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Dynamic Equations
on Time Scales and
Applications
This book presents the theory of dynamic equations on time scales and applications, providing
an overview of recent developments in the foundations of the field as well as its applications. It
discusses the recent results related to the qualitative properties of solutions like existence and
uniqueness, stability, continuous dependence, controllability, oscillations, etc.
• Presents cutting-edge research trends of dynamic equations and recent advances in contemporary research on the topic of time scales
• Connects several new areas of dynamic equations on time scales with applications in different
fields
• Includes mathematical explanation from the perspective of existing knowledge of dynamic
equations on time scales
• Offers several new recently developed results, which are useful for the mathematical modeling
of various phenomena
• Useful for several interdisciplinary fields like economics, biology, and population dynamics
from the perspective of new trends
The text is for postgraduate students, professionals, and academic researchers working in the
fields of Applied Mathematics.
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Taylor & Francis
Taylor & Francis Group
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Dynamic Equations
on Time Scales and
Applications
Edited by
Ravi P. Agarwal, Bipan Hazarika,
and Sanket Tikare
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contributors
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ISBN: 978-1-032-74004-1 (hbk)
ISBN: 978-1-032-74162-8 (pbk)
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DOI: 10.1201/9781003467908
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Contents
Preface
ix
Editors
xiii
Contributors
xv
1 Elements of Time Scales Calculus
Ravi P. Agarwal, Bipan Hazarika, and Sanket Tikare
1.1 History and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Delta Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Delta Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Nabla Calculus Essentials . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2 First-Order Functional Dynamic Equations
Svetlin G. Georgiev and Sanket Tikare
2.1 Functional Dynamic Equations–Basic Concepts, Existence and Uniqueness
Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Classification of Functional Dynamic Equations . . . . . . . . . . . .
2.1.2 The Picard–Lindelöf Theorem . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Existence and Uniqueness Theorems . . . . . . . . . . . . . . . . . .
2.1.4 Continuous Dependence on Initial Data . . . . . . . . . . . . . . . .
2.2 Uniform Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Uniformly Asymptotical Stability . . . . . . . . . . . . . . . . . . . . . . .
2.4 Global Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Asymptotic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Exponential Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Positive Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Iterated Oscillation Criteria for First-Order Functional Dynamic Equations
2.9 Oscillations of the Solutions of First-Order Functional Dynamic Equations
with Several Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Nonoscillations of First-Order Functional Dynamic Equations with Several
Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
3
8
19
25
34
39
39
39
41
45
48
49
60
64
71
83
87
103
113
118
3 Foundations of Linear Control Theory on Time Scales
129
Tom Cuchta and Nick Wintz
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.2 Linear Deterministic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.2.1 Controllability and Observability . . . . . . . . . . . . . . . . . . . . 130
3.2.2 Linear Time Varying . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.2.3 Switched Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
v
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3.3
3.4
3.5
3.6
Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Realizations, Stability, and Stabilizability . . . . . . . . . . . . . . .
3.3.3 Lyapunov Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Linear Quadratic Regulator . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Linear Quadratic Tracking (LQT) . . . . . . . . . . . . . . . . . . .
Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stochastic Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137
137
139
141
147
147
149
153
158
164
4 Optimal Control Theory for Dynamic Equations
171
Iguer Luis Domini dos Santos and Sanket Tikare
4.1 Optimal Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.2.1 Δ-Measurable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.2.2 Δ-Integrable Functions . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.2.3 Absolutely Continuous Functions . . . . . . . . . . . . . . . . . . . . 174
4.2.4 Set-Valued Functions and Measurability . . . . . . . . . . . . . . . . 175
4.2.5 Control Processes for (P2 ) . . . . . . . . . . . . . . . . . . . . . . . . 176
4.2.6 Δ-Measurable Selection . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.4 Proofs of the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5 Controllability of Dynamic Equations
192
Hugo Leiva and Cosme Duque
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.2 Controllability of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . 192
5.3 Controllability of Dynamic Equations with Memory . . . . . . . . . . . . . 195
5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.5 Controllability of a Semilinear Neutral Dynamic Equation with Impulses and
Nonlocal Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5.6 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
5.7 Conclusion and Final Remark . . . . . . . . . . . . . . . . . . . . . . . . . 209
214
6 Delayed Dynamic Equations with sp -Terms
Mohssine Es-saiydy and Mohamed Zitane
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6.2 General Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
6.2.1 Doubly Weighted Pseudo Almost Periodic Functions on Time Scales 215
6.2.2 Stepanov-like Almost Periodic Functions . . . . . . . . . . . . . . . . 216
6.3 Doubly Weighted Stepanov-like Pseudo Almost Periodic Functions on Time
Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6.4 Doubly Weighted Pseudo Almost Periodic Solution on Time Scales . . . . 223
6.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
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Contents
vii
7 Integro-Dynamic System with Stepanov-like Coefficients
231
Soniya Dhama
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
7.3 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
7.4 Stability of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
7.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8 Terminal Value Problems for Discrete Fractional Relaxation Equations 249
Sangeeta Dhawan and Jagan Mohan Jonnalagadda
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
8.3 Construction of the Green Functions . . . . . . . . . . . . . . . . . . . . . . 259
8.4 Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
8.5 Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9 Diamond-Alpha Hardy–Copson Type Dynamic Inequalities-I
271
Zeynep Kayar and Billur Kaymakçalan
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
9.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
9.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
9.4 Diamond-Alpha Hardy–Copson Type Dynamic Inequalities . . . . . . . . . 275
9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
10 Diamond-Alpha Hardy–Copson Type Dynamic Inequalities-II
290
Zeynep Kayar and Billur Kaymakçalan
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
10.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
10.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
10.4 Complementary Diamond-Alpha Hardy–Copson Type Dynamic Inequalities 297
10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
11 Fishing Model with Feedback Control on Time Scales
310
Mahammad Khuddush
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
11.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
11.3 Persistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
11.4 Almost Periodic Solutions and Stability Analysis . . . . . . . . . . . . . . . 315
11.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
11.6 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 320
12 Some Geometric Properties of Dual Space on Time Scales
325
Hatice Kusak Samanci
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
12.2 The Dual Numbers, Dual Vectors and Dual Space on the Time Scales . . . 326
12.2.1 The Inner Product and Norm of the Dual Numbers on the Time Scale 327
12.2.2 Module-D on the Time scales . . . . . . . . . . . . . . . . . . . . . . 328
12.3 Dual Directional Derivative on the Time Scale . . . . . . . . . . . . . . . . 330
12.4 Dual Vector Field on the Time Scales . . . . . . . . . . . . . . . . . . . . . 335
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12.5 The Taylor Expansion of Dual Analytic Function on the Time Scales . . .
12.6 Dual Derivative Mapping on the Time Scales . . . . . . . . . . . . . . . . .
336
337
13 Serret–Frenet Frame of a Curve Parametrized by Time Scales: A Brief
Survey
341
Hatice Kusak Samanci
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
13.2 The Discrete Frenet Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
13.3 The Frenet Frame of a Curve Parametrized by Time Scales . . . . . . . . . 342
14 Applications of Time Scales in Nature: A Brief Survey
349
Jervin Zen Lobo, Svetlin G. Georgiev, and Sanket Tikare
14.1 Motivating Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
14.1.1 El Nin̄o Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
14.1.2 Growth of a Plant Species . . . . . . . . . . . . . . . . . . . . . . . . 349
14.1.3 Tumor Growth Model on Time Scales . . . . . . . . . . . . . . . . . 350
14.2 A COVID-19 Model on Time Scales . . . . . . . . . . . . . . . . . . . . . . 351
14.2.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . 351
14.2.2 A Nonautonomous Model for COVID-19 on Time Scales . . . . . . . 354
14.2.3 Endurance and Extinction of COVID-19 Infection . . . . . . . . . . 356
14.2.4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
14.3 Delayed Predator–Prey System on Time Scales . . . . . . . . . . . . . . . . 367
14.3.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . 367
14.3.2 Existence of Periodic Solutions . . . . . . . . . . . . . . . . . . . . . 369
14.3.3 An Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
14.4 Existence of Periodic Solutions for an Ecological Model on Time Scales . . 375
14.4.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . 375
14.4.2 Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
14.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
15 Applications of Time Scales in Economics: A Brief Survey
382
Jervin Zen Lobo, Svetlin G. Georgiev, and Sanket Tikare
15.1 HMMS Models on Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . 382
15.1.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . 382
15.1.2 Statement of HMMS Model . . . . . . . . . . . . . . . . . . . . . . . 384
15.1.3 Analysis of the HMMS Model . . . . . . . . . . . . . . . . . . . . . . 385
15.2 Dynamic Optimization Problems on Multiple Time Scales . . . . . . . . . . 387
15.2.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . 387
15.2.2 Dynamic Maximization Utility Problem . . . . . . . . . . . . . . . . 388
15.2.3 Consumption Paths on Various Time Scales . . . . . . . . . . . . . . 389
15.3 Applications of Calculus of Variations in Behavioral Economics . . . . . . . 392
15.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
15.3.2 The Cake-Eating Problem . . . . . . . . . . . . . . . . . . . . . . . . 392
15.3.3 The Household Problem . . . . . . . . . . . . . . . . . . . . . . . . . 400
15.4 Qualitative Analysis of a Solow Model on Time Scales . . . . . . . . . . . . 404
15.4.1 The Solow Model on Time Scales . . . . . . . . . . . . . . . . . . . . 404
15.4.2 Improved Solow Model on Time Scales . . . . . . . . . . . . . . . . . 407
15.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
Index
415
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Preface
In the pursuit of understanding the fundamental mechanisms governing the natural world,
scientists and researchers have turned their attention to time scales as a crucial aspect of observation. Nature operates on a diverse range of time scales, from the minuscule oscillations
of subatomic particles to the grand celestial motions that span billions of years. Dynamic
equations on time scales offer a powerful framework to investigate these phenomena with a
unified approach.
Dynamic equations on time scales provide a groundbreaking mathematical tool that
merges the theories of continuous and discrete systems. Traditionally, applied science has
relied on differential equations to describe continuous processes and difference equations
for discrete events. However, many natural processes possess both continuous and discrete
characteristics, and conventional methods struggle to encapsulate their complexities adequately.
Time scales encompass an assortment of mathematical structures, including continuous
(real) time, discrete (integer) time, and more intricate hybrid time scales. By leveraging
the unifying nature of dynamic equations on time scales, researchers can overcome the
limitations of traditional approaches and develop a deeper understanding of the underlying
dynamics present in nature.
The application of dynamic equations on time scales has led to groundbreaking insights
in various fields:
1. Biological Systems: Studying biological processes requires a delicate balance between continuous physiological changes and discrete events, such as cell division or
gene expression. Dynamic equations on time scales offer valuable insights into the
behavior of biological systems, helping researchers comprehend intricate phenomena
like circadian rhythms, population dynamics, and epidemic spread.
2. Ecological Interactions: Ecological systems are governed by an interplay of continuous environmental factors and discrete ecological events. Time scale dynamic models
enable ecologists to analyze the stability of ecosystems, predator–prey relationships,
and the effects of climate change on biodiversity.
3. Quantum Mechanics: At the smallest scales of nature, quantum phenomena unfold
with discrete quantum jumps and continuous wave-like behaviors. Dynamic equations
on time scales offer a promising framework for understanding quantum systems with
a more holistic perspective.
4. Astrophysics and Cosmology: Time scales vary significantly in astrophysical phenomena, from the rapid evolution of stars to the gradual changes in the cosmos. The
application of dynamic equations on time scales allows astrophysicists to explore celestial dynamics, galaxy formation, and cosmic evolution.
As we delve deeper into the mysteries of nature, dynamic equations on time scales continue to be a driving force in shaping our understanding of the world around us. Researchers
across various disciplines are harnessing this powerful tool to unlock new insights and embrace the intricacies of natural processes. The topic of dynamic equations on time scales
ix
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x
has witnessed a tremendous growth in the recent time. The subject of interest has recently
been introduced at various level in the curricula of several universities. We believe that a
wide audience of specialists such as, mathematicians, physicists, engineers, biologists, and
economists should find this book immensely useful. The idea of this book was discussed by
the editors in the International Conference on Nonlinear Analysis and Applications (ICNAA 2022) which was held at Assam Don Bosco University, Sonapur Campus, Assam, India
during November, 22–23, 2022. Motivated by recent increased activity of research on time
scales and closed related areas, it was decided to come up with a book that pulls together
under one cover, the excellent contributions focusing on the contemporary trends in the
field.
In this book, we delve deeper into some interesting aspects of the subject and its applications in the day-to-day world. The key topics include Functional dynamic equations,
Control theory on time scales, almost and pseudo almost periodic solutions, Discrete fractional equations, diamond-alpha inequalities, Geometric properties of dual space on time
scales, survey on applications on time scales in nature and in economics. The book consists
of 15 original research chapters reflecting the advances in subject of dynamic equations
on time scales and their applications in allied areas. They are written by well-recognized
researchers in the field of dynamic equations with associated applications.
In Chapter 1, the basic concepts, definitions, and results of time scales calculus are
discussed. This material will be used in subsequent chapters. A brief historical note and
objective of the subject are covered.
Chapter 2 is devoted to the study of the first-order functional dynamic equations on
time scales. First, the authors formulate basic concepts about first-order functional dynamic
equations on time scales. Then, the existence and uniqueness theorems for such equations
are proved. After that, continuous dependence of the solutions on initial data is investigated.
