Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales [1 ed.] 9783030421168

Table of contents :
Preface......Page 6
Contents......Page 10
1 Introduction to Stability and Boundedness in Dynamical Systems......Page 14
1.1 Introduction......Page 15
1.2 Introduction to Lyapunov Functions......Page 23
1.3 Delay Dynamic Equations......Page 43
1.4 Resolvent for Volterra Integral Dynamic Equations......Page 46
1.4.1 Existence of Resolvent: Lp Case......Page 51
1.4.2 Shift Operators......Page 56
1.4.3 Continuity of r(t,s)......Page 59
1.5 Periodicity on Time Scales......Page 60
1.6 Open Problems......Page 62
2 Ordinary Dynamical Systems......Page 64
2.1 Boundedness......Page 65
2.1.1 Applications......Page 68
2.2 Exponential Stability......Page 71
2.2.1 Applications......Page 74
2.3 Perturbed Systems......Page 78
2.4 Delay Dynamical Systems......Page 81
2.4.1 Delay Dynamical Systems-Linear Case......Page 82
2.4.2 Delay Dynamical Systems-Nonlinear Case......Page 87
2.4.3 Applications......Page 96
2.4.4 Perturbed Vector Nonlinear Delayed Systems......Page 101
2.5 Open Problems......Page 103
3.1 Uniform Boundedness and Uniform Ultimate Boundedness......Page 104
3.2 Applications to Volterra Integro-Dynamic Equations......Page 112
3.3 More on Boundedness......Page 117
3.3.1 Applications to Nonlinear Volterra Integro-DynamicEquations......Page 120
3.4 Exponential Stability......Page 125
3.4.1 Applications to Volterra Integro-Dynamic Equations......Page 129
3.5 Open Problems......Page 136
4 Volterra Integro-Dynamic Equations......Page 138
4.1 Generalized Gronwall's Inequality and Functional Bound......Page 139
4.2 Principal Matrix Solutions......Page 157
4.2.1 Existence and Uniqueness......Page 159
4.2.2 Existence of Principal Matrix......Page 165
4.2.3 Variation of Parameters......Page 167
4.3.1 Introduction......Page 171
4.3.2 New Resolvent Equation and Variation of Parameters......Page 173
4.3.3 Uniform Stability......Page 175
4.3.4 Applications and Comparison......Page 178
4.4 Necessary and Sufficient Conditions via Lyapunov Functionals......Page 182
4.4.1 Applications and Comparison......Page 188
4.5 Classification of Solutions......Page 190
4.5.1 Classifications of Positive Solutions......Page 191
4.5.2 Applications......Page 201
4.5.3 Classifications of Negative Solutions......Page 203
4.6 Open Problems......Page 205
5.1 Various Results Using Lyapunov Functionals......Page 207
5.1.1 Necessary and Sufficient Conditions: The Scalar Case......Page 210
5.1.2 Necessary and Sufficient Conditions: The Vector Case......Page 223
5.2 Shift Operators and Stability in Delay Systems......Page 227
5.2.1 Generalized Delay Functions......Page 229
5.2.2 Exponential Stability via Lyapunov Functionals......Page 233
5.2.3 Instability......Page 243
5.2.4 Applications and Comparison......Page 245
5.3 Open Problems......Page 248
6 Volterra Integral Dynamic Equations......Page 249
6.1 The Resolvent Method......Page 250
6.1.1 Existence and Uniqueness......Page 255
6.2 Continuation of Solutions......Page 256
6.3 Lyapunov Functionals Method......Page 265
6.3.1 Applications to Particular Time Scales......Page 273
6.4 Open Problems......Page 274
7 Periodic Solutions: The Natural Setup......Page 276
7.1 Neutral Nonlinear Dynamic Equation......Page 277
7.1.1 Existence of Periodic Solutions......Page 278
7.2 Nonlinear Dynamic Equation......Page 285
7.2.1 Existence of Periodic Solutions......Page 286
7.2.2 Stability......Page 292
7.3 Delayed Dynamic Equations......Page 296
7.3.1 Stability......Page 297
7.3.2 Periodicity......Page 299
7.4 Integro-Dynamic Equation......Page 300
7.4.1 Existence of Periodic Solutions......Page 301
7.4.2 Applications to Scalar Equations......Page 305
7.5 Schaefer Theorem and Infinite Delay Volterra Integro-Dynamic Equations......Page 313
7.5.1 Existence of Periodic Solutions......Page 314
7.5.2 Application to Infinite Delay System......Page 316
7.6 Connection Between Boundedness and Periodicity......Page 320
7.6.1 Existence......Page 321
7.6.2 Connection Between Boundedness and Periodicity......Page 324
7.7 Almost Linear Volterra Integro-Dynamic Equations......Page 328
7.7.1 Periodic Solutions......Page 329
7.8 Almost Automorphic Solutions of Delayed Neutral DynamicSystems on Time Scales......Page 332
7.8.1 Almost Automorphy Notion on Time Scales......Page 334
7.8.2 Exponential Dichotomy and Limiting Results......Page 336
7.8.3 Existence Results......Page 343
7.9 Large Contraction and Existence of Periodic Solutions......Page 353
7.9.1 Existence of Periodic Solution......Page 355
7.9.2 Classification of Large Contractions......Page 360
7.10 Open Problem......Page 363
8 Periodicity Using Shift Periodic Operators......Page 364
8.1 Periodicity in Shifts......Page 365
8.1.1 Periodicity in Quantum Calculus......Page 369
8.1.1.1 Connection Between Two q-Periodicity Notions on qN0......Page 370
8.2 Floquet Theory on Time Scales Periodic in Shifts δ......Page 373
8.2.1 Floquet Theory Based on New Periodicity Concept:Homogeneous Case......Page 374
8.2.2 Floquet Theory Based on New Periodicity Concept:Nonhomogeneous Case......Page 382
8.2.3 Floquet Multipliers and Floquet Exponents of UnifiedFloquet Systems......Page 385
8.3 Stability Properties of Unified Floquet Systems......Page 391
8.4 Existence of Periodic Solutions in Shifts δ for Neutral Nonlinear Dynamic Systems......Page 403
8.4.1 Existence of Periodic Solutions......Page 404
8.5 Open Problems......Page 414
References......Page 415
Index......Page 425

Citation preview

Murat Adıvar Youssef N. Raffoul

Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales

Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales

Murat Adıvar • Youssef N. Raffoul

Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales

Murat Adıvar Broadwell College of Business and Economics Fayetteville State University Fayetteville, NC, USA

Youssef N. Raffoul Department of Mathematics University of Dayton Dayton, OH, USA

ISBN 978-3-030-42116-8 ISBN 978-3-030-42117-5 (eBook) https://doi.org/10.1007/978-3-030-42117-5 Mathematics Subject Classification (2010): 34NXX, 34N05, 34C25, 34K20, 34K37, 34K38, 34K40, 39A05, 39A06, 39A12, 39A60 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to the memory of my father Ali Akın Adıvar and my mother Muzaffer Adıvar who have worked tirelessly and relentlessly to provide their children with better lives.

Dedicated to my wife Burcu Adıvar, who has always supported my ambition: you are always and will be my perfect love and perfect role model to our children Buket and Aliakın. — Murat Adıvar

To my strong and beautiful wife Nancy and my wonderful children Hannah, Venicia, Paul, Joseph, and Daniel. — Youssef N. Raffoul

Preface

The book provides an integrated approach to the study of the qualitative theory of dynamical systems on time scales. It is aimed at postgraduate students and researchers with basic knowledge in calculus, differential equations, difference equations real analysis, and familiarity with time scales. The area of time scales is relatively new and advanced topics such as existence of solutions, boundedness, stability, and the existence of periodic solutions are far from being sufficiently developed. One of the most efficient approaches in analyzing the before mentioned topics is the use of Lyapunov functions/functionals. This book should serve as an encyclopedia on the construction of Lyapunov functionals in analyzing solutions of dynamical systems on time scales. The book is suitable for a graduate course in the format of graduate seminars or as special topics course on dynamical systems. Motivated by recent increased activity of research on time scales, this book will provide a systematic approach to the study of the qualitative theory of boundedness, periodicity and stability of Volterra integro-dynamic equations on time scales. Stability theory is an area that will continue to be of great interest to researchers for a long time because of the usefulness it demonstrates in real-life applications. The authors will mainly use fixed point theory and Lyapunov functionals to arrive at major results. The book is mainly devoted to functional dynamical systems on time scales with applications to Volterra integro-dynamic equations. Good percentage of the contents of this book is new. Also, the authors consider many other time scales as application to the theory and perform interesting and difficult calculations on exotic time scales to help the reader better understand the concept of dynamical systems and make it easier for the interested reader to carry on with new research. Researchers and graduate students who are interested in the method of Lyapunov functions/functionals, in the study of boundedness of solutions, in the stability of the zero solution, or in the existence of periodic solutions should be able to use this book as a primary reference and as a resource of latest findings. This book contains many open problems and should be of great benefit to those who are pursuing research in vii

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dynamical systems or in Volterra integro-dynamic equations on time scales with or without delays. Great efforts were made to present detailed proofs of theorems so that the book would be self-contained. In addition to the new results in the book, the construction of Lyapunov functions/functionals takes the center stage in arriving at most results. Time scale systems might best be understood as the continuum bridge between discrete time and continuous time systems. In 1988, Stephen Hilger introduced the theory of time scales in his PhD dissertation [88] (see also [89]). The main purpose was to create a theory that unifies dynamical systems on different domains, which he called “time scales.” Thus, “calculus” on time scales had to be developed from scratch so that the unified theory can be applied to the desired domain which can be hybrid of discrete or continuous domains. Since the introduction of time scales, many books have been written on the development of its calculus (see for instance [50] and [51]). In this book, the emphasis is on the construction of Lyapunov functionals, an area that plays a major role in the qualitative analysis of solutions of dynamical systems. The excitement of working with differential systems on time scales comes from its applications in real-life situations. Recently, time scales have been used in models in areas such as economics, linear networks, population dynamics, and minimization as well as maximization in the form of calculus of variations, and we refer to [33, 39, 85], and [92]. Chapter 1 includes a brief introduction to time scale calculus and the development of Lyapunov functions on time scales. To effectively use Lyapunov function, the authors develop the Δ-derivative of absolute value of functions. New and general theorems in terms of wedges in which we derive specific conditions for stability of the zero solution and the boundedness of all solutions for regular and nonfunctional dynamical systems are stated and proved. In addition, we introduce delay dynamic equations and the resolvent for Volterra integral dynamic equations. We will introduce the notion of shift that we make use of in Chap. 5 and utilize it along with the notion of resolvent and develop new results concerning Volterra dynamic equations. Toward the end of the chapter, we introduce the notion of periodicity and end the chapter with open problems. Throughout Chap. 2, we derive results concerning the qualitative analysis of ordinary dynamical systems by proving general theorems that assume the existence of Lyapunov functions. Finite delay dynamical systems, linear and nonlinear are discussed in details via the use of Lyapunov functionals. Chapter 3 is of abstract in nature. It is devoted to functional dynamic systems. Functional systems are general in nature and encompass results concerning existence of periodic solutions, boundedness and stability for autonomous and nonautonomous dynamical systems with finite or infinite delay, and Volterra integro-dynamical systems. Our general theorems will require the construction of suitable Lyapunov functionals, a task that is difficult but possible. General theorems concerning boundedness and stability of functional dynamical systems on time scales are almost nonexistent, and this chapter is aimed at rectifying the problem. The theorems enable us to

Preface

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qualitatively analyze the theory of boundedness, uniform ultimate boundedness, and stability of solutions of vectors and scalar Volterra integro-dynamic equations. We end the chapter with open problems. Chapter 4 is exclusively devoted to the study of various types of Volterra integrodynamic equations with or without delays. Exotic Lyapunov functionals are constructed and used in obtaining stability and instability of the zero solution and boundedness of all solutions. In addition, our delays are of the general forms and defined by what we call shift operators. Shift operators are easily defined on all type of general time scales and without the requirement that the particular time scale be additive. Also, upper and lower bounds on solutions of integro-dynamic equations are studied via the use of the resolvent. Variant forms of Gronwall’s inequality on time scales are developed. We end the chapter by classifying solutions of Volterra integro-dynamic equations by appealing to variant types of fixed point theorems and conclude the chapter with open problems. Necessary and sufficient conditions for boundedness, stability, and exponential stability for Volterra integro-dynamic equations are the main topics of Chap. 5. Most of the work will rely on the construction of exotic Lyapunov functionals. The necessary and sufficient conditions will be extended to cover vector Volterra integro-dynamic equations. Lastly, we take a scalar dynamic equation with delay operator and rewrite in the form of Volterra integro-dynamic equation and use Lyapunov functionals to study exponential stability and instability. Results concerning the construction of our exotic Lyapunov functionals of this chapter are new and should serve as the foundation for future research in the subject. The chapter is ended with open problems. Chapter 6 is totally new and devoted to the study Volterra integral dynamic equations using the concept of resolvent of Sect. 1.4.1. In such study, we use the resolvent combined with Lyapunov functionals. In Sect. 6.1.1, we make use of the resolvent along with fixed point theory and show the existence and uniqueness of solutions of nonlinear Volterra integral dynamic equations. Later in the chapter, we consider nonlinear Volterra integral dynamic equations and construct suitable Lyapunov functional to obtain results regarding boundedness and stability of the zero solution. Chapter 7 is devoted to the study of periodic solutions based on the definition of periodicity given by Kaufmann and Raffoul [99]. The chapter begins with an introduction to the concept of periodicity and its application to finite delay neutral nonlinear dynamical equations and to infinite delay Volterra integro-dynamic equations of variant forms, by appealing to Schaefer’s fixed point theorem. The a-priori bound will be obtained by the aid of Lyapunov-like functionals. Effort will be made to expose the improvement when we set the time scales to be the set of reals or integers. Then we proceed to study the existence of periodic solutions of almost linear Volterra integro-dynamic equations by utilizing Krasnosel’ski˘ı fixed point theorem. We will be examining the relationship between boundedness and the existence of periodic solutions. We give existence results for almost automorphic solutions of delayed neutral dynamic systems. Lastly, we introduce the concept of large contrac-

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tion to improve some of the results and prove a general theorem in which we classify the functions satisfying the criteria of large contraction. The chapter concludes with open problems and contains new results. Unlike Chap. 7 where we required the time scale to be additive, Chap. 8 is devoted to the study of periodic solutions on general time scales. The chapter is mainly concerned with the study of periodic solutions of functional delay dynamical systems with finite and infinite delays. The periodic delays are expressed in terms of shift operators (see Definition 1.4.4) that were developed by Adıvar in [4]. In addition, the chapter establishes the unification of Floquet theory for homogeneous and nonhomogeneous dynamical systems. We move to considering neutral dynamical systems using Floquet theory and Krasnosel’ski˘ı fixed point theorem. The existence of almost automorphic solutions of delayed neutral dynamic system using exponential dichotomy and Krasnosel’ski˘ı fixed point theorem will be studied. Part of this chapter is new. The chapter concludes with multiple open problems. The book is the result of published and new research that took place over the span of many years. There are many researchers, in one way or another that led to the writing of this book. The authors would like to thank Allan C. Peterson and Martin Bohner for their extensive research and advancement of time scales in North America. Youssef N. Raffoul extends his sincere thanks to his co-author Murat Adıvar for his deep insight and developing the notion of periodicity using shift operators. Raffoul would also like to thank current and previous chairs of the Department of Mathematics at the University of Dayton for their unwavering support. Lastly, Raffoul extends his sincere gratitude to Notre Dame University, Beirut Lebanon, and especially Dr. George Eid for hosting him for many years. Murat Adıvar thanks the Fayetteville State University; in particular, Dr. Pamela Jackson and Dr. Constance Lightner for their valuable and continued support for his research and for the efforts to develop new education programs that are changing lives of the next generations in North Carolina. Fayetteville, NC, USA Dayton, OH, USA

Murat Adıvar Youssef N. Raffoul

Contents

1

Introduction to Stability and Boundedness in Dynamical Systems 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Introduction to Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Delay Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Resolvent for Volterra Integral Dynamic Equations . . . . . . . . . . . . . . 1.4.1 Existence of Resolvent: L p Case . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Shift Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Continuity of r(t, s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Periodicity on Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 10 30 33 38 43 46 47 49

2

Ordinary Dynamical Systems 2.1 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Exponential Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Perturbed Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Delay Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Delay Dynamical Systems-Linear Case . . . . . . . . . . . . . . . . . . 2.4.2 Delay Dynamical Systems-Nonlinear Case . . . . . . . . . . . . . . . 2.4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Perturbed Vector Nonlinear Delayed Systems . . . . . . . . . . . . . 2.5 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 52 55 58 61 65 68 69 74 83 88 90

3

Functional Dynamical Systems 91 3.1 Uniform Boundedness and Uniform Ultimate Boundedness . . . . . . . 91 3.2 Applications to Volterra Integro-Dynamic Equations . . . . . . . . . . . . . 99

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3.3

3.4 3.5

More on Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.3.1 Applications to Nonlinear Volterra Integro-Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Exponential Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.4.1 Applications to Volterra Integro-Dynamic Equations . . . . . . 116 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4

Volterra Integro-Dynamic Equations 125 4.1 Generalized Gronwall’s Inequality and Functional Bound . . . . . . . . . 126 4.2 Principal Matrix Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.2.1 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.2.2 Existence of Principal Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4.2.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.3 Necessary and Sufficient Conditions for Stability via Resolvent . . . . 158 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.3.2 New Resolvent Equation and Variation of Parameters . . . . . . 160 4.3.3 Uniform Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.3.4 Applications and Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.4 Necessary and Sufficient Conditions via Lyapunov Functionals . . . . 169 4.4.1 Applications and Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.5 Classification of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 4.5.1 Classifications of Positive Solutions . . . . . . . . . . . . . . . . . . . . . 178 4.5.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 4.5.3 Classifications of Negative Solutions . . . . . . . . . . . . . . . . . . . . 190 4.6 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

5

Exotic Lyapunov Functionals for Boundedness and Stability 195 5.1 Various Results Using Lyapunov Functionals . . . . . . . . . . . . . . . . . . . 195 5.1.1 Necessary and Sufficient Conditions: The Scalar Case . . . . . 198 5.1.2 Necessary and Sufficient Conditions: The Vector Case . . . . . 211 5.2 Shift Operators and Stability in Delay Systems . . . . . . . . . . . . . . . . . . 215 5.2.1 Generalized Delay Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.2.2 Exponential Stability via Lyapunov Functionals . . . . . . . . . . . 221 5.2.3 Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 5.2.4 Applications and Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 233 5.3 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

6

Volterra Integral Dynamic Equations 237 6.1 The Resolvent Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 6.1.1 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 6.2 Continuation of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 6.3 Lyapunov Functionals Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 6.3.1 Applications to Particular Time Scales . . . . . . . . . . . . . . . . . . 261 6.4 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

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7

Periodic Solutions: The Natural Setup 265 7.1 Neutral Nonlinear Dynamic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 266 7.1.1 Existence of Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . 267 7.2 Nonlinear Dynamic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 7.2.1 Existence of Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . 275 7.2.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 7.3 Delayed Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 7.3.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 7.3.2 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 7.4 Integro-Dynamic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 7.4.1 Existence of Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . 290 7.4.2 Applications to Scalar Equations . . . . . . . . . . . . . . . . . . . . . . . 294 7.5 Schaefer Theorem and Infinite Delay Volterra Integro-Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 7.5.1 Existence of Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . 303 7.5.2 Application to Infinite Delay System . . . . . . . . . . . . . . . . . . . . 305 7.6 Connection Between Boundedness and Periodicity . . . . . . . . . . . . . . . 309 7.6.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 7.6.2 Connection Between Boundedness and Periodicity . . . . . . . . 313 7.7 Almost Linear Volterra Integro-Dynamic Equations . . . . . . . . . . . . . . 317 7.7.1 Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 7.8 Almost Automorphic Solutions of Delayed Neutral Dynamic Systems on Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 7.8.1 Almost Automorphy Notion on Time Scales . . . . . . . . . . . . . . 323 7.8.2 Exponential Dichotomy and Limiting Results . . . . . . . . . . . . . 325 7.8.3 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 7.9 Large Contraction and Existence of Periodic Solutions . . . . . . . . . . . 342 7.9.1 Existence of Periodic Solution . . . . . . . . . . . . . . . . . . . . . . . . . 344 7.9.2 Classification of Large Contractions . . . . . . . . . . . . . . . . . . . . 349 7.10 Open Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

8

Periodicity Using Shift Periodic Operators 353 8.1 Periodicity in Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 8.1.1 Periodicity in Quantum Calculus . . . . . . . . . . . . . . . . . . . . . . . 358 8.2 Floquet Theory on Time Scales Periodic in Shifts δ± . . . . . . . . . . . . . 362 8.2.1 Floquet Theory Based on New Periodicity Concept: Homogeneous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 8.2.2 Floquet Theory Based on New Periodicity Concept: Nonhomogeneous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 8.2.3 Floquet Multipliers and Floquet Exponents of Unified Floquet Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 8.3 Stability Properties of Unified Floquet Systems . . . . . . . . . . . . . . . . . . 380

xiv

Contents

8.4

8.5

Existence of Periodic Solutions in Shifts δ± for Neutral Nonlinear Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 8.4.1 Existence of Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . 393 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

References

405

Index

415

Chapter 1

Introduction to Stability and Boundedness in Dynamical Systems

Summary In this chapter, we provide a brief introduction to time scale calculus and introduce fundamental concepts that we need throughout this book. In Sect. 1.2 based on the work of Peterson and Tisdell (J Differ Equ Appl 10(13–15):1295–1306, 2004), we introduce the concept of Lyapunov functions for ordinary dynamical equations on time scales and prove general theorems in terms of wedges in which we derive specific conditions for stability of the zero solution and the boundedness of all solutions. We proceed to prove new and general theorems in terms of wedges and the existence of Lyapunov functions that satisfy certain conditions and give necessary conditions for stability and instability of the zero solution. In Sect. 1.4 we furnish all the details in proving the existence of the resolvent of Volterra integral dynamic equations by appealing to the results of Adıvar and Raffoul (Bull Aust Math Soc 82(1):139–155, 2010). In addition, we will introduce the notion of shift that we make use of in Chaps. 5 and 8, and utilize the notion of resolvent and develop new results concerning Volterra dynamic equations. We end the chapter by introducing the notion of periodicity. In the famous paper (Kuchment, Floquet theory for partial differential equations, Birkhäuser Verlag, Basel, 1993) of Raffoul and Kaufmann, the notion of periodicity on time scales was first introduced. Later on, Adıvar (Math Slovaca 63(4):817–828, 2013) generalized the definition of periodicity to general time scales by introducing the concept of periodic shift operator. We end the chapter with some interesting and meaningful open problems that should be somewhat challenging. In this chapter, we provide a brief introduction to time scale calculus and introduce fundamental concepts that we need throughout this book. In Sect. 1.2 based on the work of Peterson and Tisdell [131], we introduce the concept of Lyapunov functions for ordinary dynamical equations on time scales and prove general theorems in terms of wedges in which we derive specific conditions for stability of the zero solution and the boundedness of all solutions. We proceed to prove new and general theorems in terms of wedges and the existence of Lyapunov functions that satisfy certain conditions and give necessary conditions for stability and instability of the zero solution. In Sect. 1.4 we furnish all the details in proving the existence of the © Springer Nature Switzerland AG 2020 M. Adıvar, Y. N. Raffoul, Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales, https://doi.org/10.1007/978-3-030-42117-5_1

1

2

1 Introduction to Stability and Boundedness in Dynamical Systems

resolvent of Volterra integral dynamic equations by appealing to the results of [12]. In addition, we will introduce the notion of shift that we make use of in Chaps. 5 and 8, and utilize the notion of resolvent and develop new results concerning Volterra dynamic equations. We end the chapter by introducing the notion of periodicity. In the famous paper [100] of Raffoul and Kaufmann, the notion of periodicity on time scales was first introduced. Later on, Adıvar, in [6] generalized the definition of periodicity to general time scales by introducing the concept of periodic shift operator. We end the chapter with some interesting and meaningful open problems that should be somewhat challenging.

1.1 Introduction In this section, we provide some fundamental concepts and theorems of time scale calculus. Time scale calculus was introduced by Stefan Hilger in his Ph.D. dissertation [88] (see also [89]). In the last two decades, the theory of time scales became a very popular topic of research in mathematics. For some other applications of time scale calculus in Operations Research, Optimization, Control Theory, Robotics, Economics, and Population Dynamics see [7, 8, 28, 33, 39, 85, 91, 92], and [128]. Throughout this book, it is assumed that the reader is familiar with the calculus of time scales and this book is by no means should be considered an introductory course on the calculus of time scales. For a comprehensive review of time scale calculus we refer to [50, 51] and references therein. Definition 1.1.1 (Time Scale). A time scale (or measure chain) is a nonempty closed subset of the real numbers R. Throughout this book we denote a time scale by T and assume that T has the topology inherited from the standard topology on R. The most well-known examples of time scales are the real numbers R for continuous calculus, the integer numbers Z for discrete calculus, the discrete time scale hZ, and the quantum numbers qN := {q n : n ∈ N} ∪ {0} for q > 1 for quantum calculus (see [96]), and the set qZ := {q n : n ∈ Z} ∪ {0}

where

q > 1.

The points of a time scale T are classified by using jump operators defined below: Definition 1.1.2 (Jump Operators and Graininess Function). The forward and backward jump operators σ : T → T and ρ : T → T are defined by σ(t) := inf {τ ∈ T : τ > t} and ρ(t) := sup {τ ∈ T : τ < t} , respectively, where we use the convention that inf ∅ = sup T (i.e., σ(tmax ) = tmax ) when T has a maximum tmax and sup ∅ = inf T (i.e., ρ(tmin ) = tmin ), when T has a minimum tmin . Finally, the graininess (or step-size) function μ : T → [0, ∞) is defined by

1.1 Introduction

3

μ(t) := σ(t) − t. The following table illustrates the jump operators and the graininess function on several time scales. T σ(t) ρ(t) μ(t) R t t 0 Z t+1 t−1 1 Z q qt t/q (q − 1)t For a function f : T → R, we define the function f σ : T → R by f σ (t) := f (σ(t)) for all t ∈ T, that is f σ is the composite function f ◦ σ. The jump operators σ and ρ then allow the classification of points in a time scale in the following way: If σ(t) > t, then we say that the point t is right-scattered; while if σ(t) = t, then we say the point t is right-dense. Similarly, if ρ(t) < t, then we say the point t is left-scattered; while if ρ(t) = t, then we say the point t is left dense. An isolated time scale T is a time scale consisting only of right scattered points. T = hZ, T = N, and T = qZ are examples of isolated time scales. Next we define the Δ-derivative of a function on time scales. Definition 1.1.3 (Delta Derivative). Let the subset Tκ of the time scale T be defined by  T\(ρ(sup T), sup T] if sup T < ∞ κ . T := T if sup T = ∞ A function f : T → R is said to be Δ-differentiable at t ∈ Tκ if for all ε > 0 there exists U a neighborhood of t such that for some α, the inequality | f σ (t) − f (s) − α(σ(t) − s)| < |σ(t) − s| holds for all s ∈ U. Then we write α = f Δ (t). In the following, we present some useful relationships for the Δ-derivative. Theorem 1.1.1 ([50, Theorem 1.16]). Let f : T → R be a function. Then we have the following: 1. If f is Δ-differentiable at a right scattered point t ∈ T, then f Δ (t) =

f σ (t) − f (t) . μ(t)

2. If f is Δ-differentiable at a right-dense point t ∈ T, then f Δ (t) = lim s→t

f (t) − f (s) . t−s

4

1 Introduction to Stability and Boundedness in Dynamical Systems

3. If f is Δ-differentiable at any point t, then f σ (t) = f (t) + μ(t) f Δ (t). If T = R, then f Δ = f  is the derivative used in standard calculus. If T = Z, then f Δ = Δ f is the forward difference operator used in difference equations. If T = qZ , then f Δ = Dq f is the q-derivative used in q-difference equations, where Dq f (t) :=

f (qt) − f (t) . (q − 1)t

Next we give the product and quotient rules for the Δ-derivative. Theorem 1.1.2 ([50, Theorem 1.20]). Let f : T → R and g : T → R be two Δ-differentiable functions at t. Then: 1. The product f g : T → R is Δ-differentiable with ( f g)Δ (t) = f Δ (t)g(t) + f σ (t)g Δ (t) = f Δ (t)g σ (t) + f (t)g Δ (t). 2. If g(t)g σ (t)  0, then

f g

(Product rule)

is Δ-differentiable with

 Δ f f Δ (t)g(t) − f (t)g Δ (t) . (t) = g g(t)g σ (t)

(Quotient rule)

Throughout this book we use the notation [a, b]T to indicate time scale interval defined by (1.1.1) [a, b]T := [a, b] ∩ T. The intervals (a, b)T , [a, b)T , and (a, b]T are defined similarly. In addition, through out the book, every time we mention an initial time t0 , then t0 ∈ T. Lemma 1.1.1 ([51, Corollaries 1.15–16]). Let f : T → R be a continuous function on [a, b]T that is Δ-differentiable on [a, b)T . (i) If f Δ (t) = 0 for all t ∈ [a, b)T , then f is a constant function on [a, b]T . (ii) f is increasing, decreasing, nondecreasing, and nonincreasing on [a, b]T if f Δ (t) > 0, f Δ (t) < 0, f Δ (t) ≥ 0, and f Δ (t) ≤ 0 for all t ∈ [a, b)T , respectively. Theorem 1.1.3 (Chain Rule [50, Theorem 1.93]). Assume ν : T → R is strictly ˜ increasing and T˜ := ν(T) is a time scale. Let w : T˜ → R. If ν Δ (t) and w Δ (ν(t)) exist for t ∈ Tκ , then ˜ (w ◦ ν)Δ = (w Δ ◦ ν)ν Δ . Definition 1.1.4 (rd-Continuity). The function f : T → R is called right-dense continuous (rd-continuous) if f is continuous at every right-dense point t ∈ T and lims→t − f (s) exists and is finite, at every left-dense point t ∈ T.

1.1 Introduction

5

Throughout the book we denote by Cr d = Cr d (T, R) the set of all r d-continuous functions. Definition 1.1.5 (Antiderivative). A function F : T → R is called an antiderivative of f : T → R provided F Δ (t) = f (t) holds for all t ∈ Tκ . Hereafter, we list some theorems that will be needed in the further analysis throughout the book. Theorem 1.1.4 (Antiderivative [50, Theorem 1.74]). Every rd-continuous function f : T → R has an antiderivative F. In particular if t0 ∈ T, then the function F defined by ∫ t

F(t) :=

f (s)Δs

for

t∈T

(1.1.2)

t0

is an antiderivative of f . Theorem 1.1.5 (Cauchy Integral). Let F : T → R be an antiderivative of f : T → R. The Cauchy integral of f is then defined by ∫

b

f (t)Δt = F(b) − F(a), where a, b ∈ T.

(1.1.3)

a

If sup T = ∞, then the improper integral is defined by ∫ ∞ f (t)Δt = lim F(b) − F(a). b→∞

a

Using (1.1.3) we state the following theorem. Theorem 1.1.6 ([50, Theorem 1.77]). If a, b, c ∈ T, α, β ∈ R, and f , g ∈ Cr d , then ∫b ∫b ∫b 1. a (α f (t) + βg(t))Δt = α a f (t)Δt + β a g(t)Δt ∫a ∫b 2. a f (t)Δt = − b f (t)Δt ∫c ∫b ∫b 3. a f (t)Δt = a f (t)Δt + c f (t)Δt ∫b 4. If f (t) ≥ 0 for all t ∈ [a, b)T , then a f (t)Δt ≥ 0 ∫a 5. a f (t)Δt = 0 6. If | f (t)| ≤ g(t) for all t ∈ [a, b)T , then ∫ b  ∫ b    f (t)Δt  ≤ g(t)Δt.  a

a

Theorem 1.1.7 ([50, Theorem 1.79]). Let a, b ∈ T and f ∈ Cr d . 1. If T = R, then

∫ a

b

∫ f (t)Δt = a

b

f (t)dt

6

1 Introduction to Stability and Boundedness in Dynamical Systems

2. If [a, b]T consists only of right scattered points, then ⎧ t ∈[a,b) μ(t) f (t) if a < b ∫ b ⎪ T ⎨ ⎪ if a = b f (t)Δt = 0 ⎪ a ⎪ − t ∈[b,a) μ(t) f (t) if b < a T ⎩ Theorem 1.1.8 ([50, Theorem 1.75]). If f ∈ Cr d ([50, Definition 1.58]) and t ∈ Tκ , then ∫ σ(t)

f (s)ds = μ(t) f (t).

t

We shall need the following theorem at several occasions in our further work. Theorem 1.1.9 (Substitution [50, Theorem 1.98]). Assume ν : T → R is strictly increasing and T˜ := ν(T) is a time scale. If f : T → R is an rd-continuous function and ν is differentiable with rd-continuous derivative, then for a, b ∈ T, ∫ a

b

f (s)ν Δ (s) Δs =

∫

ν(b)

ν(a)

˜ f (ν −1 (s)) Δs.

(1.1.4)

Lemma 1.1.2 (Integration by Parts [50, Theorem 1.77, (v)]). If a, b ∈ T, α ∈ R, and f , g ∈ Cr d , then ∫

b

f (σ(t))g Δ (t)Δt = ( f g)(b) − ( f g)(a) −

a

∫

b

f (t)g Δ (t)Δt.

a

To differentiate the Lyapunov functionals given in further sections we will employ the following theorem. Theorem 1.1.10 (Derivative of Integral [50, Theorem 1.117]). Let a ∈ Tκ , b ∈ T and assume k : T × Tκ → R is continuous at (t, t), where t ∈ Tκ with t > a. Also assume that k (t, .) is rd-continuous on [a, σ (t)]T . Suppose that for each ε > 0 there exists a neighborhood U of t, independent of τ ∈ [t0, σ (t)]T , such that    k (σ (t) , τ) − k (s, r) − k Δ (t, τ) (σ (t) − s) ≤ ε |σ (t) − s| for all s ∈ U, where k Δ denotes the derivative of k with respect to the first variable. Then ∫ t ∫ t Δ k (t, τ) Δτ implies g (t) = k Δ (t, τ) Δτ + k (σ (t) , t) g (t) := a

∫ h (t) :=

t

a

b

k (t, τ) Δτ implies hΔ (t) =

∫

b

k Δ (t, τ) Δτ − k (σ (t) , t) .

t

We shall invoke Jensen’s inequality [50, Theorem 6.17] in the proof of some of our results later on in the book.

1.1 Introduction

7

Theorem 1.1.11 (Jensen’s Inequality). Let a, b ∈ T and c, d ∈ R. If g : [a, b]T → [c, d] is rd-continuous and F : (c, d) → R is continuous and convex, then

F 

∫b

∫b

a

a

g(t)Δt  ≤ b−a 

F(g(t))Δt b−a

.

Definition 1.1.6 (Regressive Function). A function h : T → R is said to be regressive provided 1 + μ(t)h(t)  0 for all t ∈ Tκ . The set of all regressive rd-continuous functions h : T → R is denoted by R while the set of all positively regressive functions is given by R + = {h ∈ R : 1 + μ(t)h(t) > 0 for all t ∈ T}. Under the “circle plus” addition on R defined by (p ⊕ q)(t) := p(t) + q(t) + μ(t)p(t)q(t) for all t ∈ T. R is an Abelian group [50, Exercise 2.26], where the additive inverse of p, denoted by p, is defined by ( p)(t) :=

−p(t) 1 + μ(t)p(t)

for all t ∈ Tκ .

