j, i, j = 1, 2 . . ., without loss of generality, let i > j , we have sup (t,x)∈T×S





f (t+αi , x)−f (t+αj , x) =



sup (t,x)∈T×S



f (t+αi −αj , x)−f (t, x) ≥ ε0 ,

and α



α







α −αj

d(Tf i , Tf j ) = d(Tf i

, Tf ) ≥ ε0 .

Therefore, there is no uniformly convergent subsequence of {f (t + αn ), Tfαn } for (t, x) ∈ T × S, this is a contradiction. Thus, f ∈ P Crd (T × Ω, X) is rd-piecewise almost periodic in t uniformly for x ∈ Ω. This completes the proof. 

Now, by Lemmas 4.4 and 4.5, we can obtain the following second equivalent definition of uniformly rd-piecewise almost periodic functions on time scales. Definition 4.8 (The Second Definition) The function ϕ ∈ P Crd (T × Ω, X) is said to be rd-piecewise almost periodic in t uniformly for x ∈ Ω with respect to  a sequence T ∈ B if for every sequence of real numbers {sm } ⊂ Π there exists  a subsequence {sn }, sn = smn such that {θsn φ} uniformly converges on P Crd (T ×  Ω, X) × B, where φ = ϕ(t, x), T ).

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249

For the convenience of discussion, we shall consider the second definition for uniformly rd-piecewise almost periodic functions on time scales. Based on the second concept of uniformly rd-piecewise almost periodic functions on time scales, we will give a sufficient and necessary condition to guarantee that a function ϕ ∈ P Crd (T × Ω, X) is rd-piecewise almost periodic in t uniformly for x ∈ Ω. Theorem 4.2 The function ϕ ∈ P Crd (T × Ω, X) is rd-piecewise almost periodic in t uniformly for x ∈ Ω with respect to a sequence T ∈ B if and only if from every     pair of sequence α , β one can extracts common subsequences α ⊂ α , β ⊂ β such that Sα+β φ = Sα Sβ φ

(4.8)

  exists pointwise, where φ = ϕ(t, x), T . 



Proof Let (4.8) exist pointwise, γ ⊂ Π be a sequence, such that for γ ⊂ γ , Sγ φ exists. If Sγ φ is uniform, we are done. If not, we can find ε > 0 and sequence  β ⊂ γ , β ⊂ γ such that 

d(Tnβ , Tnβ ) < ε,

(4.9)

but 

sup β

β



ϕ(t + βn , x) − ϕ(t + βn , x) := M ≥ ε > 0,

(t,x)∈(T\Fε (s(Tn ∪Tn )))×S β

β



where Tn and Tn are the points of discontinuity of functions ϕ(t + βn , x), ϕ(t +  βn , x), n = 1, 2, . . . , respectively. Obviously, M is a positive finite number. Next, we claim that there exists a sequence α = {αn } ⊂ Π and a point t0 ∈ T such that 

lim ϕ(t0 + αn + βn , x) − ϕ(t0 + αn + βn , x) := M ≥ ε > 0

n→∞

(4.10)

In the below two cases, if T contains at least a right or left dense point t0 , then there exists a sequence {tn } such that tn → t0 as n → ∞. If any point in T is isolated, that is, there is no right or left dense point, then we take a sequence {tn }, tn = t0 , to guarantee that tn → t0 as n → ∞. Case 1 If there exists t ∗ ∈ T such that 

ϕ(t ∗ + βn , x) − ϕ(t ∗ + βn , x) = M,

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4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

then there exists a sequence α = {αn } = {t ∗ − tn } ⊂ Π and one can easily see that αn → t ∗ − t0 as n → ∞. That is t0 + αn → t ∗ as n → ∞. Therefore, the claim is valid. Case 2 If there exists t ∗ ∈ T such that 

lim ϕ(t + βn , x) − ϕ(t + βn , x) = M,

t→t ∗

since M is finite positive number, one can get that t ∗ = ∞. Then there exists a sequence α = {αn } such that αn → t ∗ − t0 = ∞ as n → ∞. Therefore, the claim is also valid. Thus, according to the above discussion, (4.9) implies that there exists a sequence α and a point t0 ∈ T such that (4.10) holds. Then, for the sequence α there exists common subsequence α1 ⊂ α, β1 ⊂ β, β2 ⊂ β such that Sα1 +β1 φ = R1 , Sα1 +β2 φ = R2 ,   exist pointwise, where Rj = rj (t, x), Pj , rj ∈ P Crd , Pj is an equipotentially almost periodic sequence, j = 1, 2. From (4.8), we get R1 = Sα1 +β1 φ = Sα1 Sβ1 φ = Sα1 Sγ φ = Sα1 Sβ2 φ = Sα1 +β2 φ = R2 ,

(4.11)

  for (t, x) ∈ T\Fε (s(P1 ∪ P2 )) × S. On the other hand, from (4.10) it follows that r1 (t0 , x) − r2 (t0 , x) > 0, which is a contradiction of (4.11).  Let ϕ(t, x) be rd-piecewise almost periodic in t uniformly for x ∈ Ω and if α ⊂    Π and β ⊂ Π are given, we take subsequences α ⊂ α , β ⊂ β successively, such that they are common subsequences and Sα φ = φ1 , Sβ φ1 = φ2 and Sα+β φ = φ3 , where φj = (ϕj , Tj ), φj ∈ P Crd (T × Ω, X) × B∗ , j = 1, 2, 3 exist uniformly for t ∈ T\Fε s(T1 ∪ T2 ∪ T3 ) , where B ∗ ⊂ B is an equipotentially almost periodic sequences family. For any ε > 0 and each compact subset S of Ω, then ϕ(t + αn + βn , x) − ϕ3 (t, x)
0, there exists a relatively dense set Γε ⊂ Π such that if τ ∈ Γε , then φ(t + τ ) − φ(t) < ε for all t ∈ T satisfying the condition |t − ti | > ε, i ∈ Z. The number τ is called an ε-almost period of φ. We denote by AP (T, X) the space of all rd-piecewise almost periodic functions. Obviously, the space AP (T, X) endowed with norm defined by φ∞ = supt∈T φ(t) for any φ ∈ AP (T, X) is a Banach space. Throughout the rest of the paper, let UPCrd (T, X) be the space of all functions φ ∈ P Crd (T, X) such that φ satisfies the condition (2) in Definition 4.9. Definition 4.10 f ∈ P Crd (T × Ω, X) is said to be rd-piecewise almost periodic in t uniformly in x ∈ Ω if for each compact set K ⊆ Ω, {f (·, x) : x ∈ K} is uniformly bounded, and given ε > 0, there is a relatively dense set Γε such that f (t + τ, x) − f (t, x) < ε for all x ∈ K, τ ∈ Γε , and t ∈ T, |t − ti | > ε. Denote by AP (T × Ω, X) the set of all such functions. Let U be the set of all functions ρˆ : T → (0, ∞) which are positive and locally Δ-integrable over T. For a given r ∈ [0, ∞)Π and ∀t0 ∈ T, set  m(r, ρ, ˆ t0 ) :=

t0 +r

t0 −r

ρ(s)Δs ˆ

(4.13)

for each ρˆ ∈ U . Remark 4.3 In (4.13), if T = R, taking t0 = 0, one can easily get  m(r, ρ) ˆ :=

r −r

ρ(s)Δs ˆ

if T = Z, taking t0 = 0, one has the following m(r, ρ) ˆ =

r−1 

ρ(k). ˆ

k=−r

Theorem 4.5 For any r ∈ Π and a set {tn } := Λ∗ ⊂ [t0 −r, t0 +r)T , where t0 ∈ T, there exists C1 (Λ∗ ), C2 (Λ∗ ) > 0 such that C1 (Λ∗ )

 tj ∈[t0 −r,t0 +r)T

ρ(t ˆ j) ≤

 tj ∈[t0 −r,t0 +r)T

¯ˆ j ) ≤ C2 (Λ∗ ) ρ(t

 tj ∈[t0 −r,t0 +r)T

ρ(t ˆ j ),

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4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

    ¯ˆ j ) = sup ρ(t) where ρ(t ˆ j ) = inf ρ(t) ˆ : t ∈ [tj , tj +1 )T and ρ(t ˆ : t ∈ [tj , tj +1 )T . Proof Since ρˆ : T → (0, ∞) is locally Δ-integrable over T, then there exists a partition Λ∗ as follows t0 − r := t1 < t2 < . . . < tn−1 < tn := t0 + r such that 

 tj ∈[t0 −r,t0 +r)T

 ρ(t ˆ j ) (tj +1 − tj ) < ε(Λ∗ ).



¯ˆ j ) − ρ(t

tj ∈[t0 −r,t0 +r)T

Denote δ(Λ∗ ) := inf(tj +1 − tj ), then 



¯ˆ j ) − ρ(t

tj ∈[t0 −r,t0 +r)T

ρ(t ˆ j)
0,   1 μΔ Mr,ε,t0 (φ) = 0, r→∞ m(r, ρ, ˆ t0 ) lim

  where r ∈ Π and Mr,ε,t0 (φ) := t ∈ [t0 − r, t0 + r)T : φ(t) ≥ ε . Proof (a) Necessity. For contradiction, suppose that there exists ε0 > 0 such that   1 μΔ Mr,ε0 ,t0 (φ) = 0. r→∞ m(r, ρ, ˆ t0 ) lim

Then there exists δ > 0 such that for every n ∈ N,   1 μΔ Mrn ,ε0 ,t0 (φ) ≥ δ for some rn > n, where rn ∈ Π. m(rn , ρ, ˆ t0 )

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4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

So we get 1 m(rn , ρ, ˆ t0 ) =

1 m(rn , ρ, ˆ t0 )

 

t0 +r t0 −r

φ(s)ρ(s)Δs ˆ φ(s)ρ(s)Δs ˆ

Mrn ,ε0 ,t0 (φ)

 1 φ(s))ρ(s)Δs ˆ m(rn , ρ, ˆ t0 ) [t0 −r,t0 +r)T \Mrn ,ε0 ,t0 (φ)  1 φ(s)ρ(s)Δs ˆ ≥ m(rn , ρ, ˆ t0 ) Mrn ,ε0 ,t0 (φ)  ε0 ≥ φ(s)ρ(s)Δs ˆ ≥ ε0 δγ , m(rn , ρ, ˆ t0 ) Mrn ,ε0 ,t0 (φ) +

ˆ This contradicts the where γ = infs∈T ρ(s).  assumption.  (b) Sufficiency. Assume that lim m(r,1ρ,t Mr,ε,t0 (φ) = 0. Then for every ε > μ Δ ˆ 0) r→∞ 0, there exists r0 > 0 such that for every r > r0 ,   1 ε μΔ Mr,ε,t0 (φ) < , m(r, ρ, ˆ t0 ) KM where M := supt∈T φ(t) < ∞ and K := supt∈T ρ(t) ˆ < ∞. Now, we have 1 m(r, ρ, ˆ t0 )



t0 +r t0 −r

φ(s)ρ(s)Δs ˆ =

1 m(r, ρ, ˆ t0 )  +

 φ(s)ρ(s)Δs ˆ Mr,ε,t0 (φ)



φ(s)ρ(s)Δs ˆ

[t0 −r,t0 +r)T \Mr,ε,t0 (φ)

≤

MK μΔ (Mr,ε,t0 (φ)) m(r, ρ, ˆ t0 )  ε ρ(s)Δs ˆ ≤ 2ε. + m(r, ρ, ˆ t0 ) [t0 −r,t0 +r)T \Mr,ε,t0 (φ)

 t0 +r Therefore, limr→∞ m(r,1ρ,t φ(s)ρ(s)Δs ˆ ˆ 0 ) t0 −r ˆ This completes the proof. P AP0 (T, ρ).

=

0, that is, φ

∈ 

Lemma 4.7 P AP0 (T, ρ) ˆ is a translation invariant set of BPCrd (T, X) with respect to Π if ρˆ ∈ UB , i.e., for any s ∈ Π , one has φ(t + s) := θs φ ∈ P AP0 (T, ρ) ˆ if ρˆ ∈ UB .

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257

Proof For any s ∈ Π, φ ∈ P AP0 (T, ρ), ˆ ε > 0, r > 0, we have  Mr,ε,t0 (Ts φ) = t  = t  = t  ⊆ t

∈ [t0 − r, t0 + r)T : Ts (t) ≥ ε



∈ [t0 − r, t0 + r)T : φ(t + s) ≥ ε



∈ [t0 − r + s, t0 + r + s)T : φ(t) ≥ ε



 ∈ [t0 − r − |s|, t0 + r + |s|)T : φ(t) ≥ ε .

Hence   1 μΔ Mr,ε,t0 (Ts φ) m(r, ρ, ˆ t0 )   1 μΔ Mr+|s|,ε,t0 (Ts φ) ≤ m(r, ρ, ˆ t0 ) =

  1 m(r + |s|, ρ, ˆ t0 ) μΔ Mr+|s|,ε,t0 (φ) . m(r, ρ, ˆ t0 ) m(r + |s|, ρ, ˆ t0 )

Since φ ∈ P AP0 (T, ρ), ˆ then by Lemma 4.6, we have   1 Mr+|s|,ε,t0 (φ) → 0, r → ∞. m(r + |s|, ρ, ˆ t0 ) Furthermore, limr→∞

m(r+|s|,ρ,t ˆ 0) m(r,ρ,t ˆ 0)

= 1, thus

  1 μΔ Mr,ε,t0 (Ts (φ)) → 0, r → ∞. m(r, ρ, ˆ t0 ) Again, using Lemma 4.6, one can get θs φ ∈ P AP0 (T, ρ) ˆ for any s ∈ Π . This completes the proof. 

Definition 4.12 The sequence {φ˜ n }, φ˜ n = (ϕn (t), Tn ) ∈ P Crd (T, X) × B is ˜ φ˜ = (ϕ(t), T ), (ϕ(t), T ) ∈ P Crd (T, X) × B, if and only if for convergent to φ, any ε > 0 there exists n0 > 0 such that n ≥ n0 implies d(T , Tn ) < ε, ϕn (t) − ϕ(t) < ε uniformly for t ∈ T\Fε (s(Tn ∪ T )), d(·, ·) is an arbitrary distance in B. Now, we shall consider the following second definition for rd-piecewise almost periodic functions. Definition 4.13 The function ϕ ∈ P Crd (T, X) is said to be rd-piecewise almost periodic with respect to a sequence from the set T ∈ B if for every sequence of   real numbers {sm } ⊂ Π there exists a subsequence {sn }, sn = smn such that θsn φ˜ is uniformly convergent on P Crd (T, X) × B.

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4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

By Definition 4.13, one can easily get the following lemma: Lemma 4.8 Let φ ∈ AP (T, X). Then the range of φ, φ(T), is a relatively compact subset of X. Lemma 4.9 If f = g + φ with g ∈ AP (T, X), and φ ∈ P AP0 (T, ρ), ˆ where ρˆ ∈ UB , then g(T) ⊂ f (T). 

Proof By the definition of f , the result is obvious.

Lemma 4.10 The decomposition of a weighted piecewise pseudo almost periodic function according to AP ⊕ P AA0 is unique for any ρˆ ∈ UB . Proof Assume that f = g1 + φ1 and f = g2 + φ2 . Then (g1 − g2 ) + (φ1 − φ2 ) = 0. Since g1 − g2 ∈ AP (T, X), and φ1 − φ2 ∈ P AP0 (T, ρ), ˆ in view of Lemma 4.9, we deduce that g1 − g2 = 0. Consequently, φ1 − φ2 = 0, i.e. φ1 = φ2 . This completes the proof. 

Theorem 4.6 For ρˆ ∈ UB , (W P AP (T, ρ), ˆ  · ∞ ) is a Banach space. Proof Assume that {fn }n∈N is a Cauchy sequence in W P AP (T, ρ). ˆ We can write uniquely fn = gn + φn . Using Lemma 4.9, we see that gp − gq ∞ ≤ fp − fq ∞ , from which we deduce that {gn }n∈N is a Cauchy sequence in AP (T, X). Hence, φn = fn − gn is a Cauchy sequence in P AP0 (T, ρ). ˆ We deduce that gn → g ∈ AP (T, X), φn → φ ∈ P AP0 (T, ρ), ˆ and finally fn → g + φ ∈ W P AP (T, ρ). ˆ This completes the proof. 

Definition 4.14 Let ρ1 , ρ2 ∈ U∞ . One says that ρ1 equivalent ρ2 , written ρ1 ∼ ρ2 if ρ1 /ρ2 ∈ UB . Theorem 4.7 Let ρ1 , ρ2 W P AP (T, ρ2 ).

∈

U∞ . If ρ1

∼

ρ2 , then W P AP (T, ρ1 )

=

Proof Assume that ρ1 ∼ ρ2 . There exist a, b > 0 such that aρ1 ≤ ρ2 ≤ bρ1 . So am(r, ρ1 , t0 ) ≤ m(r, ρ2 , t0 ) ≤ bm(r, ρ1 , t0 ), where r ∈ Π , and a 1 b m(r, ρ1 , t0 )



t0 +r

t0 −r

φ(s)ρ1 (s)Δs ≤ ≤

1 m(r, ρ2 , t0 )



1 b a m(r, ρ1 , t0 )

t0 +r t0 −r



φ(s)ρ2 (s)Δs

t0 +r t0 −r

φ(s)ρ1 (s)Δs.



  Lemma 4.11 If g ∈ AP (T × X, X) and α ∈ AP (T, X), then G(t) := g ·, α(·) ∈ AP (T, X). This completes the proof.

4.2 Weighted Piecewise Pseudo Almost Periodic Functions on CCTS

259

Proof Let T = {ti }, φ˜ = (g(t, x), T ) ∈ AP (T × X, X) × B, from every sequence ∞ {sn }∞ n=1 ⊂ Π , we can extract a subsequence {τn }n=1 such that φ˜ ∗ := (g ∗ (t, x), T ∗ ) = lim θτn φ˜ = lim (g(t + τn , x), T − τn ), n→∞

n→∞

uniformly exists on P Crd (T × X, X) × B. Since α ∈ AP (T, X), one can extract  {τn } ⊂ {τn } such that ˜ = lim (g(t + τn , α(t + τn )), T − τn )

lim θ φ n→∞ τn 

n→∞





= lim (g(t + τn , α ∗ (t)), T − τn ) = (g ∗ (t, α ∗ (t)), T ∗ ). n→∞

Hence, G ∈ AP (T, X). This completes the proof.



Theorem 4.8 Let f = g + φ ∈ W P AP (T × X, ρ), ˆ where g ∈ AP (T × X, X), φ ∈ P AP0 (T × X, ρ), ˆ ρˆ ∈ UB and the following conditions hold: (i) {f (t, x) : t ∈ T, x ∈ K} is bounded for every bounded subset K ⊆ Ω. (ii) f (t, ·), g(t, ·) are uniformly continuous in each bounded subset of Ω uniformly in t ∈ T.   Then f ·, h(·) ∈ W P AP (T, ρ) ˆ if h ∈ W P AP (T, ρ) ˆ and h(T) ⊂ Ω. Proof We have f = g +φ, where g ∈ AP (T×X, X) and φ ∈ P AP0 (T×X, ρ) ˆ and h = φ1 + φ2 , where φ1 ∈ AP (T, X) and φ2 ∈ P AP0 (T, ρ). ˆ Hence, the function f (·, h(·)) can be decomposed as f (·, h(·)) = g(·, φ1 (·)) + f (·, h(·)) − g(·, φ1 (·)) = g(·, φ1 (·)) + f (·, h(·)) − f (·, φ1 (·)) + φ(·, φ1 (·)). By Lemma 4.11, g(·, φ1 (·)) ∈ AP (T, X). Now, consider the function Ψ (·) := f (·, h(·)) − f (·, φ1 (·)). ˆ it is sufficient to show that Clearly, Ψ ∈ BPCrd (T, X). For Ψ to be in P AP0 (T, ρ), lim

r→∞

1 μΔ (Mr,ε,t0 (Ψ )) = 0. m(r, ρ, ˆ t0 )

Let K be a bounded subset of Ω such that φ(T) ⊆ K, φ1 (T) ⊆ K. By (ii), f (t, ·) is uniformly continuous in φ( T) uniformly in t ∈ T, we see that for given ε > 0, there exists δ > 0 such that y1 , y2 ∈ K and y1 − y2  < δ implies that f (t, y1 ) − f (t, y2 ) < ε, t ∈ T.

260

4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

Thus, for each t ∈ T, φ2 (t) < δ implies for all t ∈ T, f (t, h(t)) − f (t, φ1 (t)) < ε, where φ2 (t) = h(t) − φ1 (t). For r > 0 and any fixed t0 ∈ T, let Mr,δ,t0 (φ2 ) = {t ∈ [t0 − r, t0 + r)T : φ2  ≥ δ}, we can obtain     1 1 μΔ Mr,ε,t0 (Ψ (t)) = μΔ Mr,ε,t0 (f (t, h(t)) − f (t, φ1 (t))) m(r, ρ, ˆ t0 ) m(r, ρ, ˆ t0 ) 1 μΔ (Mr,δ,t0 (h(t) − φ1 (t))) m(r, ρ, ˆ t0 )   1 μΔ Mr,δ,t0 (φ2 (t)) . = m(r, ρ, ˆ t0 ) ≤

Now since φ2 ∈ P AP0 (T, ρ), ˆ Lemma 4.6 yields that   1 μΔ Mr,ε,t0 (φ2 (t)) = 0, r→∞ m(r, ρ, ˆ t0 ) lim

this implies that Ψ ∈ P AP0 (T, ρ). ˆ Finally, we need to show φ(·, φ1 (·)) ∈ P AP0 (T, ρ). ˆ Note that f = g + φ and g(t, ·) is uniformly continuous in φ1 (T) uniformly in t ∈ T. By assumption (ii), f (t, ·) is uniformly continuous in φ1 (T) uniformly in t ∈ T, so is φ. Sinceφ1 (T) is relatively compact in X, for ε > 0, there exists δ > 0 such that φ1 (T) ⊂ m k=1 Bk , where Bk = {x ∈ X : x − xk  < δ} for some xk ∈ φ1 (T) and φ(t, φ1 (t)) − φ(t, xk )
0, then {ϕ(τk )} is an almost periodic sequence. 

Proof Construct a sequence {τk } ⊂ T satisfying the condition 



τk = τk ,

if τk is a left scattered point,

(4.15)



τk = τk − 2ε1 , if τk is a left dense point,

k ∈ Z. For ε1 , one can choose the number r and q such that ϕ(t + r) − ϕ(t) < ε1  q and |τk − r| < ν, 0 < ν < ε1 , for all |t − τk | > ε1 , t ∈ T, k ∈ Z. Since −ν <  τk+q − τk − r < ν and (4.15), then 0 < 2ε1 − ν ≤ τk+q − τk − r < 2ε1 + ν < 3ε1 .     Thus, if t , t belong to the same interval of continuity with |t − t | < 3ε1 implies   ϕ(t ) − ϕ(t ) < o(3ε1 ), assuming that 2o(3ε1 ) + ε1 < ε < θ , we find that 





ϕ(τk+q ) − ϕ(τk ) ≤ ϕ(τk+q ) − ϕ(τk + r) + ϕ(τk + r) − ϕ(τk ) 

+ϕ(τk ) − ϕ(τk ) < 2o(3ε1 ) + ε1 < ε. This completes the proof.



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4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

Lemma 4.13 A necessary and sufficient condition for a bounded sequence {an } to be in P AP0 (Z, ρ) ˆ is that there exists a uniformly continuous function f ∈ P AP0 (T, ρ) ˆ and a discretization partition {tn } such that f (tn ) = an , n ∈ Z, ρˆ ∈ UB . Proof (a) Necessity. By taking fixed r ∈ Π and without gernerality let 

 ti(n) , ti(n)+1 , . . . , tj (n) ⊆ [t0 + nr, t0 + (n + 1)r)T ,

where i(n), j (n) are mappings  ≥ i(n).  from Z to Z and j (n) Now, we rewrite the set ti(n) , ti(n)+1 , . . . , tj (n) as 

 tn(1) , tn(2) , . . . , tnk0 (n) = {tn(h) : 1 ≤ h ≤ k0 (n)}, (h)

we obtain that t0 + nr ≤ tn < t0 + (n + 1)r. Next, we define a function (h)

f (t) = an(h) + (t − tn(h) )(an+1 − an(h) ), t0 + nr ≤ t < t0 + (n + 1)r, (h)

(h)

where t ∈ T, n ∈ Z, t0 ∈ T, r ∈ Π , and the corresponding an = f (tn ) for ˆ all 1 ≤ h ≤ k0 (n), and it is uniformly continuous on T. Now f ∈ P AP0 (T, ρ) since 1 m(kr, ρ, ˆ t0 )



t0 +kr t0 −kr

f (s)ρ(s)Δs ˆ

k−1  t0 +(j +1)r  1 (h) (h) (h) (h) aj + (s − tj )(aj +1 − aj )ρ(s)Δs ˆ = m(kr, ρ, ˆ t0 ) t0 +j r j =−k

≤

  t0 +(j +1)r k−1   1 (h) ¯ (h) (h) (h) (h) ˆ j )r + aj +1 − aj  (s − tj )ρ(s)Δs ˆ aj ρ(t m(kr, ρ, ˆ t0 ) t0 +j r j =−k

≤

(h) k−1  (ak(h)  + a−k )r 2 1 (h) ¯ (h) ρ¯ˆ raj ρ(t ˆ j )+ m(kr, ρ, ˆ t0 ) m(kr, ρ, ˆ t0 ) j =−k

≤

 (h)

k−1 

1 (h)

tj ∈[t0 −kr,t0 +kr)T

=

C2 k−1  j =−k

ρ(t ˆ j ) j =−k

k−1 

¯ˆ (h) ) j =−k ρ(t j

(h)

¯ˆ rf (tj )ρ(t j )+ (h)

(h)

rf (tj )ρ(t ˆ j )+

(h)

(h) ak(h)  + a−k 

m(kr, ρ, ˆ t0 )

(h) ak(h)  + a−k 

m(kr, ρ, ˆ t0 )

r 2 ρ¯ˆ

r 2 ρ¯ˆ → 0, as k → ∞.

4.2 Weighted Piecewise Pseudo Almost Periodic Functions on CCTS

263

(b) Sufficiency. Let 0 < ε < 1, there exists δ > 0 such that for t ∈ (tn1 − δ, tn1 )T , n1 ∈ Z, we have ˆ n1 ), n1 ∈ Z. f (t)ρ(t) ˆ ≥ (1 − ε)f (tn1 )ρ(t Without loss of generality, let tn ≥ 0, t−n < 0, n ∈ Z+ , there exists r ∈ Π ∩R+ (h) such that t0 + nr ≤ tn(h) < t0 + (n + 1)r, t0 − nr ≤ t−n < t0 + (−n + 1)r and + {t−n , tn : n ∈ Z } be a discrete partition. Therefore,  t0 +nr  t (h) n−1 f (t)ρ(t)Δt ˆ ≥ f (t)ρ(t)Δt ˆ (h)

t0 −nr

≥

≥

t−n

n−1  

t0 +j r

j =−n+2 t0 +(j −1)r n−1  j =−n+2

n−1  

f (t)ρ(t)Δt ˆ ≥

(h) (h) δ(1 − ε)f (tj −1 )ρ(t ˆ j −1 )

t0 +j r

j =−n+2 t0 +j r−δ

f (t)ρ(t)Δt ˆ

n−1 

≥ δ(1 − ε)

j =−n+2

(h)

(h)

f (tj −1 )ρ(t ˆ j −1 ),

so one can obtain 1 m(rn , ρ, ˆ t0 ) ≥

≥

1 m(rn , ρ, ˆ t0 )



t0 +nr t0 −nr n−1 

j =−n+2

j =−n+2

¯ˆ h ) j =−n+2 ρ(t j −1

C2

n−1  j =−n+2

f (tj(h) ˆ j(h) −1 )ρ(t −1 )

n−1 

1

≥

f (tj(h) ˆ j(h) −1 )ρ(t −1 )

n−1 

1 n−1 

f (t)ρ(t)Δt ˆ

(h)

ρ(t ˆ j −1 ) j =−n+2

(h)

(h)

f (tj −1 )ρ(t ˆ j −1 ).

(4.16)

Since f ∈ P AP0 (T, ρ), ˆ it follows from the inequality (4.16) that f (tn ) = an ∈ P AP0 (Z, ρ). ˆ This completes the proof. 

By Lemma 4.13, we can obtain the following theorem: Theorem 4.9 A necessary and sufficient condition for a bounded sequence {an } to be in W P AP (Z, ρ) ˆ is that there exists a uniformly continuous function f ∈ W P AP (T, ρ) ˆ such that f (tn ) = an , tn ∈ T, n ∈ Z, ρˆ ∈ UB .

264

4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

Theorem 4.10 Assume that ρˆ ∈ UB and the sequence of vector-valued functions {In }n∈Z is weighted pseudo almost periodic, i.e., for any x ∈ Ω, {In (x), n ∈ Z} is weighted pseudo almost periodic sequence. Suppose {In (x) : n ∈ Z, x ∈ K} is bounded for every bounded subset K ⊆ Ω, In (x) is uniformly continuous in x ∈ Ω uniformly ˆ ∩ UPCrd (T, X) such that h(T) ⊂ Ω, then   in n ∈ Z. If h ∈ W P AP (T, ρ) In h(tn ) is weighted pseudo almost periodic. Proof Fix h ∈ W P AP (T, ρ) ˆ ∩ UPCrd (T, X). First we show h(ti ) is weighted pseudo almost periodic. Since h = φ1 + φ2 , where φ1 ∈ AP (T, X), φ2 ∈ P AP0 (T, ρ). ˆ It follows from Lemma 4.12 that the sequence φ1 (tn ) is almost periodic. By Lemma 4.13, φ2 (tn ) ∈ P AP0 (Z, ρ), ˆ then h(tn ) is weighted pseudo almost periodic.   Now, we show In φ(tn ) is weighted pseudo almost periodic. By taking fixed r ∈ Π and without gernerality let   ti(n) , ti(n)+1 , . . . , tj (n) ⊆ [t0 + nr, t0 + (n + 1)r)T , where i(n), j (n) are mappings  ≥ i(n).  from Z to Z and j (n) Now, we rewrite the set ti(n) , ti(n)+1 , . . . , tj (n) as   (1) (2) tn , tn , . . . , tnk0 (n) = {tn(l) : 1 ≤ l ≤ k0 (n)},   we obtain that t0 + nr ≤ tn(l) < t0 + (n + 1)r and the corresponding In h(tn(l) ) =  (l) (l)  I tn , h(tn ) for all 1 ≤ l ≤ k0 (n). Let I (t, x) = In (x) + (t − tn(l) )[In+1 (x) − In (x)], t0 + nr ≤ t < t0 + (n + 1)r, n ∈ Z, r ∈ Π, Φ0 (t) = h(tn(l) ) + (t − tn(l) )[h(tn+1 ) − h(tn )], t0 + nr ≤ t < t0 + (n + 1)r, n ∈ Z, r ∈ Π. Since In , h(tn ) are two weighted pseudo almost periodic sequences, by Lemma 4.13 and Theorem 4.9, we know that I ∈ W P AP (T × Ω, ρ), ˆ Φ0 ∈ W P AP (T, ρ). ˆ For every t ∈ T, there exists a number n ∈ Z such that |t − tn(l) | ≤ r, I (t, x) ≤ In (x) + |t − tn(l) |[In+1 (x) + In (x)] ≤ (1 + r)In (x) + rIn+1 (x). Since {In (x) : n ∈ Z, x ∈ K} is bounded for every bounded set K ⊆ Ω, {I (t, x) : t ∈ T, x ∈ K} is bounded for every bounded set K ⊆ Ω. For every x1 , x2 ∈ Ω, we have

4.3 Weighted Piecewise Pseudo Double-Almost Periodic Functions on ACCTS

265

I (t, x1 ) − I (t, x2 ) ≤ In (x1 ) − In (x2 ) + |t − tn(l) |[In+1 (x1 ) − In+1 (x2 ) +In (x1 ) − In (x2 )] ≤ (1 + r)In (x1 ) − In (x2 ) + rIn+1 (x1 ) − In+1 (x2 ). Noting that In (x) is uniformly continuous in x ∈ Ω uniformly in n ∈ Z, we then get  that I (t,  x) is uniformly in x ∈ Ω uniformly in t ∈ T. Then by Theorem 4.8, I ·, Φ0 (·) ∈ W P AP (T, X). Again, using Lemma 4.13 and Theorem 4.9, we  (l) (l)  have  that I tn , Φ0 (tn ) is a weighted pseudo almost periodic sequence, that is, In h(tn ) is weighted pseudo almost periodic. This completes the proof. 

From Theorem 4.10, one can easily get the following corollary: Corollary 4.2 Assume the sequence of vector-valued functions {Ii }i∈Z is weighted pseudo almost periodic, ρˆ ∈ UB , and there is a number L > 0 such that Ii (x) − Ii (y) ≤ Lx − y ˆ such that h(T) ⊂ Ω, for all x, y ∈ Ω, i ∈ Z. If h ∈ W P AP (T, ρ) ˆ ∩ UPCrd (T, ρ) then Ii h(ti ) is weighted pseudo almost periodic.

4.3 Weighted Piecewise Pseudo Double-Almost Periodic Functions on ACCTS Throughout this section, we shall assume that T is an almost complete-closed time scale and denote by X a Banach space; let B be the set consisting of all sequences {tk }k∈Z such that θ = infk∈Z (tk+1 − tk ) > 0. For {tk }k∈Z ∈ B, let BPCrd (T, X) be the space formed by all bounded rd-piecewise continuous functions φ : T → X such that φ(·) is continuous at t for any t ∈ {tk }k∈Z and φ(tk ) = φ(tk− ) for all k ∈ Z; let Ω be a set of X and BPCrd (T × Ω, X) be the space formed by all bounded piecewise continuous functions φ : T × Ω → X such that for any x ∈ Ω, φ(·, x) ∈ BPCrd (T, X) and for any t ∈ T, φ(t, ·) is continuous at x ∈ Ω. For convenience, we denote the space of all rd-piecewise continuous functions PCrd (T, X) and PCεrd (T, X) := {f |RT (τ,ε) : f ∈ PCrd (T, X)}. Notice that the set RT (τ, ε) is defined by Definition 2.28. Now, we introduce some definitions which will be used to introduce the concept of weighted piecewise pseudo double-almost periodic functions on ACCTS.   Let Bε = {tki ,ε } ⊂ {tk } : tki ∈ RT (τ, ε), tki ,ε < tki+1 ,ε , i ∈ Z, lim tki ,ε = ∞ , i→∞

Definition 4.15 Let {tki ,ε } ∈ Bε , i ∈ Z. We say {tki ,ε } where

j tki ,ε

= tki+j ,ε − tki ,ε , i, j ∈ Z.

j {tki ,ε }

is a ε-derived sequence of

266

4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

Definition 4.16 For any ε2 > ε1 > 0, let Γ ⊂ Πε1 be a set of real numbers and j {tki ,ε1 } ∈ Bε1 . We say {tki ,ε1 }, i, j ∈ Z is equipotentially double-almost periodic on an almost-complete closedness time scale T if for r ∈ Γ , there exists at least one integer q such that q

|tki ,ε1 − r| < ε2 , for all i ∈ Z. In the following, we introduce the concept of piecewise continuous doublealmost periodic functions on ACCTS: Definition 4.17 Let T be an almost-complete closedness time scale and assume that j {tki ,ε1 } ∈ Bε1 satisfying the ε1 -derived sequence {tki ,ε1 }, i, j ∈ Z, is equipotentially ε1 almost periodic. We call a function ϕ ∈ PCrd (T, Rn ) double-almost periodic if: 



(i) for any ε > 0, there is a positive number δ = δ(ε) such that if the points t and t     belong to the same interval of continuity and t , t ∈ RT (τ, ε1 )\Bε1 , |t −t | <   δ, then ϕ(t ) − ϕ(t ) < ε; (ii) for any ε2 > ε1 > 0, there is a relative dense set Γ of ε2 -almost periods such that if τ ∈ Γ ⊂ Πε1 , then ϕ(t + τ ) − ϕ(t) < ε2 for all t ∈ RT (τ, ε1 ) which satisfy the condition |t − tki ,ε1 | > ε2 , i ∈ Z. We denote by DAP(T, X) the space of all rd-piecewise double-almost periodic functions. Obviously, for any fixed ε > 0, the space DAPε (T, X) := {f |RT (τ,ε) : f ∈ DAP(T, X)} endowed with norm φε = supt∈RT (τ,ε) φ(t) for any φ ∈ DAPε (T, X) is a Banach space. We also denote by UPC(T, X) the space of all functions φ ∈ PCrd (T, X) such that φ satisfies condition (i) in Definition 4.17 and UPCε (T, X) := {f |RT (τ,ε) : f ∈ UPC(T, X)}. Now BPCrd (T, X) denotes the space of all bounded rd-piecewise functions and BPCεrd (T, X) := {f |RT (τ,ε) : f ∈ BPCrd (T, X)}. Similarly, we can also introduce the concept of uniformly piecewise doublealmost periodic functions on almost-complete closedness time scales as follows: Definition 4.18 Let T be an almost-complete closedness time scale and assume that j {tki ,ε1 } ∈ Bε1 satisfying the ε1 -derived sequence {tki ,ε1 }, i, j ∈ Z, is equipotentially double-almost periodic. We call a function f ∈ PCεrd1 (T × Ω, X) rd-piecewise double-almost periodic in t uniformly in x ∈ Ω if: (i) for each compact set K ⊆ Ω, {f (·, x) : x ∈ K} is uniformly bounded;  (ii) for any ε > 0, there is a positive number δ = δ(ε) such that if the points t and     t belong to the same interval of continuity and t , t ∈ RT (τ, ε1 )\Bε1 , |t −    t | < δ, then f (t , x) − f (t , x) < ε for all x ∈ K; (iii) for any ε2 > ε1 > 0, there is relative dense set Γ of ε2 -almost periods such that if τ ∈ Γ ⊂ Πε1 , then f (t + τ, x) − f (t, x) < ε2 for all t ∈ RT (τ, ε1 ), x ∈ K, which satisfy the condition |t − tki ,ε1 | > ε2 , i ∈ Z.

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Now, let U be the set of all functions ρ˜ : T → (0, ∞) which are positive and locally Δ-integrable over T and let U ε := ρ| ˜ RT (τ,ε) : ρ˜ ∈ U }. For a given r1 , r2 ∈ RT (τ, ε), r2 > r1 , we set  m(r1 , r2 , ρ) ˜ := ρ(s)Δs ˜ [r1 ,r2 )RT (τ,ε)

 ε := ρ˜ ∈ U ε : lim m(r , r , ρ) for each ρ˜ ∈ U ε . Let Dr := r2 − r1 and U˜ ∞ 1 2 ˜ = Dr →∞  ∞ ,  ε ε U∞ = ρ˜ ∈ U˜ ∞ : ρ(s) ˜ ≡ 0 for all s ∈ (t − δ, t + δ)RT (τ,ε) ,  where t ∈ RT (τ, ε), δ > 0 ,  ε : ρ˜ is bounded and UBε := ρ˜ ∈ U∞

inf

s∈RT (τ,ε)

 ρ(s) ˜ >0 .

ε ⊂ U ε . Now, for ρ˜ ∈ U ε , we define It is clear that for any ε > 0, UBε ⊂ U∞ ∞

 WPDAPε0 (T, ρ) ˜ := φ ∈ BPCεrd (T, X) : 1 lim Dr →∞ m(r1 , r2 , ρ) ˜

 [r1 ,r2 )RT (τ,ε)

 φ(s)ρ(s)Δs ˜ =0 .

Similarly, we define  ˜ := Φ ∈ BPCεrd (T × Ω, X) : WPDAPε0 (T × X, ρ) 1 Dr →∞ m(r1 , r2 , ρ) ˜ lim

 [r1 ,r2 )RT (τ,ε)

Φ(s, x)ρ(s)Δs ˜ =0

 uniformly with respect to x ∈ K, where K is an arbitrary compact subset of Ω .