The other sections are devoted to uniform stability, uniformly asymptotical stability, global
stability, asymptotic stability and exponential stability of the solutions of some classes of
first-order functional dynamic equations on time scales. Different criteria for oscillations of
the solutions of first-order functional dynamic equations with several delays are also given.
Chapter 3 provides a survey of the literature for control theory topics on time scales.
This survey includes theory and applications such as controllability and observability of
systems in time scales, linear optimal control, the Kalman filter, and nonlinear control.
The topics of stability analysis, optimization, and stochastic time scales are also covered.
Numerous examples from the literature are included.
Chapter 4 investigates the existence of solutions to two types of optimal control problems
in time scales, which are governed by nonlinear first-order dynamic equations on time scales.
Chapter 5 aims to introduce the concepts of exact controllability, approximate controllability and approximate controllability on free times on time scales. The characterizations
of the controllability for linear systems and sufficient conditions for the study of the controllability of semilinear systems which are obtained perturbing the linear system under a
nonlinear term are presented. Further, the controllability of systems governed by dynamic
equation with memory is also studied.
Chapter 6 makes use of the measure theory on time scales to introduce doubly weighted
Stepanov-like pseudo almost periodic functions on time scales and investigate their important properties, including the composition theorem and invariant by translation. Further,
the existence and uniqueness results are obtained for doubly weighted pseudo almost periodic solutions of a class of delayed nonautonomous dynamic equations, including doubly
weighted Stepanov-like pseudo almost periodic forcing components. A numerical example
with simulation is included to demonstrate the viability and efficacy of the abstract results.
Chapter 7 investigates the existence and uniqueness implications of the doubly weighted
pseudo almost automorphic solution for the neutral functional integro-dynamic system on
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Preface
xi
time scales using the fixed point hypothesis. The global stability of the solution is also
obtained. A numerical example is included to demonstrate the suitability of the fictitious
results.
Chapter 8 deals with the construction of Green’s functions associated with the two
classes of nabla fractional terminal value problems. Both Riemann–Liouville and Caputo’s
nabla fractional differences are used. A few of the essential properties are derived, which
are required to establish sufficient conditions for the existence of solutions to these nabla
fractional terminal value problems.
Chapter 9 discusses the novel diamond alpha Hardy–Copson type inequalities. These
inequalities are established using a new technique which do not use the integration by parts
or fundamental theorems of calculus.
Chapter 10 also discusses the novel diamond alpha Hardy–Copson type inequalities.
The approach is similar to those in Chapter 9. Further, the complementary diamond alpha
Hardy–Copson type inequalities by choosing the exponent as ζ < 0 are also derived.
Chapter 11 systematically investigates the existence of positive almost periodic solutions
of a fishing model with feedback control on time scales. The uniform asymptotic stability of
the considered model is demonstrated with the help of an appropriate Lyapunov functional.
A numerical example with simulation is included to demonstrate the viability and efficacy
of the derived results.
Chapter 12 introduces new notion of differential geometry on time scales. The following
geometric properties of dual spaces on time scathe are studied. Derivatives of dual analytical
functions, the dual Taylor formula, the dual tangent vector, derivative by dual direction,
dual derivative mapping, vector field in dual space.
Chapter 13 is devoted to obtain a partial solution regarding the Serret–Frenet frame
of the curves in the time scale, which is still an open problem. Depending on the frame
obtained, the curvature and torsion of the curve has been calculated and some geometric
comments have been obtained.
Chapter 14 surveys several applications of the time scales theory to biology and population dynamics, which also includes COVID-19 model. The conditions for permanence
and extinction of an infection for a nonautonomous epidemic model defined on an arbitrary
time scale are studied. This chapter also discusses the existence of periodic solutions of certain systems like the (i) delayed periodic predator–prey dynamic systems; (ii) Kolmogorov
predator–prey dynamic systems; and (iii) a stage-structure ecological model on time scales.
Suitable examples at the end of each system are provided to illustrate the main results.
Chapter 15 surveys several applications of the time scales theory in Economics. The
optimal production and inventory paths of HMMS-type models on time scale domains is
investigated. This chapter also covers the study of a perfect-foresight utility maximization
problem for a finitely–lived consumer while making very general assumptions about the
timing of consumption and investing decisions. Further, employing the theory of calculus
of variations on time scales, two interesting problems, namely, the household problem and
the cake-eating problem are addressed. The qualitative analysis of a Solow model on time
scales is discussed.
Each chapter of the current book has been carefully peer-reviewed by the team of experts.
The proof-reading has been done by all the authors and the editors.
Finally, the editors wish to express their appreciation to Dr. Jervin Zen Lobo, St. Xavier’s
College, Goa, India for a careful reading of the final manuscript.
Ravi P. Agarwal, Texas A&M University-Kingsville, USA
Bipan Hazarika, Gauhati University, India
Sanket Tikare, Ramniranjan Jhunjhunwala College, India
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Editors
Ravi P. Agarwal
In the past 50 years, Ravi P. Agarwal has been one of the most famous, influential, and
successful figures in the field of Mathematics. In academia, people typically find their place
either as good researchers, mentors, editors, or managers, Dr. has fulfilled his each of these
role. He is an excellent researcher, a very successful supervisor, founder of several esteemed
scientific journals, and a perfect administrator. He was born in Moradabad (India) on July
10, 1947. He earned his Master’s degree from Agra University in 1969, ranking 2nd among
over 500 in that year. In 1973, he earned his Ph.D. (Mathematics) at the Indian Institute
of Technology in Madras, India, which is one of the highest-ranking universities in India.
Since then, for the past 50 years, Dr. Agarwal has been actively involved in research and
pedagogical activities. He has made significant contributions in various research fields, including Numerical Analysis, Differential and Difference Equations, Inequalities, and Fixed
Point Theorems. He is an excellent scholar, a dedicated teacher, and a prolific researcher. He
has published 50 research monographs, and more than 1,800 publications (with almost 700
mathematicians all over the world) in prestigious national and international mathematics
journals. Dr. Agarwal has previously worked either as a full-time faculty member or as a
visiting professor and scientist in universities in several countries, including India, Germany,
Italy, Canada, Australia, USA, Singapore, and Japan. He has held several positions including
Visiting Professor, Visiting Scientist, and Professor at various universities in different parts
of the world. Specifically, from 1980 to 1981, he was awarded the most prestigious Alexander
Von Humboldt Foundation Fellowship to work with Prof. Dr. G. Hammerlin at the LudwigMaximilians Universitat, Munchen, Germany. From 1981 to 1982, he worked as a Visiting
Professor with Prof. Roberto Conti, at the Instituto Matematico, Firenze, Italy. He has
been ranked as a Highly Cited Researcher for 14 consecutive years, and has also been recognized as one of the “World’s Most Influential Scientific Minds” in 2014 and 2015 by Thomas
Reuters/Clarivate Analytics (world’s most prestigious scientific organization). In 2021, he
was listed among world’s top 2% Scientists by Stanford University (https://1drv.ms/x/s!Ag47qfFs0OhgFG7xpiw6uqkDeM?e=K0hNpf), his ranking is 6734 out of 186178. In 2022, Dr.
Agarwal has been ranked 20th among 100 Top Mathematics Scientists by Research.com
(https://research.com/scientists-rankings/mathematics). According to Google Scholar, Dr.
Agarwal has been cited more than 70,000 times. On MathSciNet, his work has been cited
more than 17,700 times by 7,400 scientists. Dr. Agarwal is the recipient of several notable
honors and awards, including Doctor Honoris Causa (2015 by the University of Constanta,
Romania), Professor Honoris Causa (2015 by the University of Cluj, Romania), Honorary
Doctorate (2017 by the University of Nis, Serbia), Doctor Honoris Causa (2017 by Plovdiv University, Bulgaria), Doctor Honoris Causa (2017 by Constantin Brancusi, University
of Targu-Jiu, Romania), Doctor Honoris Causa (2017 by University of Oradea, Romania,
2017), Doctor Honoris Causa (by 1 December 1918, University of Alba Lulia, Romania),
Honorary Doctorate (2019 by Istanbul Gelisim University, Turkey), and Doctor Honoris
Causa (2019 by Transilvania University of Brasov, Romania). He was also nominated as a
possible candidate for the Banco Bilbao Vizcaya Argentaria (BBVA) Foundation Frontiers
of Knowledge Award, an international award scheme recognizing significant contributions in
xiii
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the areas of scientific research and cultural creation. Many BBVA award winners are Nobel
laureates, including Stephen Hawkings. Florida Institute of Technology (USA) and King
Abdulaziz University have recently offered him a Distinguished University Professorship
of Mathematics. Dr. Agarwal has provided significant service to Texas A&M UniversityKingsville. Under his leadership during 2011–2019, the Department of Mathematics consistently progressed in education and preparation of students, and in new directions of
research. His overall impact to the university is considerable. Many scholars from different
countries, including China, Turkey, India, and Korea have come to work under his guidance.
The majority of these visiting post-doctoral scholars were sent to work under Dr. Agarwal
by their employing institutions for at least one year. Dr. Agarwal regularly presents his research at invited talks/colloquiums (over 100 institutions worldwide). He has been invited to
give plenary/keynote lectures at international conferences in the USA, Russia, Italy, India,
Portugal, Poland, Spain, Vietnam, China, Taiwan, Korea, Malaysia, Thailand, Romania,
Saudi Arabia, Germany, Canada, Singapore, Turkey, Ukraine, Greece, and Japan. He has
served as Editor/Honorary Editor or Associate Editor in over 40 Journals and has published
20 books as editor. Dr. Agarwal has also organized International Conferences. In summary
of these few inadequate paragraphs, Dr. Ravi P. Agarwal is a visionary scientist, educator,
mentor, organizer, and administrator who has contributed to the world through his long
service, dedication, and tireless efforts.
Bipan Hazarika is currently a Professor in the Department of Mathematics at Gauhati
University, Guwahati, Assam, India. Before joining Gauhati University he worked as a Professor at Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh, India from
2005 to 2018. He received his Ph.D. degree from Gauhati University, Guwahati, India. His
main research interests includes the field of sequences spaces, summability theory, applications fixed point theory, fuzzy analysis, function spaces of nonabsolute integrable functions.
He has published over 200 research papers in several international journals. He is regular
reviewer of more than 50 different journals published from Springer, Elsevier, Taylor and
Francis, Wiley, IOS Press, World Scientific, American Mathematical Society, De Gruyter,
etc. He has published books on Differential Equations, Differential Calculus and Integral
Calculus. He is an editorial board member of more than five International Journals and
Guest Editor of special the issue “Sequence spaces, Function spaces and Approximation
Theory”, Journal of Function Spaces. He has edited five books on different topics of mathematics, out of them two are published by CRC Press and three are published by Springer.
Sanket Tikare is an Assistant Professor in the Department of Mathematics, at Ramniranjan Jhunjhunwala College, Mumbai, India, which is an empowered autonomous college
affiliated with the University of Mumbai, India. He did his M.Sc. and Ph.D. in Mathematics,
both from Shivaji University, Kolhapur, India in the years 2008 and 2012 respectively. He
is specialized in Applied Analysis. His research area of interest includes time scale theory,
dynamic equations on time scales, differential equations, difference equations, and fractional
equations. He has published several research papers in international journals. He has collaborated with several eminent researchers from various institutes around the globe. He is a
life member of the Indian Mathematical Society, a life member of the International Society
of Difference Equations. Sanket Tikare is a regular reviewers of many highly–reputed international journals. Moreover, he has written reviews on several articles for Mathematical
Reviews (AMS) and Zentralblatt Math (Germany). He has delivered invited talks at 20
international conferences. He has also delivered several contributed talks at various conferences in India and abroad.
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Contributors
Ravi P. Agarwal
Department of Mathematics
Texas A&M University
Kingsville, TX, USA
Tom Cuchta
Marshall University
Huntington, WV, USA
Soniya Dhama
Department of Mathematical Sciences
Rajiv Gandhi Institute of Petroleum
Technology
Jais Amethi, Uttar Pradesh, India
Sangeeta Dhawan
Department of Mathematics
Birla Institute of Technology and Science
Pilani
Hyderabad, Telangana, India
Cosme Duque
Departamento de Matemáticas
Universidad de Los Andes
Mérida, Venezuela
Mohssine Es-saiydy
Department of Mathematics
Moulay Ismaïl University
Meknès, Morocco
Svetlin G. Georgiev
Department of Mathematics
Sorbonne University
Paris, France
Bipan Hazarika
Department of Mathematics
Gauhati University
Guwahati, Assam, India
Jagan Mohan Jonnalagadda
Department of Mathematics
Birla Institute of Technology and Science
Pilani
Hyderabad, Telangana, India
Zeynep Kayar
Department of Mathematics
Van Yüzüncü Yıl University
Van, Turkey
Billur Kaymakçalan
Department of Mathematics
Çankaya University
Ankara, Turkey
Mahammad Khuddush
Department of Mathematics
Chegg India Pvt. Ltd.