Definition 1.1.7 (Exponential Function). Let p ∈ R and s ∈ T. The exponential function e p (·, s) is defined by ∫ t  ξμ(τ) (p(τ)) Δτ , t ∈ T, (1.1.5) e p (t, s) = exp s

where ξh (z) is the so-called cylinder transformation defined by 1 Log(1 + zh) if h > 0 . ξh (z) := h 0 if h = 0 It is well known that if p ∈ R + , then e p (t, s) > 0 for all t ∈ T. Also, the exponential function y(t) = e p (·, t0 ) is the solution to the initial value problem y Δ = p(t)y, y(t0 ) = 1. Other properties of the exponential function are given in the following lemma. Lemma 1.1.3 ([50, Theorem 2.36]). Let p, q ∈ R. Then 1. e0 (t, s) ≡ 1 and e p (t, t) ≡ 1; 2. e p (σ(t), s) = (1 + μ(t)p(t))e p (t, s); 1 ; 3. e p (t, s) = e p (t,s)

1 4. e p (t, s) = e p (s,t) = e p (s, t); 5. e p (t, s)e p (s, r) = e p (t, r); 6. e p (t, s)eq (t, s) = e p ⊕q (t, s);

8

1 Introduction to Stability and Boundedness in Dynamical Systems e p (t,s) eq (t,s)

= e p q (t, s); Δ 1 = − e σpp(t) 8. e p (·,s) (·,s) .

7.



Remark 1.1.1. It is clear that the exponential function is equal to e p (t, s) = e

∫t s

p(τ)dτ

, eα (t, s) = eα(t−s), eα (t, 0) = eα(t)

for s, t ∈ R, where α ∈ R is a constant and p : R → R is a continuous function if T = R. Similarly, e p (t, s) =

t−1 

[1 + p(τ)], eα (t, s) = (1 + α)t−s, eα (t, 0) = (1 + α)t

τ=s

for s, t ∈ Z, with α < t, where α  −1 is a constant and p : Z → R is a sequence satisfying p(t)  −1 for all t ∈ Z if T = Z. Theorem 1.1.12 ([50, Theorem 6.1]). Let y, f ∈ Cr d and p ∈ R + . Then y Δ (t) ≤ p(t)y(t) + f (t) for all t ∈ T implies

∫ y(t) ≤ y(t0 )e p (t, t0 ) +

t

e p (t, σ(τ)) f (τ)Δτ

t0

for all t ∈ T. Next we state Gronwall’s inequality . Theorem 1.1.13 (Gronwall’s Inequality [50, Theorem 6.4]). Let y, f ∈ Cr d and p ∈ R +, p ≥ 0. Then ∫ t y(t) ≤ f (t) + y(τ)p(τ)Δτ for all t ∈ T t0

implies

∫ y(t) ≤ f (t) +

t

e p (t, σ(τ)) f (τ)p(τ)Δτ for all t ∈ T.

t0

Corollary 1.1.1. Suppose the hypothesis of Theorem 1.1.13 hold with f (t) = Q where Q ∈ R, then y(t) ≤ Qe p (t, t0 ) for all t ∈ T. Theorem 1.1.14 (Bernoulli’s Inequality [50, Theorem 6.2]). Let α ∈ R with α ∈ R + . Then eα (t, t0 ) ≥ 1 + α(t − s) for all t ≥ s.

1.1 Introduction

9

It follows from Theorem 1.1.14 that for any time scale, if the constant λ ∈ R + , then 0 < e λ (t, t0 ) ≤

1 for all t ≥ t0 . 1 + λ(t − t0 )

It follows that lim e λ (t, t0 ) = 0.

t→∞

In particular, if T = R, then e λ (t, t0 ) = e−λ(t−t0 ) , and if T = Z+ , then e λ (t, t0 ) = (1 + λ)−(t−t0 ) . For the growth of generalized exponential functions on time scales we refer to Bodine and Lutz [43]. We apply the following Cauchy–Schwartz inequality in [50, Theorem 6.2] to prove results throughout the book. Theorem 1.1.15. Let a, b ∈ T. For rd-continuous f , g : [a, b]T → R we have   ∫ b  ∫ b ∫ b | f (t)g(t)|Δt ≤ | f (t)| 2 Δt |g(t)| 2 Δt . a

a

a

To be able to prove the existence of periodic solutions we will use Schaefer‘s fixed point theorem, which we state below. Theorem 1.1.16 (Schaefer [149]). Let (B, | · |) be a normed linear space, and H a continuous mapping of B into B which is compact on each bounded subset of B. Then either (i) the equation x = λH x has a solution for λ = 1, or (ii) the set of all solutions x, for 0 < λ < 1, is unbounded. Next, we state Krasnosel’ski˘ı’s fixed point theorem. Theorem 1.1.17 (Krasnosel’ski˘ ı [153]). Let M be a closed convex nonempty subset   of a Banach space B,  ·  . Suppose that A and B map M into B such that (i) x, y ∈ M, implies Ax + By ∈ M, (ii) A is compact and continuous, (iii) B is a contraction mapping. Then there exists z ∈ M with z = Az + Bz. The following lemma comes into play in few places in the book. Lemma 1.1.4 (Young’s Inequality [166]). For any two nonnegative real numbers w and z, we have we z f + , with 1/e + 1/ f = 1. wz ≤ e f

10

1 Introduction to Stability and Boundedness in Dynamical Systems

1.2 Introduction to Lyapunov Functions In this section we define what Peterson and Tisdell [131] call a type I Lyapunov function and summarize some of their results and examples that we will refer to in the later chapters. Just a reminder that the notation [t0, ∞)T := [t0, ∞) ∩ T, where t0 ∈ T. We begin by considering the boundedness and uniqueness of solutions to the first-order dynamic equation x Δ = f (t, x), t ≥ 0,

(1.2.1)

x(t0 ) = x0, t0 ≥ 0, x0 ∈ R,

(1.2.2)

subject to the initial condition

where here x(t) ∈ Rn , f : [0, ∞)T × D → Rn where D ⊂ Rn and open and f is a continuous, nonlinear function and t is from a so-called “time scale” T. Throughout this section we assume 0 ∈ T (for convenience) and that f (t, 0) = 0, for all t in the time scale interval [0, ∞)T , and call the zero function the trivial solution of (1.2.1). Equation (1.2.1) subject to (1.2.2) is known as an initial value problem (IVP) on time scales. If T = R, then x Δ = x , and (1.2.1) and (1.2.2) become the following IVP for ordinary differential equations: x  = f (t, x), t ≥ 0, x(t0 ) = x0, t0 ≥ 0.

(1.2.3) (1.2.4)

Recently, Raffoul [134] used Lyapunov-type functions to formulate some sufficient conditions that ensure all solutions to (1.2.3) and (1.2.4) are bounded, while in a more classical setting, Hartman [86, Chapter 3] employed Lyapunov-type functions to prove that solutions to (1.2.3) and (1.2.4) are unique. Definition 1.2.1. We say V : Rn → R+ is a “type I” Lyapunov function on Rn provided n  V(x) = Vi (xi ) = V1 (x1 ) + . . . + Vn (xn ), i=1

where each Vi :

R+

→

R+

is continuously differentiable and Vi (0) = 0.

The following Chain Rule shall be very useful throughout the remainder of this book. Its proof can be found in Keller [104] and Potzsche [132]. See also Bohner and Peterson [50], Theorem 1.90. Theorem 1.2.1. Let V : R → R be continuously differentiable and suppose that x : T → R is delta differentiable. Then V ◦ x is delta differentiable and

1.2 Introduction to Lyapunov Functions

[V(x(t))]Δ =

∫

1

11

   V  x(t) + hμ (t) x Δ (t) dh x Δ (t).

0

Definition 1.2.2. V :

Rn

→ R is a “type I” function when

V(x) =

n 

Vi (xi ) = V1 (x1 ) + . . . + Vn (xn ),

i=1

where each Vi : R → R is continuously differentiable. Now assume that V : Rn → R is a type I function and x is a solution to (1.2.1). Consider  n Δ n   Δ [V(x(t))] = Vi (xi (t)) = [Vi (xi (t))]Δ, i=1

= =

n ∫ 

i=1

1

0 i=1  ∫ n 1  0

i=1

∫ =

1

 Vi(xi (t) + hμ(t)xiΔ (t))dh xiΔ (t), Vi (xi (t)

 + hμ(t) fi (t, x(t))) dh fi (t, x(t)), 

∇V(x + hμ(t) f (t, x))dh f (t, x)

0

where ∇ = (∂/∂ x1, . . . , ∂/∂ xn ) is the gradient operator. This motivates us to define V : T × Rn → R by either of the following identities: ∫ 1   V(t, x) = ∇V(x + hμ(t) f (t, x))dh f (t, x) 0

=

n ∫  i=1

0

1

 Vi (xi + hμ(t) fi (t, x)) dh fi (t, x)

n ⎧ ⎪ ⎨ i=1 {Vi (xi + μ(t) fi (t, x)) − Vi (xi )} /μ(t), when μ(t)  0, ⎪ = ⎪ ⎪ n V (xi ) fi (t, x), when μ(t) = 0. ⎩ i=1 i

(1.2.5)

If, in addition to the above, V : Rn → [0, ∞), then we call V a type I Lyapunov function. Sometimes the domain of V will be a subset D of Rn . Note that V = V(x) and even if the vector field associated with the dynamic equation is autonomous, then V still depends on t (and x of course) when the graininess function of T is nonconstant. Definition 1.2.3. We say that a type I Lyapunov functional V : [0, ∞)T ×Rn → [0, ∞) is negative definite if V(t, x)  0 for x  0, x ∈ Rn , V(t, x) = 0 for x = 0 and along  x) ≤ 0. If the condition V(t,  x) ≤ 0 does not the solutions of (1.2.1) we have V(t, n hold for all (t, x) ∈ T × R , then the Lyapunov functional is said to be nonnegative definite.

12

1 Introduction to Stability and Boundedness in Dynamical Systems

 x) for each of the following Using the above formulas we can easily calculate V(t, examples: n 1/2 Example 1.2.1. Let V(x) = i=1 xi , for x ∈ D, where D = {x ∈ Rn : xi > 0, xi + μ(t) fi (t, x) ≥ 0, i = 1, 2, · · · , n} . Then  n √ ⎧ ⎪ ⎨ i=1 fi (t, x)/[ xi + μ(t) fi (t, x) + xi ], when μ(t)  0 ⎪  x) = V(t, . ⎪ ⎪ n fi (t, x)/2√ xi, when μ(t) = 0 ⎩ i=1 n Example 1.2.2. Let V(x) = i=1 ai xi2 , for x ∈ Rn and ai > 0, i = 1, 2, · · · , n. For x ∈ Rn , we define the associated weighted vector by w(x) :=< a1 x1, a2 x2, · · · , an xn > . Then

 x) = 2w(x) · f (t, x) + μ(t)w( f (t, x)) · f (t, x). V(t, n 2 In particular, if V(x) =  x 2 = i=1 xi , then  x) = 2x · f (t, x) + μ(t) f (t, x) 2 . (1.2.6) V(t, n Example 1.2.3. Let V(x) = i=1 ai xi4 for x ∈ Rn and ai > 0, i = 1, 2, · · · , n. Then n 4 4 ⎧ ⎪ ⎨ i=1 ai [(xi + μ(t) fi (t, x)) − xi ]/μ(t), when μ(t)  0 ⎪  V(t, x) = . ⎪ ⎪ n 4ai x 3 fi (t, x), when μ(t) = 0 i ⎩ i=1 n Example 1.2.4. Let V(x) = i=1 ai xi4/3 for x ∈ Rn and ai > 0, i = 1, 2, · · · , n. Then  x) = V(t,

n 4/3 4/3 ⎧ ⎪ ⎨ i=1 ai [(xi + μ(t) fi (t, x)) − xi ]/μ(t), when μ(t)  0 ⎪ ⎪ ⎪ n 4a x 1/3 f (t, x)/3, ⎩ i=1 i i i

.

when μ(t) = 0

Example 1.2.5. For Lyapunov functions which may not be power functions, let V(x) =

n ∫  i=1

xi

pi (u)du,

0

where each pi : R → R+ is continuous. Then  x) = V(t,

n ∫  i=1

1

pi (xi + hμ(t) fi (t, x)) fi (t, x)dh.

0

= P(t, x) · f (t, x),

1.2 Introduction to Lyapunov Functions

13

where ∫

1

P(t, x) :=

∫ p1 (x1 + hμ(t) f1 (t, x))dh, . . . ,

0

1

 pn (xn + hμ(t) fn (t, x))dh .

0

Note that if T = R, then P(t, x) = P(x) = (p1 (x1 ), . . . , pn (xn )). We let x(t) = x(t, t0, x0 ) denote a solution of the initial value problem (IVP) (1.2.1), with x(t0 ) = x0, t0 ≥ 0, x0 ∈ R. Next, we give a formula for the Δ derivative of |x| Δ, that we use throughout this book. x(t)  d Details can be found in [10]. One can easily find dt |x(t)| = |x(t) | x (t) by using the equation x 2 (t) = |x(t)| 2 and the product rule in real case. However, on time scales, this is not the case which is due to the product rule. We have the following lemma. Lemma 1.2.1. For any t ∈ T, we have |x| Δ =

x + xσ Δ x for x  0, |x| + |x σ |

(1.2.7)

Proof. Let f (t) = |x(t)|. Then we have f (t) f (t) = x(t)x(t), which implies that  Δ  Δ f (t) f (t) = x(t)x(t) . Using the product rule we obtain  Thus



   f (t) + f (σ(t) f Δ (t) = x(t) + x(σ(t) x Δ (t).

   |x(t)| + |x(σ(t)| |x(t)| Δ = x(t) + x(σ(t) x(t)Δ .

Suppressing t from the argument and solving for |x| Δ we arrive at formula (1.2.7). This completes the proof. Notice that the coefficient of x Δ in (1.2.7) depends not only on the sign of x(t) but also on that of x σ (t). Therefore, the equality |x| Δ = |xx | x Δ holds only if x x σ ≥ 0 and x  0. Let us keep this case distinct from the case x x σ < 0 by separating the time scale T into two parts as follows (see also ([10])) T−x : = {s ∈ T : x (s) x σ (s) < 0} , T+x : = {s ∈ T : x (s) x σ (s) ≥ 0} .

(1.2.8)

Note that the set T−x consists only of right scattered points of T. To see the relation between |x| Δ and |xx | x Δ we prove the next result.

14

1 Introduction to Stability and Boundedness in Dynamical Systems

Lemma 1.2.2 ([10, Lemma 5]). Let x  0 be Δ-differentiable, then  x(t) x Δ (t) if t ∈ T+x . |x (t)| Δ = |x(t)2 | x(t) Δ − μ(t) |x (t)| − |x(t) | x (t) for t ∈ T−x In particular,

x Δ x x ≤ |x| Δ ≤ − x Δ (t) for all t ∈ T−x . |x| |x|

Proof. If t ∈ T+x , then x x σ ≥ 0, and from (1.2.7) we obtain |x| Δ = x  0. Let t ∈ T−x . Then t is right scattered (i.e., μ(t) > 0) and 

since

xσ |x σ |

x |x|

Δ

x Δ |x | x

since

 x 1 xσ − μ |x σ | |x| 2 x =− μ |x| =

= − |xx | x for t ∈ T−x . Applying product rule, the equality |x| =  Δ σ x x x+ xΔ |x| |x| x Δ 2 x x− x =− μ |x| |x| 2 x Δ x = − |x| − μ |x|

|x| Δ =

(1.2.9)

x |x | x

gives



(1.2.10)

in which the last equality yields |x| Δ ≤ − |xx | x Δ, and the second equality implies  x  −2x − μx Δ |x| μ x = (−x σ − x) |x| μ x x Δ ≥ (x σ − x) = x μ |x| |x|

|x| Δ =

since x x σ < 0 for t ∈ T−x . This completes the proof. We can combine the inequalities given in Lemma 1.2.2 as follows. Remark 1.2.1. If x x σ  0, then x Δ xσ x ≤ |x| Δ ≤ σ x Δ for t ∈ T. |x| |x |

(1.2.11)

1.2 Introduction to Lyapunov Functions

15

Observe that if x x σ  0 and t ∈ T+x , then

x |x | − |xx |

=

xσ |x σ | , and therefore, (1.2.11) gives xσ |x σ | and the inequality (1.2.11) is

= |x| Δ . Moreover, if t ∈ T−x , then = equivalent to (1.2.9). For x ∈ Rn , ||x|| denotes the Euclidean norm of x. For any n × n matrix A, define the norm of A by | A| = sup{| Ax| : ||x|| ≤ 1}. x Δ |x | x

Definition 1.2.4. A solution x(t) of (1.2.1) is said to be bounded if for any t0 ∈ [0, ∞)T and number r there exists a number α(t0, r) depending on t0 and r such that ||x(t, t0, x0 )|| ≤ α(t0, r) for all t ≥ t0 and x0, |x0 | < r. It is uniformly bounded if α is independent of the initial time t0 . Definition 1.2.5. Let x(t) be the solution of (1.2.1) with respect to initial condition x0 and y(t) be the solution of (1.2.1) with respect to initial condition y0 . The solution x(t) is then said to be stable, if, whenever > 0 is given, there exists δ( ) for which ||x(t) − y(t)|| < , whenever ||x0 − y0 || < δ. Let t0 ∈ [0, ∞)T and consider x Δ = 1, x(t0 ) = x0 .

(1.2.12)

It is easy to check that x(t) = x0 + (t − t0 ) is the solution of (1.2.12). If y(t) is another solution with y(t0 ) = y0, then we have y(t) = y0 + (t − t0 ). For any > 0, let δ = . Then ||x(t) − y(t)|| = ||x0 + (t − t0 ) − y0 − (t − t0 )|| = ||x0 − y0 || < whenever, ||x0 − y0 || < δ. Hence, the solution x(t) is stable, but unbounded. This simple example shows that the properties of boundedness of all solutions and stability of a solution do not coincide. Throughout this book we denote wedges by Wi, i = 1, 2, 3, . . . such that Wi [0, ∞)T → [0, ∞) be continuous with Wi (0) = 0, Wi (r) strictly increasing, and Wi (r) → ∞ as r → ∞. Theorem 1.2.2. Suppose there exist a “type I” function V(t, x) and W1 such that W1 (||x||) ≤ V(t, x), t ≥ t0

(1.2.13)

 x) ≤ 0, V(t,

(1.2.14)

 x) is given by (1.2.5) and where V(t, W1 (||x||) → ∞, as ||x|| → ∞.

(1.2.15)

Assume for any initial time t0 with x(t0 ) = x0 , and V(t0, x0 ) is bounded, then solutions of (1.2.1) are bounded.

16

1 Introduction to Stability and Boundedness in Dynamical Systems

Proof. Let r0 be any positive constant such that ||x0 || ≤ r0 . By (1.2.15) and since V(t0, x0 ) is bounded, there exists a function α(t0, r0 ) such that  V(t0, x0 ) ≤ W1 α(t0, r0 )). Utilizing conditions (1.2.13) and (1.2.14) we have   W1 (||x||) ≤ V(t, x) ≤ V(t0, x0 ) ≤ W1 α(t0, r0 ) .

(1.2.16)

Taking the inverse in (1.2.16) we arrive at ||x(t, t0, x0 )|| ≤ α(t0, r0 ). This completes the proof. For more on such results we refer to [90]. Example 1.2.6. Consider the scalar nonlinear Volterra integro-dynamic equation x Δ (t) = a(t)x(t) + b(t)

x(t) , t ∈ [0, ∞)T . ∫t 1 + 0 x 2 (s)Δs

Let the function α : T → R be defined by  2 − μ(t) − a(t) for t ∈ [0, ∞)T−x α(t) := . a(t) for t ∈ [0, ∞)T+x

(1.2.17)

(1.2.18)

If there exists a β > 0 such that α(t) + |b(t)| ≤ −β for all t ∈ [0, ∞)T, then all solutions of (1.2.17) are bounded. To see this, consider the Lyapunov function V(t, x) = |x(t)|. Then along the solutions of (1.2.17) we have by using Lemma 1.2.2  x) = x x Δ (t) V(t, |x| " ! x x(t) = a(t)x(t) + b(t) ∫t |x| 1 + 0 x 2 (s)Δs " ! x2 1 = a(t) + b(t) ∫t |x| 1 + x 2 (s)Δs 0

≤ (a(t) + |b(t)|) |x(t)| = (α (t) + |b(t)|) |x(t)| ≤ −β|x(t)| for t ∈ [0, ∞)T+x . On the other hand, for t ∈ [0, ∞)T−x , we have

1.2 Introduction to Lyapunov Functions

17

 x) = − 2 |x (t)| − x (t) x Δ (t) V(t, μ (t) |x (t)| " ! x 2 x(t) =− a(t)x(t) + b(t) |x (t)| − ∫t μ (t) |x| 1 + 0 x 2 (s)Δs " ! 2 x2 1 =− a(t) + b(t) |x (t)| − ∫t μ (t) |x| 1 + 0 x 2 (s)Δs ! " 2 1 = |x (t)| − − a(t) − b(t) ∫t μ (t) 1 + 0 x 2 (s)Δs   2 ≤ − − a(t) + |b(t)| |x(t)| μ (t) = (α (t) + |b(t)|) |x(t)| ≤ −β|x(t)|. Clearly, W1 (||x||) = ||x|| → ∞, as ||x|| → ∞. Thus the results follow from Theorem 1.2.2. Theorem 1.2.2 has limitations as it cannot be applied to Volterra integro-dynamic equations of the form ∫ t x Δ (t) = a(t)x(t) + b(t, s)x(s)Δs, t ∈ [0, ∞)T . 0

To see this, using the same V(t, x) as in the previous example, we arrive at ∫ t  V(t, x) ≤ α(t)|x(t)| + |b(t, s)||x(s)|Δs, 0

where α(t) is given by (1.2.18). It is clear that nothing can be deduced from the above inequality based on Theorem 1.2.2. In Chap. 3 we will develop theorems that will handle such equations. The rest of the materials in this section is new and belongs to the authors. The next lemma is new and necessary for the upcoming theorem. Lemma 1.2.3. Let the function p : [0, ∞)T → R be regressive; 1 + μ(t)p(t)  0. Then ∫ t  ∫ t  ξμ(τ) p(τ)Δτ ≤ exp |p(τ)|Δτ . (1.2.19) e p (t, s) = exp s

s

Proof. Assume 1 + μ(u)p(u) > 0 and define f : (−1, ∞) → R by f (x) = x − log(1 + x).

18

1 Introduction to Stability and Boundedness in Dynamical Systems

Then we have f (x) ≥ 0 for all x > −1. By definition, we have  p(u) if μ(u) = 0   ξμ(u) p(u) = log 1+μ(u)p(u) if μ(u) > 0. μ(u) Thus for μ > 0 and 1 + μ(u)p(u) > 0 we have that     log 1 + μ(u)p(u) ξμ(u) p(u) = μ(u)   f μ(u)p(u) = p(u) − μ(u) ≤ p(u). Thus e p (t, s) = exp

∫ s

t

 ∫ ξμ(τ) p(τ)Δτ ≤ exp

t

 |p(τ)|Δτ .

s

Next we assume 1 + μ(u)p(u) < 0 and define f : (−∞, −1) → R by f (x) = |x| − log |1 + x|. Then we have f (x) ≥ 0 for all x < −1. By [50, Theorem 2.44] we know that if 1 + μ(u)p(u) < 0 on Tk , then e p (t, s) = α(t, t0 )(−1)nt for all t ∈ [0, ∞)T where  ∫ t log 1 + μ(s)p(s)  Δs > 0 α(t, t0 ) = exp μ(s) t0 

and nt =

|[t0, t)| if t ≥ t0 |[t, t0 )| if t < t0 .

That is e p (t, t0 ) = α(t, t0 ). Since,     log 1 + μ(s)p(s) f μ(s)p(s) = |p(s)| − μ(s) μ(s) ≤ |p(s)|, we have that α(t, t0 ) ≤ exp

∫

t

 |p(s)|Δs .

t0

That is e p (t, t0 ) ≤ exp

∫ t0

This completes the proof.

t

 |p(s)|Δs .

1.2 Introduction to Lyapunov Functions

19

Theorem 1.2.3. Let the function h : [0, ∞)T → [0, ∞) be regressive; (1 + μ(t)h(t)  0). Given x0  0 suppose there exists a “type I” function V(t, x) and such that for t ∈ [t0, ∞)T W1 (||x||) ≤ V(t, x),  x) ≤ h(t)V(t, x) + g(t), V(t,

(1.2.20)

 x) is given by (1.2.5) . Then all solutions of (1.2.1) are bounded prowhere V(t, ∫ ∞  ∫ ∞ vided that exp |h(t)|Δt ≤ M1 and |g(t)|Δt ≤ M2 for positive constants M1 and M2 .

0

0

Proof. Using Lemma 1.2.3 and an application of the variation of parameters formula on (1.2.20) give ∫ t eh (t, σ(τ))g(τ)Δτ V(t, x) ≤ V(t0, x0 )eh (t, t0 ) + t0  ∫ t |g(t)|Δt ≤ M1 V(t0, x0 ) + t0

≤ M1 [V(t0, x0 ) + M2 ] , since e p (t, σ(s)) ≤ exp at

∫ t σ(s)

 |p(s)|Δs . Thus from the above inequality we arrive

  $ 1/2 # . ||x|| ≤ W1−1 M1V(t0, x0 ) + M1 M2

(1.2.21)

This completes the proof. Remark 1.2.2. It is clear from (1.2.21) that if V(t0, x0 ) ≤ C(|x0 |) where C is a positive constant independent of t0 , then solutions of (1.2.1) are uniformly bounded . We have the following application. Example 1.2.7. Consider the scalar dynamical equation x Δ (t) = a(t)x(t) + f (t), x(t0 ) = x0, t ∈ [t0, ∞)T with t0 ≥ 0.

(1.2.22)

For the rest of the example we write a and f ; we suppress t in both functions. Let V(t, x) = x 2 . By Theorem 1.1.1 (3) we have along the solutions of (1.2.22) that      x) = 2a + μ(t)a x 2 + 1 + μ(t)a 2x f + μ(t) f 2 . V(t, (1.2.23) We notice that



   1 + μ(t)a 2x f ≤ | 1 + μ(t)a |2|x|| f | 1/2 | f | 1/2    ≤ | 1 + μ(t)a | x 2 | f | + | f |).

Thus, (1.2.23) reduces to    x) = 2a + μ(t)a + |1 + μ(t)a|| f | x 2 + |1 + μ(t)a|| f | + μ(t) f 2 . V(t,

(1.2.24)

20

1 Introduction to Stability and Boundedness in Dynamical Systems

Let h(t) := 2a + μ(t)a + |1 + μ(t)a|| f | and g(t) := |1 + μ(t)a|| f | + μ(t) f 2 then (1.2.24) is equivalent to  x) ≤ h(t)V(t, x) + g(t). V(t, If we let a(t) = f (t) =

1 , (t + 1)3

then we have h(t) =

2μ (t) μ(t) 1 2μ(t) 3 + + and g(t) = + (t + 1)3 (t + 1)3 (t + 1)6 (t + 1)3 (t + 1)6

(1.2.25)

3 1 1. If T = R, we have μ(t) = 0 and hence h(t) = (t+1) 2 and g(t) = (t+1)2 . Moreover, ∫ ∞ ∫ ∞ |h(t)|Δt = 3 and |g(t)|Δt = 1 and by Theorem 1.2.3 and Remark 1.2.2 0

0

all solutions of (1.2.22) are uniformly bounded. 2. If T = Z, we have μ(t) = 1 and hence ∫

∞

|h(t)|Δt =

0

and

∫

∞   k=0

∞

|g(t)|Δt =

0

5 1 π6 < ∞, + = 5ζ(3) + 3 6 945 (k + 1) (k + 1)

∞   k=0

1 2 2π 6 < ∞, + = ζ(3) + 945 (k + 1)3 (k + 1)6

where ζ(s) is Riemann zeta function. By Theorem 1.2.3 and Remark 1.2.2 all solutions of (1.2.22) are uniformly bounded. % 3. If T = P1,1 = [2k, 2k + 1], then k ∈Z



& t + 1 if t ∈ k ∈Z {2k + 1} & t if t ∈ k ∈Z [2k, 2k + 1)



& 1 if t ∈ &k ∈Z {2k + 1} . 0 if t ∈ k ∈Z [2k, 2k + 1)

σ(t) = and hence μ(t) = For any ξ : P1,1 → R we have ∫ t

By (1.2.25),

σ(t)

 ξ (t) Δt =

& ξ (t) if t&∈ k ∈Z {2k + 1} . 0 if t ∈ k ∈Z [2k, 2k + 1)

1.2 Introduction to Lyapunov Functions

h(t) =

21

3 2μ (t) μ(t) 1 2μ (t) + + and g(t) = + . 3 3 3 6 (t + 1) (t + 1) (t + 1) (t + 1) (t + 1)6

Then ∫

∞

∞ ∫ 

|h (t)| Δt =

0

k=0

∞ ∫ 

+

k=0

3 4

=

|g (t)| Δt =

0

k=0

+

∞ ∫  k=0

= =

1 4

2k+2

2k+1

k=0 7π 4

960

2k+1

2k

∞ 

2k+1



∞ ∫ 2k+2  3 3 Δt + Δt (t + 1)3 (t + 1)3 k=0 2k+1  2 1 + Δt 3 (t + 1) (t + 1)6

 1 5 1 − Δt + 4 4 (2k + 1) (2k + 2) (2k + 2)3 k=0 ∞

1 (2k + 2)6

π6 7π 4 5ζ(3) + + 0, there is a δ = δ(ε) > 0 such that |x(t0 )| < δ implies |x(t, t0, x0 )| < ε. It is uniformly stable (US) if δ is independent of t0 . Definition 1.2.9. The zero solution of (1.2.1) is asymptotically stable (AS) if its (S) and |x(t, t0, x0 )| → 0, as t → ∞. Definition 1.2.10. The zero solution of (1.2.1) is uniformly asymptotically stable (UAS) if it is (US) and there that for each μ > 0 ' exists a γ > 0 with the property ( there exists T > 0 such that t0 ≥ 0, with |x0 | < γ, t ≥ t0 +T implies |x(t, t0, φ)| < μ. Theorem 1.2.4. Assume f (t, 0) = 0 and there is a Lyapunov function V for (1.2.1) (see Definition 1.2.6.) 1. If V is positive definite, then x = 0 is stable. 2. If V is positive definite and decrescent, then x = 0 is uniformly stable.  x) is negative definite, then x = 0 3. If V is positive definite and decrescent, and V(t, is uniformly asymptotically stable . 4. If D = Rk and if V is radially unbounded, then all solutions of (1.2.1) are bounded.

1.2 Introduction to Lyapunov Functions

23

Proof.  x) ≤ 0, V is right-dense continuous in x, V(t, 0) = 0, and W1 (|x|) ≤ 1. We have V(t, V(t, x). Let > 0 and t0 ≥ 0 be given. We must find δ such that |x0 | < δ and t ≥ t0 imply |x(t, t0, x0 )| < . (Throughout these proofs we assume is small enough so that |x(t, t0, x0 )| < implies that x ∈ D.) As V is right-dense continuous in x and V(t, 0) = 0 there is a δ > 0 such that |x0 | < δ implies V(t0, x0 ) < W1 ( ). Thus, if t ≥ t0 and |x0 | < δ and x = x(t, t0, x0 ), we have W1 (|x(t)|) ≤ V(t, x) ≤ V(t0, x0 ) < W1 ( ), or |x(t)| < as required. 2. For a given we select a δ > 0 such that W2 (δ) < W1 ( ) where W1 (|x|) ≤ V(t, x) ≤ W2 (|x|). If t0 ≥ 0, we have W1 (|x(t)|) ≤ V(t, x) ≤ V(t0, x0 ) ≤ W2 (|x0 |) < W2 (δ) < W1 ( ), or |x(t)| < as required. 3. Let = 1, and find δ of uniform stability and call it η. Let γ be given. We must find T > 0 such that |x0 | < η, n0 ≥ 0, and t ≥ t0 + T imply |x(t, t0, x0 )| < γ. Pick μ > 0 with W2 (μ) < W1 (γ), so that there is t1 ≥ n0 with |x(t1 )| < μ, then, for n ≥ t1 , we have W1 (|x(t)|) ≤ V(t, x) ≤ V(t1, x1 ) ≤ W2 (|x1 |) < W2 (δ) < W1 (γ),  x) ≤ −W3 (|x|), so as long as |x(t)| > μ, then V(t, x) ≤ or |x(t1 )| < γ. Since V(t, −W3 (μ); thus ∫ t V(t, x(t)) ≤ V(t0, x0 ) − W3 (|x(s)|)Δs t0

≤ W2 (|x0 |) − W3 (μ)(t − t0 ) ≤ W2 (η) − W3 (μ)(t − t0 ), which vanishes at t = t0 +

W2 (η) ≥ t0 + T, W3 (μ)

W2 (η) 2 (η) where T ≥ W W3 (μ) . Hence, if T > W3 (μ) , then |x(t)| > μ fails, and we have |x(t)| < γ for all t ≥ t0 + T . This proves (UAS). 4. Since V is radially unbounded, we have V(t, x) ≥ W1 (|x|) → ∞ as |x| → ∞. Thus, given t0 ≥ 0, and x0 , there is an r > 0 with W1 (r) > V(n0, x0 ). Hence, if t ≥ t0 and x(t) = x(t, t0, x0 ), then

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1 Introduction to Stability and Boundedness in Dynamical Systems

W1 (|x(t)|) ≤ V(t, x(t)) ≤ V(t0, x0 ) < W1 (r), or |x(n)| < r. The proof of Theorem 1.2.4 is complete. According to Theorem 1.2.4, all solutions of (1.2.1) are bounded and its zero solution is (UAS).  x) ≤ 0 is not enough to drive solutions to zero. In the next example we show that V(t, Lemma 1.2.4. Let T be any time scale and g : T → R be Δ-differentiable with g(t)  0 for all t ∈ T and the function p : [0, ∞)T → R, defined by p(t) := 2

 Δ 2 g (t) g Δ (t) + μ (t) , g (t) g (t)

(1.2.26)

be regressive. Then e p (t, t0 ) =

g 2 (t) . g 2 (t0 )

Proof. Let y(t) = g 2 (t) /g 2 (t0 ). Then by using g σ (t) = g (t) + μ (t) g Δ (t), we have y Δ (t) =

1 g 2 (t

0)

[g (t) + g σ (t)] g Δ (t) '

( g (t) + g (t) + μ (t) g Δ (t) g Δ (t) 0)   Δ 2 2 Δ g (t) g (t) g (t) + μ (t) = 2 g (t) g (t) g 2 (t0 ) =

1

g 2 (t

= p(t)y(t). The proof follows from the uniqueness of the solution e p (t, t0 ) of the following initial value problem: y Δ = p(t)y, y (t0 ) = 1 (see [50], Theorem 2.33). Example ∫ ∞ 1.2.8. Let g : [0, ∞)T → (0, 1] be a Δ-differentiable function with g(0) = 1 and 0 g(s)Δs < ∞. Suppose that the function p(t) defined by (1.2.26) is regressive. We wish to construct a function V(t, x) = a(t)x 2 with a(t) > 0 and with the derivative of V along any solution of  Δ g (t) Δ x (1.2.27) x = g (t) satisfying

.

V(t, x) = −x 2 . .

We shall, thereby, see that V ≥ 0 and V negative definite do not imply that solutions tend to zero, because x(t) = g(t) is a solution of (1.2.27). To this end, we compute

1.2 Introduction to Lyapunov Functions

25

 x) = aσ (t) [x (t) + x σ (t)] x Δ (t) + aΔ (t) x 2 (t) V(t,  !   Δ 2" g (t) g Δ (t) σ Δ + μ (t) = a (t) 2 + a (t) x 2 (t) g (t) g (t) where we used (1.2.27) and x σ (t) = x (t) + μ (t) x Δ (t). Setting .

V(t, x) = −x 2 (t) we have

!

 Δ 2" g (t) g Δ (t) + μ (t) a (t) = −a (t) 2 − 1. g (t) g (t) Δ

σ

Using variation of parameters formula and Lemma 1.2.4, the equality e p (t, s) =   −1 e p (t, s) , we get ∫ t e p (t, s)Δs a (t) = e p (t, 0)a(0) − 0 ∫ t 2 g (s) a (0) − Δs, = 2 2 g (t) 0 g (t) where p(t) is given by (1.2.26).