We are now ready to introduce the sets WPDAPε (T, ρ) ˜ and WPDAPε (T × X, ρ) ˜ of weighted pseudo double-almost periodic functions on ACCTS:  WPDAPε (T, ρ) ˜ = f = g + φ ∈ PCεrd (T, X) : g ∈ DAPε (T, X)  and φ ∈ WPDAPε0 (T, ρ) ˜ ,  ˜ = f = g + φ ∈ PCεrd (T × X, X) : g ∈ DAPε (T × X, X) WPDAPε (T × X, ρ)  and φ ∈ WPDAPε0 (T × X, ρ) ˜ .

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4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

ε1 Lemma 4.14 Let φ ∈ BPCεrd1 (T, X). Then, φ ∈ WPDAPε01 (T, ρ) ˜ where ρ˜ ∈ U∞ if and only if for every ε2 > ε1 > 0,

1 Dr →∞ m(r1 , r2 , ρ) ˜

 ρ(t)Δt ˜ = 0,

lim

Mr1 ,r2 ,ε2 (φ)

  and Mr1 ,r2 ,ε2 (φ) := t ∈ [r1 , r2 )RT (τ,ε1 ) : φ(t) ≥ ε2 . Proof (a) Necessity. For contradiction, suppose that there exists ε20 > ε10 > 0 such that 

1 Dr →∞ m(r1 , r2 , ρ) ˜

ρ(t)Δt ˜ = 0.

lim

Mr

0 1 ,r2 ,ε2

(φ)

Then, there exists δ > 0 such that 1 n1 n2 m(r1 , r2 , ρ) ˜

 ρ(t)Δt ˜ ≥ δ, M

n n (φ) r1 1 ,r2 2 ,ε20

where r1n1 ≤ #rn1 $ := n1 , r2n2 ≥ %r2n2 & := n2 and r1n1 ≤ r2n2 , r1n1 , r2n2 ∈ RT (τ, ε10 ). Thus, we have 1 m(r1n1 , r2n2 , ρ) ˜ 1 = n1 n2 m(r1 , r2 , ρ) ˜



n

r1 1

1 n1 n2 m(r1 , r2 , ρ) ˜

ε20 ≥ n1 n2 m(r1 , r2 , ρ) ˜

φ(s)ρ(s)Δs ˜



φ(s)ρ(s)Δs ˜ M

n n (φ) r1 1 ,r2 2 ,ε20



1 + n1 n2 m(r1 , r2 , ρ) ˜ ≥

n

r2 2

[r1 ,r2 )R

0 \Mr n1 ,r n2 ,ε0 (φ) T (τ,ε1 ) 1 2 2



φ(s)ρ(s)Δs ˜

φ(s)ρ(s)Δs ˜ M



n n (φ) r1 1 ,r2 2 ,ε20

M

n n (φ) r1 1 ,r2 2 ,ε20

ρ(s)Δs ˜ ≥ ε20 δ,

and this contradicts the assumption. (b) Sufficiency. Assume that 1 lim Dr →∞ m(r1 , r2 , ρ) ˜

 ρ(t)Δt ˜ = 0. Mr1 ,r2 ,ε2 (φ)

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Then, for every ε2 > ε1 > 0, there exists r0 > 0 such that for every Dr > r0 , 1 m(r1 , r2 , ρ) ˜

 ρ(t)Δt ˜ < Mr1 ,r2 ,ε2 (φ)

ε2 , M

where M := supt∈RT (τ,ε1 ) φ(t) < ∞. Now, we have  r2 1 φ(s)ρ(s)Δs ˜ m(r1 , r2 , ρ) ˜ r1  1 φ(s)ρ(s)Δs ˜ = m(r1 , r2 , ρ) ˜ Mr1 ,r2 ,ε2 (φ)   + φ(s)ρ(s)Δs ˜ [r1 ,r2 )RT (τ,ε1 ) \Mr1 ,r2 ,ε2 (φ)

M ≤ m(r1 , r2 , ρ) ˜



ρ(t)Δt ˜ Mr1 ,r2 ,ε2 (φ)

ε2 + m(r1 , r2 , ρ) ˜



[r1 ,r2 )RT (τ,ε1 ) \Mr1 ,r2 ,ε2 (φ)

ρ(s)Δs ˜ ≤ 2ε2 .

Therefore, it follows that 1 lim Dr →∞ m(r1 , r2 , ρ) ˜

 [r1 ,r2 )RT (τ,ε1 )

φ(s)ρ(s)Δs ˜ = 0,

that is, φ ∈ W P DAP0ε1 (T, ρ). ˜ This completes the proof.



Lemma 4.15 Let T be an almost-complete closedness time scale. Then, ε , WPDAPε0 (T, ρ) ˜ is an translation almost-closed set of BPCεrd (T, X) if ρ˜ ∈ U∞ ε ε i.e., φ(t + s) := θs φ ∈ WPDAP0 (T, ρ), ˜ t ∈ RT (τ, ε), s ∈ Πε , if ρ˜ ∈ U∞ . Proof For φ ∈ WPDAPε01 (T, ρ), ˜ ε2 > ε1 > 0, r1 , r2 ∈ RT (τ, ε1 ), we have  Mr1 ,r2 ,ε2 (θs φ) = t  = t  = t  ⊆ t

∈ [r1 , r2 )RT (τ,ε1 ) : θs (t) ≥ ε2



∈ [r1 , r2 )RT (τ,ε1 ) : φ(t + s) ≥ ε2



∈ [r1 + s, r2 + s)RT (τ,ε1 ) : φ(t) ≥ ε2  ∈ [˜r1 , r˜2 )RT (τ,ε1 ) : φ(t) ≥ ε2 ,



where r˜1 , r˜2 ∈ RT (τ, ε1 ) and r˜1 < r1 + s and r˜2 > r2 + s. Let Dr˜ := r˜2 − r˜1 , so Dr → +∞ implies Dr˜ → +∞. Hence, it follows that

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4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

1 m(r1 , r2 , ρ) ˜

 ρ(t)Δt ˜ ≤ Mr1 ,r2 ,ε2 (θs φ)

1 m(r1 , r2 , ρ) ˜

 ρ(t)Δt ˜ Mr˜1 ,˜r2 ,ε2 (θs φ)

1 ˜ m(˜r1 , r˜2 , ρ) = m(r1 , r2 , ρ) ˜ m(˜r1 , r˜2 , ρ) ˜

 ρ(t)Δt. ˜ Mr1 ,r2 ,ε2 (φ)

Since φ ∈ WPDAPε01 (T, ρ), ˜ then we have 1 m(˜r1 , r˜2 , ρ) ˜ Further, lim

Dr →∞

 ρ(t)Δt ˜ → 0, Dr → ∞. Mr˜1 ,˜r2 ,ε2 (φ)

m(˜r1 , r˜2 , ρ) ˜ = 1, and thus m(r1 , r2 , ρ) ˜ 1 m(r1 , r2 , ρ) ˜

 ρ(t)Δt ˜ → 0, Dr → ∞. Mr1 ,r2 ,ε2 (θs (φ))

Again, we have θs φ ∈ WPDAPε01 (T, ρ). ˜ This completes the proof. Bε



Let T , P ∈ and let sε (T ∪ P ) : → be a map such that the set sε (T ∪ P ) forms a strictly increasing sequence for any fixed ε > 0. Let D ⊂ RT (τ, ε), and we introduce the notations D ξ = {t + ξ : t ∈ D}, Fξ (D) = D ∩ D ξ .   We denote by φ˜ = ϕ(t), T the element from the space PCεrd (T, X) × Bε , and for ˜ we consider the sets every sequence of real numbers {sn }, n = 1, 2, . . . with θsn φ, ε −s ε −s n n } ⊂ PCrd × B , where T = {tki − sn : i ∈ Z, n = 1, 2, . . .}. {ϕ(t + sn ), T Next, we introduce the convergent form of piecewise functions on almostcomplete closedness time scales: Bε

Bε

Definition 4.19 (Almost-Uniform Convergence for Piecewise Functions) Let closedness time scale. The sequence {φ˜ n}, φ˜ n = T be an almost-complete  ε1 ˜ φ˜ = ϕ(t), T , ∈ PC ϕ (t), T (T, X) × Bε1 is almost convergent to φ, rd  n n ϕ(t), T ∈ PCεrd1 (T, X) × Bε1 , if and only if for any ε2 > ε1 > 0 there exists n0 > 0 such that n ≥ n0 implies ˜ , Tn ) < ε2 , ϕn (t) − ϕ(t) < ε2 d(T   ˜ ·) is an arbitrary distance uniformly for t ∈ RT (τ, ε1 )\Fε2 sε1 (Tn ∪ T ) , here d(·, in Bε1 . Remark 4.5 The convergence described by Definition 4.19 is a distinct convergence form on ACCTS, which will contribute to studying approximation of functions under the time scale that is almost the same as each other under translations. Traditionally, researchers analyze functions on complete-closed time scales, which provides no difficulties in introducing and studying functions, especially for some important and basic functions like almost periodic functions, almost automorphic

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271

functions and so on because these types of functions are defined by their translations on the domain and their domain is at least a complete-closed time scale. However, ACCTS are a class of necessary and important time variable forms that exist. Obviously, if let ε1 → 0 in Definition 4.19, then one can easily obtain the convergence form for piecewise functions on CCTS. In view of Definition 4.19, we now introduce the second definition of piecewise continuous double-almost periodic functions on ACCTS: Definition 4.20 Let T be an almost-complete closedness time scale. The function ϕ ∈ PCεrd (T, X) is said to be rd-piecewise continuous double-almost periodic with respect to a sequence from the set T ∈ Bε if for every sequence of real numbers   {sm } ⊂ Πε there exists a subsequence {sn }, sn = smn and a sequence {A−sn } such that the limit set T0 of {T−sn \A−sn } exists and θsn φ˜ is uniformly convergent on PCrd (T0 , X) × B0 , where B0 := T0 ∩ B. From Definition 4.20, the following lemma is immediate: ε Lemma 4.16   Let φ ∈ DAP (T, X). Then for any ε > 0, the almost range of φ, φ RT (τ, ε) , is relatively compact subset of X.

Lemma 4.17 If f = g + φ with g ∈ DAPε (T, X), and φ ∈ WPDAPε0 (T, ρ), ˜ where     ε ρ˜ ∈ U∞ , then g RT (τ, ε) ⊂ f RT (τ, ε)) . Proof By definition of f , the result is obvious.



Lemma 4.18 For any ε > 0, the decomposition of a weighted piecewise pseudo double-almost periodic function according to DAPε ⊕ WPDAPε0 is unique for any ε . ρ˜ ∈ U∞ Proof Assume that f = g1 + φ1 and f = g2 + φ2 . Then, (g1 − g2 ) + (φ1 − φ2 ) = 0. Since g1 − g2 ∈ DAPε (T, X), and φ1 − φ2 ∈ WPDAPε0 (T, ρ), ˜ in view of Lemma 4.17, we find that g1 − g2 = 0. Consequently, φ1 − φ2 = 0, i.e. φ1 = φ2 . This completes the proof. 

ε , (WPDAPε (T, ρ), Theorem 4.11 For ρ˜ ∈ U∞ ˜  · ε ) is a Banach space for any ε > 0.

Proof Assume that {fn }n∈N is a Cauchy sequence in WPDAPε (T, ρ). ˜ We can write fn = gn + φn uniquely. Using Lemma 4.17, we have gp − gq ε ≤ fp − fq ε , from which we deduce that {gn }n∈N is a Cauchy sequence in DAPε (T, X). Hence, φn = fn − gn is a Cauchy sequence in WPDAPε0 (T, ρ). ˜ Thus, gn → g ∈ DAPε (T, X), φn → φ ∈ WPDAPε0 (T, ρ), ˜ and finally fn → g + φ ∈ WPDAPε (T, ρ). ˜ This completes the proof. 

ε . We say that ρ˜ is ε-equivalent to ρ˜ , written as Definition 4.21 Let ρ˜1 , ρ˜2 ∈ U∞ 1 2 ε ε ρ˜1 ∼ ρ˜2 if ρ˜1 /ρ˜2 ∈ UB . ε . If ρ˜ ε ρ˜ , then WPDAPε (T, ρ˜ ) = Theorem 4.12 Let ρ˜1 , ρ˜2 ∈ U∞ 1 ∼ 2 1 ε WPDAP (T, ρ˜2 ).

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4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

Proof Assume that ρ˜1 ∼ε ρ˜2 . Then, there exist a, b > 0 such that a ρ˜1 ≤ ρ˜2 ≤ bρ˜1 . Thus, for any r1 , r2 ∈ RT (τ, ε), we have am(r1 , r2 , ρ˜1 ) ≤ m(r1 , r2 , ρ˜2 ) ≤ bm(r1 , r2 , ρ˜1 ), and 1 a b m(r1 , , r2 , ρ˜1 )



r2

r1

1 φ(s)ρ˜1 (s)Δs ≤ m(r1 , r2 , ρ˜2 ) ≤



1 b a m(r1 , r2 , ρ˜1 )

r2

φ(s)ρ˜2 (s)Δs

r1



r2

φ(s)ρ˜1 (s)Δs.

r1



This completes the proof.

Lemma  4.19 If g ∈ DAP (T × X, X) and α ∈ DAP (T, X), then G(t) := g ·, α(·) ∈ DAPε (T, X). ε

ε

Proof Let T = {tki ,ε } ⊂ Bε , φ˜ = (g(t, x), T ) ∈ DAPε (T × X, X) × Bε , and ∞ from every sequence {sn }∞ n=1 ⊂ Πε , we can extract a subsequence {τn }n=1 and a −τ sequence {A−τn } such that the limit set T0 of {T n \A−τn } exists and     φ˜ ∗ := g ∗ (t, x), T ∗ = lim θτn φ˜ = lim g(t + τn , x), T −τn , n→∞

n→∞

uniformly exists on PCεrd (T0 × X, X) × Bε . Since α ∈ DAPε (T, X), we can extract  {τn } ⊂ {τn } such that     lim θτ  φ˜ = lim g(t + τn , α(t + τn )), T −τn

n→∞

n

n→∞

     = lim g(t + τn , α ∗ (t)), T −τn = g ∗ (t, α ∗ (t)), T ∗ .

n→∞

Hence, G ∈ DAPε (T, X). This completes the proof.



Theorem 4.13 Let f = g + φ ∈ WPDAPε (T × X, ρ), ˜ where g ∈ DAPε (T × X, X), ε ε φ ∈ WPDAP0 (T × X, ρ), ˜ ρ˜ ∈ U∞ and the following conditions hold:   (i) f (t, x) : t ∈ RT (τ, ε), x ∈ K is bounded for every bounded subset K ⊆ Ω; (ii) f (t, ·), g(t, ·) are uniformly continuous in each bounded subset of Ω uniformly in t ∈ RT (τ, ε).     Then, f ·, h(·) ∈ WPDAPε (T, ρ) ˜ if h ∈ WPDAPε (T, ρ) ˜ and h RT (τ, ε) ⊂ Ω. Proof For any ε1 > 0, we have f = g + φ, where g ∈ DAPε1 (T × X, X) and φ ∈ WPDAPε01 (T × X, ρ) ˜ and h = φ1 + φ2 , where φ1 ∈ DAPε1 (T, X) and φ2 ∈  ε1 ˜ Hence, the function f ·, h(·) can be decomposed as WPDAP0 (T, ρ).

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        f ·, h(·) = g ·, φ1 (·) + f ·, h(·) − g ·, φ1 (·)         = g ·, φ1 (·) + f ·, h(·) − f ·, φ1 (·) + φ ·, φ1 (·) . By Lemma 4.19, g(·, φ1 (·)) ∈ DAPε1 (T, X). Now, consider the function     Ψ (·) := f ·, h(·) − f ·, φ1 (·) . ˜ for any ε2 > ε1 > 0, Clearly, Ψ ∈ BPCεrd1 (T, X). For Ψ to be in WPDAPε01 (T, ρ), it is sufficient to show that  1 lim ρ(t)Δt ˜ = 0. Dr →∞ m(r1 , r2 , ρ) ˜ Mr1 ,r2 ,ε2 (Ψ )

    Let K be a bounded subset of Ω such that φ RT (τ,ε1 ) ⊆ K, φ1 RT (τ, ε1 ) ⊆ K. From (ii), f (t, ·) is uniformly continuous in φ1 RT (τ, ε1 ) uniformly in t ∈ RT (τ, ε1 ). Hence, for a given ε2 > ε1 > 0, there exists δε2 > 0 such that y1 , y2 ∈ K and y1 − y2  < δε2 implies that f (t, y1 ) − f (t, y2 ) < ε2 , t ∈ RT (τ, ε1 ).

Thus, for each t ∈ RT (τ, ε1 ), φ2 (t) < δε2 implies that uniformly in t ∈ RT (τ, ε1 ), '    ' 'f t, h(t) − f t, φ1 (t) ' < ε2 , where φ2 (t) = h(t) − φ1 (t). For r1 , r2 ∈ RT (τ, ε1 ), r2 > r1 , let Mr1 ,r2 ,δε2 (φ2 ) = {t ∈ [r1 , r2 )RT (τ,ε1 ) : φ2  ≥ δε2 }, and we obtain 1 m(r1 , r2 , ρ) ˜ = ≤

1 m(r1 , r2 , ρ) ˜ 1 m(r1 , r2 , ρ) ˜

1 = m(r1 , r2 , ρ) ˜

 ρ(t)Δt ˜ 

Mr1 ,r2 ,ε2 (Ψ (t))

ρ(t)Δt ˜ 

Mr1 ,r2 ,ε2 (f (t,h(t))−f (t,φ1 (t)))

ρ(t)Δt ˜ 

Mr1 ,r2 ,δε (h(t)−φ1 (t)) 2

ρ(t)Δt. ˜ Mr1 ,r2 ,δε (φ2 (t)) 2

˜ Lemma 4.14 yields that Now, since φ2 ∈ WPDAPε01 (T, ρ), 1 lim Dr →∞ m(r1 , r2 , ρ) ˜

 ρ(t)Δt ˜ = 0, Mr1 ,r2 ,ε2 (φ2 (t))

which confirms that Ψ ∈ WPDAPε01 (T, ρ). ˜

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4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

  Finally, we will show that φ ·, φ1 (·) ∈ WPDAPε01 (T, ρ). ˜ Note that f = g + φ and g(t, ·) is uniformly continuous in φ1 RT (τ, ε1 ) uniformly in t ∈ RT (τ, ε1 ).  By assumption (ii), f (t, ·) is uniformly continuous in φ1 RT (τ, ε1 ) uniformly in   ε2 t ∈ RT (τ, ε1 ), so is φ. Since φ1 RT (τ, ε1 ) is relatively compact in X, for > 2ε   m ε2 2 ε1 > 0, there exists δε2 > 0 such that φ1 R  T (τ, ε1 ) ⊂ k=1 Bk , where Bk = {x ∈ X : x − xk  < δε2 } for some xk ∈ φ1 RT (τ, ε1 ) , and ε2 (4.17) , φ1 (t) ∈ Bkε2 , t ∈ RT (τ, ε1 ). 2   : φ1 (t) ∈ Bkε2 is open and It is easy to see that the set Ukε2 := t ∈ RT (τ, ε1 )  ε2 ε2 for any r1 , r2 ∈ RT (τ, ε1 ), we have [r1 , r2 )RT (τ,ε1 ) ⊆ m k=1 Uk . We define V1 =  ε2 ε2 ε2 U1ε2 , Vkε2 = Ukε2 \ k−1 i=1 Ui , 2 ≤ k ≤ m. Then, it is clear that Vi ∩ Vj = ∅ if i = j, 1 ≤ i, j ≤ m. Thus, we have φ(t, φ1 (t)) − φ(t, xk )
0, q infi tki ,ε = θε > 0, then {ϕ(tki ,ε )} is a double-almost periodic sequence for {tki ,ε } ⊂ ε B. 

Proof For ε1 > 0, we construct a sequence {tki ,ε1 } ⊂ RT (τ, ε1 ) satisfying the condition  tki ,ε1 = tki ,ε1 , if tki ,ε1 is a left scattered point in RT (τ, ε1 ), (4.18)  tki ,ε1 = tki ,ε1 − 2ε1 , if tki ,ε1 is a left dense point in RT (τ, ε1 ), where i ∈ Z. For ε2 > ε1 > 0, we choose numbers r ∈ Πε1 , q ∈ Z such that  q ϕ(t + r) − ϕ(t) < ε2 and |tki ,ε1 − r| < ν, 0 < ν < ε2 , for all |t − tki ,ε1 | > ε2 , t ∈ RT (τ, ε1 ), i ∈ Z. Since −ν < tki+q ,ε1 − tki ,ε1 − r < ν in view of (4.18), we find    that 0 < 2ε2 − ν ≤ tki+q ,ε1 − tki ,ε1 − r < 2ε2 + ν < 3ε2 . Thus, if t , t belong to 







the same interval of continuity with |t − t | < 3ε2 then ϕ(t ) − ϕ(t ) < o(3ε2 ).  Now, assuming that 2o(3ε2 ) + ε2 < ε2 < θε1 , we find 





ϕ(tki+q ,ε1 ) − ϕ(tki ,ε1 ) ≤ ϕ(tki+q ,ε1 ) − ϕ(tki ,ε1 + r) + ϕ(tki ,ε1 + r) − ϕ(tki ,ε1 ) 



+ϕ(tki ,ε1 ) − ϕ(tki ,ε1 ) < 2o(3ε2 ) + ε2 < ε2 .

This completes the proof.



Lemma 4.21 A necessary and sufficient condition for a bounded sequence {an } to be in WPDAPε0 (Z, ρ) ˜ is that there exists a uniformly continuous function f ∈ WPDAPε0 (T, ρ) ˜ and a discretization partition {tn,ε } ⊂ RT (τ, ε) such that f (tn,ε ) = an , n ∈ Z, ρ˜ ∈ UBε . Proof (a) Necessity. Let r1 , r2 ∈ RT (τ, ε), r2 > r1 , and we partition the interval [r1 , r2 )∩ RT (τ, ε) as follows: rn1 := r1 ≤ #r1 $ := n1 < rn1 +1 < . . . < rn2 −1 < %r2 & := n2 ≤ r2 := rn2 .

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4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

Denote ξ = maxj {rj +1 − rj }, and define a function f (t) = aj + (t − rj )(aj +1 − aj ), t ∈ [rj , rj +1 )RT (τ,ε) , n1 ≤ j ≤ n2 , j ∈ Z. It is obviously uniformly continuous on RT (τ, ε). To show f ∈ WPDAPε0 (T, ρ) ˜ it suffice to note that 1 m(r1 , r2 , ρ) ˜



r2

f (s)ρ(s)Δs ˜

r1

n 2 −1  rj +1 1 aj + (s − rj )(aj +1 − aj )ρ(s)Δs ˜ = m(r1 , r2 , ρ) ˜ rj j =n1

≤

  rj +1 n 2 −1  1 ¯˜ j )ξ + aj +1 − aj  (s − rj )ρ(s)Δs ˜ aj ρ(t m(r1 , r2 , ρ) ˜ rj j =n1

≤

n 2 −1 2 1 ¯˜ j ) + (an2  + an1 ξ ρ¯ ξ aj ρ(t m(r1 , r2 , ρ) ˜ m(r1 , r2 , ρ) ˜ j =n1



≤

n 2 −1

1

tj ∈[r1 ,r2 ]RT (τ,ε)

=

C2 n 2 −1 j =n1

ρ(t ˜ j)

n 2 −1

¯˜ j ) j =n1 ρ(t

j =n1

¯˜ j ) + an2  + an1  ξ 2 ρ¯ ξ f (tj )ρ(t m(r1 , r2 , ρ) ˜

¯˜ j ) + an2  + an1  ξ 2 ρ¯ → 0, as n2 − n1 → +∞, ξ f (tj )ρ(t m(r1 , r2 , ρ) ˜

¯˜ j ) = sup{ρ(t) where ρ(t ˜ : t ∈ [rj , rj +1 )RT (τ,ε) } and ρ¯ = supt∈RT (τ,ε) ρ(t). ˜ (b) Sufficiency. For any tn1 ,ε1 < tn2 ,ε1 , we partition the interval [tn1 ,ε1 , tn2 ,ε1 )RT (τ,ε1 ) as follows: tn1 ,ε1 ≤ #tn1 ,ε1 $ := n1 < tn1 +1,ε1 < . . . < tn2 −1,ε1 < %tn2 ,ε1 & := n2 ≤ tn2 ,ε1 . Let 1 > ε2 > ε1 > 0, there exists a δε2 > 0 such that for t ∈ (tj,ε1 − δε2 , tj,ε1 )RT (τ,ε1 ) , j ∈ Z and n1 ≤ j ≤ n2 , we have ˜ j,ε1 ), n1 ≤ j ≤ n2 . f (t)ρ(t) ˜ ≥ (1 − ε2 )f (tj,ε1 )ρ(t

4.3 Weighted Piecewise Pseudo Double-Almost Periodic Functions on ACCTS

277

Hence, 

tn2 ,ε1

f (t)ρ(t)Δt ˜ ≥

tn1 ,ε1

 n2 

tj,ε1

f (t)ρ(t)Δt ˜ ≥

j =n1 +1 tj −1,ε1

≥

n2 

 n2 

tj,ε1

j =n1 +1 tj,ε1 −δε2

f (t)ρ(t)Δt ˜

δε2 (1 − ε2 )f (tj,ε1 )ρ(t ˜ j,ε1 )

j =n1 +1 n2 

≥ δε2 (1 − ε2 )

f (tj,ε1 )ρ(t ˜ j,ε1 ),

j =n1 +1

which implies that 1 m(tn1 ,ε1 , tn2 ,ε1 , ρ) ˜ ≥ δε2 (1 − ε2 )



tn2 ,ε1

f (t)ρ(t)Δt ˜

tn1 ,ε1

n2  1 f (tj,ε1 )ρ(t ˜ j,ε1 ) m(tn1 ,ε1 , tn2 ,ε1 , ρ) ˜ j =n1 +1

≥ δε2 (1 − ε2 )  ≥ δε2 (1 − ε2 ) ×

n2 

n2 

1 tj ∈[tn1 ,ε1 ,tn2 ,ε1 )RT (τ,ε1 )

C2



¯˜ j,ε ) ρ(t 1 j =n

f (tj )ρ(t ˜ j)

1 +1

1 tj,ε1 ∈[tn1 ,ε1 ,tn2 ,ε1 )RT (τ,ε1 )

ρ(t ˜ j,ε1 )

f (tj,ε1 )ρ(t ˜ j,ε1 ).

(4.19)

j =n1 +1

From (4.19) and f ∈ WPDAPε01 (T, ρ), ˜ it follows that f (tn,ε1 ) = an ∈ WPDAPε01 (Z, ρ). ˜ This completes the proof. 

From Lemma 4.21, we have the following result directly: Theorem 4.14 A necessary and sufficient condition for a bounded sequence {an } to be in WPDAPε (Z, ρ) ˜ is that there exists a uniformly continuous function f ∈ WPDAPε (T, ρ) ˜ and a discretization partition {tn,ε } ⊂ RT (τ, ε) such that f (tn,ε ) = an , n ∈ Z, ρ˜ ∈ UBε . ε and the sequence of vector-valued functions Theorem 4.15 Assume that ρ ∈ U∞ {Ik }k∈Z is weighted pseudo-almost periodic, i.e., for any x ∈ Ω, {Ik (x), k ∈ Z} is weighted pseudo-almost periodic sequence. Suppose that {Ik (x) : k ∈ Z, x ∈ K} is bounded for every bounded subset K ⊆ Ω, Ik (x) is uniformly continuous  in x ∈ Ω ε ε uniformly in k ∈ Z. If h ∈ WPDAP (T, ρ) ˜ ∩ UPC (T, X) such that h RT (τ, ε) ⊂   Ω, then Ik h(tki ,ε ) is weighted pseudo double-almost periodic.

278

4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

Proof Fix h ∈ WPDAPε (T, ρ) ˜ ∩ UPCε (T, X). First we will show that h(tki ,ε ) is weighted pseudo double-almost periodic. Since h = φ1 + φ2 , where φ1 ∈ ˜ it follows from Lemma 4.20 that the DAPε (T, X) and φ2 ∈ WPDAPε0 (T, ρ), sequence φ1 (tki ,ε ) is double-almost periodic. To show h(tki ,ε ) is weighted pseudo double-almost periodic, we need to show that φ2 (tki ,ε ) ∈ WPDAPε0 (Z, ρ). ˜ From the assumption, h, φ1 ∈ UPCε (T, X), so is φ2 . For any tkn1 ,ε1 , tkn2 ,ε1 ∈ Bε1 , we partition the interval [tkn1 ,ε1 , tkn2 ,ε1 )RT (τ,ε1 ) , and we repeat the same proof process of Sufficiency for Lemma 4.21, so we have φ2 (tki ,ε1 ) ∈ WPDAPε01 (Z, ρ). ˜ Hence, h(tki ,ε1 ) is weighted pseudo double-almost periodic.   Next, we will show that Ik h(tki ,ε ) is weighted pseudo double-almost periodic. Let I (t, x) = In (x) + (t − tn,ε )[In+1 (x) − In (x)], Φ0 (t) = h(tn,ε ) + (t − tn,ε )[h(tn+1,ε ) − h(tn,ε )], where ξ is as in Lemma 4.21 and t ∈ [rn , rn+1 )RT (τ,ε) , kn1 ≤ n ≤ kn2 , n ∈ Z. Since {In } is weighted pseudo almost periodic sequence and {h(tn,ε )} is weighted pseudo double-almost periodic sequence, by Lemma 4.21 and Theorem 4.14, it follows that I ∈ WPDAPε (T × Ω, ρ), ˜ Φ0 ∈ WPDAPε (T, ρ). ˜ For every t ∈ RT (τ, ε), there exists a integer kn1 ≤ n ≤ kn2 , n ∈ Z such that |t − tn,ε | ≤ ξ , and hence I (t, x) ≤ In (x) + |t − tn,ε |[In+1 (x) + In (x)] ≤ (1 + ξ )In (x) + ξ In+1 (x).

Since {In (x) : kn1 ≤ n ≤ kn2 , n ∈ Z, x ∈ K} is bounded for every bounded set K ⊆ Ω, I (t, x) : t ∈ RT (τ, ε), x ∈ K is bounded for every bounded set K ⊆ Ω. For every x1 , x2 ∈ Ω, we have I (t, x1 ) − I (t, x2 ) ≤ In (x1 ) − In (x2 ) + |t − tn,ε |[In+1 (x1 ) − In+1 (x2 ) +In (x1 ) − In (x2 )] ≤ (1 + ξ )In (x1 ) − In (x2 ) + ξ In+1 (x1 ) − In+1 (x2 ). From the fact that Ik (x) is uniformly continuous in x ∈ Ω uniformly in k ∈ Z, it follows that I (t, x) is uniformly continuous in x ∈ Ω uniformly in t ∈ RT (τ, ε).   Thus, by Theorem 4.13, I ·, Φ0 (·)  ∈ WPDAPε (T, ˜ Again, using Lemma 4.21  ρ). and Theorem 4.14, we find that I tki ,ε , Φ0 (tki ,ε ) is a weighted pseudo doublealmost periodic sequence, that is, Ik h(tki ,ε ) is weighted pseudo double-almost periodic. This completes the proof. 

From Theorem 4.15, the following corollary follows:

4.3 Weighted Piecewise Pseudo Double-Almost Periodic Functions on ACCTS

279

Corollary 4.4 Assume that the sequence of vector-valued functions {Ik }k∈Z is weighted pseudo-almost periodic, ρ˜ ∈ UBε and there is a number L > 0 such that Ik (x) − Ik (y) ≤ Lx − y ε ˜ ∩ UPCε (T, ρ) ˜ such that for  all x, y ∈ Ω, k ∈ Z. If h ∈ WPDAP (T, ρ) h RT (τ, ε) ⊂ Ω, then Ik (h(tki ,ε )) is weighted pseudo double-almost periodic.

In the following, we show a criterion for a relatively compact set in PCεrd (T, X). Let h˜ 0 : T → R be a continuous function such that h˜ 0 (t) ≥ 1 for all t ∈ T and ˜h0 (t) → ∞ as |t| → ∞. Then, the space   φ(t) =0 PCεh0 (T, X) = φ ∈ PCεrd (T, X) : lim |t|→∞ h0 (t) endowed with the norm φεh0 = h˜ 0 |RT (τ,ε) .

φ(t) is a Banach space, where h0 (t) = t∈RT (τ,ε) h0 (t) sup

Theorem 4.16 A set B ⊆ PCεh0 (T, X) is relatively compact if and only if φ(t) = 0 uniformly for φ ∈ B. h0 (t) (2) Bt = {φ(t) : φ ∈ B} is relatively compact in X for every t ∈ RT (τ, ε). (3) The set B is equicontinuous on each interval (tki ,ε , tki+1 ,ε )RT (τ,ε) , i ∈ Z. (1) lim|t|→∞

ε Proof Let PC0,ε rd (T, X) = {φ ∈ PCrd (T, X) : lim|t|→∞ φ(t) = 0}. It is clear 0,ε that PCrd (T, X) is isometrically isomorphic with the space PCεh0 (T, X). In order to

prove Theorem 4.16, we only need to show that B∗ ⊆ PC0,ε rd (T, X) is a relatively compact set if and only if

(a) lim|t|→∞ f (t) = 0 uniformly for f ∈ B∗ . (b) B∗t = {f (t) : f ∈ B∗ } is relatively compact in X for every t ∈ RT (τ, ε). (c) The set B∗ is equicontinuous on each interval (tki ,ε , tki+1 ,ε )RT (τ,ε) , i ∈ Z. (a) Sufficiency. By (a), for any ε2 > ε1 > 0, there exists δ1ε2 > 0 such that f (t) < ε2 , |t| > δ1ε2 , f ∈ B∗ . By (c), for the above ε2 , there exists δ : 0 < δε2 < δ1ε2 , such that 



t , t ∈ (tki ,ε1 , tki+1 ,ε1 )RT (τ,ε1 ) , i ∈ Z, 



|t − t | < δε2 implies 



f (t ) − f (t ) < ε2 , ∀f ∈ B∗ .

(4.20)

280

4 Piecewise Almost Periodic Functions and Generalizations on Translation Time. . .

For the interval [−δ1ε2 , δ1ε2 ]RT (τ,ε1 ) , there exists a set S = {s1 , s2 , . . . , sq } ⊂ [−δ1ε2 , δ1ε2 ]RT (τ,ε1 ) , sj = tki ,ε1 , j = 1, 2, . . . , q such that |t − sj | < δε2 and f (t) − f (sj ) < ε2 , j = 1, 2, . . . , q, f ∈ B∗ .

(4.21)

For any sequence {fk : k ≥ 1} ⊆ B∗ , by (b), we can extract a subsequence that converges at each point t ∈ RT (τ, ε1 ). Since S is finite, for the above ε2 > ε1 > 0, there exists n0 ∈ N, m, n ≥ n0 such that fm (t) − fn (t) < ε2 , t ∈ S.

(4.22)

Hence, for t ∈ [−δ1ε2 , δ1ε2 ]RT (τ,ε1 ) , by (4.21) and (4.22), it follows that fm (t)−fn (t) ≤ fm (t)−fm (sj )+fm (sj )−fn (sj )+fn (sj )−fn (t) < 3ε2 . For |t| > δ1ε2 , by (4.20), we have |fm (t) − fn (t) < 2ε2 . Thus, {fk : k ≥ 1} is almost uniformly convergent on RT (τ, ε1 ), and hence 1 B∗ ⊆ PC0,ε rd (T, X) is a relatively compact set. 1 (b) Necessity. Since B∗ ⊆ PC0,ε rd (T, X) is relatively compact, for any ε2 > ε1 > 0, there exist a finite number of functions f1 , f2 , . . . , fm of B∗ such that f − fj ε1 < ε2 , j = 1, 2, . . . , m, f ∈ B∗ .

(4.23)

This finite set of functions f1 , f2 , . . . , fm is equicontinuous, that is, for the   above ε2 > ε1 > 0, there exists a number δ2ε2 > 0 such that t , t ∈     (tki ,ε1 , tki+1 ,ε1 )RT (τ,ε1 ) , i ∈ Z, |t − t | < δ2ε2 implies that fj (t ) − fj (t ) < ∗ ε2 . Now, using (4.23), for any f ∈ B , we have 















f (t )−f (t ) ≤ f (t )−fj (t )+fj (t )−fj (t )+fj (t )−f (t ) < 3ε2 , which shows (c). Since fj ∈ B∗ , for the above ε2 > ε1 > 0, there exist numbers νjε2 > 0 such that fj (t) < ε2 , |t| > νjε2 , j = 1, 2, . . . , m.

(4.24)

4.3 Weighted Piecewise Pseudo Double-Almost Periodic Functions on ACCTS

281

ε2 Let δ3ε2 = max{ν1ε2 , . . . , νm }, by (4.23) and (4.24), for any f ∈ B∗ , it follows that

f (t) ≤ f (t) − fj (t) + fj (t) < 2ε2 , |t| > δ3ε2 , which shows (a). Since B∗ is relatively compact, for any sequence {fk : k ≥ 1} ⊆ B∗ , there exists a subsequence that converges almost uniformly on RT (τ, ε1 ). Fix t ∈ RT (τ, ε1 ), in the sequence {fk (t) : k ≥ 1} ⊆ X, there exists a convergent subsequence. Therefore, for fixed t ∈ RT (τ, ε1 ), the set {f (t) : f ∈ B∗ } is relatively compact, which shows (b). This completes the proof. 

Chapter 5

Almost Automorphic Functions and Generalizations on Translation Time Scales

5.1 Almost Automorphic Functions on CCTS The notion of almost automorphic functions came from the research of differential geometry (see Veech [210]). The almost automorphic dynamical behavior of solutions for dynamic equations is an important research field (see Chang et al.[83, 84, 86, 87], Diagana et al.[115–117, 119], Ezzinbi et al.[129–131], N’Guérékata [190–192], Shen et al.[205], Zhao et al.[251]). Naturally, this notion was also generalized to pseudo almost automorphic functions (see Chang et al.[79, 81], Diagana et al.[118, 120, 121], Ezzinbi et al.[127, 128], Liang et al.[181], Xiao et al.[247]), weighted pseudo almost automorphic functions (see Chang et al.[82, 85]), and Stepanov-like (weighted pseudo) almost automorphic functions (see Chang et al.[80, 88], Fan et al.[132], N’Guérékata et al.[193], Zhang et al.[252]) which are extensively used to investigate the dynamical behavior of solutions for almost automorphic dynamic equations. To unify discrete and continuous dynamic equations on this topic and extend these results to hybrid domains, Wang, Agarwal, O’Regan and N’Guérékata introduced the almost automorphic functions on different types of time scales (see Agarwal et al.[16], Kéré et al.[154], Mophou et al.[185], Milcé et al.[187], N’Guérékata et al.[194, 195], Wang et al.[236, 239, 241]) and these notions were extended to weighted pseudo almost automorphic functions and applied to study abstract dynamic equations on time scales (see Wang et al.[238, 242]). In this section, the concept of almost automorphic functions on CCTS is introduced and some of their basic properties are presented. Definition 5.1 (i) A rd-continuous function f : R → X is said to be almost automorphic if for  every sequence of real numbers {sn } ⊂ Π there exists a subsequence {sn } such that

© Springer Nature Switzerland AG 2020 C. Wang et al., Theory of Translation Closedness for Time Scales, Developments in Mathematics 62, https://doi.org/10.1007/978-3-030-38644-3_5

283

284

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

lim lim f (t + sn − sm ) = f (t)

m→∞ n→∞

for each t ∈ T. (ii) A rd-continuous function f : R × X → X is said to be almost automorphic in t uniformly for all x ∈ B, where B is any bounded subset of X, if for every  sequence of real numbers {sn } ⊂ Π there exists a subsequence {sn } such that lim lim f (t + sn − sm , x) = f (t, x)

m→∞ n→∞

for each t ∈ T. This definition can also be reformulated as follows: Definition 5.2 (i) Let f : T → X be a bounded rd-continuous function. We say that f is almost automorphic if for every sequence of real numbers {sn }∞ n=1 ⊂ Π, we can extract a subsequence {τn }∞ ⊂ Π such that: n=1 g(t) = lim f (t + τn ) n→∞

(5.1)

is well defined for each t ∈ T and lim g(t − τn ) = f (t)

n→∞

(5.2)

for each t ∈ T. Denote by AA(T, X) the set of all such functions or short for AA(T). (ii) A rd-continuous function f : T × X → X is said to be almost automorphic if f (t, x) is almost automorphic in t ∈ T uniformly for all x ∈ B, where B is any bounded subset of X. Denote by AA(T × X, X) the set of all such functions or short for AA(T × X). If f, g : R × X → X are almost automorphic in t uniformly for x ∈ B. Let λ ∈ R. Then the following results are obvious: (i) (ii) (iii) (iv)

f + g ∈ AA(T × X); λf ∈ AA(T × X); fλ0 ∈ AA(T × X) where fλ0 (t) := f (t + λ0 ) for λ0 ∈ Π . The range of f , that is, R(f ) := {f (t) : t ∈ T} is relatively compact.