Visakhapatnam, Andhra Pradesh, India
Hugo Leiva
University Yachay Tech
Imbabura, Ecuador
Jervin Zen Lobo
Department of Mathematics
St. Xavier’s College
Mapusa, Goa, India
Hatice Kusak Samanci
Department of Mathematics
Faculty of Sciences
Bitlis Eren University
Bitlis, Turkey
Iguer Luis Domini dos Santos
Departamento de Matemática
UNESP-Univ Estadual Paulista
lha Solteria, São Paulo, Brazil
Sanket Tikare
Department of Mathematics
Ramniranjan Jhunjhunwala College
Mumbai, Maharashtra, India
xv
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Nick Wintz
Department of Mathematics
Lindenwood University
St. Charles, MO, USA
Mohamed Zitane
Department of Mathematics
Moulay Ismaïl University
Meknès, Morocco
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Chapter 1
Elements of Time Scales Calculus
Ravi P. Agarwal, Bipan Hazarika, and Sanket Tikare
1.1
History and Objectives
Calculus has been separated into multiple distinct theories such as differential calculus,
difference calculus, quantum calculus, and many others. The central concept of all these
theories is ‘Change’, but in various contexts. Time scale calculus bridges the gap between
continuous and discrete calculus and expands both theories. The objective of establishing
the time scale calculus is not only to exclusively unify discrete and continuous processes
but also to treat more general dynamic systems involving both discrete and continuous
time elements. Traditionally, there are two separate dominated mathematical modelling:
differential equations and difference equations. Theory of differential equations models phenomena that are entirely continuous, whereas theory of difference equations works well for
phenomena that are entirely discrete. There are certain natural phenomena that are of
hybrid continuous-discrete behaviour. That is, they involve both continuous and discrete
data simultaneously, for example, in the population dynamics, the plant grows continuously
during the months of spring and summer, and they die at the beginning of autumn while
the seeds remain in the ground, and then the plant grows in a new season, giving rise to
a nonoverlapping population. However, when the behaviour is sometimes continuous and
sometimes discrete, both these models are deficient. The occurrence of both continuous and
discrete behaviour created the need for a different type of mathematical model. Therefore,
it is best to have a mathematical rigorous way that considers both processes simultaneously.
This is the motivation behind the concept of dynamic equations on time scales.
For the study of dynamic equations on time scales, first we have to develop calculus on time
scales. So, in this chapter, we will discuss the basic notions of time scale calculus which play
a foundational role in the subsequent chapters.
The notion of time scale (which is a special case of a measure chain) first appeared in
1988 in a Ph.D. thesis [11] of Stefan Hilger. The novel idea of time scale is rooted in
the fact that many results of differential equations and that of difference equations are
analogous and some of the results are totally different. The theory of time scales appropriately unifies continuous and discrete calculus into a single framework, called ‘time scales
calculus’.
At the beginning, the founder of the theory of time scales, Stefan Hilger, published
several interesting and thought-provoking research papers. Among them, [5, 6, 12–14] are
significant. During that period, very few researchers were working on time scales. But the
topic received significant attention to researchers around the globe, when the two monographs [7, 8] by Martin Bohner and Allan Peterson were published. So far, a great deal
DOI: 10.1201/9781003467908-1
1
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of work has been done in the field of time scales. This is mainly because of the following
reasons:
1. Unifies the discrete and continuous cases;
2. Unifies other different cases, depending on the time scale set;
3. Has applications in quantum calculus (when T = q Z ∪{0}, q > 1); economics, biological
modelling, and so on.
In fact, it is applicable in almost every field where hybrid modelling is required. Below we
consider three different dynamic models and demonstrate that how time scale theory is used
to unify and extend the continuous, discrete, and continuous-discrete dynamic processes.
Model 1: A radioactive material, such as the isotope thorium-234, disintegrates at a rate
proportional to the amount currently present. If Q(t) is the amount present at time t, then
dQ
= −rQ,
dt
(1.1)
where r > 0 is the decay rate.
Model 2: If y(0) rupees are invested at an annual interest rate of 7% compounded quarterly,
then y(t), the value of the investment after t quarters of a year is
y(t + 1) − y(t) = (0.0175)y(t)
(1.2)
for t = 0, 1, 2, . . ..
Model 3: Let N (t) be the number of plants of one particular species at time t in a certain
area. By experiments we know that N grows exponentially according to N = N during
the months of April until September. At the beginning of October, all plants die, but seeds
remain in the ground and start growing again at the beginning of April, with N now being
doubled. So we have the following model:
N (t) = N (t)
(1.3)
N (2k + 2) − N (2k + 1) = N (2k + 1)
(1.4)
for t ∈ [2k, 2k + 1) and
for k = 0, 1, 2, . . ..
From the above models, we see the following.
Here, from Table 1.1, we see that the domain of these three models are different. However,
they all have at least one thing in common. That is, they are closed subset of R. This
observation demonstrates the premise for time scale calculus.
TABLE 1.1:
Summary of all three
models.
Model
Model 1
Model 2
Model 3
Domain
The set of real numbers R
The set of whole numbers N0
∪∞
k=0 [2k, 2k + 1], k ∈ N0
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1.2
Time Scales
A time scale T is an arbitrary nonempty closed subset of R, with the subspace topology
inherited from the standard topology of R. For example, R, N, Z, q N0 , [0, 1] ∪ [2, 3], and the
Cantor set are time scales while Q, R \ Q, (0, 1) are not time scales.
Some of the concrete examples of time scale calculus are the followings.
(i) For T = R, we get ordinary differential calculus and Ordinary Differential Equations
(ODE) theory;
(ii) For T = Z, we get difference calculus and ordinary difference equations theory;
(iii) For T = q N0 , we get quantum calculus and q-difference equations theory;
(iv) For T = S: sequence of points of R with varying step size, we get numerical scheme
with varying step size;
(v) For T = P: sequence of closed intervals, we get differential equations with impulses.
Even by taking the Cantor set as a time scale T, one can develop the calculus on the Cantor
set, which is again an interesting theory whose applications are not so obvious.
Since time scale T is a subset of R, naturally one would ask: Are ‘Dynamic equations on
time scales’ a nontrivial generalization of ordinary differential equations? Surprisingly, the
answer is ‘no’. Discretization of ordinary differential equations (with constant or variable
step-size) lead to dynamic equations on discrete time scale. On the other hand, dynamic
equations on time scales can be represented as restrictions of ordinary differential equations
on nonempty closed subsets of R. But depending on how this representability is defined,
there are considerable difficulties (as we can see difficulties with the chain rule). One of the
possibilities of how to overcome such technical difficulties is to use an idea of embedding, the
idea that solutions of dynamic equations on T ⊂ R are projections of solutions of suitable
ordinary differential equations to T. This study is done in [1, 2, 9, 10]. While doing this
one should keep in mind that the resulting ordinary differential equations are not uniquely
determined.
The order structure and topological structure are very much essential to determine the whole
calculus. The time scale carries an order as well as topological structure in a canonical way.
Some of the consequences of the embedding of T in R and the closedness of time scale T
are given next.
1. Algebraic Structure: There is, in general, no algebraic structure on T.
2. Order Structure: A time scale T may be bounded above or below. All order theoretical notions such as bounds, least upper bound, greatest lower bound, and intervals
are available in time scale T as they are in R. Particularly, the supremum and infimum
of a bounded subset of time scale T may belong to T and not necessarily to the subset
itself.
3. Topological Structure: Since the time scale T is closed, the closedness in T coincides
with closedness in R. The same is true for compactness, i.e., the compactness in
T coincides with compactness in R. However, a problem arises with the notion of
openness. Any subset A of T which is open in R is also open in T. The reverse is not
true, in general. For example, take T = Z. Then any subset, in the induced topology,
is open in T but not open in R. We will take care of this by distinguishing between
R-openness and T-openness.
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4. The Order Topology The Hausdorff topology on T which is generated by open
intervals
(r, s) := {t ∈ T : r < t < s}, r, s ∈ T ∪ {±∞}
is known as order topology.
• For T = R, this is the usual Euclidean topology.
• For T = Z, the order topology reduces to the discrete topology where every
subset is open.
5. Connectedness It is well known that a closed set in R may not be connected. Therefore a time scale T may or may not be connected. But, to be able to properly define
the notion of derivative on time scale T, connectedness is very essential. To overcome
this topological deficiency, we introduce the concept of jump operators.
6. Jump Operators
The mapping σ : T → T defined by σ(t) := inf{s ∈ T : s > t} is called as the forward
jump operator and ρ : T → T defined by ρ(t) := sup{s ∈ T : s < t} is called as the
backward jump operator. To ensure the soundness of the above definitions, we adopt
the convention that σ(M ) := M if T has maximum M and ρ(m) := m if T has
minimum m. The forward jump operator σ results the next point in the time scale
while the backward jump operator ρ results the previous point in the time scale. We
note that σ is nondecreasing right continuous and ρ is nondecreasing left continuous.
Using these jump operators, we precisely classify the points of a time scale T as follows.
A point t ∈ T is called
• right-scattered (rs) if σ(t) > t;
• left-scattered (ls) if ρ(t) < t;
• right-dense (rd) if σ(t) = t < sup T;
• left-dense (ld) if ρ(t) = t > inf T;
• dense if σ(t) = t = ρ(t);
• an isolated if ρ(t) < t < σ(t).
Remark 1.2.1. Illustration of a natural classification of the points in a time scale T.
t1
•
•
•
t3
•
•
•
t2
t4
•
•
Here t1 is dense point (both left and right dense), t2 is left-dense and right-scattered point,
t3 is left-scattered and right-dense point, and t4 is an isolated (both left and right scattered)
point.
Example 1.2.1. If T = R, then for t ∈ R, we obtain
σ(t) := inf{s ∈ R : s > t} = inf(t, +∞) = t
for all
t ∈ T,
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Elements ofBy
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and
ρ(t) := sup{s ∈ R : s < t} = sup(−∞, t) = t
for all
t ∈ T.
Thus, σ(t) = t = ρ(t), that is, t is dense point.
Example 1.2.2. If T = Z, then, for t ∈ Z, we obtain
σ(t) := inf{s ∈ Z : s > t} = inf{t + 1, t + 2, . . .} = t + 1
for all
t ∈ T,
ρ(t) := sup{s ∈ Z : s < t} = sup{t − 1, t − 2, . . .} = t − 1
for all
t ∈ T.
σ(t) := inf{s ∈ hZ : s > t} = inf{t + h, t + 2h, . . .} = t + h
for all
t ∈ T,
ρ(t) := sup{s ∈ hZ : s < t} = sup{t − h, t − 2h, . . .} = t − h
for all
t ∈ T.
and
Thus, ρ(t) < t < σ(t), that is, t is an isolated point.
Example 1.2.3. If T = hZ (for h > 0), then
and
Thus, ρ(t) < t < σ(t), that is, t is an isolated point.
Example 1.2.4. If T = [0, 1] ∪ N, then we can see that
t
for t ∈ [0, 1),
σ(t) =
t+1
for t ∈ N
while
ρ(t) =
t
t−1
for t ∈ [0, 1],
for t ∈ N \ {1}.
Example 1.2.5. If T = [0, 1], then as per T = R, we obtain σ(t) = t = ρ(t). Note that
maxT = 1, so that σ(1) = 1 and minT = 0 so that ρ(0) = 0.
Example 1.2.6. (Elegant time scale) Let T = {0} ∪ {1, 21 , 13 , . . .}. Then we have the following.
• At the point t = 0,
σ(0) = inf{s ∈ T : s > 0} = inf{1, 1/2, 1/3, . . .} = 0
and
ρ(0) = sup{s ∈ T : s < 0} = sup ∅ = 0.
• Since maxT = 1, we obtain σ(1) = 1 and for t ∈ {1/n}∞
n=2 ,
σ(t) = inf{1/2, , 1/3, . . . , 1/(n − 1)} =
t
.
1−t
• For t ∈ {1/n}∞
n=1 ,
ρ(t) = sup{1/(n + 1), 1/(n + 2), . . .} =
t
1
=
.
n+1
1+t
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⎧
⎪
⎨0
It is then clear that
t
σ(t) = 1−t
⎪
⎩
1
if t = 0,
if t ∈ {1/n}∞
n=2 ,
if t = 1
and similarly,
ρ(t) =
0
t
1+t
if t = 0,
if t ∈ {1, 1/2, 1/3, . . .} = {1/n}∞
n=1 .
Remark 1.2.2. The calculus on time scale is easier to handle in scattered points when it
comes to topological and analytical considerations while it is easier to handle in right-dense
points in an algebraic context. However, at points which are at the same time left-dense and
right-scattered (or right-dense and left-scattered), the combination of jump operators and
topology leads to additional difficulties.
The calculus on time scale includes the difference calculus as well as differential calculus.
But, unlike differential calculus, the difference calculus is not symmetric in time. So, we
break up the time symmetry at this point by positive time direction and negative time
direction. For this, we define the graininess functions accordingly as follows.
Definition 1.2.1. The forward graininess function μ : T → [0, ∞) is defined by
μ(t) := σ(t) − t.
Definition 1.2.2. The backward graininess function ν : T → [0, ∞) is defined by
ν(t) := t − ρ(t).
Example 1.2.7. For T = R, μ(t) = 0 = ν(t) for all t ∈ R.
For T = Z, μ(t) = 1 = ν(t) for all t ∈ Z. Here, for both R and Z, the graininess μ is
constant.
Remark 1.2.3. We notice that
1. the graininess function measure the distance between two consecutive elements of T.
2. the graininess function may not be constant and not even continuous.
Below we give examples of time scale with nonconstant graininess.
Example 1.2.8. Let q > 1 be a real number and consider the time scale
T := {q k : k ∈ Z} ∪ {0}.