∫t Since 0 < g (t) ≤ 1 and g is in L 1 [0, ∞)T , we have 0 g 2 (s) Δs < ∞ (see Theorem 5.52, [51]). That means, we may choose a(0) large enough so that a(t) > 1 on  x) is negative definite do not imply [0, ∞)T .Thus we have shown that V ≥ 0 and V(t, that solutions tend to zero. Notice that V is not decrescent. That is, there is no wedge W2 with V(t, x) ≤ W2 (|x|). In the next theorem we obtain necessary conditions for the instability of the zero solution of (1.2.1). Definition 1.2.11. The zero solution of (1.2.1) is unstable if there is an ε > 0, and t0 ≥ 0, such that for any δ > 0 there is an x0 with |x0 | < δ and there is a t1 > t0 such that |x(t1, t0, x0 )| ≥ ε. Theorem 1.2.5. Let the function V : [0, ∞)T × D → R, be right-dense continuous in  x) with respect to (1.2.1) is given by (1.2.5) which is locally Lipschitz in x and V(t, x such that (1.2.28) W1 (|x|) ≤ V(t, x) ≤ W2 (|x|), and along the solutions of (1.2.1) we have  x) ≥ W3 (|x|). V(t,

(1.2.29)

Then the zero solution of (1.2.1) is unstable. Proof. Suppose not, then for = min{1, d(0, ∂D)} we can find a δ > 0 such that |x0 | < δ and t ≥ 0 imply that |x(t, 0, x0 )| < . We may pick x0 in such a way so that

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1 Introduction to Stability and Boundedness in Dynamical Systems

|x0 | = δ/2 and find γ > 0 with W2 (γ) = W1 (δ/2). Then for x(t) = x(t, 0, x0 ) we have  x) ≥ 0 so that V(t, W2 (|x(t)|) ≥ V(t, x(t)) ≥ V(0, x0 ) ≥ W1 (δ/2) = W2 (γ) from which we conclude that γ ≤ |x(t)| for t ≥ 0. Thus  x) ≥ W3 (|x(t)|) ≥ W3 (γ). V(t, Thus, W2 (|x(t)|) ≥ V(t, x(t)) ≥ V(0, x0 ) + tW3 (γ), from which we conclude that |x(t)| → ∞, which is a contradiction. This completes the proof. In the next example we show that stability or instability is inherently dependent on the chosen time scale. Example 1.2.9. Let T be any time scale that includes 0 and consider the autonomous system x Δ = −x − xy y Δ = −y + x 2 for t ∈ [0, ∞)T . Let

(1.2.30)

V(x, y) = x 2 + y 2 .

Then we have along the solutions of (1.2.30) that  x) = 2x · f (t, x) + μ(t) f (t, x) 2 V(t, = 2x(−x − xy) + 2y(−y + x 2 ) 2  2  + μ(t) − x − xy + μ(t) − y + x 2   = −2(x 2 + y 2 ) + μ(t) x 2 + y 2 + x 2 y 2 + x 4 .

(1.2.31)

Thus, for T = R we have μ(t) = 0 and from the above inequality we see that the zero solution of (1.2.30) is (UAS) by Theorem 1.2.4. On the other hand, if T = Z, then μ(t) = 1 and  x) = −(x 2 + y 2 ) + x 2 (x 2 + y 2 ) V(t, ≥ (x 2 + y 2 )(x 2 − 1) > 0 on the set D = {(x, y) ∈ R2 : |x| > 1}. Thus, the zero solution is unstable by Theorem 1.2.5. Next we let T = hN0 = {0, 2h, . . .}, then μ(t) = h and in this case system (1.2.30) takes the form

1.2 Introduction to Lyapunov Functions

27

Δx(k) = −x(k) − x(k)y(k) h Δy(k) = −y(k) + x 2 (k), h k = 0, h, 2h, · · · Then for h ≥ 2, we have from (1.2.31) that  x) = −2(x 2 + y 2 ) + h(x 2 + y 2 + x 2 y 2 + x 4 ) V(t, ≥ x2 y2 + x4 ≥ 0 and hence the zero solution is unstable by Theorem 1.2.5. Assume the function a(t) is right-dense continuous and that a(t) > 0 for all t ∈ [0, ∞)T and consider the two-dimensional system x Δ = −a(t)x + a(t)y y Δ = −a(t)x − a(t)y.

(1.2.32)

1 − μ(t)a(t) > 0,

(1.2.33)

Theorem 1.2.6. Assume then (0, 0) of (1.2.32) is (UAS). Proof. Let V(x, y) = x 2 + y 2 . Then we have along the solutions of (1.2.32) that  x) = 2x · f (t, x) + μ(t) f (t, x) 2 V(t, = 2x(−a(t)x + a(t)y) + 2y(−a(t)x − a(t)y)  2  2 + μ(t) − a(t)x + a(t)y + μ(t) − a(t)x − a(t)y   = −2a(t)(x 2 + y 2 ) + 2μ(t) a2 (t)x 2 + a2 (t)y 2 ' ( = −2a(t)(x 2 + y 2 ) 1 − μ(t)a(t) . Thus, by Theorem 1.2.4, (0, 0) of (1.2.32) is (UAS). In the investigation of stability and boundedness of nonlinear nonautonomous systems, the best approach is to study equations that are related in some way to the linear equations whose behavior is known to be covered by the known theory. To see this, let Cr d (T, R) be the space of all right-dense continuous functions from T into R and let f : [t0, ∞)T × R → Rn with f (t, 0) = 0 and consider the nonlinear dynamical system (1.2.34) x Δ = A(t)x(t) + f (t, x), x(t0 ) = x0, t ∈ [t0, ∞)T, where A(t) is n × n matrix such that all of its entries belong to Cr d (T, R) and x is n × 1 vector. If D is a matrix, then |D| means the sum of the absolute values of the elements. For what to follow we write x for x(t).

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1 Introduction to Stability and Boundedness in Dynamical Systems

Stability results can be found in [71] for the case f (t, x) = 0. The next theorem shows that the stability of (1.2.34) depends only on the stability of linear part when the function f is uniformly bounded in t and x. Theorem 1.2.7. Assume there exists a n × n positive definite constant symmetric matrix B such that for positive constants γ1, γ2, 2|B|| f (t, x)| ≤ γ1 |x|

(1.2.35)

2μ(t)| f (t, x)||BA| ≤ γ2 |x|

(1.2.36)

and uniformly for t ∈ [t0, ∞)T . If there exists positive constant ξ, where − βξ2 ∈ R + and AT B + BA + μ(t)AT BA ≤ −ξ I,

(1.2.37)

where −ξ + γ1 + γ2 ≤ 0, then the zero solution of (1.2.34) is uniformly asymptotically stable. Proof. Let the matrix B be defined by (1.2.37) and define V(t, x) = xT Bx.

(1.2.38)

Here xT x = x 2 = (x12 + x22 + · · · + xn2 ). Using the product rule given in [50] we have along the solutions of (1.2.34) that  x) = (x Δ )T Bx + (x σ )T Bx Δ V(t, = (x Δ )T Bx + (x + μ(t)x Δ )T Bx Δ = (x Δ )T Bx + xT Bx Δ + μ(t)(x Δ )T Bx Δ ' ( = x AT B + BA + μ(t)AT BA xT + 2xT B f (t, x) + μ(t) f T (t, x)BAx + μ(t)xT AT B f (t, x) + μ(t) f T (t, x)B f (t, x) ( ' = x AT B + BA + μ(t)AT BA xT + 2xT B f (t, x) + 2μ(t) f T (t, x)BAx + μ(t) f T (t, x)B f (t, x) ≤ −ξ |x| 2 + 2|x||B|| f (t, x)| + 2μ(t)| f (t, x)||BA||x| + μ(t)|B| f 2 (t, x) ≤ (−ξ + γ1 + γ2 )|x| 2 ≤ 0. Since the matrix B is constant and symmetric positive definite, there are positive constants α2, β2 and such that α2 xT x ≤ xT Bx ≤ β2 xT x.

1.2 Introduction to Lyapunov Functions

29

This implies that the Lyapunov function V is positive definite and decrescent, and  x) is negative definite. Thus, by Theorem 1.2.4 the zero solution is uniformly V(t, asymptotically stable. For good reference on how to go about finding the matrix B on time scales, we refer to [71]. For more on the stability of the dynamical system x Δ = A(t)x(t), we refer the reader to [71] and to the book [121]. When Lyapunov functionals are used to study the behavior of solutions of functional dynamic equations we often end up with a pair of inequalities of the form ∫ t V(t, x(·)) = W1 (x(t)) + K(t, s)W2 (x(s))Δs, s=0

and

·

V(t, x(·)) ≤ −W3 (x(t)) + F(t). The above two inequalities are rich in information regarding the qualitative behavior of the solutions. However, getting such information will require deep knowledge in the analysis of Lyapunov functionals. In Chap. 3, we will develop general theorems to deal with such inequalities that arise from the assumption of the existence of a Lyapunov functional. To be specific, we consider the nonlinear Volterra integrodynamic equation ∫ t x Δ (t) = a(t) x(t) + B(t, s) f (s, x(s))Δs + g(t, x(t)), t ≥ 0, (1.2.39) 0

where a(t) is continuous for t ≥ 0 and B(t, s) is right-dense continuous for 0 ≤ s ≤ t < ∞. We assume f (t, x) and g(t, x) are continuous in x and t and satisfy |g(t, x)| ≤ γ1 (t) + γ2 (t) |x(t)|,

(1.2.40)

| f (t, x)| ≤ γ(t) |x(t)|,

(1.2.41)

where γ(t) and γ2 (t) are positive and bounded, and γ1 (t) is nonnegative and bounded. Define ∫ t∫ ∞ |B(u, s)|Δu| f (s, x(s))|Δs, (1.2.42) V(t, x(.)) = |x(t)| + k 0

t

and let α(t) be given by (1.2.18). Then by Theorem 1.1.10 and Lemma 1.2.2, we have along the solutions of (1.2.39) that ∫ ∞ Δ  V(t, x) = |x (t)| + k |B(u, t)|Δu| f (t, x(t))| ∫ −k 0

σ(t)

t

|B(t, s)|| f (s, x(s))|Δs

30

1 Introduction to Stability and Boundedness in Dynamical Systems

∫ t ≤ α(t)|x(t)| + |B(t, s)|| f (s, x(s))|Δs + |g(t, x(t))| 0 ∫ t ∫ ∞ |B(u, t)|Δu| f (t, x(t))| − k |B(t, s)|| f (s, x(s))|Δs +k σ(t) 0 ∫ ∞ # $ ≤ α(t) + γ2 (t) + k |B(u, t)|Δuγ(t) |x(t)| ∫ + (1 − k)

σ(t)

t

|B(t, s)|| f (s, x(s))|Δs + γ1 (t)

0

≤ −η|x(t)| + γ1 (t),

(1.2.43)

provided that #

∫ α(t) + γ2 (t) + k

∞

σ(t)

$ |B(u, t)|Δuγ(t) ≤ −η < 0,

γ1 (t) is bounded and k > 1 for positive constant η, where α is as in (1.2.18). It will be shown in Chap. 3 that all solutions of (1.2.39) are uniformly bounded.

1.3 Delay Dynamic Equations In this brief section we introduce delay dynamic equations and prove a general theorem in term of the existence of Lyapunov functionals and show that the zero solution is (US). Thus, we consider the general dynamic system with delay x Δ (t) = f (t, x(δ(t))), t ∈ [t0, ∞)T

(1.3.1)

on an arbitrary time scale T which is unbounded above and 0 ∈ T. Here the function f is rd-continuous, where x ∈ Rn and f : [t0, ∞)T × Rn → Rn with f (t, 0) = 0. The delay function δ : [t0, ∞)T → [δ(t  T is strictly increasing, invertible, and delta  0 ), ∞) differentiable such that δ(t) < t, δΔ (t) < ∞ for t ∈ T, and δ(t0 ) ∈ T. For each t0 ∈ T and for a given r d-continuous initial function ψ : [δ(t0 ), t0 ]T → Rn , we say that x(t) := x(t; t0, ψ) is the solution of (1.3.1) if x(t) = ψ(t) on [δ(t0 ), t0 ]T and satisfies (1.3.1) for all t ≥ t0 . We define Et0 = [δ(t0 ), t0 ]T which we call the the initial interval. For x ∈ Rn , |x| denotes the Euclidean norm of x. For any n × n matrix A, | A| will denote any compatible norm so that | Ax| ≤ | A||x|. Let CH denote the set of rd-continuous functions φ : [δ(t0 ), t]T → Rn and φ = sup{|φ(s)| : δ(t0 ) ≤ s ≤ t} < H for some positive constant H.

1.3 Delay Dynamic Equations

31

Definition 1.3.1. The zero solution of (1.3.1) is stable if for each ε > 0 and each t0 ≥ 0, there exists a δ1 = δ1 (ε, t0 ) > 0 such that [φ ∈ Et0 → Rn, φ ∈ CH : φ < δ1 < H] implies |x(t, t0, φ)| < ε for all t0 ≥ 0. Definition 1.3.2. The zero solution of (1.3.1) is uniformly stable (US) if it is stable and δ1 is independent of t0 . For the next theorem, if φ ∈ CH , then we define |||φ||| =

n ∫ # i=1

t0

δ(t0 )

φ2i (s)Δs

$ 1/2 ,

where φ(t) = (φ1 (t), · · · , φn (t)) and φi is the ith component of φ. Theorem 1.3.1. Suppose H > 0 and there is a scalar functional V(t, φ) that is r d-continuous in φ and locally Lipschitz in φ when t ≥ t0 and φ ∈ CH . If V satisfies W(|φ(0)|) ≤ V(t, φ) ≤ W1 (|φ(0)|) + W2 (|||φ|||), and

 φ) ≤ 0, V(t,

then the zero solution of (1.3.1) is (US). Proof. Let > 0 be given such that < H and choose δ1 > 0 so that W1 (δ1 ) + W2 ([δ12 n(t0 − δ(t0 ))]1/2 ) < W( ). Then, for an initial function φ ∈ CH satisfying ||φ|| < δ1, we have along the solutions  φ) ≤ 0 and hence for x(t) = x(t, t0, φ) we have that of (1.3.1) V(t, W(|x(t)|) ≤ V(t, xt (t0, φ)) ≤ V(t0, φ) ≤ W1 (|φ(0)|) + W2 (|||φ|||) ≤ W1 (δ1 ) + W2 ([δ12 n(t0 − δ(t0 ))]1/2 ) < W( ). Thus, the zero solution (1.3.1) is (US). This completes the proof. Next we display an example with two different types of Lyapunov functionals to illustrate that the end results and conditions are dependent on the suitable Lyapunov functionals. We begin by considering the delay dynamical equation x Δ (t) = ax(t) + bx(δ(t)), t ∈ [t0, ∞)T,

(1.3.2)

where δ(t) enjoys the same properties as stated in this section for Eq. (1.3.1). We have the following propositions. Proposition 1.3.1. Suppose δΔ (t) ≥

1 |b + 2abμ(t)| + b2 μ(t), 2

(1.3.3)

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1 Introduction to Stability and Boundedness in Dynamical Systems

and a + μ(t)a2 + 1 +

1 |b + 2abμ(t)| ≤ 0, 2

(1.3.4)

then the zero solution of (1.3.2) is (US). Proof. Let V(t, x) =

x 2 (t) + 2

∫

t

δ(t)

x 2 (s)Δs.

Then along the solutions of (1.3.2) we have by using (1.3.3) and (1.3.4) that  x) V(t,

=

 2 x(t)x Δ (t) + μ(t) ax(t) + bx(δ(t)) + x 2 (t) − x 2 (δ(t))δΔ (t)

=

(a + a2 μ(t) + 1)x 2 (t) + (b + 2abμ(t))x(t)x(δ(t)) + μ(t)b2 x 2 (δ(t)) − x 2 (δ(t))δΔ (t) 1 |b + 2abμ(t)|)x 2 (t) 2

≤

(a + a2 μ(t) + 1 +

+

1 ( |b + 2abμ(t)| + μ(t)b2 )x 2 (δ(t)) − x 2 (δ(t))δΔ (t) ≤ 0. 2

Thus, by Theorem 1.3.1 the zero solution of (1.3.2) is (US). This completes the proof. 1 , condition (1.3.4) requires that We note for the special case b = a and μ(t) = − 2a a ≤ −2. In the next proposition we consider a different suitable Lyapunov functional and deduce that the conditions are less stringent. In preparation for the next result, define the function α : T → R by  2 − μ(t) − a for t ∈ [0, ∞)T−x α(t) := a for t ∈ [0, ∞)T+x Proposition 1.3.2. Suppose

δΔ (t) ≥ |b|

(1.3.5)

α(t) + 1 ≤ 0

(1.3.6)

and for all t ∈ T, then the zero solution of (1.3.2) is (US). Proof. Let

∫ V(t, x) = |x(t)| +

t

δ(t)

|x(s)|Δs.

Let t ∈ [0, ∞)T+x . Then along the solutions of (1.3.2) we have by using (1.3.5) and (1.3.6) that  x) = x(t) x Δ (t) + |x(t)| − |x(δ(t))|δΔ (t) V(t, |x(t)| = (a + 1)|x| + (|b| − δΔ (t)) ≤ (α(t) + 1)|x(t)| ≤ 0.

1.4 Resolvent for Volterra Integral Dynamic Equations

33

On the other hand, for t ∈ [0, ∞)T−x , we have by using (1.3.5) and (1.3.6) that  x) = − 2 |x(t)| − x(t) x Δ (t) + |x(t)| − |x(δ(t))|δΔ (t) V(t, μ(t) |x(t)| 2 − a + 1)|x(t)| + (|b| − δΔ (t)) = (− μ(t) ≤ (α(t) + 1)|x(t)| ≤ 0. Thus, by Theorem 1.3.1 the zero solution of (1.3.2) is (US). This completes the proof. We note that (1.3.5) and (1.3.6) are less stringent which is due to the new Lyapunov functional.

1.4 Resolvent for Volterra Integral Dynamic Equations In this section we develop the concept of the resolvent for Volterra integral dynamic equations that we use in later chapter. Materials of this section can be found in [12]. Results of this section will be applied in Chap. 5 to prove boundedness of solutions of Volterra dynamic equations, using Lyapunov functionals. In [124, Chapter IV], the existence of resolvent kernel r(t, s) corresponding to integral equations of the form ∫ t a(t, s)x(s)ds x(t) = f (t) + 0

is discussed by giving several theorems that provide sufficient conditions on a(t, s). In this paper, we extend the theory established in [124, Chapter IV] to the Volterra integral dynamic equation on time scales ∫ t x(t) = f (t) + a(t, s)x(s)Δs, 0

scales, in which integral and summation equations are included as special cases. For brevity we assume familiarity with basic notions of time scale theory. A comprehensive review on Δ-derivative and Δ-Riemann integral can be found in [50]. In [46, pp. 157–163] a theory for a Lebesgue Δ-integration has been established by means of Lebesgue Δ-measure, denoted μΔ , with the following properties. Theorem 1.4.1 ([46, Theorem 5.76]). For each s0 ∈ Tκ = T\{max T}, the single point set {s0 } is Δ-measurable, and its measure is given by μΔ {s0 } = σ(s0 ) − s0 = μ(s0 ). Theorem 1.4.2 ([46, Theorem 5.77]). If a, b ∈ T and a ≤ b, then μΔ {[a, b)T } = b − a, μΔ {(a, b)T } = b − σ(a). If a, b ∈ Tκ and a ≤ b, then μΔ {(a, b]T } = σ(b) − σ(a), μΔ {[a, b]T } = σ(b) − a.

34

1 Introduction to Stability and Boundedness in Dynamical Systems

Furthermore, it has been concluded that all theorems of the general Lebesgue integral theory, including the Lebesgue Dominated Convergence theorem, hold also for the Lebesgue Δ-integral on time scales. The next theorem gives the relationship between Riemann and Lebesgue integrals on time scales. Theorem 1.4.3 ([51, Theorem 5.81]). Let f be a bounded real-valued function defined on [a, b]T . If f is Riemann Δ-integrable from a to b, then f is Lebesgue Δ-integrable on [a, b) and ∫

b

∫ f (t)Δt =

[a,b)

a

f μΔ .

Definition 1.4.1. We say that a property holds almost everywhere (a.e.) in S ⊂ T if the set of all right dense points of S for which the property does not hold is a null set, i.e., is a set with Δ-measure zero. The following theorem gives a criteria for Δ–Riemann integrability of a function. Theorem 1.4.4. Let f be a bounded function defined on the finite closed interval [a, b]T of the time scale T. Then f is Riemann Δ-integrable from a to b if and only if f is continuous a.e. in [a, b)T . Let T1 and T2 be two time scales. In [49], double Δ-integral of a multivariable function f : T1 × T2 → R over a rectangular region [a, b)T1 × [c, d)T2 has been introduced. The following result provides sufficient conditions for interchanging the order of integration over a rectangular region. Lemma 1.4.1. Let f be Δ-integrable over R = [a, b)T1 × [c, d)T2 and suppose that the single integrals ∫ b f (x, y)Δ1 x K(y) = ∫

a

I(x) =

d

f (x, y)Δ2 y

c

exists for each x ∈ [a, b)T1 and y ∈ [c, d)T2 . Then ∬

∫ f (x, y)Δ1 xΔ2 y =

b

∫ Δ1 x

a

R

∫ = c

d

f (x, y)Δ2 y

c d

∫ Δ2 y

b

f (x, y)Δ1 x.

a

where Δ1 and Δ2 indicate the derivatives on T1 and T2 , respectively. Multiple Δ-integration over more general sets has been defined in [48, Definitions 4.13 and 4.15].

1.4 Resolvent for Volterra Integral Dynamic Equations

35

Theorem 1.4.5 ([48, Theorem 4.30]). Let ϕ : [a, b]Tk → T2k and ψ : [a, b]Tk → T2k 1 1 be two continuous functions such that ϕ(t) < ψ(t) for all t ∈ [a, b]Tk . Let E be a 1 bounded set in T1 × T2 given by E = {(t, s) ∈ T1 × T2 : a ≤ t < b, ϕ(t) ≤ s < ψ(t)}. Then E is Jordan Δ-measurable, and if f : E → R is Δ-integrable over E if the single integral ∫ ψ(t) f (t, s)Δ2 s ϕ(t)

exists for each t ∈ [a, b)T1 , then the iterated integral ∫

b

∫ Δ1 t

ψ(t)

ϕ(t)

a

f (t, s)Δ2 s

exists and we have ∬

∫ f (t, s)Δ1 tΔ2 s =

b

∫ Δ1 t

a

E

ψ(t)

ϕ(t)

f (t, s)Δ2 s.

Now we use some properties of multiple Δ-integral to construct resolvent equations corresponding to linear and nonlinear systems of integro-dynamic equations. Linearity and additivity properties of multiple integral on time scales can be found in [49] and [48]. Let t0 ∈ Tκ be a fixed point and let T > t0 be given. Denote by IT the closed interval [t0, T]T . For our further computations, it is essential to know when the formula ∫ t ∫ u ∬ f (s, u)ΔsΔu = Δu f (s, u)Δs t0

E1

∫

t0 t

∫

(1.4.1)

E1 := {(s, u) ∈ IT × IT : t0 ≤ s < u, t0 ≤ u < t} .

(1.4.2)

t0

Δs

t

f (s, u)Δu, t ∈ IT

=

σ(s)

holds. Here, E1 is the triangular region given by

Evidently, E1 is a bounded subset of [t0, T)T × [t0, T)T . Let t ∈ IT be fixed. For the existence of iterated integrals in (1.4.1) it is natural to ask that all integrals ∫ u ∫ t K(u) := f (s, u)Δs, J(s, t) := f (s, u)Δu (1.4.3) t0

σ(s)

exist for each u ∈ [t0, t)T and s ∈ [t0, t)T . Inspired by the method in [97, Lemma 1], we obtain the following result which enables us to interchange the order of integration over a triangular region.

36

1 Introduction to Stability and Boundedness in Dynamical Systems

Lemma 1.4.2. Suppose that f : IT × IT → R is Δ-integrable over E1 and that the single integrals (1.4.3) exist for each u ∈ [t0, t)T and s ∈ [t0, t)T . If the function J(s, t) defined in (1.4.3) satisfies the conditions of Theorem 1.4.5, then (1.4.1) holds for all t ∈ IT . Proof. First, from (1.4.2) and Theorem 1.4.5 we deduce that the set E1 is Jordan Δ-measurable and the double integral ∬ f (s, u)ΔsΔu E1

exists. Existence of single integrals in (1.4.3) and Theorem 1.4.5 imply existence of iterated integrals and the following equality: ∫ t ∫ u ∬ f (s, u)ΔsΔu = Δu f (s, u)Δs, t ∈ IT . t0

E1

t0

Theorem 1.1.10 guarantees that the function ∫ t ∫ u ∫ t ∫ h(t) := Δu f (s, u)Δs − Δs t0

t0

t0

is Δ-differentiable and ∫ t ∫ f (s, t)Δs − hΔ (t) = t0

∫ =

t0

t

σ(t)

σ(t) t

t0

σ(s)

∫ f (t, u)Δu − t0

∫ f (s, t)Δs −

t

t

f (s, u)Δu, t ∈ IT .

∫ Δs

t

σ(s)

f (s, u)Δu

Δt

f (s, t)Δs = 0 for all t ∈ [t0, T)T .

Applying [51, Corollary 1.15] we conclude that h is constant. On the other hand, since h(t0 ) = 0, we have h(t) = 0 for all t ∈ IT . The proof is complete. First, we consider the linear system of integral dynamic equations of the form ∫ t x(t) = f (t) + a(t, s)x(s)Δs. (1.4.4) 0

The corresponding resolvent equation associated with a(t, s) is given by ∫ t r(t, s) = −a(t, s) + r(t, u)a(u, s)Δu. σ(s)

(1.4.5)

If a is scalar valued, then so is r. If a is n × n matrix, then so is r. If Eq. (1.4.5) has a solution r(t, s) and all necessary integrals make sense, then the solution of the linear system (1.4.4) may be written in terms of f as follows:

1.4 Resolvent for Volterra Integral Dynamic Equations

∫

t

x(t) = f (t) −

37

r(t, u) f (u)Δu.

(1.4.6)

0

To see this we multiply both sides of (1.4.4) by r(t, s) to obtain ∫ t ∫ t r(t, u)x(u)Δu − r(t, u) f (u)Δu 0 0 ∫ u ∫ t r(t, u) a(u, s)x(s)ΔsΔu = 0 0  ∫ t ∫ t r(t, u)a(u, s)Δu x(s)Δs = σ(s)

0

∫ =

t

[r(t, s) + a(t, s)] x(s)Δs,

0

which implies

∫ −

t

∫ r(t, u) f (u)Δu =

0

t

a(t, s)x(s)Δs,

(1.4.7)

0

and hence (1.4.6). Equation (1.4.7) also shows that (1.4.6) implies (1.4.4). Second, we consider the nonlinear system of integro-dynamic equations of the form ∫ t * x (t) = f (t) + a(t, s) {* x (s) + G(s, * x (s))} Δs, (1.4.8) 0

where G(t, * x ) indicates the higher order terms of * x . (1.4.8) can be rewritten as ∫ t * x (t) = F(t) + a(t, s)* x (s)Δs, 0

∫

where F(t) = f (t) +

t

a(t, s)G(s, * x (s))Δs.

0

If the existence of resolvent r(t, s) is known, we get by (1.4.6) that ∫ t * x (t) = F(t) − r(t, s)F(s)Δs 0 ∫ t a(t, s)G(s, * x (s))Δs = f (t) + 0   ∫ s ∫ t r(t, s) f (s) + a(s, u)G(u, * x (u))Δu Δs − 0 0 ∫ t ∫ t r(t, s) f (s)Δs + a(t, s)G(s, * x (s))Δs = f (t) − 0 0 ∫ t ∫ t r(t, s)a(s, u)Δs G(u, * x (u))Δu − 0

σ(u)

38

1 Introduction to Stability and Boundedness in Dynamical Systems

∫ t = f (t) − r(t, s) f (s)Δs 0  ∫ t ∫ t r(t, s)a(s, u)Δs G(u, * x (u))Δu a(t, u) − + 0

∫

= f (t) −

σ(u)

t

∫

r(t, s) f (s)Δs −

0

t

r(t, u)G(u, * x (u))Δu

0

Thus, from (1.4.6) we obtain resolvent equation corresponding to (1.4.8) as follows ∫ t * x (t) = x(t) − r(t, u)G(u, * x (u))Δu. (1.4.9) 0

One may easily show that (1.4.9) implies (1.4.8). Equation (1.4.9) is called variation of constants form of Eq. (1.4.8).

1.4.1 Existence of Resolvent: L p Case In this section, we study existence of resolvent r(t, s) corresponding to the linear integral dynamic equation (1.4.4) (so, existence of resolvent in nonlinear case remains as an open problem). We also show by Theorems 1.4.6 and 1.4.7 that ∫ t ∫ t r(t, u)a(u, s)Δu = a(t, u)r(u, s)Δu. σ(s)

σ(s)

This will enable us to rewrite (1.4.5) as ∫ r(t, s) = −a(t, s) +

t

σ(s)

a(t, u)r(u, s)Δu.

(1.4.10)

Let the set Ω be given by Ω := {(t, s) ∈ IT × IT : t0 ≤ s ≤ t ≤ T }. Hereafter, we let 1 < p < ∞ and assume that 1/p + 1/q = 1. For any n × n matrix A we denote by | A| the matrix norm | A| = sup  Ax , |x | ≤1

where u indicates the vector norm of u. Let us define the functions ∫ T ∫ t A(t) := |a(t, s)| q Δs, B(t) := |a(s, t)| p Δs, t ∈ IT t0

t

(1.4.11)

1.4 Resolvent for Volterra Integral Dynamic Equations

∫

and

t

c(t, s) :=

39

A(u) p/q Δu, (t, s) ∈ Ω.

(1.4.12)

s

Next, we define a class of n × n matrix valued functions α : Ω → Rn×n satisfying the following conditions: C.1 α(t, s) is measurable in (t, s) ∈ Ω with ∫ T α(t, s) = t0 a.e. when σ(s) > t. C.2 For almost all t in IT , the integral t |α(t, s)| q Δs exists, and for almost all s in 0 ∫T IT , the integral t |α(t, s)| p Δt exists. 0 , p/q , q/p ∫ T +∫ T ∫ T +∫ T C.3 The numbers t Δt and t Δs are |α(t, s)| q Δs |α(t, s)| p Δt t t 0 0 0 0 both finite. Definition 1.4.2. We say that an n×n matrix valued function α(t, s) is of type (L p, T) if and only if the conditions (C.1–C.3) hold. Example 1.4.1. Any function α(t, s) which is continuous in (t, s) for (t, s) ∈ Ω is of type (L p, T) for all p > 1 and T > t0 . Definition 1.4.3. An n × n matrix valued function α(t, s) is said to be of type LL p if and only if it is of type (L p, T) for each T > t0 . Let the kernel a(t, s) be of type (L p, T). Define the sequence {rn (t, s)} s ∈N by r1 (t, s) := a(t, s), ∫ t a(t, u)rn (u, s)Δu rn+1 (t, s) := σ(s)

(1.4.13) (1.4.14)

for (t, s) ∈ Ω and rn (t, s) = 0 for t0 ≤ t < σ(s) ≤ T. The following lemma plays a substantial role in obtaining inequality (1.4.17). Lemma 1.4.3. Let 1 < p < ∞ and the kernel a(t, s) be of type (L p, T). Then c(t, s)n−1 {c(t, s)n }Δt ≥ A(t) p/q n! (n − 1)!

(1.4.15)

holds for all positive integer n > 1 and (t, s) ∈ Ω. Proof. We use the formula -

f n+1 (t)

.Δ

 =

n 

/ f (t)k f (σ(t))n−k

f Δ (t),

(1.4.16)

k=0

(see [50, Exercise 1.23]). Since a(t, s) is of type (L p, T), it is obvious that c(t, s) is Δ-differentiable in its both parameters and cΔt (t, s) = A(t) p/q, cΔs (t, s) = −A(s) p/q .

40

1 Introduction to Stability and Boundedness in Dynamical Systems

On the other hand, it follows from (1.4.12) that c is increasing in t and decreasing in s, i.e., c(σ(t), s) ≥ c(t, s) and c(t, σ(s)) ≤ c(t, s) for all t ∈ T. Thus, from (1.4.16) we obtain /  n−1  1 {c(t, s)n }Δt k n−k−1 = c(t, s) c(σ(t), s) cΔ (t, s) (n − 1)! (n − 1)! k=0 /  n−1  1 k n−k−1 ≥ c(t, s) c(t, s) A(t) p/q (n − 1)! k=0 = nA(t) p/q

c(t, s)n−1 . (n − 1)!

This completes the proof. Lemma 1.4.4. Let 1 < p < ∞ and the kernel a(t, s) be of type (L p, T). Then for each positive integer n ≥ 1, the function rn (t, s) is of type (L p, T). Moreover, for each nonnegative integer n ≥ 0 and for (t, s) ∈ Ω |rn+2 (t, s)| ≤ A(t)1/q B(s)1/p {c(t, s)n /n!}1/p

(1.4.17)

holds. Proof. If t0 ≤ t < σ(s) ≤ T for some (t, s) ∈ Ω, then rn+2 (t, s) = 0 and (1.4.17) holds. Suppose σ(s) ≤ t for all (t, s) ∈ Ω. We proceed by induction. From Hölder’s inequality we find ∫ t |a(t, u)| |a(u, s)| Δu |r2 (t, s)| ≤ σ(s)

∫ ≤

t

 1/q ∫ |a(t, u)| Δu q

s

t

 1/p |a(u, s)| Δu p

s

≤ A(t)1/q B(s)1/p . This shows that the kernel r2 satisfies (1.4.17) and is of type (L p, T). Suppose that r1, r2, . . . , rn+1 are all kernels of type (L p, T) and that (1.4.17) holds for n − 1. It follows from (1.4.14) and (1.4.15) that ∫ t |rn+2 (t, s)| = |a(t, u)| |rn+1 (u, s)| Δu σ(s)

∫ ≤

t

 1/q ∫ |a(t, u)| Δu q

s

 ∫ ≤ A(t)1/q B(s) s

t

 1/p |rn+1 (u, s)| Δu p

s t

A(u) p/q

c(u, s)n−1 Δu (n − 1)!

 1/p

1.4 Resolvent for Volterra Integral Dynamic Equations

∫ ≤ A(t)

1/q

B(s)

t

41

[c(t, s) /n!]

1/p

n

Δu

 1/p Δu

s

= A(t)1/q B(s)1/p {c(t, s)n /n!}1/p . Thus (1.4.17) is satisfied. Using (1.4.11) and (1.4.17) we conclude that rn+2 is of type (L p, T). The proof is complete. Theorem 1.4.6. If 1 < p < ∞ and the kernel a(t, s) be of type (L p, T), then there exists a kernel r(t, s) of type (L p, T) which solves the resolvent equation (1.4.10) a.e. in (t, s) ∈ Ω. Proof. Let rn be defined by (1.4.13) and (1.4.14). Let r(t, s) := −

∞ 

rn (t, s), for (t, s) ∈ Ω

(1.4.18)

n=1

and r(t, s) = 0 whenever t0 ≤ t < σ(s) ≤ T. From (1.4.17) we obtain |r(t, s)| ≤ |a(t, s)| + A(t)1/q B(s)1/p

∞ 

{α n /n!}1/p ,

(1.4.19)

n=2

where

∫ α=

T

A(u) p/q Δu.

t0

For any n > α we have 

α n+1 n! n α (n + 1)!

 1/p =

+ α , 1/p < 1, n+1

the series in (1.4.19) convergences by ratio test. Since A(t), B(s), and a(t, s) are finite, r(t, s) is almost everywhere well defined and measurable in (t, s) for (t, s) ∈ Ω. By (1.4.19) we deduce that r is of type (L p, T). Finally we resort to Lebesgue Dominated Convergence Theorem to obtain  ∞ / ∫ t ∫ t  a(t, u)r(u, s)Δu = a(t, u) − rn (u, s) Δu σ(s)

σ(s)

=− =−

∞ ∫  n=1 ∞ 

n=1 t

σ(s)

a(t, u)rn (u, s)Δu

rn+1 (t, s)

n=1

= r(t, s) + a(t, s). This shows that r defined in (1.4.18) solves Eq. (1.4.10).

42

1 Introduction to Stability and Boundedness in Dynamical Systems

Lemma 1.4.5. If a(t, s) is of type (L p, T), then for any positive integers v and w with v + w = n + 1, ∫ rn+1 (t, s) =

t

σ(s)

rw (t, u)rv (u, s)Δu.