Remark 5.1 If f and g are the functions defined by Definition 5.2, then not only f ∞ < ∞ and f ∞ = f ∞ but also R(g) ⊆ R(f ). Theorem 5.1 Fix f ∈ AA(T). If h : Π → R and h ∈ L1 (R), then their convolution defined by  (f ∗ h)(t) :=

+∞

−∞

f (t − s)h(s)ΔΠ s.

5.1 Almost Automorphic Functions on CCTS

285

belongs to AA(T), where ΔΠ denotes the Δ-differential over the time scale Π . 

Proof Let {sn } ⊂ Π be an arbitrary sequence. Now since f is almost automorphic, there exists {sn } a subsequence such that g(t − s) := lim f (t − s + sn ) n→∞

and f (t − s) = lim g(t − s − sn ) n→∞

for all t ∈ T, s ∈ Π . Consider  (f ∗ h)(t + sn ) :=

+∞

f (t − s + sn )h(s)ΔΠ s, for all t ∈ T.

−∞

Clearly, f (t − s + sn )h(s) ≤ f ∞ |h(s)| for all t ∈ T, s ∈ Π. Since h ∈ L1 (R) and according to the Lebesgue Dominated Convergence Theorem, we have  +∞ lim (f ∗ h)(t + sn ) = lim f (t − s + sn )h(s)ΔΠ s n→∞

 =

−∞ n→∞ +∞ −∞

g(t − s)h(s)ΔΠ s := (g ∗ h)(t) for each t ∈ T.

Similarly, consider  (g ∗ h)(t − sn ) :=

+∞

−∞

g(t − s − sn )h(s)ΔΠ s

for all t ∈ T. Clearly, g(t − s − sn ) ≤ g∞ |h(s)| for all t ∈ T, s ∈ Π . By the Lebesgue Dominated Convergence Theorem, it follows that  lim (g ∗ h)(t − sn ) =

n→∞

lim g(t − s − sn )h(s)ΔΠ s

−∞ n→∞

 =

+∞

+∞

−∞

f (t − s)h(s)ΔΠ s := (f ∗ h)(t)

for each t ∈ T. Thus, f ∗ h is almost automorphic. This completes the proof.



286

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

Theorem 5.2 Let {fn } ⊂ AA(X) be a sequence such that there exists f ∈ Crd (T, X) with fn − f ∞ → 0 as n → ∞. Then f ∈ AA(X). 

Proof From Definition 5.2, let {τn } be an arbitrary sequence of real numbers. Then  we can extract a subsequence {τn } of {τn } ⊂ Π such that lim fi (t + τn ) = gi (t),

n→∞

(5.3)

for each i = 1, 2, . . . , pointwise. We claim that the sequence of functions {gi } is a Cauchy sequence. In fact, we can obtain gi (t) − gj (t) ≤ gi (t) − fi (t + τn ) + fi (t + τn ) − fj (t + τn ) +fj (t + τn ) − gj (t).

(5.4)

Let ε > 0. By the uniform convergence of {fn } there exists a positive integer N such that for all i, j > N implies fi (t + τn ) − fj (t + τn ) < ε. By using (5.3), (5.4) and the completeness of the space X, we can deduce the pointwise convergence of the sequence {fn }, say to a function f . Now, we claim that limn→∞ f (t + τn ) = g(t) and limn→∞ g(t − τn ) = f (t) pointwise on T. Let ε > 0, there exists some positive integer M such that f (t + τn ) − fM (t + τn ) < ε and gM (t) − g(t) < ε pointwise so that f (t + τn ) − g(t) ≤ 2ε + fM (t + τn ) − gM (t) pointwise, since for each M, there exists some positive integer K = K(t, M) such that fM (t + τn ) − gM (t) < ε, then we can obtain f (t + τn ) − g(t) ≤ 3ε for n ≥ N0 , where N0 is some positive integer depending on t and ε. Similarly, the same step can be applied for limn→∞ g(t − τn ) = f (t) pointwise on T, thus, we can obtain the desired result. This completes the proof. 

  Theorem 5.3 The space AA(T),  · ∞ is a Banach space. Proof Clearly, AA(T) ⊂ BCrd (T, X).  Now Theorem  5.2 yields AA(T) is a closed subspace of BCrd (T, X). Therefore, AA(T),  · ∞ is a Banach space. The proof is completed. 

Theorem 5.4 If f ∈ AA(T), then  F (t) :=

t

f (s)Δs, t0 ∈ T,

t0

belongs to AA(T) if and only if the range of F is relatively compact in X.

5.1 Almost Automorphic Functions on CCTS

287 

Proof If F ∈ AA(T), then from any given sequence {sn } ⊂ Π there exists a subsequence {sn } such that (i) limn→∞ f (t + sn ) = g(t) pointwise on T; (ii) limn→∞ g(t − sn ) = f (t) pointwise on T; (iii) limn→∞ F (t0 + sn ) = C1 in X. Now  F (t + sn ) =

t0 +sn

 f (s)Δs +

t0

t+sn

t0 +sn



t

f (s)Δs = F (t0 + sn ) +

f (s + sn )Δs.

t0

According to Lebesgue’s theorem, we have 

t

lim F (t + sn ) = C1 +

n→∞

g(s)Δs := G(t),

t0

noting that G(t) ≤ F ∞ , thus we may assume without any loss of generality that limn→∞ G(t0 − sn ) := C2 in X, then  G(t − sn ) =

t0 −sn

 g(s)Δs +

t0

t−sn

t0 −sn

 g(s)Δs = G(t0 − sn ) +

t

g(s − sn )Δs

t0

and  lim G(t − sn ) = C2 +

n→∞

t

f (s)Δs := H (t).

t0

For any function f0 : T → X}, let Af0 = {f0 (t) : t ∈ T}. Then A can be applied AH (t) = C2 + AF (t) = 2C2 + F (t). For any positive integer n we have An F (t) = nC2 + F (t). If C2 = 0, we get a contradiction. Hence, C2 = 0 and An F (t) = F (t), which proves the almost automorphy of F (t). This completes the proof. 

From Theorem 5.4, the following theorem can also be proved. Theorem 5.5 Let f ∈ AA(T) and suppose that the Banach space X is uniformly convex. Then  t F (t) := f (s)Δs, t0 ∈ T, t0

belongs to AA(T) if and only if F is bounded.

288

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

Theorem 5.6 Let f : T × X → X be an almost automorphic function. Assume that u → (t, u) is Lipschitzian uniformly in t ∈ T, that is, there exists L > 0 such that f (t, x) − f (t, y) ≤ Lx − y for all (x,y) ∈ X  × X and t ∈ T. If ϕ ∈ AA(T), then F : T → X defined by F (·) := f ·, ϕ(·) belongs to AA(T). 

Proof Let {sn } ⊂ Π be an arbitrary sequence. By Definition 5.2, we can extract a subsequence {sn } such that limn→∞ f (t + sn , x) = g(t, x) for each t ∈ T and x ∈ B; limn→∞ g(t − sn , x) = f (t, x) for each t ∈ T and x ∈ B; limn→∞ ϕ(t + sn ) = ψ(t) for each t ∈ T; limn→∞ ψ(t + sn ) = ϕ(t) for each t ∈ T.   Letting G(t) := g t, ψ(t) , it is sufficient to show that for each t ∈ T, (i) (ii) (iii) (iv)

lim F (t + sn ) = G(t),

n→∞

and lim G(t − sn ) = F (t).

n→∞

In fact, we can obtain '    ' F (t + sn ) − G(t) ≤ 'f t + sn , ϕ(t + τn ) − f t + sn , ψ(t) ' '    ' +'f t + sn , ψ(t) − g t, ψ(t) ' '    ' ≤ Lϕ(t + sn ) − ψ(t) + 'f t + sn , ψ(t) − g t, ψ(t) '. It follows from (iii) that lim ϕ(t + sn ) − ψ(t) = 0.

n→∞

Similarly, from item (i) we have lim f (t + sn , ψ(t)) − g(t, ψ(t)) = 0

n→∞

and hence limn→∞ F (t + sn ) = G(t). Using the same argument above we obtain that lim G(t − sn ) = F (t),

n→∞

which yields F is almost automorphic. This completes the proof.



5.1 Almost Automorphic Functions on CCTS

289

The following theorem is a generalization of Theorem 5.6. Theorem 5.7 Let f : T × X → X be an almost automorphic function. Assume that u → (t, u) is uniformly continuous on every bounded subset B ⊂ X uniformly   for t ∈ T, that is, for each ε > 0, there exists δ > 0 such that x , y ∈ B and   f (t, x) − f (t.y) < ε for all t ∈ T. If ϕ ∈ AA(T), then x − y  < δ implies  F (·) := f ·, ϕ(·) belongs to AA(T). 

Proof Let {sn } ⊂ Π be an arbitrary sequence. By Definition 5.2, we can extract a subsequence {sn } such that limn→∞ f (t + sn , x) = g(t, x) for each t ∈ T and x ∈ B; limn→∞ g(t − sn , x) = f (t, x) for each t ∈ T and x ∈ B; limn→∞ ϕ(t + sn ) = ψ(t) for each t ∈ T; limn→∞ ψ(t + sn ) = ϕ(t) for each t ∈ T.   Letting G(t) := g t, ψ(t) , it is sufficient to show that for each t ∈ T, (i) (ii) (iii) (iv)

lim F (t + sn ) = G(t),

n→∞

and lim G(t − sn ) = F (t).

n→∞

In fact, we can obtain '    ' F (t + sn ) − G(t) ≤ 'f t + sn , ϕ(t + τn ) − f t + sn , ψ(t) ' '    ' +'f t + sn , ψ(t) − g t, ψ(t) '. Since ϕ is almost automorphic, then both ϕ and ψ are bounded. Thus, we can choose a bounded subset B of X such that ϕ(t) ∈ B and ψ(t) ∈ B for all t ∈ T. It follows from (iii) and the uniform continuity of f in x ∈ B that '    ' lim 'f t + sn , ϕ(t + sn ) − f t + sn , ψ(t) ' = 0.

n→∞

Similarly, from item (i) we have '    ' lim 'f t + sn , ψ(t + sn ) − g t + sn , ψ(t) ' = 0.

n→∞

and hence limn→∞ F (t + sn ) = G(t). Using the same argument above we obtain that lim G(t − sn ) = F (t),

n→∞

which yields F is almost automorphic. This completes the proof.



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5 Almost Automorphic Functions and Generalizations on Translation Time Scales

5.2 Almost Automorphic Functions on Semigroups Induced by CCTS In Definitions 5.1 and 5.2, it is easy to observe that f (t) and g(t) have the same domain R, so the limit equalities (5.1) and (5.2) are meaningful. However, this will not be the case if f is defined on R+ ⊂ R. In fact, if we consider f : R+ → X when (5.1) holds for the sequence {sn } ⊂ R+ , then (5.2) will lose its sense since g(t) and f (t) enjoy the same domain R+ and obviously for some t ∈ R+ , t − sn ∈ R− is out of the domain of g for each sn ∈ R+ that is large enough. To solve the above problem, it is necessary to study almost automorphy of functions on semigroups since the sequence (sn  ) in Definition 5.1 is taken from a group (R, +) according to Definition 1.2.1 in the literature [210], then (5.2) is meaningful, but for the semigroup (R+ , +), (5.2) will lose its sense since the inverse element of sn may not exist. In fact, consider the classical example of an almost automorphic function: f (t) = cos

2 + sin

√

1 2t + sin

√ , 7t



obviously, for any sequence {αn } ⊂ [a, +∞) (a ∈ R+ ), there exists a subsequence {αn } and a sequence {α˜ n } ⊂ [a, +∞) depending on {αn } such that lim lim f (t + αm + α˜ n ) = f (t),

n→∞ m→∞

which gives an opportunity to judge the almost automorphy of f directly on [a, +∞) rather than on R when the function is defined on an infinite subinterval of R. Motivated by the above discussion, in this section, we will provide some basic concepts and theorems which can be used to study the almost automorphic functions on semigroups and generalize the results to a more general setting of time scales. Based on these results, we introduce the concepts of Bochner and Bohr almost automorphic functions on semigroup induced by time scales and their equivalence is proved.

5.2.1 Bochner Almost Automorphic Functions on Semigroups In this subsection, we present Bochner’s definition of almost automorphic function on positive direction semigroups induced by time scales (note that one can also obtain the related definitions and properties of almost automorphic function on negative semigroups induced by time scales in the similar way). We denote by (X,  · ) the Banach space over the field C of complex numbers and x∞ = supt∈T x(t) for any x ∈ X. The notation BC(T, X) is given to denote the set of all bounded continuous functions from T to X. For convenience, given a subsequence

5.2 Almost Automorphic Functions on Semigroups Induced by CCTS

291

{αn } of a semigroup elements such that limn→∞ f (t + αn ) exists pointwise on a semigroup induced by the complete-closed time scale, we shall write g = Tα f . Given subsequences {αn }, {βn }, . . . , {γn } the expression Tγ . . . Tβ Tα f will be taken to mean limi→∞ . . . limj →∞ limk→∞ f (t + αk + βj + . . . + γi ) the limits existing pointwise on a semigroup induced by the complete-closed time scale. + Definition   5.3 Let T be a+ complete-closed time scale. If Π := [0, +∞)Π ∈ ∅, {0} , then we say (Π , +) is a positive  direction semigroup induced by the time scale T; if Π − := (−∞, 0]Π ∈ ∅, {0} , then we say (Π − , +) is a negative direction semigroup induced by the time scale T.  + Remark 5.2 According to Definition 5.3, let T1 = +∞ k=0 [2k, 2k + 1], then Π = − + {2n : n ∈ N}. However, Π1 = {0}, so Π1 is a positive direction semigroup induced  by the positive direction complete-closed time scale T1 . Conversely, let T2 = 0k=−∞ [2k, 2k + 1], then Π2− = {2n : n ∈ N− }, where N− = {−n, n ∈ N}. However, Π2+ = {0}, so Π − is a negative direction semigroup induced by the negative direction complete-closed time scale T2 .  From Remark 5.2, we can observe that T = T1 ∪ T2 = +∞ k=−∞ [2k, 2k + 1] and its period set Π = Π1+ ∪ Π2− , obviously, (Π, +) forms an Abelian group and T is a periodic time scale exactly. In fact, we can obtain the following theorem.

Theorem 5.8 Let T be a complete-closed time scale. Then T is a periodic time scale if and only if (Π, +) is an Abelian group. Proof If T is a periodic time scale, then by Definition 2.26 (d), for any τ1 , τ2 , τ3 ∈ Π , we obtain that (1) for all τ1 , τ2 ∈ Π , we have τ1 + τ2 ∈ Π ; (2) for all τ1 , τ2 , τ3 ∈ Π , we have (τ1 + τ2 ) + τ3 = τ1 + (τ2 + τ3 ); (3) there exists an element 0 ∈ Π , such that for all elements τ ∈ Π , the equation 0 + τ = τ + 0 = τ holds; (4) for each τ ∈ Π , there exists an element −τ ∈ Π such that τ + (−τ ) = 0, where 0 is the identity element; (5) for all τ1 , τ2 ∈ Π , we have τ1 + τ2 = τ2 + τ1 . Hence, (Π, +) is an Abelian group. On the other hand, if (Π, +) is an Abelian group, then by (2.29), for any τ ∈ Π \{0}, there exists −τ ∈ Π \{0} such that T−τ ⊆ T and Tτ ⊆ T, which indicates that T−τ ∪ Tτ ⊆ T, i.e., for any t ∈ T, we have t ± τ ∈ T. Hence, T is periodic. This completes the proof. 

Definition 5.4 Let T be a positive direction complete-closed time scale and (Π + , +) be a semigroup. A function f : T → X is said to be almost automorphic   function on the semigroup (Π + , +) if for any sequence α = {αn }n∈N ⊂ Π + of semigroup elements, there is a subsequence α = {αn }n∈N and a sequence {α˜ n } ⊂ Π + depending on {αn } such that for each t ∈ T both lim f (t + αn ) = Tα f = g(t)

n→∞

and lim g(t + α˜ n ) = Tα˜ g = f (t)

n→∞

(5.5)

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5 Almost Automorphic Functions and Generalizations on Translation Time Scales

hold on (Π + , +) for some function g ∈ BC(T, X). Remark 5.3 Let T = R+ , then Π + = R+ ∪ {0}, we can obtain the case of almost automorphic functions on positive direction continuous semigroup which is the new case compared to Definition 1.2.1 from the literature [210]. In fact, the equality (1.2.2) in Definition 1.2.1 in [210] may be invalid on a semigroup since there may be no inverse element α −1 for a semigroup. Remark 5.4 When convergence in (5.5) is required to be uniform and let T = R+ , then Π + = R+ ∪ {0}, we can obtain that f is almost periodic on the semigroup in the sense of Bochner-von Neumann which is the new case compared to the case from the literature [189]. From Definition 5.4, the following equivalent definition will immediately follow. Definition 5.5 Let T be a positive direction complete-closed time scale and (Π + , +) be a semigroup. A function f : T → X is said to be almost automorphic   function on the semigroup (Π + , +) if for any sequence α = {αn }n∈N ⊂ Π + of semigroup elements, there is a subsequence α = {αn }n∈N and a sequence {α˜ n } ⊂ Π + depending on α such that for each t ∈ T the equality lim lim f (t + αm + α˜ n ) = Tα˜ Tα f = f (t)

n→∞ m→∞

holds on (Π + , +). Let AP = AP (Π + ) be the set of almost automorphic functions on the positive direction semigroup (Π + , +) related to a positive direction complete-closed time scale T. Similarly, let AN = AN (Π − ) be the set of almost automorphic functions on the negative direction semigroup (Π + , −) related to a negative direction completeclosed time scale T; let AB = AB (Π ) be the set of almost automorphic functions on the Abelian group (Π, +) related to a bi-direction complete-closed time scale T. Lemma 5.1 If f ∈ AP , then the double limit lim f (t + αm + α˜ n ) = f (t).

m→∞ n→∞





Proof Since f ∈ AP , then by Definition 5.4, for any sequence α = {αn }n∈N ⊂ Π + of semigroup elements, there is a subsequence α = {αn }n∈N and a sequence {α˜ n } depending on α such that for any ε > 0, there exists N > 0 such that n > N implies f (t + αn ) − g(t) < ε and n > M implies g(t + α˜ n ) − f (t) < ε for each t ∈ T. Hence, when n, m > max{N, M}, we can obtain f (t +αm + α˜ n )−f (t) = f (t +αm + α˜ n )−g(t + α˜ m )+g(t + α˜ m )−f (t) < 2ε. Thus we can obtain the desired results.



5.2 Almost Automorphic Functions on Semigroups Induced by CCTS

293

Theorem 5.9 If f ∈ AP , then there exists a sequence β ⊂ Π + such that Tβ f = f pointwise. 

Proof By Lemma 5.1 and Definition 5.5, there exists a subsequence {αn } ⊂ {αn } and a sequence {α˜ n } depending on α such that lim f (t + αm + α˜ n ) = lim lim f (t + αm + α˜ n ) = Tα˜ Tα f = f.

m→∞ n→∞

n→∞ m→∞

Particularly, we can take {αnk } ⊂ {αn } such that lim f (t + αm + α˜ n ) = lim f (t + αmk + α˜ nk ) = Tβ f = f,

m→∞ n→∞

k→∞

where βk = αmk + α˜ nk . This completes the proof.



Remark 5.5 For convenience of discussion, we can also take βk ≥ αnk if {αn } ⊂ Π + in the proof of Theorem 5.9. Theorem 5.10 The following properties will hold for AP : If f1 , f2 ∈ AP , then f1 + f2 , f1 · f2 ∈ AP . If f ∈ AP and λ ∈ C, then λf ∈ AP . The constant λ is in AP . If f ∈ AP , then f (t) is uniformly bounded on the semigroup (Π + , +). If {fn } ⊂ AP is a sequence such that limn→∞ fn = f holds uniformly on (Π + , +) for some function f , then f ∈ AP . (v) If f ∈ AP and τ ∈ Π + , then f (· + τ ) ∈ AP .

(i) (ii) (iii) (iv)

Proof From the Definition 5.4, the results (i) and (ii) quite easily follow. Now, we prove (iii). For contradiction. Assume that (iii) does not hold, then there would be a sequence {αn } ⊂ Π + and t0 ∈ T such that limn→∞ f (t0 +αn ) = +∞. Since there is no subsequence of {αn } such that (5.5) hold with g(t0 ) finite, then (iii) must be valid. Next, we prove (iv). Since f is the uniform limit of bounded functions, then f is   uniformly bounded. Therefore, for any given sequence α = {αn } ⊂ Π + , there will exist a subsequence α and a sequence α˜ ⊂ Π + depending on α such that Tα f and Tα˜ Tα f exist pointwise. Now, we claim that limm→∞ limn→∞ f (t+αn +α˜ m ) = f (t) for each t ∈ T. In fact, for any given ε > 0, there is N > 0 such that n > N implies supt∈T fn (t)−f (t) < ε. Let β be a subsequence of α such that limk→∞ fn (t +βk ) exists and limm→∞ limk→∞ fn (t + βk + β˜m ) = fn (t) for each t ∈ T, where β˜ is determined by β. It follows from Tα˜ Tα f = Tβ˜ Tβ f that Tα˜ Tα f − Tβ˜ Tβ fn  = Tβ˜ Tβ f − Tβ˜ Tβ fn  ≤ Tβ˜ Tβ fn − f ∞ < ε, which yields that Tα˜ Tα f − f  ≤ Tα˜ Tα f − Tβ˜ Tβ fn  + Tβ˜ Tβ fn − f  ≤ ε + ε = 2ε. By taking ε → 0, we can get the result (iv).

294

5 Almost Automorphic Functions and Generalizations on Translation Time Scales 

Finally, we will prove (v). Since f ∈ AP , then for any sequence α ⊂ Π + ,  there is a subsequence α ⊂ α and a sequence α˜ that depending on α such that Tα˜ Tα f (t ∗ ) = f (t ∗ ) for each t ∗ ∈ T. Thus, for t ∗ = t + τ , we have Tα˜ Tα f (t + τ ) = f (t + τ ). This completes the proof. 

Similar to Definition 5.4, we can also introduce the following definition. Definition 5.6 Let T be a negative direction complete-closed time scale and (Π − , +) be a semigroup. A function f : T → X is said to be almost automorphic   function on the semigroup (Π − , +) if for any sequence α = {αn }n∈N ⊂ Π − of semigroup elements, there is a subsequence α = {αn }n∈N and a sequence {α˜ n } ⊂ Π − depending on α such that for each t ∈ T both lim f (t + αn ) = Tα f = g(t)

n→∞

and lim g(t + α˜ n ) = Tα˜ g = f (t)

n→∞

hold on (Π − , +) for some function g ∈ BC(T, X). Theorem 5.11 The following properties will hold for AN : If f1 , f2 ∈ AN , then f1 + f2 , f1 · f2 ∈ AN . If f ∈ AN and λ ∈ C, then λf ∈ AN . The constant λ is in AN . If f ∈ AN , then f (t) is uniformly bounded on the semigroup (Π − , +). If {fn } ⊂ AN is a sequence such that limn→∞ fn = f holds uniformly on (Π − , +) for some function f , then f ∈ AN . (v) If f ∈ AN and τ ∈ Π − , then f (· + τ ) ∈ AN .

(i) (ii) (iii) (iv)

Proof The proof is similar to the proof of Theorem 5.10, so we omit it here.



In the following, we will establish some results when T is a bi-direction complete-closed time scale and investigate the relationship of function f from the spaces AP , AN and AB . Theorem 5.12 Let T be a bi-direction complete-closed time scale. Then f ∈ AP if and only if f ∈ AN . 

Proof Since f ∈ AP , then for any subsequence {αn } ⊂ Π + , there exists a  subsequence α = {αn } ⊂ α and a sequence α˜ ⊂ Π + depending on α such that lim lim f (t + αn + α˜ m ) = f (t).

m→∞ n→∞

Because T is a bi-direction complete-closed time scale, according to Theorem 5.8, (Π, +) is an Abelian group. Now, we choose the subsequence α(1) = {αn(1) k } and

5.2 Almost Automorphic Functions on Semigroups Induced by CCTS

295

α(2) = {αnk } of α such that αnk −αnk ⊂ Π + . Then for the sequence {αnk −αnk } ⊂ (1) (2) (1) (2) Π + , there exists a subsequence β = {βnk1 } = {αnk1 − αnk1 } ⊂ {αnk − αnk } and a (1) (2) sequence β˜ = {β˜nk } = {α˜ nk − α˜ nk } that is determined by β such that (2)

(1)

1

lim

1

(2)

(1)

(2)

1

lim f (t + βnk1 + β˜nk2 )

k2 →∞ k1 →∞

= lim

lim f (t + αn(1) − αn(2) + α˜ n(1) − α˜ n(2) ) = f (t) k k k k

= lim

lim f (t + αn(1) + α˜ n(1) − αn(2) − α˜ n(2) ) k k k k

= lim

lim f (t − αn(2) − α˜ n(2) ), k k

k2 →∞ k1 →∞ k2 →∞ k1 →∞ k2 →∞ k1 →∞

1

1

1

1

2

2

1

2

2

2

where {−α˜ nk } ⊂ Π − . Hence, f ∈ AN . Conversely, in the similar way, we can also prove f ∈ AP if f ∈ AN , we will not repeat the proof process here. Therefore, we can obtained the desired result. 

(2)

Let Π˜ ± be Π + or Π − and Π be an Abelian group, we will introduce the following concept of almost automorphic functions on groups and obtain the relationship of such a type of functions on groups and on semigroups. Definition 5.7 Let T be a bi-direction complete-closed time scale and (Π˜ ± , +) be a semigroup induced by T. A function f : T → X is said to be almost automorphic   function on the group (Π, +) if for any sequence α = {αn }n∈N ⊂ Π˜ ± of semigroup elements, there is a subsequence α = {α}n∈N such that for each t ∈ T both lim f (t + αn ) = Tα f = g(t)

n→∞

and lim g(t − αn ) = T−α g = f (t)

n→∞

hold on (Π˜ ± , +) for some function g ∈ BC(T, X). From the Definition 5.7, we can also adopt the group (Π, +) to an equivalent definition of the almost automorphic functions on groups as follows. Definition 5.8 Let T be a bi-direction complete-closed time scale and (Π, +) be a group induced by T. A function f : T → X is said to be almost automorphic   function on the group (Π, +) if for any sequence α = {αn }n∈N ⊂ Π of semigroup elements, there is a subsequence α = {α}n∈N such that for each t ∈ T both lim f (t + αn ) = Tα f = g(t)

n→∞

296

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

and lim g(t − αn ) = T−α g = f (t)

n→∞

hold on (Π, +) for some function g ∈ BC(T, X). Remark 5.6 Let {αn } be the sequence in Definition 5.7 or 5.8. For any sequence {αn± } := {α1 , −α1 , α2 , −α2 , . . . , αn , −αn }, there exists a subsequence {αnk , −αmk } := {βk } (where nk , mk ∈ Z+ , k ∈ {1, 2, . . .}) such that lim lim f (t + βm − βn ) = f (t),

n→∞ m→∞

In fact, this result is easy to follow since for any {αn } ⊂ Π there exists a subsequence β ⊂ α such that T−β f = Tβ f = g pointwise. Also, we can provide the following equivalent concept of Definition 5.7. Definition 5.9 Let T be a bi-direction complete-closed time scale and (Π + , +) be a semigroup induced by T. A function f : T → X is said to be almost automorphic   function on the group (Π, +) if for any sequence α = {αn }n∈N ⊂ Π + of semigroup elements, there is a subsequence α = {α}n∈N such that for each t ∈ T the equality lim lim f (t + αm − αn ) = Tα T−α f = f (t)

n→∞ m→∞

holds on (Π + , +). Lemma 5.2 If f ∈ AB , then the double limit lim f (t + αm − αn ) = f (t).

m→∞ n→∞

Proof By using Definition 5.7, the proof is similar to Lemma 5.1, we omit it here. 

Lemma 5.3 If f ∈ AB , then there exist a sequence β ⊂ Π + such that Tβ f = f and T−β f = f pointwise. 

Proof By Lemma 5.2 and Definition 5.9, there exists a subsequence {αn } ⊂ {αn } such that lim f (t + αm − αn ) = lim lim f (t + αm − αn ) = Tα T−α f = f.

m→∞ n→∞

n→∞ m→∞

5.2 Almost Automorphic Functions on Semigroups Induced by CCTS

297

Particularly, we can take {αnk } ⊂ {αn } such that lim f (t + αm − αn ) = lim f (t + αmk − αnk ) = Tβ f = f,

m→∞ n→∞

k→∞

where βk = αmk − αnk . Now, we can choose the required sequences {αmk } and {αnk } such that βk ∈ Π + or choose the proper sequences {αmk } and {αnk } such that βk ∈ Π − . This completes the proof. 

Theorem 5.13 Let T be a bi-direction complete-closed time scale. Then f ∈ AP if and only if f ∈ AB . 

Proof Let f ∈ AP , then for any sequence {αn } ⊂ Π + , there is a subsequence  {αn } ⊂ {αn } and a sequence α˜ ⊂ Π + depending on α such that Tα Tα˜ f = f pointwise. By Theorem 5.12, we also have f ∈ AN , then for any sequence   {αn } ⊂ Π + , there is a subsequence {αn } ⊂ {αn } and α˜ depending on α such that T−α T−α˜ f = f pointwise. Hence, Tα Tα˜ T−α T−α˜ f = f pointwise. Then Tα T−α˜ f exists pointwise. In fact, if Tα T−α˜ f does not exist pointwise, then there exist some t0 ∈ T such that f (t0 ) = Tα Tα˜ T−α T−α˜ f = Tα˜ T−α (Tα T−α˜ f ) does not exist, this is a contradiction. So we can obtain f (t) = lim lim lim lim f (t + αn − α˜ m + α˜h − αω ) ω→∞ h→∞ m→∞ n→∞

= lim lim lim f (t + αn − α˜ m + α˜ h − αω ) ω→∞ m→∞ n→∞ h→∞

= lim

lim

= lim

lim f (t + αnk1 − αnk3 ) = Tβ T−β f pointwise,

lim f (t + αnk1 − α˜ nk2 + α˜ nk2 − αnk3 )

k3 →∞ k2 →∞ k1 →∞ k3 →∞ k1 →∞

where {αnk := βk } ⊂ {αn }. Hence, f ∈ AB .  On the other hand, if f ∈ AB , then for any sequence {αn } ⊂ Π˜ + , there is a  subsequence {αn } ⊂ {αn } such that limn→∞ limm→∞ f (t + αm − αn ) = f (t). By Lemma 5.3, we can obtain that there exists a sequence β = {βnk } ⊂ Π + and βnk ≥ αnk for each k such that limk→∞ f (t + βnk ) = f (t). Therefore, lim lim f (t + αm − αn ) = lim

lim f (t + αnk1 − αnk2 + βnk2 ) = f (t),

k2 →∞ k1 →∞

n→∞ m→∞

which implies that lim

lim f (t + αnk1 + β˜nk2 ) = f (t),

k2 →∞ k1 →∞

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5 Almost Automorphic Functions and Generalizations on Translation Time Scales

where β˜nk2 = βnk2 − αnk2 ⊆ Π + and it is determined by α. Hence, f ∈ AP . This completes the proof. 

By Theorems 5.12 and 5.13, the following corollary is obvious. Corollary 5.1 Let T be a bi-direction complete-closed time scale. Then f ∈ AP or f ∈ AN if and only if f ∈ AB . Theorem 5.14 Let T be a positive (or negative) direction complete-closed time scale. If f ∈ AP (or f ∈ AN ), then Tα f = g (or T−α f = g) for some sequence α ⊂ Π + implies there exist a sequence α˜ depending on α such that Tα˜ g = f (or T−α˜ g = f ). Moreover, assume T is bi-direction, if f ∈ AB and Tα f = g, then T−α g = f . Proof For contradiction. Assume that Tα˜ g = f does not hold, there must exist some ε0 > 0 and t0 ∈ T and a subsequence β˜ = {β˜j } of α˜ such that g(t0 + β˜j )−f (t0 ) ≥ ε0 for each j ∈ N. Noticing that Tβ f = g (β ⊂ α), then for any subsequence γ of β and γ˜ determined by γ , it cannot be Tγ˜ Tγ f = Tγ˜ Tβ f = Tγ˜ g = f , this is a contradiction since f ∈ AP . For f ∈ AN and α ∈ Π + , the similar proof process can be applied again. Moreover, if T is bi-direction and f ∈ AB , assume that T−α g = f does not hold, there must exist some ε0 > 0 and t0 ∈ T and a subsequence β = {βj } of α such that g(t0 − βj ) − f (t0 ) ≥ ε0 for each j ∈ N. Noticing that T−β f = g, then for any subsequence γ of β, it cannot be T−γ Tγ f = T−γ Tβ f = T−γ g = f , this is a contradiction since f ∈ AB . This completes the proof. 

5.2.2 Bohr Almost Automorphic Functions on Semigroups In this subsection, we introduce and discuss the notion of a Bohr almost automorphic function on semigroups related to time scales. We denote the set {1, 2, . . . , m} by Λ. Definition 5.10 A subset E of a semigroup Π + induced by time  scales is relatively dense if there exists elements s1 , s2 , . . . , sm in Π + such that i∈Λ (si + E) = Π + , where si + E = {si + e : e ∈ E}. Remark 5.7 Note that Definition 5.10 can be extended to the relatively dense set E of a group Πinduced by time scales, i.e., if there exists elements s1 , s2 , . . . , sm in Π such that i∈Λ (si + E) = Π , where si + E = {si + e : e ∈ E}. Hence, for any s1∗ , s2∗ ∈ Π , there exist some τ1 , τ2 ∈ E and si , sj such that s1∗ − s2∗ = sj − si + τ1 − τ2 since s1∗ = sj + τ1 and s2∗ = si + τ2 .

(5.6)

5.2 Almost Automorphic Functions on Semigroups Induced by CCTS

299

Definition 5.11 A subset E of Π + is called relatively dense if there exists a positive number L ∈ Π + such that [a, a + L]Π + ∩ E = ∅ for all a ∈ Π + . The number L is called the inclusion length. Theorem 5.15 Definition 5.10 is equivalent to Definition 5.11. Proof Let E be a relatively dense set that satisfies Definition 5.10, for contradiction, assume that E does not satisfy Definition 5.11, then there exists some a0 ∈ Π + , for any L ∈ Π + , we have [a0 , a0 + L]Π + ∩ E = ∅, which implies that for any s ∈ Π + , we have [a0 + s, a0 + L + s]Π + ∩ (s + E) = ∅, so there exists no elements s1 , s2 , . . . , sm in Π + such that i∈Λ (si +E) = Π + , this contradicts Definition 5.10. On the other hand, let E be a relatively dense set that satisfies Definition 5.11, then there exists a positive number L ∈ Π + such that [a, a + L]Π + ∩ E = ∅ for all a ∈ Π + , thus we have  (s + E) = Π + . (5.7) s∈[0,L]Π +

According to the Finite Covering Theorem, there exist some open intervals (a1 , b1 )Π + , (a2 , b2 )Π + , . . . , (am−1 , bm−1 )Π + m−1 such that [0, L]Π + ⊆ i=1 (ai , bi )Π + . Now, we can choose the suitable numbers s1 , s2 , . . . , sm−1 , sm = 0 ∈ Π + such that s1 + E ⊇ (a1 , b1 )Π + , s2 + E ⊇ (a2 , b2 )Π + , . . . , sm−1 + E ⊇ (am−1 , bm−1 )Π + . Hence, from (5.7), we obtain 

(s + E) = Π + ,

(5.8)

s∈∪m i=1 (si +E)

so (5.8) yields

m

i=1 (si

+ E) = Π + . This completes the proof.



Π+

Definition 5.12 A bounded function f on a semigroup is said to be positive direction Bohr almost automorphic if for each finite set NT ⊂ T and prescribed ε > 0 there is a set Bε = Bε (NT ) ⊂ Π + such that (i) Bε is relatively dense. (ii) If τ ∈ Bε , then maxt∈NT |f (t + τ ) − f (t)| < ε. (iii) If τ1 , τ2 ∈ Bε , then maxt∈NT |f (t + τ1 + τ2 ) − f (t)| < 2ε. Lemma 5.4 Let f be a positive direction Bochner almost automorphic function on a semigroup Π + . If NT ⊂ T is finite and ε > 0 is given, the set   Cε (NT ) = τ ∈ Π + : max |f (t + τ ) − f (t)| < ε t∈NT

is relatively dense.

(5.9)

300

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

Proof For the sake of contradiction, assume that there exists a finite set NT ⊂ T and an ε > 0 such that the set Cε (NT ) of (5.9) Then there  is not relatively dense. + . Now, we will do not exist s1 , s2 , . . . , sm ∈ Π + such that m (s + C ) = Π i ε i=1 construct a sequence {τn } ⊂ Π + and {τ˜n } ⊂ Π + determined by {τn } such that τ˜n + τk ∈ Cε for k < n. For n = 1 let τ1 be an arbitrary element of Π + . Now, we select τ1 , τ2 , . . . , τn ∈  Π + such that τ˜m +τl ∈ Cε , m > l, there will exist an element τ˜ = τ˜n+1 with τ˜n+1 ∈ nj=1 (τj +Cε ). Then we also have τ˜n+1 +τk ∈ Cε for k ≤ n. From this constructed sequence we can extract a subsequence {τi∗ } = τ ∗ , τi∗ = τni and a sequence τ˜ ∗ ⊂ τ˜ depending on τ ∗ such that Tτ ∗ f (t) = g(t) and Tτ˜ ∗ g(t) = f (t) for t ∈ NT . Let τ˜j∗ = τ˜nj be selected so that maxt∈NT |g(t + τ˜j∗ )−f (t)| < ε/2, and then choose τi∗ = τni with nj > ni such that max |f (t + τi∗ + τ˜j∗ ) − f (t)| ≤ max |f (t + τi∗ + τ˜j∗ ) − g(t + τ˜j∗ )|

t∈NT

t∈NT

+ max |g(t + τ˜j∗ ) − f (t)| < ε/2 + ε/2 = ε. t∈NT

Hence τi∗ + τ˜j∗ = τni + τ˜nj ∈ Cε which is a contradiction with the construction of the sequence because nj > ni . Thus, Cε (NT ) must be relatively dense. This completes the proof. 