Then, for t = q m ,
σ(t) = inf{q n : n ∈ [m + 1, ∞)} = q m+1 = qq m = qt
and σ(0) = 0. So, we obtain
μ(t) = (q − 1)t
for all
t ∈ T.
Here the graininess μ is a nonconstant function of t.
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Example 1.2.9. For T = {0} ∪ {1/n}∞
n=1 , we found that
⎧
⎪
if t = 0,
⎨0
t
σ(t) = 1−t
if t ∈ {1/n}∞
n=2 ,
⎪
⎩
1
if t = 1.
⎧
⎪
⎨0 2
Then
t
⎪ 1−t
μ(t) =
⎩
0
if t = 0,
if t ∈ {1/n}∞
n=2 ,
if t = 1.
Here the graininess μ is a nonconstant function of t.
Example 1.2.10. Let a, b > 0 be two real numbers and consider the time scale
Pa,b := ∪∞
k=0 [k(a + b), k(a + b) + a] .
Then
σ(t) =
and
if t ∈ ∪∞
k=0 [k(a + b), k(a + b) + a) ,
if t ∈ ∪∞
k=0 {k(a + b) + a}
t
t+b
μ(t) =
0
b
if t ∈ ∪∞
k=0 [k(a + b), k(a + b) + a),
if t ∈ ∪∞
k=0 {k(a + b) + a}.
Here the graininess μ is a nonconstant noncontinuous function of t.
Example 1.2.11. Some frequently used time scales are mentioned in the following table.
TABLE 1.2:
Some examples of
time scales.
T
μ(t)
σ(t)
ρ(t)
R
0
t
t
Z
1
t+1
t−1
hZ
h
t+h
t−h
N
(q − 1)t
qt
t
q
q
2N
N20
√
t
2 t+1
√
2t
( 2 + 1)2
√
t
2
( 2 − 1)2
Example 1.2.12. It is enough to recognize that, for connected points, the forward and
backward jump operators return the same element of the time scale T that was drawn from
the domain. However, for nonconnected points, the forward and backward jump operators
returns the next and previous elements of the time scale T, respectively. This motivates us
to say that the operators σ and ρ are, in general, not opposite of each other. Given a time
scale T, below we give the conditions which guarantee σ(ρ(t)) = t and ρ(σ(t)) = t.
1. If t is an isolated, then σ(ρ(t)) = t and ρ(σ(t)) = t.
2. If t is right-scattered and left-dense, then σ(ρ(t)) = σ(t) = t and ρ(σ(t)) = t.
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3. If t is right-dense and left-scattered, then σ(ρ(t)) = t and
ρ(σ(t)) = ρ(t) = t.
4. If t is dense, then σ(ρ(t)) = σ(t) = t and ρ(σ(t)) = ρ(t) = t.
Remark 1.2.4. Notice, in the definitions of σ and ρ, that both σ(t) and ρ(t) are in T when
t ∈ T. This is due to fact that T is a closed subset of R.
The principle of mathematical induction is a basic tool which is used in many proofs.
Below we state the induction principle for time scales and its dual version.
Theorem 1.2.1. (Induction Principle) Let t0 ∈ T and assume that
{A(t) : t ∈ [t0 , ∞)T }
is a family of statements satisfying
(i) A(t0 ) is true.
(ii) If t ∈ [t0 , ∞)T is right-scattered and A(t) is true, then A(σ(t)) is true.
(iii) If t ∈ [t0 , ∞)T is right-dense and A(t) is true, then there exists a neighbourhood N of
t such that A(s) is true for all s ∈ N ∩ (t, ∞)T .
(iv) If t ∈ (t0 , ∞)T is left-dense and A(s) is true for s ∈ [t0 , t)T , then A(t) is true.
Then A(t) is true for all t ∈ [t0 , ∞)T .
Remark 1.2.5. For T = N, the conditions (iii) and (iv) disappear and Theorem 1.2.1
becomes the well-known principle of mathematical induction.
Theorem 1.2.2. (Dual Version of Induction Principle) Let t0 ∈ T and assume that
{A(t) : t ∈ (−∞, t0 ]T }
is a family of statements satisfying
(i) A(t0 ) is true.
(ii) If t ∈ (−∞, t0 ]T is left-scattered and A(t) is true, then A(ρ(t)) is true.
(iii) If t ∈ (−∞, t0 ]T is left-dense and A(t) is true, then there exists a neighbourhood N of
t such that A(s) is true for all s ∈ N ∩ (−∞, t)T .
(iv) If t ∈ (−∞, t0 )T is right-dense and A(s) is true for s ∈ [t0 , t)T , then A(t) is true.
Then A(t) is true for all t ∈ (−∞, t0 ]T .
1.3
Delta Differentiation
We are aware that differentiation is a kind of linear approximation and the topological
structure plays a vital role in defining the notion of derivative. Also, since time scales lack
in openness, we require additional tools to pursue differentiation on time scale. In this
process, the maximum element which is left-scattered in a time scale T plays significant role
in several aspects. Such a points are called degenerate. All other points of time scale T are
then called nondegenerate. These points play an extraordinary role in the development of
calculus on time scales. To identify the nondegenerate points of T, we define the κ-operator
are follows.
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Definition 1.3.1. For a time scale T, we define the new set Tκ as
Tκ := {t ∈ T : t is nonmaximum or left-dense}.
In more generality, we let
T :=
κ
T \ (ρ(sup T), sup T]
T
if sup T < ∞,
if sup T = ∞.
The κ-operator cuts off an eventually existing isolated maximum of T.
To save on notation, we introduce the following function.
Definition 1.3.2. If f : T → R, then we define f σ : T → R by f σ (t) := f (σ(t)) for all
t ∈ T. That is, f σ = f ◦ σ.
Definition 1.3.3. For intervals in a time scale we will use the following notation. Let
a, b ∈ T. We define
[a, b]T := [a, b] ∩ T
[a, b)T := [a, b) ∩ T
and so on. A neighbourhood of a point t0 ∈ T is an interval
Nδ := (t0 − δ, t0 + δ)T , δ > 0.
We now introduce the notion of derivative on time scale.
Definition 1.3.4. Assume that f : T → R and let t ∈ Tκ . Define f Δ (t) to be the number
(if it exists) with the following property: for given ε > 0 there exists a neighbourhood Nδ of
t such that
(1.5)
|[f σ (t) − f (s)] − f Δ (t)[σ(t) − s]| ≤ ε|σ(t) − s| for all s ∈ N.
We say that f is delta differentiable on Tκ provided that f Δ (t) exists for all t ∈ Tκ . The
function f Δ : Tκ → R is called the ‘delta derivative’ of f on Tκ .
Remark 1.3.1. For time scales R and Z, the delta derivative f Δ (t) is uniquely determined.
Δ
In fact, for R, f Δ (t) = df
dt (t) and for Z, f (t) = f (t + 1) − f (t). Also, for an arbitrary time
Δ
scale T, the delta derivative f (t) is uniquely determined at any nondegenerate point t in
time scale T. But, if t ∈ T is a degenerate point (left-scattered maximum), then any α ∈ R
will be the delta derivative of f at t. Thus, the delta derivative is unique everywhere except
at left-scattered maximum on time scale T. This can be seen in the following example.
Example 1.3.1. Suppose T is a time scale such that sup T < ∞ and f Δ (t) is defined at
point t ∈ T \ Tκ . Then the unique point t ∈ T \ Tκ is sup T. Hence, for any ε > 0, there is
a neighbourhood N = {t} of t such that
f (σ(t)) = f (s) = f (σ(sup T)) = f (sup T),
s ∈ N.
Thus, for each α ∈ R and for all s ∈ N , we have
|f (σ(t)) − f (s) − α(σ(t) − s)| = |f (sup T) − f (sup T) − α(sup T − sup T)|
≤ ε|σ(t) − s|.
That is, each α ∈ R is the delta derivative of f if t ∈
/ Tκ .
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Example 1.3.2. Let f (t) = t2 , t ∈ T. Then f Δ (t) = t + σ(t), t ∈ Tκ .
From (1.5), we consider
|[f σ (t) − f (s)] − f Δ (t)[σ(t) − s]| = |(σ(t))2 − s2 − (t + σ(t))(σ(t) − s)|
= |(s − t)(σ(t) − s)|.
(1.6)
Now, given ε > 0, we construct a neighbourhood N of t in the following way. Let N :=
(t − δ, t + δ)T and choose δ(ε) := ε. Thus, for all s ∈ N , we have |s − t| ≤ ε. Hence we can
make (1.6) smaller than ε|σ(t) − s| and the result follows.
Theorem 1.3.1. (Properties of Delta Derivative) Let f : T → R be a function and t ∈ Tκ .
Then we have the following.
1. If f is delta differentiable at t, then f is continuous at t.
2. If f is continuous at right-scattered point t, then f is delta differentiable at t with
f Δ (t) =
f (σ(t)) − f (t)
.
σ(t) − t
(1.7)
3. If f is right-dense, then f is delta differentiable at t if and only if,
lim
s→t
f (t) − f (s)
t−s
exists and is finite. In this case
f (t) − f (s)
.
t−s
(1.8)
f (σ(t)) = f (t) + μ(t)f Δ (t).
(1.9)
f Δ (t) = lim
s→t
4. If f is delta differentiable at t, then
This is known as the ‘simple useful formula’ (suf ).
Example 1.3.3. Let T = {n/2 : n ∈ N} and f : T → R be defined by f (t) := t2 . Then f is
continuous at all points in T. Also, all points in T are isolated. In particular, all points are
right-scattered. We also have
σ(t) = t +
1
1
and μ(t) = σ(t) − t = .
2
2
By (1.7), we have
f Δ (t) = (t2 )Δ
f (σ(t)) − f (t)
σ(t) − t
(t + 1/2)2 − t2
=
1/2
1
= 2t + .
2
=
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√
√
Example 1.3.4. Let T = { n : n ∈ N0 } = {0, 1, 2, . . .} and f : T → R be defined√by
f (t) := t2 . Then all points in T are right-scattered. Let us first find σ(t). For t = n,
n = t2 and so
σ(t) = inf{s ∈ T : s > t}
√
√
= inf{ n + 1, n + 2, . . .}
√
= n+1
= t2 + 1
and then μ(t) =
√
t2 + 1 − t. Thus, by (1.7), we have
f Δ (t) = (t2 )Δ
f (σ(t)) − f (t)
σ(t) − t
√
( t2 + 1)2 − t2
= √
t2 + 1 − t
= t + t2 + 1.
=
√
Example 1.3.5. Let T = { 3 n : n ∈ N0 } and f : T → R √
be defined by f (t) := √
t3 . Then all
3 3
points in this time scale are right-scattered with σ(t) = t + 1 and μ(t) = 3 t3 + 1 − t.
Thus, by (1.7), we have
f Δ (t) = (t3 )Δ
√
( 3 t3 + 1)3 − t3
= √
3 3
t +1−t
1
= √
3 3
t +1−t
3
3
= (t + 1)2/3 + t t3 + 1 + t2 .
n
Example 1.3.6. Let T = {tn = (1/2)2 : n ∈ N0 } ∪ {0, −1}. We notice that σ(0) = 0, i.e.,
0 is right-dense point in T. Also, σ(tn ) → 0 as n → ∞, and hence lims→0 σ(s) = σ(0).
Therefore, σ is continuous at 0. Next, by using (1.8), we have
√
σ(σ(0)) − σ(s)
σ(s)
s
1
= lim
= lim
= lim √ = ∞.
lim
s→0
s→0
s→0
s→0
σ(0) − s
s
s
s
Hence the jump operator σ is not delta differentiable at 0.
The next theorem shows that delta differentiation is linear and fulfils slightly modified
product and quotient rules.
Theorem 1.3.2. (Delta Differentiation Rules) Assume that f, g : T → R are both delta
differentiable at t ∈ Tκ . Then we have the following.
1. The sum f + g : T → R is delta differentiable at t and
(f + g)Δ (t) = f Δ (t) + g Δ (t).
2. For α ∈ R, the function αf : T → R is delta differentiable at t and (αf )Δ (t) = αf Δ (t).
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3. The product f g : T → R is delta differentiable at t and
(f g)Δ (t) = f (t)g Δ (t) + f Δ (t)g(σ(t)) = f Δ (t)g(t) + f (σ(t))g Δ (t).
4. If g(t)g(σ(t)) = 0, then the quotient fg : T → R is delta differentiable at t, and
f
g
Δ
(t) =
f Δ (t)g(t) − f (t)g Δ (t)
.
g(t)g(σ(t))
Example 1.3.7. Let T be an arbitrary time scale and the function f : T → R defined by
f (t) = t2 . Then f (t) = g(t)h(t), where g(t) = t = h(t). Since g Δ (t) = 1 = hΔ (t) and
h(σ(t)) = σ(t), using the product rule, we have
f Δ (t) = (g(t)h(t))Δ
= g(t)hΔ (t) + g Δ (t)h(σ(t))
= (t · 1) + (1 · σ(t))
= t + σ(t).
Thus, (t2 )Δ = t + σ(t).
Remark 1.3.2. In fact, for any delta differentiable function f , we have
(f 2 )Δ = (f · f )Δ = (f + f σ )f Δ .
Example 1.3.8. Let T be an arbitrary time scale and f (t) = 1/t. Then, we write f (t) =
g(t)
h(t) , where g(t) = 1 and h(t) = t. Using the quotient rule, we have
f Δ (t) =
g
h
Δ
(t)
g Δ (t)h(t) − g(t)hΔ (t)
h(t)h(σ(t))
(0 · t) − (1 · 1)
=
tσ(t)
1
.