(1.4.20)

Proof. For n = 1, the proof is trivial. Let (1.4.20) be true for w0 + v0 ≤ n, n ≥ 1. Given v, w ≥ 1 with v + w = n + 1 define ∫ t rw (t, u)rv (u, s)Δu. I(w, v) = s

We have ∫

t

∫



u

rw (t, u) a(u, z)rv−1 (z, s)Δz Δu s s ∫ t∫ t rw (t, u)a(u, z)rv−1 (z, s)ΔuΔz = s z ∫ t = rw+1 (t, z)rv−1 (z, s)Δz

I(w, v) =

s

= I(w + 1, v − 1). Hence, we arrive at I(1, n) = I(2, n − 1) = I(3, n − 3) . . . = I(n, 1), which proves the result for n + 1. As a consequence we have the following theorem. Theorem 1.4.7. If a(t, s) is a kernel of type LL p , then there exists a kernel r(t, s) of type LL p such that r satisfies both resolvent equations (1.4.5) and (1.4.10) for almost all (t, s) in the region Ω. Proof. Let r(t, s) be defined by (1.4.18). We know from Theorem 1.4.6 that r(t, s) is of type (L p, T) for each T > t0 and solves the resolvent equation (1.4.10) a.e. in (t, s) ∈ Ω. On the other hand, we get by (1.4.18) and (1.4.20) that / ∫ t   ∫ t ∞ r(t, u)a(u, s)Δu = rn (t, u) a(u, s)Δu − σ(s)

σ(s)

=− =−

∞ ∫  n=1 ∞ 

n=1 t

σ(s)

rn (t, u)a(u, s)Δu

rn+1 (t, s)

n=1

= r(t, s) + a(t, s),

1.4 Resolvent for Volterra Integral Dynamic Equations

43

where Lebesgue Dominated Convergence Theorem enables us to interchange summation and integration. Hence, r(t, s) solves Eq. (1.4.5). The proof is complete.

1.4.2 Shift Operators Next, we give a generalized version of shift operators (see [12]). A limited version of shift operators can be found in [4]. Definition 1.4.4 (Shift Operators). Let T∗ be a nonempty subset of the time scale T including a fixed number t0 ∈ T∗ such that there exist operators δ± : [t0, ∞)T × T∗ → T∗ satisfying the following properties: P.1 The functions δ± are strictly increasing with respect to their second arguments, i.e., if (T0, t), (T0, u) ∈ D± := {(s, t) ∈ [t0, ∞)T × T∗ : δ± (s, t) ∈ T∗ } , then T0 ≤ t < u implies δ± (T0, t) < δ± (T0, u). P.2 If (T1, u), (T2, u) ∈ D− with T1 < T2 , then δ− (T1, u) > δ− (T2, u), and if (T1, u), (T2, u) ∈ D+ with T1 < T2 , then δ+ (T1, u) < δ+ (T2, u). P.3 If t ∈ [t0, ∞)T , then (t, t0 ) ∈ D+ and δ+ (t, t0 ) = t. Moreover, if t ∈ T∗ , then (t0, t) ∈ D+ and δ+ (t0, t) = t holds. P.4 If (s, t) ∈ D± , then (s, δ± (s, t)) ∈ D∓ and δ∓ (s, δ± (s, t)) = t, respectively. P.5 If (s, t) ∈ D± and (u, δ± (s, t)) ∈ D∓ , then (s, δ∓ (u, t)) ∈ D± and δ∓ (u, δ± (s, t)) = δ± (s, δ∓ (u, t)), respectively. Then the operators δ− and δ+ associated with t0 ∈ T∗ (called the initial point) are said to be backward and forward shift operators on the set T∗ , respectively. The variable s ∈ [t0, ∞)T in δ± (s, t) is called the shift size. The values δ+ (s, t) and δ− (s, t) in T∗ indicate s units translation of the term t ∈ T∗ to the right and left, respectively. The sets D± are the domains of the shift operators δ± , respectively. Hereafter, we shall denote by T∗ the largest subset of the time scale T such that the shift operators δ± : [t0, ∞)T × T∗ → T∗ exist. Example 1.4.2. Let T = R and t0 = 1. The operators  t/s if t ≥ 0 δ− (s, t) = , for s ∈ [1, ∞) st if t < 0

(1.4.21)

44

1 Introduction to Stability and Boundedness in Dynamical Systems



and δ+ (s, t) =

st if t ≥ 0 , for s ∈ [1, ∞) t/s if t < 0

(1.4.22)

are backward and forward shift operators (on the set R∗ = R− {0}) associated with the initial point t0 = 1. In the table below, we state different time scales with their corresponding shift operators. T R Z

t0 T∗ 0 R 0 Z

qZ ∪ {0} 1 qZ N1/2

0 N1/2

δ− (s, t) δ+ (s, t) t−s t+s t−s t+s t st s √ √ t 2 − s2 t 2 + s2

The proof of the next lemma is a direct consequence of Definition 1.4.4. Lemma 1.4.6. Let δ− and δ+ be the shift operators associated with the initial point t0 . We have (i) δ− (t, t) = t0 for all t ∈ [t0, ∞)T . (ii) δ− (t0, t) = t for all t ∈ T∗ . (iii) If (s, t) ∈ D+ , then δ+ (s, t) = u implies δ− (s, u) = t. Conversely, if (s, u) ∈ D− , then δ− (s, u) = t implies δ+ (s, t) = u. (iv) δ+ (t, δ− (s, t0 )) = δ− (s, t) for all (s, t) ∈ D+ with t ≥ t0 . (v) δ+ (u, t) = δ+ (t, u) for all (u, t) ∈ ([t0, ∞)T × [t0, ∞)T ) ∩ D+ . (vi) δ+ (s, t) ∈ [t0, ∞)T for all (s, t) ∈ D+ with t ≥ t0, . (vii) δ− (s, t) ∈ [t0, ∞)T for all (s, t) ∈ ([t0, ∞)T × [s, ∞)T ) ∩ D−, . (viii) If δ+ (s, .) is Δ-differentiable in its second variable, then δ+Δt (s, .) > 0. (ix) δ+ (δ− (u, s), δ− (s, v)) = δ− (u, v) for all (s, v) ∈ ([t0, ∞)T × [s, ∞)T ) ∩ D− and (u, s) ∈ ([t0, ∞)T × [u, ∞)T ) ∩ D− . (x) If (s, t) ∈ D− and δ− (s, t) = t0 , then s = t. Proof. (i) is obtained from P.3 to P.5 since δ− (t, t) = δ− (t, δ+ (t, t0 )) = t0 for all t ∈ [t0, ∞)T . (ii) is obtained from P.3 to P.4 since δ− (t0, t) = δ− (t0, δ+ (t0, t)) = t for all t ∈ T∗ . Let u := δ+ (s, t). By P.4 we have (s, u) ∈ D− for all (s, t) ∈ D+ , and hence, δ− (s, u) = δ− (s, δ+ (s, t)) = t. The latter part of (iii) can be done similarly. We have (iv) since P.3 and P.5 yield δ+ (t, δ− (s, t0 )) = δ− (s, δ+ (t, t0 )) = δ− (s, t).

1.4 Resolvent for Volterra Integral Dynamic Equations

45

P.3 and P.5 guarantee that t = δ+ (t, t0 ) = δ+ (t, δ− (u, u)) = δ− (u, δ+ (t, u)) for all (u, t) ∈ ([t0, ∞)T × [t0, ∞)T ) ∩ D+ . Using (iii) we have δ+ (u, t) = δ+ (u, δ− (u, δ+ (t, u))) = δ+ (t, u). This proves (v). To prove (vi) and (vii) we use P.1–P.2 to get δ+ (s, t) ≥ δ+ (t0, t) = t ≥ t0 for all (s, t) ∈ ([t0, ∞) × [t0, ∞)T ) ∩ D+ and δ− (s, t) ≥ δ− (s, s) = t0 for all (s, t) ∈ ([t0, ∞)T × [s, ∞)T ) ∩ D− . Since δ+ (s, t) is strictly increasing in its second variable we have (viii) by Lemma 1.1.1. (ix) is proven as follows: from P.5 and (v) we have δ+ (δ− (u, s), δ− (s, v)) = δ− (s, δ+ (v, δ− (u, s))) = δ− (s, δ− (u, δ+ (v, s))) = δ− (s, δ+ (s, δ− (u, v))) = δ− (u, v) for all (s, v) ∈ ([t0, ∞)T × [s, ∞)T ) ∩ D− and (u, s) ∈ ([t0, ∞)T × [u, ∞)T ) ∩ D− . Suppose (s, t) ∈ D− = {(s, t) ∈ [t0, ∞)T × T∗ : δ− (s, t) ∈ T∗ } and δ− (s, t) = t0 . Then by P.4 we have t = δ+ (s, δ− (s, t)) ∈ δ+ (s, t0 ) = s. This is (x). The proof is complete. Notice that the shift operators δ± are defined once the initial point t0 ∈ T∗ is known. For instance, we choose the initial point t0 = 0 to define shift operators δ± (s, t) = t ± s on T = R. However, if we choose λ ∈ (0, ∞) as the initial point, then the new shift operators associated with λ are defined by * δ± (s, t) = t ∓ λ ± s. In terms of δ± the new shift operators * δ± can be given as follows: * δ± (s, t) = δ∓ (λ, δ± (s, t)). Example 1.4.3. In the following, we give some particular time scales with shift operators associated with different initial points to show the change in the formula of shift operators as the initial point changes.

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1 Introduction to Stability and Boundedness in Dynamical Systems

T = N1/2 T = hZ t0 0 λ 0 hλ √ √ 2 2 2 2 2 δ− (s, t) t − s t + λ − s t − s t + hλ − s √ √ δ+ (s, t) t 2 + s2 t 2 − λ2 + s2 t + s t − hλ + s λ ∈ Z+ , N1/2 =

-√

T = 2N 1 2λ t/s 2λ ts−1 ts 2−λ ts

. n : n ∈ N , 2N = {2n : n ∈ N}, and hZ = {hn : n ∈ Z}.

1.4.3 Continuity of r(t, s) In this section, we are concerned with the existence of continuous function r(t, s) on Ω which solves the resolvent equations (1.4.5) and (1.4.10). Let t0 ∈ T∗ be a nonnegative initial point and let δ± (s, t) be the shift operators associated with t0 . Hereafter, we will suppose that T is a time scale such that the right shift operator δ+ (s, t) on T is Δ-differentiable in its second variable with r dcontinuous derivative. Substituting δ+ (σ(s), t) for t in Eq. (1.4.10) we get ∫ r(δ+ (σ(s), t), s) = −a(δ+ (σ(s), t), s) +

δ+ (σ(s),t)

σ(s)

a(δ+ (σ(s), t), v)r(v, s)Δv. (1.4.23)

Since σ(s) ≤ u ≤ δ+ (σ(s), t) implies t0 ≤ δ− (σ(s), σ(s)) ≤ δ− (σ(s), u) ≤ δ− (t0, t) = t, we obtain by using (1.4.23) and Theorem 1.1.9 that r(δ+ (σ(s), t), s) = −a(δ+ (σ(s), t), s) ∫ t a(δ+ (σ(s), t), δ+ (σ(s), u))δ+Δt (σ(s), t)r(δ+ (σ(s), u), s)Δu. + t0

(1.4.24) The advantage of considering resolvent equation of the form (1.4.24) is that the variable s appears in the equation only as a parameter. If the role of s is suppressed, then (1.4.24) turns into ∫ t as (t, u)R(u)Δu, (1.4.25) R(t) = −A(t) + t0

where R(t) = r(δ+ (σ(s), t), s), A(t) = a(δ+ (σ(s), t), s) and as (t, u) = a(δ+ (σ(s), t), δ+ (σ(s), u))δ+Δt (σ(s), t), for (t, s) ∈ Ω.

(1.4.26)

1.5 Periodicity on Time Scales

47

Let β be a constant and let . denote the Euclidean norm on Rn . Define JT := [a, b]T . We will consider the space C(JT, Rn ) of continuous functions endowed with a suitable norm  x(t) ; or  x 0 := sup  x(t) .  x β := sup e t ∈JT β (t, a) t ∈JT It was proven in [156, Lemma 3.3] that  x β is a norm and is equivalent to the supremum norm  x 0 . Also, it was concluded that (C(JT, Rn ),  x β ) is a Banach space. We will denote this Banach space by P. To prove the next result we will employ the following theorem. Theorem 1.4.8. Consider the linear integral dynamic equation ∫ t x(t) = f (t) + B(t, s)x(s)Δs, t ∈ JT . α

(1.4.27)

Let B : JT × JT → Rn×n be matrix valued function which is continuous in its first variable and rd-continuous in its second variable and let f : JT → Rn be continuous. Then (1.4.27) has a unique solution in P. In addition, if a sequence of functions {xi } is defined inductively by choosing any x0 ∈ C(IT, Rn ) and setting ∫ t xi+1 (t) = f (t) + B(t, s)xi (s)Δs, t ∈ JT, α

then the sequence {xi } converges uniformly on IT to the unique solution x of (1.4.27). Theorem 1.4.9. If a(t, s) is continuous in (t, s) for (t, s) ∈ Ω, then there exists a continuous function r(t, s) on Ω which solves the resolvent equations (1.4.5) and (1.4.10). Proof. We can conclude from Theorem 1.4.8 that for each fixed s ≥ 0 there exists a unique solution R(t; s) of (1.4.25) on t0 ≤ t ≤ δ− (σ(s), T). Moreover, R(t; s) is continuous in the pair (t, s). Let us define r(t, s) = R(δ− (σ(s), t); s) for (t, s) ∈ Ω. Evidently, r solves (1.4.10). Since a is continuous in (t, s) for (t, s) ∈ Ω, from Theorem 1.4.7 there is an L 2 function k(t, s) which solves (1.4.5) and (1.4.10). By uniqueness of resolvent k(t, s) among L 2 functions, we have k = r. Thus, r also solves Eq. (1.4.5).

1.5 Periodicity on Time Scales In this chapter we introduce a new periodicity concept on time scales by using shift operators. All the definitions and examples in this section can be found in [6]. In the last two decades, theory of time scales has become a very useful tool for the unification of difference and differential equations under dynamic equations on

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1 Introduction to Stability and Boundedness in Dynamical Systems

time scales (see [4–13, 15–101], and references therein). To be able to investigate the notion of periodicity of the solutions of dynamic equations on time scales researchers had to first introduce the concept of periodic time scales and then define what it meant for a function to be periodic on such a time scale. Kaufmann and Raffoul [99] were the first to define the notion of time scale and subsequently, the notion of periodic functions. Later on, the concept was slightly modified in [32, 41], and [100]. To be more specific, we restate the following definitions and introductory examples which can be found in [99]. The following definition is due to Atıicı et al. [32] and Kaufmann and Raffoul [99]. Definition 1.5.1. A time scale T is said to be periodic if there exists a P > 0 such that t ± P ∈ T for all t ∈ T. If T  R, the smallest positive P is called the period of the time scale. Example 1.5.1. The following time scales are periodic. (i) T = Z has period P = 1, (ii) T = hZ has period P = h, (iii) T = & R, (iv) T = ∞ i=−∞ [(2i − 1)h, 2ih], h > 0 has period P = 2h, (v) T = {t = k − q m : k ∈ Z, m ∈ N0 } where, 0 < q < 1 has period P = 1. Definition 1.5.2. Let T  R be a periodic time scale with period P. We say that the function f : T → R is periodic with period T if there exists a natural number n such that T = nP, f (t ± T) = f (t) for all t ∈ T and T is the smallest number such that f (t ± T) = f (t). If T = R, we say that f is periodic with period T > 0 if T is the smallest positive number such that f (t ± T) = f (t) for all t ∈ T. There is no doubt that a time scale T which is periodic in the sense of Definition 1.5.1 must satisfy t ± P ∈ T for all t ∈ T (1.5.1) for a fixed P > 0. This property obliges the time scale to be unbounded from above and below. However, these two restrictions prevent us from investigating the periodic solutions of q-difference equations since the time scale qZ = {q n : q > 1 is constant and n ∈ Z} ∪ {0} , which is neither closed under the operation t ± P for a fixed P > 0 nor unbounded below. Hereafter, we introduce a new periodicity concept on time scales which does not oblige the time scale to be closed under the operation t ± P for a fixed P > 0 or to be unbounded. We define our new periodicity concept with the aid of shift operators which are first defined in [4] and then generalized in [12]. In the following we propose a new periodicity notion which does not oblige the time scale to be closed under the operation t ± P for a fixed P > 0 or to be unbounded.

1.6 Open Problems

49

Definition 1.5.3 (Periodicity in Shifts). Let T be a time scale with the shift operators δ± associated with the initial point t0 ∈ T∗ . The time scale T is said to be periodic in shifts δ± if there exists a p ∈ (t0, ∞)T∗ such that (p, t) ∈ D∓ for all t ∈ T∗ . Furthermore, if P := inf {p ∈ (t0, ∞)T∗ : (p, t) ∈ D∓ for all t ∈ T∗ }  t0, then P is called the period of the time scale T. The following example indicates that a time scale, periodic in shifts, does not have to satisfy (1.5.1). That is, a time scale periodic in shifts may be bounded. Example 1.5.2. The following time scales are not periodic in the sense of Definition 1.5.1 but periodic with respect to the periodicity in shifts given in Definition 1.5.3. √ √  2 ⎧ ⎪ t± P if t > 0 ⎪ ⎪ ⎨ ⎪ - 2 . if t = 0 , P = 1, t0 = 0, 1. T1 = ±n : n ∈ Z , δ± (P, t) = ±P √ ⎪ √ 2 ⎪ ⎪ ⎪ − −t ± P if t < 0 ⎩ 2. T2 =qZ , δ± (P, t) = P±1 t, P = q, t0 = 1, ' ( 3. T3 =∪n∈Z 22n, 22n+1 , δ± (P, t) = P±1 t, P = 4, t0 = 1, + n , q 4. T4 = 1+q n : q > 1 is constant and n ∈ Z ∪ {0, 1}, (

δ± (P, t) =

q

) (

t P

ln 1−t ±ln 1−P  ln q 

1+q



ln

(

) (

)  

t ±ln P 1−t 1−P ln q

)

,

P=

q . 1+q

Notice+ that the time scale T4 in Example , 1.5.2 is bounded above and below and qn T∗4 = 1+q : q > 1 is constant and n ∈ Z . n

1.6 Open Problems Open Problem 1 Show the existence of resolvent r(t, s) of type LL 1 such that (1.4.5) and (1.4.10) are satisfied. Open Problem 2 For φ ∈ X, use the norm ||φ|| = max Ae B (s, 0)|φ(s)|, t ∈[0,T ]T

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1 Introduction to Stability and Boundedness in Dynamical Systems

for A and B are positive constants to be determined so the conclusion of Theorem 6.1.5 can be strengthened to T = min{a, b/M }. Open Problem 3 (Extension of Theorem 1.3.1) Theorem 1.6.1. Suppose H > 0 and there is a scalar functional V(t, φ) that is r d-continuous in φ and locally Lipschitz in φ when t ≥ t0 and φ ∈ CH . If V W(|φ(0)|) ≤ V(t, φ) ≤ W1 (|φ(0)|) + W2 (|||φ|||), and

 φ) ≤ −W3 (|φ(0)|), V(t,

then the zero solution of (1.3.1) is (UAS).

Chapter 2

Ordinary Dynamical Systems

Summary In this chapter, we analyze boundedness and stability of solutions of ordinary dynamical systems. We will develop general theorems in which Lyapunov functions play an important role. We begin by considering some of the boundedness results of Peterson and Tisdell (J Differ Equ Appl 10(13–15):1295–1306, 2004) and then the results on exponential stability of Peterson and Raffoul (Adv Differ Equ 2005(2):133–144, 2005). We define suitable Lyapunov-type functions on time scales and then formulate appropriate inequalities that guarantee the boundedness of solutions and the exponential stability of the zero solution. In the second part of the chapter, we turn our attention to finite delay dynamical systems, linear and nonlinear, where Lyapunov functionals are the main tools for proving stability, uniform stability, and asymptotic stability of the zero solution. Some of the results of the chapter are new and the rest can be found in Raffoul and Ünal (J Nonlinear Sci Appl 7(6):422– 428, 2014) and Ünal and Raffoul (Int J Differ Equ 2013:764389, 10, 2013; Int J Math Comput 26(4):62–73, 2015) and the references therein. For more on the existence and uniqueness of solutions for ordinary dynamic equations on time scales, we refer to Dai and Tisdell (Int J Differ Equ 1(1):1–17, 2006), Erbe and Peterson (Nonlinear Anal 69(7):2303–2317, 2008), and Tisdell (Cubo 8(3):11–24, 2006). In this chapter, we analyze boundedness and stability of solutions of ordinary dynamical systems. We will develop general theorems in which Lyapunov functions play an important role. We begin by considering some of the boundedness results of [131] and then the results on exponential stability of [130]. We define suitable Lyapunov-type functions on time scales and then formulate appropriate inequalities that guarantee the boundedness of solutions and the exponential stability of the zero solution. In the second part of the chapter, we turn our attention to finite delay dynamical systems, linear and nonlinear, where Lyapunov functionals are the main tools for proving stability, uniform stability, and asymptotic stability of the zero solution. Some of the results of the chapter are new and the rest can be found in [146, 158], and [159] and the references therein. For more on the existence and

© Springer Nature Switzerland AG 2020 M. Adıvar, Y. N. Raffoul, Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales, https://doi.org/10.1007/978-3-030-42117-5_2

51

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2 Ordinary Dynamical Systems

uniqueness of solutions for ordinary dynamic equations on time scales, we refer to [74, 81], and [155].

2.1 Boundedness We begin by considering the boundedness and uniqueness of solutions to the firstorder dynamic equation x Δ = f (t, x), t ∈ [0, ∞)T,

(2.1.1)

subject to the initial condition x(t0 ) = x0, t0 ∈ [0, ∞)T, x0 ∈ R,

(2.1.2)

where x(t) ∈ Rn , f : [0, ∞)T × D → Rn where D ⊂ Rn and open and f is a continuous, nonlinear function. Equation (2.1.1) subject to (2.1.2) is known as an initial value problem (IVP) on time scales. For T = R, recently, Raffoul [134] used Lyapunov-type functions to formulate some sufficient conditions that ensure all solutions to (2.1.1), (2.1.2) are bounded, while in a more classical setting, Hartman [86, Chapter 3] employed Lyapunov-type functions to prove that solutions to (2.1.1), (2.1.2) are unique. Motivated by [86] and [134] (see also references therein), we investigate the boundedness and uniqueness of solutions of the system of dynamic equations (2.1.1), (2.1.2) in the more general time scale setting. Most of the results of this section are published in [131]. Definition 2.1.1. We say solutions x of the IVP (2.1.1), (2.1.2) t0 ≥ 0, x0 ∈ Rn are uniformly bounded provided there is a constant C = C(x0 ) which may depend on x0 but not on t0 such that  x(t) ≤ C for all t ∈ [t0, ∞)T . Throughout this chapter, we assume that 0 is in the time scale T. We begin with the following theorem. Theorem 2.1.1. Assume D ⊂ Rn and there exists a type I Lyapunov function V : D → [0, ∞) such that for all (t, x) ∈ [0, ∞)T × D: V(x) → ∞, as  x → ∞; V(x) ≤ φ( x);  x) ≤ ψ( x) + L ; V(t, 1 + μ(t) ψ(φ−1 (V(x))) + V(x) ≤ γ;

(2.1.3) (2.1.4) (2.1.5) (2.1.6)

2.1 Boundedness

53

where φ, ψ are functions such that φ : [0, ∞)T → [0, ∞), ψ : [0, ∞)T → (−∞, 0], ψ is nonincreasing, and φ−1 exists; L and γ are nonnegative constants. Then all solutions of the IVP (2.1.1), (2.1.2) that stay in D are uniformly bounded. Proof. Let x be a solution to the IVP (2.1.1), (2.1.2) that stays in D for all t ≥ t0 ≥ 0. Consider V(x(t))e1 (t, t0 ), where e1 (t, t0 ) is the unique solution to the IVP x Δ = x, x(t0 ) = 1. Since p = 1 ∈ R + , e1 (t, t0 ) is well defined and positive on T. Now consider  x(t))eσ (t, t0 ) + V(x(t))eΔ (t, t0 ), [V(x(t))e1 (t, t0 )]Δ = V(t, 1 1 ψ( x(t)) + L (1 + μ(t))e1 (t, t0 ) + V(x(t))e1 (t, t0 ), by (2.1.5), ≤ 1 + μ(t) = (ψ( x(t)) + L + V(x(t)))e1 (t, t0 ), ≤ (ψ(φ−1 (V(x(t))) + V(x(t)) + L)e1 (t, t0 ), by (2.1.4), ≤ (γ + L)e1 (t, t0 ), by (2.1.6). Integrating both sides from t0 to t we obtain V(x(t))e1 (t, t0 ) ≤ V(x0 ) + (γ + L)(e1 (t, t0 ) − 1), where x0 = x(t0 ). It follows that V(x(t)) ≤ V(x0 )/e1 (t, t0 ) + (γ + L), ≤ V(x0 ) + (γ + L), for all t ∈ [t0, ∞)T .

(2.1.7)

Thus by (2.1.3), V(x(t)) ≤ V(x0 ) + γ + L implies that  x(t) ≤ R for some R > 0, and R will depend on V(x0 ) and γ and L. Hence all solutions of the IVP (2.1.1), (2.1.2) that stay in D are uniformly bounded. This concludes the proof. The next theorem is a special case of Theorem 2.1.1. Theorem 2.1.2. Assume D ⊂ Rn and there exists a type I Lyapunov function V : D → [0, ∞) such that for all (t, x) ∈ [0, ∞)T × D: V(x) → ∞, as  x → ∞; V(x) ≤ λ2  x q ; r  x) ≤ −λ3  x + L ; V(t, 1 + M μ(t) V(x) − V r/q (x) ≤ γ;

(2.1.8) (2.1.9) (2.1.10) (2.1.11)

where λ2 , λ3 , q, r are positive constants; L and γ are nonnegative constants, and r/q M := λ3 /λ2 . Then all solutions of the IVP (2.1.1) and (2.1.2) that stay in D are uniformly bounded.

54

2 Ordinary Dynamical Systems r/q

Proof. Note that M := λ3 /λ2 Consider

∈ R +, so e M (t, t0 ) is well defined and positive.

[V(x(t))e M (t, t0 )]Δ .

Following the steps in the proof of Theorem 2.1.1 we obtain V(x(t)) ≤ V(x0 ) + (γ + L)/M, for all t ∈ [t0, ∞)T,

(2.1.12)

with a bound on solutions following from (2.1.8). As a consequence of Theorem 2.1.2 we state the following corollary. Corollary 2.1.1. Assume that the conditions of Theorem 2.1.2 are satisfied with (2.1.8) replaced with λ1  x p ≤ V(x), (2.1.13) where λ1 and p are positive constants. Then all solutions of the IVP (2.1.1), (2.1.2) that stay in D satisfy −1/p

 x(t) ≤ λ1

(V(x0 ) + (γ + L)/M)1/p , for all t ∈ [t0, ∞)T .

(2.1.14)

Proof. Let x be a solution of the IVP (2.1.1) and (2.1.2) that stays in D. Then (2.1.12) and (2.1.13) imply that λ1  x(t) p ≤ V(x(t)) ≤ V(x0 ) + (γ + L)/M, and (2.1.14) follows. This concludes the proof. Theorem 2.1.3. Assume D ⊂ Rn and there exists a type I Lyapunov function V : D → [0, ∞) such that for all (t, x) ∈ [0, ∞)T × D: V(x) → ∞, as  x → ∞;  x) ≤ −λ3V(x) + L ; V(t, 1 + λ3 μ(t)

(2.1.15) (2.1.16)

where λ3 > 0 and L ≥ 0 are constants. Then all solutions of the IVP (2.1.1) and (2.1.2) that stay in D are uniformly bounded. Proof. Let x be a solution to the IVP (2.1.1), (2.1.2) that stays in D for all t ∈ [t0, ∞)T . Since λ3 ∈ R + , eλ3 (t, t0 ) is well defined and positive. Now consider  x(t))eσ (t, t0 ) + V(x(t))λ3 eλ3 (t, t0 ), [V(x(t))eλ3 (t, t0 )]Δ = V(t, λ3  x(t))(1 + λ3 μ(t))eλ3 (t, t0 ) + V(x(t))λ3 eλ3 (t, t0 ), = V(t, ≤ (−λ3V(x(t)) + L)eλ3 (t, t0 ) + V(x(t))λ3 eλ3 (t, t0 ), by (2.1.16), = Leλ3 (t, t0 ). Integrating both sides from t0 to t we obtain V(x(t))eλ3 (t, t0 ) ≤ V(x0 ) + Leλ3 (t, t0 )/λ3,

2.1 Boundedness

55

and therefore V(x(t)) ≤ V(x0 )/eλ3 (t, t0 ) + L/λ3, ≤ V(x0 ) + L/λ3 .

(2.1.17)

Thus by (2.1.15), V(x(t)) ≤ V(x0 ) + L/λ3 implies that  x(t) ≤ R for some R > 0, and R will depend on V(x0 ), L and λ3 . Hence all solutions of the IVP (2.1.1), (2.1.2) that stay in D are uniformly bounded. This concludes the proof. Corollary 2.1.2. Assume that the conditions of Theorem 2.1.3 are satisfied with (2.1.15) and replaced with (2.1.18) λ1  x p ≤ V(x), where λ1 and p are positive constants. Then all solutions of the IVP (2.1.1) and (2.1.2) that stay in D satisfy −1/p

 x(t) ≤ λ1

(V(x0 ) + L/λ2 )1/p .

(2.1.19)

Proof. Let x be a solution of the IVP (2.1.1), (2.1.2) that stays in D. Then (2.1.17) and (2.1.18) imply that λ1  x(t) p ≤ V(x(t)) ≤ V(x0 ) + L/λ2, and (2.1.19) follows. This concludes the proof.

2.1.1 Applications In this section, we apply the results of Sect. 2.1 to some initial value problems on time scales and we will compare our results with the existing literature when the time scale is the set of reals and positive integers. We begin with the following example. Example 2.1.1. Consider the IVP x Δ = ax + bx 1/3, x(t0 ) = x0,

(2.1.20)

where a, b are constants, x0 ∈ Rn , and t0 ∈ [0, ∞)≈ . If there is a constant λ3 > 0 such that 4 1 (2.1.21) (2a + a2 μ(t) + + μ(t))(1 + λ3 μ(t)) ≤ −λ3, 3 3 and   |b + abμ(t)| 3 + μ(t)|b| 3 (1 + λ3 μ(t)) ≤ M, for some constant M ≥ 0 and all t ∈ [0, ∞)T , then all solutions to (2.2.13) are uniformly bounded.

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2 Ordinary Dynamical Systems

Proof. We shall show that under the above assumptions, the conditions of Theorem 2.1.2 are satisfied. Choose D = R and V(x) = x 2 so q = 2, λ2 = 1 and (2.1.8) holds. Now from (1.2.6)  x) V(t, = 2x · f (t, x) + μ(t) f (t, x) 2, = 2x(ax + bx 1/3 ) + μ(t)(ax + bx 1/3 )2, = (2a + a2 μ(t))x 2 + 2(b + abμ(t))x 4/3 + b2 μ(t)x 2/3,    4/3 3/2 3 (x ) |b + abμ(t)| 3 (x 2/3 )3 (b2 ) 2 2 2 ≤ (2a + a μ(t))x + 2 + + 3 , + μ(t) 3/2 3 3 2 ≤ (2a + a2 μ(t) +

4 μ(t) 2 2 2 + )x + |b + abμ(t)| 3 + μ(t)|b| 3, 3 3 3 3

where we have made use of Young’s inequality (see Lemma 1.1.4) twice. Dividing and multiplying the right-hand side by (1 + λ3 μ(t)) we see that (2.2.7) holds under the above assumptions with r = 2 and γ = 0. Therefore all the conditions of Theorem 2.1.2 are satisfied and we conclude that all solutions to (2.1.20) are uniformly bounded. Case 1. If T = R ( [134]), then μ(t) = 0 and (2.1.21) reduces to 2a + 4/3 ≤ −λ3 . If a ≤ −2/3, then we take λ3 = −(2a + 4/3) > 0 and we can choose M = |b| 3 , concluding that all solutions to (2.1.20) are uniformly bounded. Case 2. If T = hN0 = {0, h, 2h, . . .}, then μ(t) = h and (2.2.14) reduces to (2a + a2 h + 4/3 + h/3) ≤ −λ3 /(1 + λ3 h). Therefore we want to find those h > 0 such that ha2 + 2a + (4 + h)/3 < 0. Now the polynomial p(a) := ha2 + 2a + (4 + h)/3 will have distinct real roots

 √ √ a1 (h) = (− 3 − 3 − 4h − h2 )/( 3h)  √ √ a2 (h) = (− 3 + 3 − 4h − h2 )/( 3h) √ √ if 0 < h < 7 − 2. Therefore if 0 < h < 7 − 2 and a1 (h) < a < a2 (h), then A := ha2 + 2a + (4 + h)/3 < 0. Now, for such an h, let λ3 be defined by −λ3 (1 + λ3 h) = A < 0,

2.1 Boundedness

57

that is λ3 := −A/(1 + hA). √ Therefore if 0 < h < 7 − 2, then for a1 (h) < a < a2 (h) all solutions are uniformly bounded by Theorem 2.1.2. Remark 2.1.1. It is interesting to note that  √ √ lim a2 (h) = lim (− 3 + 3 − 4h − h2 )/( 3h) = −2/3, h→0+

and

h→0+

 √ √ lim a1 (h) = lim (− 3 − 3 − 4h − h2 )/( 3h) = −∞,

h→0+

h→0+

recalling that if T = R then for −∞ < a ≤ −2/3, then all solutions are uniformly bounded. Example 2.1.2. Consider the following system of IVPs for t ≥ t0 ≥ 0 x1Δ = −ax1 + ax2,

(2.1.22)

x2Δ

= −ax1 − ax2,

(2.1.23)

(x1 (t0 ), x2 (t0 )) = (c, d),

(2.1.24)

for certain constants a > 0; c and d. If there is a constant λ3 > 0 such that for all t ∈ [0, ∞)T λ3 /(1 + λ3 μ(t)) ≤ 2a(1 − aμ(t)), (2.1.25) then all solutions to (2.1.22)–(2.1.24) are uniformly bounded. Proof. We will show that, under the above assumptions, the conditions of Theorem 2.1.3 are satisfied. Choose D = R2 and V(x) =  x 2 = x12 + x22 so (2.1.15) holds. From (1.2.6) we see that  x) = 2x · f (t, x) + μ(t) f (t, x) 2, V(t, = −2a(1 − aμ(t)) x 2, ≤ −λ3  x 2 /(1 + λ3 μ(t)), by (2.1.25), = −λ3V(x)/(1 + λ3 μ(t)). Hence (2.1.16) holds under the above assumptions with L = 0. Therefore all the conditions of Theorem 2.1.3 are satisfied and we conclude that all solutions to (2.1.22)– (2.1.24) are uniformly bounded. In fact, if there is a constant K such that 0 ≤ aμ(t) ≤ K < 1

(2.1.26)

for all t ∈ [0, ∞), then (2.1.25) will hold. Case 1. If T = R, then μ(t) = 0 and (2.1.26) will hold for any 0 ≤ K < 1 which, in turn, will make (2.1.25) hold and we conclude that all solutions are uniformly bounded.

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2 Ordinary Dynamical Systems

Case 2. If T = {Hn }0∞ defined by H0 = 0, Hn =

n 

1/r, n ∈ N,

r=1

then μ(t) = 1/(n + 1) and (2.1.26) will hold when a < 1 which, in turn, will make (2.1.25) hold and we conclude that all solutions are uniformly bounded. Case 3. If T = hN0 , then μ(t) = h and (2.1.26) will hold when ah < 1 which, in turn, will make (2.1.25) hold and we conclude that all solutions are uniformly bounded. Remark 2.1.2. By using standard methods [50], the system (2.1.22)–(2.1.24) has solutions x1 (t) = c1 e−a+ia (t, t0 ) + c2 e−a−ia (t, t0 ), x2 (t) = ac1 (−1 + i)e−a+ia (t, t0 ) − ac2 (1 + i)e−a−ia (t, t0 ), and for T = hN0 , we see by closely investigating these exponential functions that when h ≤ 1/a all solutions are uniformly bounded and when h ≥ 1/a all nontrivial solutions are unbounded. It is interesting to note that even though the eigenvalues of the coefficient matrix   −a a A= , (2.1.27) −a −a are complex with negative real parts, our system is not stable when ah > 1.