Lemma 5.5 Let f be a positive direction Bochner almost automorphic function on Π + . For any given ε > 0 and a finite set NT ⊂ T, there exists δ > 0 and a finite superset MT of NT such that whenever σ, τ ∈ Cδ (MT ), then σ + τ ∈ Cε (NT ). Proof For contradiction. Assume that for some finite set NT ⊂ Π + and ε > 0 and every finite superset MT of NT there must exist σ, τ ∈ Cδ (MT ) such that σ + τ ∈ Cε (NT ). Now, we choose a sequence {δn } of positive real numbers so that ∞ n=1 δn < ∞. Next, we will construct a sequence {MnT } of finite supersets of NT . First, we ∗ choose a finite set M1∗ ⊆ Π + , and then let Mk+1 = Mk∗ + Mk∗ for k = 1, 2, . . .. T ∗ . = MkT + Mk∗ + Mk∗ = MkT + Mk+1 Then we set M1T = NT + M1∗ and Mk+1 T For M1 is a finite superset of NT and so by assumption there exists (σ1 , τ1 ) (σ1 , τ1 ∈ M1∗ ) with σ1 , τ1 ∈ Cδ1 (M1T ) such that σ1 + τ1 ∈ Cε (NT ). Having T chosen M1∗ , . . . , Mk∗ and (σ1 , τ1 ), . . . , (σk , τk ) we set Mk+1 = NT + Mk∗ + Mk∗ , T then Mk+1 is a finite superset of NT , and again by assumption there is (σk+1 , τk+1 ) T ) and σ such that σk+1 , τk+1 ∈ Cδk+1 (Mk+1 k+1 + τk+1 ∈ Cε (NT ). The construction the proceeds by induction.  ∗ ∗ Now, let Π0+ = ∞ k=1 Mk . By the construction of Mk , one will observe that if + ∗ ∗ v, w ∈ Π0 , then v, w ∈ Mk for k sufficiently large then v + w ∈ Mk+1 ⊂ Π0+ . + + Hence Π0 is a sub-semigroup of Π , and obviously f is also a positive direction almost automorphic function when it is restricted to Π0+ . In the next step, we will construct a sequence {αk } of thesub-semigroup elements. Let α1 = τ1 , α2 = σ1 . T For k ≥ 1, we define α2k+1 = k+1 n=1 τn , α2k+2 = α2k−1 + σk+1 . If t ∈ Mk , then T t + α2k−1 ∈ Mk+1 . We can obtain

5.2 Almost Automorphic Functions on Semigroups Induced by CCTS

301

f (t + α2k+1 ) − f (t + α2k+2 ) ≤ f (t + α2k−1 + τk+1 ) − f (t + α2k−1 ) +f (t + α2k−1 ) − f (t + α2k−1 + σk+1 ) ≤ δk+1 + δk+1 = 2δk+1 . Hence, it is sufficient to prove that limk→∞ f (t + α2k+1 ) = g(t) exists, then lim f (t + αk ) = g(t)

k→∞

T follows. Let t ∈ MkT and k < j , then t + α2(k+i)+1 ∈ Mk+i+2 . Thus,

f (t + α2k+1 ) − f (t + α2j +1 ) ≤

j −k−1 

f (t + α2(k+i)+1 ) − f (t + α2(k+i+1)+1 )

i=1

=

j −k−1 

f (t + α2(k+i)+1 ) − f (t + α2(k+i)+1 + τk+i+2 )

i=1

≤

j −k−1 

δk+i+2 .

i=0

So let k → ∞, it immediately follows that limk→∞ f (t + α2k+1 ) = g(t), which indicates that limk→∞ f (t + αk ) = g(t). Since f is a positivedirection almost T automorphic function, so limk→∞ g(t + αk ) = f (t) for each t ∈ ∞ i=1 Mk . Next, we choose sufficiently large k so that max |g(t + α2k ) − f (t) < ε/4,

t∈NT

(5.10)

and then choose j > k so that max |f (t + α2k + α2k+1 ) − g(t + α2k ) < ε/4,

t∈NT

(5.11)

so (5.10) and (5.11) yield that max f (t + α2k + α2k+1 ) − f (t) < ε/4 + ε/4 = ε/2.

t∈NT

(5.12)

 Now, α2k = τ1 +τ2 +. . .+τk−1 +σk = k−1 n=1 τn +σk and α2j +1 = τ1 +. . .+τk+1 = j +1 k−1 j +1 n=1 τn + n=1 τn , since j > k, then α2k + α2j +1 = n=1 τn + σk . Hence, we can obtain

302

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

f (t + α2k + α2j +1 ) − f (t + σk+1 + τk+1 ) ≤ f (t + σk+1 + τk+1 ) − f (t + σk+1 + τk+1 + τk+1 ) ≤ f (t + σk+1 + τk+1 + τk+1 ) − f (t + σk+1 + τk+1 + τk+1 + τk+2 ) +f (t + σk+1 + τk+1 + τk+1 + τk+2 ) −f (t + σk+1 + τk+1 + τk+1 + τk+2 + τk+3 ) +... +f (t + σk+1 + τk+1 + . . . + τj ) − f (t + σk+1 + τk+1 + . . . + τj + τj +1 ) ≤

j −k−1 

f (t + σk+1 + τk+1 + . . . + τk+i+1 )

i=0

−f (t + σk+1 + τk+1 + . . . + τk+i+1 + τk+i+2 ).

(5.13)

∗ T , then t +σk+1 +τk+1 +. . .+τk+i+1 ∈ Mk+2+i , Since τk+1 +. . .+τk+i+1 ∈ Mk+2+i so (5.13) yields that

max f (t + α2k + α2j +1 ) − f (t + σk+1 + τk+1 ) ≤

t∈NT

j −k−1 

δk+i+2 .

(5.14)

i=0

Let k be so large that and (5.14) that

∞

n=k δn

< ε/2 since

∞

n=0

< ∞. It follows from (5.12)

max f (t + σk+1 + τk+1 ) − f (t) < ε,

t∈NT

which indicates that σk+1 + τk+1 ∈ Cε (NT ), this is a contradiction with the construction of the sequence (σn , τn ). This completes the proof. 

By Lemma 5.5, we can obtain the following result which will be used later. Corollary 5.2 Let f, NT , ε > 0 be as in Lemma 5.5. If for any integer n > 0 there exists a finite superset MT of NT and a δ > 0 such that τ1 , τ2 , . . . , τn ∈ Cδ (MT ),  ζ then ni=1 τi i ∈ Cε (NT ) for any choices of ζi = 0, 1. Proof This proof can be given by induction on n. For n = 1, by Lemma 5.5, we can choose a set MT ⊃ NT and a δ > 0. Obviously, if τ ∈ Cδ (MT ), then τ ∈ Cε (NT ) for ζ1 = 0, 1 since 0 ∈ Cδ (MT ). Assume that the corollary is valid for some integer n, and let M1T ⊃ NT and δ1 > 0 be chosen by Lemma 5.5 so that whenever τ, σ ∈ Cδ1 (M1T ), then σ + τ ∈ Cε (NT ). Through Lemma 5.5 again let M2T ⊃ M1T and δ2 > 0 be selected so that whenever σ, τ ∈ Cδ2 (M2T ), then σ + τ ∈ Cδ1 (M1T ). By the induction assumption we choose a set MT ⊃ M2T and  ζ δ > 0 such that whenever τ1 , τ2 , . . . , τn ∈ Cδ (MT ), then ni=1 τi i ∈ Cδ2 (M2T ) for any ζi = 0, 1. Let τ1 , . . . , τn , τn+1 be the elements of Cδ (MT ) and assume

5.3 Equivalence of Bochner and Bohr Almost Automorphy on Semigroup. . . ζ

ζ







ζ

303 ζ



ζ

n+1 n+1 γ = τ1 1 + . . . + τn+1 = γ + γ , where γ = τ1 1 + . . . + τn n , γ = τn+1   T T with ζi = 0, 1. Then we obtain γ ∈ Cδ2 (M2 ) and γ ∈ Cδ2 (M2 ). Therefore,   γ = γ + γ ∈ Cδ1 (M1T ), then γ ∈ Cε (NT ). Then by induction we can complete the proof of this corollary. 

5.3 Equivalence of Bochner and Bohr Almost Automorphy on Semigroup Related to Time Scales In this section, we will prove the equivalence of Bochner and Bohr almost automorphy on semigroup related to time scales. Based on this theorem, we can investigate the almost automorphic mild solution of evolution equations on semigroups induced by time scales in the future. For convenience, denote {s1 , s2 , . . . , sm } := SE in Definition 5.10. Lemma 5.6 For any α ∈ Π , there exist some τ1 , τ2 , τ3 ∈ E and si , sj ∈ SE such that α + τ3 = sj − si + τ1 + τ2 . Particularly, if α ∈ Π + and sj = si , then α + τ3 = τ1 + τ2 . Proof In (5.6) from the Remark 5.7, let s1∗ = α and s2∗ = τ1 , it immediately follows α + τ3 = sj − si + τ1 + τ2 . Moreover, if sj = si , then it follows α + τ3 = τ1 + τ2 . The proof is completed.



Theorem 5.16 A function f on semigroup Π + is a positive direction Bochner almost automorphic function if and only if it is a positive direction Bohr almost automorphic function. 

Proof Let f be a positive direction Bohr almost automorphic function and α =  {αk } be a sequence of semigroup elements. From Definition 5.12, f is bounded on the semigroup Π + , and so there is a subsequence {αk } = α and α˜ determined by α such that Tα f = g and Tα˜ g = h exist. Fixing t ∈ T and ε > 0, we choose a set Bε = Bε (0, t) fulfilling the conditions (i)–(iii) of Definition 5.12. Since Bε is relatively dense, there exist elements s1 , s2 , . . . , sm of Π + such that each s ∈ Π + may be written s = τ + sj , where τ ∈ Bε and 1 ≤ j ≤ m. For each k denote αk = τk + sj and α˜ k = τ˜k + sj , j = j (k). Then there are finitely many sj , so by Lemma 5.6, there will be a subsequence β = {βk } and β˜ = {β˜k }, βk = (1) (2) (1) (2) τk + sj0 = τk + τk , β˜k = τ˜k + sj0 = τ˜k + τ˜k where j0 is independent of (1) (2) (1) (2) k and τk , τk , τ˜k , τ˜k ∈ Bε , such that Tβ Tβ˜ f = h. Now, let k1 and k2 be so sufficiently large that f (t + βk1 + β˜k2 ) − h(t) < ε.

(5.15)

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5 Almost Automorphic Functions and Generalizations on Translation Time Scales

(1) (2) (1) (2) Since βk1 + β˜k2 = τk1 + τk1 + τ˜k2 + τ˜k2 and adopting (iii) of Definition 5.12, we obtain

f (t + βk1 + β˜k2 ) − f (t) (1)

(2)

(1)

(2)

(1)

(2)

(1)

(2)

= f (t + τk1 + τk1 + τ˜k2 + τ˜k2 ) − f (t) (1)

(2)

≤ f (t + τk1 + τk1 + τ˜k2 + τ˜k2 ) − f (t + τk1 + τk1 ) (1)

(2)

+f (t + τk1 + τk1 ) − f (t) ≤ 2ε + 2ε = 4ε.

(5.16)

Thus (5.15) and (5.16) yield that h(t) − f (t) < 5ε. Therefore f = h and f is a positive direction Bochner almost automorphic function. On the other hand, if f is a positive direction Bochner almost automorphic function, then f is bounded by Theorem 5.10. Given a finite set NT ⊂ T and ε > 0 choose a finite superset MT of NT and δ > 0 for n = 2 in Corollary 5.2. Define Bε (NT ) = Cδ (MT ). Then by Lemma 5.4 we have Bε is relatively dense. If τ ∈ Bε , then τ ∈ Cε (NT ), meanwhile if σ, τ ∈ Bε then σ + τ ∈ Cε (NT ) from Corollary 5.2. Hence, f completely fulfill all the properties of Definition 5.12, i.e., f is a positive direction almost automorphic function under Bohr sense. This completes the proof. 

5.4 Weighted Pseudo Almost Automorphic Functions on CCTS Let U be the set of all functions ρˆ : T → (0, ∞) which are positive and locally integrable over T. Now for ρˆ ∈ U∞ define   t0 +r 1 ˆ := f ∈ BCrd (T, X) : lim f (s)ρ(s)Δs ˆ = 0, P AA0 (T, ρ) r→∞ m(r, ρ, ˆ t0 ) t0 −r  t0 ∈ T, r ∈ Π . ˆ as the collection of all functions F : T×X → Similarly, we define P AA0 (T×X, ρ) X continuous with respect to its two arguments and F (·, y) is bounded for each y ∈ X, and 1 r→∞ m(r, ρ, ˆ t0 )



lim

uniformly for y ∈ X, where r ∈ Π .

t0 +r t0 −r

F (s, y)ρ(s)Δs ˆ =0

5.4 Weighted Pseudo Almost Automorphic Functions on CCTS

305

We are now ready to introduce the sets W P AA(T, ρ) ˆ and W P AA(T × X, ρ) ˆ of weighted pseudo almost automorphic functions:   W P AA(T, ρ) ˆ = f = g+φ ∈ BCrd (T, X) : g ∈ AA(T, X) and φ ∈ P AA0 (T, ρ) ˆ ;

 W P AA(T × X, ρ) ˆ = f = g + φ ∈ BCrd (T × X, X) : g ∈ AA(T × X, X)  and φ ∈ P AA0 (T × X, ρ) ˆ . From the Definition of W P AA(T, X), one can easily show the following lemma: Lemma 5.7 If f = g + φ with g ∈ AA(T, X), and φ ∈ P AA0 (T, ρ) ˆ where ρ ∈ U∞ , then g(T) ⊂ f (T). Theorem 5.17 Assume that P AA0 (T, ρ) ˆ is translation invariant. Then the decomposition of a weighted pseudo almost automorphic function as AA ⊕ P AA0 is unique for any ρˆ ∈ U∞ . Proof Assume that f = g1 + φ1 and f = g2 + φ2 . Then 0 = (g1 − g2 ) + (φ1 − φ2 ). Since g1 − g2 ∈ AA(T, X), and φ1 − φ2 ∈ P AA0 (T, ρ), ˆ in view of Lemma 5.7, we deduce that g1 − g2 = 0. Consequently, φ1 − φ2 = 0, that is, φ1 = φ2 . The proof is complete. 

Theorem ˆ is translation invariant and ρˆ ∈ U∞ .  5.18 Assume thatP AA0 (T, ρ) Then W P AA(T, ρ), ˆ  · ∞ is a Banach space. Proof Assume that {fn }n∈N is a Cauchy sequence in W P AA(T, ρ). ˆ We can write uniquely fn = gn + φn . Using Lemma 5.7, we see that: gp − gq ∞ ≤ fp − fq ∞ , from which we deduce that {gn }n∈N is a Cauchy sequence in the Banach space AA(T, X). So, φn = fn − gn is also a Cauchy sequence in the Banach space P AA0 (T, ρ). ˆ We can deduce that gn → g ∈ AA(T, X), φn → φ ∈ P AA0 (T, ρ), ˆ and finally fn → g + φ ∈ W P AA(T, ρ). ˆ The proof is complete. 

Definition 5.13 Let ρˆ1 , ρˆ2 ∈ U∞ . One says that ρˆ1 equivalent to ρˆ2 , denoting this as ρˆ1 ∼ ρˆ2 if ρρˆˆ1 ∈ UB . 2

Let ρˆ1 , ρˆ2 , ρˆ3 ∈ U∞ . It is clear that ρˆ1 ≺ ρˆ1 (reflexivity); if ρˆ1 ≺ ρˆ2 , then ρˆ2 ≺ ρˆ1 (symmetry), and if ρˆ1 ≺ ρˆ2 and ρˆ2 ≺ ρˆ3 , then ρˆ1 ≺ ρˆ3 (transitivity). So, ≺ is a binary equivalence relation on U∞ . Theorem 5.19 Let ρˆ1 , ρˆ2 W P AA(T, ρˆ2 ).

∈

U∞ . If ρˆ1

∼

ρˆ2 , then W P AA(T, ρˆ1 )

=

Proof Assume that ρˆ1 ∼ ρˆ2 . There exists a > 0, b > 0 such that a ρˆ1 ≤ ρˆ2 ≤ bρˆ1 . So, am(r, ρˆ1 , t0 ) ≤ m(r, ρˆ2 , t0 ) ≤ bm(r, ρˆ1 , t0 ),

306

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

where r ∈ Π , and a 1 b m(r, ρˆ1 , t0 )



t0 +r

φ(s)ρˆ1 (s)Δs ≤

t0 −r

≤

1 m(r, ρˆ2 , t0 )



1 b a m(r, ρˆ1 , t0 )

t0 +r t0 −r



φ(s)ρˆ2 (s)Δs

t0 +r t0 −r

φ(s)ρˆ1 (s)Δs. 

The proof is completed.

Lemma 5.8 Let f ∈ BCrd (T, X). Then f ∈ P AA0 (T, ρ) ˆ where ρˆ ∈ U∞ if and only if for every ε > 0, 1 r→+∞ m(r, ρ, ˆ t0 )

 ρ(t)Δt ˆ = 0,

lim

Mr,ε (f )

where r ∈ Π and Mr,ε (f ) := {t ∈ [t0 − r, t0 + r)T : f (t) ≥ ε}. Proof (a) Necessity: By contradiction, we suppose that there exists ε0 > 0 such that 1 r→∞ m(r, ρ, ˆ t0 )

 ρ(t)Δt ˆ = 0.

lim

Mr,ε (f )

Then there exists δ > 0 such that for every n ∈ N, 1 m(rn , ρ, ˆ t0 )

 ρ(t)Δt ˆ ≥δ Mrn ,ε (f )

for some rn > n, where rn ∈ Π. So we get  t0 +rn  1 1 f (s)ρ(s)Δs ˆ = f (s)ρ(s)Δs ˆ m(rn , ρ, ˆ t0 ) t0 −rn m(rn , ρ, ˆ t0 ) Mrn ,ε (f ) 0  1 f (s)ρ(s)Δs ˆ + m(rn , ρ, ˆ t0 ) ([t0 −r,t0 +r)T )\Mrn ,ε (f ) 0  1 f (s)ρ(s)Δs ˆ ≥ m(rn , ρ, ˆ t0 ) Mrn ,ε (f ) 0  ε0 ≥ ρ(t)Δt ˆ ≥ ε0 δ, m(rn , ρ, ˆ t0 ) Mrn ,ε (f )

, which contradicts the assumption.

5.4 Weighted Pseudo Almost Automorphic Functions on CCTS

307

(b) Sufficiency: Assume that 

1 r→∞ m(r, ρ, ˆ t0 )

ρ(t)Δt ˆ = 0.

lim

Mr,ε (f )

Then for every ε > 0, there exists r0 > 0 such that for every r > r0 , 1 m(r, ρ, ˆ t0 )

 ρ(t)Δt ˆ < Mr,ε (f )

ε , M

where M := supt∈T f (t) < ∞. Now, we have 1 m(r, ρ, ˆ t0 )  +



t0 +r

t0 −r

1 f (s)ρ(s)Δs ˆ = m(r, ρ, ˆ t0 )  f (s)ρ(s)Δs ˆ

 f (s)ρ(s)Δs ˆ Mr,ε (f )

([t0 −r,t0 +r)T )\Mr,ε (f )

≤

M m(r, ρ, ˆ t0 )



ρ(t)Δt ˆ Mr,ε (f )

ε + m(r, ρ, ˆ t0 )



[t0 −r,t0 +r)T \Mr,ε (f )

ρ(s)Δs ˆ ≤ 2ε.

Therefore, 1 lim r→∞ m(r, ρ, ˆ t0 )



t0 +r

t0 −r

f (s)ρ(s)Δs ˆ = 0,

that is f ∈ P AA0 (T, ρ). ˆ 

The proof is completed. By Lemma 5.8, the following corollary is immediate.

Corollary 5.3 Let f ∈ BCrd (T, X). Then f ∈ P AA0 (T, ρ) ˆ where ρˆ ∈ UB if and only if for every ε > 0,   1 μΔ Mr,ε (f ) = 0, r→+∞ m(r, ρ, ˆ t0 ) lim

where r ∈ Π and Mr,ε (f ) := {t ∈ [t0 − r, t0 + r)T : f (t) ≥ ε}. We make the following assumptions (H1) f (t, x) is uniformly continuous in any bounded subset K ⊂ X for all t ∈ T. (H2) g(t, x) is uniformly continuous in any bounded subset K ⊂ X for all t ∈ T.

308

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

Theorem 5.20 Let f = g +φ ∈ W P AA(T×X, ρ) ˆ where g ∈ AA(T×X, X), φ ∈ P AA0 (T × X, ρ), ˆ ρ ˆ ∈ U and assume that (H1) and (H2) are satisfied. Then ∞   L(·) := f ·, h(·) ∈ W P AA(T, ρ) ˆ if h ∈ W P AA(T, ρ). ˆ Proof We have f = g + φ where g ∈ AA(T × X, X) and φ ∈ P AA0 (T × X, ρ) ˆ and h = μ0 + ν0 where μ0 ∈ AA(T, X) and ν0 ∈ P AA0 (T, ρ). ˆ Now let us write L(·) = g(·, μ0 (·)) + f (·, h(·)) − g(·, μ0 (·)) = g(·, μ0 (·)) + f (·, h(·)) − f (·, μ0 (·)) + φ(·, μ0 (·)). By Theorem 5.7, g(·, μ0 (·)) ∈ AA(T, X). Consider now the function Ψ (·) := f (·, h(·)) − f (·, μ0 (·)). ˆ it is sufficient to show Clearly Ψ (·) ∈ BCrd (T, X). For Ψ to be in P AA0 (T, ρ), that  1 lim ρ(t)Δt ˆ = 0. r→∞ m(r, ρ, ˆ t0 ) Mr,ε (Ψ ) By Lemma 5.7, μ(T) ⊂ h(T) which is a bounded set. Using assumption (H1) with K = h(T), we say that for every ε > 0, there exists δ > 0 such that x, y ∈ K, x − y < δ ⇒ f (t, x) − f (t, y) < ε, t ∈ T. Thus we can obtain   1 1 ρ(t)Δt ˆ = ρ(t)Δt ˆ m(r, ρ, ˆ t0 ) Mr,ε (Ψ ) m(r, ρ, ˆ t0 ) Mr,ε (f (t,h(t))−f (t,μ0 (t)))  1 ρ(t)Δt ˆ ≤ m(r, ρ, ˆ t0 ) Mr,ε (h(t)−μ0 (t))  1 ρ(t)Δt. ˆ = m(r, ρ, ˆ t0 ) Mr,ε (ν0 (t)) ˆ then by Lemma 5.8, Now since ν0 ∈ P AA0 (T, ρ), 1 lim r→∞ m(r, ρ, ˆ t0 )

 ρ(t)Δt ˆ = 0. Mr,ε (ν0 (t))

Consequently, 1 lim r→∞ m(r, ρ, ˆ t0 ) Thus, Ψ ∈ P AA0 (T, X).

 ρ(t)Δt ˆ = 0. Mr,ε (Ψ )

5.4 Weighted Pseudo Almost Automorphic Functions on CCTS

309

Finally, we need to show that φ(·, μ0 (·)) ∈ P AA0 (T, ρ). ˆ Note that φ(t, μ0 (t)) is uniformly continuous on [t0 − r, t0 + r)T , and that μ0 ([t0 − r, t0 + r)T ) is compact since μ0 is continuous on T as an almost automorphic function. Thus given ε > 0, m  Bk where Bk = {x ∈ X : there exists δ > 0 such that μ0 ([t0 − r, t0 + r)T ) ⊂ k=1

x − xk  < δ} for some xk ∈ μ0 ([t0 − r, t0 + r)T ), and

ε , μ0 (t) ∈ Bk , t ∈ [t0 − r, t0 + r)T . 2

φ(t, μ0 (t)) − φ(t, xk )
0. For {tk }k∈Z ∈ B, let BP Cld (T, X) be the space formed by all bounded piecewise ld-continuous functions φ : T → X such that φ(·) is ldcontinuous at t for any t ∈ {tk }k∈Z and φ(tk ) = φ(tk− ) for all k ∈ Z; let Ω be a subset of X and BP Cld (T × Ω, X) be the space formed by all bounded piecewise functions which are ld-continuous in t, φ : T × Ω → X such that for any x ∈ Ω, φ(·, x) ∈ BP Cld (T × X, X). For any t ∈ T, φ(t, ·) is continuous at x ∈ Ω. Let U P Cld (T, X) be the space of all functions ϕ ∈ P Cld (T, X) such that φ satisfies the condition: for any ε > 0, there exists a positive number δ = δ(ε) such   that if the left-dense points t , t belong to the same interval of continuity of ϕ and     |t − t | < δ implies ϕ(t ) − ϕ(t ) < ε. Let T , P ∈ B and let s(T ∪ P ) : B → B be a map such that the set s(T ∪ P ) forms a strictly increasing sequence. For D ⊂ R and ε > 0, we introduce the

5.5 Weighted Piecewise Pseudo Almost Automorphic Functions

311

notations θε (D) = {t +ε : t ∈ D}, Fε (D) = D ∩{θε (D)}. Denote by φ˜ = (ϕ(t), T ) the element from the space P Cld (T, X) × B. For every sequence of real numbers {sn }, n = 1, 2, . . . with θsn φ˜ := (ϕ(t + sn ), T − sn ), we shall consider the sets {ϕ(t + sn ), T − sn } ⊂ P Cld × B, where T − sn = {tk − sn : k ∈ Z, n = 1, 2, . . .}. j

Definition 5.15 Let {tk } ∈ B, k ∈ Z. We say {tk } is a derivative sequence of {tk } and j

tk = tk+j − tk , k, j ∈ Z. j

j

Definition 5.16 Let tk = tk+j − tk , k, j ∈ Z. We say {tk }, k, j ∈ Z is equipotentially almost automorphic on a periodic time scale T if for any sequence  {sn } ⊂ Z, there exists a subsequence {sn } such that s



lim tk n = γk

n→∞

is well defined for each k ∈ Z and 

−sn

lim γk

n→∞

= tk

for each k ∈ Z. Definition 5.17 A function φ ∈ P Cld (T, X) is said to be piecewise ld-continuous almost automorphic (short for ld-piecewise almost automorphic) if the following conditions are fulfilled: (i) Let T = {tk } is an equipotentially almost automorphic sequence. (ii) Let ϕ ∈ P Cld (T, X) be a bounded function with respect to a sequence T = {tk }. We say that ϕ is ld-piecewise almost automorphic if from every sequence ∞ {sn }∞ n=1 ⊂ Π , we can extract a subsequence {τn }n=1 such that     φ˜ ∗ = ϕ ∗ (t), T ∗ = lim ϕ(t + τn ), T − τn = lim θτn φ˜ n→∞

n→∞

is well defined for each t ∈ T and     φ˜ = ϕ(t), T = lim ϕ ∗ (t − τn ), T ∗ + τn = lim θ−τn φ˜ ∗ n→∞

n→∞

for each t ∈ T. Denote by AApl (T, X) the set of all such functions. (iii) A bounded function f ∈ P Cld (T × X, X) with respect to a sequence T = {tk } is said to be ld-piecewise uniformly almost automorphic if f (t, x) is ldpiecewise automorphic in t ∈ T uniformly in x ∈ B, where B is any bounded subset of X. Denote by AApl (T × X, X) the set of all such functions.

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5 Almost Automorphic Functions and Generalizations on Translation Time Scales

Similarly, we can also introduce the concept of piecewise almost automorphic functions which belong to P Crd (T, X). Let U be the set of all functions ρˆ : T → (0, ∞) which are positive and locally ∇-integrable over T. For a given r ∈ [0, ∞)Π and ∀t0 ∈ T, set  m(r, ρ, ˆ t0 ) :=

t0 +r t0 −r

ρ(s)∇s ˆ

(5.18)

for each ρˆ ∈ U . Remark 5.8 In (5.18), if T = R, t0 = 0, one can easily get  m(r, ρ, ˆ t0 ) :=

r −r

ρ(s)ds ˆ

if T = Z, t0 = 0, one has the following r 

m(r, ρ, ˆ t0 ) =

ρ(k). ˆ

k=−r+1

Define U∞

  := ρˆ ∈ U : lim m(r, ρ, ˆ t0 ) = ∞ , r→∞

  UB := ρˆ ∈ U∞ : ρˆ is bounded and inf ρ(s) ˆ >0 . s∈T

It is clear that UB ⊂ U∞ ⊂ U . Now for ρˆ ∈ U∞ define  pl P AA0 (T, ρ) ˆ : = φ ∈ BP Cld (T, X) : lim  ∀t0 ∈ T, r ∈ Π .

 t0 +r 1 φ(s)ρ(s)∇s ˆ = 0, r→∞ m(r, ρ, ˆ t0 ) t0 −r

Similarly, we define  pl ˆ : = Φ ∈ BP Cld (T × Ω, X) : P AA0 (T × X, ρ) 1 r→∞ m(r, ρ, ˆ t0 ) lim



t0 +r t0 −r

Φ(s, x)ρ(s)∇s ˆ =0

 uniformly with respect to x ∈ K, ∀t0 ∈ T, r ∈ Π .

5.5 Weighted Piecewise Pseudo Almost Automorphic Functions

313

We are now ready to introduce the sets W P AApl (T, ρ) ˆ and W P AApl (T×X, ρ) ˆ of piecewise ld-continuous weighted pseudo almost automorphic functions:   pl W P AApl (T, ρ) ˆ = f = g + φ ∈ P Cld (T, X) : g ∈ AApl (T, X) and φ ∈ P AA0 (T, ρ) ˆ ,  W P AApl (T × X, ρ) ˆ = f = g + φ ∈ P Cld (T × X, X) : g ∈ AApl (T × X, X)  pl and φ ∈ P AA0 (T × X, ρ) ˆ . pl

Lemma 5.9 Let φ ∈ BP Cld (T, X). Then φ ∈ P AA0 (T, ρ) ˆ where ρˆ ∈ UB if and only if for every ε > 0,   1 μ∇ Mr,ε,t0 (φ) = 0, r→∞ m(r, ρ, ˆ t0 ) lim

  where r ∈ Π and Mr,ε,t0 (φ) := t ∈ (t0 − r, t0 + r]T : φ(t) ≥ ε . Proof (a) Necessity: For contradiction, suppose that there exists ε0 > 0 such that lim

r→∞

  1 μ∇ Mr,ε0 ,t0 (φ) = 0. m(r, ρ, ˆ t0 )

Then there exists δ > 0 such that for every n ∈ N,   1 μ∇ Mrn ,ε0 ,t0 (φ) ≥ δ for some rn > n, where rn ∈ Π. m(rn , ρ, ˆ t0 ) So we get 1 m(rn , ρ, ˆ t0 )



t0 +r t0 −r

φ(s)ρ(s)∇s ˆ =

1 m(rn , ρ, ˆ t0 ) + ×

 φ(s)ρ(s)∇s ˆ Mrn ,ε0 ,t0 (φ)

1 m(rn , ρ, ˆ t0 )  (t0 −r,t0 +r]T \Mrn ,ε0 ,t0 (φ)

1 ≥ m(rn , ρ, ˆ t0 ) ε0 ≥ m(rn , ρ, ˆ t0 )

φ(s)ρ(s)∇s ˆ



φ(s)ρ(s)∇s ˆ 

Mrn ,ε0 ,t0 (φ)

φ(s)ρ(s)∇s ˆ Mrn ,ε0 ,t0 (φ)

≥ ε0 δγ , where γ = infs∈T ρ(s). ˆ This contradicts the assumption.

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5 Almost Automorphic Functions and Generalizations on Translation Time Scales

(b) Sufficiency: Assume that lim m(r,1ρ,t μ (Mr,ε,t0 (φ)) = 0. Then for every ε > ˆ 0) ∇ r→∞ 0, there exists r0 > 0 such that for every r > r0 , 1 ε μ∇ (Mr,ε,t0 (φ)) < , m(r, ρ, ˆ t0 ) KM ˆ < ∞. where M := supt∈T φ(t) < ∞ and K := supt∈T ρ(t) Now, we have 1 m(r, ρ, ˆ t0 )



t0 +r t0 −r

φ(s)ρ(s)∇s ˆ =

1 m(r, ρ, ˆ t0 )  +

 φ(s)ρ(s)∇s ˆ Mr,ε,t0 (φ)

(t0 −r,t0 +r]T \Mr,ε,t0 (φ)

≤

 φ(s)ρ(s)∇s ˆ

MK μ∇ (Mr,ε,t0 (φ)) m(r, ρ, ˆ t0 )  ε ρ(s)∇s ˆ + m(r, ρ, ˆ t0 ) (t0 −r,t0 +r]T \Mr,ε,t0 (φ)

≤ 2ε. Therefore, limr→∞ m(r,1ρ,t ˆ 0) This completes the proof.

 t0 +r t0 −r

pl

φ(s)ρ(s)∇s ˆ = 0, that is, φ ∈ P AA0 (T, ρ). ˆ 

Similar to the proof of Lemma 5.8, one can easily obtain the following theorem: pl

Theorem 5.21 Let φ ∈ BP Cld (T, X). Then φ ∈ P AA0 (T, ρ) ˆ where ρˆ ∈ U∞ if and only if for every ε > 0, 1 lim r→∞ m(r, ρ, ˆ t0 )

 ρ(t)∇t ˆ = 0, Mr,ε,t0 (φ)

  where r ∈ Π and Mr,ε,t0 (φ) := t ∈ (t0 − r, t0 + r]T : φ(t) ≥ ε . pl

ˆ is a translation invariant set of BP Cld (T, X) with Lemma 5.10 P AA0 (T, ρ) pl ˆ respect to Π if ρˆ ∈ U∞ , i.e., for any s ∈ Π , one has φ(t+s) := θs φ ∈ P AA0 (T, ρ) if ρˆ ∈ U∞ . pl

Proof For any s ∈ Π, φ ∈ P AA0 (T, ρ), ˆ ε > 0, r > 0, we have  Mr,ε,t0 (Ts φ) = t  = t  = t  ⊆ t

∈ (t0 − r, t0 + r]T : Ts (t) ≥ ε



∈ (t0 − r, t0 + r]T : φ(t + s) ≥ ε



∈ (t0 − r + s, t0 + r + s]T : φ(t) ≥ ε



 ∈ (t0 − r − |s|, t0 + r + |s|]T : φ(t) ≥ ε .

5.5 Weighted Piecewise Pseudo Almost Automorphic Functions

315

Hence  1 ρ(t)∇t ˆ ≤ ρ(t)∇t ˆ m(r, ρ, ˆ t0 ) Mr+|s|,ε,t0 (Ts φ) Mr,ε,t0 (Ts φ)  1 m(r + |s|, ρ, ˆ t0 ) ρ(t)∇t. ˆ = m(r, ρ, ˆ t0 ) m(r + |s|, ρ, ˆ t0 ) Mr+|s|,ε,t0 (φ) 1 m(r, ρ, ˆ t0 )



pl

Since φ ∈ P AA0 (T, ρ), ˆ then by Theorem 5.21, we have 1 m(r + |s|, ρ, ˆ t0 ) Furthermore, limr→∞

 ρ(t)∇t ˆ → 0, r → ∞. Mr+|s|,ε,t0 (φ)

m(r+|s|,ρ,t ˆ 0) m(r,ρ,t ˆ 0)

1 m(r, ρ, ˆ t0 )

= 1, thus

 ρ(t)∇t ˆ → 0, r → ∞. Mr,ε,t0 (Ts (φ)) pl

Again, using Theorem 5.21, one can get θs φ ∈ P AA0 (T, ρ) ˆ for any s ∈ Π . This completes the proof. 

By Definition 5.17, one can easily get the following lemma: Lemma 5.11 Let φ ∈ AApl (T, X). Then the range of φ, φ(T), is a relatively compact subset of X. pl

Lemma 5.12 If f = g + φ with g ∈ AApl (T, X), and φ ∈ P AA0 (T, ρ), ˆ where ρˆ ∈ U∞ , then g(T) ⊂ f (T). Lemma 5.13 The decomposition of a piecewise ld-continuous weighted pseudo pl almost automorphic function according to AApl ⊕ P AA0 is unique for any ρˆ ∈ U∞ . Proof Assume that f = g1 + φ1 and f = g2 + φ2 . Then (g1 − g2 ) + (φ1 − φ2 ) = 0. pl ˆ in view of Lemma 5.12, Since g1 − g2 ∈ AApl (T, X), and φ1 − φ2 ∈ P AA0 (T, ρ), we deduce that g1 − g2 = 0. Consequently, φ1 − φ2 = 0, i.e. φ1 = φ2 . This completes the proof. 

  pl Theorem 5.22 For ρˆ ∈ U∞ , W P AA (T, ρ), ˆ  · ∞ is a Banach space. ˆ We can write Proof Assume that {fn }n∈N is a Cauchy sequence in W P AApl (T, ρ). uniquely fn = gn +φn . Using Lemma 5.12, we see that gp −gq ∞ ≤ fp −fq ∞ , from which we deduce that {gn }n∈N is a Cauchy sequence in AApl (T, X). Hence, pl ˆ We deduce that gn → g ∈ φn = fn − gn is a Cauchy sequence in P AA0 (T, ρ). pl ˆ and finally fn → g +φ ∈ W P AApl (T, ρ). ˆ AApl (T, X), φn → φ ∈ P AA0 (T, ρ), This completes the proof. 

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5 Almost Automorphic Functions and Generalizations on Translation Time Scales

Theorem 5.23 Let ρˆ1 , ρˆ2 ∈ U∞ . If ρˆ1 ∼ ρˆ2 . Then W P AApl (T, ρˆ1 ) = W P AApl (T, ρˆ2 ). Proof Assume that ρˆ1 ∼ ρˆ2 . There exist a, b > 0 such that a ρˆ1 ≤ ρˆ2 ≤ bρˆ1 . So am(r, ρˆ1 , t0 ) ≤ m(r, ρˆ2 , t0 ) ≤ bm(r, ρˆ1 , t0 ), where r ∈ Π , and 1 a b m(r, ρˆ1 , t0 )



t0 +r t0 −r

φ(s)ρˆ1 (s)∇s ≤ ≤

1 m(r, ρˆ2 , t0 )



1 b a m(r, ρˆ1 , t0 )

t0 +r t0 −r



φ(s)ρˆ2 (s)∇s

t0 +r t0 −r

φ(s)ρˆ1 (s)∇s. 

This completes the proof.

Lemma  5.14 If g ∈ AApl (T × X, X) and α ∈ AApl (T, X), then G(·) := g ·, α(·) ∈ AApl (T, X).   Proof Let T = {ti }, φ˜ = g(t, x), T ∈ AApl (T×X, X)×B, from every sequence ∞ {sn }∞ n=1 ⊂ Π , we can extract a subsequence {τn }n=1 such that     φ˜ ∗ := g ∗ (t, x), T ∗ = lim θτn φ˜ = lim g(t + τn , x), T − τn n→∞

n→∞

uniformly exists on P Cld (T × X, X) × B. Since α ∈ AApl (T, X), one can extract  {τn } ⊂ {τn } such that     lim θτ  φ˜ = lim g(t + τn , α(t + τn )), T − τn

n→∞

n

n→∞

     = lim g(t + τn , α ∗ (t)), T − τn = g ∗ (t, α ∗ (t)), T ∗ . n→∞

Hence, G ∈ AApl (T, X). This completes the proof.