=−
tσ(t)
=
Thus,
1 Δ
1
= − tσ(t)
.
t
Remark 1.3.3. In fact, for any delta differentiable function f , we have
Δ
1
fΔ
= − σ.
f
ff
Remark 1.3.4. If both f and g are twice delta differentiable at t ∈ Tκ , then the product
f g may not be twice delta differentiable at t. For example, the identity function f (t) = t is
twice delta differentiable at 0 but f (t)f (t) = t · t = t2 is not twice delta differentiable at 0.
Because (t2 )Δ = t + σ(t) and the jump function σ is not always delta differentiable. This
can be seen from Example 1.3.6.
Theorem 1.3.3. Let f : T → R be delta differentiable at t ∈ Tκ . Then
(f n+1 )Δ (t) = f Δ (t)
n
f k (t)(f σ )n−k (t).
k=0
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σ
(t)+f (t)
Δ
(t) for f (t) = 0, t ∈ T.
Theorem 1.3.4. For f : T → R, we have |f (t)|Δ = |ff(t)|+|f
σ (t)| f
Theorem 1.3.5. Let f : T → R be a nonzero delta differentiable function. Then
f (t)
f Δ (t)
if t ∈ {t ∈ T : f (t)f σ (t) ≥ 0},
Δ
|f (t)| = |f (t)|
(t) Δ
2
|f (t)| − |ff (t)|
f (t) if t ∈ {t ∈ T : f (t)f σ (t) < 0}.
− μ(t)
In particular,
f Δ
f
f (t) ≤ |f |Δ (t) ≤ − f Δ (t) for all t ∈ {t ∈ T : f (t)f σ (t) < 0}.
|f |
|f |
Remark 1.3.5. If f f σ = 0, then
f Δ
fσ
f (t) ≤ |f |Δ (t) ≤ σ f Δ (t) for t ∈ T.
|f |
|f |
Theorem 1.3.6. (Delta Derivatives of Power Functions) Let α ∈ R be a constant and
m ∈ N. Then we have the following.
1. If f (t) = (t − a)m , then
f Δ (t) =
m−1
(t − a)k (σ(t) − a)m−1−k .
k=0
2. If g(t) = (t − a)−m , then
g Δ (t) = −
m−1
k=0
1
.
(t − a)m−k (σ(t) − a)k+1
We now define higher-order delta derivatives in the natural way.
Definition 1.3.5. (Higher-Order Delta Derivatives) For a given function f : T → R, we
define the second-order delta derivative of f , denoted by f ΔΔ , provided that f Δ is delta
differentiable on Tκ , by
2
2
f ΔΔ := (f Δ )Δ : Tκ → R, where Tκ := (Tκ )κ .
n
n
Similarly, we define higher-order delta derivatives f Δ : Tκ → R.
Remark 1.3.6. For t ∈ T, we denote σ 2 (t) = σ(σ(t)) and ρ2 (t) = ρ(ρ(t)), and in general,
for n ∈ N, σ n (t) = σ(σ n−1 (t)) and ρn (t) = ρ(ρn−1 (t)). Also, we set σ 0 (t) = ρ0 (t) = t,
0
0
f Δ = f , and Tκ = T.
Remark 1.3.7. In computing the higher-order delta derivatives, care is needed. Because a
function f may be once delta differentiable but not twice delta differentiable at t ∈ T. For
example, f (t) = t2 , t ∈ T. In particular as
σ
(f g)ΔΔ = (f Δ g + f σ g Δ )Δ = f ΔΔ g + f Δ g Δ + f σΔ g Δ + f σσ g ΔΔ .
In computing the nth delta derivative of product of two functions f, g : T → R, certain
conditions are essentially required. This is given in the following theorem, called the Leibniz
formula.
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Theorem 1.3.7. (The Leibniz Formula) Let Sk be the set consisting of all possible strings
(n)
of length n, containing exactly k times σ and n − k times Δ. If f Λ exists for all Λ ∈ Sk ,
then
⎛
⎞
n
⎜ n
⎟ k
(f g)Δ =
f Λ ⎠ gΔ .
⎝
k=0
(n)
Λ∈Sk
Theorem 1.3.8. (Mean Value Theorem) Let T be a time scale and a, b ∈ T, a < b. Suppose
f : [a, b]T → R be a continuous function such that f Δ exists at each point of [a, b)T . Then
there exist α1 , α2 ∈ [a, b)T such that
f Δ (α1 )(b − a) ≤ f (b) − f (a) ≤ f Δ (α2 )(b − a).
Theorem 1.3.9. Let T be a time scale and a, b ∈ T, a < b. Suppose
f : [a, b]T → R is a continuous function such that f Δ exists at each point of [a, b)T . If
f Δ (t) = 0 for all t ∈ [a, b)T , then f is constant on [a, b]T .
Theorem 1.3.10. Let f : T → R be such that f Δ ≥ 0. Then f is nondecreasing on T.
The chain rule from the ordinary calculus states that: If f, g : R → R are such that g is
differentiable at t and f is differentiable at g(t), then
(f ◦ g) (t) = f (g(t))g (t).
(1.10)
Unfortunately, this rule does not hold for all time scales as seen through the following
example.
Example 1.3.9. Assume f, g : Z → Z are defined by f (t) := t2 and g(t) := 2t. Then we
see that
(f ◦ g)Δ (t) = 8t + 4 and f Δ (g(t))g Δ (t) = 8t + 2.
Thus,
(f ◦ g)Δ (t) = f Δ (g(t))g Δ (t)
for all t ∈ Z.
The following version of the chain rule is sometimes useful in time scale calculus.
Theorem 1.3.11. (Chain Rule) Assume that g : R → R is continuous, g : T → R is delta
differentiable on Tκ , and f : R → R is continuously differentiable. Then there exists c ∈
[t, σ(t)] with
(1.11)
(f ◦ g)Δ (t) = f (g(c))g Δ (t), t ∈ T.
Example 1.3.10. Let T = Z, f (t) := 2t2 +1, and g(t) := t2 . Then g : R → R is continuous,
g : T → R is delta differentiable on Tκ , and f : R → R be continuously differentiable, and
σ(t) = t + 1. Then g Δ (t) = 2t + 1. Taking t = 1, we see that [1, σ(1)] = [1, 2] and
f (g(c))g Δ (1) = 4g(c)(2(1) + 1)
= 12g(c)
= 12c2 .
Also,
(f ◦ g)(t) = f (g(t)) = 2(g(t))2 + 1 = 2(t2 )2 + 1 = 2t4 + 1
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and
(f ◦ g)Δ (t) = (f ◦ g)(t + 1) − (f ◦ g)(t)
= f (g(t + 1)) − f (g(t))
= f ((t + 1)2 ) − f (t2 )
= 2((t + 1)2 )2 + 1 − [2t4 + 1]
= 2(t + 1)4 − 2t4
= 2(4t3 + 6t2 + 4t + 1).
Δ
2
Thus,
(f ◦ g) (1) = 2(4 + 6 + 4 + 1) = 30. Hence, from (1.11), we get 30 = 12c and
c = 52 ∈ [1, 2].
The following chain rule is due to Christian Pötzsche which calculates (f ◦ g)Δ , where
f : R → R and g : T → R.
Theorem 1.3.12. (Chain Rule) Let f : R → R be continuously differentiable and suppose
g : T → R is delta differentiable on Tκ . Then f ◦ g : T → R is delta differentiable on Tκ and
1
f (g(t) + hμ(t)g Δ (t))dh g Δ (t)
(1.12)
(f ◦ g)Δ (t) =
0
holds for t ∈ T .
κ
Remark 1.3.8. Note that for T = R, the chain rule (1.12) is identical to the chain rule
known from ordinary calculus (1.10).
Example 1.3.11. Define g : Z → R and f : R → R by g(t) := t2 and
f (x) := exp(x), respectively. Then
g Δ (t) = (t + 1)2 − t2 = 2t + 1 and f (x) = exp(x).
Now
1
f (g(t) + hμ(t)g (t))dh g Δ (t)
0
Δ
1
2
=
exp(t + h(2t + 1))dh (2t + 1)
0
1
= (2t + 1) exp(t2 )
exp(h(2t + 1))dh
0
h=1
1
exp(h(2t + 1))
= (2t + 1) exp(t )
2t + 1
h=0
2
= exp(t2 )(exp(2t + 1) − 1)
and
(f ◦ g)Δ (t) = Δf (g(t))
= f (g(t + 1)) − f (g(t))
= exp((t + 1)2 ) − exp(t2 )
= exp(t2 + 2t + 1) − exp(t2 )
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= exp(t2 )(exp(2t + 1) − 1).
Thus,
1
(f ◦ g)Δ (t) =
f (g(t) + hμ(t)g Δ (t))dh g Δ (t)
0
which is (1.12).
Example 1.3.12. Let x : T → R be delta differentiable function. We shall compute [x2 (t)]Δ
by Theorem 1.3.12. Here we take g(t) = x(t) and f (x) = x2 . Then we have g Δ (t) = xΔ (t)
and f (t) = 2t. Therefore
[x2 (t)]Δ = (f ◦ g)Δ (t)
1
Δ
=
f (g(t) + hμ(t)g (t))dh g Δ (t)
0
1
Δ
2(x(t) + hμ(t)x (t))dh xΔ (t)
=
0
1
1
xΔ (t)
= 2 x(t)h + h2 μ(t)xΔ (t)
2
h=0
μ(t) Δ
Δ
= 2 x(t) +
x (t) x (t).
2
Comparing this with the product rule, we find that
[x(t)x(t)]Δ = (2x(t) + μ(t)xΔ (t))xΔ (t).
Thus, we can see that the two methods (chain rule and product rule) yield the same result.
The another version of chain rule which is used in deriving the formula for delta derivative
of inverse function and the substitution rule for delta integrals (which will be discussed
later).
Theorem 1.3.13. (Chain Rule) Assume that f : T → R is strictly increasing delta differentiable on Tκ and T̃ = f (T) is a time scale. Let g : T̃ → R be delta differentiable at f (t) ∈ T̃.
Then
(g ◦ f )Δ = (g Δ̃ ◦ f )f Δ ,
where Δ is represents the delta differentiation in T and Δ̃ is represents the delta differentiation in T̃.
Remark 1.3.9. Eventually, we observe that there is no natural chain rule for time scales
due to the fact that the functions under consideration are mapping between spaces of different
categories, i.e., domain is closed subset of R but the codomain or range may not be a closed
subset of R.
Theorem 1.3.14. (Delta Derivative of the Inverse) Assume that f : T → R is strictly
increasing delta differentiable at t ∈ Tκ such that f Δ (t) = 0 and T̃ = f (T) is a time scale.
Then
1
(f −1 )Δ̃ ◦ f (t) = Δ , t ∈ Tκ ,
f (t)
where Δ is represents the delta differentiation in T and Δ̃ is represents the delta differentiation in T̃.
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The following theorem is proved in [15] which is an essential tool in deriving the formula
for delta derivatives of some basic functions on arbitrary time scales.
Theorem 1.3.15. Let f : R → R be differentiable. If f : T → R is delta differentiable at
t ∈ Tκ , then
1
f Δ (t) =
f (t + τ μ(t))dτ,
(1.13)
0
where μ is the graininess function of time scale T.
In what follows, T is time scale and μ is its graininess function, c and k are constants,
and n ∈ N. The equalities given in examples 1.3.13 and 1.3.14 hold for t ∈ Tκ and can be
derived using (1.13).
Example 1.3.13. (Delta Derivative of Basic Functions)
1.
(k)Δ = 0
(1.14)
2. For k > 0,
(k t )Δ =
3.
((t + k)n )Δ =
k μ(t) − 1 t
k
μ(t)
(1.15)
n n
μ(t)n−i (t + k)i−1
i
−
1
i=1
(1.16)
Example 1.3.14. (Delta Derivative of Roots)
1.
√
Δ
( t) =
2.
√
( k + tn ) Δ =
3.
n
t
√
Δ
k + ct
t + μ(t) −
μ(t)
√
t
k + (t + μ(t))n −
μ(t)
(1.17)
√
k + tn
√
n
k + c(t + μ(t)) ( i=0 ni μ(t)n−i ti ) − k + ct tn
=
μ(t)
(1.18)
(1.19)
Example 1.3.15. (Delta Derivative of Logarithms)
1.
n
Δ
(ln(t )) =
n ln 1 + μ(t)
t
μ(t)
(1.20)
2.
(ln(kt + c))Δ =
ln 1 + kμ(t)
kt+c
μ(t)
(1.21)
Example 1.3.16. (Delta Derivative of Exponentials)
1.
(ekt )Δ =
ekμ(t) − 1 kμ(t)
e
μ(t)
(1.22)
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2.
n
i=0
n kt Δ
(t e ) =
n
n−i i
t
i μ(t)
μ(t)
ekμ(t) − tn kt
e
(1.23)
Example 1.3.17. (Delta Derivative of Trigonometric Functions)
1.
(sin t)Δ =
sin t (cos μ(t) − 1) + cos t sin μ(t)
μ(t)
(1.24)
(cos t)Δ =
cos t (cos μ(t) − 1) + sin t sin μ(t)
μ(t)
(1.25)
2.
Example 1.3.18. (Delta Derivative of Products of Trigonometric Functions and Monomials)
1.