2.2 Exponential Stability In this section we extend the results of Sect. 2.1 to study the exponential stability of the ordinary dynamical systems (2.1.1) and (2.1.2). Again, we will furnish general theorems and for applications will rely on the construction of suitable Lyapunov functions. Most of the results of this section can be found in [130]. Throughout this section we assume 0 and t0 are in T (for convenience) and that f (t, 0) = 0, for all t and call the zero function the trivial solution of (2.1.1). In addition, x(t, t0, x0 ) denotes a solution of the initial value problem (2.1.1), (2.1.2). We begin with the following definition concerning exponential stability. Definition 2.2.1. We say the trivial solution of (2.1.1) is exponentially stable on [0, ∞)T if there exists a positive constant d, a constant C ∈ R+ , and an M > 0 such that for any solution x(t, t0, x0 ) of the IVP (2.1.1), (2.1.2), t0 ≥ 0, x0 ∈ Rn ,    x(t, t0, x0 ) ≤ C ||x0 ||, t0 (e M (t, t0 ))d, for all t ∈ [t0, ∞)T, where  ·  denotes the Euclidean norm on Rn . The trivial solution of (2.1.1) is said to be uniformly exponentially stable on [0, ∞)T if C is independent of t0 .

2.2 Exponential Stability

59

Note that if T = R, then (e λ (t, t0 ))d = e−λd(t−t0 ) and if T = Z+ , then (e λ (t, t0 ))d = (1 + λ)−dλ(t−t0 ) . Theorem 2.2.1. Assume D ⊂ Rn contains the origin and there exists a type I Lyapunov function V : D → [0, ∞) such that for all (t, x) ∈ [0, ∞)T × D: W( x) ≤ V(x) ≤ φ( x);  x) ≤ ψ( x) − L(M δ)(t)e δ (t, 0) ; V(t, 1 + μ(t)M ψ(φ−1 (V(x))) + MV(x) ≤ 0;

(2.2.1) (2.2.2) (2.2.3)

where W, φ, ψ are continuous functions such that φ, W : [0, ∞) → [0, ∞), ψ : [0, ∞) → (−∞, 0], ψ is nonincreasing, φ and W are strictly increasing; L ≥ 0, δ > M > 0 are constants. Then all solutions of the IVP (2.1.1), (2.1.2) that stay in D satisfy   ||x(t)|| ≤ W −1 (V(x0 ) + L)e M (t, t0 ) , for all t ≥ t0 . Proof. Let x be a solution to (2.1.1), (2.1.2) that stays in D for all t ≥ 0. Consider  x(t))eσ (t, 0) + V(x(t))eΔ (t, 0) [V(x(t))e M (t, 0)]Δ = V(t, M M  by (2.2.2)  ≤ ψ( x(t)) − L(M δ)(t)e δ (t, 0) e M (t, 0) + MV(x(t))e M (t, 0)   = ψ( x(t)) − L(M δ)(t)e δ (t, 0) + MV(x(t)) e M (t, 0)  by (2.2.1)  ≤ ψ(φ−1 (V(x(t)))) + MV(x(t)) − L(M δ)(t)e δ (t, 0) e M (t, 0) by (2.1.6)

≤

−L(M δ)(t)e δ (t, 0)e M (t, 0)

by Lemma1.1.3

=

−L(M δ)(t)e M δ (t, 0).

Integrating both sides from t0 to t with x0 = x(t0 ), we obtain, for t ∈ [t0, ∞)T, V(x(t))e M (t, 0) ≤ V(x0 )e M (t0, 0) − Le M δ (t, 0) + Le M δ (t0, 0) ≤ V(x0 )e M (t0, 0) + Le M δ (t0, 0) ≤ (V(x0 ) + L) e M (t0, 0). It follows that for t ∈ [t0, ∞)T , V(x(t)) ≤ (V(x0 ) + L) e M (t0, 0)e M (t, 0) = (V(x0 ) + L)e M (t, t0 ).

(2.2.4)

Thus by (2.2.1),    x(t) ≤ W −1 (V(x0 ) + L)e M (t, t0 ) , This concludes the proof.

t ∈ [t0, ∞))T .

(2.2.5)

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2 Ordinary Dynamical Systems

The next theorem is a special case of Theorem 2.2.1 for certain functions φ and ψ. Theorem 2.2.2. Assume D ⊂ Rn contains the origin and there exists a type I Lyapunov function V : D → [0, ∞)T such that for all (t, x) ∈ [0, ∞)T × D: λ1 (t) x p ≤ V(x) ≤ λ2 (t) x q ; r  x) ≤ −λ3 (t) x − L(M δ)(t)e δ (t, 0) ; V(t, 1 + M μ(t)

(2.2.6) (2.2.7)

V(x) − V r/q (x) ≤ 0;

(2.2.8)

where λ1 (t), λ2 (t), and λ3 (t) are positive functions, where λ1 (t) is nondecreasing; p, q, 3 (t) r are positive constants; L is a nonnegative constant, and δ > M := inf [λ λ(t)] r /q > 0. t ≥0

2

Then the trivial solution of (2.1.1), (2.1.2) is exponentially stable on [0, ∞)T .

Proof. As in the proof of Theorem 2.2.1, let x be a solution to (2.1.1), (2.1.2) that 3 (t) stays in D for all t ≥ 0. Since M = inf [λ λ(t)] r /q > 0, e M (t, 0) is well defined and positive. Since

λ3 (t) [λ2 (t)]r /q

t ≥0

2

≥ M, we have

 x(t))eσ (t, 0) + V(x(t))eΔ (t, 0) [V(x(t))e M (t, 0)]Δ = V(t, M M  by (2.2.7)  ≤ − λ3 (t) x(t) r − L(M δ)(t)e δ (t, 0) e M (t, 0) + MV(x(t))e M (t, 0)  by (2.2.6)  −λ3 (t) r/q ≤ V (x(t)) + MV(x(t)) − L(M

δ)(t)e (t, 0) e M (t, 0)

δ [λ2 (t)]r/q   ≤ M(V(x(t)) − V r/q (x(t))) − L(M δ)(t)e δ (t, 0) e M (t, 0) by (2.2.8)

≤

−L(M δ)(t)e M δ (t, 0).

Integrating both sides from t0 to t with x0 = x(t0 ), and by invoking condition (2.2.6) and the fact that λ1 (t) ≥ λ1 (t0 ) we obtain   1/p (t) (V(x0 ) + L)e M (t, t0 )   1/p −1/p , ≤ λ1 (t0 ) (V(x0 ) + L)e M (t, t0 ) −1/p

 x(t) ≤ λ1

(2.2.9) for all t ≥ t0 . (2.2.10)

This concludes the proof.

Remark 2.2.1. In Theorem 2.2.2, if λi (t) = λi, i = 1, 2, 3 are positive constants, then the trivial solution of (2.1.1), (2.1.2) is uniformly exponentially stable on [0, ∞)T . The proof of this remark follows from Theorem 2.2.2 by taking δ > [λ λ]3r /q and M=

λ3 [λ2 ]r /q

.

The next theorem is an extension of Theorem 2.1 in [68].

2

2.2 Exponential Stability

61

Theorem 2.2.3. Assume D ⊂ Rn contains the origin and there exists a type I Lyapunov function V : D → [0, ∞) such that for all (t, x) ∈ [0, ∞)T × D: λ1  x p ≤ V(x),

(2.2.11)

 x) ≤ −λ3V(x) − L(ε δ)(t)e δ (t, 0) ; V(t, 1 + ε μ(t)

(2.2.12)

where λ1, λ3, p, δ > 0, L ≥ 0 are constants and 0 < ε < min{λ3, δ}. Then the trivial solution of (2.1.1), (2.1.2) is uniformly exponentially stable on [0, ∞)T . Proof. Let x be a solution of the IVP (2.1.1) and (2.1.2) that stays in D for all t ∈ [0, ∞)T . Since ε ∈ R + , eε (t, 0) is well defined and positive. Now consider  x(t))eεσ (t, 0) + εV(x(t))eε (t, 0), [V(x(t))eε (t, 0)]Δ = V(t,  by (2.2.12)  ≤ − λ3V(x(t)) − L(ε δ)(t)e δ (t, 0) eε (t, 0) + εV(x(t))eε (t, 0) = eε (t, 0)[εV(x(t)) − λ3V(x(t)) − L(ε δ)(t)e δ (t, 0)] ≤ −eε (t, 0)L(ε δ)(t)e δ (t, 0) = −L(ε δ)(t)eε δ (t, 0). Integrating both sides from t0 to t we obtain V(x(t))eε (t, 0) ≤ V(x0 )eε (t0, 0) − Leε δ (t, 0) + Leε δ (t0, 0) ≤ V(x0 )eε (t0, 0) + Leε δ (t0, 0)   ≤ V(x0 ) + L eε (t0, 0). Dividing both sides of the above inequality by eε (t, 0), we obtain   V(x(t)) ≤ V(x0 ) + L eε (t0, 0)e ε (t, 0)   = V(x0 ) + L e ε (t, t0 ). This completes the proof.

2.2.1 Applications We now present some examples to illustrate the theory developed in Sect. 2.2. Example 2.2.1. Consider the IVP x Δ = ax + bx 1/3 e δ (t, 0), x(t0 ) = x0,

(2.2.13)

where δ > 0, a, b are constants, x0 ∈ R, and t0 ∈ [0, ∞)T . If there is a constant 0 < M < δ such that

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2 Ordinary Dynamical Systems

(2a + a2 μ(t) + 1)(1 + M μ(t)) ≤ −M,

(2.2.14)

and 

 2 |2b + 2abμ(t)| 3 2 3/2 (μ(t)b ) + (1 + M μ(t)) ≤ −L(M δ)(t), 3 3

(2.2.15)

for some constant L ≥ 0 and all t ∈ [0, ∞), then the trivial solution of (2.2.13) is uniformly exponentially stable. Proof. We shall show that under the above assumptions, the conditions of Remark 3.4.2 are satisfied. Choose D = R and V(x) = x 2 , then (2.2.6) holds with p = q = 2, λ1 = λ2 = 1. Now from (1.2.6)  x) = 2x · f (t, x) + μ(t) f (t, x) 2 V(t, = 2x(ax + bx 1/3 e δ (t, 0)) + μ(t)(ax + bx 1/3 e δ (t, 0))2 ≤ (2a + a2 μ(t))x 2 + |2b + 2abμ(t)|x 4/3 e δ (t, 0) + b2 μ(t)x 2/3 (e δ (t, 0))2 .

(2.2.16)

To further simplify the above inequality, we make use of Young’s inequality Lemma 1.1.4, with e = 3/2 and f = 3. Thus |2b + 2abμ(t)| 3 (e δ (t, 0))3 $ 3/2 3 3 2 |2b + 2abμ(t)| (e δ (t, 0))3 = x2 + , 3 3

x 4/3 |2b + 2abμ(t)|e δ (t, 0) ≤

# (x 4/3 )3/2

+

and

 x 2/3 b2 μ(t)(e δ (t, 0))2 ≤

(x 2/3 )3 3

+

b2 μ(t)(e δ (t, 0))2

 3/2

3/2 x2 2 + (μ(t)b2 )3/2 (e δ (t, 0))3 . = 3 3

Thus, putting everything together we arrive at  x) ≤ (2a + μ(t)a2 + 1)x 2 V(t,  2 |2b + 2abμ(t)| 3 (μ(t)b2 )3/2 + + (e δ (t, 0))3 3 3 ≤ (2a + μ(t)a2 + 1)x 2  2 |2b + 2abμ(t)| 3 (μ(t)b2 )3/2 + + e δ (t, 0). 3 3

(2.2.17)

Dividing and multiplying the right-hand side by (1 + M μ(t)) we see that (2.2.7) holds under the above assumptions with r = 2 (note that λ3 = M). Also, since r = q =

2.2 Exponential Stability

63

2, (2.2.8) is satisfied. Therefore all the hypotheses of Remark 2.2.1 are satisfied and we conclude that the trivial solution of (2.2.13) is uniformly exponentially stable. We next look at the three special cases of (2.2.13) when T = R, T = N0 and when T = hN0 = {0, h, 2h, . . .}. Case 1. If T = R, then μ(t) = 0, and it is easy to see that if we assume that 8 |b | 3 , a < − 12 , then (2.2.14) is true if we take M = −(2a + 1) > 0. For L = 3(δ−M) condition (2.2.15) is satisfied. Hence in this case we conclude that if a < − 12 and δ > −(2a + 1), then the trivial solution to (2.2.13) is uniformly exponentially stable. Case 2. If T = N0 , then μ(t) = 1 and condition (2.2.14) cannot be satisfied for positive M. To get around this we will adjust the steps leading to inequality (2.2.17) as follows: (e δ (t, 0))3 $ 3/2 3 2 |2b + 2abμ(t)| = |2b + 2abμ(t)|x 2 + (e δ (t, 0))3, 3 3

x 4/3 |2b + 2abμ(t)|e δ (t, 0) ≤ |2b + 2abμ(t)|

# (x 4/3 )3/2

+

and x 2/3 b2 μ(t)(e δ (t, 0))2 ≤ b2 μ(t)

# (x 2/3 )3

+

(e δ (t, 0))2 )3/2 $ 3/2

3 2 2 x = b2 μ(t) + μ(t)b2 (e δ (t, 0))3 . 3 3

Hence, inequality (2.2.17) becomes 2  x) ≤ (2a + μ(t)a2 + 2 |2b + 2abμ(t)| + μ(t)b )x 2 V(t, 3 3 # |2b + 2abμ(t)| + 2 μ(t)b2 $ 3 e δ (t, 0). + 3

Now, if T = N0 , then μ(t) = 1 and so from this last inequality, given δ > 0, we want to find 0 < M < δ, and L ≥ 0 such that (2a + a2 +

b2 2 |2b + 2ab| + )(1 + M) ≤ −M, 3 3

(2.2.18)

and

|2b + 2ab| + 23 b2 δ−M (1 + M) ≤ −L(M δ)(t) = L. 3 1+δ Note that condition (2.2.18) is satisfied for all M > 0 sufficiently small if 2a + a2 +

b2 2 |2b + 2ab| + < 0. 3 3

(2.2.19)

(2.2.20)

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2 Ordinary Dynamical Systems

For such a 0 < M < δ if we take L=

2(3|b||1 + a| + b2 )(1 + M)(1 + δ) , 9(δ − M)

then (2.2.15) is satisfied (note for each δ > 0 we can find such a M so our result holds for all δ). In conclusion, we have for the case T = N0 that if (2.2.20) holds, then the trivial solution of (2.2.13) is uniformly exponentially stable. In particular if a = − 45 and b = 15 , then (2.2.20) is satisfied. Case 3. If T = hN0 = {0, h, 2h, . . .}, then μ(t) = h, and in this case by (2.2.14) and (2.2.15) we want to find 0 < M < δ and L ≥ 0 such that (2a + a2 h + 1) ≤ −M/(1 + M h)

(2.2.21)

and |2b + 2abh| 3 + 23 (hb2 )3/2 δ−M (1 + hM) ≤ −L(M δ)(t) = L. 3 1 + hδ

(2.2.22)

Note that (2.2.21) is satisfied for all M > 0 sufficiently small provided h > 0 satisfies ha2 + 2a + 1 < 0. Now the polynomial p(a) := ha2 + 2a + 1 will have distinct real roots √ a1 (h) = (−1 − 1 − h)/h √ a2 (h) = (−1 + 1 − h)/h if 0 < h < 1. Therefore if 0 < h < 1 and a1 (h) < a < a2 (h), then ha2 + 2a + 1 < 0 as desired. Now, for such an h, if we let  $ # 3 2 2 3/2 + |2b+2abh | (1 + M h) 9 (hb ) 3 (1 + δh), L= δ−M then (2.1.25) is satisfied. Putting this all together we get that if 0 < h < 1 and a1 (h) < a < a2 (h), then the trivial solution of (2.2.13) is uniformly exponentially stable.

2.3 Perturbed Systems

65

Remark 2.2.2. It is interesting to note that √ lim a2 (h) = lim (−1 + 1 − h)/h = −1/2,

h→0+

and

h→0+

√ lim a1 (h) = lim (−1 − 1 − h)/h = −∞,

h→0+

h→0+

recalling that if T = R then for −∞ < a < −1/2 the zero solution to (2.2.13) is uniformly exponentially stable.

2.3 Perturbed Systems In Sect. 1.2 we considered the perturbed nonlinear system x Δ = A(t)x(t) + f (t, x), x(t0 ) = x0, t ∈ [t0, ∞)T

(2.3.1)

where f : [t0, ∞)T × R → Rn with f (t, 0) = 0 and consider the nonlinear dynamical system where A(t) is n × n matrix such that all of its entries belong to Cr d (T, R) and x is n × 1 vector. If D is a matrix, then |D| means the sum of the absolute values of the elements. For what to follow we write x for x(t). In Chap. 1, Theorem 1.2.7 we proved the zero solution of (2.3.1) is (UAS). In the next theorem we show the exponential stability of the zero solution provided that the perturbation term f is small enough. We begin with the following theorem. Theorem 2.3.1. Assume there are positive constants α2, β2 and k × k positive definite constant symmetric matrix B such that α2 xT x ≤ xT Bx ≤ β2 xT x. Also, assume that

| f (t, x)| = 0, |x|

(2.3.3)

μ(t)| f (t, x)||BA| = 0, |x|

(2.3.4)

lim

|x |→0

and lim

|x |→0

(2.3.2)

uniformly for t ∈ [t0, ∞)T . If there exists positive constant ξ, where − βξ2 ∈ R + and AT B + BA + μ(t)AT BA ≤ −ξ I,

(2.3.5)

then the zero solution of (2.3.1) is uniformly exponentially asymptotically stable. Proof. Let the matrix B be defined by (2.3.5) and define V(t, x) = xT Bx.

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2 Ordinary Dynamical Systems

Here xT x = x 2 = (x12 + x22 + · · · + xk2 ). Using the product rule given in Theorem 1.1.2 we have along the solutions of (2.3.1) that  x) = (x Δ )T Bx + (x σ )T Bx Δ V(t, = (x Δ )T Bx + (x + μ(t)x Δ )T Bx Δ = (x Δ )T Bx + xT Bx Δ + μ(t)(x Δ )T Bx Δ ' ( = x AT B + BA + μ(t)AT BA xT + 2xT B f (t, x) + μ(t) f T (t, x)BAx + μ(t)xT AT B f (t, x) + μ(t) f T (t, x)B f (t, x) ' ( = x AT B + BA + μ(t)AT BA xT + 2xT B f (t, x) + 2μ(t) f T (t, x)BAx + μ(t) f T (t, x)B f (t, x) ≤ −ξ |x| 2 + 2|x||B|| f (t, x)| + 2μ(t)| f (t, x)||BA||x| + μ(t)|B| f 2 (t, x) (2.3.6) by (2.3.5). Conditions (2.3.3) and (2.3.4) allow us to find γ > 0 small enough so that |x| ≤ γ, for t ∈ [t0, ∞)T imply that 2|B|| f (t, x)| ≤

ξ ξ ξ |x|, 2μ(t)| f (t, x)||BA| ≤ |x|, μ(t)|B| f 2 (t, x) ≤ |x| 2 . (2.3.7) 4 4 4

Substituting (2.3.7) into (2.3.6) we arrive at  x) ≤ − ξ xT x V(t, 4 ξ ≤ − 2 xT Bx 4β ξ = − 2 V(t, x) 4β

(2.3.8)

by (2.3.2) and (2.3.8). It is easy to see that (2.3.8) gives V(t, x) ≤ V(t0, x(t0 ))e−

ξ β2

(t, t0 )

(2.3.9)

Thus, by (2.3.8) we have that α2 xT x ≤ V(t, x) ≤ V(t0, x(t0 ))e− ≤ β2 xT (t0 )x(t0 )e−

ξ β2

ξ β2

(t, t0 )

(t, t0 )

Since, xT x = |x| 2 , we obtain from the last inequality that |x(t)| ≤

 1 β |x0 | e− ξ (t, t0 ) 2 . α β2

2.3 Perturbed Systems

67

Example 2.3.1. Let T be a time scale such that 0 ≤ μ(t) < 45 , μ(t)|g(t)| ≤ M1 and |g(t)| ≤ M2, for all t ∈ [t0, ∞)T , for positive constants M1 and M2 . We consider the nonlinear system     g(t) x12 + x22 −2 1 x+ √ xΔ = (2.3.10) 2 2 −1 −2 2 x1 + x2 Choose B = I. Then AT B + BA + μ(t)AT BA = (5μ(t) − 4)I ≤ −ξ I, for some positive ξ. Since B = I, condition (2.3.2) is satisfied with α2 = β2 = 1. Thus, − βξ2 = −ξ. In

addition, 1 − μ(t)ξ > 0, which implies that − βξ2 ∈ R + . Also, it is easy to check that conditions (2.3.3) and (2.3.4) are satisfied. Thus, we have shown that the zero solution of the system (2.3.10) is uniformly exponentially stable. Next we display couple time scales that satisfy the requirement of Example 2.3.1. n 1 1. Let H0 = 0, and Hn = k=1 2k , for n ∈ N. Consider the time scale T1 = {Hn : n ∈ N0 }, 1 then μ(Hn ) = Hn+1 − Hn = 2n+2 . 4 2. For a, b > 0, with b < 5 , we consider the time scale

T2 = ∪∞ k=0 [k(a + b), k(a + b) + a]. 

Then μ(t) =

0, b,

t ∈ ∪∞ k=0 [k(a + b), k(a + b) + a] ∞ t ∈ ∪k=0 {k(a + b) + a}.

Example 2.3.2. Let T be a time scale such that μ(t)|g(t)||BA| ≤ M1 and |g(t)| ≤ M2, for all t ∈ [t0, ∞)T , for positive constants M1 and M2 . Consider the nonlinear system     g(t) x12 + x22 −1 0 xΔ = (2.3.11) x+ √ 2 2 0 −a(t) 2 x1 + x2 Choose B = I. Then

 μ(t) − 2 0 A B + BA + μ(t)A BA = := A∗ 0 (μ(t)a(t) − 2)a(t) T

Now for any 2 × 2 matrix



T



d d D = 11 12 d21 d22



to be negative definite, we must have −d11 > 0 and det(D) > 0. Hence, 2 − μ(t) > 0, which implies that 0 < μ(t) < 2. Moreover, det(A∗ ) = 2 (μ(t) − 2)((μ(t)a(t) − 2)a(t)) > 0, for a(t) > 1 and μ(t) < . As before, since a(t)

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2 Ordinary Dynamical Systems

B = I, condition (2.3.2) is satisfied with α2 = β2 = 1. Thus, − βξ2 = −ξ. In addition, 2 , we 1 − μ(t)ξ > 0, which implies that − βξ2 ∈ R + . Thus for a(t) > 1 and , μ(t) < a(t) have the zero solution of (2.3.11) is uniformly exponentially stable.

2.4 Delay Dynamical Systems We consider variation of the general dynamic system x Δ (t) = G(t, x(t), x(δ(t))), t ∈ [t0, ∞)T

(2.4.1)

on an arbitrary time scale T which is unbounded above and 0 ∈ T. Here the function G is rd-continuous, where x ∈ Rn and G : [t0, ∞)T × Rn × Rn → Rn with G(t, 0, 0) = 0. the delay function δ : [t0, ∞)T → [δ(t 0 ), ∞) T is strictly increasing, invertible, and delta differentiable such that δ(t) < t, δΔ (t) < ∞ for t ∈ T, and δ(t0 ) ∈ T. For each t0 ∈ T and for a given r d-continuous initial function ψ := [δ(t0 ), t0 ]T → Rn , we say that x(t) := x(t; t0, ψ) is the solution of (2.4.1) if x(t) = ψ(t) on [δ(t0 ), t0 ]T and satisfies (2.4.1) for all t ≥ t0 . We define Et0 = [δ(t0 ), t0 ]T which we call the initial interval. For x ∈ Rn , |x| denotes the Euclidean norm of x. For any n × n matrix A, | A| will denote any compatible norm so that | Ax| ≤ | A||x|. Let C(t) denote the set of rd-continuous functions φ : [δ(t0 ), t]T → Rn and φ = sup{|φ(s)| : δ(t0 ) ≤ s ≤ t}. Definition 2.4.1. The zero solution of (2.4.1) is stable if for each ε > 0 and each t0 ≥ 0, there exists a δ = δ(ε, t0 ) > 0 such that [φ ∈ Et0 → Rn, φ ∈ C(t) : φ < δ] implies |x(t, t0, φ)| < ε for all t0 ≥ 0. Definition 2.4.2. The zero solution of (2.4.1) is uniformly stable (US) if it is stable and δ is independent of t0 . Definition 2.4.3. The zero solution of (2.4.1) is asymptotically stable (AS) if it is stable and if for each t0 ≥ 0 there is an η > 0 such that [φ ∈ Et0 → Rn, φ ∈ C(t) : φ < η] implies that any solution of (2.4.1) |x(t, t0, φ)| → 0 as t → ∞. If this is true for any t0 and any η, then x = 0 is globally asymptotically stable (GAS). Definition 2.4.4. The zero solution of (2.4.1) is uniformly asymptotically stable (UAS) if it is uniformly stable and if there exists an η > 0 and for each ν > 0 there exists a T > t0 such that [t0 ≥ 0, φ ∈ Et0 → Rn, φ ∈ C(t) : φ < η, t ≥ T] implies that any solution of (2.4.1) satisfies |x(t, t0, φ)| < ν. Definition 2.4.5. We say that the zero solution of (2.4.1) is exponentially asymptotically stable on [δ(t0 ), ∞)T if there exists a positive constant d, a constant C ∈ R+ , and an M > 0 such that for any solution x(t, t0, φ) of (2.4.1),

2.4 Delay Dynamical Systems

69

   x(t, t0, x0 ) ≤ C |φ|, t0 (e−M (t, t0 ))d,

for all t ∈ [t0, ∞)T,

  where C |φ|, t0 is a constant depending on |φ| and t0 , φ is a given continuous and bounded initial function. It is uniformly asymptotically stable if C is independent of t0 .

2.4.1 Delay Dynamical Systems-Linear Case In this section we will use fixed point theory and study the stability of the totally delayed dynamical linear system x Δ (t) = −a(t)x(δ(t))δΔ (t),

(2.4.2)

where δ : [t0, ∞)T → [δ(t0 ), ∞)T is strictly increasing, invertible, and Δ-differentiable delay function having the following properties:  0. Now, given any δ > 0, with  y0  < δ and W(r, y(·)) > 0 so that if y (r) is a solution of (5.1.11), we have 2 ∫r

  y (r) − L(r, τ)y(τ)Δτ  ≥ W(r, y(·)) r0   ∫r ≥ W(r0, y0 ) + α y 2 (τ)Δτ. r0

∫r We will show that y(r) is unbounded. If y(r) is bounded, then as

L(r, τ) Δτ is ro

∫r

L(r, τ)y(τ)Δτ is bounded, this implies that y 2 (r) is in L 1 (T0 ).

bounded, we have r0

Using Theorem 1.1.14, we have 2 ∫r

  L(r, τ)  y(τ) Δτ  r0  2 ∫r

 1/2 1/2 =  L(r, τ) L(t, τ)  y(s) Δs r0  ∫r ∫r ≤ L(r, τ) Δs L(r, τ) y 2 (τ)Δτ r0

r0

∫r

≤

∫r

L(r, τ) Δτ t0

* l(r, σ(τ))y 2 (τ)Δτ.

r0

The last integral is the convolution of an L 1 (T0 ) function with a function tending to zero. Thus the integral tends to zero as r → ∞ and hence ∫r L(r, τ)y(τ)Δτ → 0 as r → ∞. r0

5.1 Various Results Using Lyapunov Functionals

Since

207

< < < < ∫r < < < y(r) − < ≥ [W(r0, y0 )]1/2, L(r, τ)y(τ)Δτ < < < < r0

then for sufficiently large T,  y(r) ≥ α for some α > 0 and all r ≥ T . This contradicts the fact that y 2 (t) is in L 1 (T0 ) . Thus, y(r) is unbounded and the zero solution of (5.1.11) is unstable, which is the contradiction to the fact that the zero solution is stable. Hence, B (r) < 0 which completes our proof. Example 5.1.2. For T0 = qN ∪ hN∪ {0} , h ≥ 2, 0 < q < 1, consider the following Volterra integro-dynamic equation: y Δ (r) = A (r) y (r) +

∫r K (r, τ) y (τ) Δτ,

(5.1.18)

r0

where

⎧ 1 − 2h2 ⎪ ⎪ , for all r ∈ qN ∪ {0} , ⎨ ⎪ 2h3 A (r) = 1 − (r + 2h) (r + h) ⎪ ⎪ , for all r ∈ hN, ⎪ ⎩ h (r + h) (r + 2h) N ⎧ ⎪ ⎨ 0, for r ∈ q ∪ {0} , ⎪ 2 K (r, τ) = ⎪ ⎪ hr (r + h) (r + 2h) , for r ∈ hN. ⎩

Let us consider the following function: ⎧ 1 ⎪ N ⎪ ⎨ 3 , for r ∈ q ∪ {0} , ⎪ 2h L (r, τ) = 1 ⎪ ⎪ ⎪ hr (r + h) , for r ∈ hN. ⎩ It is easy to see that Δr L (r, τ) = K (r, τ) and B (r) := A (r) − L (σ (r) , r) −1 , = h so that (5.1.18) may be written as equation −y (r) y (r) = − Δr h Δ

∫r r0

y (τ) Δτ. hr (r + h)

It follows from (5.1.19) that B (r) =

1 . h

(5.1.19)

208

5 Exotic Lyapunov Functionals for Boundedness and Stability

Furthermore, for every r ∈ T0 ∫r L (r, τ) Δτ ≤

1 = J < 1. h2

(5.1.20)

r0

Since μ (r) ≤ h for all r ∈ T0, by using (5.1.20), we obtain ∫r

∫∞ L (r, τ) Δτ +

L (u, τ) Δu + μ (r) B22

σ(r)

r0

≤

1 + h2

∫∞

L (u, τ) Δu +

σ(r)

1 h

1 1 1 + < 2. ≤ 2+ h (r + h) h h 1 for all r ∈ T0, then all the assumptions of Theohr (r + h) rem 5.1.3 are satisfied and hence the zero solution of (5.1.18) is stable.

If we take * l(r, σ(τ)) =

In the next theorem we address the asymptotic stability of the zero solution of (5.1.11). Assume for the present time A (r) and K (r, τ) are continuously differentiable and that both ∫∞

∫∞ K (u, τ) Δu, and Δr K (u, r) Δu,

r

r

exist. Theorem 5.1.4. Suppose (5.1.13) hold and there are constants B1, B2, J, N, and R with R < 2 such that for r ∈ T0 , one has (i) −B2 ≤ B(r) ≤ −B1 < 0, ∫r (ii) L(r, τ) Δτ ≤ J < 1 r0

∫r

∫∞ L(u, r) Δu + μ(t)B22 ≤ RB1 /B2 ,

L(r, τ) Δτ +

(iii) r0

∫r

σ(r) ∫∞

K(r, τ) Δτ + |

(iv) r0

K(u, r)Δu| ≤ N,

σ(r)

then the zero solution of (5.1.11) is asymptotically stable. Furthermore, every solution y(r) of (5.1.11) is in L 2 (T0 ). In addition, if

5.1 Various Results Using Lyapunov Functionals

209

  (v) μ(r) 2BΔ + B2 + μ2 (r)(BΔ )2 + 2μ(r)BΔ B + < Δ < < < < B − BL + μ2 (r)(BΔ )2 − μ(r)BΔ L < + < B + μ(r)BΔ < ∫r (vi) K(r, τ), then

K(r, τ) Δτ ≤ RB1, r0

∫∞ Δr K(r, τ) Δτ, and Δr K(u, r) Δu are bounded,

r0

y Δ (r)

∫r

r

→ 0 as r → ∞ and

y Δ (r)

is in L 2 (T0 ) .

Proof. Let V be given by (5.1.15). The stability of the zero solution readily follows from Theorem 5.1.3. Moreover, V Δ (r, y(·)) ≤ −βy 2 (r), β > 0 for all r ∈ T0 . As V(r, y(·)) ≥ 0, we have y(r) in L 2 (T0 ) . To show that zero solution is asymptotically stable, we first observe from (iv) that L(σ(r), r) is bounded and since B(r) is bounded, hence A(r) is bounded. Now (5.1.11) yields that < Δ < < y (r)< =  A(r)  y(r) +

∫r K(r, τ)  y(τ) Δτ,

(5.1.21)

r0

and using (iv), that y Δ (r) is bounded. Since y 2 (r) is in L 1 (T0 ) and [y 2 (r)]Δ = 2y(r)y Δ (r) + μ(r)[y Δ (r)]2 is bounded, it follows that y(r) → 0 as r → ∞. Hence, zero solution is asymptotically stable. To show that y Δ (r) is in L 2 (T0 ) , differentiate (5.1.14), we obtain y ΔΔ (r) = (B(r)y(r))Δ ⎡ ⎤ ∫r ⎢ ⎥ +Δr ⎢⎢ L(σ(r), r)y(r) + K(r, τ)y(τ)Δτ ⎥⎥ . ⎢ ⎥ r0 ⎣ ⎦ Let B = B(r), L = L(σ(r), r) and consider the functional V(r, y(·),

y Δ (·))

∫r =

(By(r))2 ∫∞ +C

+M

E(r, τ)  y(τ) 2 Δτ r0

y 2 (τ)Δτ. r

The -derivative along a solution y(r) of (5.1.14) for r ∈ T0 satisfies V Δ (r, y(·), y Δ (·)) = 2(By(r))(By(r))Δ + μ(r)[(By(r))Δ ]2 ∫r 2 2 − C y (r) + M E(σ(r), r)y (r) + M Δr E(r, τ)  y(τ) 2 Δτ r0

210

5 Exotic Lyapunov Functionals for Boundedness and Stability

= 2By(r)By Δ (r) + 2By(r)BΔ y(r) + 2μ(r)By(r)BΔ y Δ (r) + μ(r)[(By(r))Δ ]2 − C y 2 (r) + M E(σ(r), r)y 2 (r) ∫r + M Δr E(r, τ)  y(τ) 2 Δτ r0

and substituting the value of By(r) from Eq. (5.1.14), we obtain after some calculations V Δ (r, y(·), y Δ (·)) ≤ [2B + μ(r)(2QΔ + B2 + μ2 (r)(BΔ )2 + 2μ(r)BΔ B) + ||BΔ − BL + μ2 (r)(BΔ )2 − μ(r)BΔ L|| ∫r ||K(r, τ)||Δτ](y Δ (r))2 + ||B + μ(r)BΔ || · r0 Δ

+ [||B − BL + μ (r)(BΔ )2 − μ(r)BΔ L|| ∫r Δ ||K(r, τ)||Δτ + μ(r)(BΔ )2 + M E(σ(r), r) − C]y 2 (r)] + ||B || 2

r0 Δ

∫r

Δ

+ (||B + μ(r)B || + ||B || + M)

Δr E(r, τ)||y(τ)|| 2 Δτ. r0

By (v), BΔ is bounded. Thus, by choosing C and M sufficiently large, we obtain #   V Δ (r, y(·), y Δ (·)) ≤ 2B + μ(r) 2BΔ + B2 + μ2 (r)(BΔ )2 + 2μ(r)BΔ B < < < < + < BΔ − BL + μ2 (r)(BΔ )2 − μ(r)BΔ L < + < B + μ(r)BΔ < ∫r ×

K(r, τ)]Δτ(y Δ (r))2

r0

≤ [−2B1 + RB1 ](y Δ (r))2 = −α(y Δ (r))2 . As V(r, y(·), y Δ (·)) ≥ 0, we have y Δ (r) in L 2 (T0 ) . To show y Δ (r) → 0, we differentiate (5.1.11) and obtain y ΔΔ (r) = A(r)y Δ (r) + AΔ (r)y(r) + μ(r)AΔ (r)y Δ (r) ∫r +K(σ(r), r)y(r) + Δt K(r, τ)y(τ)Δτ. r0

Since

∫∞ B(r) = A(r) − L(σ(r), t) = A(r) + σ(r)

K(u, r)Δu,

5.1 Various Results Using Lyapunov Functionals

211

we have Δ

∫∞

Δ

B (r) = A (r) − K(σ(r), σ(r)) +

Δr K(u, r)Δu.