Theorem 5.24 Let f = g + φ ∈ ∈ AApl (T × X, X), pl ˆ ρˆ ∈ UB and the following conditions hold: φ ∈ P AA0 (T × X, ρ),   (i) f (t, x) : t ∈ T, x ∈ K is bounded for every bounded subset K ⊆ Ω. (ii) f (t, ·), g(t, ·) are uniformly continuous in each bounded subset of Ω for all t ∈ T.   Then f ·, h(·) ∈ W P AApl (T, ρ) ˆ if h ∈ W P AApl (T, ρ) ˆ and h(T) ⊂ Ω. W P AApl (T × X, ρ), ˆ where g

pl

ˆ Proof We have f = g + φ, where g ∈ AApl (T × X, X) and φ ∈ P AA0 (T × X, ρ) pl ˆ Hence, the and h = φ1 + φ2 , where φ1 ∈ AApl (T, X) and φ2 ∈ P AA0 (T, ρ). function f (·, h(·)) can be decomposed as

5.5 Weighted Piecewise Pseudo Almost Automorphic Functions

317

        f ·, h(·) = g ·, φ1 (·) + f ·, h(·) − g ·, φ1 (·)         = g ·, φ1 (·) + f ·, h(·) − f ·, φ1 (·) + φ ·, φ1 (·) .   By Lemma 5.14, g ·, φ1 (·) ∈ AApl (T, X). Now, consider the function     Ψ (·) := f ·, h(·) − f ·, φ1 (·) . pl

Clearly, Ψ ∈ BP Cld (T, X). For Ψ to be in P AA0 (T, ρ), ˆ it is sufficient to show that   1 μ∇ Mr,ε,t0 (Ψ ) = 0. r→∞ m(r, ρ, ˆ t0 ) lim

Let K be a bounded subset of Ω such that φ(T) ⊆ K, φ1 (T) ⊆ K. By (ii), f (t, ·) is uniformly continuous in φ(T) uniformly in t ∈ T, we see that for given ε > 0, there exists δ > 0 such that y1 , y2 ∈ K and y1 − y2  < δ implies that f (t, y1 ) − f (t, y2 ) < ε, t ∈ T. Thus, for each t ∈ T, φ2 (t) < δ implies for all t ∈ T, '    ' 'f t, h(t) − f t, φ1 (t) ' < ε,  where φ2 (t) = h(t) − φ1 (t). For r > 0 and any fixed t0 ∈ T, let Mr,δ,t0 (φ2 ) = t ∈ (t0 − r, t0 + r]T : φ2  ≥ δ , we can obtain     1 1 μ∇ Mr,ε,t0 (Ψ (t)) = μ∇ Mr,ε,t0 (f (t, h(t)) − f (t, φ1 (t))) m(r, ρ, ˆ t0 ) m(r, ρ, ˆ t0 )   1 μ∇ Mr,δ,t0 (h(t) − φ1 (t)) ≤ m(r, ρ, ˆ t0 )   1 μ∇ Mr,δ,t0 (φ2 (t)) . = m(r, ρ, ˆ t0 ) pl

ˆ Lemma 5.9 yields that Now since φ2 ∈ P AA0 (T, ρ),   1 μ∇ Mr,ε,t0 (φ2 (t)) = 0, r→∞ m(r, ρ, ˆ t0 ) lim

pl

ˆ this implies that Ψ ∈ P AA0 (T, ρ).   pl ˆ Note that f = g + φ and Finally, we need to show φ ·, φ1 (·) ∈ P AA0 (T, ρ). g(t, ·) is uniformly continuous in φ1 (T) for all t ∈ T. By assumption (ii), f (t, ·) is uniformly continuous in φ1 (T) for all t ∈ T, so is φ. Since φ1 (T) is relatively

318

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

compact  in X, for ε > 0, there  exists δ > 0 such that φ1 (T) ⊂ Bk = x ∈ X : x − xk  < δ for some xk ∈ φ1 (T) and

m

k=1 Bk ,

where

ε , φ1 (t) ∈ Bk , t ∈ T. (5.19) 2   It is easy to see that the set Uk := t ∈ T : φ1 (t) ∈ Bk is open and φ1 (T) = m k=1 Uk . Define φ(t, φ1 (t)) − φ(t, xk )
0, q ∈ Z, then ϕ(tk ) is a ld-piecewise almost automorphic sequence in X. j

Proof Let tk = tk+j − tk , k, j ∈ Z. Obviously, from the definition of Π , it is easy j to know that tk ∈ Π . Since ϕ ∈ P Cld (T, X) is an almost automorphic function and {tk } ⊂ T is equipotentially almost automorphic, from Definitions 5.16 and 5.17, for  any sequence {sn } ⊂ Z, we have there exists a subsequence {sn } such that        s  s s  lim ϕ(tk+s  ), T − tk n = lim ϕ(tk + tk n ), T − tk n = ϕ ∗ (tk ), T ∗ n n→∞ n→∞   = ϕ(tk + γk ), T − γk

and     −s  −s  lim ϕ ∗ (tk−s  ), T ∗ + tk n = lim ϕ(tk−s  + γk−s  ), T − γk−s  + tk n n n n n n→∞ n→∞   = ϕ(tk ), T .

Hence, {ϕ(tk )} is an almost automorphic sequence in X. This completes the proof. 

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5 Almost Automorphic Functions and Generalizations on Translation Time Scales

Lemma 5.16 A necessary and sufficient condition for a bounded sequence {an } to pl ˆ is that there exists a piecewise uniformly ld-continuous function be in P AA0 (Z, ρ) pl ˆ and a discretization partition {tn } such that f (tn ) = an , n ∈ f ∈ P AA0 (T, ρ) Z, ρˆ ∈ UB . Proof Necessity: By taking fixed r ∈ Π and without generality let   ti(n) , ti(n)+1 , . . . , tj (n) ⊆ (t0 + nr, t0 + (n + 1)r]T , where i(n), j (n) are mappings  ≥ i(n).  from Z to Z and j (n) Now, we rewrite the set ti(n) , ti(n)+1 , . . . , tj (n) as  (1) (2) k0 (n)  (H ) = {tn+1 tn+1 , tn+1 , . . . , tn+1 : 1 ≤ H ≤ k0 (n)}, (H )

we obtain that t0 + nr < tn+1 ≤ t0 + (n + 1)r. We define a function (H )

(H )

(H )

f (t) = an+1 + (t − tn+1 )(an+1 − an(H ) ), t0 + nr < t ≤ t0 + (n + 1)r, t ∈ T, r ∈ Π, n ∈ Z, fixed t0 ∈ T, (H )

(H )

then the corresponding an+1 = f (tn+1 ) for all 1 ≤ H ≤ k0 (n) and it is piecewise pl

ˆ since uniformly ld-continuous on T. f ∈ P AA0 (T, ρ) 1 m(kr, ρ, ˆ t0 )



t0 +kr

t0 −kr

f (s)ρ(s)∇s ˆ

k−1  t0 +(j +1)r  1 (H ) (H ) (H ) (H ) = aj +1 + (s − tj +1 )(aj +1 − aj )ρ(s)∇s ˆ m(kr, ρ, ˆ t0 ) t0 +j r j =−k

≤

  t0 +(j +1)r k−1   1 (H ) ¯ (H ) (H ) (H ) (H ) ˆ j +1 )r + aj +1 − aj  (s − tj +1 )ρ(s)∇s ˆ aj +1 ρ(t m(kr, ρ, ˆ t0 ) t0 +j r j =−k

≤

(H ) (H ) k−1  (ak  + a−k )r 2 1 (H ) ¯ (H ) raj +1 ρ(t ˆ j +1 ) + ρˆ0 m(kr, ρ, ˆ t0 ) m(kr, ρ, ˆ t0 ) j =−k

≤

 (H )

k−1 

1 (H )

tj +1 ∈(t0 −kr,t0 +kr]T

=

C2 k−1  j =−k

ρ(t ˆ j +1 ) j =−k

k−1 

¯ˆ (H ) ) j =−k ρ(t j +1

(H ) ¯ˆ (H ) ) + rf (tj +1 )ρ(t j +1

(H ) ¯ˆ (H ) ) + rf (tj +1 )ρ(t j +1

(H )

ak

(H )

ak

(H )

 + a−k |

m(kr, ρ, ˆ t0 )

r 2 ρˆ0

(H )

 + a−k 

m(kr, ρ, ˆ t0 )

r 2 ρˆ0 → 0, as k → ∞,

5.5 Weighted Piecewise Pseudo Almost Automorphic Functions

321

where ¯ˆ (H ) ) = ρ(t j +1

sup

s∈(t0 +j r,t0 +(j +1)r]T

ρ(s), ˆ ρˆ0 = sup ρ(t) ˆ t∈T

and ) ρ(t ˆ j(H +1 ) =

inf

s∈(t0 +j r,t0 +(j +1)r]T

ρ(s). ˆ 

Sufficiency: Let 0 < ε < 1, there exists δ > 0 such that for t ∈ (tn − δ, tn )T , n ∈ Z, we have 

ˆ n ), n ∈ Z. f (t)ρ(t) ˆ ≥ (1 − ε)f (tn )ρ(t Without loss of generality, let tn ≥ 0, t−n < 0, n ∈ Z+ , there exists r ∈ Π ∩ R+ (H ) (H ) such that t0 + nr < tn+1 ≤ t0 + (n + 1)r, t0 − nr < t−n+1 ≤ t0 + (−n + 1)r and + {t−n , tn : n ∈ Z } be a discrete partition. Therefore, 

t0 +nr t0 −nr

≥

≥

≥

 f (t)ρ(t)∇t ˆ ≥

(H )

tn

f (t)ρ(t)∇t ˆ

(H )

t−n+1

n−2  

t0 +(j +1)r

j =−n+1 t0 +j r n−2  

f (t)ρ(t)∇t ˆ

t0 +(j +1)r

j =−n+1 t0 +(j +1)r−δ n−2 

f (t)ρ(t)∇t ˆ

(H )

j =−n+1

n−2 

(H )

δ(1 − ε)f (tj +1 )ρ(t ˆ j +1 ) ≥ δ(1 − ε)

j =−n+1

(H )

(H )

f (tj +1 )ρ(t ˆ j +1 ),

so one can obtain  t0 +nr n−2  1 1 (H ) (H ) f (t)ρ(t)∇t ˆ ≥ δ(1 − ε) f (tj +1 )ρ(t ˆ j +1 ) m(nr, ρ, ˆ t0 ) t0 −nr m(nr, ρ, ˆ t0 ) j =−n+1 ≥ δ(1 − ε)

n−2 

1 n−2  j =−n+1

¯ˆ (H ) ) j =−n+1 ρ(t j +1 1

≥ δ(1 − ε) C2

n−2  j =−n+1

(H )

(H )

f (tj +1 )ρ(t ˆ j +1 )

n−2  (H ) ρ(t ˆ j +1 ) j =−n+1

(H )

(H )

f (tj +1 )ρ(t ˆ j +1 ),

(5.20)

322

5 Almost Automorphic Functions and Generalizations on Translation Time Scales pl

since f ∈ P AA0 (T, ρ), ˆ it follows from the inequality (5.20) that f (tn ) = an ∈ pl ˆ This completes the proof.

 P AA0 (Z, ρ). By Lemma 5.16, we get the following theorem: Theorem 5.26 A necessary and sufficient condition for a bounded sequence {an } ˆ is that there exists a uniformly ld-continuous function to be in W P AApl (Z, ρ) f ∈ W P AApl (T, ρ) ˆ and a discretization partition {tn } such that f (tn ) = an , n ∈ Z, ρˆ ∈ UB . Theorem 5.27 Assume that ρˆ ∈ UB and the sequence of vector-valued functions {Ii }i∈Z is weighted pseudo almost automorphic, i.e., for any x ∈ Ω, {Ii (x), i ∈ Z} is weighted pseudo almost automorphic sequence. Suppose {Ii (x) : i ∈ Z, x ∈ K} is bounded for every bounded subset K ⊆ Ω, Ii (x) is uniformly continuous in x ∈ Ω pl uniformly ˆ ∩ U P Cld (T, X) such that h(T) ⊂ Ω,  in i ∈ Z. If h ∈ W P AA (T, ρ) then Ii h(ti ) is a weighted pseudo almost automorphic sequence. ˆ ∩ U P Cld (T, X). First we show h(ti ) is weighted Proof Fix h ∈ W P AApl (T, ρ) pseudo almost automorphic. Since h = φ1 + φ2 , where φ1 ∈ AApl (T, X), φ2 ∈ pl ˆ It follows from Lemma 5.15 that the sequence φ1 (ti ) is almost P AA0 (T, ρ). automorphic. By Theorem 5.26, h(ti ) is weighted pseudo almost automorphic sequence.   Now, we show Ii φ(ti ) is weighted pseudo almost periodic. By taking fixed r ∈ Π and without generality let   ti ∗ (i) , ti ∗ (i)+1 , . . . , tj ∗ (i) ⊆ (t0 + ir, t0 + (i + 1)r]T , from Z to Z and j ∗(i) ≥ i ∗ (i). where i ∗ (i), j ∗ (i) are mappings  Now, we rewrite the set ti ∗ (i) , ti ∗ (i)+1 , . . . , tj ∗ (i) as  (1) (2) k0 (i)  (L) = {ti+1 : 1 ≤ L ≤ k0 (i)}, ti+1 , ti+1 , . . . , ti+1  (L)  (L) we obtain that t0 + ir < ti+1 ≤ t0 + (i + 1)r and the corresponding Ii+1 h(ti+1 ) =   (L) (L) I ti+1 , h(ti+1 ) for all 1 ≤ L ≤ k0 (i). Let  (L)  I (t, x) = In (x) + (t − ti+1 ) Ii+1 (x) − Ii (x) ,  (L) (L)  Φ0 (t) = h(ti+1 ) + (t − ti+1 ) h(ti+1 ) − h(ti ) , where t0 + ir < t ≤ t0 + (i + 1)r, i ∈ Z, r ∈ Π . Since Ii , h(ti ) are two weighted pseudo almost automorphic, by Lemma 5.16 and Theorem 5.26, we know that I ∈ W P AApl (T×Ω, ρ), ˆ Φ0 ∈ W P AApl (T, ρ). ˆ For every t ∈ T, there exists a number (L) i ∈ Z such that |t − ti+1 | ≤ r,

5.6 Local Almost Automorphic Functions on Changing-Periodic Time Scales

323

 (L)  I (t, x) ≤ Ii (x) + |t − ti+1 | Ii+1 (x) + Ii (x) ≤ (1 + r)Ii (x) + rIi+1 (x). Since {Ii (x) : i ∈ Z, x ∈ K} is bounded for every bounded set K ⊆ Ω, {I (t, x) : t ∈ T, x ∈ K} is bounded for every bounded set K ⊆ Ω. For every x1 , x2 ∈ Ω, we have (L)

I (t, x1 ) − I (t, x2 ) ≤ Ii (x1 ) − Ii (x2 ) + |t − ti+1 |[Ii+1 (x1 ) − Ii+1 (x2 ) +Ii (x1 ) − Ii (x2 )] ≤ (1 + r)Ii (x1 ) − Ii (x2 ) + rIi+1 (x1 ) − Ii+1 (x2 ). Noting that Ii (x) is uniformly continuous in x ∈ Ω for all i ∈ Z, then we get  that I(t, x) is uniformly in x ∈ Ω for all t ∈ T. Then by Theorem 5.24, I ·, Φ0 (·) ∈ W P AA pl (T, ρ). ˆ Again, using Lemma 5.16 and Theorem 5.26, we have that I t , Φ (t ) is a weighted pseudo almost autmorphic sequence, that is, i 0 i   Ii h(ti ) is weighted pseudo almost automorphic sequence. This completes the proof. 

From Theorem 5.27, one can easily get the following corollary: Corollary 5.7 Assume the sequence of vector-valued functions {Ii }i∈Z is weighted pseudo almost automorphic sequence, ρˆ ∈ UB and there is a number L > 0 such that Ii (x) − Ii (y) ≤ Lx − y ˆ ∩ U P Cld (T, ρ) ˆ such that h(T) ⊂ for all x, y ∈ Ω, i ∈ Z. If h ∈ W P AApl (T, ρ) Ω, then Ii h(ti ) is a weighted pseudo almost automorphic sequence.

5.6 Local Almost Automorphic Functions on Changing-Periodic Time Scales It is clear that periodic time scales are regular, so we can obtain some of their nice properties (such as translation closedness property), and on them introduce and study well defined functions. However, most of time scales are not periodic or complete-closed; thus, it is necessary and meaningful to find an effective way to build bridges between periodic time scales and an arbitrary time scale. This will easily allow us to extract results known for periodic time scales to other types of time scales, and vice versa. Having this in mind, in this section, based on changing-periodic time scales, we shall introduce “local-almost automorphic” functions and study some of their related properties on arbitrary time scales with bounded graininess function μ.

324

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

5.6.1 Local Semigroup on Changing-Periodic Time Scales Based on Definitions 2.34 and 2.36 in Sect. 2.6, we will present some basic properties of semigroups induced by changing-periodic time scales. Definition 5.18 Let T be a changing-periodic time scales, i.e., there exists a index function τt for all t ∈ T such that t + τt ∈ T. The set Sτt := {τ : t + τ ∈ Tτt , ∀t ∈ Tτt } is called the invariant translation number set of the sub-timescale Tτt .   Definition 5.19 If Sτ+t := Sτt ∩ [0, +∞) ∈ {0}, ∅ , then we say Sτ+t is a positivedirection semigroup induced by the changing-periodic time scale T; if Sτ−t := Sτt ∩   (−∞, 0] ∈ {0}, ∅ , then we say Sτ−t is a negative-direction semigroup induced by T. Example 5.1 Let a changing-periodic time scale be the following:     +∞  2 T = 2k, k + 1 : k ∈ Z 11

i=−∞

! 2 2 i, i + 1 . 3 3

The time scale can be easily decomposed into the sub-timescales as follows:  T1 = {2k : k ∈ Z}, T2 =

 +∞  2 k + 1 : k ∈ Z , T3 = 11

i=−∞

! 2 2 i, i + 1 . 3 3

Obviously, T = T1 ∪ T2 ∪ T3 and hence there exists a index function τt as follows:

τt =

⎧ ⎪ ⎪ ⎨1, 2, ⎪ ⎪ ⎩3,

t ∈ T1 , t ∈ T2 , t ∈ T3 .

According to Theorem 2.31 (Decomposition Theorem of Time Scales), we have

Sτt =

⎧ ⎪ ⎪ ⎨S1 = {τ : t + τ ∈ T1 , ∀t ∈ T1 } = {2n : n ∈ Z},

2 S2 = {τ : t + τ ∈ T2 , ∀t ∈ T2 } = { 11 n : n ∈ Z}, ⎪ ⎪ ⎩S = {τ : t + τ ∈ T , ∀t ∈ T } = { 2 n : n ∈ Z}. 3 3 3 3

Hence, the positive-direction semigroup induced by T is:

Sτ+t =

⎧ ⎪ ⎪S1+ = {τ ∈ Z+ : t + τ ∈ T1 , ∀t ∈ T1 } = {2n : n ∈ Z+ }, ⎨

2 n : n ∈ Z+ }, S2+ = {τ ∈ Z+ : t + τ ∈ T2 , ∀t ∈ T2 } = { 11 ⎪ ⎪ ⎩S + = {τ ∈ Z+ : t + τ ∈ T , ∀t ∈ T } = { 2 n : n ∈ Z+ }. 3 3 3 3

5.6 Local Almost Automorphic Functions on Changing-Periodic Time Scales

325

Moreover, the negative-direction semigroup induced by T is:

Sτ−t =

⎧ ⎪ ⎪S1− = {τ ∈ Z− : t + τ ∈ T1 , ∀t ∈ T1 } = {2n : n ∈ Z− }, ⎨

2 S2− = {τ ∈ Z− : t + τ ∈ T2 , ∀t ∈ T2 } = { 11 n : n ∈ Z− }, ⎪ ⎪ ⎩S − = {τ ∈ Z− : t + τ ∈ T , ∀t ∈ T } = { 2 n : n ∈ Z− }. 3 3 3 3

Example 5.2 Some changing-periodic time scales may only possess one-way semigroups. Let T=

 ∞ k=1

1 1 3 k, k + 3 3 4

!    ∞ i=1

2 2 1 − i, − i − 13 13 7

! .

The time scale can be easily decomposed into the sub-timescales as follows: ! ! ∞ ∞   1 1 3 2 1 2 k, k + T1 = − i, − i − , T2 = . 3 3 4 13 13 7 k=1

i=1

Obviously, T = T1 ∪ T2 and hence there exists a index function τt as follows:  τt =

1,

t ∈ T1 ,

2,

t ∈ T2 .

According to Theorem 2.31 (Decomposition Theorem of Time Scales), we have Sτ+t = S1+ = {τ ∈ Z+ : t + τ ∈ T1 , ∀t ∈ T1 } = Sτ−t = S2− = {τ ∈ Z− : t + τ ∈ T2 , ∀t ∈ T2 } =

 

 1 n : n ∈ Z+ , 3

 2 n : n ∈ Z− . 13

Note that for T1 , S1− = {0}; for T2 , S2+ = {0}. Hence, there exist only one-way semigroups for this changing-periodic time scale. By using the basic knowledge of Π -semigroup from Sect. 3.4, we can introduce the concept of local Π -semigroup attached with translation direction on changingperiodic time scales. Definition 5.20 Let T be a changing-periodic time scale, and {Tτ } be a family of bounded linear operators on Banach space X. If for all τ1 , τ2 ∈ Sτ+t (or Sτ−t ) the following holds: Tτ1 +τ2 = Tτ1 Tτ2 ,

(5.21)

326

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

then {Tτ : τ ∈ Sτ+t } (or {Tτ : τ ∈ Sτ−t }) is called a local one-parameter operator semigroup; if (5.21) holds for all τ ∈ Sτt , we call {Tτ : τ ∈ Sτt } a local oneparameter operator group. Definition 5.21 Let T be a periodic time scale, and {Tτ : τ ∈ Sτ+t } (or {Tτ : τ ∈ Sτ−t }) be an operator group on a Banach space X, i.e., Tτ1 Tτ2 = Tτ1 +τ2 ,

τ1 , τ2 ∈ Sτ+t (or τ1 , τ2 ∈ Sτ−t ),

T0 = I.

If for every τ0 ≥ 0 (or τ ≤ 0) and any ε > 0, there is a neighborhood U of τ0 (i.e., U = (τ0 − δ, τ0 + δ) ∩ Sτ+t (or U = (τ0 − δ, τ0 + δ) ∩ Sτ−t ) for some δ > 0) such that Tτ x − Tτ0 x < ε

for all τ ∈ U,

then we call {Tτ : τ ∈ Sτ+t } (or {Tτ : τ ∈ Sτ−t }) the local strong-continuous operator semigroup or the local Π -semigroup.   Definition 5.22 If Sτ+t ∈ {0}, ∅ , then we say Π -semigroup {Tτ : τ ∈ Sτ+t } is a positive-direction local Π -semigroup on the changing-periodic time scale T; if   Sτ−t ∈ {0}, ∅ , then we say Π -semigroup {Tτ : τ ∈ Sτ−t } is a negative-direction local Π -semigroup on T. Remark 5.9 Note that local Π -semigroups on changing-periodic time scales are operator semigroups on the sub-timescale Tτt , so the semigroups induced by time scales and Π -semigroups on time scales are completely different concepts through comparing Definitions 5.19 and 5.22. Remark 5.10 In fact, the Π -semigroup for time scales proposed in [219] is more general than the C0 -semigroup on time scales proposed in [145] since the condition ∀t1 , t2 ∈ T, t1 ± t2 ∈ T is not required for the Π -semigroup, i.e., the condition that T is closed for addition and subtraction can be removed. Since changing-periodic time scales can be decomposed into countable sub-timescales with the properties of CCTS, all the properties of the Π -semigroup on CCTS can be naturally extended to its sub-timescales when the Decomposition of Theorem of Time Scales is applied.

5.6.2 Local Almost Automorphic Functions on Translation Invariant Sub-Timescales   Throughout this subsection, we assume that Sτ±t ∈ {0}, ∅ and introduce the concepts of local-almost automorphic functions on changing-periodic time scales. Definition 5.23 Let X be a Banach space and T be a changing-periodic time scale.

5.6 Local Almost Automorphic Functions on Changing-Periodic Time Scales

327

(i) Let f : T → X be a bounded continuous function. We say that f is localalmost automorphic if for every adaption factor sequence {snτ }∞ n=1 ⊂ Sτt ⊂ Π, we can extract a subsequence {τnτ }∞ such that n=1 g(t) = lim f (t + τnτ ) n→∞

is well defined for each t ∈ T, and lim g(t − τnτ ) = f (t)

n→∞

for each t ∈ T. We shall denote by AAL (T, X) the set of all such functions. (ii) A continuous function f : T × B → X is said to be local-almost automorphic if f (t, x) is local-almost automorphic in t ∈ T uniformly for all x ∈ B, where B is any bounded subset of X or B = X. We shall denote by AAL (T × X, X) the set of all such functions. Remark 5.11 From Definition 5.23 and the property of Sτt , we can obtain the local almost automorphy on the periodic sub-timescale Tτt . In fact, as a consequence of Definition 5.23, let D ⊂ X be an open set and S is a compact subset of D, we have the following proposition: Proposition 5.1 Let T be a changing-periodic time scale. If f ∈ C(T × D, X) is a local-almost automorphic in t uniformly for x ∈ D, then f ∈ C(T0 ×D, X) is localalmost automorphic in t uniformly for x ∈ D, where T0 ⊂ T is a changing-periodic time scale. Proof Let f ∈ C(T × D, X) be uniformly local-almost automorphic, then by  Definition 5.23, for any adaption factors sequence (α τ ) ⊂ Sτt ⊂ Π , there exists a  subsequence α τ ⊂ (α τ ) such that Tα τ f (t, x) = g(t, x) and T−α τ g(t, x) = f (t, x) for each t ∈ T uniformly for x ∈ S, where S is any compact set in D. Thus it follows that Tα τ f (t, x) = g(t, x) and T−α τ g(t, x) = f (t, x) for each t ∈ T0 uniformly for x ∈ S. This completes the proof. 

Definition 5.24 Let f, g ∈ C(T × D, X) be uniformly local-almost automorphic and T be a changing-periodic time scale We say f and g are synchronously local-almost automorphic if f, g are almost automorphic on the same periodic subtimescales of T. In what follows, we will introduce the concept of combinable-almost automorphic functions on changing-periodic time scales. Definition 5.25 Let T be a changing-periodic time scale. If there exists an ωi0 periodic sub-timescale set {Ti0 }i0 ∈I such that the period set {ωi0 }i0 ∈I has a lowest common multiple ω and f is almost automorphic on Ti0 for each i0 , where I is a combinable index number set, then we say f is a combinable-almost automorphic function on T or f is almost automorphic on the ω-periodic sub-timescale

328

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

 Ti0 . Further, if i0 ∈I Ti0 = T, then f is called globally combinable-almost automorphic function on T.



i0 ∈I

Corollary 5.8 The function f is globally combinable-almost automorphic on T if and only if f is almost automorphic on the periodic time scale T.  Proof Since Ti is an ωi -periodic sub-timescale, from T = i∈I Ti , it follows that T is an ω-periodic time scale, where ω is a lowest common multiple of {ωi }i∈I and I is a combinable index number set. Hence, the desired result follows from Definition 5.25. This completes the proof. 

Remark 5.12 From Corollary 5.8 it follows that the concept of almost automorphic functions on periodic time scales is equivalent to the concept of globally combinable-almost automorphic functions on changing-periodic time scales.

5.6.3 Local Pseudo Almost Automorphic Functions on Sub-CCTS In this section, we introduce the new concept of local pseudo almost automorphic functions on changing-periodic time scales and obtain some basic properties. Definition 5.26 Each sub-timescale of a changing-periodic time scale is called a complete-closed sub-timescale, short for Sub-CCTS. Definition 5.27 Let X be a Banach space and T be a changing-periodic time scale. (i) Let f : T → X be a bounded continuous function. We say that f is localalmost automorphic if for every adaption factor sequence {snτ }∞ n=1 ⊂ Sτt , we can extract a subsequence {τnτ }∞ such that n=1 g(t) = lim f (t + τnτ ) n→∞

is well defined for each t ∈ T and a sequence {τ˜nτ } ⊂ Sτt depending on {τnτ } such that f (t) = lim g(t + τ˜nτ ) n→∞

is well defined for each t ∈ T. We shall denote by LAA(T, X) the set of all such functions. (ii) A continuous function f : T × B → X is said to be local-almost automorphic if f (t, x) is local-almost automorphic in t ∈ T uniformly for all x ∈ B, where B is any bounded subset of X or B = X. We shall denote by LAA(T × X, X) the set of all such functions. Definition 5.27 also has the following equivalent form.

5.6 Local Almost Automorphic Functions on Changing-Periodic Time Scales

329

Definition 5.28 Let X be a Banach space and T be a changing-periodic time scale and f : T → X be a bounded continuous function. We say that f is local-almost automorphic if for every adaption factor sequence {snτ }∞ n=1 ⊂ Sτt , we can extract a τ } ⊂ S depending on {τ τ } such that and a sequence { τ ˜ subsequence {τnτ }∞ τ t n n n=1 f (t) = lim lim f (t + τnτ + τ˜mτ ) m→∞ n→∞

is well defined for each t ∈ T.   Definition 5.29 If Sτ+t ∈ {0}, ∅ , then we say f is positive-direction local  almost automorphic; if Sτ−t ∈ {0}, ∅ , then we say f is negative-direction   local-almost automorphic; if Sτ±t ∈ {0}, ∅ , then we say f is bi-direction localalmost automorphic. Definition 5.30 Let f, g ∈ C(T × D, En ) be uniformly local-almost automorphic and T be a changing-periodic time scale. We say f and g are synchronously local-almost automorphic if f, g are almost automorphic on the same periodic subtimescales of the changing-periodic time scale T. Remark 5.13 Throughout this section, we always assume that all the functions on the same changing-periodic time scale are synchronously local-almost automorphic, i.e, all the functions satisfy Definition 5.30. Definition 5.31 (i) The set of bounded continuous functions with local vanishing mean value is defined as   1 t0 +T LAA0 (T, X) = φ ∈ BCrd (T, X) : lim φ(s)Δτs s = 0, T →∞ T t0  where t0 ∈ T, T ∈ Sτt0 . (ii) Similarly we define LAA0 (T × X, X) to be the collection of all functions f ∈ BCrd (T × X, X) satisfying 1 lim T →∞ T



t0 +T

t0

f (s, x)Δτs s = 0, where t0 ∈ T, T ∈ Sτt0 ,

uniformly for x in any bounded subset of X.

  Remark 5.14 In Definition 5.31, for any t0 ∈ T, if Sτ±t ∈ {0}, ∅ , i.e, Sτ+t = 0 0     Sτt0 ∩ [0, +∞) ∈ {0}, ∅ and Sτ−t = Sτt0 ∩ (−∞, 0] ∈ {0}, ∅ , then Tτt0 is a 0 bi-direction periodic sub-timescale. Hence, we obtain 1 lim T →+∞ T



t0 +T t0

1 φ(s)Δτs s = 0, lim T →+∞ −T



t0 −T

t0

φ(s)Δτs s = 0,

330

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

so, 1 T →+∞ 2T



t0 +T

lim

t0

φ(s)Δτs s = 0,

1 T →+∞ 2T



lim

t0

t0 −T

φ(s)Δτs s = 0.

Thus we obtain 1 T →+∞ 2T



lim

=

1 T →+∞ 2T



t0 +T t0 −T t0 +T

lim

t0

φ(s)Δτs s 1 T →+∞ 2T

φ(s)Δτs s + lim



t0 t0 −T

φ(s)Δτs s = 0.

Therefore, if Tτt0 is a bi-direction periodic sub-timescale, then the space LAA0 is as follows:   t0 +T 1 LAA0 (T, X) = φ ∈ BCrd (T, X) : lim φ(s)Δτs s = 0, T →∞ 2T t0 −T  where t0 ∈ T, T ∈ Sτt0 . Now, we introduce the set LP AA(T, X) and LP AA(T × X, X) of local pseudo almost automorphic functions:   LP AA(T, X) = f = g+φ ∈ BCrd (T, X) : g ∈ LAA(T, X) and φ ∈ LAA0 (T, X) ;

 LP AA(T × X, X) = f = g + φ ∈ BCrd (T × X, X) : g ∈ LAA(T × X, X)  and φ ∈ LAA0 (T × X, X) . From the Definition of LP AA(T, X), one can easily show the following theorem: Theorem 5.28 For any t ∈ T, let Ω ⊂ Tτt , f = g + φ be a local pseudo almost automorphic function. Then we have {g(t) : t ∈ Ω} ⊂ {f (t) : t ∈ Ω}.

(5.22)

Proof We claim that LAA0 (T, X) is a closed sub-space. In fact, for any {φm } ⊂ LAA0 (T, X) and φm → φ, m → +∞, we obtain 1 lim T →∞ T



t0 +T

t0

1 φ(s)Δτs s ≤ lim T →∞ T



t0 +T

t0

1 + lim T →∞ T



φm (s) − φ(s)Δτs s

t0 +T

t0

φm (s)Δτs s = 0,

5.6 Local Almost Automorphic Functions on Changing-Periodic Time Scales

331

which implies that LAA0 (T, X) is a closed sub-space. Hence, LAA0 (T, X) is a Banach space. Therefore, we have LP AA(T, X) = LAA(T, X) ⊕ LAA0 (T, X), which implies that (5.22) holds. This completes the proof. 

Corollary 5.9 The decomposition of a local pseudo almost automorphic function is unique. Proof Suppose f ∈ LP AA(T, X) has two decompositions, that is f = g1 + φ1 = g2 + φ2 , where g1 , g2 ∈ LAA(T, X) and φ1 , φ2 ∈ LAA0 (T, X). Then (g1 − g2 ) + (φ1 − φ2 ) = f − f = 0. Using Theorem 5.28, we know (g1 − g2 )(t) = 0 for each t ∈ Ω. Thus, g1 = g2 . Therefore, φ1 = φ2 . Hence, the decomposition of f is unique. This completes the proof. 

In the following, using the knowledge of Δ-measurability on time scales, we can obtain the following lemma. Lemma 5.17 Let f ∈ BCrd (T, X). Then f ∈ LAA0 (T, X) if and only if for any ε > 0, lim

T →∞

  1 μΔτt MT ,ε (f ) = 0 0 T

and   MT ,ε (f ) := t ∈ [t0 , t0 + T )Tτt : f (t) ≥ ε, t0 ∈ T, T ∈ Sτt0 , 0

where μΔτt (·) denotes the Δ-measurability on the periodic sub-timescale Tτt0 of 0 the time scale T.   Proof Without loss of generality, we assume that for t0 ∈ T, Sτ+t ∈ {0}, ∅ , i.e., 0 Tτt0 is a positive-direction periodic sub-timescale. (i) Necessity: By contradiction suppose that there exists ε0 > 0 such that   1 μΔτt MT ,ε0 (f ) = 0. 0 T →∞ T lim

Then there exists δ > 0 such that for every n ∈ N, Tn ∈ Sτt0 ,   1 μΔτt MTn ,ε0 (f ) ≥ δ for some Tn > n. Thus we have 0 Tn

332

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

1 Tn



t0 +Tn

t0

f (s)Δτs s

 1 f (s)Δτs s + f (s)Δτs s Tn [t0 ,t0 +Tn )Tτ \MTn ,ε0 (f ) MTn ,ε0 (f ) t   1 ε0 ≥ f (s)Δτs s ≥ Δτ s ≥ ε0 δ. Tn MTn ,ε0 (f ) Tn MTn ,ε0 (f ) s

1 = Tn



This contradicts the assumption. (ii) Sufficiency: From the statement of Lemma 5.17 it is clear that f  ≤ M, M is some constant. For any ε > 0, there exists T0 > 0 such that for T > T0 ,   1 ε μΔτt MT ,ε (f ) < . 0 T M +1 Then     1 t0 +T 1 f (s)Δτs s = f (sΔτs s + f (s)Δτs s T t0 T MT ,ε (f ) [t0 ,t0 +T )Tτt \MT ,ε (f ) ≤

 1 1 M T − μΔτt (MT ,ε (f )) · μΔτt (MT ,ε (f )) + 0 0 T T M +1

≤

1 Mε + . M +1 M +1

Hence,  lim

t0 +T

T →∞ t0

f (s)Δτs s = 0.

That is, f ∈ LAA0 (T, X).

  A similar argument can be supplied for the case Sτ−t ∈ {0}, ∅ , i.e., Tτt0 is a 0   negative-direction periodic sub-timescale and the case Sτ±t ∈ {0}, ∅ , i.e., Tτt0 is 0 a bi-direction periodic sub-timescale, and one can easily obtain the same result (we omit the details). This completes the proof. 

Theorem 5.29 Let f = g + φ ∈ LP AA(T × X, X) with g ∈ LAA(T × X, X), φ ∈ LAA0 (T × X, X). Assume: (i) g(t, x) is uniformly continuous on any bounded subset uniformly for t ∈ T. (ii) There exists a nonnegative function ∗ ∈ Lp (Tτt )(1 ≤ p ≤ ∞) such that f (t, x) − f (t, y) ≤ ∗ (t)x − y for all x, y ∈ X and t ∈ Tτt .

5.6 Local Almost Automorphic Functions on Changing-Periodic Time Scales

333

  If x ∈ LP AA(T, X), then f ·, x(·) ∈ LP AA(T, X), where 



∗

L (Tτt0 ) :=  : Tτt → R : p

 [ (s)] Δτs s < ∞ ∗

T τt

p

and 

∗

 

Lp (T

τt )

=

T τt

∗

1

[ (s)] Δτs s p

p

.

  Proof Without loss of generality, we assume that for t ∈ T, Sτ+t ∈ {0}, ∅ , i.e., Tτt is a positive-direction periodic sub-timescale. Since f ∈ LP AA(T × X, X) and x ∈ LP AA(T, X), we have by definition that f = g + φ and x = α + β, where g ∈ LAA(T × X, X), φ ∈ LAA0 (T × X, X), α ∈ LAA(T, X), β ∈ LAA0 (T, X). Thus the function f can be decomposed as           f t, x(t) = g t, α(t) + f t, x(t) − f t, α(t) + φ t, α(t) . Let   G(t) = g t, α(t) ;

      Φ(t) = f t, x(t) − f t, α(t) + φ t, α(t) .

From g(·, x) ∈ LAA(T×X, X), then for every sequence of real numbers {snτ }∞ n=1 ⊂ such that: Sτt , we can extract a subsequence {τnτ }∞ n=1 g ∗ (t, x) = lim g(t + τnτ , x) n→∞

τ is well defined for each t ∈ T, and a sequence {τ˜nτ }∞ n=1 depending on {τn } such that

g(t, x) = lim g ∗ (t + τ˜nτ , x). n→∞

is well defined for each t ∈ T. In view of assumption (i) and α ∈ LAA(T, X), one  τ ∞ can extract {τn }∞ n=1 ⊂ {τn }n=1 such that          lim g t + τn , α(t + τn ) = lim g t + τn , α ∗ (t) = g ∗ t, α ∗ (t) ,

n→∞

n→∞





and a sequence {τ˜n } depending on {τn } such that          lim g ∗ t + τ˜n , α ∗ (t + τ˜n ) = lim g ∗ t + τ˜n , α(t) = g t, α(t) .

n→∞

n→∞

  Hence, G(·) ∈ LAA(T, X). To show that f ·, x(·) ∈ LP AA(T, X), it is sufficient to prove that Φ(·) ∈ LAA0 (T, X).

334

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

    ∈ LAA0 (T,'X). Clearly,   First, we prove that f ·, x(·) − f ·, α(·) f t, x(t) '−f t, α(t) is bounded and rd-continuous. We can assume 'f t, x(t) − f t, α(t) ' ≤ M, ∀t ∈ Tτt . Since x(t), α(t) are bounded, we can choose a bounded subset Bτt ⊂ Tτt such that x(Tτt ), α(Tτt ) ⊂ Bτt , where x(Tτt ), α(Tτt ) denote the value field of x, α under Tτt . Under assumption (ii), for a given ε > 0, x − y ≤ ε, implies that f (t, x) − f (t, y) < ε∗ (t),

for all t ∈ Tτt .