(t sin(kt))Δ =
(t + μ(t))(sin(kt) cos(kμ(t)) + cos(kt) sin(kμ(t))) − t sin(kt)
μ(t)
(1.26)
(t cos(kt))Δ =
(t + μ(t))(cos(kt) cos(kμ(t)) − sin(kt) sin(kμ(t))) − t cos(kt)
μ(t)
(1.27)
2.
Example 1.3.19. (Delta Derivative of Products of Exponentials and Trigonometric Functions)
1.
(ekt sin(ct))Δ =
(cos(ct) sin(cμ(t)) + sin(ct) cos(cμ(t)))ekμ(t) − sin(ct) kt
e
μ(t)
(1.28)
(ekt cos(ct))Δ =
(cos(ct) cos(cμ(t)) − sin(ct) sin(cμ(t)))ekμ(t) − cos(ct) kt
e
μ(t)
(1.29)
2.
Example 1.3.20. (Delta Derivative of Hyperbolic Functions)
1.
(sinh(kt))Δ =
(ek(t+μ(t)) − ekt ) − (e−k(t+μ(t)) − e−kt )
2μ(t)
(1.30)
(cosh(kt))Δ =
(ek(t+μ(t)) − ekt ) + (e−k(t+μ(t)) − e−kt )
2μ(t)
(1.31)
(e2k(t+μ(t)) − e2kt ) − (e−2k(t+μ(t)) − e−2kt )
4μ(t)
(1.32)
2.
3.
(sinh(kt) cosh(kt))Δ =
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1.4
Delta Integration
To introduce the concept of integration in time scales, we need to define some new
function spaces, which are essential prerequisite to describe the types of functions that are
delta integrable.
Definition 1.4.1. A function f : T → R is called regulated provided
• at all right-dense point in T, the right-sided limit of f exists (and is finite);
• at all left-dense points in T, the left-sided limit of f exists (and is finite).
Let us see the following interesting example which shows that the jump function σ is,
in general, not continuous.
Example 1.4.1. Let T := tn = − n1 : n ∈ N ∪ N0 . That is,
1 1 1
T = . . . , −1, − , − , − , . . . , 0, 1, 2, 3, . . . .
2 3 4
Then, tn → 0 as n → ∞. We notice that σ(0) = 1 and
σ(tn ) = tn+1 = −
1
→ 0 = 1 = σ(0),
n+1
n → ∞.
Hence, lims→0 σ(s) = σ(0) and so σ is not continuous at 0. However, we note that the jump
function σ is continuous at right-dense points and lims→t− σ(s) exists at left-dense points
t ∈ T.
The above example of time scale motivates us to introduce the new concept of continuity,
called rd-continuity. The notion of rd-continuity is useful in obtaining an appropriate class
of functions having delta antiderivatives. In fact, the class of rd-continuous functions turns
out to be a ‘natural’ class within the context of calculus on time scales.
Definition 1.4.2. A function f : T → R is said to be rd-continuous if it is continuous at
every right-dense points in T and its left sided limits exist at left-dense points in T. The set
of rd-continuous function f : T → R is denoted by Crd (T, R).
On the same line, we can define the notion of ld-continuity.
Definition 1.4.3. A function f : T → R which is continuous at every left-dense points in
T and its right-sided limits exist at right-dense points in T is called ld-continuous function.
The set of ld-continuous function f : T → R is denoted by Cld (T, R).
Example 1.4.2. For T = [0, 1] ∪ N, the graininess function μ : T → R is rd-continuous but
not continuous at 1.
Remark 1.4.1. In general, if f : T → R is continuous, then f : T → R is rd-continuous.
If T contains points which are either left-dense and right-scattered or right-dense and leftscattered, then f : T → R may be rd-continuous but not continuous. If T contains points
which are either dense or isolated points, then f : T → R is continuous if and only if
f : T → R is rd-continuous. Thus, for the usual time scales T = R or T = Z, the notion of
rd-continuity coincides with that of continuity.
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The main interest in Crd functions is that they are delta integrable (as we will see
later in Theorem 1.4.4). In the following theorem, we list some useful results concerning
rd-continuous functions and regulated functions.
Theorem 1.4.1. Assume that f : T → R.
1. If f is continuous, then f is rd-continuous.
2. If f is rd-continuous, then f is regulated.
3. The jump operator σ is rd-continuous.
4. If f is regulated or rd-continuous, then so is f σ .
5. Assume f is continuous. If g : T → R is regulated or rd-continuous, then so is f ◦ g.
Another important concept that is required for defining the delta integral is given below.
Definition 1.4.4. A continuous function f : T → R is called delta pre-differentiable on T
(with region of differentiability D), if
• D ⊂ Tκ ,
• Tκ \ D is countable and contains no right-scattered points of T,
• f is delta differentiable on D.
That is, a continuous f : T → R is delta pre-differentiable on T if it is delta differentiable
everywhere with the exception of countable number of right-dense points.
Definition 1.4.5. A continuous f : T → R is called delta differentiable on T, if it is predifferentiable with D = Tκ .
Remark 1.4.2. If T has only isolated points, then D = Tκ . If T is an arbitrary interval in
R, then a function f : T → R which is delta differentiable on T, is delta differentiable in the
interior of T and delta differentiable with respect to one-side in end points belonging to T.
The main interest in the notion of delta pre-differentiability is that it assures the existence
of delta antiderivatives. This can be seen in Theorem 1.4.4.
The following theorem is analogous to the mean value theorem which is useful for establishing the existence of delta pre-antiderivatives and delta antiderivatives.
Theorem 1.4.2. (Mean Value Theorem) Let f, g : T → R be pre-differentiable with D. If
|f Δ (t)| ≤ g Δ (t) for all t ∈ D,
then
|f (s) − f (r)| ≤ g(s) − g(r) for all r, s ∈ T, r ≤ s.
Corollary 1.4.1. Let f, g be delta pre-differentiable with region D. Then we have the following.
1. If U is a compact interval with end points r, s ∈ T, then
|f (s) − f (r)| ≤ |s − r| sup |f Δ (t)|.
t∈U κ ∩D
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2. If f Δ (t) = 0 for all t ∈ D, then f is constant on D.
3. If f Δ (t) = g Δ (t) for all t ∈ D, then f (t) = g(t)+K for all t ∈ D, where K is constant.
Theorem 1.4.3. (Existence of Delta Pre-antiderivatives) If f : T → R is a regulated function, then there exists a function F which is delta pre-differentiable with region D such
that
F Δ (t) = f (t) hold for all t ∈ D.
Definition 1.4.6. (Delta Pre-antiderivative) Let f : T → R be a regulated function. Any
function F as in Theorem 1.4.3 is called a delta pre-antiderivative of f .
We define the indefinite delta integral of a regulated function f : T → R by
f (t)Δt = F (t) + C,
where C is a constant and F is a delta pre-antiderivative of f . We define the Cauchy delta
integral by
r
f (t)Δt = F (r) − F (s) for all r, s ∈ T.
s
Now, we define the concept of delta antiderivative as follows.
Definition 1.4.7. (Delta Antiderivative) Let f : T → R be a regulated function. A delta
differentiable function F : T → R is said to be delta antiderivative of f , if
F Δ (t) = f (t) hold for all t ∈ T.
From continuous
analysis, it is well-known result that the function F : T → R defined by
t
F (t) = τ f (s)Δs is differentiable at t ∈ R with derivative f (t), provided f is continuous at
t. Nevertheless, we already know that possibly F is not delta differentiable at the right-dense
points of Tκ \ D. Thus, in order to guarantee the delta differentiability of F , it is required to
assume the continuity of f only at right-dense points, that is, rd-continuity of f . We then
arrive at the conclusion that: Every rd-continuous function has delta antiderivative. This is
essentially given in the following theorem.
Theorem 1.4.4. (Existence of Delta Antiderivatives) If f : T → R is rd-continuous, then
f has delta antiderivative F : T → R defined by
t
F (t) :=
f (s)Δs t, t0 ∈ T with t0 ≤ t.
t0
That is,
t
Δ
F (t) =
Δ
f (s)Δs
= f (t)
t ∈ T.
t0
A summary of delta antiderivatives is given below in the table.
TABLE 1.3:
Summary of delta antiderivatives.
F is delta pre-antiderivative of f
F is delta antiderivative of f
Definition
f regulated & F Δ = f on D
f regulated & F Δ = f on Tκ
Existence
f regulated
f rd-continuous
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Theorem 1.4.5. If f : T → R is rd-continuous, then
σ(t)
f (s)Δs = μ(t)f (t), t ∈ Tκ .
t
Theorem 1.4.6. (Delta Integration Rules) Let f, g : T → R be rd-continuous functions and
a, b, c ∈ T, and α ∈ R. Then we have the following.
a
1. a f (t)Δt = 0;
b
a
2. a f (t)Δt = − b f (t)Δt;
b
b
b
3. a (f (t) + g(t))Δt = a f (t)Δt + a g(t)Δt;
b
b
4. a αf (t)Δt = α a f (t)Δt;
b
c
b
5. a f (t)Δt = a f (t)Δt + c f (t)Δt;
b
b
6. a f (σ(t))g Δ (t)Δt = f (b)g(b) − f (a)g(a) − a f Δ (t)g(t)Δt;
b
b
7. a f (t)g Δ (t)Δt = f (b)g(b) − f (a)g(a) − a f Δ (t)g(σ(t))Δt;
b
b
8. If |f (t)| ≤ g(t) on [a, b)T , then a f (t)Δt ≤ a g(t)Δt;
9. If f (t) ≥ 0 on [a, b)T , then
b
a
f (t)Δt ≥ 0.
Some special cases of the delta integral are now presented.
Theorem 1.4.7. Let f : T → R be an rd-continuous function and a, b ∈ T.
b
b
1. If T = R, then a f (t)Δt = a f (t)dt, where the integral on right side is the usual
Riemann integral from continuous calculus.
2. If T consists of only isolated points between a and b, then
⎧
⎪
if a < b,
b
⎨ t∈[a,b) μ(t)f (t)
f (t)Δt = 0
if a = b,
⎪
a
⎩ − t∈[b,a) μ(t)f (t)
if a > b.
3. If T = hZ := {hk : k ∈ Z}, where h > 0, then
⎧b/h−1
⎪
⎨ k=a/h f (hk)h
f (t)Δt = 0
⎪
a
⎩ a/h−1
− k=b/h f (hk)h
b
if a < b,
if a = b,
if a > b.
Example 1.4.3. Let a, b ∈ T and T be arbitrary time scale. Then
b
1Δs = b − a,
a
where we take f (t) = 1.
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TABLE 1.4:
Some properties of two most important time scales.
Time scale T
R
Z
Backward jump operator ρ(t)
t
t−1
Forward jump operator σ(t)
t
t+1
Graininess μ(t)
0
1
Derivative f Δ (t)
b
Integral a f (t)Δt
f (t)
b
a
rd-continuous f
Δf (t)
f (t)dt
b−1
t=a f (t) (if a < b)
continuous f
Example 1.4.4. Evaluate
any f
t
s Δs
0
for 0, t ∈ T. This is actually not a trivial delta integral. However, something we do know is
that (by the product rule) (t2 )Δ = t + σ(t) and hence
t
t
(s2 )Δ Δs =
(s + σ(s)) Δs
t
t
s Δs +
σ(s) Δs.
t 2 − 02 =
0
0
0
Then
t
0
t
s Δs = t2 −
0
(1.33)
σ(s) Δs.
0
Alternately, we could have used integration by parts. Now, let us investigate a number of
different time scales.
(i) Let T = R. For t ∈ R, σ(t) = t and hence, by (1.33)
t
t
s Δs = t2 −
0
s Δs =
0
1 2
t .
2
(ii) Let T = Z. For t ∈ Z, σ(t) = t + 1 and hence, by (1.33)
t
t
s Δs = t2 −
0
= t2 −
(s + 1) Δs
0
t−1
(s + 1)
s=0
= t2 −
t−1
s=0
= t2 −
t−1
s−
t−1
1
s=0
s1 − t
(here · represents the falling factorial power)
s=0
= t2 −
s=t
1 2
s
−t
2
s=0
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= t2 − t(t − 1) − t
2
t2
t
=
− .
2
2
(iii) Let T = hZ, h > 0. Then we have σ(t) = t + h and μ(t) = h and hence, by (1.33)
t
t
s Δs = t2 −
σ(s) Δs
0
0
t
2
(s + h)Δs
=t −
0
t/h−1
= t2 −
h(sh + h)
s=0
⎡
= t2 − h2 ⎣
t/h−1
s+
s=0
$
=t −h
2
2
= t2 − h2
1⎦
s=0
2 t/h
s
2
⎤
t/h−1
%
t/h
+ [s]0
0
t/h(t/h − 1)
t
+
.
2
h
This agrees with part (ii) when h = 1.
(iv) Let T = [0, 1] ∪ [2, 3]. Then for t ∈ T,
t
σ(t) =
t+1
if t ∈ [0, 1) ∪ [2, 3],
if t = 1.
Now, by using (1.33), for t ∈ [0, 1],
t
t
t
1
s Δs = t2 −
σ(s) Δs = t2 −
s Δs = t2
2
0
0
0
and for t ∈ [2, 3],
t
t
(s2 )Δ Δs =
2
which gives
(s + σ(s))Δs
2
t
t
s Δs = t2 − 4 −
2
s ds =
2
1 2
t − 2.