σ(r)

Thus, by (v) and (vi), AΔ (r) is bounded and hence by (5.1.21) and (5.1.1), y ΔΔ (r) is bounded; therefore, ((y Δ (r))2 )Δ = 2y Δ (r)y ΔΔ (r) + μ(r)[y ΔΔ (r)]2 is bounded. As (y Δ (r))2 is in L 1 (T0 ), we have y Δ (r) → 0 as r → ∞. This completes the proof.

5.1.2 Necessary and Sufficient Conditions: The Vector Case Now we extend the results of Sect. 5.1.1 to the system of Volterra equations; that is, n ≥ 1. Owing to the greater complexity of systems over scalars, it seems preferable to reduce the generality of A and K. As we have mentioned before in the previous chapters, constructing Lyapunov functionals for vector equations is almost impossible nevertheless, obtaining necessary and sufficient conditions. Thus, we consider ∫r

Δ

K (r, τ) y (τ) Δτ,

y (r) = Ay (r) +

(5.1.22)

r0

in which A is constant n × n matrix and K is n × n matrix of functions continuous on Ω. We suppose that there is a symmetric matrix D which satisfies the equation AT D + DA = −I.

(5.1.23)

Theorem 5.1.5. Suppose (5.1.23) holds for some symmetric matrix D and that there is a constant M > 0 such that ∫r

 D (1 + μ(r)  A) K (r, τ) Δτ + μ(r)A2 + E(σ(r), r) r0   ≤ M < 1. Then the zero solution of (5.1.22) is stable if and only if D is positive definite. Proof. We consider the functional ∫r V(r, y(·)) = y (r)Dy(r) + D

E (r, τ)  y(τ) 2 Δτ.

T

r0

The -derivative of V(r, y(·)) along a solution y(r) of (5.1.22) satisfies

212

5 Exotic Lyapunov Functionals for Boundedness and Stability

∫r

 V Δ (r, y(·)) = 2yT (r)D  Ay(r) + K(r, τ)y(s)Δτ  r0   + D E(σ(r), r)  y(r) 2 T ∫r ∫r

  + μ(r)  Ay(r) + K(r, τ)y(τ)Δτ  D  Ay(r) + K(t, τ)y(τ)Δτ  r0 r0     r ∫ + D Δr E(r, τ)  y(τ) 2 Δτ r0

∫r = y (r)DAy(r) + y A Dy(r) + 2y (r)D T

T

T

K(r, τ)y(τ)Δτ

T

r0

+ D E(σ(r), r)  y(r) + μ(r)yT (r)AT DAy(r) ∫r ∫r T T + 2μ(r) y (τ)K (r, τ)ΔτDAy(r) + μ(r) yT (τ)K T (r, τ)Δτ 2

r0

r0

∫r ×D

∫r K(r, τ)y(τ)Δτ + D

r0

Δr E(r, τ)  y(τ) 2 Δτ. r0

Using (5.1.23), we may write V Δ (r, y(·)) ≤ −  y(r) 2 + |D| E(σ(r), r)  y(r) 2 2 ∫r

 + μ(r) |D|  K(r, τ) |y(τ)| Δτ  + (1 + μ(r)  A) D r0  ∫r × K(r, τ) ( y(r) 2 +  y(τ) 2 )Δτ + μ(r) D  A 2  y(r) 2 r0

∫r − D

(1 + μ(r) { A + C(r)}) K(r, τ)  y(τ) 2 Δτ r0

+ μ(r) D  A 2  y(r) 2 . Thus, Δ



∫r

V (r, y(·)) ≤ ||D||E(σ(r), r)||y(r)|| + μ(r)||D|| C(r) 2

||K(r, τ)||||y(τ)|| 2 Δτ r0

+ μ(r)||D|||| A|| 2 ||y(r)|| 2 − ||y(r)|| 2 + (1 + μ(r)|| A||)||D||



5.1 Various Results Using Lyapunov Functionals

213

∫r ×

||K(r, τ)||(||y(t)|| 2 + ||y(τ)|| 2 )Δτ r0

∫r − ||D||

(1 + μ(r)(|| A|| + C(r))||K(r, τ)||||y(τ)|| 2 Δτ r0

∫r = [−1+||D||(1+μ(r)|| A||

||K(r, τ)||Δτ+μ(r)A2 +E(σ(r), r)]||y(r)|| 2 r0

≤ [−1 + M]||y(r)|| = −α||y(r)|| 2, 2

where α = −1 + M > 0. Now, if D is positive definite, then yT (r)Dy(r) > 0, for all y  0 and hence V(r, y(·)) is positive definite and V Δ (r, y(·)) is negative definite it follows that zero solution of (5.1.22) is stable. Conversely, suppose that zero solution of (5.1.22) is stable but D is not positive definite. Then there is an y0  0 such that y0T Dy0 ≤ 0. If y0T Dy0 = 0, then along the solution y(r) of (5.1.22), we have V(r0, y0 ) = y0T Dy0 = 0, V(r, y (·)) ≤ −α  y (r) 2 . It follows that for some r1 > r0, V(r1, y(·)) < 0. Thus yT (r1 )Dy(r1 ) < 0. Hence, yT (r)Dy(r) is not always positive for y (r)  0, and we may suppose there is an y0  0 such that y0T Dy0 < 0. Let ε = 1. Since zero solution of (5.1.22) is stable, then there is a δ > 0 and y0  0 such that  y0  < δ and  y(r) < ε for all r ∈ T0 . We may choose y0 such that  y0  < δ and y0T Dy0 < 0. Moreover, we have ∫r yT (r)Dy(r)

≤ V(r, y(·)) ≤ V(r, y (·)) − α r0

∫r = y0T Dy0 − α

 y(τ) 2 Δτ

 y(τ) 2 Δτ.

r0

We show that y(r) is bounded away from zero. Suppose that it is not true, then there is a sequence {rn } tending to infinity monotonically such that y(rn ) → 0, a contradiction to yT (r)Dy(r) ≤ y0T Dy0 < 0. Thus there is β > 0 with |y(r)| 2 ≥ β so that yT (r)Dy(r) ≤ y0T Dy0 − αβr, implying that  y(r) → ∞ as r → ∞. This contradicts  y(r) < 1 and completes the proof. We now prove instability result for the zero solution of (5.1.22). For this suppose D be a positive definite symmetric matrix satisfying AT D + DA = I.

(5.1.24)

214

5 Exotic Lyapunov Functionals for Boundedness and Stability

Theorem 5.1.6. Suppose that (5.1.24) holds for some symmetric matrix D and that there is a constant M > 0 such that ∫r

 D (1 + μ(r)  A) K(r, τ) Δτ + μ(r)A2 + E(σ(r), r) ≤ M < 1. r0   Then the zero solution of (5.1.22) is completely unstable. Proof. Consider the functional ∫r W(r, y(·)) = yT (r)Dy(r) − D

E(r, τ)  y(τ) 2 Δτ. r0

The -derivative of W(r, y(·)) along a solution y(r) of (5.1.22) satisfies ∫r

 W Δ (r, y(·)) = 2yT (r)D  Ay(r) + K(r, τ)y(τ)Δτ  r0   T ∫r

 2 − D E(σ(r), r)  y(r) + μ(t)  Ay(r) + K(r, τ)y(τ)Δτ  r0   ∫r

 ×D  Ay(r) + K(r, τ)y(τ)Δτ  r0   ∫r − D Δr E(r, τ)  y(τ) 2 Δτ. r0

Similarly, as in Theorem 5.1.5, we obtain the following estimate: W Δ (r, y(·)) ≥ ζ y 2 (r), where ζ = [1 − M] > 0. Choose y0 so that W(r, y0 ) > 0. Then D  y(r) 2 ≥ yT (r)Dy(r) ≥ W(r, y(·)) ∫r ≥ W(r0, y0 ) + ζ  y(τ) 2 Δτ, r0

for all r ∈ T0 . As  y(r) 2 ≥ W(r0, y(·))/D , we have D  y(r) 2 ≥ W(r0, y0 ) +

ζW(r0, y0 )(r − r0 ) . D

5.2 Shift Operators and Stability in Delay Systems

215

If r0 = 0, y0  0, then '  y(r) ≥ 2

y0T Dy0

( +(

D

ζ D

2

( ' ) y0T Dy0 r.

The proof is now complete.

5.2 Shift Operators and Stability in Delay Systems In this section, we use what we call (Adıvar–Raffoul) shift operator so that general delay dynamic equations of the form x Δ (t) = a(t)x(t) + b(t)x(δ− (h, t))δ−Δ (h, t), t ∈ [t0, ∞)T can be analyzed with respect to stability and existence of solutions. By means of the shift operators we define a general delay function opening an avenue for the construction of Lyapunov functional on time scales. Thus, we use Lyapunov’s direct method to obtain inequalities that lead to stability and instability. Therefore, we extend and unify stability analysis of delay differential, delay difference, delay hdifference, and delay q-difference equations which are the most important particular cases of our delay dynamic equation. Lyapunov functionals are widely used in stability analysis of differential and difference equations . However, the extension of utilization of Lyapunov functionals in dynamical systems on time scales has been lacking behind due to the constrained presented by the particular time scale. For example, in delay differential equations, a suitable Lyapunov functional will involve a term with double integrals, in which one of the integral’s lower limit is of the form t + s. Such a requirement will restrict the time scale that can be considered. For example, in [15] the authors improved the results of [163] by considering the delay differential equation of the form x (t) = a(t)x(t) + b(t)x(t − h(t)), 0 < h(t) ≤ r0 .

(5.2.1)

On the other hand, stability analysis of delay difference equations of the form x(t + 1) = a(t)x(t) + b(t)x(t − τ), τ ∈ Z+ is treated in [15, 38], and [139]. The reader has noticed by now that the above two equations require the time scale to be additive. Moreover, in [15] the authors use the following Lyapunov functional 

∫

V(t) = x(t) + ∫ +λ

0

−h

∫

t

2

b(s + h)x(s)ds

t−h t

t+s

b2 (z + h)x 2 (z)dzds

216

5 Exotic Lyapunov Functionals for Boundedness and Stability

to study the exponential stability of the zero solution of (5.2.1). In this work we do not adopt this type of Lyapunov functional since it requires the time scale to be additive. To circumvent such requirements, we use the operators δ− and δ+ associated with t0 ∈ T∗ (called the initial point) and they are called backward and forward shift operators on the set T∗ , respectively. The δ− and δ+ are the same of Definition 1.4.4 of Sect. 1.4.2 and satisfy the same P.1–P.5. The sets D± are the domains of the shift operators δ± , respectively. Most of the results of this section can be found in [14]. In Sect. 1.4.2 we introduced the basic notion of shift operators that were created by the authors. There are many time scales that are not additive. To be more specific, the √ -√ . time scales q Z = {0} ∪ {q n : n ∈ Z}, N = n : n ∈ N are not additive. However, √ δ± (s, t) = ts±1 and δ± (s, t) = t 2 ± s2 are the shift operators defined on q Z and √ N, respectively. It turns out that we need the notion of shift operators to avoid closeness assumption on the time scale. That is, to include more time scales in the investigation. Shift operators were first introduced in [4] to obtain function bounds for convolution type Volterra integro-dynamic equations on time scales. However, the time scales considered in [4] is restricted to the ones having an initial point t0 ∈ T so that there exist the shift operators defined on [t0, ∞)T . Afterwards, in [12] definition of shift operators was extended so that they are defined on the whole time scale T. In this section, our new and generalized shift operators include positive and negative values. However, in this section since we are dealing with stability, we needed to introduce the concept of sticky point and prove new results of the shift operators that are directly related to the concept of sticky point. Thus we have the following definition. Definition 5.2.1. Let T be a time scale. A point t ∗ ( t0 ) ∈ T is said to be a sticky point of T if δ± (s, t ∗ ) = t ∗ for all s ∈ [t0, ∞)T with (s, t ∗ ) ∈ D± . Throughout this section we will denote by t ∗ and T∗ the sticky point and the largest subset of T without sticky point, respectively. Corollary 5.2.1. A sticky point t ∗ cannot be included in the interval [t0, ∞)T . Proof. If t ∗ ∈ [t0, ∞)T is a sticky point, then t ∗ = δ− (t ∗, t ∗ ) = t0 ∈ T∗ = T− {t ∗ } . This leads to a contradiction. Example 5.2.1. Let T = R and t0 = 1. The operators  t/s if t ≥ 0 δ− (s, t) = , for s ∈ [1, ∞) st if t < 0 

and δ+ (s, t) =

st if t ≥ 0 , for s ∈ [1, ∞) t/s if t < 0

are left and right shift operators associated with the initial point t0 = 1. Also, t ∗ = 0 is a sticky point (i.e., T∗ = R− {0}) sinceδ± (s, 0) = 0 for all s ∈ [1, ∞). In the following we give further time scales with their corresponding shift operators.

5.2 Shift Operators and Stability in Delay Systems

T R Z Z q ∪ {0} N1/2

t0 t ∗ 0 N/A 0 N/A 1 0

T∗ R Z qZ

0 N/A N1/2

217

δ− (s, t) δ+ (s, t) t−s t+s t−s t+s t st √ s √ 2 2 2 t − s t + s2

It is clear that the shift operators δ− and δ+ satisfy the hypothesis of Lemma 1.4.6. In Sect. 2.4, we assumed that the delay function δ : [t0, ∞)T → [δ(t0 ), ∞)T is surjective, strictly increasing and satisfies the following properties. δ(t) < t,

δΔ (t) < ∞.

5.2.1 Generalized Delay Functions We introduce the delay function on time scales that will be used for the construction of the Lyapunov functional when dealing with exponential stability. Definition 5.2.2. Let T be a time scale that is unbounded above and t0 ∈ T∗ an element such that there exist the shift operators δ± : [t0, ∞)×T∗ → T∗ associated with t0 . Suppose that h ∈ (t0, ∞)T is a constant such that (h, t) ∈ D± for all t ∈ [t0, ∞)T , the function δ− (h, t) is differentiable with an r d-continuous derivative, and δ− (h, t) maps [t0, ∞)T onto [δ− (h, t0 ), ∞)T . Then the function δ− (h, t) is called the delay function generated by the shift δ− on the time scale T. It is obvious from P.2 and (iii) of Lemma 1.4.6 that δ− (h, t) < δ− (t0, t) = t for all t ∈ [t0, ∞)T .

(5.2.2)

Notice that δ− (h, .) is strictly increasing and it is invertible. Hence, by P.4–5 δ−−1 (h, t) = δ+ (h, t). Hereafter, we shall suppose that T is a time scale with the delay function δ− (h, .) : [t0, ∞)T → [δ− (h, t0 ), ∞)T , where t0 ∈ T is fixed. Denote by T1 and T2 the sets T1 = [t0, ∞)T and T2 = δ− (h, T1 ).

(5.2.3)

Evidently, T1 is closed in R. By definition we have T2 = [δ− (h, t0 ), ∞)T . Hence, T1 and T2 are both time scales. Let σ1 and σ2 denote the forward jump operators on the time scales T1 and T2 , respectively. By (5.2.2–5.2.3) T1 ⊂ T2 ⊂ T. Thus, σ(t) = σ2 (t) for all t ∈ T2 andσ(t) = σ1 (t) = σ2 (t) for all t ∈ T1 . That is, σ1 and σ2 are the restrictions of the forward jump operator σ : T → T to the time scales T1 and T2 , respectively, i.e.,σ1 = σ| T1 and σ2 = σ| T2 . Recall that the Hilger derivatives Δ, Δ1 , and Δ2 on the time scales T, T1 , and T2 are defined in terms of the forward jumps σ, σ1 , and σ2 , respectively. Hence, if f is a differentiable function at t ∈ T2 , then we have f Δ2 (t) = f Δ1 (t) = f Δ (t), for all t ∈ T1 . Similarly, if a, b ∈ T2 are two points with a < b and if f is a r d-continuous function on the interval (a, b)T2 , then

218

5 Exotic Lyapunov Functionals for Boundedness and Stability

∫

b

∫ f (s)Δ2 s =

a

b

f (s)Δs.

(5.2.4)

a

The next result is essential for future calculations. Lemma 5.2.1. The delay function δ− (h, t) preserves the structure of the points in T1 . t) = 4 t implies σ2 (δ− (h, 4 t)) = δ− (h, 4 t).and σ1 (4 t) > 4 t implies σ2 (δ− (h, 4 t) > That is,σ1 (4 t). δ− (h, 4 Proof. By definition σ1 (t) ≥ t for all t ∈ T1 . Thus,δ− (h, σ1 (t)) ≥ δ− (h, t). Since σ2 (δ− (h, t)) is the smallest element satisfyingσ2 (δ− (h, t)) ≥ δ− (h, t), we get δ− (h, σ1 (t)) ≥ σ2 (δ− (h, t)) for all t ∈ T1 .

(5.2.5)

If σ1 (4 t) = 4 t, then we have δ− (h, 4 t) = δ− (h, σ1 (4 t)) ≥ σ2 (δ− (h, 4 t)). That is, t) = σ2 (δ− (h, 4 t)). If σ1 (4 t) > 4 t, then (4 t, σ1 (4 t))T1 = (4 t, σ1 (4 t))T =  and δ− (h, 4 t)) > δ− (h, 4 t). Suppose the contrary. That is δ− (h, 4 t) is right dense; namely δ− (h, σ1 (4 t)) = δ− (h, 4 t). This along with (5.2.5) implies(δ− (h, 4 t), δ− (h, σ1 (4 t)))T2  . σ2 (δ− (h, 4 t), δ− (h, σ1 (4 t)))T2 . Since δ− (h, t) is strictly increasing in Pick one element s ∈ (δ− (h, 4 t))T1 such that δ− (h, t) = s. t and invertible, there should be an element t ∈ (4 t, σ1 (4 t) must be right scattered. This leads to a contradiction. Hence, δ− (h, 4 Using the preceding lemma and applying the fact that σ2 (u) = σ(u) for all u ∈ T2 we arrive at the following result. Corollary 5.2.2. We have δ− (h, σ1 (t)) = σ2 (δ− (h, t)) for all t ∈ T1 . Thus, δ− (h, σ(t)) = σ(δ− (h, t)) for all t ∈ T1 .

(5.2.6)

By (5.2.6) we have δ− (h, σ(s)) = σ(δ− (h, s)) for all s ∈ [t0, ∞)T . Substituting s = δ+ (h, t) we obtain δ− (h, σ(δ+ (h, t))) = σ(δ− (h, δ+ (h, t))) = σ(t). This and (iv) of Lemma 1.4.6 imply σ(δ+ (h, t)) = δ+ (h, σ(t)) for all t ∈ [δ− (h, t0 ), ∞)T . Example 5.2.2. In the following, we give some time scales with their shift operators: T R Z

h δ− (h, t) δ+ (h, t) ∈ R+ t − h t+h ∈ Z+ t − h t+h

qZ ∪ {0} ∈ qZ+ N1/2

t ht h √ √ 2 2 2 ∈ Z+ t − h t + h2

5.2 Shift Operators and Stability in Delay Systems

219

Example 5.2.3. There is no delay function δ− (h, .) : [0, ∞)*T → [δ− (h, 0), ∞)T on the time scale * T= (−∞, 0] ∪ [1, ∞). Suppose the contrary that there exists such a delay function on * T. Then since 0 is right scattered in * T1 := [0, ∞)*T the point δ− (h, 0) must be right scattered in * T2 = [δ− (h, 0), ∞)T , i.e., σ2 (δ− (h, 0)) > δ− (h, 0). Since σ2 (t) = σ(t) for all t ∈ [δ− (h, 0), 0)T , we have σ(δ− (h, 0)) = σ2 (δ− (h, 0)) > δ− (h, 0). That is, δ− (h, 0) must be right scattered in * T. However, in * T we have δ− (h, 0) < 0, that is, δ− (h, 0) is right dense. This leads to a contradiction. The next result is essential for future calculations. Theorem 5.2.1 (Leibnitz-Type Formula). Let δ− (h, .) be a delay function on the time scale T which is unbounded above. Suppose that the functions f : [t0, ∞)T × f (t, s) := f (t, δ− (h, s))δ−Δ (h, s) are continuous at (t, t) where t ∈ [t0, ∞)T → R and * f Δ (t, .) are rd-continuous at [δ_(h, t0 ), σ(t)]T . [t0, ∞)T . Also suppose that f Δ (t, .) and * Then we have ∫ t Δ f (t, s)Δs = f (σ(t), t) − f (σ(t), δ− (h, t))δ−Δ (h, t) δ− (h,t)

∫ +

t

δ− (h,t)

f Δ (t, s)Δs.

(5.2.7)

Proof. First, since the operator δ : [t0, ∞)T → [δ(t0 ), ∞)T is strictly increasing, it is bijection. By using the substitution theorem with ν(t) = δ− (h, t) we get ∫ b ∫ δ− (h,b) * f (t, s) Δ2 s f (t, s)Δs = δ− (h,a)

a

for any a, b ∈ T1 = [t0, ∞)T where Δ2 is the derivative on T2 = [δ_(h, t0 ), ∞)T . We obtain by (5.2.4) that ∫ t ∫ t f (t, s)Δs = f (t, s)Δ2 s δ− (h,t)

δ− (h,t) δ− (h,t0 )

∫ =

δ− (h,t) t0

∫ =

∫ f (t, s)Δ2 s +

* f (t, s)Δs +

t

∫

t

δ− (h,t0 )

t

δ− (h,t0 )

f (t, s)Δ2 s

f (t, s)Δ2 s.

From theorem [50, Theorem 1.117], we get the derivatives ∫

t

δ− (h,t0 )

f (t, s)Δ2 s

Δ

∫ =

t

δ− (h,t0 )

f Δt (t, s)Δs + f (σ(t), t)

220

5 Exotic Lyapunov Functionals for Boundedness and Stability

and ∫

t0

* f (t, s)Δs

Δ

∫

t0

=

t

* f (σ(t), t) f Δt (t, s)Δs − *

t

∫ =

δ− (h,t0 )

δ− (h,t)

f Δt (t, s)Δ2 s − f Δt (σ(t), δ− (h, t))δ−Δ (h, t).

and hence ∫

t

δ− (h,t)

f (t, s)Δs

Δ

= f (σ(t), t) − f (σ(t), δ− (h, t))δ−Δ (h, t) ∫ +

t

f Δ (t, s)Δ2 s.

δ− (h,t)

Since [δ− (h, t0 ), t)T1 ⊂ T2 for any t ∈ T1 , we get ∫ t ∫ t Δ f (t, s)Δ2 s = δ− (h,t)

(5.2.8)

f Δ (t, s)Δs.

δ− (h,t)

by using (5.2.4). This and (5.2.8) yield (5.2.7). Theorem 5.2.2. Let k be an r d-continuous function. Then ∫

∫

t

δ− (h,t)

Δs

t

∫ k(u)Δu =

∫

t

δ− (h,t)

s

Δu

σ(u)

δ− (h,t)

k(u)Δs.

(5.2.9)

Proof. Substituting ∫ f (s) = s − δ− (h, t), into the formula ∫ a

z

t

g(s) =

k(u)Δu s

∫ f (σ(x))g(x)Δx = [ f (x)g(x)]za −

and using Lemma 1.4.6 we get ∫ t ∫ ∫ t Δs k(u)Δu = δ− (h,t)

s

=

t

δ− (h,t) ∫ t δ− (h,t)

z

f Δ (x)g(x)Δx

a

[σ(s) − δ− (h, t)] k(s)Δs ∫ Δu

σ(u)

δ− (h,t)

k(u)Δs.

(5.2.10)

5.2 Shift Operators and Stability in Delay Systems

221

5.2.2 Exponential Stability via Lyapunov Functionals Let T be a time scale having a delay function δ− (h, t) where h ≥ t0 and t0 ∈ T is nonnegative and fixed. We consider the equation x Δ (t) = a(t)x(t) + b(t)x(δ− (h, t))δ−Δ (h, t), t ∈ [t0, ∞)T and assume that

(5.2.11)

 Δ  δ (h, t) ≤ M < ∞ for all t ∈ [t0, ∞)T . −

(5.2.12)

Let ψ: [δ− (h, t0 ), t0 ]T → R be r d-continuous and let x(t) := x(t, t0, ψ) be the solution of Eq. (5.2.11) on [t0, ∞)T with x(t) = ψ(t) on [δ− (h, t0 ), t0 ]T . Let ϕ = sup {|ϕ(t)| : t ∈ [δ− (h, t0 ), t0 )T }. Observe that using (5.2.7), Eq. (5.2.11) can be rewritten as follows: ∫

Δ

x (t) = Q(t)x(t) −

t

δ− (h,t)

b(δ+ (h, s))x(s)Δs

Δt

,

(5.2.13)

where Q(t) := a(t) + b(δ+ (h, t)) and Δt indicates the delta derivative with respect to t. Lemma 5.2.2. Let ∫t A(t) := x(t) +

b(δ+ (h, s))x(s)Δs

(5.2.14)

δ− (h,t)

and β(t) := t − δ− (h, t).

(5.2.15)

Assume that there exists a λ > 0 such that −

λδ−Δ (h, t) ≤ Q(t) ≤ −λ [β(t) + μ(t)] b(δ+ (h, t))2 − μ(t)Q2 (t) (5.2.16) β(t) + λ [β(t) + μ(t)]

for all t ∈ [t0, ∞)T . If ∫t V(t) = A(t) + λ 2

δ− (h,t)

∫t Δs

b(δ+ (h, u))2 x(u)2 Δu

(5.2.17)

s

then, along the solutions of Eq. (5.2.11) we have V Δ (t) ≤ Q(t)V(t) for all t ∈ [t0, ∞)T .

(5.2.18)

222

5 Exotic Lyapunov Functionals for Boundedness and Stability

Proof. It is obvious from (5.2.13) and (5.2.14) that AΔ (t) = Q(t)x(t). Then by (5.2.7) and the formula A(σ(t)) = A(t) + μ(t)A(t) we have V Δ (t) = [A(t) + A(σ(t))] AΔ (t) + λ − λδ−Δ (h, t)

∫

∫

σ(t)

t σ(t)

δ− (h,t)

b(δ+ (h, u))2 x(u)2 Δu

b(δ+ (h, u))2 x(u)2 Δu + λ (t − δ− (h, t)) b(δ+ (h, t))2 x(t)2

= [2A(t) + μ(t)Q(t)x(t)] Q(t)x(t) − λδ−Δ (h, t)

∫

σ(t)

δ− (h,t)

b(δ+ (h, u))2 x(u)2 Δu

+ λ [β(t) + μ(t)] b(δ+ (h, t))2 x(t)2 . Using the identity 2

t

∫  2 2  2A(t)x(t) = x (t) + A (t) −  b(δ+ (h, s))x(s)Δs δ− (h,t) 

(5.2.19)

and condition (5.2.16) we have V Δ (t) = Q(t)V(t) + R(t) ' ( + x 2 (t) λ (β(t) + μ(t)) b(δ+ (h, t))2 + Q(t) + μ(t)Q2 (t) ≤ Q(t)V(t) + R(t),

(5.2.20)

where 2

∫  R(t) = −Q(t)  b(δ+ (h, s))x(s)Δs δ− (h,t)  ∫ σ(t) − λδ−Δ (h, t) b(δ+ (h, u))2 x(u)2 Δu t

∫t − λQ(t)

δ− (h,t) ∫t

Δs

b(δ+ (h, u))2 x(u)2 Δu.

δ− (h,t)

(5.2.21)

s

Hereafter, we will show that (5.2.16) implies R(t) ≤ 0. This and (5.2.20) will enable us to derive the desired inequality (5.2.18). First we have ∫

σ(t)

δ− (h,t)

∫ b(δ+ (h, u))2 x(u)2 Δu ≥

t

δ− (h,t)

b(δ+ (h, u))2 x(u)2 Δu.

(5.2.22)

5.2 Shift Operators and Stability in Delay Systems

223

From Hölder’s inequality we get 2

∫

∫    b(δ (h, s))x(s)Δs ≤ β(t) b(δ+ (h, s))2 x(s)2 Δs. +   δ− (h,t) δ− (h,t)  t

t

(5.2.23)

On the other hand, (5.2.9) yields ∫t

∫t Δs

δ− (h,t)

∫t b(δ+ (h, u)) x(u) Δu = 2

σ(u) ∫

Δu

2

δ− (h,t)

s

b(δ+ (h, u))2 x(u)2 Δs

δ− (h,t)

∫t [σ(u) − δ− (h, t)] b(δ+ (h, u))2 x(u)2 Δu

= δ− (h,t)

∫t b(δ+ (h, u))2 x(u)2 Δu.

≤ [β(t) + μ(t)] δ− (h,t)

(5.2.24) Substituting (5.2.23) and (5.2.24) into (5.2.21) and using (5.2.22) together with δ−Δ (h, t) > 0 we deduce that -

R(t) ≤ − (β(t) + [λβ(t) + μ(t)]) Q(t) +

λδ−Δ (h, t)

.

∫t b(δ+ (h, s))2 x(s)2 Δs.

δ− (h,t)

Hence, using the left-hand side of (5.2.16) we arrive at the inequality R(t) ≤ 0. The proof is complete. In preparation for the next result, we remind the following result in Remark 4.1.1. Lemma 5.2.3. If ϕ ∈ R + , then ∫ 0 < eϕ (t, s) ≤ exp

t

 ϕ(r)Δr

(5.2.25)

s

for all t ∈ [s, ∞)T . Theorem 5.2.3. Let a ∈ R + and Q ∈ R. Suppose the hypothesis of Lemma 5.2.2. If there exists an α ∈ (t0, h)T such that (a, t) ∈ D± for all t ∈ [t0, ∞)T and δ− (h, t) ≤

δ− (α, t) + δ− (h, δ− (α, t)) for all t ∈ [α, ∞)T, 2

(5.2.26)

(5.2.27)

224

5 Exotic Lyapunov Functionals for Boundedness and Stability

then any solution x(t) = x(t, t0, ϕ) of (5.2.11) satisfies the exponential inequalities = > ∫ ? 1 δ− (α, t ) 2 Q(s)Δs  V(t0 )e 2 t0 (5.2.28) |x(t)| ≤  1 1 − ξ(t) for all t ∈ [α, ∞)T and |x(t)| ≤ ψ e

∫t t0

a(s)Δs



 ∫s ∫ t   − a(u)Δu b(s) t   Δs 1+M  1 + μ(s)a(s)  e 0

(5.2.29)

t0

for all t ∈ [t0, α)T , where M is as defined by (5.2.12), ξ(t) := 1 +

λΛ(t) > 1, β(t)

and Λ(t) := δ− (h, t) − δ− (h, δ− (α, t)). Proof. Since t0 < α < h the condition (5.2.27) implies δ− (h, t) < δ− (α, t) for all t ∈ [α, ∞)T

(5.2.30)

0 < Λ(t) ≤ δ− (α, t) − δ− (h, t) for all t ∈ [α, ∞)T .

(5.2.31)

and Let V(t) be defined by (5.2.17). First we get by (5.2.10), (5.2.17), and (5.2.30–5.2.31) that ∫t V(t) ≥ λ

∫t Δs

δ− (h,t) ∫ t

=λ

≥λ

δ− (h,t) ∫ t δ− (α,t)

b(δ+ (h, u))2 x(u)2 Δu s

[σ(u) − δ− (h, t)] b(δ+ (h, u))2 x(u)2 Δu [σ(u) − δ− (h, t)] b(δ+ (h, u))2 x(u)2 Δu ∫

≥ λ [δ− (α, t) − δ− (h, t)]

t

δ− (α,t)

b(δ+ (h, u))2 x(u)2 Δu.

This along with (5.2.31) yields ∫ V(t) ≥ λΛ(t) for all t ∈ [α, ∞)T . Similarly, we get

t

δ− (α,t)

b(δ+ (h, u))2 x(u)2 Δu

(5.2.32)

5.2 Shift Operators and Stability in Delay Systems

∫ V(δ− (α, t)) ≥ λ

δ− (α,t)

δ− (h,δ− (α,t)) δ− (α,t)

225

[σ(u) − δ− (h, δ− (α, t))] b(δ+ (h, u))2 x(u)2 Δu

∫ ≥λ

δ− (h,t)

∫

≥ λΛ(t)

[σ(u) − δ− (h, δ− (α, t))] b(δ+ (h, u))2 x(u)2 Δu

δ− (α,t)

b(δ+ (h, u))2 x(u)2 Δu

δ− (h,t)

(5.2.33)

for all t ∈ [α, ∞)T since δ− (α, t) ≤ δ− (t0, t) = t. Utilizing (5.2.17), (5.2.32), and (5.2.33) we obtain ∫ t V(t) + V(δ− (α, t)) ≥ A(t)2 + λΛ(t) b(δ+ (h, u))2 x(u)2 Δu ∫ + λΛ(t)

δ− (α,t)

δ− (α,t)

δ− (h,t)

≥ A(t)2 + λΛ(t)

b(δ+ (h, u))2 x(u)2 Δu

∫

t

δ− (h,t)

b(δ+ (h, u))2 x(u)2 Δu

(5.2.34)

for all t ∈ [α, ∞)T . Substituting (5.2.23) and (5.2.14) into (5.2.34) we find   1 V(t) + V(δ− (α, t)) ≥ 1 − x 2 (t) ξ(t) ⎡ ⎤2 ∫t ⎢ 1 ⎥ 

 ⎢ ⎥ + ⎢ b(δ+ (h, u))x(u)Δu⎥ x(t) + ξ(t)  ⎢ ξ(t) ⎥ ⎢ ⎥ δ− (h,t) ⎦ ⎣  1 ≥ 1− (5.2.35) x 2 (t) ξ(t) for all t ∈ [α, ∞)T . Since V Δ (t) ≤ 0, we get by (5.2.35) that   1 1− x 2 (t) ≤ V(t) + V(δ− (α, t)) ≤ 2V(δ− (α, t)) ξ(t)

(5.2.36)

for all t ∈ [α, ∞)T . Multiplying (5.2.18) by e Q (σ(s), t0 ) and integrating the resulting inequality from t0 to t we derive ∫ t ' Δ ( 0≥ V (s) − Q(s)V(s) e Q (σ(s), t0 )Δs ∫ =

t0 t

t0

' (Δ V(s)e Q (s, t0 ) Δs

= V(t)e Q (t, t0 ) − V(t0 ). That is,

(5.2.37)

226

5 Exotic Lyapunov Functionals for Boundedness and Stability

V(t) ≤ V(t0 )eQ (t, t0 ) for all t ∈ [t0, ∞)T .

