Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download By email at student.support@hotmail.com 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. Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download By email at student.support@hotmail.com Taylor & Francis Taylor & Francis Group http://taylorandfrancis.com Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download By email at student.support@hotmail.com Dynamic Equations on Time Scales and Applications Edited by Ravi P. Agarwal, Bipan Hazarika, and Sanket Tikare Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download By email at student.support@hotmail.com Designed cover image: Vasilius/ShutterStock First edition published 2025 by CRC Press 2385 NW Executive Center Drive, Suite 320, Boca Raton FL 33431 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2025 selection and editorial matter, Ravi P. Agarwal, Bipan Hazarika and Sanket Tikare; individual chapters, the contributors Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact mpkbookspermissions@tandf.co.uk Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. ISBN: 978-1-032-74004-1 (hbk) ISBN: 978-1-032-74162-8 (pbk) ISBN: 978-1-003-46790-8 (ebk) DOI: 10.1201/9781003467908 Typeset in Computer Modern font by KnowledgeWorks Global Ltd. Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download By email at student.support@hotmail.com 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 Get Complete eBook Download By email at student.support@hotmail.com Get email at student.support@hotmail.com vi Complete eBook Download ByContents 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 Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download By email at student.support@hotmail.com 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 Get Complete eBook Download By email at student.support@hotmail.com Get email at student.support@hotmail.com viii Complete eBook Download ByContents 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 Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download By email at student.support@hotmail.com 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 Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download ByPreface email at student.support@hotmail.com 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 Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download By email at student.support@hotmail.com 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 Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download By email at student.support@hotmail.com Taylor & Francis Taylor & Francis Group http://taylorandfrancis.com Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download By email at student.support@hotmail.com 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 Get Complete eBook Download By email at student.support@hotmail.com Get xiv Complete eBook Download By email Editors at student.support@hotmail.com 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. Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download By email at student.support@hotmail.com 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 Get Complete eBook Download By email at student.support@hotmail.com Get email at student.support@hotmail.com xvi Complete eBook Download By Contributors Nick Wintz Department of Mathematics Lindenwood University St. Charles, MO, USA Mohamed Zitane Department of Mathematics Moulay Ismaïl University Meknès, Morocco Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download By email at student.support@hotmail.com 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 Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download emailScales at student.support@hotmail.com 2 Dynamic Equations By on Time and Applications 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 Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email atCalculus student.support@hotmail.com3 Elements ofBy Time Scales 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. Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download emailScales at student.support@hotmail.com 4 Dynamic Equations By on Time and Applications 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, Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email atCalculus student.support@hotmail.com5 Elements ofBy Time Scales 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 Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download emailScales at student.support@hotmail.com 6 Dynamic Equations By on Time and Applications ⎧ ⎪ ⎨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. Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email atCalculus student.support@hotmail.com7 Elements ofBy Time Scales 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. Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download emailScales at student.support@hotmail.com 8 Dynamic Equations By on Time and Applications 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. Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email atCalculus student.support@hotmail.com9 Elements ofBy Time Scales 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 . Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 10 Complete eBook Dynamic EquationsBy on email Time Scales and Applications 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 = Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email student.support@hotmail.com Elements of By Time ScalesatCalculus 11 √ √ 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). Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 12 Complete eBook Dynamic EquationsBy on email Time Scales and Applications 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 Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email student.support@hotmail.com Elements of By Time ScalesatCalculus 13 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. Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 14 Complete eBook Dynamic EquationsBy on email Time Scales and Applications (n) 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 Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email student.support@hotmail.com Elements of By Time ScalesatCalculus 15 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 ) Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 16 Complete eBook Dynamic EquationsBy on email Time Scales and Applications = 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. Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email student.support@hotmail.com Elements of By Time ScalesatCalculus 17 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) Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 18 Complete eBook Dynamic EquationsBy on email Time Scales and Applications 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 = Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email student.support@hotmail.com Elements of By Time ScalesatCalculus 19 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. Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 20 Complete eBook Dynamic EquationsBy on email Time Scales and Applications 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 Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email student.support@hotmail.com Elements of By Time ScalesatCalculus 21 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 Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 22 Complete eBook Dynamic EquationsBy on email Time Scales and Applications 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. Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email student.support@hotmail.com Elements of By Time ScalesatCalculus 23 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 Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 24 Complete eBook Dynamic EquationsBy on email Time Scales and Applications 1 = 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 = Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email student.support@hotmail.com Elements of By Time ScalesatCalculus 25 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, Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 26 Complete eBook Dynamic EquationsBy on email Time Scales and Applications 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 Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email student.support@hotmail.com Elements of By Time ScalesatCalculus 27 = = 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+ . Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 28 Complete eBook Dynamic EquationsBy on email Time Scales and Applications 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. Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email student.support@hotmail.com Elements of By Time ScalesatCalculus 29 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. Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 30 Complete eBook Dynamic EquationsBy on email Time Scales and Applications 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. Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email student.support@hotmail.com Elements of By Time ScalesatCalculus 31 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! Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 32 Complete eBook Dynamic EquationsBy on email Time Scales and Applications 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) . Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email student.support@hotmail.com Elements of By Time ScalesatCalculus 33 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. Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 34 Complete eBook Dynamic EquationsBy on email Time Scales and Applications 1 , 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) Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email student.support@hotmail.com Elements of By Time ScalesatCalculus 35 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. Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 36 Complete eBook Dynamic EquationsBy on email Time Scales and Applications 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 . Get Complete eBook Download By email at student.support@hotmail.com Get Complete eBook Download email student.support@hotmail.com Elements of By Time ScalesatCalculus 37 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). Bibliography [1] L. Adamec, A remark on matrix equation on small time scales, Journal of Difference Equations and Applications, 10(12) (2004) 1107–1117. [2] L. 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An introduction with applications, Birkhäuser, Boston, 2001. [8] M. Bohner and A. Peterson, editors., Advances in dynamic equations on time scales, Birkhäuser, Boston, 2003. [9] B. M. Garay and S. Hilger, Embeddability of time scale dynamics in ODE dynamics, Nonlinear Analysis: Theory, Methods & Applications, 47(2) (2001) 1357–1371. Get Complete eBook Download By email at student.support@hotmail.com Get Download at student.support@hotmail.com 38 Complete eBook Dynamic EquationsBy on email Time Scales and Applications [10] B. M. Garay and J. Várdai, Interpolation in dynamic equations on time scales, Journal of Difference Equations and Applications, 13(8–9) (2007) 847–854. [11] S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmanningfaltigkeiten, Ph.D. thesis, Universität Würzburg, Universität Würzburg, 1988. [12] S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results in Mathematics, 18(1–2) (1990) 18–56. [13] S. Hilger, Differential and difference calculus—unified!, Nonlinear Analysis: Theory, Methods & Applications, 30(5) (1997) 2683–2694. [14] S. Hilger, Special functions, Laplace and Fourier transform on measure chains, Special Issue on “Discrete and Continuous Hamiltonian Systems”, edited by R. P. Agarwal and M. Bohner, Dynamic Systems and Applications, 8(3-4) (1999) 471–488. [15] D. F. M. Torres, A table of derivatives, on arbitrary time scales, Global and Stochastic Analysis, 4(2) (2017) 199–205. Get Complete eBook Download By email at student.support@hotmail.com