2
So, for t ∈ [0, 1] ∪ [2, 3], we have
1
2
t
t
s Δs =
s Δs +
s Δs +
s Δs
0
0
1
1
t
s Δs + 1 · 1 +
=
0
2
s Δs
2
1
1
+ 1 + t2 − 2
2
2
1 2 1
= t − .
2
2
=
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1.5
Dynamic Equations
We now have an understanding of the general time scale calculus. Next, we analyse
dynamic equations on time scales. Such equations are generalizations of
• ordinary differential equations and
• difference equations.
We are always interested in discovering properties of solutions to dynamic equations. In
this process, we try to find answers to several important questions, some of them are the
following.
• Do such equations have a solution?
• If so, then is the solution unique?
• How can we find the solution(s)?
• How can we approximate the solution?
• When is the solution stable?
• What other characteristics of the solution can we uncover?
Definition 1.5.1. Let D be subset of T and f : D × R2 → R. We call the equation
xΔ = f (t, x, xσ ),
t ∈ Tκ ,
(1.34)
a first-order dynamic equation.
We generally subject (1.34) to an initial condition such as
x(t0 ) = x0 ∈ R,
t0 ∈ T.
(1.35)
The pair (1.34)–(1.35) is known as the dynamic initial value problem (DIVP). A solution of
DIVP (1.34)–(1.35) is a function x : T → R which satisfies (1.34) and (1.35) and (t, x(t)) ∈
D.
Example 1.5.1. Consider the DIVP
xΔ = x, t ∈ Tκ ,
x(0) = 1, 0 ∈ T.
(1.36)
For T = R, the DIVP (1.36) becomes
x = x,
x(0) = 1,
which has the solution x = et .
For T = N0 , the DIVP (1.36) becomes
Δx(t) = x(t), t = 0, 1, 2, . . . ,
x(0) = 1,
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and we can solve this recursively to get
x(t + 1) = 2x(t)
x(1) = 2x(0) = 2
x(2) = 2x(1) = 4
.. ..
.=.
x(t) = 2t .
This motivates the question: What type of function solves (1.36) for an arbitrary time scale
T? To answer this question we identify the similarity of two functions, viz., et and 2t and
define the unique solution to (1.36) on T (with 0 ∈ T) by
t
1
log(1 + μ(τ ))Δτ , t ∈ T.
(1.37)
e(t, 0) := exp
0 μ(τ )
First, we see that how this solution compare with our solutions in T = R and T = N0 .
For T = R, μ ≡ 0. So letting μ → 0 in (1.37), we obtain
t
1
log(1 + μ)dτ
e(t, 0) = lim exp
μ→0
0 μ
t
log(1 + μ)
= exp
dτ
lim
μ→0
μ
$0
%
1
t
= exp
lim
0 μ→0
t
= exp
1+μ
1
dτ
1dτ
0
= exp(t)
= et .
Now, for T = N0 , μ ≡ 1. Therefore, from (1.37), we get
t
1
log(1 + 1)Δτ
e(t, 0) = exp
0 1
t
= exp
log(2)Δτ
0
$
%
t−1
= exp log(2)
1
τ =0
= exp[t log 2]
= elog 2 × elog 2 × . . . × elog 2 (t times)
= 2 × 2 × × . . . × 2 (t times)
= 2t .
Thus, the function e(t, 0) defined in (1.37) agrees with our solutions in T = R and T = N0 .
Now, we show that e(t, 0) solves DIVP (1.36) for an arbitrary T. We shall look at the rightscattered case only and show that [e(t, 0)]Δ = e(t, 0). Using Theorem 1.3.1, part 2, we can
write
[e(t, 0)]Δ
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=
=
e(σ(t), 0) − e(t, 0)
μ(t)
&
' &
'
σ(t) log(1+μ(τ ))
t
))
exp 0
Δτ − 0 log(1+μ(τ
Δτ
μ(τ )
μ(τ )
exp
=
&
t log(1+μ(τ ))
Δτ
μ(τ )
0
$
e(t, 0)
=
exp
μ(t)
$
σ(t)
t
μ(t)
'
&
'
&
'
σ(t) log(1+μ(τ ))
t
))
exp t
Δτ − exp 0 log(1+μ(τ
Δτ
μ(τ )
μ(τ )
μ(t)
%
%
log(1 + μ(τ ))
Δτ − 1 .
μ(τ )
(1.38)
By using Theorem 1.4.5, it follows that
$
%
σ(t)
log(1 + μ(τ ))
exp
Δτ = 1 + μ(τ ).
μ(τ )
t
Hence (1.38) becomes
[e(t, 0)]Δ =
e(t, 0)
[μ(t) + 1 − 1] = e(t, 0).
μ(t)
Thus, we conclude that e(t, 0) really solves DIVP (1.36).
Example 1.5.2. Consider the DIVP
xΔ = −x,
t ∈ Tκ ,
x(0) = x0 ,
0 ∈ T.
(1.39)
For T = R, the DIVP (1.39) becomes
x = −x,
x(0) = x0 ,
which has the solution x = x0 e−t .
However, for discrete time scale, notice that in contrast to differential equations, the solutions of difference equations are, in general, not uniquely determined in backward time
direction. In fact, for T = N0 , we see that DIVP (1.39) has infinitely many solutions, if
x0 = 0 and has no solution, if x0 = 0. This situation arises because of the jumps which
appear in the time scale N0 . Thus, when jumps exist in the time scale, the continuation of
solutions in the backward direction is either not possible in unique manner or not possible at
all. Hence we have to find a condition which guarantees the unique backward continuation
of solution of dynamic equation. This is necessitates the following concept.
Definition 1.5.2. (Regressive Function) A function p : T → R is said to be regressive if
1 + μ(t)p(t) = 0 for all t ∈ T.
The set of all regressive rd-continuous functions p : T → R is denoted by R(T, R) or R(T)
or R.
Definition 1.5.3. (Positively Regressive Function) A function p : T → R is said to be
positively regressive if 1 + μ(t)p(t) > 0 for all t ∈ T.
The set of all positively regressive rd-continuous functions p : T → R is denoted by R+ (T, R)
or R+ (T) or R+ .
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Remark 1.5.1. If p : T → R is an rd-continuous function such that p(t) ≥ 0 for all t ∈ T,
then p ∈ R+ .
Example 1.5.3. Let p : T → R be defined as p(t) := cos t. Since p is continuous, we have
p ∈ Crd (T, R).
If T = R, then for all t ∈ Tκ
1 + μ(t)p(t) = 1 + 0 · cos t = 1 = 0.
So, p ∈ R.
If T = Z, then for all t ∈ Tκ
1 + μ(t)p(t) = 1 + cos t
which equals 0 for t = ±π, ±3π, . . ., but these t ∈
/ Z. So, p ∈ R. In fact, p ∈ R+ .
Example 1.5.4. If p(t) = −t, then p is not regressive on Z.
Definition 1.5.4. For p, q ∈ R(T, R), we define
p ⊕ q := p + q + μpq,
p :=
−p
,
1 + μp
p q := p ⊕ (q).
We notice that the set R is an Abelian group under the ‘circle plus’ addition ⊕. This group,
(R, ⊕) is called the regressive group.
Definition 1.5.5. (Exponential Function) Let p ∈ R. Then the exponential function on T,
denoted by ep (t, t0 ), is defined by
'
&
⎧
t Log(1+μ(τ )p(τ ))
⎪
Δτ
, μ > 0,
exp
⎪
μ(τ
)
t
0
⎨
ep (t, t0 ) :=
&
'
⎪
⎪
⎩exp t p(τ )Δτ ,
t0
(1.40)
μ = 0,
where Log is the principal logarithm function.
Remark 1.5.2. In fact, if p ∈ R, then the first-order dynamic equation
xΔ = p(t)x,
t ∈ Tκ ,
is called regressive.
We will now consider the more general DIVP
xΔ = p(t)x, t ∈ Tκ ,
x(t0 ) = 1,
t0 ∈ T,
(1.41)
where p ∈ Crd (T, R). The unique solution to DIVP (1.41) is given by ep (t, t0 ).
Remark 1.5.3. Let p ∈ R be constant. Then we have the followings.
• If p ≥ 0, then ep (t, s) ≥ 1 for t ≥ s and s, t ∈ T.
• If T = Z, then ep (t, t0 ) = (1 + p)t−t0 for all t ∈ T.
• If T = R, then ep (t, t0 ) = ep(t−t0 ) for all t ∈ T.
From Definition 1.5.5 and Remark 1.5.3, we have the following result.
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Theorem 1.5.1. If p ∈ R, then
ep (σ(t), t) = 1 + μ(t)p(t)
for
t ∈ Tκ .
Theorem 1.5.2. (Properties of Exponential Function) Let p, q ∈ R be two regressive and
rd-continuous functions and t0 , t, s ∈ T. Then
1. e0 (t, t0 ) = 1 and ep (t, t) = 1;
2. ep (σ(t), t0 ) = (1 + μ(t))ep (t, t0 );
1
3. ep (t,t
= ep (t, t0 );
0)
4. ep (t, t0 ) = ep (t10 ,t) = ep (t0 , t);
5. ep (t, t0 )ep (t0 , s) = ep (t, s);
6. ep (t, t0 )eq (t, t0 ) = ep⊕q (t, t0 );
e (t,t )
7. epq (t,t00 ) = epq (t, t0 ).
Theorem 1.5.3. (Delta Derivative of Exponential Function) Let p, q ∈ R and t0 , t, s ∈ T.
Then
e (t,t )
p
0
1. eΔ
pq (t, t0 ) = (p(t) − q(t)) eσ (t,t0 ) ;
q
2. eΔ
p (t, t0 ) = p(t)ep (t, t0 );
3. (ep (t0 , t))Δ = −p(t)ep (t0 , σ(t)).
Theorem 1.5.4. Let p ∈ R and s, t, t0 ∈ T. Then
t
p(t)ep (t0 , σ(τ ))Δτ = ep (t0 , s) − ep (t0 , t).
s
Theorem 1.5.5. (Sign of Exponential Function) Assume that p ∈ R and t0 ∈ T.
1. If 1 + μp > 0 on Tκ , then ep (t, t0 ) > 0 for all t ∈ T;
2. If 1 + μp < 0 for some Tκ , then ep (t, t0 )ep (σ(t), t0 ) < 0;
3. If 1 + μp < 0 for all Tκ , then ep (t, t0 ) changes sign at every point t ∈ T.
Theorem 1.5.6. Let p ∈ R([0, ∞)T ). Then
t
|p(τ )|Δτ
ep (t, s) ≤ exp
for all s, t ∈ [0, ∞)T .
s
Example 1.5.5. Consider the following DIVP:
xΔ = ax, t ∈ Tκ ,
x(0) = 1,
0 ∈ T,
a < −1,
(1.42)
For T = Z, we see that the solution to DIVP (1.42) changes sign at every point in Z. That
is, for each n ∈ Z, x(n)x(n + 1) < 0.
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Here p(t) ≡ a and t0 = 0. So our solution (by definition) to DIVP (1.42) is
x(t) = ep (t, t0 ) = ea (t, 0),
and by Theorem 1.5.2,
x(σ(t)) = ep (σ(t), t0 ) = ea (σ(t), 0) = (1 + aμ(t))ea (t, 0).
Then
x(t)x(σ(t)) = ea (t, t0 )ea (σ(t), t0 )
= ea (t, t0 )(1 + aμ(t))ea (t, 0)
= [ea (t, 0)]2 (1 + aμ(t))
= [ea (t, 0)]2 (1 + a)
< 0 whenever a < −1.
For arbitrary time scale T, 1 + μ(t)a < 0, then solutions to DIVP (1.42) oscillate at each
point.
We now quickly check our solutions to DIVP (1.42) for R.
For T = R, the DIVP (1.42) becomes
x = ax, a < −1,
x(0) = 1, 0 ∈ T,
which has the solution x = eat . Note that there are no oscillations in these solutions.
Example 1.5.6. Let T = N0 and p(t) = α, a constant with α ∈ R. Then eα (t, 0) = (1+α)t .
To show this, we try to solve the corresponding DIVP
xΔ = αx,
(1.43)
x(0) = 1
recursively. Using the simple useful formula, we get
x(t + 1) = x(σ(t)) = x(t) + xΔ (t)
= x(t) + αx(t)
= (1 + α)x(t).
Hence
x(t) = (1 + α)t x(0) = (1 + α)t .
Thus, the solution to (1.43) is x(t) = (1 + α)t and as we know that the exponential function
eα (t, 0) is the unique solution, both these functions must be identical.
Example 1.5.7. Consider T := {n2 : n ∈ N0 }. We claim that
√ √
e1 (t, t0 ) = 2 t ( t)!.
(1.44)
We show that (1.44) satisfies the DIVP
xΔ = 1x,
x(0) = 1.
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Now, it is clear that x(0) = (20 )(0!) = 1. Notice that all the points in T are right-scattered,
so
x(σ(t)) − x(t)
xΔ (t) =
.
σ(t) − t
We need to find σ(t). Let t = n2 . Then
σ(t) = inf{s ∈ T : s > t}
σ(n2 ) = inf{s ∈ T : s > n2 }
= inf{(n + 1)2 , (n + 2)2 , . . .}
= (n + 1)2 .
√
So σ(t) = σ(n2 ) = (n + 1)2 = ( t + 1)2 . Then
x(σ(t)) − x(t)
σ(t) − t
√
√ √
2 σ(t) ( σ(t))! − 2 t ( t)!