(5.2.38)

Combining (5.2.36) and (5.2.38) we arrive at x 2 (t) ≤ 

2 1−

1 ξ(t)

 V(t0 )eQ (δ− (α, t), t0 )

for all t ∈ [α, ∞)T . The hypothesis Q ∈ R and the condition (5.2.16) guarantee that Q(t) ∈ R + . Thus, (5.2.25) implies = > ∫ ? 1 δ− (α, t ) 2 Q(s)Δs 2 t0   V(t0 )e |x(t)| ≤ 1 1 − ξ(t) for all t ∈ [α, ∞)T . Multiplying (5.2.11) by e a (σ(t), t0 ) and integrating the resulting equation from t0 to t we have ∫ t b(s) ea (t, s)x(δ− (h, s))δ−Δs (h, s)Δs. (5.2.39) x(t) = x(t0 )ea (t, t0 ) + t0 1 + μ(s)a(s) Since δ− (h, t) < δ− (α, t) ≤ δ− (α, α) = t0 for all t ∈ [t0, α)T , (5.2.25) along with Eq. (5.2.39) yields  ∫ t b(s) ea (t0, s)ψ(δ− (h, s))δ−Δs (h, s)Δs |x(t)| = ea (t, t0 ) ψ(t0 ) + t0 1 + μ(s)a(s)  ∫  ∫t ∫ t   t a(u)Δu b(s) a(s)ds   ≤ ψ e t0 +M Δs  1 + μ(s)a(s)  e s t0  ∫s  ∫ t ∫t   − a(u)Δu b(s) a(s)ds t   ≤ ψ e t0 Δs . 1+M  1 + μ(s)a(s)  e 0 t0

The proof is complete. Notice that Theorem 5.2.3 does not work for the time scales in which (t0, h)T = . For instance, let T = Z, t0 = 0, δ− (h, t) = t − h, and h = 1. It is obvious that (t0, h)Z = (0, 1)Z = . That is, there is no α so that (5.2.26) and (5.2.27) hold. In preparation for the proof of the next theorem we give the following lemma. Lemma 5.2.4. Let T be a time scale and t0 a fixed point. Suppose that the shift operators δ± (h, t) associated with the initial point t0 are defined on T. Suppose also that there is a delay function δ− (h, t) defined on T. If (t0, h)T = , then the time scale T is isolated (i.e., T consists only of right scattered points). Moreover, σ(t) = δ+ (h, t)

(5.2.40)

5.2 Shift Operators and Stability in Delay Systems

227

for all t ∈ [δ− (h, t0 ), ∞)T or equivalently σ(δ− (h, t)) = t

(5.2.41)

for all t ∈ [t0, ∞)T . Proof. Suppose that (t0, h)T = . Define δ+0 (h, t0 ) = t0 and δ+k (h, t0 ) = δ+ (h, δ+k−1 (h, t0 )) for k ∈ Z+ . Since δ+ (h, t) is surjective and strictly increasing we have       δ+k−1 (h, t0 ), δ+k (h, t0 ) = δ− h, δ+k−2 (h, t0 ), δ+k−1 (h, t0 ) , for k = 2, 3, . . . . T

T

Thus, one can show by induction that   δ+k−1 (h, t0 ), δ+k (h, t0 ) =  for all k ∈ Z+ . T

(5.2.42)

  That is, σ δ+k−1 (h, t0 ) = δ+k (h, t0 ) for k ∈ Z+ . On the other hand, we can write k−1 k [t0, ∞)T = ∪∞ k=1 [δ+ (h, t0 ), δ+ (h, t0 ))T .

Hence, for any t ∈ [t0, ∞)T there is a k0 ∈ Z+ so that t ∈ [δ+k0 −1 (h, t0 ), δ+k0 (h, t0 ))T . By (5.2.42) we have t = δ+k0 −1 (h, t0 ). This shows that σ(t) = σ(δ+k0 −1 (h, t0 )) = δ+k0 (h, t0 ) = δ+ (h, δ+k0 −1 (h, t0 )) = δ+ (h, t) for all t ∈ [δ− (h, t0 ), ∞)T . This along with σ(δ− (h, t)) = δ− (h, σ(t)) yields (5.2.41). The proof is complete. Theorem 5.2.4. Let a ∈ R + , Q ∈ R. Assume the hypothesis of Lemma 5.2.2 . If (t0, h)T = , then any solution x(t) = x(t, t0, ϕ) of (5.2.11) satisfies the exponential inequality   ∫ 1 t 1 Q(s)Δs 1+ V(t0 )e 2 t0 |x(t)| ≤ λ for all t ∈ [t0, ∞)T . Proof. Let H be defined by ∫t H(t) =

∫t Δs

δ− (h,t)

b(δ+ (h, u))2 x(u)2 Δu. s

From (5.2.10), (5.2.23), and (5.2.41), we get ∫ t H(t) = [σ(u) − δ− (h, t)] b(δ+ (h, u))2 x(u)2 Δu δ− (h,t)

≥ [σ(δ− (h, t)) − δ− (h, t)]

∫

t

δ− (α,t)

b(δ+ (h, u))2 x(u)2 Δu

(5.2.43)

228

5 Exotic Lyapunov Functionals for Boundedness and Stability

∫ = β(t)

t

δ− (α,t)

b(δ+ (h, u))2 x(u)2 Δu 2

t

∫   ≥ b(δ+ (h, u))x(u)Δu . δ− (h,t) 

Hence, by (5.2.17) we have V(t) = A2 (t) + λH(t) 2

∫

 ≥  x(t) + b(δ+ (h, s))x(s)Δs δ− (h,t)   t

2

t

∫   +λ b(δ+ (h, u))x(u)Δu δ− (h,t)    1 2 = 1− x (t) 1+λ ⎤2 ⎡ ∫t ⎥ ⎢ 1 √

 ⎥ ⎢ b(δ+ (h, u))x(u)Δu⎥ + ⎢√ x(t) + 1 + λ  ⎥ ⎢ 1+λ ⎥ ⎢ δ− (h,t) ⎦ ⎣  1 ≥ 1− x 2 (t). 1+λ

This along with (5.2.38) yields  |x(t)| ≤

1+

 ∫ 1 t 1 Q(s)Δs . V(t0 )e 2 t0 λ

The proof is complete. In the next corollary, we summarize the results obtained in Theorems 5.2.3 and 5.2.4. Corollary 5.2.3. Assume the hypothesis of Lemma 5.2.2. Let a ∈ R + and Q ∈ R. Suppose that there exists a λ > 0 such that (5.2.16) holds for all t ∈ [t0, ∞)T . 1. If there exists an α ∈ (t0, h)T such that (5.2.26) and (5.2.27) hold, then any solution x(t) = x(t, t0, ϕ) of (5.2.11) satisfies = > ∫ δ− (α, t ) ? 2 −1 [λ(β(s)+μ(s))b(δ+ (h,s))2 +μ(s)Q2 (s)]Δs  V(t0 )e 2 t0 |x(t)| ≤  1 1 − ξ(t)

5.2 Shift Operators and Stability in Delay Systems

229

Thus, if ∫

δ− (α,t)

lim

t→∞

t0

'

( λ (β(s) + μ(s)) b(δ+ (h, s))2 + μ(s)Q2 (s) Δs = ∞,

then the zero solution of Eq. (5.2.11) is exponentially stable. 2. If (t0, h)T = , then any solution x(t) = x(t, t0, ϕ) of (5.2.11) satisfies   ∫t 1 −1 [λ(β(s)+μ(s))b(σ(s))2 +μ(s)Q2 (s)]Δs . 1+ V(t0 )e 2 t0 |x(t)| ≤ λ Thus, if ∫

t

lim

t→∞

'

( λ (β(s) + μ(s)) b(σ(s))2 + μ(s)Q2 (s) Δs = ∞,

t0

then the zero solution of Eq. (5.2.11) is exponentially stable. Now we offer some applications in the form of a theorem. Theorem 5.2.5. Define a continuous function η(t) ≥ 0 by η(t) :=

1+λ

ea (t, t0 )

∫t

e (δ (h, s), t0 )Δs δ− (h,t) a +

.

(5.2.44)

Suppose that a ∈ R + and that |b(t)| − λη σ (t)δ−Δ (h, t) ≤ 0

(5.2.45)

holds for all t ∈ [t0, ∞)T . Then any solution of Eq. (5.2.11) satisfies the inequality |x(t)| ≤ V(t0, xt0 )eγ (t, t0 ) for all t ∈ [t0, ∞)T , ∫

where V(t0, xt0 ) := |x(t0 )| + λη(t0 )

t0

δ− (h,t0 )

(5.2.46)

|x(s)| Δs,

* = max {1, M },and M is as in (5.2.12). * σ (t), M γ(t) := a(t) + λ Mη Proof. For convenience define ∫ ζ(t) := 1 + λ

t

δ− (h,t)

ea (δ+ (h, s), t0 )Δs.

Then ζ Δ (t) = λea (δ+ (h, t), t0 ) − ea (t, t0 )δ−Δ (h, t) ' ( = λea (t, t0 ) ea (δ+ (h, t), t) − δ−Δ (h, t) .

(5.2.47)

230

5 Exotic Lyapunov Functionals for Boundedness and Stability

This and a differentiation of (5.2.44) yield   ea (t, t0 ) aζ(t) − ζ Δ (t) ηΔ (t) = ζ(t) ζ σ (t)   aζ(t) + aμ(t)ζ Δ (t) − aμ(t)ζ Δ (t) − ζ Δ (t) = η(t) ζ(t) + μ(t)ζ Δ (t)  ζ(t) ζ Δ (t) = a(t)η(t) − (1 + μ(t)a(t)) η(t) σ ζ (t) ζ(t) = a(t)η(t) − η σ (t)

ζ Δ (t) ζ(t)

= a(t)η(t) + λη σ (t)η(t)δ−Δ (h, t) − λη σ (t)ea (δ+ (h, t), t) $ # * σ (t) , ≤ η(t) a(t) + λ Mη

(5.2.48)

where we also used ζ σ (t) = ζ(t) + μ(t)ζ Δ (t) and (1 + μ(t)a(t)) η(t)

ζ(t) = η σ (t). ζ σ (t) ∫

Define V(t, xt ) := |x(t)| + λη(t)

t

δ− (h,t)

|x(s)| Δs.

(5.2.49)

x(t) Δ Let t ∈ T+ ∩ [t0, ∞)T . Then we have |x(t)| Δ = |x(t) | x (t). Differentiating (5.2.49) and utilizing (5.2.45) and (5.2.48) we arrive at ∫ t V Δ (t, xt ) = |x(t)| Δ + ληΔ (t) |x(s)| Δs δ− (h,t)

' ( + λη σ (t) |x(t)| − |x(δ− (h, t))| δ−Δ (h, t) # $∫ t x(t) Δ σ * x (t) + λη(t) a(t) + λ Mη (t) ≤ |x(s)| Δs |x(t)| δ− (h,t) ' ( + λη σ (t) |x(t)| − |x(δ− (h, t))| δ−Δ (h, t)     * σ (t) |x(t)| + |b(t)| − λδ−Δ (h, t)η σ (t) |x(δ− (h, t))| = a(t) + λ Mη # $∫ t σ * + λη(t) a(t) + Mη (t) |x(s)| Δs δ− (h,t)

≤ γ(t)V(t, xt ). 2 Similarly, if t ∈ T− ∩ [t0, ∞)T , then |x(t)| Δ = − μ(t) |x(t)| −

# $∫ * σ (t) V Δ (t, xt ) ≤ |x(t)| Δ + η(t) a(t) + λ Mη

t

δ− (h,t)

' ( + λη σ (t) |x(t)| − |x(δ− (h, t))| δ−Δ (h, t)

x(t) Δ |x(t) | x (t).

|x(s)| Δs

Hence,

5.2 Shift Operators and Stability in Delay Systems

231

  2 * σ (t) |x(t)| − a(t) + λ Mη ≤ − μ(t)   + |b(t)| − λδ−Δ (h, t)η σ (t) |x(δ− (h, t))| # $∫ t σ * + λη(t) a(t) + Mη (t) |x(s)| Δs δ− (h,t)

$∫ # σ σ * * ≤ a(t) + λ Mη (t) |x(t)| + λη(t) a(t) + λ Mη (t) 



t

δ− (h,t)

|x(s)| Δs

= γ(t)V(t, xt ). since 1 + μ(t)a(t) > 0 implies − Thus,

2 − a(t) < a(t). μ(t)

V Δ (t, xt ) ≤ γ(t)V(t, xt ) for all t ∈ [t0, ∞)T .

(5.2.50)

An integration of (5.2.50) and applying the fact that V(t, xt ) ≥ |x(t)| we arrive at the desired result.

5.2.3 Instability This subsection is devoted to the study of instability of the zero solution of (5.2.11). Again, the results will be established by positive definite Lyapunov functional. Theorem 5.2.6. Suppose there exists positive constant D such that β(t) < D ≤

Q(t) b(δ+ (h, t))2

(5.2.51)

for all t ∈ [t0, ∞)T , where β(t) is as defined in (5.2.15). Let the function A be defined by (5.2.14). If ∫ V(t) = A(t)2 − D

t

δ− (h,t)

b(δ+ (h, s))2 x(s)2 Δs,

(5.2.52)

then along the solutions of Eq. (5.2.11) we have V Δ (t) ≥ Q(t)V(t) for all t ∈ [t0, ∞)T . Proof. Let V be defined by (5.2.52). Using (5.2.19) and (5.2.23) we obtain V Δ (t) = [A(t) + A(σ(t))] AΔ (t) − Db(δ+ (h, t))2 x(t)2 + Db(t)2 x(δ− (h, t))2 δ−Δ (h, t) ≥ [2A(t) + μ(t)Q(t)x(t)] Q(t)x(t) − Db(δ+ (h, t))2 x(t)2

(5.2.53)

232

5 Exotic Lyapunov Functionals for Boundedness and Stability

≥ 2Q(t)A(t)x(t) − Db(δ+ (h, t))2 x(t)2 2⎤ ⎡ ∫t ⎢ ⎥

 ⎢ ⎥ = Q(t) ⎢ x 2 (t) + A2 (t) −  b(δ+ (h, s))x(s)Δs ⎥ ⎢ ⎥ ⎢ ⎥ δ− (h,t)  ⎦ ⎣ − Db(δ+ (h, t))2 x(t)2 ( ' ≥ Q(t)V(t) + Q(t) − Db(δ+ (h, t))2 x(t)2 .

This along with (5.2.51) implies (5.2.53). Theorem 5.2.7. Suppose all hypotheses of Theorem 5.2.6 hold. Suppose also that β(t) is bounded above by β0 with 0 < β0 < D. Then the zero solution of Eq. (5.2.11) is unstable, provided that ∫ t lim b(δ+ (h, s))2 Δs = ∞. t→∞

t0

Proof. As we did in (5.2.37), an integration of (5.2.53) from t0 to t gives V(t) ≥ V(t0 )eQ (t, t0 ) for all t ∈ [t0, ∞)T .

(5.2.54)

Let V(t) be given by (5.2.52). Then ∫ V(t) = x(t)2 + 2x(t) ∫ −D

t

δ− (h,t)

∫

t

δ− (h,t)

b(δ+ (h, s))x(s)Δs +

t

δ− (h,t)

b(δ+ (h, s))2 x(s)2 Δs.

2 b(δ+ (h, s))x(s)Δs (5.2.55)

Let C := D − β0 . Then from !@

β0 K− C

we have 2K L ≤

 C L β0

"2 ≥ 0,

β0 2 C 2 K + L . C β0

With this in mind we arrive at ∫ t β0 2 x (t) 2 |x(t)| |b(δ+ (h, s))| |x(s)| Δs ≤ C δ− (h,t) 2 ∫ t C + b(δ+ (h, s))x(s)Δs . β0 δ− (h,t)

5.2 Shift Operators and Stability in Delay Systems

233

A substitution of the above inequality into (5.2.55) yields  ∫ t 2  β0 C V(t) ≤ 1 + b(δ+ (h, s))x(s)Δs x(t)2 + (1 + ) C β0 δ− (h,t) ∫ t −D b(δ+ (h, s))2 x(s)2 Δs δ− (h,t)

2 ∫ t D D x(t)2 + b(δ+ (h, s))x(s)Δs C β0 δ− (h,t) ∫ t −D b(δ+ (h, s))2 x(s)2 Δs. =

δ− (h,t)

Using (5.2.23) we find V(t) ≤

D x(t)2 . C

By (5.2.51), (2.4.5), and (5.2.54) we get @ C V(t0 )eQ (t, t0 ) |x(t)| ≥ D    ∫ t C V(t0 ) 1 + ≥ Q(s)Δs D t0  ∫ t  2 ≥ CV(t0 ) b(δ+ (h, s)) Δs . t0

This completes the proof. We end this section by comparing the results of this section to the existing ones. In [11]

5.2.4 Applications and Comparison In [11] by means of Lyapunov’s direct method the authors investigated the stability analysis of the delay dynamic equation x Δ (t) = a(t)x(t) + b(t)x(δ(t))δΔ (t),

(5.2.56)

where a : T → R and b : T → R are functions and a ∈ R + . Moreover, the delay function δ : [t0, ∞)T → [δ(t0 ), ∞)T is surjective, strictly increasing, and supposed to have the following properties: δ(t) < t,

δΔ (t) < ∞, δ ◦ σ = σ ◦ δ.

It is concluded in [11, Theorem 6] that

234

5 Exotic Lyapunov Functionals for Boundedness and Stability

|b(t)| ≤ N and a(t) < −N

(5.2.57)

are the sufficient conditions guaranteeing stability of the zero solution of Eq. (5.2.56). Next, we furnish an example to show that Theorem 5.2.3 allows us to relax condition (5.2.57) that leads to exponential stability of zero solution Eq. (5.2.56). Example 5.2.4. Let T = R, a(t) = 1, b(t) = − 32 , δ(t) = t − 13 , and N = 1. It is obvious that the condition (5.2.57) does not hold. So, [11, Theorem 6] does not imply the stability of the zero solution of the delayed differential equation x (t) = x(t) −

1 3 x(t − ). 2 3

On the other hand, setting T = R, λ = 13 , and δ− (h, t) = t − into (5.2.58) and the condition (5.2.16) becomes −

(5.2.58) 1 3

Eq. (5.2.11) turns

1 3 ≤ Q(t) ≤ − b(δ+ (h, t))2, 4 9

which holds for all t ∈ [0, ∞) since Q(t) = a(t) + b(δ+ (h, t)) = − 12 . One may easily verify that condition (5.2.27) is satisfied for δ− (α, t) = t − 16 and δ− (h, t) = t − 13 . Thus, we conclude the exponential stability of the zero solution of (5.2.58) by Corollary 5.2.3. Now, let us consider the totally delayed equation x Δ (t) = b(t)x(δ− (h, t))δ−Δ (h, t), t ∈ [t0, ∞)T .

(5.2.59)

We observe the following by combining Corollary 5.2.3 and Theorem 5.2.5. Remark 5.2.1. Let b ∈ R. Suppose that there exists a λ > 0 such that λδ−Δ (h, t) ≤ b(δ+ (h, t)) ≤ −b(δ+ (h, t))2 [λβ(t) + (1 + λ)μ(t)] , β(t) + λ [β(t) + μ(t)] (5.2.60) holds for all t ∈ [t0, ∞)T . −

1. If there exists an α ∈ (t0, h)T such that (5.2.26) and (5.2.27) hold and if ∫

δ− (α,t)

lim

t→∞

t0

[λβ(s) + (1 + λ)μ(s)] b(δ+ (h, s))2 Δs = ∞,

then the zero solution of Eq. (5.2.59) is exponentially stable. 2. If (t0, h)T =  and if ∫ t lim [λβ(s) + (1 + λ)μ(s)] b(σ(s))2 Δs = ∞, t→∞

t0

then the zero solution of Eq. (5.2.59) is exponentially stable.

(5.2.61)

5.2 Shift Operators and Stability in Delay Systems

235

3. Suppose that a ∈ R + and that |b(t)| − λη σ (t)δ−Δ (h, t) ≤ 0 holds for all t ∈ [t0, ∞)T , where η(t) :=

1 . 1 + λβ(t)

Then any solution of Eq. (5.2.11) satisfies the inequality 1

|x(t)| ≤ V(t0, xt0 )e 2

∫t t0

γ(s)Δs

eγ (t, t0 ) for all t ∈ [t0, ∞)T , ∫

where V(t0, xt0 ) := |x(t0 )| + λη(t0 )

t0

δ− (h,t0 )

|x(s)| Δs,

* = max {1, M },and M is as in (5.2.12). * σ (t), M γ(t) := λ Mη In [11, Theorem 7], the authors utilized fixed point theory and deduced that the conditions p(t) := b(δ+ (h, t))  0 for all t ∈ [t0, ∞)T, (5.2.62) lim e p (t, t0 ) = 0,

t→∞

and ∫

t

δ− (h,t)

∫

t

|p(s)| Δs + t0

∫ | p(s)| e p (t, s)

s

δ− (h,s)

 |p(u)| Δu Δs ≤ N < 1

(5.2.63)

lead to stability of x(t, t0 ; ψ) of Eq. (5.2.59). Notice that [11] generalizes all the results of [139]. Moreover, Wang (see [163, Corollary 1]) proposed the inequality −

1 ≤ a(t) + b(t + h) ≤ −hb2 (t + h) 2h

(5.2.64)

as sufficient condition for uniform asymptotic stability of the zero solution of the delay differential equation x (t) = a(t) + b(t)x(t − h), h > 0. It can be easily seen that the conditions (5.2.63–5.2.64) are not satisfied for the data given in the following example. Example 5.2.5. Let a(t) = 0, T = R, δ− (h, t) = t − h, and p < 0 be fixed. Then Eq. (5.2.11) becomes x (t) = b(t)x(t − h). We can simplify condition (5.2.63) as follows:

236

5 Exotic Lyapunov Functionals for Boundedness and Stability

h |p| (2 − e pt ) ≤ N < 1.

(5.2.65)

9 If h = 23 , and b(t) = − 10 , then (5.2.62) implies

h |p| (2 − e pt ) = for all t ≥ − 10 9 ln

 

other hand, for h =

1 3 2 3

 9 3 2 − e− 10 t ≥ 1 5

 1.22. Thus, the condition (5.2.65) does not hold. On the

and λ = 32 , condition (5.2.60) turns into −

9 ≤ b(δ+ (h, t)) ≤ −b(δ+ (h, t))2 . 10

9 The last inequality holds for b(t) = − 10 . In addition, setting δ− (α, t) = t − 13 one may easily verify that conditions (5.2.26), (5.2.27), and (5.2.61) are satisfied. Hence, the first part of Remark 5.2.1 yields exponential stability while [11, Theorem 7] and [163, Corollary 1] cannot.

5.3 Open Problems Open Problem 1. Extend the results of this chapter to the delayed Volterra integro-dynamic equation where the delay operator is as defined in Sect. 5.2.2 ∫ t a(t, s) f (y(s))Δs, t ∈ [δ− (h, t), ∞)T . (5.3.1) x Δ (t) = h(t)x(t) + δ− (h,t)

Chapter 6

Volterra Integral Dynamic Equations

Summary In this chapter, we apply the concept of resolvent that we developed in Sect. 1.4.1 for vector Volterra integral dynamic equations and show the boundedness of solutions. The resolvent is an abstract term which makes it difficult, if not impossible, to make efficient use of it. However, by the help of Lyapunov functionals and variation of parameters, we will be able to verify all the conditions that are related to the resolvent. In Sect. 6.1.1 we make use of the resolvent along with fixed point theory and show the existence and uniqueness of solutions of nonlinear Volterra dynamic equations. Later in the chapter, we consider nonlinear Volterra integral dynamic equations and construct suitable Lyapunov functional to obtain results regarding boundedness and stability of the zero solution. Contents of this chapter are totally new and not published anywhere else except those of Sect. 6.3 that can be found in Adıvar and Raffoul (Appl Math Comput 273:258–266, 2016). In this chapter, we apply the concept of resolvent that we developed in Sect. 1.4.1 for vector Volterra integral dynamic equations and show the boundedness of solutions. The resolvent is an abstract term which makes it difficult, if not impossible, to make efficient use of it. However, by the help of Lyapunov functionals and variation of parameters, we will be able to verify all the conditions that are related to the resolvent. In Sect. 6.1.1 we make use of the resolvent along with fixed point theory and show the existence and uniqueness of solutions of nonlinear Volterra dynamic equations. Later in the chapter, we consider nonlinear Volterra integral dynamic equations and construct suitable Lyapunov functional to obtain results regarding boundedness and stability of the zero solution. Contents of this chapter are totally new and not published anywhere else except those of Sect. 6.3 that can be found in [17].

© Springer Nature Switzerland AG 2020 M. Adıvar, Y. N. Raffoul, Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales, https://doi.org/10.1007/978-3-030-42117-5_6

237

238

6 Volterra Integral Dynamic Equations

6.1 The Resolvent Method Consider the integral dynamic equation ∫ t x(t) = f (t) + C(t, s)x(s)Δs, t ∈ [0, ∞)T

(6.1.1)

0

where x and f are n-dimension vectors, n ≥ 1, and C is an n × n matrix. To clear any confusion, we note that the integral term in (6.1.1) could have been started at any initial time t0 ≥ 0. For x ∈ Rn , |x| denotes the Euclidean norm of x. As before, the time scale interval [0, ∞)T := {t ∈ T : 0 ≤ t < ∞}. For any n × n matrix A, define the norm of A by | A| = sup{| Ax| : |x| ≤ 1}. Let X denote the vector space of bounded functions set φ : [0, ∞)T → Rn and φ = sup{|φ(s)| : s ≥ 0}. In Sect. 1.4.1, we showed that the resolvent equation associated with (6.1.1) is given by ∫ t r(t, s) = −C(t, s) + r(t, u)C(u, s)Δu, (6.1.2) σ(s)

and the solution of (6.1.1) is given by the variation of parameters formula ∫ t x(t) = f (t) − r(t, u) f (u)Δu.

(6.1.3)

0

In this section we assume that 0 ∈ T, and hence, [0, ∞)T  ∅, and it should cause no confusion to start the integral in (6.1.1) from any initial time t0 instead of 0. Thus, if the time scale does not include zero, then any initial time t0 ∈ T will do the job. Now we use the resolvent equation and the corresponding variation of parameters formula, along with fixed point theory to show the existence and uniqueness of solution of the linear Volterra integral dynamic equation (6.1.1). Theorem 6.1.1. Assume the existence of two positive constants K and α ∈ (0, 1) such that a ∈ X we have ∫ t |C(t, s)|Δs ≤ α, (6.1.4) | f (t)| ≤ K and sup t ∈[0,∞)T

0

then there is a unique bounded solution of (6.1.1). Proof. Define a mapping D : X → X, by ∫ (Dφ)(t) = f (t) +

t

C(t, s)φ(s)Δs. 0

It is clear that (X,  · ) is a Banach space. Now for φ ∈ X, with φ ≤ q for positive constant q we have that (Dφ) ≤ K + αq.

6.1 The Resolvent Method

239

Thus D : X → X. Left to show that D defines a contraction mapping on X. Let φ, ϕ ∈ X. Then ∫ t |C(t, s)|Δsφ − ϕ (Dφ) − (Dϕ) ≤ sup t ∈[0,∞)T

0

≤ αφ − ϕ. Hence, D is a contraction, and by the contraction mapping principle it has a unique solution in X that solves (6.1.1). This completes the proof. In preparation for the next result we define M to be the Banach space of bounded continuous functions on IT = [t0, ∞)T endowed with the supremum norm, || . ||, where ||x|| := supt ∈IT |x(t)|. One can easily see that (M, || . ||) is a Banach space. In the next theorem, we extend Perron’s Theorem for integral equation over the reals to an arbitrary time scale. Its application is essential for the proof of our next results. For the continuous case of Perron’s theorem we refer the reader to [58, p. 114]. The next theorem was first published in [17]. and C : IT × IT → R be a continuous Theorem 6.1.2 (Perron). Let IT := [t0, ∞) ∩ ∫T t real valued function on t0 ≤ s ≤ t < ∞. If t R(t, s) f (s)Δs ∈ M for each f ∈ M, 0 ∫t then there exists a positive constant K such that t |R(t, s)| Δs < K, for all t ∈ IT . 0

Proof. Define the mapping Tt : M → R, by ∫ t Tt f = R(t, s) f (s) Δs. t0

For each fixed t, it follows that ∫ |Tt f | ≤

t

|R(t, s)|| f (s)| Δs ≤ tKt || f ||,

0

where Kt = sup |R(t, s)|. Moreover, for every f ∈ M, there exists a positive t0 ≤s ≤t

constant, say K f such that |Tt f | ≤ K f . This shows that the family of operators {Tt } is pointwise bounded. Thus, by the uniform boundedness principle there exists K > 0 such that for all t ≥ t0 and for all f ∈ M we have |Tt f | ≤ K || f ||; that is ∫ t |R(t, s)|| f (s)| Δs ≤ K || f ||, t0

t ≥ t0 and for all f ∈ M. Now for a fixed t define ft (s) :=



sgn {R(t, s)} for t0 ≤ s ≤ t . 0 for s > t

Also, for r > t0 , define the function

240

6 Volterra Integral Dynamic Equations

∫

s+1/(r−t0 )

ft,r (s) := (r − t0 )

ft (ξ)dξ, s ≥ t0 .

s

  Since | ft (ξ)| ≤ 1 for all ξ, we have  ft,r (s) ≤ 1 for all s. Moreover, ft,r (s) is continuous on s ≥ t0, and lim+ ft,r (s) = ft (s) r→t0

at all points s of continuity of ft (s). The points of discontinuity are at most points s where R(t, s) changes∫ sign and has Lebesque Δ-measure zero. Since t |R(t, s)|| ft,r (s)| ≤ |R(t, s)| and t |R(t, s)| Δs < ∞, we apply the Lebesque domi0 nated convergence theorem on time scales (see [48]), to conclude that ∫ t ∫ t lim+ R(t, s) ft,r (s) Δs = R(t, s) ft (s)Δs. r→t0

t0

t0

Then, considering all this we arrive at ∫ ∫ t r(t, s) Δs ≤ t0

t

R(t, s) ft (s) Δs

t0

= lim+ r→0

∫

t

r(t, s) ft,r (s) Δs

t0

≤K < < since ft,r ∈ M and < ft,r < = 1. Theorem 6.1.3. Suppose r(t, s) satisfies (6.1.2) and that f ∈ X. Then every solution x(t) of (6.1.1) is bounded if and only if ∫ t sup |r(t, s)|Δs < ∞ (6.1.5) t ∈[0,∞)T

0

holds. Proof. Suppose (6.1.5) hold. Then, using (6.1.3), it is trivial to show that x(t) is bounded. If x(t) and f (t) are bounded, then from (6.1.3), we have ∫ t |r(t, s)|| f (s)|Δs ≤ β 0

for some positive constant β and the proof follows from Theorem 6.1.2. Theorem 6.1.4. Let C be a k × k matrix. Assume the existence of a constant α ∈ (0, 1) such that ∫ t

max

t ∈[0,∞)

0

|C(t, s)|Δs ≤ α,

(6.1.6)

6.1 The Resolvent Method

241

(i) If f ∈ X so is the solution x of (6.1.1).

∫

T

(ii) Suppose, in addition, that for each T > 0 we have |C(t, s)|Δs → 0 as ∫ t0 t → ∞. If f (t) → 0 as t → ∞, then so does x(t) and r(t, s) f (s)Δs. 0 ∫ t α (iii) . |r(t, s)|Δs ≤ 1−α 0 Proof. The proof of (i) is the same as the proof of Theorem 6.1.1. For the proof of (ii) we define the set M = {φ : [0, ∞)T → Rn | |φ(t)| → 0, as t → ∞}. For φ ∈ M, define the mapping Q by 

 Qφ (t) = f (t) +

∫

t

C(t, s)φ(s)Δs. 0

Then

    Qφ (t) ≤ | f (t)| +

∫

t

|C(t, s)φ(s)|Δs.

0

We already know that f (t) → 0 as t → ∞. Given an ε > 0 and φ ∈ M, find T such that |φ(t)| < ε if t ≥ T and find d with ∫ T |φ(t)| ≤ d for all t ≥ T . For this fixed T, find ε η > T such that t ≥ η implies that |C(t, s)|Δs ≤ . Then t ≥ η implies that d 0 ∫ t ∫ T ∫ t |C(t, s)|Δs ≤ |C(t, s)|Δs + |C(t, s)|Δs 0

0

T

≤ (dε)/d + αε < 2ε. Thus, Q : M → M and the fixed point satisfies x(t) → 0, as t → ∞, for every vector function f ∈ M. Using (6.1.3) we have ∫ t r(t, s)a(s)Δs = x(t) − f (t) → 0, as t → ∞. 0

This completes the proof of (ii). Using (6.1.2) and (6.1.6) we have by changing of order of summations ∫ t ∫ t ∫ t∫ t |r(t, s)||Δs ≤ |C(t, s)|Δs + |r(t, u)||C(u, s)|ΔuΔs 0

0

∫

σ(s)

0

∫

t

t

|C(t, s)|Δs + |r(t, u)|Δu 0 0 ∫ t |r(t, u)|Δu. ≤ α+α

≤

0

∫ 0

t

|C(u, s)|Δs

242

6 Volterra Integral Dynamic Equations

Therefore,

∫ (1 − α)

t

|r(t, s)|Δs ≤ α.

0

That is,

∫

t

sup

t ∈[0,∞)

α . 1−α

|r(t, s)|Δs ≤

0

Example 6.1.1. Suppose there is a function d : [0, ∞)T → (0, 1], with d(t) ↓ 0 with ∫ t sup |C(t, s)|(d(s)/d(t))Δs ≤ α, α ∈ (0, 1) (6.1.7) t ∈[0,∞)T

0

and | f (t)| ≤ kd(t)

(6.1.8)

for some positive constant k. Then the unique solution x(t) of (6.1.1) is bounded and ∫ t

goes to zero as t approaches infinity. Moreover,

r(t, s) f (s)Δs → 0, as t → ∞.

0

Proof. Let M = {φ : [0, ∞)T → Rn | |φ|d ≤ 

sup

t ∈[0,∞)T

|φ(t)| < ∞}. |d(t)|



Then M, | · |r is a Banach space. For φ ∈ M, define the mapping Q by 

 Qφ (t) = f (t) +

∫

t

C(t, s)φ(s)Δs. 0

Then, ∫ t     Qφ (t)/d(t) ≤ | f (t)|/d(t) + |C(t, s)|(d(s)/d(t))|φ(s)|/d(s)Δs 0 ∫ t |C(t, s)|(d(s)/d(t))Δs ≤ k + |φ|d 0

≤ k + α|φ|d, which shows that Qφ ∈ M. Let φ, η ∈ M, then we readily have that          Qφ (t) − Qη (t)/d(t) ≤ α|φ − η|d, and so we have that Q is a contraction on M and therefore it has a unique fixed |x(t) | point x(t) in M that solves (6.1.1). Moreover, supt ∈[0,∞) |d(t) | < ∞, implies that |x(t)| ≤ k ∗ d(t) → 0, as t → ∞. Also by (6.1.8) we have | f (t)| → 0, as t → ∞ and hence using (6.1.3) we have

6.1 The Resolvent Method

∫

t

243

|r(t, s) f (s)|Δs ≤ | f (t)| + |x(t)| → 0, as t → ∞.

0

This completes the proof.

6.1.1 Existence and Uniqueness In this section we consider the system of nonlinear Volterra dynamic equation of the form ∫ t

x(t) = f (t) +

g(t, s, x(s))Δs

(6.1.9)

0

in which x is an n vector, f : [0, ∞)-T → Rn, and g : π × Rn → Rn is Cr d in t and . s and continuous in x, where π = (t, s) ∈ [0, ∞)T × [0, ∞)T : 0 ≤ s ≤ t < ∞ . Due to the nonlinearity of the function g, the resolvent method of Sect. 1.4 is not applicable to (6.1.9). Instead, we will use the contraction mapping principle and show the existence of solutions of (6.1.9) over a short interval, say [0, T]T ⊂ [0, ∞)T . Theorem 6.1.5. Suppose there are positive constants a, b, and α ∈ (0, 1). Suppose (a) f is continuous on [0, a]T ⊂ [0, ∞)T, (b) g is continuous on . U = (t, s, x) : (t, s) ∈ [0, ∞)T × [0, ∞)T, 0 ≤ s ≤ t < ∞ , and |x − f (t)| ≤ b , (c) g satisfies a Lipschitz condition with respect to x on U   g(t, s, x) − g(t, s, y) ≤ L|x − y| for (t, s, x), (t, s, y) ∈ U. Then there is a unique solution of (6.1.9) on [0, T]T ⊂ [0, ∞)T, where * c}, T := min{a, b/ M,   * := maxU g(t, s, x), and c := α/L for fixed α. M Proof. Let Ωb denote the space of Cr d functions φ : [0, T]T → Rn, such that ||φ − f || = max {|φ(t) − f (t)|} ≤ b, t ∈[0,T ]T

where for Ψ ∈ Ωb and the norm || · || is taken to be and Ψ = maxt ∈[0,T ]T {|Ψi (t)|}. Clearly, the norm defines the metric ρ. Let φ ∈ Ωb and define an operator D : Ωb → Ωb, by ∫ t

D(φ)(t) = f (t) +

g(t, s, φ(s))Δs.