Since β(·) ∈ LAA0 (T, X), Lemma 5.17 yields that   1 μΔτt MT ,ε (β(t)) = 0. T →∞ T lim

Thus for any t0 ∈ Tτt , T ∈ Sτt0 , we obtain 1 T =

1 T +

≤ Case 1:



t0 +T t0

'    ' 'f t, x(t) − f t, α(t) 'Δτ t t

MT ,ε (β(t))

1 T



[t0 ,t0 +T )Tτt \MT ,ε (β(t))

'    ' 'f t, x(t) − f t, α(t) 'Δτ t t 

t0 +T t0

∗ (t)Δτt t.

if p = 1, we obtain that 

t0 +T t0

∗ (t)Δτt t ≤

ε T

 T τt

∗ (t)Δτt t ≤

ε∗ Lp (Tτt ) T

.

if p = ∞, we obtain that ε T

Case 3:

'    ' 'f t, x(t) − f t, α(t) 'Δτ t t

  ε M μΔτt MT ,ε (β(t)) + T T

ε T Case 2:





t0 +T t0

∗ (t)Δτt t ≤ ε∗ L∞ (Tτt ) .

if 1 < p < ∞, then

  t +T  1   t +T 1  0 0 ε∗ Lp (Tτt ) p q ε t0 +T ∗ ε  (t)Δτt t ≤ [∗ (t)]p Δτt t Δτt t ≤ , 1− 1 T t0 T t0 t0 T q

where q = p(p − 1)−1 . Hence, we obtain

5.6 Local Almost Automorphic Functions on Changing-Periodic Time Scales

1 T →∞ T



335

'    ' 'f t, x(t) − f t, α(t) 'Δτ t = 0. t

t0 +T

lim

t0

  Next, we show that φ ·, α(·) ∈ LAA0 (T, X). Let ε > 0. Since g(t, x) is uniformly continuous in any bounded subset uniformly for t ∈ T, there exists a δ > 0 such that g(t, x)−g(t, y) ≤ ε for all x, y ∈ Bτt with x −y ≤ δ. Let δ0 = min{ε, δ}. Then   φ(t, x) − φ(r, x) ≤ f (t, x) − f (t, y) + g(t, x) − g(t, y) ≤ ε ∗ (t) + 1 , for all x, y ∈ Bτt with x − y ≤ δ0 . Set I = α([t0 , t0 + T )Tτt ), where T ∈ Sτt . Then I is compact in R since the image of a compact set under a continuous mapping is compact. Therefore, one can find finite number of open balls Ok , (k = 1, 2, . . . , m) with center xk ∈ I and m  Ok and radius δ0 small enough such that I ⊂ k=1

'  '    'φ t, α(t) − φ(t, xk )' ≤ ε ∗ (t) + 1 , α(t) ∈ Ok , t ∈ [t0 , t0 + T )T . τt Suppose φ(t, xp ) = max {φ(t, xk )}, where p is an index number among 1≤k≤m

{1, 2, . . . , m}. The set Bτkt = {t ∈ [t0 , t0 + T )Tτt : α(t) ∈ Ok } is open in m  [t0 , t0 + T )Tτt = Bτkt . Let k=1

Eτ1t = Bτ1t , Eτkt = Bτkt \

k−1 

Bτjt , (2 ≤ k ≤ m).

j =1 j

Then Eτi t ∩ Eτt = ∅ when i = j, 1 ≤ i, j ≤ m. Notice that 1 T



t0 +T

t0

φ(t, α(t)Δτt t =

1 T

 m  k=1

'  ' 'φ t, α(t) 'Δτ t t

Bτkt

m  '  '   1  'φ t, α(t) − φ(t, xk )' + φ(t, xk ) Δτ t ≤ t k T Bτ k=1

t

m  m   ∗  1  1  ε  (t) + 1 Δτt t + φ(t, xk )Δτt t ≤ T T Bτk Bτk k=1

≤ε+

ε T

k=1

t



t0 +T t0

∗ (t)Δτt t +

Using the same discussion as above, we obtain

1 T



t0 +T

t0

t

φ(t, xp )Δτt t.

336

5 Almost Automorphic Functions and Generalizations on Translation Time Scales

1 T →∞ T



t0 +T

lim

t0

'  ' 'φ t, α(t) 'Δτ t = 0. t

  G(·) ∈ LAA(T, X) and Φ(·) ∈ That is, φ ·, α(·) ∈ LAA0 (T,  X). Hence,  LAA0 (T, X). This means that f ·, x(·) ∈ LP AA(T, X).   A similar argument can be supplied for the case Sτ−t ∈ {0}, ∅ , i.e., Tτt is a   negative-direction periodic sub-timescale and the case Sτ±t ∈ {0}, ∅ , i.e., Tτt is a bi-direction periodic sub-timescale, and one can easily obtain the same result (we omit the details). This completes the proof. 

Chapter 6

Nonlinear Dynamic Equations on Translation Time Scales

6.1 Almost Periodic Generalized Solutions for Dynamic Equations on CCTS Consider the following nonlinear dynamic equation x Δ = f (t, x),

(6.1)

where f (t, x) ∈ Crd (T × En , En ). Let Ω = {x(t) : x(t) is a bounded solution to (6.1)}. Definition 6.1 If Ω = φ, then λ = infx∈Ω x exists, λ is called the least-value of solutions to (6.1). If there exists ϕ(·) ∈ Ω such that ϕ = λ, then ϕ(t) is called a minimum norm solution to (6.1), where  ·  = supt∈T | · |. Lemma 6.1 If f ∈ Crd (T × S, En ) is bounded on T × S and (6.1) has a bounded solution ϕ(t) such that {ϕ(t), t ∈ T} ⊂ S and 0 ∈ S, then there must have a minimum norm solution to (6.1). Proof Define: λ = inf{x : x(t) is a solution to (6.1) and for all t ∈ T, x ∈ S},   λn = inf sup |x(t)| : x(t) is a solution to (6.1) and for all t ∈ T, x ∈ S . |t|≤n

It is easy to see that 0 ≤ λ < +∞, λn ≤ λn+1 , lim λn = λ. n→+∞

Let xn (t)(n = 1, 2, . . . , ) be solutions to (6.1) and satisfy sup |xn (t)| ≤

|t|≤n

1 + λn . n

© Springer Nature Switzerland AG 2020 C. Wang et al., Theory of Translation Closedness for Time Scales, Developments in Mathematics 62, https://doi.org/10.1007/978-3-030-38644-3_6

337

338

6 Nonlinear Dynamic Equations on Translation Time Scales

Since |f (t, x)| is bounded on T×S, according to Corollary 2.14, we see that {xn (t)} is sequentially compact and there must exist subsequence {nk } ⊂ {n} such that {xnk (t)} uniformly converges to some function y(t). By Corollary 2.14, y(t) is a solution to (6.1). For any fixed t and sufficiently large k, we have |xnk (t)| ≤

1 + λnk . nk

Let k → +∞, we have |y(t)| ≤ λ. Since {y(t) : t ∈ T} ⊂ S, we can get supt∈T |y(t)| = y = λ. The proof is completed. 

Lemma 6.2 If f ∈ Crd (T × S, En ) is almost periodic in t uniformly for x ∈ En , S = {ϕ(t) : t ≥ t0 } and (6.1) has a bounded solution ϕ(t) on [t0 , ∞)T , then (6.1) has a bounded solution ψ(t) on T and {ψ(t), t ∈ T} ⊂ S. 



Proof In fact, we may take α = {αk } ⊂ Π such that



lim α n→+∞ k

= +∞ and

Tα  f (t, x) = f (t, x) holds uniformly on T × S. For any fixed a, consider the  interval (a, ∞)T and ϕk (t) = ϕ(t + αk ). We obtain that for k sufficiently large,  {ϕk } is defined on (a, ∞)T and is a solution to x Δ = f (t + αk , x), and {ϕk (t)} is uniformly bounded. Then let α be a sequence which goes to −∞, according to  Corollary 2.14, there must exist α ⊂ α such that Tα ϕ(t) = ψ(t) holds uniformly on T and for all t ∈ T, we have ψ(t) ∈ S. Since Tα f (t, x) = f (t, x), by Lemma 3.6, ψ(t) is a solution to (6.1). This completes the proof. 

Lemma 6.3 Let f ∈ Crd (T × En , En ) be almost periodic in t uniformly for x ∈ En . If (6.1) has a minimum norm solution, then for any g ∈ H (f ), the following equation x Δ = g(t, x)

(6.2)

has the same least-value of solutions as that to (6.1). Proof Let ϕ(t) be the minimum norm solution to (6.1) and λ is the least-value.  Since g ∈ H (f ), there exists a sequence α ∈ Π such that Tα  f (t, x) = g(t, x)  holds uniformly on T × S. From Corollary 2.14, there exists α ⊂ α such that Tα ϕ(t) = ψ(t) holds uniformly on T. By Lemma 3.6, ψ(t) is a solution to (6.2).  For |ϕ(t)| ≤ λ, we have |ψ(t)| ≤ λ, thus, λ = ψ(t) ≤ λ. Since ϕ(t) = T−α ψ(t)     and |ψ(t)| ≤ λ , we have |ϕ(t)| ≤ λ , thus, λ = ϕ(t) ≤ λ . Therefore, λ = λ . The proof is completed. 

From the process of the proof of Lemma 6.3, one can easily get Lemma 6.4 Assume that ϕ(t) is a minimum norm solution to (6.1) and there exists  a sequence α ⊆ Π such that Tα  f (t, x) = g(t, x) exists uniformly on T × S and  there exists a subsequence α ⊂ α such that Tα ϕ(t) = ψ(t) holds uniformly on T. Then ψ(t) is a minimum norm solution to (6.2).

6.1 Almost Periodic Generalized Solutions for Dynamic Equations on CCTS

339

Lemma 6.5 Let f ∈ Crd (T × En , En ) be almost periodic in t uniformly for x ∈ En and for every g ∈ H (f ), (6.2) has a unique minimum norm solution. Then these minimum norm solutions are almost periodic on T. Proof For a fixed g ∈ H (f ), (6.2) has the unique minimum norm solution ψ(t). Since g(t, x) is almost periodic in t uniformly for x ∈ En , we have for any     sequences α , β ⊆ Π , there exist common subsequences α ⊂ α , β ⊂ β such that Tα+β g(t, x) = Tα Tβ g(t, x) holds uniformly on T × S and Tα Tβ ψ(t), Tα+β ψ(t) hold uniformly on T. It follows from Lemmas 6.3 and 6.4 that Tα Tβ ψ(t) and Tα+β ψ(t) are minimum norm solutions to the following equation: x Δ = Tα+β g(t, x). Since the minimum norm solution is unique, we have Tα Tβ ψ(t) = Tα+β ψ(t). Therefore, ψ(t) is almost periodic. The proof is completed. 

We will now discuss the linear almost periodic dynamic equation: x Δ = A(t)x + f (t)

(6.3)

and its associated homogeneous equation: x Δ = A(t)x,

(6.4)

where A ∈ Crd (T, Rn×n ) is an almost periodic matrix function and f ∈ Crd (T, Rn ) is an almost periodic vector function. Definition 6.2 If B ∈ H (A), we say that y Δ = B(t)y

(6.5)

is a homogeneous equation in the hull of (6.3). Definition 6.3 If B ∈ H (A) and g ∈ H (f ), we say that y Δ = B(t)y + g(t)

(6.6)

is an equation in the hull of (6.3). Definition 6.4 Let A(t) be n × n rd-continuous matrix function on T, the linear dynamic equation: x Δ (t) = A(t)x(t)

(6.7)

340

6 Nonlinear Dynamic Equations on Translation Time Scales

is said to admit an exponential dichotomy on T if there exist positive constants K, α, a projection P and the fundamental solution matrix X(t) of (6.7), satisfying 

|X(t)P X−1 (s)| ≤ Keα (t, s), s, t ∈ T, t ≥ s, |X(t)(I − P )X−1 (s)| ≤ Keα (s, t), s, t ∈ T, t ≤ s.

(6.8)

Lemma 6.6 If A ∈ Crd (T, Rn×n ) is an almost periodic matrix function and x(t) is an almost periodic solution of the homogeneous linear almost periodic dynamic equation x Δ = A(t)x, then inft∈T |x(t)| > 0 or x(t) ≡ 0. Proof If inft∈T |x(t)| = 0, there exists {tn } ⊂ T such that |x(tn )| → 0 as n → ∞,   and by x ∈ H (x), then there exists α ⊂ Π such that limn→+∞ x(t + αn ) = x(t) for all t ∈ T, this implies that for any ε > 0, there exists N > 0 so that n > N  implies |x(tn + αn ) − x(tn )| < 2ε . Furthermore, since x(t) is almost periodic on T, it is uniformly rd-continuous on T, then we can take t0 ∈ T with |t0 − tn | < δ such   that |x(t0 + αn ) − x(tn + αn )| < 2ε . Therefore, for sufficiently large n ∈ N, we have     |x(t0 + αn ) − x(tn )| ≤ |x(t0 + αn ) − x(tn + αn )| + |x(tn + αn ) − x(tn )| < 2ε + 2ε < ε.  Hence, we can obtain that |x(t0 + αn )| → 0 as n → +∞. Since A(t) is also almost  periodic on T, there exists a sequence α ⊂ α such that Tα A(t) = B(t),

Tα x(t) = y(t),

T−α B(t) = A(t),

T−α y(t) = x(t)

hold uniformly on T, according to Lemma 3.6, we have y Δ (t) = (Tα x(t))Δ = ( lim x(t+αn ))Δ = lim x Δ (t+αn ) = lim A(t+αn )x(t+αn ), n→∞

n→∞

n→∞

that is, y(t) is a solution to the following equation: y Δ = B(t)y satisfying the initial condition: y(t0 ) = Tα x(t0 ) = lim x(t0 + αn ) = 0. n→+∞

Hence, y(t) = y(t0 )eB (t, t0 ) ≡ 0, therefore, x(t) = T−α y(t) ≡ 0. The proof is completed.



6.1 Almost Periodic Generalized Solutions for Dynamic Equations on CCTS

341

Lemma 6.7 Suppose that (6.4) has an almost periodic solution x(t) and inft∈T |x(t)| > 0. If (6.3) has bounded solution on [t0 , ∞)T , then (6.3) has an almost periodic solution. Proof According to Lemmas 6.1 and 6.2, we obtain that there is a minimum norm solution to (6.3) on T and for every pair of Tα A(t) = B(t) and Tα f (t) = g(t), (6.6) has a minimum norm solution. Next, we will show that the minimum norm solution to (6.6) is unique. For a fixed pair of B(t) and g(t), we consider (6.6). Let (6.6) have two different minimum x1 (t) and x2 (t), and their least-values are equal to λ.  norm solutions  Since 12 x1 (t) − x2 (t) is a bounded non-trivial solution to (6.5), by the condition of Lemma 6.7, there exists a real number ρ0 > 0 such that 1 inf x1 (t) − x2 (t) ≥ ρ0 > 0. t∈T 2 Now by the parallelogram law, we have 1      x1 (t) + x2 (t) 2 + 1 x1 (t) − x2 (t) 2 = 1 |x1 (t)|2 + |x2 (t)|2 ≤ λ2 , 2 2 2 and noting that 1 Δ 1 1 x1 (t) + x2 (t) = x1Δ + x2Δ 2 2 2  1  1 = B(t)x1 + g(t) + B(t)x2 + g(t) 2 2 1  = B(t) (x1 + x2 ) + g(t), 2   so 12 x1 (t) + x2 (t) is a solution to (6.6), thus 1  ) x1 (t) + x2 (t) < λ2 − ρ 2 < λ. 2 This is a contradiction. The proof is completed.



Lemma 6.8 If every bounded solution of a homogeneous equation in the hull of (6.3) is almost periodic, then all bounded solutions of (6.3) are almost periodic. Proof According to Lemma 6.6, we obtain that every non-trivial bounded solution of equations in the hull of (6.3) satisfies inft∈T |x(t)| > 0. From Lemma 6.7 it follows that if (6.3) has bounded solutions on T, then (6.3) must have an almost periodic solution ψ(t). If ϕ(t) is an arbitrary bounded solution of (6.3), then η(t) = ψ(t) − ϕ(t) is a bounded solution of its associated homogeneous equation (6.4), and it is almost periodic. Thus, ϕ(t) is almost periodic. This completes the proof. 

342

6 Nonlinear Dynamic Equations on Translation Time Scales

Lemma 6.9 If a homogeneous equation in the hull of (6.3) has the unique bounded solution x(t) ≡ 0, then (6.3) has a unique almost periodic solution. Proof Let ψ(t), ϕ(t) be two bounded solutions to (6.3), then x(t) = ϕ(t) − ψ(t) is a solution of a homogeneous equation in the hull of (6.3), since x(t) ≡ 0, we have that ϕ(t) ≡ ψ(t). Thus, by Lemma 6.8, (6.3) has a unique almost periodic solution. This completes the proof. 

Similar to the proof of Lemma 7.4 in [73], one can easily prove that Lemma 6.10 Let P be a projection and X a differentiable invertible matrix such that XP X−1 is bounded on T. Then there exists a differentiable matrix S such that XP X−1 = SP S −1 for all t ∈ T and S, S −1 are bounded on T. In fact, there is an S of the form S = XQ−1 , where Q commutes with P . Lemma 6.11 If (6.4) has an exponential dichotomy and X(t) is the fundamental solution matrix of (6.4), C non-singular, then X(t)C has an exponential dichotomy with the same projection P if and only if CP = P C. Proof Let ξ be a vector, from (6.8), we know that |X(t)(I − P )X−1 (s)ξ | ≤ |ξ |Keα (s, t),

t ≤ s.

Take ξ = X(s)η, where η is a vector, then |X(s)η| ≥ K −1 eα (t, s)|X(t)(I − P )η|,

s ≥ t.

If (I − P )η = 0, then |X(s)η| → +∞, s → +∞; If (I − P )η = 0, then for t ≥ s, we have |X(t)η| = |X(t)P η| = |X(t)P X−1 (s)ξ | ≤ Keα (t, s)|ξ |. Thus, |X(t)η| → 0 as t → +∞. Similarly, if P η = 0, then |X(t)η| → +∞ as s → +∞; if P η = 0, then for s ≥ t, we have |X(t)η| = |X(t)(I − P )η| = |X(t)(I − P )X−1 (s)ξ | ≤ Keα (s, t)|ξ |. Thus, |X(t)η| → 0 as t → −∞. Since X(t)C has an exponential dichotomy with the same projection P , X(t)CP is bounded when t → +∞ and X(t)C(I − P ) is bounded when t → −∞. Hence, we can get (P − I )CP = 0 and P [C(I − P )] = 0. Therefore, P C = P CP = CP . Conversely, if CP = P C, then |X(t)Pi X−1 (s)| = |X(t)CPi C −1 X−1 (s)|, i = 1, 2 where P = P1 , I − P = P2 . Thus, X(t)C and X(t) admit exponential dichotomies with the same projection P . The proof is completed. 

Lemma 6.12 Suppose that A ∈ Crd (T, Rn×n ) is an almost periodic matrix function and (6.4) has an exponential dichotomy. Then for every B ∈ H (A), (6.5) has an exponential dichotomy with the same projection P and the same constants K, α. Proof We invoke Lemma 6.10 where X is the fundamental solution matrix satisfying (6.8). Let Q and S be given the same as those in Lemma 6.10 and Tα A = B uniformly on T. For any given t0 ∈ T, let Xn (t) = X(t + αn )Q−1 (t0 + αn ), then

6.1 Almost Periodic Generalized Solutions for Dynamic Equations on CCTS

343

it is a fundamental solution matrix to x Δ = A(t + αn )x, by Lemma 6.11, it has an exponential dichotomy with the same projection P and the same constants since Q−1 and P are commutable. Now, since there exist bounded S(t0 + αn ) and S −1 (t0 + αn ), then we may take subsequences Xn (t0 ) and Xn−1 (t0 ) which are convergent. Without changing notation we may assume that Xn (t0 ) → Y0 and hence Xn−1 (t0 ) → Z0 where Z0 = Y0−1 . However, for a suitable subsequence Xn (t), it converges to a solution of x Δ = Bx uniformly on T. Let this solution be Y , then Y (t0 ) = Y0 that is non-singular and clearly Y satisfies (6.8) since Xn does for all n. The proof is completed. 

Lemma 6.13 If the homogeneous equation (6.4) has an exponential dichotomy, then (6.4) has only one bounded solution x(t) ≡ 0. Proof Let X(t) be the fundamental solution matrix to (6.4). For any sequence α ⊂ Π , denote An = A(t + αn ), Xn (t) = X(t + αn ). Since the homogeneous equation (6.4) has an exponential dichotomy, it is easy to see that there exists a ¯ constant M such that Xn (t) ≤ M and XΔ (t) = An (t)Xn (t) ≤ AM, where  ¯ A = supt∈T ||A(t)||. Therefore, by Corollary 2.14, there exists {αnk } := α ⊂ α such that {Xnk } converges uniformly on T and limn→+∞ X(t +αn ) exists uniformly on T. So X(t) is almost periodic. Since the homogeneous equation (6.4) has an exponential dichotomy, then inft∈T x(t) = 0, from Lemma 6.6, x(t) ≡ 0. This completes the proof. 

Lemma 6.14 If the homogeneous equation (6.4) has an exponential dichotomy, then all equations in the hull of (6.4) have only one bounded solution x(t) ≡ 0. Proof By Lemma 6.12, all equations in the hull of (6.4) have an exponential dichotomy, according to Lemma 6.13, all equations in the hull of (6.4) have only one bounded solution x(t) ≡ 0. This completes the proof. 

Theorem 6.1 Let A ∈ Crd (T, Rn×n ) be an almost periodic matrix function and f ∈ Crd (T, Rn ) be an almost periodic vector function. If (6.4) admits an exponential dichotomy, then (6.3) has a unique almost periodic solution  x(t) =

t −∞

X(t)P X−1 (σ (s))f (s)Δs −



+∞

X(t)(I − P )X−1 (σ (s))f (s)Δs,

t

where X(t) is the fundamental solution matrix of (6.4). Proof First, we prove that x(t) is a bounded solution of system (6.3). In fact, x Δ (t) − A(t)x(t)  t P X−1 (σ (s))f (s)Δs + X(σ (t))P X −1 (σ (t))f (t) = X Δ (t) −∞

−X Δ (t)

 +∞ t

(I − P )X −1 (σ (s))f (s)Δs + X(σ (t))(I − P )X −1 (σ (t))f (t)

344

6 Nonlinear Dynamic Equations on Translation Time Scales

−A(t)X(t)

 t −∞

P X−1 (σ (s))f (s)Δs + A(t)X(t)

 +∞ t

(I − P )X −1 (σ (s))f (s)Δs

= X(σ (t))(P + I − P )X −1 (σ (t))f (t) = f (t)

and  x = sup t∈T

t

X(t)P X

−∞

−1



+∞

(σ (s))f (s)Δs −

X(t)(I − P )X

−1

t

(σ (s))f (s)Δs

 +∞    t ≤ sup eα (t, σ (s))Δs + eα (σ (s), t)Δs Kf  −∞ t t∈T   1 1 2 + μα ≤ − Kf , kf  = α α α where  ·  = supt∈T | · |. Next, we will show that x(t) is almost periodic. By Lemma 6.14, all equations in the hull of (6.4) have only one bounded solution x(t) ¯ ≡ 0. Thus, for given     α , β , we can find common subsequences α ⊂ α , β ⊂ β so that Tα+β A = Tα Tβ A, Tα+β f = Tα Tβ f, y = Tα+β x, and z = Tα Tβ x exist uniformly on T. But y − z = Tα+β x − Tα Tβ x ≡ 0 since y − z is the bounded solution to all equations in the hull of (6.4). Hence, Tα+β x = Tα Tβ x, from Theorem 3.17 , x ∈ A P(T)n . This completes the proof. 

Example 6.1 Consider the following dynamic equation on a periodic time scale T x Δ (t) = Ax(t) + f (t),

(6.9)

where   −6 0 A= , 0 −6

√  sin √3t , f (t) = cos 2t 

and μ(t) =

1 . 6

Obviously, I + μ(t)A is invertible for all t ∈ T, so A ∈ R. We claim that the homogeneous equation of (6.9) admits an exponential dichotomy. In fact, the eigenvalues of the coefficient matrix in (6.9) are λ1 = λ2 = −6, according to Lemma 1.17, the P -matrices are given by   10 P0 = I = 01



and

 00 P1 = (A − λ1 I )P0 = A + 6I = . 00

We want to choose r1Δ = −6r1 , r1 (t0 ) = 1

and

r2Δ = r1 − 6r2 , r2 (t0 ) = 0.

6.1 Almost Periodic Generalized Solutions for Dynamic Equations on CCTS

345

Solving the first IVP for r1 we get r1 (t) = e−6 (t, t0 ). Solving the second IVP, i.e., r2Δ = −6r2 + e−6 (t, t0 ), r2 (t0 ) = 0, we obtain  r2 = e−6 (t, t0 )

t t0

Δs . 1 − 6μ(s)

Using Lemma 1.17, we get 

 10 eA (t, t0 ) = r1 (t)P0 + r2 (t)P1 = e−6 (t, t0 ) . 01 Thus, |X(t)P0 X

−1

'    ' √ ' 10 10 ' ' ' ≤ 2e3 (t, s). (s)| = 'e−6 (t, t0 ) e−6 (s, t0 ) ' 01 01

√ Then we can take K = 2, α = 3 such that the homogeneous equation of (6.9) admits an exponential dichotomy. Finally, according to Theorem 6.1, we can obtain the unique almost periodic solution of (6.9):  x(t) =

t

−∞



t

X(t)P0 X

−1



+∞

(σ (s))f (s)Δs − 

10 = e−6 (t, σ (s)) 01 −∞

X(t)(I − P0 )X−1 (σ (s))f (s)Δs

t

√  sin √3s Δs. cos 2s



6.1.1 Almost Periodic Solutions for Delay Dynamic Equations In this section, we consider the following almost periodic dynamic equation with delays on a periodic time scale T: x Δ (t) = A(t)x(t) +

n    f t, x(t − τi (t)) , i=1

(6.10)

346

6 Nonlinear Dynamic Equations on Translation Time Scales

where A ∈ Crd (T, Rn×n ) is an almost periodic matrix function on T, for every i = 1, 2, . . . , n, τi ∈ Crd (T, Π ) is almost periodic on T and f ∈ Crd (T × Rn , Rn ) is almost periodic uniformly in t for x ∈ Rn . Theorem 6.2 Suppose that the following hold: (H1 ) x Δ (t) = A(t)x(t) admits an exponential dichotomy on T with positive constants K and α. α (H2 ) There exists M < (2+μα)Kn such that |f (t, x) − f (t, y)| ≤ M|x − y| for n t ∈ T, x, y ∈ R . Then system (6.10) has a unique almost periodic solution. Proof For any ϕ ∈ A P(T)n , consider the following equation x Δ (t) = A(t)x(t) +

n 

f (t, ϕ(t − τi (t))).

(6.11)

i=1

According to Theorem 6.1, (6.11) has a unique solution T ϕ ∈ A P(T)n and  T ϕ(t) =

t −∞



X(t)P X−1 (σ (s))

f (s, ϕ(s − τi (s)))Δs

i=1 +∞

−

n 

X(t)(I − P )X−1 (σ (s))

t

n 

f (s, ϕ(s − τi (s)))Δs.

i=1

Define a mapping T : A P(T)n → A P(T)n by setting (T ϕ)(t) = xϕ (t), ∀ x ∈ A P(T)n . From (H1 ), we have |X(t)P X−1 (s)| ≤ Keα (t, s), s, t ∈ T, t ≥ s, |X(t)(I − P )X−1 (s)| ≤ Keα (s, t), s, t ∈ T, t ≤ s. For any ϕ, ψ ∈ A P(Tn ), we have  T ϕ − T ψ ≤

t

X(t)P X −1 (σ (s))

−∞

 −

n 

|f (s, ϕ(s − τi (s))) − f (s, ψ(s − τi (s)))|Δs

i=1 +∞

X(t)(I − P )X −1 (σ (s))

t

×

n  i=1

|f (s, ϕ(s − τi (s))) − f (s, ψ(s − τi (s)))|Δs

6.1 Almost Periodic Generalized Solutions for Dynamic Equations on CCTS

 ≤ ≤

t

−∞

 Kα (t, σ (s))Δs +

+∞

t

Kα (σ (s), t)Δs

! n

347

Mϕ − ψ

i=1

2 + μα ¯ KnMϕ − ψ, α

where μ¯ = supt∈T μ(t). Since (H2 ), T is a contractive operator. Therefore, (6.10) has a unique almost periodic solution. 

6.1.2 Pseudo Almost Periodic Solutions for Dynamic Equations Considering the non-autonomous equation: x Δ = A(t)x + F (t)

(6.12)

and its associated homogeneous equation: x Δ = A(t)x,

(6.13)

where the n × n coefficient matrix A(t) is rd-continuous on T and column vector F = (f1 , f2 , . . . , fn )T is in En . Define F  = supt∈T |F (t)|. We will call A(t) almost periodic if all the entries are almost periodic. Lemma 6.15 Let α > 0. For fixed s ∈ T, the following holds: eα (t, s) → 0, t → +∞. Proof If μ(t) > 0, since α ∈ R + , we have 1 + μ(t)  α = 1 + μ(t)

1 −α = < 1. 1 + μ(t)α 1 + μ(t)α

Thus, α ∈ R + and it is easy to have Log(1 + μ(t)  α) ∈ R for all t ∈ T. So ξμ(t) (α) = Hence

Log(1 + μ(t)  α) 0 t

Then the following linear system   x Δ (t) = diag − c1 (t), −c2 (t), . . . , −cn (t) x(t)

(6.14)

admits an exponential dichotomy on T, where m(ci ) denote the mean-value of ci , i = 1, 2, . . . , n. Proof Since the linear system (6.14) has a unique solution x(t) = x0 e−c (t, t0 ),

6.1 Almost Periodic Generalized Solutions for Dynamic Equations on CCTS

351

  where x(t0 ) = x0 , −c = diag − c1 (t), −c2 (t), . . . , −cn (t) , we demonstrate x(t) admits an exponential dichotomy on T. According to Theorems 3.21 and 3.27, one has 1 m(ci ) = lim T →∞ T



t+T

1 ci (s)Δs = lim T →∞ T

t



t0 +T

ci (s)Δs > 0, t0 ∈ T.

t0

Hence there exists T0 > 0, when T > T0 , one can obtain 1 T



t0 +T

ci (s)Δs > t0



1 1 m(ci ) = 2 T

t0 +T

t0

1 m(ci )Δs, 2

that is 1 T



t0 +T

t0

  1 ci (s) − m(ci ) Δs > 0, 2

thus, for T > T0 , we have ci (t) > 12 m(ci ). Case 1:

If μ(η) > 0, η ∈ [s, t]T , s, t ∈ T, one can obtain 1−

i) μ(t) m(c 2 i) 1 + μ(t) m(c 2

> 1 − μ(t)

m(ci ) > 1 − μ(t)ci (t), 2

then 

t

s

log(1 − μ(η)ci (η)) Δη ≤ μ(η)



log(1 −

t

m(ci ) 2 m(c ) 1+μ(η) 2 i

μ(η)

)

μ(η)

s

Δη,

thus, we have  exp s

t



log(1 − μ(η)ci (η)) Δη ≤ exp μ(η)



t

log(1 −

m(ci ) 2 m(c ) 1+μ(η) 2 i

μ(η)

μ(η)

s

)

 Δη ,

that is e−ci (t, s) ≤ e m(ci ) (t, s). 2

Case 2:

If μ(η) = 0, η ∈ [s, t]T , s, t ∈ T, one can easily get 

e−ci (t, s) = exp

t s





−ci (η)Δη ≤ exp s

t

 m(ci ) Δη = e m(ci ) (t, s). − 2 2

352

6 Nonlinear Dynamic Equations on Translation Time Scales

Then set P = I , and we have |X(t)P X−1 (s)| = |x0 e−c (t, t0 )I x0−1 e−c (s, t0 )| ≤ Ke M (t, s), 2

where K ≥ 1, M = min1≤i≤n {m(c1 ), m(c2 ), . . . , m(cn )}. Therefore, x(t) admits an exponential dichotomy with P = I on T. This completes the proof. 

Example 6.2 Consider the following dynamic equation on a periodic time scale T x Δ (t) = Ax(t) + F (t),

(6.15)

where 1 ⎞ 3t + ⎜ tσ (t) ⎟ , F (t) = ⎝ √ 1 ⎠ cos 2t + tσ (t) ⎛

  −3 0 A= , 0 −3

sin

√

and μ(t) =

1 . 3

Obviously, I + μ(t)A is invertible for all T, so A ∈ R. By Lemma 6.16, we obtain that the homogeneous equation of (6.15) admits an exponential dichotomy with √ P = √ ˜ P(T)2 . In fact, sin 3(·), cos 2(·) ∈ I on T. Moreover, we can check F ∈ PA A P(T) and 1 lim T →∞ T



t0 +T

t0 −T

1 1 Δt = − lim T →∞ T tσ (t)

1 t0 +T =0 t t0 −T

˜ P0 (T). By Theorems 6.3 and 1.17, one can obtain which implies 1/(·)σ (·) ∈ PA a unique pseudo almost periodic solution  x(t) =

t

−∞

X(t)P X−1 (σ (s))F (s)Δs −



+∞

X(t)(I − P )X−1 (σ (s))F (s)Δs

t

⎞ ⎛ √   sin 3s + 1 10 ⎜ sσ (s) ⎟ Δs. = e−3 (t, σ (s)) ⎝ √ 1 ⎠ 0 1 −∞ cos 2s + sσ (s) 

t

6.2 Weighted Pseudo Almost Periodic Solutions Under Π-Semigroup In this section, we will consider linear abstract dynamic equations on time scales which are based on the Π -semigroup. Suppose that X(t) is the fundamental solution of the linear system: x Δ = Ax,

(6.16)

6.2 Weighted Pseudo Almost Periodic Solutions Under Π -Semigroup

353

where A is the infinitesimal generator of a Π -semigroup that satisfies all the conditions in Theorem 3.42 and x : T → X, where X is a Banach space. Next, we introduce the following definition: Definition 6.5 Equation (6.16) is said to be admitted exponential dichotomy if there is a projection P and positive numbers α > 0 and β ≥ 1 such that X(t)P X−1 (s) ≤ βeα (t, s), t ≥ s,

(6.17)

X(t)(I − P )X−1 (s) ≤ βeα (s, t), s ≥ t.

(6.18)

Let F : T → X and consider the dynamic equation: x Δ = Ax + F (t).

(6.19)

In view of Theorem 6.1, we state the following theorem: Theorem 6.5 Let A be the infinitesimal generator of a Π -semigroup and all the conditions in Theorem 3.42 are satisfied, and F ∈ Crd (T, X) is almost periodic. If (6.16) admits an exponential dichotomy, then (6.19) has a unique almost periodic solution  t  +∞     −1 x(t) = X(t)P X σ (s) F (s)Δs − X(t)(I − P )X−1 σ (s) F (s)Δs, −∞

t

where X(t) is the fundamental solution of (6.16). From Theorem 6.5, the following corollary follows immediately. Corollary 6.1 Suppose (6.16) admits exponential dichotomy, that is, there exist constants α > 0, β ≥ 1 such that (6.17) and (6.18) hold. Then for each almost periodic function F ∈ Crd (T, X), both 

t −∞

  X(t)P X−1 σ (s) F (s)Δs

and 

∞

  X(t)(I − P )X−1 σ (s) F (s)Δs

t

are almost periodic. In what follows, we will provide sufficient conditions which ensure the existence and uniqueness of weighted pseudo almost periodic solutions of the following neutral dynamic equations:     x Δ (t) = Ax(t) + F Δ t, x(t − g(t)) + G t, x(t), x(t − g(t)) , t ∈ T,

(6.20)

354

6 Nonlinear Dynamic Equations on Translation Time Scales

where T is an invariant time scale under translations and A is the infinitesimal generator of a Π -semigroup that satisfies all the conditions in Theorem 3.42, F : T × X → X is almost periodic in t uniformly for x ∈ X, G : T × X × X → X is almost periodic in t uniformly for (x, y) ∈ X × X, g : T → Π . Using the properties of weighted pseudo almost periodic functions in the previous sections and the exponential dichotomy of a linear dynamic equations together with Krasnoselskii’s fixed point theorem, we will obtain some sufficient conditions that guarantee the existence of weighted pseudo almost periodic solutions of (6.20). Lemma 6.17 ([75]) Let M be a closed convex nonempty subset of X. Suppose that B and C map M into X such that (i) x, y ∈ M implies Bx + Cy ∈ M, (ii) C is continuous and CM is contained in a compact set, (iii) B is a contraction mapping. Then there exists z ∈ M with z = Bz + Cz. With respect to (6.20), we shall assume the following conditions: (H1 ) There exist positive numbers LF , LG such that for each xi , yi ∈ X, i = 1, 2, and all t ∈ T, F (t, x1 ) − F (t, x2 ) ≤ LF x1 − x2 

(6.21)

and G(t, x1 , y1 ) − G(t, x2 , y2 ) ≤ LG (x1 − x2  + y1 − y2 ;

(6.22)

(H2 ) A is the infinitesimal generator of the Π -semigroup and all the conditions in Theorem 3.42 are satisfied, and F ∈ W P AP (T × X, X, ρ) ˆ and G ∈ W P AP (T × X × X, X, ρ); ˆ (H3 ) Equation (6.16) admits exponential dichotomy, that is, there exists constants α > 0, β ≥ 1 such that (6.17) and (6.18) hold; (H4 ) The weight ρˆ : T → (0, ∞) is continuous and ! ! ρ(s ˆ + τ) m(r + τ, ρ, ˆ t0 ) < ∞, lim sup 0, we have 0≤

1 m(r, ρ, ˆ t0 )

1 = m(r, ρ, ˆ t0 ) 1 = m(r, ρ, ˆ t0 ) 1 ≤ m(r, ρ, ˆ t0 )

 

t0 +r t0 −r

φ(t − u)ρ(t)Δt ˆ

t0 +r−u t0 −r−u



φ(t)ρ(t ˆ + u)Δt

t0 +r+u

t0 −r−u



t0 +r+u t0 −r−u

 φ(t)ρ(t ˆ + u)Δt −

t0 +r+u t0 +r−u

 φ(t)ρ(t ˆ + u)Δt

φ(t)ρ(t ˆ + u)Δt

and for each u ∈ Π, u < 0, we have 0≤

1 m(r, ρ, ˆ t0 )

1 = m(r, ρ, ˆ t0 ) 1 = m(r, ρ, ˆ t0 ) 1 ≤ m(r, ρ, ˆ t0 )

 

t0 +r t0 −r

φ(t − u)ρ(t)Δt ˆ

t0 +r−u t0 −r−u



φ(t)ρ(t ˆ + u)Δt

t0 +r−u

t0 −r+u



t0 +r−u t0 −r+u

 φ(t)ρ(t ˆ + u)Δt −

t0 −r−u t0 −r+u

 φ(t)ρ(t ˆ + u)Δt

φ(t)ρ(t ˆ + u)Δt.

Thus it follows that 0≤

1 m(r, ρ, ˆ t0 )



t0 +r t0 −r

φ(t−u)ρ(t)Δt ˆ ≤

1 m(r, ρ, ˆ t0 )



t0 +r+|u| t0 −r−|u|

φ(t)ρ(t+u)Δt. ˆ

ˆ we find Now from condition (H4 ) and the fact that φ ∈ P AP0 (T, X, ρ), 1 lim r→∞ m(r, ρ, ˆ t0 )



t0 +r+|u|

t0 −r−|u|

φ(t)ρ(t ˆ + u)Δt

ˆ u + u) 1 m(r + |u|, ρ, ˆ t0 ) ρ(ξ r→∞ m(r, ρ, ˆ t0 ) ρ(ξ ˆ u ) m(r + |u|, ρ, ˆ t0 )

≤ lim



t0 +r+|u| t0 −r−|u|

φ(t)ρ(t)Δt ˆ = 0,

where ξu ∈ (t0 − r − |u|, t0 + r + |u|)T . Hence, we have 1 lim r→∞ m(r, ρ, ˆ t0 )



t0 +r t0 −r

φ(t − u)ρ(t)Δt ˆ = 0,

356

6 Nonlinear Dynamic Equations on Translation Time Scales

that is, t → φ(t − u) ∈ P AP0 (T, X, ρ). ˆ Therefore, the space P AP0 (T, X, ρ) ˆ is translation invariant. This completes the proof. 

To prove our results, we define a mapping Φ as follows   (Φu)(t) = F t, u(t − g(t)) + 

∞

−



t

−∞

    X(t)P X−1 σ (s) G s, u(s), u(s − g(s)) Δs

    X(t)(I − P )X−1 σ (s) G s, u(s), u(s − g(s)) Δs,

t

where X(t) is the fundamental solution of (6.16). Lemma 6.19 Φu is weighted pseudo almost periodic if u is weighted pseudo almost periodic. Proof Let u(t) be weighted pseudo almost periodic. Now  from (H1 ), (H4 ), Theorem 3.59 and Lemma 6.18, it is clear that F t, u(t −g(t)) and G t, u(t), u(t −  g(t)) are also weighted pseudo almost periodic. Now we will show that  t     X(t)P X−1 σ (s) G s, u(s), u(s − g(s)) Δs H1 (t) = −∞

is weighted pseudo almost periodic. Let   G t, u(t), u(t − g(t)) = G1 (t) + φ(t), ˆ Then where G1 ∈ A P(T, X) and φ ∈ P AP0 (T, X, ρ).  H1 (t) =

t −∞

  X(t)P X−1 σ (s) G1 (s)Δs +



t

−∞

  X(t)P X−1 σ (s) φ(s)Δs.