=
σ(t) − t
√ √
( √
)
t
2 ( t)! 2( t + 1) − 1
√
=
1+2 t
√ √
= 2 t ( t)!
= x(t).
xΔ (t) =
Therefore, (1.44) has been proved true.
Example 1.5.8. Let T = Z and p ∈ R. Then
*
(1 + p(s)) = (1 + p)t−t0 .
ep (t, t0 ) =
s∈[t0 ,t)
Example 1.5.9. Let T = hZ, h > 0 and p ∈ R. Then
*
(1 + hp(s)) = (1 + ph)(t−t0 )/h .
ep (t, t0 ) =
s∈[t0 ,t)
Example 1.5.10. Let T = q N0 , q > 1 and p ∈ R. Then
*
(1 + (q − 1)sp(s))
ep (t, t0 ) =
t > t0 .
s∈[t0 ,t)
Example 1.5.11. Let T = 2N0 and p ∈ R. Then
*
(1 + p(s))
ep (t, t0 ) =
t > t0 .
s∈[t0 ,t)
Example 1.5.12. Let T = {Hn : n ∈ N0 }, where Hn are the harmonic numbers defined by
n
H0 = 0 and Hn = k=1 k1 , n ∈ N. If p ∈ R, p ≥ 0 is some constant, then
n+p
(n + p)(n − 1 + p) . . . (1 + p)
=
.
ep (t, t0 ) =
n
n!
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Theorem 1.5.7. Let p ∈ R. Then the unique solution of DIVP
xΔ = p(t)x, t ∈ Tκ ;
x(t0 ) = x0 ∈ R, t0 ∈ T
(1.45)
is given by
x(t) = ep (t, t0 )x0 .
Theorem 1.5.8. Let p ∈ R and f ∈ Crd (T, R). Then the unique solution of DIVP
xΔ = p(t)x + f (t), t ∈ Tκ ,
(1.46)
x(t0 ) = x0 ∈ R, t0 ∈ T
is given by
t
x(t) = x0 ep (t, t0 ) +
ep (t, σ(s))f (s)Δs.
t0
Example 1.5.13. Let T be an arbitrary time scale. Consider the DIVP
xΔ = 2x + 1, t ∈ Tκ ,
x(t0 ) = 1, t0 ∈ T.
Here p = 2, f = 1 and x0 = 1. This DIVP is of the form (1.46) whose unique solution is
given by
t
1
x(t) = e2 (t, t0 ) 1 +
Δs
σ
t0 e2 (s, t0 )
$
s=t %
−1
= e2 (t, t0 ) 1 +
2e2 (s, t0 ) s=t0
3
1
−
.
= e2 (t, t0 )
2 2e2 (t, t0 )
Thus,
3
1
x(t) = e2 (t, t0 )
−
.
2 2e2 (t, t0 )
Theorem 1.5.9. Let p ∈ R. Then the unique solution of DIVP
xΔ = −p(t)xσ , t ∈ Tκ ,
x(t0 ) = x0 ∈ R, t0 ∈ T
(1.47)
is given by
x(t) =
1
x0 .
ep (t, t0 )
Theorem 1.5.10. Let p ∈ R. Then the unique solution of DIVP
xΔ = −p(t)xσ + f (t), t ∈ Tκ ,
x(t0 ) = x0 ∈ R, t0 ∈ T
is given by
x(t) =
t
x0 + t0 ep (s, t0 )f (s)Δs
ep (t, t0 )
(1.48)
.
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Example 1.5.14. Let T be an arbitrary time scale. Consider the DIVP
xΔ = −xσ + 1, t ∈ Tκ ,
x(t0 ) = 2, t0 ∈ T.
See that this DIVP is of the form (1.48) with p(t) = 1, f (t) = 1, x0 = 2. Then, according
to Theorem 1.5.10, its unique solution is given by
t
x(t0 ) + t0 ep (s, t0 )f (s)Δs
x(t) =
ep (t, t0 )
t
2 + t0 e1 (s, t0 ) · 1Δs
=
e1 (t, t0 )
s=t
2 + [e1 (s, t0 )]s=t0
e1 (t, t0 )
1 + e1 (t, t0 )
=
.
e1 (t, t0 )
=
Thus,
x(t) =
1 + e1 (t, t0 )
.
e1 (t, t0 )
Example 1.5.15. Let T be an arbitrary time scale. Consider the DIVP
xΔ = −txσ + t, t ∈ Tκ ;
x(t0 ) = 2, t0 ∈ T.
See that this DIVP is of the form (1.48) with f (t) = t, x0 = 2, p(t) = t, thus p = id, identity
map. Then, according to Theorem 1.5.10, its unique solution is given by
t
x(t0 ) + t0 ep (s, t0 )f (s)Δs
x(t) =
ep (t, t0 )
t
2 + t0 eid (s, t0 )sΔs
=
eid (t, t0 )
s=t
2 + [eid (s, t0 )]s=t0
eid (t, t0 )
1 + eid (t, t0 )
.
=
eid (t, t0 )
=
Thus,
x(t) =
1 + eid (t, t0 )
.
eid (t, t0 )
Example 1.5.16. Let T be an arbitrary time scale. Consider the DIVP
⎧
1
⎨xΔ = −txσ +
, t ∈ Tκ ,
eid (t, t0 )
⎩
x(t0 ) = 0, t0 ∈ T.
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, x0 = 0, p(t) = t, thus
See that the above DIVP is of the form (1.48) with f (t) = eid (t,t
0)
p = id, the identity map. Then, according to Theorem 1.5.10, its unique solution is given
by
t
1
0 + t0 eid (s, t0 ) eid (s,t
Δs
0)
x(t) =
eid (t, t0 )
t
Δs
= t0
eid (t, t0 )
t − t0
.
=
eid (t, t0 )
Thus,
x(t) =
1.6
t − t0
.
eid (t, t0 )
Nabla Calculus Essentials
The concept of nabla derivative and its associated calculus is developed very similar to
that of delta case. See [3, 4] for primary work on the topic of nabla calculus. In this section,
we give essential notions from nabla calculus. To meet the requirement, for a given time
scale T, we need a derived set Tκ which we define as follows.
Definition 1.6.1. Let T be a time scale. We define the set Tκ = T \ {m}, where m is a
right-scattered minimum element of T.
Definition 1.6.2. (Nabla Derivative) Assume that f : T → R and let t ∈ Tκ . Define f ∇ (t)
to be the number (if it exists) with the following property: for given ε > 0 there exists a
neighbourhood Nδ of t such that
|[f ρ (t) − f (s)] − f ∇ (t)[ρ(t) − s]| ≤ ε|ρ(t) − s| for all s ∈ N.
(1.49)
We say that f is nabla differentiable on Tκ provided that f ∇ (t) exists for all t ∈ Tκ . The
function f ∇ : Tκ → R is called the ‘nabla derivative’ of f on Tκ .
Remark 1.6.1. For time scales R and Z, the nabla derivative f ∇ (t) is uniquely determined.
∇
In fact, for R, f ∇ (t) = df
dt (t), which is an ordinary derivative of f and for Z, f (t) =
f (t) − f (t − 1), which is an ordinary backward difference of f . Also, for an arbitrary time
scale T, the nabla derivative f Δ (t) is uniquely determined at t ∈ Tκ .
Theorem 1.6.1. (Properties of Nabla Derivative) Let f : T → R be a function and t ∈ Tκ .
Then we have the following.
1. If f is nabla differentiable at t, then f is continuous at t.
2. If f is continuous at left-scattered point t, then f is nabla differentiable at t with
f ∇ (t) =
f (t) − f (ρ(t))
.
t − ρ(t)
(1.50)
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3. If f is left-dense, then f is nabla differentiable at t if and only if,
lim
s→t
f (t) − f (s)
t−s
exists and is finite. In this case
f (t) − f (s)
.
t−s
(1.51)
f (ρ(t)) = f (t) − ν(t)f ∇ (t).
(1.52)
f ∇ (t) = lim
s→t
4. If f is nabla differentiable at t, then
This is known as the ‘simple useful formula’ (suf ) for nabla calculus.
Theorem 1.6.2. (Nabla Differentiation Rules) Assume that f, g : T → R are both nabla
differentiable at t ∈ Tκ . Then we have the following.
1. The sum f + g : T → R is nabla differentiable at t and
(f + g)∇ (t) = f ∇ (t) + g ∇ (t).
2. For α ∈ R, the function αf : T → R is nabla differentiable at t and (αf )∇ (t) = αf ∇ (t).
3. The product f g : T → R is nabla differentiable at t and
(f g)∇ (t) = f (t)g ∇ (t) + f ∇ (t)g(ρ(t)) = f ∇ (t)g(t) + f (ρ(t))g ∇ (t).
4. If g(t)g(ρ(t)) = 0, then the quotient fg : T → R is nabla differentiable at t, and
f
g
∇
(t) =
f ∇ (t)g(t) − f (t)g ∇ (t)
.
g(t)g(ρ(t))
Definition 1.6.3. (Nabla Antiderivative) Let f : T → R be a given function. A nabla
differentiable function F : T → R is said to be ‘nabla antiderivative’ of f , if
F ∇ (t) = f (t) hold for all t ∈ Tκ .
Definition 1.6.4. (Cauchy Nabla Integral) Let F : T → R be a nabla antiderivative of
f : T → R. The Cauchy nabla integral of f is defined by
t
f (τ )∇τ = F (t) − F (a), t, t0 ∈ T.
t0
Theorem 1.6.3. If f : T → R is ld-continuous, then
t
f (s)∇s = ν(t)f (t), t ∈ Tκ .
ρ(t)
Theorem 1.6.4. (Nabla Integration Rules) Let f, g : T → R be ld-continuous functions and
a, b, c ∈ T, and α ∈ R. Then we have the following.
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1.
a
a
f (t)∇t = 0;
a
f (t)∇t = − b f (t)∇t;
b
b
b
3. a (f (t) + g(t))∇t = a f (t)∇t + a g(t)∇t;
b
b
4. a αf (t)∇t = α a f (t)∇t;
c
b
b
4. a f (t)∇t = a f (t)∇t + c f (t)∇t;
b
b
5. a f (ρ(t))g ∇ (t)∇t = f (b)g(b) − f (a)g(a) − a f ∇ (t)g(t)∇t;
b
b
6. a f (t)g ∇ (t)∇t = f (b)g(b) − f (a)g(a) − a f ∇ (t)g(ρ(t))∇t;
b
b
7. If |f (t)| ≤ g(t) on (a, b]T , then a f (t)∇t ≤ a g(t)∇t;
2.
b
a
8. If f (t) ≥ 0 on (a, b]T , then
b
a
f (t)∇t ≥ 0.
Some special cases of the nabla integral are now presented.
Theorem 1.6.5. Let f : T → R be an ld-continuous function and a, b ∈ T. Then an analogous result holds as in Theorem 1.4.7.
b
b
1. If T = R, then a f (t)∇t = a f (t)dt, where the integral on right side is the usual
Riemann integral from continuous calculus.
2. If T consists of only isolated points between a and b, then
⎧
⎪
if a < b,
b
⎨ t∈(a,b] ν(t)f (t)
f (t)∇t = 0
if a = b,
⎪
a
⎩ − t∈(b,a] ν(t)f (t)
if a > b.
3. If T = hZ := {hk : k ∈ Z}, where h > 0, then
⎧b/h
⎪
b
⎨ k=(a+h)/h f (hk)h
f (t)∇t = 0
⎪
a
⎩ a/h
− k=(b+h)/h f (hk)h
4. If T = Z, then
⎧ b
⎪
⎨ k=a+1 f (t)
f (t)∇t = 0
⎪
a
⎩ a
− k=b+1 f (t)
b
if a < b,
if a = b,
if a > b.
if a < b,
if a = b,
if a > b.
There are certain natural relationships between delta derivative and nabla derivative,
which are essentially given in the following theorem.
Theorem 1.6.6. (Relations Between Delta and Nabla Derivatives)
1. Assume that f : T → R is delta differentiable on Tκ . Then f is nabla differentiable at
t and
(1.53)
f ∇ (t) = f Δ (ρ(t))
t ∈ Tκ such that σ(ρ(t)) = t. In addition, if f Δ is continuous on Tκ , then f is nabla
differentiable at t and (1.53) holds for any t ∈ Tκ .
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2. Assume that f : T → R is nabla differentiable on Tκ . Then f is delta differentiable at
t and
f Δ (t) = f ∇ (σ(t))
(1.54)
t ∈ Tκ such that ρ(σ(t)) = t. In addition, if f ∇ is continuous on Tκ , then f is delta
differentiable at t and (1.54) holds for any t ∈ Tκ .
Theorem 1.6.7. Let t0 ∈ T. Assume that f, f Δ , f ∇ : T × T → R are continuous. Then we
have the following.
'Δ &
t
t
1. t0 f (t, s)∇s = t0 f Δ (t, s)Δs + f (σ(t), t);
2.
3.
4.
&
'
∇
t
t
f (t, s)Δs = t0 f ∇ (t, s)Δs + f (ρ(t), ρ(t));
t0
&
t
t0
'Δ
f (t, s)∇s
&
t
f (t, s)∇s
t0
=
'∇
=
t
t0
t
t0
f Δ (t, s)∇s + f (σ(t), σ(t));
f ∇ (t, s)∇s + f (ρ(t), t).
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on email
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