Since G1 (t) is almost periodic, it follows from Corollary 6.1 that 

t

−∞

X(t)P X−1 (σ (s))G1 (s)Δs

is almost periodic. Let  h(t) =

t −∞

X(t)P X−1 (σ (s))φ(s)Δs.

In order to show H1 ∈ W P AP (T, X, ρ), ˆ we only need to show that h ∈ P AP0 (T, X, ρ), ˆ that is

6.2 Weighted Pseudo Almost Periodic Solutions Under Π -Semigroup

1 lim r→∞ m(r, ρ, ˆ t0 )



t0 +r

t0 −r

357

h(t)ρ(t)Δt ˆ = 0.

It follows from (H3 ) and eα (t, σ (s)) ≤ 1 for t ≥ s that 1 m(r, ρ, ˆ t0 ) ≤

1 m(r, ρ, ˆ t0 )

≤

β m(r, ρ, ˆ t0 )

=

β m(r, ρ, ˆ t0 )



t0 +r

t0 −r



t0 +r

t0 −r



t0 +r

t0 −r



h(t)ρ(t)Δt ˆ  ρ(t)Δt ˆ  ρ(t)Δt ˆ 

∞

ΔΠ u

t −∞ t −∞

t0 +r t0 −r

0

βeα (t, σ (s))φ(s)Δs φ(s)Δs

ρ(t)φ(t ˆ − u)Δt.

Now from (H4 ) and Lemma 6.18, P AP0 (T, X, ρ) ˆ translation invariant, we find that t → φ(t − u) ∈ P AP0 (T, X, ρ) ˆ for each u ∈ Π. Thus we have 1 r→∞ m(r, ρ, ˆ t0 ) lim



t0 +r t0 −r

ρ(t)φ(t ˆ − u)Δt = 0,

for each u ∈ Π. This implies that 1 r→∞ m(r, ρ, ˆ t0 ) lim



t0 +r

t0 −r

h(t)ρ(t)Δt ˆ = 0,

that is, h ∈ P AP0 (T, X, ρ), ˆ and hence H1 ∈ W P AP (T, X, ρ). ˆ Finally, let  H2 (t) =

∞

    X(t)(I − P )X−1 σ (s) G s, u(s), u(s − g(s)) Δs.

t

ˆ Thus we have Φu ∈ In a similar way we see that H2 ∈ W P AP (T, X, ρ). W P AP (T, X, ρ) ˆ for u ∈ W P AP (T, X, ρ). ˆ This completes the proof.

 Next we will apply Krasnoselskii’s fixed point theorem. Let (Φu)(t) = (Bu)(t) + (Cu)(t) where B, C : W P AP (T, X, ρ) ˆ → W P AP (T, X, ρ) ˆ are given by   (Bu)(t) = F t, u(t − g(t))

(6.23)

358

6 Nonlinear Dynamic Equations on Translation Time Scales

and  (Cu)(t) =

t

−∞



    X(t)P X−1 σ (s) G s, u(s), u(s − g(s)) Δs ∞

−

    X(t)(I − P )X−1 σ (s) G s, u(s), u(s − g(s)) Δs. (6.24)

t

Lemma 6.20 The operator B is contraction provided LF < 1. Proof From (6.21), we have '    '   ' '  'B φ(t) − B ψ(t) ' = 'F t, φ(t − g(t)) − F t, ψ(t − g(t)) ' '    ' ≤ LF 'φ t − g(t) − ψ t − g(t) ' ≤ LF φ − ψ∞ . 

Since LF < 1, B is contraction. This completes the proof.

Lemma 6.21 The operator C is continuous and the image CD is contained in a compact set, where D = {u ∈ W P AP (T, X, ρ) ˆ : u∞ ≤ k}, and k is a fixed constant. Proof Clearly, we have  (Cu)(t) ≤

t −∞



' '  ''  'X(t)P X−1 σ (s) ''G s, u(s), u(s − g(s)) 'Δs '  ''  'X(t)(I − P )X−1 σ (s) ''G s, u(s), u(s − g(s)) 'Δs.

∞'

+ t

Thus, we obtain '  ' (Cu)(·)∞ ≤ 'G ·, u(·), u(· − g(·)) '∞ 

−∞

'  ' 'X(t)P X−1 σ (s) 'Δs

 ' 'X(t)(I − P )X−1 σ (s) 'Δs

t

'  ' ≤ 'G ·, u(·), u(· − g(·)) '∞ 

∞

+ =β

t

∞'

+





t

  βeα σ (s), t Δs





t

−∞



  βeα t, σ (s) Δs

  ' 1 ' 1 'G ·, u(·), u(· − g(·)) ' . − ∞ α α

(6.25)

To see that C is continuous, we let u, v ∈ A P(T, X). Given ε > 0, take δ = ε . Then we have 1 2LG β( α1 − α )

6.2 Weighted Pseudo Almost Periodic Solutions Under Π -Semigroup

359

(Cu)(t) − (Cv)(t)  t '  ''    ' 'X(t)P X−1 σ (s) ''G s, u(s), u(s − g(s)) − G s, v(s), v(s − g(s)) 'Δs ≤ −∞



+

∞'

  ''  'X(t)(I − P )X−1 σ (s) ''G s, u(s), u(s − g(s))

t

 ' − G s, v(s), v(s − g(s)) 'Δs  t '    '   ≤ βeα t, σ (s) LG u(s) − v(s) + LG 'u s − g(s) − v s − g(s) ' Δs −∞



+ t

∞

'      ' βeα σ (s), t LG u(s) − v(s) + LG 'u s − g(s) − v s − g(s) ' Δs 

≤ 2LG u − v∞

t −∞

  βeα t, σ (s) Δs +

 1 1 − u − v∞ < ε, = 2LG β α α 



∞ t

  βeα σ (s), t Δs



whenever u − v∞ < δ. This proves that C is continuous. For D = {u ∈ W P AP (T, X) : u∞ ≤ k}, to show that the image CD is contained in a compact set, let {un } be a sequence in D. In view of (6.22), we have ' '  '   ' 'G t, u(t), v(t) ' ≤ 'G t, u(t), v(t) − G(t, 0, 0)' + G(t, 0, 0)X X X ≤ LG (u(t)X + v(t)X ) + a ≤ LG (u∞ + v∞ ) + a ≤ LG (2k + a),

(6.26)

where a = G(t, 0, 0). From (6.25) and (6.26), we obtain  Cun ∞ ≤ LG (2k + a)β

1 1 − α α

 := L.

(6.27)

Next, we calculate (Cun )Δ (t) and show that it is uniformly bounded. Clearly,   (Cun )Δ (t) = A(Cun )(t) + G t, un (t), un (t − g(t)) . Since A is a bounded operator, there exists a positive constant K such that A ≤ K. This, when combined with (6.26) and (6.27) implies (Cun )Δ ∞ ≤ KL + LG (2k + a). Thus the sequence {(Cun )(t)} is uniformly bounded and equi-continuous. Hence, by Corollary 2.14, CD is compact. The completes the proof. 

360

6 Nonlinear Dynamic Equations on Translation Time Scales

Theorem 6.6 Suppose that (H1 )–(H4 ) hold, b = G(t, 0, 0)∞ . If there exists a constant k, such that  LF k + b + β

F (t, 0)∞ and a

 1 1 − LG (2k + a) ≤ k, α α

=

(6.28)

where α, β are constants given in (6.17) and (6.18), then (6.20) has a weighted pseudo almost periodic solution in M = {u ∈ W P AP (T, X, ρ) ˆ : u ≤ k}. Proof Note that condition (6.28) implies that LF < 1. Thus in view of Lemma 6.20 the mapping B defined by (6.23) is contraction. The mapping C defined by (6.24) is continuous by Lemma 6.21 and CM is contained in a compact set. Now, for u, v ∈ M, we have (Bu)(t) + (Cv)(t)X ' '   ≤ 'F t, u(t − g(t)) − F (t, 0)'X + F (t, 0)  t ' '  ''  'X(t)P X−1 σ (s) ''G s, v(s), v(s − g(s)) 'Δs +  +

−∞ ∞'

'  ''  'X(t)(I − P )X−1 σ (s) ''G s, v(s), v(s − g(s)) 'Δs

t



≤ LF u∞ + b + β

 1 1 − LG (2k + a) ≤ k. α α

Thus Bu + Cv ∈ M. Therefore all the conditions of Krasnoselskii’s theorem are satisfied, and as a consequence there exists a fixed point z ∈ M such that z = Bz + Cz, i.e., (6.20) has a weighted pseudo almost periodic solution in M. This completes the proof. 

Theorem 6.7 Suppose that (H1 )–(H4 ) hold. Further, suppose that  LF + 2LG β

1 1 − α α

 < 1.

(6.29)

Then (6.20) has a unique weighted pseudo almost periodic solution. Proof It follows from Lemma 6.19 that Φu maps W P AP (T, X, ρ) ˆ to W P AP (T, X, ρ). ˆ Thus for u, v ∈ W P AP (T, X, ρ), ˆ we have  Φu(t) − Φv(t) ≤ LF u − v∞ + 

t

−∞

'  ' 'X(t)P X−1 σ (s) '2LG u − v∞ Δs

∞'

− t

 ' 'X(t)(I − P )X−1 σ (s) '2LG u − v∞ Δs 

1 1 − ≤ LF + 2LG β α α

! u − v∞ .

6.2 Weighted Pseudo Almost Periodic Solutions Under Π -Semigroup

361

1 Since LF + 2LG β( α1 − α ) < 1, Φ is a contractive mapping. Therefore, Φ has ∗ a unique fixed point u ∈ W P AP (T, X, ρ). ˆ We conclude that u∗ (t) is the unique weighted pseudo almost periodic solution of (6.20). This completes the proof. 

Example 6.3 Let ρ(t) ˆ = 1 + t 2 + tσ (t) + σ 2 (t), and let T be an invariant time scale under translations and 0 ∈ T. For ρ(t), ˆ we have 

r



−r r



−r r

m(r, ρ) ˆ = = =

−r

  1 + t 2 + tσ (t) + σ 2 (t) Δt   1 + t 2 · 1 + (t + σ (t))σ (t) Δt   1 + t 2 · 1 + (t 2 )Δ σ (t) Δt =



r

−r

  1 + (t 3 )Δ Δt = 2(r + r 3 ).

Thus condition (H4 ) holds. Now consider the following perturbed dynamic equation for small ε1 and ε2 on T :     x Δ = Ax + F Δ t, x(t − g(t)) + G t, x(t), x(t − g(t)) , t ∈ T,

(6.30)

where   x x= 1 , x2   F t, x(t − g(t)) =

  −5 0 A= , 0 −5

and

μ(t) =

1 , 5



 0 √ ε1 (sin t + sin 2t + e3 (t, 0))x12 (t − g(t))

and   G t, x(t), x(t − g(t)) =



 0 √ . ε2 (cos t + cos 2t + e3 (t, 0)) − ε2 x12 (t)x2 (t)

√ √ It is clear that sin t + sin 2t and cos t + cos 2t are almost periodic and ˆ lim e3ρ(t) ˆ is bounded, and hence we have (t,0) = 0. Thus, e3 (t, 0)ρ(t)

t→+∞

1 lim r→∞ m(r, ρ) ˆ



r

1 e3 (t, 0)ρ(t)Δt ˆ = lim r→∞ 2(r + r 3) −r



r −r

e−3 (t, 0)ρ(t)Δt ˆ = 0.

Thus F ∈ W P AP (T × R2 , R2 , ρ) ˆ and G ∈ (T × R2 × R2 , R2 , ρ). ˆ Obviously, I + μ(t)A is invertible for all T, so A ∈ R. We claim that x Δ = Ax admits an exponential dichotomy. In fact, the eigenvalues of the coefficient matrix A are λ1 = λ2 = −5, and thus in Theorem 1.17, the P matrices are given by

362

6 Nonlinear Dynamic Equations on Translation Time Scales

  10 P0 = I = 01



and

 00 P1 = (A − λ1 I )P0 = A + 5I = . 00

We choose r1Δ = −5r1 , r1 (t0 ) = 1 and

r2Δ = r1 − 5r2 , r2 (t0 ) = 0.

Solving the first IVP for r1 we get r1 = e−5 (t, t0 ). Solving the second IVP, i.e., r2Δ = −5r2 + e−5 (t, t0 ), r2 (t0 ) = 0, we obtain  r2 = e−5 (t, t0 )

t t0

Δs . 1 − 5μ(s)

Now using Theorem 1.17, we get 

 10 eA (t, t0 ) = r1 (t)P0 + r2 (t)P1 = e−5 (t, t0 ) . 01 Thus, X(t)P X

−1

'    ' √ ' 10 10 ' ' ' ≤ 2e 5 (t, s). (s) = 'e−5 (t, t0 ) e−5 (s, t0 ) 2 ' 01 01

√ Thus we can take β = 2, α = 52 so that x Δ = Ax admits an exponential ˆ : u ≤ k}, where k is a fixed dichotomy. Define M = {u ∈ W P AP (T, R2 , ρ) constant. For x(t) = (x1 (t), x2 (t)), y(t) = (y1 (t), y2 (t)) ∈ M, we have '    ' 'F t, x1 (t − g(t)) − F t, x2 (t − g(t)) ' ≤ 6ε1 kx1 − x2 , '    ' 'G t, x1 (t), y1 (t − g(t)) − G t, x2 (t), y2 (t − g(t)) ' ≤ ε2 k 2 x1 − x2 . √ 5 then (6.28) Thus if we take LF = 6ε1 k, LG = ε2 k 2 , β = 2, α = 2 holds. Therefore, with these choices the conditions of Theorem 6.6 are satisfied. In conclusion, (6.30) has a weighted pseudo almost periodic solution in M. Moreover, for each positive number k, if ε1 , ε2 are small enough such that  LF + 2LG β

1 1 − α α



√ 4 + 5μ(t) < 1, = 6ε1 k + 2ε2 k 2 2 5

i.e., (6.29) is satisfied, then (6.30) has a unique weighted pseudo almost periodic solution in M.

6.3 Local-Periodic Solutions on Changing-Periodic Time Scales

363

6.3 Local-Periodic Solutions on Changing-Periodic Time Scales The topic of periodic solutions of dynamic equations has received lots of attention (see Al-Islam et al. [42], Chow [74]) and the analysis of the existence and stability of periodic solutions on time scales attracts many researchers (see Bi et al.[67], Kaufmann et al.[153], Wang et al.[221]). In the following, based on the concept of changing-periodic time scales, we introduce the following concept. Definition 6.6 Let T be a changing-periodic time scale and τt be an index function of T. A function f is said to be ωτt local-periodic on T if for any ωτt ∈ Sτt , one has f (t + ωτt ) = f (t) for all t ∈ T. Lemma 6.22 Let T be an arbitrary time scale and " T be any closed subset of T. If p ∈ Crd (T), then p ∈ Crd (" T). Proof If " T is a closed subset of T, then the set of all right and left dense points of " T are contained in the set of all right and left dense points of T, i.e., if p ∈ Crd (T), then p ∈ Crd (" T). This completes the proof. 

From the definition of changing-periodic time scales and properties, for each periodic sub-timescale of T, some of the results on classical periodic time scales can be extended directly as follows. Lemma 6.23 Let T be a changing-periodic time scale and τt be an index function, a, b ∈ Tτt , ωτt ∈ Sτt and p ∈ Crd (Tτt ) is ωτt -periodic. Then στt (t + ωτt ) = στt (t) + ωτt , ρτt (t + ωτt ) = ρτt (t) + ωτt , μτt (t + ωτt ) = μτt (t), 

b+ωτt a+ωτt

 p(t)Δτt t =

b a

p(t)Δτt t, epτ (b, a) = epτ (b + ωτt , a + ωτt ) if p ∈ Rτt ,

kpτ := epτ (t + ωτt , t) − 1, is independent of t ∈ Tτt whenever p ∈ Rτt , where στt , ρτt denote the forward jump operator and backward jump operator on Tτt , respectively. μτt is the graininess function on Tτt . p ∈ Rτt denote p is regressive on Tτt , epτ (·, t0 ) is the exponential function on the periodic sub-timescale Tτt . Lemma 6.24 Let T be a changing-periodic time scale and τt be an index function. Suppose f : T × T → R satisfies the assumptions of Theorem 1.23 and Tτt is an ωτt -periodic sub-timescale. Define 

t+ωτt

g(t) = t

f (t, s)Δτs s.

364

6 Nonlinear Dynamic Equations on Translation Time Scales

If f Δτt (t, s) denotes the derivative of f with respect to t ∈ Tτt , then  g Δτt (t) =

t+ωτt

t

f Δτt (t, s)Δτs s + f (στt (t), t + ωτt ) − f (στt (t), t).

Remark 6.1 According to Theorem 2.31, i.e., the Composition Theorem of Time Scales, all the results established in this paper are also suitable for any arbitrary time scale T with bounded graininess function μ.

6.3.1 The Clh Space on Changing-Periodic Time Scales In this section, we establish a type of local phase space for functional dynamic equations with infinite delay on changing-periodic time scales. Let f˜ be a function, and [a,ˆ b] =

 [a, b],

a < b,

[b, a],

a > b,

[a,ˇ b] =

 [a, b],

b > 0, b > a,

[b, a],

b < 0, b < a,



Sτ+t := Sτt ∩ [0, +∞), if there exits a K ∈ S˜τt and K > 0, S˜τt = Sτ−t := Sτt ∩ (−∞, 0], if there exits a K ∈ S˜τt and K < 0, ˆ we use the notation |f˜|[a,b]L

˜ ˜ := supθ∈[a,b] ˆ L |f (θ )|, where D(f ) denotes the domain of the function f˜ and [a,ˆ b]L = [a,ˆ b] ∩ D(f˜). Similarly, ˇ ˜ ˜ |f˜|[a,b]L := supθ∈[a,b] ˇ L |f (θ )|, where D(f ) denotes the domain of the function f˜ and [a,ˇ b]L = [a,ˇ b] ∩ D(f˜). Suppose T isa changing-periodic time scale. Let h ∈ Crd (Π, R+ ), h(s) > 0 for all s ∈ S˜τt , and S˜τ h(s)ΔSτs s = 1. Define t

  Clh = ϕ ∈ Crd (S˜τt , Rn ) :

S˜τt

ˆ

 ˆ h(s)|ϕ|[s,0]L ΔSτs s < ∞ ,

where |ϕ|[s,0]L = supθ∈[s,0]∩ ˆ S˜τ |ϕ(θ )|. One can easily prove that Clh is a linear t subspace of Crd (Sτt ) and BCrd (Sτt ) is a linear subspace of Clh . For ϕ ∈ Clh , define  ˆ |ϕ|lh = S˜τ h(s)|ϕ|[s,0]L ΔSτs s < ∞; then (Clh , | · |lh ) is a normed space. For t simplicity, we denote it by Clh . In the following, we will describe three types of phase spaces. Denote Sτ+t = Sτt ∩ [0, +∞), Sτ−t = Sτt ∩(−∞, 0], Λ− = Λ∩(−∞, 0], where Λ is as in Definition 2.34.

6.3 Local-Periodic Solutions on Changing-Periodic Time Scales

365

Definition 6.7

  + (i) We say Clh is a positive-direction phase space if Sτ+t ∈ {0}, ∅ and   + Clh = ϕ ∈ Crd (Sτ+t , Rn ) :

Sτ+t

 h(s)|ϕ|[0,s]L ΔSτs s < ∞ ,

where |ϕ|[0,s]L = supθ∈[0,s]∩Sτt |ϕ(θ )|.   − is a negative-direction phase space if Sτ−t ∈ {0}, ∅ and (ii) We say Clh − Clh

  − n = ϕ ∈ Crd (Sτt , R ) :

Sτ−t

h(s)|ϕ|

[s,0]L

 Δ Sτ s s < ∞ ,

where |ϕ|[s,0]L = supθ∈[s,0]∩Sτt |ϕ(θ )|. ± ± + − is a bi-direction phase space if Sτ±t = {0} and Clh = Clh ∪ Clh . (iii) We say Clh ± ± (iv) We say Clh := Ch is a Λ-phase space if T is a periodic time scale with inf T = −∞, sup T = +∞, and Ch± = Ch+ ∪ Ch− . Remark 6.2 Note that Definition 6.7 is a more general concept of phase spaces for functional dynamic equations with infinite delay than ever before. It follows from (iv) in Definition 6.7 that   Ch− = ϕ ∈ Crd (Λ− , Rn ) :

Λ−

 h(s)|ϕ|[s,0]L ΔΛ s < ∞ ,

(6.31)

where |ϕ|[s,0]L = supθ∈[s,0]∩Λ |ϕ(θ )|, here, if one let T = Λ, it follows that t1 +t2 ∈ T for any t1 , t2 ∈ T, (6.31) will turn into Ch−

  − n = ϕ ∈ Crd (T , R ) :

0

−∞

h(s)|ϕ|

[s,0]L

 Δs < ∞ ,

where |ϕ|[s,0]L = supθ∈[s,0]∩T |ϕ(θ )|, which is the phase space from [67] and just a particular case of (6.31). Lemma 6.25 For any ε > 0 and K ∈ S˜τt \{0}, there exists δ = δ(ε, K) > 0 such ˆ that, for any ϕ1 , ϕ2 ∈ Clh , if |ϕ1 − ϕ2 |lh ≤ δ, then |ϕ1 − ϕ2 |[K,0]L ≤ ε. Proof We argue by contradiction. Assume that there exist ε∗ > 0 and K ∗ ∈ S˜τt \{0} such that, for any δ > 0, there exist ϕ1δ , ϕ2δ ∈ Clh , such that |ϕ1δ − ϕ2δ |lh ≤ δ but ∗ˆ

|ϕ1δ − ϕ2δ |[K ,0]L > ε∗ . ∗  Let δ ∗ = ε2 S˜τ \[K ∗ˆ ,0] t

ϕ2∗ |lh ≤ δ ∗ but |ϕ1∗ − ϕ2∗ |

∗

∗

h(s)ΔSτs s > 0 and ϕ1∗ = ϕ1δ , ϕ2∗ = ϕ2δ . Then |ϕ1∗ −

L [K ∗ˆ ,0]

L

> ε∗ . Hence,

366

6 Nonlinear Dynamic Equations on Translation Time Scales

δ ∗ ≥ |ϕ1∗ − ϕ2∗ |lh =  =  ≥  ≥



ˆ

S˜τt

h(s)|ϕ1∗ − ϕ2∗ |[s,0]L ΔSτs s ˆ

S˜τt \[K ∗ˆ ,0]L

h(s)|ϕ1∗ − ϕ2∗ |[s,0]L ΔSτs s +



ˆ

[K ∗ˆ ,0]

L

h(s)|ϕ1∗ − ϕ2∗ |[s,0]L ΔSτs s

ˆ

S˜τt \[K ∗ˆ ,0]L

S˜τt \[K ∗ˆ ,0]L

h(s)|ϕ1∗ − ϕ2∗ |[s,0]L ΔSτs s h(s)|ϕ1∗ − ϕ2∗ |[K

∗ˆ ,0] L

Δ Sτ s s > ε ∗

 S˜τt \[K ∗ˆ ,0]L

h(s)ΔSτs s = 2δ ∗ , 

which is a contradiction. This completes the proof. Lemma 6.26 Assume that {ϕn } ⊂ Crd (Sτt , Rn ) is uniformly bounded. Then lim |ϕn − ϕ0 |lh = 0

n→∞

ˆ

if and only if for any K ∈ S˜τt \{0}, one has limn→∞ |ϕn − ϕ0 |[K,0]L = 0. Proof Necessity: By Lemma 6.25, for any ε > 0 and K ∈ S˜τt \{0}, there exists δ = δ(ε, K) such that, for any ϕ1 , ϕ2 ∈ Clh with |ϕ1 − ϕ2 |lh ≤ δ, we have |ϕ1 − ˆ ϕ2 |[K,0]L < ε. Since limn→∞ |ϕn − ϕ0 |lh = 0, there exists N ∈ N such that, for any n > N, we have |ϕn − ϕ0 |lh ≤ δ for all n ≥ N . Hence, ˆ

ˆ

|ϕn − ϕ0 |[K,0]L < ε, i.e., lim |ϕn − ϕ0 |[K,0]L = 0. n→∞

i.e., there exists H > 0 Sufficiency: Assume that {ϕn } is uniformly bounded,  such that |ϕn | ≤ H for all n ∈ N. Let ε > 0. Since S˜τ h(s)ΔSτs s = 1 < ∞, there t  exists K ∈ S˜τt \{0} such that S˜τ \[K,0] ˆ h(s)ΔSτs s < ε. Moreover, for all k ∈ N, t

ˆ

there exists Nk ∈ N such that if n ≥ Nk , then |ϕn − ϕ0 |[K,0]L ≤ ε, whence, ˆ |ϕ0 |[K,0]L ≤ H + ε, so |ϕ0 | < H + ε. Therefore, for n ≥ Nk , we have  ˆ |ϕn − ϕ0 |lh = h(s)|ϕn − ϕ0 |[s,0]L ΔSτs s S˜τt

 =

ˆ

ˆ L S˜τt \[K,0]



+

h(s)|ϕn − ϕ0 |[s,0]L ΔSτs s ˆ

ˆ L [K,0]

h(s)|ϕn − ϕ0 |[s,0]L ΔSτs s

≤ (2H + ε)ε + ε = (2H + ε + 1)ε. This completes the proof.



6.3 Local-Periodic Solutions on Changing-Periodic Time Scales

367

Since the space (Crd [a,ˆ b], Rn ) is complete when endowed with the supremum norm, one can obtain the following lemma. Lemma 6.27 (Clh , | · |lh ) is a Banach space. Proof Let {ϕn } ⊂ Clh be a Cauchy sequence. Thus {ϕn } is bounded, i.e., there exists M > 0 such that |ϕn |lh ≤ M for all n ∈ N. Let ε > 0 and K ∈ S˜τt \{0} such that  ˆ L S˜τt \[K,0]

h(s)ΔSτs s
0 such that |a(t, x)| ≤ M τ . Thus, s ∈ [t, t + Tτ 2 2 t

|fn (t)| ≤ 2M2τ |(F u)(t)| + 2M2τ ≤ 2M2τ Bωτt M1τ + 2M1τ . Consequently, v n  ≤ B ∗ B∗ =

 pτ pτt

t +ωτt

sup t≥s, t,s∈[pτt ,pτˆt +ωτt ]∩Tτt

fn (s)Δτs s, where

G∗n (t, s), G∗n (t, s) =

eaτ ∗ (s, t) n

kaτ ∗

, an∗ = an .

According to the dominated convergence theorem, we now obtain v n  = F un − F u → 0 as n → ∞, which indicates that F is continuous. Hence, our claim is true. ¯ is compact. In fact, for any u ∈ (K τ ∩ Ω), ¯ we Next, we show that F (Kδτ ∩ Ω) δ obtain  t+ωτ t |(F u)(t)| ≤ B f (s, us )Δτs s ≤ Bωτt M1τ t

and |(F u)Δt | = (a)(t)(F u)(t) +

f (t, ut ) ≤ Bωτt M1τ + M1τ . 1 + a(t)μτt (t)

¯ is uniformly bounded and equicontinuous. Hence, F (K τ ∩ Ω) ¯ is Thus, F (Kδτ ∩ Ω) δ compact by the Arzelà-Ascoli theorem. Therefore, F : Kδτ ∩ Ω¯ → Kδτ is completely continuous. The proof is completed. 

376

6 Nonlinear Dynamic Equations on Translation Time Scales

6.3.3 Positive Local-Periodic Solutions for FDEID We introduce the following notation for simplicity of the discussion: τ = lim fM 0

τ = fM ∞

t

f (t, ϕ) f (t, ϕ) , fmτ 0 = lim min , |ϕ|lh →0 t∈[pt ,ptˆ+ωτ ] |ϕ|lh |ϕ|lh t T

t

f (t, ϕ) f (t, ϕ) , fmτ ∞ = lim min , |ϕ| →∞ |ϕ|lh |ϕ|lh lh t∈[pt ,ptˆ+ωτt ]T

max

|ϕ|lh →0 t∈[pt ,ptˆ+ωτ ] t Tτ

lim

max

|ϕ|lh →∞ t∈[pt ,ptˆ+ωτ ] t Tτ

τt

τt

  Dδ τ = ϕ ∈ Clh : ϕ(θ ) ≥ δ τ |ϕ|lh for all θ ∈ S˜τt . In what follows, we will consider positive local-periodic solutions for the system (6.32). Suppose that: (H3 ) There exists K1 > 0 such that for any ϕ ∈ Dδ τ with |ϕ|lh ∈ [δ τ K1 , K1 ], f (t, ϕ) > K1 /(Aωτt ) holds. (H4 ) There exists K2 > 0 such that for any ϕ ∈ Dδ τ with |ϕ|lh ≤ K2 , f (t, ϕ) < K2 /(Bωτt ) holds. Theorem 6.9 τ = fτ (i) If (H3 ) holds and fM M∞ = 0, then (6.32) has at least two positive ωτt 0 local-periodic solutions u1 and u2 with 0 < u1  < K1 < u2 . (ii) If (H4 ) holds and fmτ 0 = fmτ ∞ = ∞, then (6.32) has at least two positive ωτt local-periodic solutions u1 and u2 with 0 < u1  < K2 < u2 .

Proof We only show (i) since the proof of (ii) can also be given in the similar way. τ = 0 that for any ε with 0 < ε ≤ 1/(Bω ), there exists r < K It follows from fM τt 0 1 0 such that f (t, ϕ) ≤ ε|ϕ|lh for ϕ ∈ Dδ τ with 0 < |ϕ|lh ≤ r0 .

(6.37)

Let Ωr0 = {u ∈ Pωτ : u < r0 }. Then for any u ∈ Kδτ ∩ ∂Ωr0 , we have u = r0 and δ τ |ut |lh ≤ δ τ u ≤ ut (θ ). Hence, ut ∈ Dδ τ and δ τ r0 ≤ |ut |lh ≤ r0 . By (6.36) and (6.37), we obtain  F u ≤ B

pt +ωτt pt

 f (s, us )Δτs s ≤ Bε

pt +ωτt

pt

|us |lh Δτs s ≤ Bεωτt u ≤ u,

thus, F u ≤ u for u ∈ Kδτ ∩ ∂Ωr0 . τ On the other hand, it follows from fM = 0 that for any ε with 0 < ε < ∞ 1/(2Bωτt ), there is N1 > K1 such that

6.3 Local-Periodic Solutions on Changing-Periodic Time Scales

377

|f (t, ϕ)| ≤ ε|ϕ|lh for ϕ ∈ Dδτ with |ϕ|lh ≥ N1 .

(6.38)

Let Ωr1 = {u ∈ Pωτ : u < r1 }, where r1 is selected such that r1 > N1 + 1 + 2Bωτt

sup

t∈[pt ,pt +ωτt ]Tτt |ϕ|lh ≤N1 , ϕ∈Dδ τ

(6.39)

f (t, ϕ).

Then, for any u ∈ Kδτ ∩ ∂Ωr1 , one has δ τ |ut |lh ≤ δ τ u ≤ ut (θ ). Hence δ τ r1 ≤ |ut |lh ≤ r1 = u. Moreover, (6.36), (6.38) and (6.39) yield that  F u ≤ B

f (s, us )Δτs s = B

t

 ≤B



t+ωτt

pt +ωτt

pt

r1 Δτ s + B 2Bωτt s





I1

f (s, us )Δτs s + B

pt +ωτt

pt

ε|us |lh Δτs s ≤

I2

f (s, us )Δτs s

r1 + Br1 εωτt < r1 = u, 2

where     I1 = s ∈ [ps , psˆ+ ωτs ]Tτ : |us |lh ≤ N1 and I2 = s ∈ [ps , psˆ+ ωτs ]Tτ : |us |lh > N1 , s

s

which implies F u ≤ u for u ∈ Kδτ ∩ ∂Ωr1 . Let ΩK1 = {u ∈ Pωτ : u < K1 } with K1 > r0 . Then, for any u ∈ Kδτ ∩ ∂ΩK1 , we obtain K1 ≥ |ut |lh ≥ δ τ K1 . By (H3 ), f (t, xt ) > K1 /(Aωτt ). From Lemma 6.31, we have  t+ωτ t AK1 ωτt F u ≥ A f (s, us )Δτs s > = K1 = u. Aωτt t This proves that F u ≥ u for u ∈ Kδτ ∩ ∂ΩK1 . According to Lemmas 6.32 and 6.33, we obtain     F : Kδτ ∩ Ω¯ r1 \ΩK1 → Kδτ and F : Kδτ ∩ Ω¯ K1 \Ωr0 → Kδτ are completely continuous. τ Therefore,  by Lemma 6.28,  points u1 and u2 which satisfy u1 ∈ Kδ ∩ F has fixed τ Ω¯ K1 \Ωr0 and u2 ∈ Kδ ∩ Ω¯ r1 \ΩK1 . That is, (6.32) has at least two positive ωτt local-periodic solutions u1 and u2 with 0 < u1  < K1 < u2 . This completes the proof. 

378

6 Nonlinear Dynamic Equations on Translation Time Scales

Lemma 6.34 Assume that (H3 ) and (H4 ) hold. Then (6.32) has at least one positive ωτt local-periodic solution u with u whose bounds is between K1 and K2 , where K1 and K2 are defined in (H3 ) and (H4 ), respectively. Proof Without loss of generality, we assume that K2 < K1 . Otherwise, one can apply Lemma 6.28 (ii) to show this lemma. Let ΩK2 = {u ∈ Pωτ : u < K2 }. Then for any u ∈ Kδτ ∩ ∂ΩK2 , by virtue of (6.36) and (H4 ), one obtains 

t+ωτt

F u ≤ B

f (s, us )Δτs s


Aωτt K1 = K1 = u, Aωτt

which implies that F u > u for u ∈ Kδτ ∩ ∂ΩK1 . Therefore, according to Lemma 6.28, we can get the desired result. This completes the proof.

 Theorem 6.10

    τ ∈ 0, 1/(Bω ) and f τ τ (i) Assume that fM τt m∞ ∈ 1/(Aδ ωτt ), ∞ . Then (6.32) 0 has at least one positive solution.  ωτt local-periodic    τ τ ∈ 1/(Aδ τ ω ), ∞ . Then (6.32) (ii) Assume that fM ∈ 0, 1/(Bω ) and f τ τt t m0 ∞ has at least one positive ωτt local-periodic solution.   τ = α ∈ 0, 1/(Bω ) and f τ = β ∈ Proof The proof of (i). Suppose that fM 1 τt 1 m∞ 0   1/(Aδ τ ωτt ), ∞ . For ε = 1/(Bωτt ) − α1 > 0, there exists a sufficiently small R1 > 0 such that for |ϕ|lh ≤ R1 , we obtain max

t∈[pt ,ptˆ+ωτt ]Tτ

t

f (t, ϕ) 1 < α1 + ε = , |ϕ|lh Bωτt

i.e., |ϕ|lh ≤ R1 and t ∈ [pt , ptˆ+ ωτt ]Tτ imply f (t, ϕ) < |ϕ|lh /(Bωτt ) ≤ t R1 /(Bωτt ). So one can obtain (H4 ) is fulfilled. For ε = β1 − 1/(Aδ τ ωτt ) > 0, there exists a sufficiently large R2 such that |ϕ|lh ≥ δ τ R2 implies min

t∈[pt ,ptˆ+ωτt ]Tτ

t

f (t, ϕ) 1 > β1 − ε = . τ |ϕ|lh Aδ ωτt

6.3 Local-Periodic Solutions on Changing-Periodic Time Scales

379

Therefore, |ϕ|lh ∈ [δ τ R2 , R2 ] and t ∈ [pt , ptˆ+ ωτt ]Tτ imply f (t, ϕ) > t R2 δ τ /(Aδ τ ωτt ) = R2 /(Aωτt ), i.e., (H3 ) is fulfilled. By Lemma 6.34, we obtain that the claim (i) is true. τ The proof of (ii). Assume that fmτ 0 = α2 ∈ (1/(Aδ τ ωτt ), ∞) and fM = β2 ∈ ∞ τ [0, 1/(Bωτt )). For any ε = α2 − 1/(Aδ ωτt ) > 0, there exists a sufficiently small R3 < δR2 such that 0 < |ϕ|lh ≤ R3 implies min

t∈[pt ,ptˆ+ωτt ]Tτ

t

f (t, ϕ) 1 > α2 − ε = . τ |ϕ|lh Aδ ωτt

This indicates that, when |ϕ|lh ∈ [δ τ R3 , R3 ] and t ∈ [pt , ptˆ+ ωτt ]Tτ , one has t f (t, ϕ) > δR3 /(Aδ τ ωτt ) = R3 /(Aωτt ), which demonstrates that (H3 ) holds. For τ ε = 1/(Bωτt ) − β2 > 0, fM = β2 means that there exists a sufficiently large ∞ R4 > R1 such that |ϕ|lh > R4 implies max

t∈[pt ,ptˆ+ωτt ]Tτ

t

f (t, ϕ) 1 < β2 + ε = . |ϕ|lh Bωτt

(6.40)

To show that (H4 ) is satisfied, in what follows, we consider two cases. Case 1:

Suppose that

f (t, ϕ) is unbounded, then there exist ϕ ∗ ∈

max

t∈[pt ,ptˆ+ωτt ]Tτ

t

Dδ τ with |ϕ ∗ |lh = R5 > R4 and t0 ∈ [pt , ptˆ+ ωτt ]Tτ such that t

f (t, ϕ) ≤ f (t0 , ϕ ∗ ) for 0 < |ϕ|lh ≤ |ϕ ∗ |lh = R5 .

(6.41)

Because |ϕ ∗ |lh = R5 > R4 , so (6.40) and (6.41) yield f (t, ϕ) ≤ f (t0 , ϕ ∗ )
0 t

such that f (t, ϕ) ≤ M5 for (t, ϕ) ∈ [pt , ptˆ+ ωτt ]Tτ × Dδ τ . t

(6.42)

Here one can select R5 such that R5 > M5 Bωτt . From (6.42), 0 < |ϕ|lh ≤ R5 and t ∈ [pt , ptˆ+ ωτt ]Tτ imply f (t, ϕ) ≤ M5 < R5 /(Bωτt ), i.e., (H4 ) holds. t According to Lemma 6.34, one can obtain (ii) holds. This completes the proof. 

380

6 Nonlinear Dynamic Equations on Translation Time Scales

Theorem 6.11

  (i) If (H4 ) is fulfilled and fmτ 0 , fmτ ∞ ∈ 1/(Aδ τ ωτt ), ∞ , then (6.32) has at least two positive ωτt local-periodic solutions u1 and u2 with 0 < u1  < K2 < u2 , where K2 is defined in (H4 ). τ ,f (ii) If (H3 ) is satisfied and fM M∞ ∈ [0, 1/(Bωτt )), then (6.32) has at least two 0 positive ωτt local-periodic solutions u1 and u2 with 0 < u1  < K1 < u2 , where K1 is defined in (H3 ).

Proof We only prove (i) since (ii) can be shown in a very similar way. From fmτ ∞ ∈   1/(Aδ τ ωτt ), ∞ and the proof of Theorem 6.10 (i), one can observe that there exists a sufficiently large R2 > K2 such that f (t, ϕ) >

R2 for ϕ ∈ Dδ τ with |ϕ|lh ∈ [δ τ R2 , R2 ]. Aωτt

Also, from fmτ 0 ∈ (1/(Aδ τ ωτt ), ∞) and the proof of Theorem 6.10 (ii), one can observe that there exists a sufficiently small R2∗ ∈ (0, K2 ) such that f (t, ϕ) >

R2∗ , for ϕ ∈ Dδ τ with |ϕ|lh ∈ [δ τ R2∗ , R2∗ ]. Aωτt

According to Lemma 6.34, one obtains (6.32) has at least two positive ωτt localperiodic solutions u1 and u2 which fulfill R2∗ < u1  < K2 < u2  < R2 . This completes the proof.



Theorem 6.12 If one of the conditions   τ (i) fmτ 0 = ∞ and fM ∈ 0, 1/(Bωτt ) , ∞   τ ∈ 0, 1/(Bω ) , (ii) fmτ ∞ = ∞ and fM τt 0   τ = 0 and f τ ∈ 1/(Aδ τ ω ), ∞ , (iii) fM τt m∞ 0   τ = 0 and fmτ 0 ∈ 1/(Aδ τ ωτt ), ∞ , (iv) fM ∞ are fulfilled, then (6.32) has at least one positive ωτt local-periodic solution. Proof If the condition in (i) is fulfilled, then one can select M6 so that M6 > 1/(Aδ τ ωτt ). Since fmτ 0 = ∞, there exists a constant r4 such that f (t, ϕ) ≥ M6 |ϕ|lh for ϕ ∈ Dδ τ with 0 < |ϕ|lh ≤ r4 .

(6.43)

Let Ωr4 = {u ∈ Pωτ : u < r4 }. Then, for any u ∈ Kδτ ∩ ∂Ωr4 , we have δ τ |ut |lh ≤ δ τ u ≤ ut (θ ). It follows that ut ∈ Dδ τ and δ τ r4 ≤ |ut |lh ≤ r4 . so (6.36) and (6.43) yield

6.3 Local-Periodic Solutions on Changing-Periodic Time Scales

 F u ≥ A t

t+ωτt



f (s, us )Δτs s ≥ AM6

381

pt +ωτt

pt

|us |lh Δτs s ≥ AM6 δ τ r4 ωτt ≥ r4 = u,

i.e., for u ∈ Kδτ ∩ ∂Ωr4 , one has F u ≥ u. τ ∈ [0, 1/(Bωτt )) and Theorem 6.10 (ii) On the other hand, it follows from fM ∞ that there exists R5 > r4 > 0 such that f (t, ϕ) < R5 /(Bωτt ) for ϕ ∈ Dδ τ with |ϕ|lh ≤ R5 . Let ΩR5 = {u ∈ Pωτ : u < R5 }. Then, for any u ∈ Kδτ ∩ ∂ΩR5 , we obtain  t+ωτ t Bωτt R5 F u ≤ B f (s, us )Δτs s < = R5 = u, Bωτt t i.e., for u ∈ Kδτ ∩ ∂ΩR5 , we can obtain F u ≤ u. Hence, from Lemma 6.28, (i) is true. By the similar discussion of those above, one can also prove the cases (ii)–(iv). The details are omitted here. This completes the proof. 

  τ τ τ Theorem 6.13 Suppose that (H4 ) is fulfilled. If fm0 = ∞, fm∞ ∈ 1/(Aδ ωτt ), ∞   or fmτ ∞ = ∞, fmτ 0 ∈ 1/(Aδ τ ωτt ), ∞ holds, then (6.32) has at least two positive ωτt local-periodic solutions u1 and u2 with 0 < u1  < K2 < u2 , where K2 is defined in (H4 ).   Proof We only choose to prove the case fmτ 0 = ∞, fmτ ∞ ∈ 1/(Aδ τ ωτt ), ∞ , because the other case can also be shown in a similar way. Let Ωr4 = {u ∈ Pωτ : u < r4 } and fmτ ∞ = α3 , where r4 < r2 . From fmτ 0 = ∞ and the proof of τ Theorem 6.12 (i), one has F u ≥ u for u ∈ Kδτ ∩ ∂Ωr4 . Let  ΩR2 = {u ∈ Pω : τ τ u < R2 } with R2 > K2 . By fm∞ = α3 ∈ 1/(Aδ ωτt ), ∞ and Theorem 6.10 (i), we obtain f (t, ϕ) >

R2 for ϕ ∈ Dδ τ with |ϕ|lh ∈ [δ τ R2 , R2 ]. Aωτt

From (H4 ) and the proof of Lemma 6.34, one can obtain that (6.32) has at least two positive ωτt local-periodic solutions u1 and u2 with 0 < u1  < K2 < u2 . This completes the proof. 

  τ τ Theorem 6.14 Assume that (H3 ) holds. If fM = 0, fM ∈ 0, 1/(Bωτt ) or ∞ 0   τ τ ∈ 0, 1/(Bω ) , then (6.32) has at least two positive ω local= 0, f fM τt τt M0 ∞ periodic solutions u1 and u2 with 0 < u1  < K1 < u2 , where K1 is defined in (H3 ). Now, we have studied the existence and the multiplicity of positive periodic solutions. Therefore, we have established the following theorem.   τ   τ Theorem 6.15 If fmτ 0 ∈ 1/(Aδ τ ωτt ), ∞ , fM ∈ 0, 1/(Bω ) or fM ∈ τ t ∞ 0   τ   τ 0, 1/(Bωτt ) , fm∞ ∈ 1/(Aδ ωτt ), ∞ or (H3 ), (H4 ) holds, then (6.32) has at   τ ,f least one positive ωτt local-periodic solution. If (H3 ), fM M∞ ∈ 0, 1/(Bωτt ) or 0

382

6 Nonlinear Dynamic Equations on Translation Time Scales

  (H4 ), fmτ 0 , fmτ ∞ ∈ 1/(Aδ τ ωτt ), ∞ is fulfilled, then (6.32) has at least two positive ωτt local-periodic solutions. In the following, we will provide the sufficient conditions for the nonexistence of local-periodic solutions for (6.32). Theorem 6.16 Let Ri , i ∈ {1, 2, 3, 4}, be as in the proof of Theorem 6.10. If fmτ 0 , fmτ ∞ ,

min

R3 ≤|ϕ|lh ≤δ τ R2

f (t, ϕ) ∈ |ϕ|lh



 1 ,∞ Aδ τ ωτt

or τ τ , fM , fM ∞ 0

max

R1 ≤|ϕ|lh ≤R4

f (t, ϕ) 1 ∈ 0, |ϕ|lh Bωτt



is fulfilled, then (6.32) has no positive ωτt local-periodic solution.   Proof We only show the first claim. Since fmτ 0 , fmτ ∞ ∈ 1/(Aδ τ ωτt ), ∞ and the proof of Theorem 6.10, one obtains that there exist R3 , R2 > 0 such that f (t, ϕ) >

1 |ϕ|lh for 0 < |ϕ|lh ≤ R3 Aδ τ ωτt

and f (t, ϕ) > Then, by

f (t,ϕ) min R3 ≤|ϕ|lh ≤δ τ R2 |ϕ|lh

f (t, ϕ) >

>

1 |ϕ|lh for |ϕ|lh ≥ δ τ R2 . Aδ τ ωτt 1 Aδ τ ωτt

, one has

1 |ϕ|lh for any |ϕ|lh ∈ (0, ∞). Aδ τ ωτt

If (6.32) has a positive ωτt local-periodic solution v, then F v = v. Hence '  t+ωτ '  t+ωτ ' ' t t 1 τ ' v = F v = ' G (t, s)f (s, vs )Δτs s ' A τ |vs |lh Δτs s '> Aδ ωτt t t  t+ωτ  t+ωτ t t 1 1 ≥ |v (0)|Δ s = |v(s)|Δτs s s τ s τ τ ω δ ω τt τt δ t t  t+ωt 1 τ ≥ δ vΔτs s = v, τ ω τt δ t which is a contradiction. The first claim is true. In a similar way, one can also prove the second claim and we omit it here. This completes the proof. 

6.3 Local-Periodic Solutions on Changing-Periodic Time Scales

383

Next, consider the system     x Δ = −a t, x(t) x σ (t) + λf (t, xt ), t ∈ T,

(6.44)

where a parameter λ is isolated from the functional f . Through the same discussion above, one can easily reach the following claim. τ ,fτ Theorem 6.17 If fmτ 0 , fmτ ∞ > 0 or fM M∞ < ∞, then (6.44) has no positive ωτt 0 local-periodic solution for sufficiently large or small λ > 0, respectively.

Remark 6.4 From the above established results, we have systematically investigated the existence and nonexistence of positive local-periodic solutions of (6.32). Actually, through exactly the same discussion, it is easy to obtain the corresponding (same) criteria for the existence and nonexistence of positive local-periodic solutions for     x Δ (t) = a t, x(t) x σ (t) − f (t, xt ). The only difference is that Gτ (t, s) should be substituted for Gτ (t, s) =

eaτ (t + ωτt , s) . kaτ

Remark 6.5 Note that one of our principal aims is not only to unify the existence of periodic solutions of some differential equations and their corresponding discrete analogues, but also to unify the existence of local-periodic solutions on an arbitrary time scales with bounded graininess function μ. If T = R and T = Z, i.e., T is the classical periodic time scale, then (6.32) reduces to    x (t) = −a t, x(t) x(t) + f (t, xt ), t ∈ R

(6.45)

and   x(n + 1) − x(n) = −a n, x(n) x(n + 1) + f (n, xn ), n ∈ Z, respectively. Another discrete form of (6.45) is   x(n + 1) − x(n) = −a n, x(n) x(n) + f (n, xn ), n ∈ Z. To unify this equation, it is sufficient to study the existence and nonexistence of local-periodic solutions for the system   x Δ (t) = a t, x(t) x(t) − f (t, xt ), t ∈ T.

(6.46)

Through exactly the same discussion as for (6.32), one can easily observe that Theorems 6.15 and 6.16 are also true for (6.46). The only difference is that

384

6 Nonlinear Dynamic Equations on Translation Time Scales

Gτ , γ1τ , γ2τ , A, B here should be substituted for Gτ (t, s) =

eaτ (t + ωτt , s) , γ1 = kaτ γ2 =

t

eβτ (t + ωτt , s);

inf

t≤s, t,s∈[pt ,ptˆ+ωτt ]Tτ

eατ (t + ωτt , s),

inf

t≤s, t,s∈[pt ,ptˆ+ωτt ]Tτ

t

γ1τ γ2τ γ1τ γ2τ , B = ; forω < 0, A = , B = . τ t kβτ kατ kατ kβτ

for ωτt > 0, A =

Moreover, based on the exact same discussion, one can also obtain the same criteria for the existence and nonexistence of local-periodic solutions for   x Δ (t) = −a t, x(t) x(t) + f (t, xt )   provided that 1/ 1 − μτs (s)a(s) is positive and bounded. For simplicity, we omit it here. Remark 6.6 Note that all theorems established are true for changing-periodic time scales, which indicates that one can extract a local-periodic solution for dynamic equations on an arbitrary time scale with a bounded graininess function μ through some index function. For example, let k ∈ Z, and consider the following time scale: T=

 +∞ 

3 1 3 (2k + 1), (2k + 1) + 2 2 12

k=−∞

!

√ √ √ ! 3 2 3 2 3 (2k + 1), (2k + 1) + . 2 2 5

   +∞ k=−∞

We denote +∞ 

T1 =

k=−∞

3 3 1 (2k + 1), (2k + 1) + 2 2 12

!

and T2 =

+∞  k=−∞

√ √ √ ! 3 2 3 2 3 (2k + 1), (2k + 1) + . 2 2 5

Then, by a direct calculation the set Λ is

6.3 Local-Periodic Solutions on Changing-Periodic Time Scales

385

  √  Λ = 3n, n ∈ Z 3 2n, n ∈ Z := S1 ∪ S2 . This time scale is a changing-periodic time scale according to Definition 2.34. Thus, for instance, by Theorems 6.15 and 6.16, one obtains that (6.32) has no positive local-periodic solution on T1 and T2 . Next, we will provide some new results on local-periodic solutions for higher dimensional dynamic systems. Consider the n-dimensional systems   XΔ (t) = −A(t)X σ (t) + G(t, Xt ),

(6.47)

where A(t) = diag[a1 (t), a2 (t), . . . , an (t)] with ai ∈ Crd , ai (t + ωτt ) = ai (t) for all i ∈ {1, 2, . . . , n}. G = (g1 , g2 , . . . , gn )T is defined on R × Clh , and G(t, φ) is rd-continuous in t and is continuous in φ with G(t + ωτt , φ) = G(t, φ). Moreover, gi (t, φ) maps bounded sets into bounded sets and gi (t, φ) ≥ 0 for φ ∈ Clh with φi (θ ) ≥ 0 and θ ∈ S˜τt . For convenience, we introduce the following notations: Gτi (t, s) =

eaτ i (s, t) kaτi

, δiτ =

δiτ 1 1 τ τ , δ = min {δ }, A = , Bi = , i i τ τ 1≤i≤n eai (ωτt , 0) 1 − δi 1 − δiτ A0 =

n 

Ai , B0 =

i=1

n 

Bi ,

i=1

  P = u ∈ Crd (Tτt , Rn ) : u(t + ωτt ) = u(t), t ∈ Tτt , ui ∈ Crd (Tτt , R) , u =

max

n 

t∈[pt ,ptˆ+ωτt ]Tτ i=1 t

|ui (t)| for u ∈ P ,

   Eδτ = {φ ∈ Clh : φi (θ ) ≥ δ τ |φi |lh , θ ∈ S˜τt , Kδτ = x ∈ P : xi (t) ≥ 0, xi (t) ≥ δ τ |xi | ,

giτM = lim 0

giτM∞ =

lim

t

gi (t, ϕ) τ gi (t, ϕ) , gim = lim min , 0 |ϕ|lh →0 t∈[pt ,ptˆ+ωτ ] |ϕ|lh |ϕ|lh t T

t

gi (t, ϕ) τ gi (t, ϕ) , gim∞ = lim min , |ϕ|lh →∞ t∈[pt ,ptˆ+ωτ ] |ϕ|lh |ϕ|lh t T

max

|ϕ|lh →0 t∈[pt ,ptˆ+ωτ ] t Tτ

max

|ϕ|lh →∞ t∈[pt ,ptˆ+ωτ ] t Tτ

τt

τt

GτM0 = max giτM , GτiM 1≤i≤n

0

∞

= max giτm , Gτim∞ = min giτm∞ . 1≤i≤n

0

1≤i≤n

Since the proofs process are very similar to those in the above section, for simplicity, we can only list some conclusions below without proofs.

386

6 Nonlinear Dynamic Equations on Translation Time Scales

Lemma 6.35 Let A, Q ∈ P and  T A(t) = diag[a1 (t), a2 (t), . . . , an (t)], Q(t) = q1 (t), q2 (t), . . . , qn (t) . Then   XΔ (t) = −A(t)X σ (t) + Q(t) has a unique ωτt local-periodic solution with the form   X(t) = x1 (t), x2 (t), . . . , xn (t) , where xi (t) =



t+ωτt

t

Gτi (t, s)qi (s)Δτs s.

For any u ∈ P , consider the equation   XΔ (t) = −A(t)X σ (t) + G(t, ut ).

(6.48)

It follows from Lemma 6.35 that the unique ωτt local-periodic solution of (6.48) is given by   Xu (t) = x1 (t), x2 (t), . . . , xn (t) , where xi (t) =

 t

t+ωτt

Gτi (t, s)gi (s, us )Δτs s.

Define the operator F : Kδτ → P with components (F1 , F2 , . . . , Fn ) by 

t+ωτt

(Fi u)(t) = t

Gτi (t, s)gi (s, us )Δτs s for u ∈ Pωτ and t ∈ Tτt .

We can easily demonstrate that X is an ωτt local-periodic solution of (6.47) if and only if X is a fixed point of F in Kδτ . It is easy to prove that F (Kδτ ) ⊂ Kδτ . Let η be a positive constant and Ω = {x ∈ P : |x| ≤ η}. Then F : Kδτ ∩ Ω¯ → Kδτ is completely continuous. In addition, based on Gτm0 , Gτm∞ , GτM∞ , GτM0 , one can also obtain exactly the same sufficient criteria (Theorems 6.15 and 6.16) for the existence and nonexistence of ωτt local-periodic solutions for (6.47). For simplicity, for example, we only provide the proof for the case where GτM0 = 0, Gτm∞ = ∞. Theorem 6.18 If GτM0 = 0 and Gτm∞ = ∞, then (6.47) has at least one positive ωτt local-periodic solution. Proof It follows from GτM0 = 0 that giτM = 0. Select ε > 0 such that εB0 ωτt < 1. 0 Then there exists a constant s1 such that gi (t, φ) ≤ ε|φ|lh , 0 < |φ|lh ≤ s1 , φ ∈ Eδ τ . Let Ω1 = {x ∈ P : |x| < s1 }. Then for any u ∈ Kδτ ∩ ∂Ω1 , one has uit (t) ≥ δ τ |uit |lh and δ τ |u| ≤ |ut |lh ≤ u, so ut ∈ Eδ τ . Therefore

6.3 Local-Periodic Solutions on Changing-Periodic Time Scales

F u =

n   t

i=1

≤

n 

t+ωτt

Gτi (t, s)gi (s, us )Δτs s ≤



pt +ωτt

Bi pt

ε|us |lh Δτs s ≤ ε

n 

pt +ωτt

Bi



pt +ωτt

Bi pt

i=1

gi (s, us )Δτs s

pt

i=1



i=1

n 

387

|u|Δτs s < |u|.

On the other hand, assume that Gτm∞ = ∞. One may take M such that Mδ τ ωτt A0 > 1 and one can observe that there exists a constant s0 with s0 > s1 such that gi (t, ϕ) > M|ϕ|lh , |ϕ|lh ≥ s0 , ϕ ∈ Eδ τ . Let s2 = s0 /δ τ and Ω2 = {x ∈ P : |x| < s2 }. Then, for any u ∈ Kδτ ∩ ∂Ω2 , we obtain ut ∈ Eδ τ and  |ut |lh =

S˜τt

sup

h(s)

n 

ˆ L i=1 θ∈[s,0]

ˆ

|uit (θ )|[s,0]L Δτs s ≥

 S˜τt

h(s)

sup

n 

ˆ L i=1 θ∈[s,0]

δ τ |uit |Δτs s

= δ |u| = δ s2 = s0 . τ

τ

Hence F u =

n   i=1

≥

n  i=1

t+ωτt

t

Gτi (t, s)gi (s, us )Δτs s ≥

n  i=1



pt +ωτt

Ai pt



pt +ωτt

Ai pt

gi (s, us )Δτs s

M|us |lh Δτs s ≥ Mδ τ |u|ωτt A0 ≥ |u|.

By (6.32), (6.41) and (6.42), one can obtain that F has a fixed point u1 ∈ Kδτ ∩ (Ω¯ 2 \Ω1 ) such that (F u1 )(t) = u1 (t) and ui1 (t) ≥ δ τ ui1  ≥ δ τ s1 > 0. Therefore, u1 is an ωτt local-periodic solution for (6.47). This completes the proof. 

Chapter 7

Impulsive Dynamic Equations on Translation Time Scales

7.1 The Cauchy Matrix and Liouville’s Formula on Time Scales For the history of Liouville’s formula of n-dimensional dynamic equations on time scales, we refer the reader to the literature Li et al.[182]. In the following, we construct a Liouville’s Formula for n × n-matrix dynamic equations on time scales through determinant algorithms and techniques rather than considering the eigenpolynomial and eigenvalue of the coefficient matrix of the dynamic equations. Particularly, if n = 2, then the Liouville’s Formula is Theorem 1.22. Lemma 7.1 Let A be an upper triangular n × n-matrix-valued function. Then A is regressive iff each diagonal element of A is regressive. Proof Let ⎡ a11 ⎢ 0 ⎢ A=⎢ . ⎣ .. 0

a12 a22 .. .

... ... .. .

⎤ a1n a2n ⎥ ⎥ .. ⎥ . . ⎦

0 . . . ann

Hence 1 + μa11 μa12 0 1 + μa22 det[I + μA] = . .. . . . 0 0

n 0   1 + μaii . = i=1 . . . 1 + μann ... ... .. .

μa1n μa2n .. .

© Springer Nature Switzerland AG 2020 C. Wang et al., Theory of Translation Closedness for Time Scales, Developments in Mathematics 62, https://doi.org/10.1007/978-3-030-38644-3_7

389

390

7 Impulsive Dynamic Equations on Translation Time Scales

Therefore A is regressive iff each diagonal element of A is regressive. This completes the proof. 

Remark 7.1 Let A : T → Rn×n be an upper triangular matrix. We obtain det A =

n 0 i=1

aii , trA =

n 

n     0 1 + μaii . aii , det I + μA =

i=1

i=1

Hence for n > 2, n n n  0 0     1 + μ trA + μdetA = 1 + μ( 1 + μaii , aii + μ aii ) = i=1

i=1

i=1

which implies that I + μA is invertible cannot be equivalent to trA + μ det A is regressive, that is, the Liouville’s Formula of Theorem 1.22 is not suitable for n ≥ 3. It is easy to check the following Δ-derivative formula of determinant function by using the determinant algorithm and Theorem 1.6. Lemma 7.2 Let A : T → Rn×n be the following function matrix: a11 a12 . .. .. . a (i−1)1 a(i−1)2 A(t) = ai1 ai2 a(i+1)1 a(i+1)2 .. .. . . an1 an2

... .. .

a1i .. .

a1(i+1) .. .

. . . a(i−1)i a(i−1)(i+1) . . . aii ai(i+1) . . . a(i+1)i a(i+1)(i+1) .. .. .. . . . . . . ani an(i+1)

... .. . ... ... ... .. . ...

a(i−1)n ain . a(i+1)n .. . ann a1n .. .

Then σ σ a11 a12 . .. .. . a σ σ a n  (i−1)1 (i−1)2 Δ Δ AΔ (t) = ai1 ai2 i=1 a (i+1)1 a(i+1)2 . .. .. . a an2 n1

σ σ . . . a1i a1(i+1) .. .. .. . . . σ σ . . . a(i−1)i a(i−1)(i+1) Δ . . . aiiΔ ai(i+1) . . . a(i+1)i a(i+1)(i+1) .. .. .. . . . . . . ani an(i+1)

σ . . . a1n .. .. . . σ . . . a(i−1)n Δ . . . ain . . . . a(i+1)n .. .. . . . . . ann

Theorem 7.1 (Liouville’s Formula) Let A ∈ R be an upper triangular n × nmatrix-valued function and assume that X is a solution of XΔ = A(t)X. Then X satisfies Liouville’s formula

7.1 The Cauchy Matrix and Liouville’s Formula on Time Scales

det X(t) = e  n i−1 1  i=1 j =1

391

 (t, t0 ) det X0 , f or t ∈ T.

1+μajj aii

Proof For the matrix A, by Lemma 7.1, we can obtain A is regressive iff each diagonal element of A is regressive. Let ⎡ a11 ⎢ 0 ⎢ A=⎢ . ⎣ ..

a12 a22 .. .

0 x11  Δ 0 det X(t) = . .. 0 Δ x11 0 = . .. 0

0 . . . ann

x12 x22 .. .

x13 . . . x23 . . . .. . . . . 0 0 ...

Δ x12 x22 .. .

0

i−1 n 0  i=1 j =1

This completes the proof.

Δ x13 x23 .. .

... ... .. .

0 ...

σ x 11 0 n  ... = . . . i=1 .. . 0 =

⎡ ⎤ x11 a1n ⎢ ⎥ a2n ⎥ ⎢ 0 .. ⎥ X = ⎢ .. ⎣ . . ⎦

... ... .. .

σ x12 σ x22 .. . ... .. . 0

... ... .. . ... .. . 0

σ Δ xjj xii

0

x12 x22 .. .

... ... .. .

⎤ x1n x2n ⎥ ⎥ .. ⎥ . ⎦

0 . . . xnn

Δ x1n x2n .. . x nn

σ x 11 0 Δ x1n n−1 .. x2n  . .. + . i=2 . . . .. xnn . 0 σ xσ x1i 1(i+1) σ xσ x2i 2(i+1) .. .. . . Δ xiiΔ xi(i+1) .. .. . . 0 0

n 0 k=i+1

xkk =

... ... .. . ... .. . ...

σ x12 σ x22 .. . ... .. . 0 σ x1n σ x2n .. . Δ xin .. . xnn

... ... .. . ... .. . 0

i−1 n 0  

σ xσ x1i 1(i+1) σ xσ x2i 2(i+1) .. .. . . Δ xiiΔ xi(i+1) .. .. . . 0 0

... ... .. . ... .. . ...

σ x1n σ x2n .. . Δ xin .. . x nn

!  1 + μajj aii det X.

i=1 j =1



Remark 7.2 Let A be a 2 × 2-matrix-valued function. Then A is regressive iff the scalar-valued function trA + μ det A is regressive (where trA denotes the trace of the matrix A, i, e., the sum of diagonal elements of A). In fact, let

392

7 Impulsive Dynamic Equations on Translation Time Scales

! a11 a12 , A= a21 a22 then we can obtain 1 + μa11 μa12 det(I + μA) = det μa21 1 + μa22

!

= (1 + μa11 )(1 + μa22 ) − μa12 μa21 = 1 + μ(a11 + a22 ) + μ2 (a11 a22 − a12 a21 ) = 1 + μ(trA + μ det A). Hence A is regressive iff the scalar-valued function trA + μdetA is regressive. Remark 7.3 Assume that for n = k, we can obtain similar characterizations of Remark 7.2. For n = k + 1, let Ak+1

! Ak 0 = , 0 a0

where Ak+1 is (k + 1) × (k + 1)-matrix-valued function, Ak is k × k-matrix-valued function. Then ! 0 I + μAk , det(I + μAk+1 ) = det k 0 1 + μa0 = (1 + μa0 ) det(I + μAk ), 1 + μ(trAk+1 + μ det Ak+1 ) = 1 + μ[trAk + a0 + μa0 det Ak ] = 1 + μtrAk + μa0 + μa0 μ det Ak . Hence there exists Ak+1 such that Ak+1 is regressive and so is trAk+1 + μ det Ak+1 . Theorem 7.2 (Liouville’s Formula) Let A ∈ R be n × n-matrix-valued matrix function and assume that X is a solution of XΔ = A(t)X. Then X satisfies Liouville’s formula n det X(t) = e  i=1

where

det Ai

(t, t0 ) det X0 , f or t ∈ T,

7.1 The Cauchy Matrix and Liouville’s Formula on Time Scales

⎡ 1 + μa11 μa12 ⎢ μa 1 + μa22 21 ⎢ ⎢ .. .. ⎢ . . ⎢ ⎢ μa(i−1)2 ⎢ μa Ai = ⎢ (i−1)1 ⎢ ai1 ai2 ⎢ ⎢ 0 0 ⎢ ⎢ .. .. ⎣ . . 0 0

... ... .. .

μa1(i−1) μa2(i−1) .. .

393

μa1i μa2i .. .

. . . 1 + μa(i−1)(i−1) μa(i−1)i ... ai(i−1) aii ... 0 0 .. .. .. . . . ... 0 0

0 0 .. .

... ... .. .

0 0 1 .. .

0 0 0 .. .

0 ...

⎤ 0 0⎥ ⎥ .. ⎥ .⎥ ⎥ ⎥ 0⎥ ⎥. 0⎥ ⎥ 0⎥ ⎥ .. ⎥ .⎦ 1

Proof For n = 2, by Remark 7.2, A is regressive implies trA+μ det A is regressive. Let ! ! a11 a12 x11 x12 A(t) = , X(t) = . a21 a22 x21 x22 Then 

Δ det X(t) = = = = = =

Δ Δ σ σ x11 x12 x11 x12 + Δ Δ x21 x22 x21 x22 Δ x + μx Δ a11 x11 + a12 x21 a11 x12 + a12 x22 x11 + μx11 12 12 + Δ Δ x21 x22 x21 x22 x11 + (a11 x11 + a12 x21 )μ x12 + (a11 x12 + a12 x22 )μ a11 det X(t) + a21 x12 + a22 x22 a21 x11 + a22 x21 1 + μa11 a12 μ a11 det X(t) + det X a21 a22   trA + μ det A det X   a11 0 1 + μa11 a12 μ + det X. 0 1 a21 a22

For n ≥ 3, let ⎡

a11 ⎢a21 ⎢ A(t) = ⎢ . ⎣ ..

a12 a22 .. .

a13 a23 .. .

... ... .. .

⎤ a1n a2n ⎥ ⎥ .. ⎥ , . ⎦

an1 an2 an3 . . . ann Hence by Lemma 7.2, we have

⎡

x11 ⎢x21 ⎢ X(t) = ⎢ . ⎣ ..

x12 x22 .. .

x13 x23 .. .

... ... .. .

⎤ x1n x2n ⎥ ⎥ .. ⎥ . . ⎦

xn1 xn2 xn3 . . . xnn

394

7 Impulsive Dynamic Equations on Translation Time Scales

x11  Δ x21 X(t) = . .. xn1

x12 x22 .. . xn2

x13 x23 .. . xn3

... ... .. . ...

σ σ σ x11 x12 x13 Δ . .. .. .. x1n . . n σ σ σ x2n  x x(i−1)2 x(i−1)3 (i−1)1 Δ .. = Δ Δ xi1 xi2 xi3 . i=1 . .. .. xnn .. . . x xn2 xn3 n1

Δ Δ Δ x11 + μx11 x12 + μx12 x13 + μx13 .. .. .. . . . n Δ Δ Δ  x(i−1)2 + μx(i−1)2 x(i−1)3 + μx(i−1)3 x(i−1)1 + μx(i−1)1 = Δ Δ Δ xi1 xi2 xi3 i=1 .. .. .. . . . x x x n1

n2

n3

n n   x11 + μ a1j xj 1 x12 + μ a1j xj 2 j =1 j =1 .. .. . . n n   n  x(i−1)1 + μ a(i−1)j xj 1 x(i−1)2 + μ a(i−1)j xj 2 = j =1 j =1 n n   i=1 aij xj 1 aij xj 2 j =1 j =1 .. .. . . xn2 xn1 i i  x +μ  a x x12 + μ a1j xj 2 11 1j j 1 j =1 j =1 .. .. . . i i   n  x(i−1)1 + μ a(i−1)j xj 1 x(i−1)2 + μ a(i−1)j xj 2 = j =1 j =1 i i i=1   aij xj 1 aij xj 2 j =1 j =1 .. .. . . xn2 xn1 =

n  i=1

Then

det(Ai X) =

n 

... .. . ... ... .. . ... ... .. . ... ... .. . ...

σ . . . x1n .. .. . . σ . . . x(i−1)n Δ . . . xin .. .. . . . . . xnn

Δ ... x1n + μx1n .. .. . . Δ . . . x(i−1)n + μx(i−1)n Δ ... xin .. .. . . ... xnn

j =1 .. . n  x(i−1)n + μ a(i−1)j xj n j =1 n  aij xj n j =1 .. . xnn i  x1n + μ a1j xj n j =1 .. . i  x(i−1)n + μ a(i−1)j xj n j =1 i  aij xj n j =1 .. . xnn x1n + μ

n 

a1j xj n

det Ai det X.

i=1

 n X(t) = e 

det Ai

 (t, t0 ) det X0 .

i=1

This completes the proof.



7.1 The Cauchy Matrix and Liouville’s Formula on Time Scales

395

Theorem 7.3 Let A be a n × n-matrix-valued function. Then A is regressive iff the n  scalar-valued function det Ai is regressive, where Ai is defined by Theorem 7.2. i=1

Proof By Theorem 7.2, for n = 2 we can obtain  a det(I + μA) = 1 + μ 11 0

 0 1 + μa11 a12 μ . + a22 1 a21

Next, let ⎡

a11 ⎢a21 ⎢ A(t) = ⎢ . ⎣ ..

a12 a22 .. .

a13 a23 .. .

... ... .. .

⎤ ⎡ a1n ⎢ a2n ⎥ ⎥ ⎢ .. ⎥ = ⎢ . ⎦ ⎣

an1 an2 an3 . . . ann

A(0)

⎤ a1n a2n ⎥ ⎥ .. ⎥ , . ⎦

an1 an2 an3 . . . ann

where A(0) is a (n − 1) × (n − 1)-valued-matrix function. Assume that (n − 1) × n−1  det Ai (n − 1)-matrix-value function is regressive iff the scalar-valued function i=1

is regressive. That is det(I + μA(0) ) = 1 + μ

n−1 

det Ai .

i=1

Hence   n−1 n−1    1+μ det Ai + det An = 1 + μ det Ai + μ det An = det I + μA(0) + μ det An i=1

  = det I + μA(0) + μ a

i=1

I + μA(0)

μa1n μa2n . . . . . . ann

an3 n1 an2 μa1n I + μA(0) μa2n I + μA(0) = + . . . 0 0 0 . . . 1 μan1 μan2 μan3 . . . μa1n I + μA(0) μa2n = det(I + μA). = . . . μa μa μan3 . . . 1 + μann n1 n2

μa1n μa2n . . . μa nn

396

7 Impulsive Dynamic Equations on Translation Time Scales

Therefore A is regressive iff the scalar-valued function

n 

det Ai is regressive. This

i=1

completes the proof.



In what follows, we will investigate the Cauchy matrix of the impulsive dynamic equations. Denote the matrix by W (t), the columns of which are the solutions of following homogeneous dynamic equations ˜ x Δ = A(t)x, t = tk , Δx = Bk x, t=t k

(7.1)

where A ∈ P Crd (T, Rn×n ) is an n × n-matrix function, Bk are constant matrices, {tk } ⊂ T are fixed times such that tk < tk+1 , k ∈ Z. Denote by W (t) the matrix, the columns of which are the solutions of (7.1) that form a fundamental solution system. The matrix W (t) is called an n×n fundamental solution matrix of (7.1) (n ≥ 2). It follows from the definition of the matrix W (t) that it satisfies the matrix impulsive dynamic equations W Δ = A(t)W, t = tk ,

˜ ΔW = Bk W. t=t k

(7.2)

A nondegenerate solution of (7.2) W (t), which satisfies the condition W (t0 ) = E is called the Cauchy matrix of (7.1) and denoted by W (t, t0 ). Lemma 7.3 Suppose the following conditions are satisfied: (a) any compact interval [a, b]T contains only a finite number of points tk ; (b) for all k ∈ Z, the matrices E + Bk are nonsingular. Then W (t, t0 )W −1 (κ, t0 ) = W (t, κ), where tj ≤ tj +s−1 < κ < tj +s < tj +k < t ≤ tj +k+1 , j, k, s ∈ Z. Furthermore, any solution of (7.1) with initial condition x(t0 , x0 ) = x0 can be written as x(t; t0 , x0 ) = W (t, t0 )x0 . Proof Let U (t, τ ) be a solution of the matrix Cauchy problem U Δ = A(t)U, U (τ, τ ) = E, i.e. the Cauchy matrix of (7.1) without impulses. Then any solution W (t) of matrix dynamic equations (7.2) can be represented as W (t) = U (t, tj +k )(E + Bj +k )U (tj +k , tj +k−1 )(E + Bj +k−1 ) . . . . . . (E + Bj )U (tj , t0 )W (t0 ), where tj −1 < t0 ≤ tj < tj +k < t ≤ tj +k+1 . In particular, for the Cauchy matrix W (t, t0 ), we have

7.1 The Cauchy Matrix and Liouville’s Formula on Time Scales k 0

W (t, t0 ) = U (t, tj +k )(E + Bj +k )

397

U (tj +ν , tj +ν−1 )(E + Bj +ν−1 )U (tj , t0 ),

ν=1

(7.3) where tj −1 < t0 ≤ tj < tj +k < t ≤ tj +k+1 . By using the Liouville formula Theorem 7.2 on time scales, we get from (7.3) that det W (t) = det U (t, tj +k ) det(E + Bj +k )

k 0

det U (tj +ν , tj +ν−1 ) det(E + Bj +ν−1 )

ν=1

× det U (tj , t0 ) det W (t0 ) n = e

det Ai

(t, tj +1 ) det(E + Bj +k )

k 0 ν=1

i=1 n × det(E + Bj +ν−1 )e 

det Ai

n e

det Ai

(tj +ν , tj +ν−1 )

i=1

(tj , t0 ) det W (t0 ),

i=1

i.e. n det W (t) = det W (t0 )e 

det Ai

(t, t0 )

i=1

k+1 0

det(E + Bj +ν−1 ),

(7.4)

ν=1

where tj −1 < t0 ≤ tj < tj +k < t ≤ tj +k+1 . Since the matrices E + Bk are nonsingular, it follows from (7.4) that the matrix W (t) is nonsingular if W (t0 ) is such. For a nonsingular matrix W (t), the inverse matrix W −1 (t) is given by the following W −1 (t) = W −1 (t0 )U −1 (tj , t0 )(E + Bj )−1 . . . . . . U −1 (tj +k , tj +k−1 )(E + Bj +k )−1 U −1 (t, tj +k ) = W −1 (t0 )U −1 (tj , t0 )

k 0

(E + Bj +ν−1 )−1 U −1 (tj +ν , tj +ν−1 )

ν=1

×(E + Bj +k )

−1

U

−1

(t, tj +k )

where tj −1 < t0 ≤ tj < tj +k < t ≤ tj +k+1 , and the product W (t)W −1 (κ) = U (t, tj +k )(E + Bj +k )U (tj +k , tj +k−1 ) . . . . . . (E + Bj +s+1 )U (tj +s+1 , tj +s )(E + Bj +s )U (tj +s , κ)W (κ)W −1 (κ)

398

7 Impulsive Dynamic Equations on Translation Time Scales

= U (t, tj +k )

s+1 0

(E + Bj +ν )U (tj +ν , tj +ν−1 )(E + Bj +s )U (tj +s , κ)

ν=k

where tj −1 < t0 ≤ tj ≤ tj +s−1 < κ < tj +s < tj +k < t ≤ tj +k+1 . In particular, from (7.3), for the Cauchy matrix W (t, t0 ), we have W (t, t0 )W −1 (κ, t0 ) = U (t, tj +k )

s+1 0

(E + Bj +ν )U (tj +ν , tj +ν−1 )

ν=k

×(E + Bj +s )U (tj +s , κ) = W (t, κ), where tj −1 < t0 ≤ tj ≤ tj +s−1 < κ < tj +s < tj +k < t ≤ tj +k+1 . Therefore, W (t, t0 )W −1 (κ, t0 ) = W (t, κ). Note that, by using the Cauchy matrix, any solution of (7.1) with initial condition x(t0 , x0 ) = x0 can be written as x(t, x0 ) = W (t, t0 )x0 . This completes the proof. 

Remark 7.4 From Lemma 7.3, we can obtain the Cauchy matrix of (7.1) as follows: ⎧ ⎪ U (t, s), t, s ∈ (tk−1 , tk ]T , ⎪ ⎪ ⎪ ⎪ + ⎪ U (t, t )(E + Bk )U (tk , s), tk−1 < s ≤ tk < t ≤ tk+1 , ⎪ k ⎪ ⎪ ⎪ + + −1 ⎪U (t, t )(E + Bk ) U (t , s), tk−1 < t ≤ tk < s ≤ tk+1 , ⎪ k k ⎪ ⎪ ⎪ ⎪ 1 ⎨U (t, t + ) i+1 (E + Bj )U (tj , tj+−1 )(E + Ai )Ui (ti , s), k W (t, s) = j =k ⎪ ⎪ ⎪ ⎪ t < s ≤ ti < tk < t ≤ tk+1 , i−1 ⎪ ⎪ ⎪ k−1 ⎪ 1 ⎪ ⎪ ⎪ U (t, ti ) (E + Bj )−1 U (tj+ , tj +1 )(E + Bk )−1 U (tk+ , s), ⎪ ⎪ ⎪ j =i ⎪ ⎪ ⎩ ti−1 < t ≤ ti < tk < s ≤ tk+1 , Then the solutions of (7.1) are in the form x(t; t0 , x0 ) = W (t, t0 )x0 , t0 ∈ T, x0 ∈ Rn . In the following, consider the system w Δ = A(t)w + f (t), t = tk ,

˜ Δw = Bk w + Ik , t=t k

(7.5)

where the matrix A ∈ P Crd (T, Rn×n ) and f ∈ P Crd (T, Rn ), Bk and the times tk are the same as in (7.1), Ik are constants, (7.5) is called linear nonhomogeneous impulsive dynamic equations. The relationship between the nonhomogeneous equations (7.5) and the corresponding homogeneous equations (7.1) is given by the following lemma.

7.2 Piecewise Almost Periodic Solutions on CCTS

399

Lemma 7.4 Suppose (a), (b) in Lemma 7.3 are satisfied and let W (t) be a fundamental solution matrix of (7.1). Then for t ≥ t0 , every solution of (7.5) is given by the formula  w(t) = W (t) c +

t t0

! W −1 (tk )(E + Bk )−1 Ik .



W −1 (σ (τ ))f (τ )Δτ +

t0