0

Since φ is Cr d we have that D(φ) is Cr d too and

244

6 Volterra Integral Dynamic Equations

∫ t    ||D(φ) − f || = max  g(t, s, φ(s))Δs t ∈[0,T ]T

0

* ≤ b. ≤ MT This shows that D maps Ωb into itself. For the contraction part, we let φ, ψ ∈ Ωb . Then ∫ t ∫ t    g(t, s, φ(s))Δs − g(t, s, ψ(s))Δs ||D(φ) − D(ψ)|| = max  t ∈[0,T ]T 0 0 ∫ t    ≤ max g(t, s, φ(s)) − g(t, s, ψ(s))Δs t ∈[0,T ]T 0 ∫ t    ≤ max L φ(s)) − ψ(s))Δs t ∈[0,T ]T 0     ≤ T max L φ(s)) − ψ(s)) t ∈[0,T ]T

= T L||φ − ψ|| ≤ cL||φ − ψ|| = α||φ − ψ||. Thus, by the contraction mapping principle, there is a unique function x ∈ Ωb with ∫ t D(x)(t) = x(t) = f (t) + g(t, s, x(s))Δs. 0

6.2 Continuation of Solutions In this section we define the concept of maximal solution of nonlinear Volterra integral dynamic equations and use a combination of Gronwall’s inequality and Lyapunov functionals to show that solutions can be continued for all time t ∈ [0, ∞)T . We begin with a definition on maximal solution. Definition 6.2.1. Let f : [0, ∞)T → R be r d-continuous and for . Υ := (t, s, x) : (t, s) ∈ [0, ∞)T × [0, ∞)T : 0 ≤ s ≤ t < ∞ and x ∈ R , let g : Υ → R. Let x(t) be a r d-continuous solution of the scalar Volterra integral dynamic equation ∫ t

x(t) = f (t) +

g(t, s, x(s))Δs

(6.2.1)

0

on [0, A]T ⊂ [0, ∞)T with the property that if y(t) is any other solution, then as long as y(t) exists and t ≤ A we have y(t) ≤ x(t). Then x(t) is called the maximal solution of (6.2.1). Minimal solutions are defined by asking y(t) ≥ x(t). Of course, if solutions are unique, then the unique solution is the maximal and the minimal solution.

6.2 Continuation of Solutions

245

Theorem 6.2.1. Let x be the maximal solution of the scalar equation ∫ t x(t) = B + p(s, x(s))Δs 0

on [0, A]T ⊂ [0, ∞)T where B is a constant, and let p : [0, A]T × R → R be r dcontinuous and p(t, ·) is nondecreasing in its second argument when t ∈ [0, A]T . If y(t) is r d-continuous scalar function on [0, A]T ⊂ [0, ∞)T satisfying ∫ t y(t) ≤ B + p(s, x(s))Δs, y0 = y(0) ≤ B, 0

then y(t) ≤ x(t) on [0, A]T ⊂ [0, ∞)T . Proof. Let

∫

t

Y (t) = y0 +

p(s, x(s))Δs, 0

so that y(t) ≤ Y (t). Then we have Y Δ (t) = p(t, y(t)) ≤ p(t, Y (t)), by monotonicity. Hence Y Δ (t) ≤ p(t, Y (t)), Y (0) = y0 . From this we conclude that Y (t) ≤ x(t) according to [Theorem 2.1, [103]] or [Theorem 6.1, [151]]. This completes the proof. In the next theorem we utilize Gronwall’s inequality to show that solutions of a nonlinear Volterra dynamic equation can be continued for all time. Theorem 6.2.2. Let g : [0, ∞)T × [0, ∞)T × R → R be r d-continuous, and suppose that for each T > 0 there is an r d-continuous scalar function P(·, T) such that P(·, T) ∈ R + (positively regressive) and

  |g(t, s, x)| ≤ P(s, T) 1 + |x| for 0 ≤ s ≤ t ≤ T .

Let f : [0, ∞)T → R be r d-continuous. If x(t) is a solution of ∫ t x(t) = f (t) + g(t, s, x(s))Δs

(6.2.2)

0

on some interval [0, α]T, then it is bounded and, hence, is continuable to all of [0, ∞)T provided that ∫ α

| f (t)| + 0

where α ∈ [0, ∞)T .

P(s, α)Δs ≤ Q

(6.2.3)

246

6 Volterra Integral Dynamic Equations

Proof. Let x(t) be a solution on [0, α)T . Then for 0 ≤ t < α, we have ∫ t |x(t)| ≤ | f (t)| + P(s, α)(1 + |x(s)|)Δs 0 ∫ t P(s, α)|x(s)|Δs. ≤Q+ 0

Thus, by Gronwall’s inequality [50] and (6.2.3) we have |x(t)| ≤ Qe P (t, 0) < ∞. This completes the proof. The following theorem is new for the continuous case, T = R, and for the discrete case, T = Z. Theorem 6.2.3. Let f : [0, ∞)T → R be r d-continuous, and C(t, s) be a scalar continuous function on [0, ∞)T × [0, ∞)T . Let f be a continuously Δ-differentiable function. In addition, we assume Ct (t, s) exists and continuous. Suppose that γ : R → R is a continuous function satisfying |γ(x(t))| ≤ ζ |x(t)| for all t ∈ T where ζ ∈ (0, 1) is a constant. Let the function β : T → R be defined by  2 −λ − μ(t) + ζ |C(σ(t), t)| for t ∈ T− β(t) := , −λ + ζ |C(σ(t), t)| for t ∈ T+

(6.2.4)

(6.2.5)

where T− : = {s ∈ [0, ∞)T : x (s) x σ (s) < 0} , T+ : = {s ∈ [0, ∞)T : x (s) x σ (s) ≥ 0} . Suppose that for each T ∈ [0, ∞)T we have ∫ T |Cu (u, t)|Δu ≤ 0. β(t) + σ(t)

(6.2.6)

Then each solution of ∫ x(t) = f (t) +

t

C(t, s)γ(x(s))Δs

(6.2.7)

0

can be continued for all future times. Proof. We show that if a solution x(t) is defined on [0, T)T , it is bounded. Let λ be a positive constant such that | f Δ (t)| ≤ λ on [0, T)T . Define the functional V(t) = V(t, x(·)) by

6.2 Continuation of Solutions

247

∫ t∫ # V(t) = e λ (t, 0) 1 + |x(t)| + 0

T

$ |Cu (u, s)|Δu|γ(x(s))|Δs .

From [10, Lemma 5] we have that  x(t) |x (t)| Δ =

Δ |x(t) | x (t) 2 − μ(t) |x (t)|

(6.2.8)

t

if t ∈ T+ −

x(t) Δ |x(t) | x (t)

for t ∈ T−

.

(6.2.9)

Thus for t ∈ T+, using the product rules, we get that ∫ t∫ T # $  x) = λe λ (t, 0) 1 + |x(t)| + V(t, |Cu (u, s)|Δu|γ(x(s))|Δs 0 t ∫ T # |Cu (u, t)|Δu|γ(x(t))| + e λ (σ(t), 0) |x(t)| Δ + ∫ −

σ(t)

t

$ |Ct (t, s)||γ(x(s))|Δs .

(6.2.10)

0

Rearranging the expression (6.2.10) gives ∫ t∫ #  x) = e λ (t, 0) λ + λ|x| + λ V(t, + + ≤ + ≤ +

$ |Cu (u, s)|Δu|γ(x(s))|Δs 0 t ∫ t # x   f Δ (t) + C(σ(t), t)γ(x) + Ct (t, s)γ(x(s))Δs e λ (σ(t), 0) |x| 0 ∫ t ∫ T $ |Cu (u, s)|Δu|γ(x(t))| − |Ct (t, s)||γ(x(s))|Δs σ(t) 0 # $ e λ (t, 0) λ + λ|x| ∫ T # $ e λ (σ(t), 0) | f Δ (t)| + ζ |C(σ(t), t)||x(t)| + ζ |Cu (u, t)|Δu|x(t)| σ(t) # $ e λ (t, 0) λ + λ|x| ∫ T # $ e λ (σ(t), 0) λ + ζ |C(σ(t), t)||x(t)| + ζ |Cu (u, t)|Δu|x(t)| , T

σ(t)

(6.2.11) where (6.2.9) is used to calculate the Δ-derivative of |x(t)|. Since we have   e λ (σ(t), 0) = 1 + μ(t) λ e λ (t, 0)  μ(t)(−λ)  = 1+ e λ (t, 0) 1 + μ(t)λ 1 e λ (t, 0), (6.2.12) = 1 + μ(t)λ

248

6 Volterra Integral Dynamic Equations

the expression (6.2.11) reduces to −λ e λ (t, 0)|x(t)| 1 + μ(t)λ ∫ T   1 + e λ (t, 0) ζ |C(σ(t), t)| + ζ |Cu (u, t)|Δu |x(t)| 1 + μ(t)λ σ(t) ∫ T   1 ≤ e λ (t, 0) β(t) + ζ |Cu (u, t)|Δu |x(t)| 1 + μ(t)λ σ(t) ≤ 0.

 x) ≤ V(t,

On the other hand, for t ∈ T− , we have ∫ t∫ #  V(t, x) = e λ (t, 0) λ + λ|x| + λ + + + ≤ + + ≤ + +

0

T

|Cu (u, s)|Δu|γ(x(s))|Δs

$

t

x (t)  Δ 2 f (t) + C(σ(t), t)γ(x(t)) e λ (σ(t), 0) − |x (t)| − μ (t) |x (t)| ∫ t  Ct (t, s)γ(x(s))Δs 0 ∫ t ∫ T $ |Cu (u, t)|Δu|γ(x(t))| − |Ct (t, s)||γ(x(s))|Δs σ(t) 0 # $ e λ (t, 0) λ + λ|x| # 2 |x(t)| + | f Δ (t)| + |C(σ(t), t)||γ(x)| e λ (σ(t), 0) − μ(t) ∫ T $ |Cu (u, s)|Δu|γ(x(t))| σ(t) # $ e λ (t, 0) λ + λ|x| # 2 1 e λ (t, 0) − |x(t)| + λ + ζ |C(σ(t), t)||x(t)| 1 + μ(t)λ μ(t) ∫ T $ ζ |Cu (u, t)|Δu|x(t)| #

σ(t)

# λ 1 2 |x(t)| + = e λ (t, 0) λ + λ|x(t)| − μ(t) 1 + μ(t)λ 1 + μ(t)λ ∫ T   $ 1 ζ |C(σ(t), t)| + ζ |Cu (u, s)|Δu |x(t)| + 1 + μ(t)λ σ(t) ∫ T  2 e λ (t, 0)  + ζ |C(σ(t), t)| + ζ −λ− |Cu (u, t)|Δu |x(t)| = 1 + μ(t)λ μ(t) σ(t) ∫ T   e λ (t, 0) β(t) + = |Cu (u, t)|Δu |x(t)| 1 + μ(t)λ σ(t) ≤ 0,

6.2 Continuation of Solutions

249

where we use (6.2.9) to calculate the Δ-derivative of |x(t)|. This shows that V is decreasing and hence for all t ∈ [0, T)T, we have that V(t) ≤ V(0) ≤ 1 + |x(0)|. This shows that V(t) is bounded on [0, T)T . Now using (6.2.8) and for t ∈ [0, T)T we arrive at |x(t)| ≤ eλ (t, 0)V(t) ≤ eλ (t, 0)V(0)   ≤ eλ (t, 0) 1 + |x(0)|   ≤ eλ (T, 0) 1 + |x(0)| < ∞. This shows x is bounded on [0, T)T and since T is arbitrary, solutions can be continued for all future times. This completes the proof. We illustrate the results of Theorems 6.1.1, 6.1.4, and 6.2.3 in the following example: Example 6.2.1. Let T = 2N0 = {2n : n = 0, 1, 2, ...} and t0 = 1. Then for t = 2n , we have σ(t) = 2n+1 and μ(t) = 2n . Setting f (t) = 1t = 2−n , s = 2m , and C(t, s) = ts2 = 2m−2n , Eq. (6.1.1) turns into the following: x(2n ) = 2−n −

n−1 

22(m−n) x (2m ) , n = 0, 1, ...,

(6.2.13)

m=1

n−1 2m−2n 2 = 1 − 2−2n+2 are bounded we (see Theorem 1.1.7). Since f (t) and m=1 have that by Theorem 6.1.1 that solutions of (6.3.19) are bounded. Similarly, for T = 2k ∈ 2N0 , we have that k−1  m=1

22(m−n) =

1 −2n 2k 2 (2 − 4) → 0, as n → ∞. 3

Thus by (ii) of Theorem 6.1.4, the solution x(t) of (6.3.19) approaches zero as ∫ t n−1  r(t, s) f (s)Δs = r(2n, 2m ) → 0, as t → ∞. t → ∞. In addition, we have that 0

m=1

Next we apply Theorem 6.2.3. If we set λ = 1, ζ = 1, and γ(x) = x, then we have by (6.2.5) that  −1 − 7(2−n−2 ) for t ∈ T− β(t) = . (6.2.14) −1 + 2−n−2 for t ∈ T+ Setting u = 2m and t = 2n one can find

250

6 Volterra Integral Dynamic Equations

|Cu (u, t)| =

t(u + σ(u)) 3 2n = . 4 23m u2 σ(u)2

For 2n+1 = σ(t) ≤ 2k = T, we have ∫

T

σ(t)

|Cu (u, t)|Δu =

k−1 3 n  −2m 2 2 = 2n−2 (2−2n − 2−2k ) = 2−n−2 − 2−2k−n−2 . 4 m=n+1

Then, the condition (6.2.6) turns into  ∫ T −1 − (6 + 2−2k )2−n−2 for t ∈ T− β(t) + |Cu (u, t)|Δu = 0 and x(t)x σ (t) is nonnegative on T+ , we have that hence,

x |x |

=

xσ |x σ | ,

and

x |x σ ||γ(x σ )| x σ γ(x σ ) C(σ(t), t)γ(x σ ) = C(σ(t), t) = C(σ(t), t) = C(σ(t), t)|γ(x σ )| σ |x| |x | |x σ | for all t ∈ T+ . Rearranging the expression (6.2.20) gives ∫ t∫ #  x) = e λ (t, 0) λ + λ|x| + λ V(t, + + ≤ + ≤ +

$ |Cu (u, s)|Δu|γ(x σ (s))|Δs 0 t ∫ t # x   f Δ (t) + C(σ(t), t)γ(x σ ) + Ct (t, s)γ(x σ (s))Δs e λ (σ(t), 0) |x| 0 ∫ t ∫ T $ |Cu (u, s)|Δu|γ(x σ (t))| − |Ct (t, s)||γ(x σ (s))|Δs σ(t) 0 # $ e λ (t, 0) λ + λ|x| ∫ T # $ e λ (σ(t), 0) | f Δ (t)| + C(σ(t), t)|γ(x σ (t))| + |Cu (u, t)|Δu|γ(x σ (t))| σ(t) # $ e λ (t, 0) λ ∫ T # $ e λ (σ(t), 0) λ + C(σ(t), t)|γ(x σ (t))| + |Cu (u, t)|Δu|γ(x σ (t))| , T

σ(t)

(6.2.21) Using (6.2.12) the expression (6.2.21) reduces to  x) ≤ V(t,

 1 e λ (t, 0) C(σ(t), t) + 1 + μ(t)λ

∫

T

σ(t)

 |Cu (u, t)|Δu |γ(x σ (t))| ≤ 0 (6.2.22)

On the other hand, for t ∈ T− , we have x x σ ≤ 0 and therefore, along with xγ(x) ≥ 0 yields

−x |x |

=

xσ |x σ | .

This

−x |x σ ||γ(x σ )| x σ γ(x σ ) C(σ(t), t)γ(x σ ) = C(σ(t), t) = C(σ(t), t) = C(σ(t), t)|γ(x σ )|. |x| |x σ | |x σ | Thus for t ∈ T− we have

252

6 Volterra Integral Dynamic Equations

∫ t∫ #  x) = e λ (t, 0) λ + λ|x| + λ V(t, + + + ≤ + + ≤ + +

0

T

|Cu (u, s)|Δu|γ(x σ (s))|Δs

$

t

x (t)  Δ 2 f (t) + C(σ(t), t)γ(x σ (t)) e λ (σ(t), 0) − |x (t)| − μ (t) |x (t)| ∫ t  Ct (t, s)γ(x σ (s))Δs 0 ∫ t ∫ T $ |Cu (u, t)|Δu|γ(x σ (t))| − |Ct (t, s)||γ(x σ (s))|Δs σ(t) 0 # $ e λ (t, 0) λ + λ|x| # 2 |x(t)| + | f Δ (t)| + C(σ(t), t)|γ(x σ (t))| e λ (σ(t), 0) − μ(t) ∫ T $ |Cu (u, s)|Δu|γ(x σ (t))| σ(t) # $ e λ (t, 0) λ + λ|x| # 2 1 e λ (t, 0) − |x(t)| + λ + C(σ(t), t)|γ(x σ (t))| 1 + μ(t)λ μ(t) ∫ T $ |Cu (u, t)|Δu|γ(x σ (t))| #

σ(t)

# λ 1 2 |x(t)| + = e λ (t, 0) λ + λ|x(t)| − μ(t) 1 + μ(t)λ 1 + μ(t)λ ∫ T   $ 1 C(σ(t), t) + |Cu (u, s)|Δu |γ(x σ (t))| + 1 + μ(t)λ σ(t) ∫ T   e λ (t, 0) C(σ(t), t) + |Cu (u, t)|Δu |γ(x σ (t))| = 1 + μ(t)λ σ(t) ≤ 0, where we use (6.2.9) to calculate the Δ-derivative of |x(t)|. The proof is completed by using the similar arguments in the proof of Theorem 6.2.3. Remark 6.2.1. In the particular case when T = R, the set T− is empty and Theorem 6.2.4 covers [57, Theorem 3.3.7]. To the best of our knowledge Theorem 6.2.4 is new for all other time scales including the discrete case, T = Z. Example 6.2.2. Let T = Z and t0 = 0. Then for t ∈ Z, we have σ(t) = t + 1 and 1 , C(t, s) = −2s−2t , and γ(x) = x 3 , Eq. (6.1.1) turns into μ(t) = 1. Setting f (t) = t+1 the following:  1 − 2s−2n x 3 (s + 1) , t = 0, 1, ..., t + 1 s=0 t−1

x(t) =

(see Theorem 1.1.7). Since

(6.2.23)

6.3 Lyapunov Functionals Method

253

C(σ(t), t) = −2−t+1 and |Ct (t, s)| =

3 s−2t 3 2 = |C(t, s)|, 4 4

we have C(σ(t), t) +

T 

|Ct (t, s)| = −2−t+1 + 2−t−2 − 2−T +t ≤ 0.

s=t+1

This shows that the condition (6.2.18) holds for T ≥ t + 1. One can similarly show that (6.2.18) holds for T < t + 1. From Theorem 6.2.4 each solution of (6.2.23) can be continued for all future times.

6.3 Lyapunov Functionals Method In this section we use the notion of the resolvent equation that was developed in Chap. 1 for Volterra integral dynamic equations and Lyapunov’s method to study boundedness and integrability of the solutions of the nonlinear Volterra integral equation on time scales. In particular, the existence of bounded solutions with various L p properties are studied under suitable conditions on the functions involved in the above Volterra integral equation. At the end of the section we display some examples on different time scales. Constructing Lyapunov functionals for integral equations has been very challenging until recently, even in the continuous case. Burton (see, e.g., [61]) was the first one to construct such functions and utilize them to qualitatively analyze solutions of integral equations. The papers of [64, 65] and [93] should be of interest when the T = R. The study of integral equations on time scales provides deeper and comprehensive understanding of traditional integral equations and summation equations. Research of Volterra integral equations on time scales is still in its beginning stage and more work on this area is desperately needed and we refer to [111]. Some of the results in this paper are new for the continuous case and all of them are new for the discrete case and are published in [17]. Throughout this section we suppose that T is a time scale that is unbounded above and denote by L p (IT ) the Banach space of real valued functions on the interval IT := [t0, ∞) ∩ T endowed by the norm ∫  f  p :=

t ∈IT

 1/p | f (t)| p Δt

(see [148]). We limit our discussion to scalar integral equations. It will be easy to prove similar theorems for vector systems. We first begin with the following lemma which enables us to change the order of integration on a triangular region. Lemma 6.3.1 (Change of Order of Integration [98, Theorem 3.2]). Let ϕ ∈ Cr d (T × T, R) be an r d-continuous function. For t, t0 ∈ Tκ we have

254

6 Volterra Integral Dynamic Equations

∫ t∫ t0

t

σ(s)

ϕ(u, s)ΔuΔs =

∫ t∫ t0

u

ϕ(u, s)ΔsΔu.

(6.3.1)

t0

Now we consider the nonlinear Volterra integral equation on time scales, ∫ t x(t) = a(t) − C(t, s)G(s, x(s))Δs, t ∈ IT,

(6.3.2)

t0

where t0 ∈ Tκ is fixed and the functions a : IT → R, C : IT × IT → R, and G : IT × R → R are supposed to be continuous. We saw in Chap. 1, Sect. 1.4 that a linear system of integral equations of the form ∫ t x(t) = a(t) − C(t, s)x(s)Δs, t0 ∈ Tκ (6.3.3) t0

corresponds to the following resolvent equation ∫ t R(t, s) = C(t, s) − R(t, u)C(u, s)Δu. σ(s)

(6.3.4)

If C(t, s) is scalar valued, then so is R(t, s). If C(t, s) is n × n matrix, then so is R(t, s). Moreover, the solution of (6.3.3) in terms of R is given by ∫ t x(t) = a(t) − R(t, u)a(u)Δu. (6.3.5) t0

The next lemma is essential to our analysis since the boundedness of the solutions depends on the integrability of the resolvent. Lemma 6.3.2. If

∫

t

sup

t ∈IT

holds, then we have

∫

t

sup

t ∈IT

|C(t, s)|Δs ≤ L < 1

t0

|R(t, s)|Δs ≤ M < ∞.

t0

Proof. Using (6.3.1) and (6.3.4) we have ∫ t ∫ t∫ ∫ t |R(t, s)| Δs ≤ |C(t, s)| Δs + t0

=

t0 ∫ t

|C(t, s)| Δs +

t0

∫

≤ L+L t0

Therefore,

t0 ∫ t t0

t

|R(t, u)|Δu.

t

σ(s)

|R(t, u)||C(u, s)|ΔuΔs ∫

|R(t, u)| t0

u

|C(u, s)|ΔsΔu

6.3 Lyapunov Functionals Method

255

∫ (1 − L)

t

|R(t, s)|Δs ≤ L.

t0

That is,

∫

t

sup

t ∈IT

|R(t, s)|Δs ≤

t0

L := M. 1−L

In preparation for the next result we define M to be the Banach space of bounded continuous functions on IT = [t0, ∞)T endowed with the supremum norm, || . ||, where ||x|| := supt ∈IT |x(t)|. One can easily see that (M, || . ||) is a Banach space. Theorem 6.3.1. Suppose R(t, s) satisfies (6.3.4) and that a ∈ M. Then every solution x(t) of (6.3.3) is bounded if and only if ∫ t sup |R(t, s)|Δs < ∞ (6.3.6) t ∈IT

t0

holds. Proof. Suppose (6.3.6) hold. Then, using (6.3.5), it is trivial to show that x(t) is bounded. If x(t) and a(t) are bounded, then from (6.3.5), we have ∫ t |R(t, s)||a(s)| Δs ≤ γ t0

for some positive constant γ and the proof follows from Theorem 6.1.2. This completes the proof. Now we turn our attention to Eq. (6.3.2). Throughout this section, we assume the functions a(t), G(t, x(t)), and C(t, s) are continuous with respect to their respective arguments. In preparation of the next result we assume the existence of a continuous function H satisfying the following conditions: H1. G(t, x(t)) = x(t) + H(t, x(t)) for each t ∈ IT and x ∈ M, H2. H(t, 0) = 0 and there is a k > 0 with |H(t, x(t)) − H(t, y(t))| ≤ k |x(t) − y(t)| for t ∈ IT and x, y ∈ M. Hence, taking (H1) into account we rewrite Eq. (6.3.2) as follows: ∫ t x(t) = a(t) − C(t, s)[x(s) + H(s, x(s))]Δs, t ∈ IT .

(6.3.7)

t0

Suppose R(t, s) satisfies (6.3.4). Then any solution x(t) of Eq. (6.3.7) is given by ∫ t x(t) = a(t) − R(t, s)[a(s) + H(s, x(s))]Δs, t ∈ IT, t0

where a, R, and H are all continuous functions.

256

6 Volterra Integral Dynamic Equations

Theorem 6.3.2. Suppose there exists a function H satisfying (H1–H2). Assume that a(t) is bounded and ∫ t

sup k

t ∈IT

|R(t, s)|Δs ≤ α < 1,

t0

where k is defined as in (H2). Then (6.3.7) has a unique bounded continuous solution. Proof. For each φ ∈ M, define ∫ (T φ)(t) = a(t) −

t

t0

R(t, s)[a(s) + H(s, φ(s))]Δs, t ∈ IT .

It follows from the continuity assumptions on a, H, and R that (T φ)(t) is continuous in t. Also, one can easily verify from the given assumptions that |(T φ)(t)| < ∞, and |(T φ)(t) − (Tψ)(t)| ≤ α||φ − ψ|| for all φ, ψ ∈ M. This shows T : M → M and T is a contraction. Therefore (6.3.7) has a unique bounded continuous solution. In next result, we suppose that the function G satisfies the following properties: G1. G(t, 0) = 0 for all t ∈ IT , G2. There exists a constant 4 k > 0 such that |G(t, x(t)) − G(t, y(t))| ≤ 4 k |x(t) − y(t)| for t ∈ IT and x, y ∈ M. Theorem 6.3.3. Suppose (G1–G2). If the function a is bounded and ∫ t sup 4 |C(t, s)|Δs ≤ 4 α < 1, k t ∈IT

t0

then there exists a unique bounded continuous solution of (6.3.2). Proof. For each φ ∈ M, define 4φ)(t) = a(t) − (T

∫ t0

t

C(t, s)G(s, φ(s))Δs, t ∈ IT .

4 is a contraction mapping. It follows from the continuity We shall show that T 4φ)(t) is continuous in t. Moreover, assumptions on a, G, and C that (T ∫ t 4 |(T φ)(t)| ≤ |a(t)| + |C(t, s)||G(s, φ(s))|Δs t0

≤ |a(t)| + 4 k4 α ||φ|| < ∞. 4φ) is bounded and T 4 : M → M. If φ, ψ ∈ M, Therefore, (T ∫ t 4 4 |(T φ)(t) − (Tψ)(t)| ≤ |C(t, s)||G(s, φ(s)) − G(s, ψ(s))|Δs t0

6.3 Lyapunov Functionals Method

257

 ∫ ≤ 4 k

 |C(t, s)| Δs ||φ − ψ||

t

t0

≤4 α ||φ − ψ||. 4 is a contraction mapping and hence T 4 has a unique fixed point, which Since 4 α < 1, T solves (6.3.2). Now we employ Lyapunov functionals to obtain sufficient conditions guaranteeing that all solutions of Eqs. (6.3.2) and (6.3.3) are in L p (IT ), p = 1, 2. The next theorem, in which we employ the identity (see [50, Theorem 1.117]) ∫

t

f (t, τ) Δτ

Δ

∫

t

=

a

f Δt (t, τ) Δτ + f (σ (t) , t) ,

a

provides sufficient conditions guaranteeing that the solution x of (6.3.2) is in L 1 (IT ). Theorem 6.3.4. Let x(t) be any solution of (6.3.2) for t ∈ IT . Suppose there exists a constant * k ≥ 0 such that |G(t, x(t))| ≤ * k |x(t)|, for all t ∈ IT and x ∈ M and that * k

(6.3.8)

∫∞ |C(u, t)| Δu ≤ * α 0 for all (t, x (t)) ∈ IT × (0, ∞). Suppose ∫ t

a(t) − k

C(t, s)a(s) Δs ≥ 0,

(6.3.11)

t0

where the constant k is defined as in (H.2). Then (6.3.3) has a unique nonnegative solution. Proof. The proof is very similar to the proof in the continuous case, and hence we omit it. For reference, we ask the reader to look at [93, Theorem 3.4]. Hereafter, we simply use the notations Ct (t, s) and Cs (t, s) to indicate the partial derivatives Ct (t, s) := lim

C(σ(t), s) − C(τ, s) , where τ → t, τ ∈ IT \ {σ(t)} σ(t) − τ

Cs (t, s) := lim

C(t, σ(s)) − C(t, τ) , where τ → s, τ ∈ IT \ {σ(s)} , σ(s) − τ

τ→t

and τ→s

respectively. Similarly, Cst (t, s) is defined as the partial derivative of Cs (t, s) with respect to t. For more on partial derivatives on time scales we refer to [23] and [47]. Theorem 6.3.6. Let x(t) be a nonnegative solution of (6.3.2) for all t ∈ IT . Assume there exist constants α, β > 0 such that for all x ∈ M and t ∈ IT 0 ≤ G(t, x(t)) ≤ αx(t)

(6.3.12)

6.3 Lyapunov Functionals Method

259

and 1 − α C(t, t)μ(t) ≥ β

(6.3.13)

i.e., −αC(t, t) is positively regressive. Let C(t, s) ≥ 0, Cs (t, s) ≥ 0, Cst (t, s) ≤ 0, and Ct (t, t0 ) ≤ 0. Then x ∈ L 2 (IT ) if a ∈ L 2 (IT ). Proof. Let x(t) be a nonnegative solution of (6.3.2) for all t ∈ IT . Define ∫ t F(t, s) := G(u, x(u))Δu s

∫

and

t

V(t) :=

Cs (t, s)F 2 (t, σ(s)) Δs + C(t, t0 )F 2 (t, t0 ).

t0

It is obvious that F(t, t) = 0, Ft (t, s) = G(t, x(t)), Fs (t, s) = −G(s, x(s)). Then, along the solutions of (6.3.2) we have ∫ t ' ( Δt V Δ (t) = Cs (t, s)F 2 (t, σ(s)) Δs + Cs (σ(t), t)F 2 (σ(t), σ(t)) t0

' ( + Ct (t, t0 )F 2 (σ(t), t0 ) + C(t, t0 )G(t, x(t)) F(σ(t), t0 ) + F(t, t0 ) ∫ t+ , Cst (t, s)F 2 (σ(t), σ(s)) + Cs (t, s)[F 2 (t, σ(s))]Δt Δs = t0 ' ( + C(t, t0 )G(t, x(t)) F(σ(t), t0 ) + F(t, t0 ) , + ' ( (6.3.14) ≤ G(t, x(t)) C(t, t0 ) F(σ(t), t0 ) + F(t, t0 ) + J(t) , where J is defined by ∫ J(t) =

t

Cs (t, s)[F(σ(t), σ(s)) + F(t, σ(s))]Δs.

t0

Using the integration by parts formula (see [50, Theorem 1.77]), we have ' ( J(t) = −C(t, t0 ) F(σ(t), t0 ) + F(t, t0 ) ∫ t C(t, s)G(s, x(s))Δs. + C(t, t)μ(t)G(t, x(t)) + 2 t0

Hence, by substituting J(t) in (6.3.14), and by making use of ∫ t a(t) − x(t) = C(t, s)G(s, x(s))Δs t0

we obtain

260

6 Volterra Integral Dynamic Equations

V Δ (t) ≤ C(t, t)μ(t)G2 (t, x(t)) + 2G(t, x(t))[a(t) − x(t)]

(6.3.15)

for all t ∈ IT . Notice that a(t) − x(t) ≥ 0 by our positivity assumptions on C and G. Thus, (6.3.12–6.3.15) imply V Δ (t) ≤ C(t, t)μ(t)α2 x 2 (t) + 2αx(t)[a(t) − x(t)] ≤ C(t, t)μ(t)α2 x 2 (t) + α[a2 (t) + x 2 (t) − 2x 2 (t)] = α[C(t, t)μ(t)α − 1]x 2 (t) + αa2 (t) ≤ −αβx 2 (t) + αa2 (t)

(6.3.16)

for all t ∈ IT . By integrating (6.3.16) from t0 to t, we arrive at the following inequality: ∫ t ∫ t 2 V(t) ≤ V(t0 ) + α a (s) Δs − αβ x 2 (s)Δs. t0

t0

This implies x ∈ L 2 (IT ) if a ∈ L 2 (IT ). We remark that in Theorem 6.3.6 the condition (6.3.12), and hence, the nonnegativity assumption for the solution x of (6.3.2) are needed to proceed from (6.3.15) to (6.3.16). If G(t, x(t)) = x(t), then without assuming nonnegativity of x we can directly obtain (6.3.16) by setting ∫ t ∫ t 2 ∫ t 2 V(t) = Cs (t, s) x(u) Δu Δs + C(t, 0) x(s) Δs . (6.3.17) t0

σ(s)

t0

Hence, we state the next corollary in which we propose a sufficient condition in order for any solution x of (6.3.3) to be in L 2 (IT ). Corollary 6.3.1. Let x(t) be any solution of (6.3.3) for all t ∈ IT . Assume there exists a constant β > 0 such that 1 − C(t, t)μ(t) ≥ β holds, i.e., −C(t, t) is positively regressive. If C(t, s) ≥ 0, Cs (t, s) ≥ 0, Cst (t, s) ≤ 0, and Ct (t, t0 ) ≤ 0, then x ∈ L 2 (IT ) if a ∈ L 2 (IT ). Proof. Let x(t) be any solution of (6.3.3) for all t ∈ IT and let the Lyapunov functional V(t) be defined as in (6.3.17). By a similar work as in (6.3.15) and (6.3.16) we get V Δ (t) ≤ [C(t, t)μ(t) − 1]x 2 (t) + a2 (t) − βx 2 (t) + a2 (t). The rest of the proof follows along the line of the proof of Theorem 6.3.6. This completes the proof. Notice that Corollary 6.3.1 covers the results of [63, Theorem 3.3] in the particular case T = R.

6.3 Lyapunov Functionals Method

261

6.3.1 Applications to Particular Time Scales It follows from the definitions of σ and μ that σ(t) = and

⎧ if T = R ⎪ ⎨t ⎪ t + 1 if T = Z ⎪ ⎪ qt if T = qN0 ⎩

⎧ if T = R ⎪ ⎨0 ⎪ if T = Z , μ(t) = 1 ⎪ ⎪ (q − 1)t if T = qN0 ⎩

where R and Z denote the sets of reals and integers respectively, and qN0 := {q n : n = 0, 1, 2, ...}, q > 1. All results obtained in the previous sections can be applied to integral equations on time scales R, Z, and qN0 , for which we obtain Volterra integral equation, Volterra summation equation, and Volterra q-integral equation, respectively. We only focus on the applications of Corollary 6.3.1 and Theorem 6.3.6. 1. (Volterra integral equation) For the time scale T = R, Eqs. (6.3.2) and (6.3.3) turn into the integral equations ∫ t x(t) = a(t) − C(t, s)G(s, x(s))ds, 0

∫ x(t) = a(t) −

t

C(t, s)x(s)ds, 0

respectively. Since we have μ(t) = 0 for all t ∈ R, we can rule out the condition (6.3.13). Hence, in the special case T = R, Corollary 6.3.1 and Theorem 6.3.6 cover the results [63, Theorem 3.3.] and [93, Theorem 3.2], respectively. Moreover, the results in Corollary 6.3.1 and Theorem 6.3.6 are new for the following cases: 2. (Summation equation) For the time scale T = Z, Eq. (6.3.2) becomes the wellknown Volterra summation equation x(n) = a(n) −

n−1 

C(n, m)G(m, x(m)), n ∈ Z,

m=0

and the condition (6.3.13) is equivalent to 1 − αC(t, t) ≥ β.

262

6 Volterra Integral Dynamic Equations

3. (q-integral equation) For the time scale T = qN0 , Eq. (6.3.2) becomes  x(t) = a(t) − (q − 1)sC(t, s)G(s, x(s)), t ∈ qN0 (6.3.18) s ∈[1,t)∩q N0

(see [2]). Observe that condition (6.3.13) is equivalent to 1 − αt(q − 1)C(t, t) ≥ β for T =qN0 . In the following, we illustrate our results on the time scale T =2N0 . Example 6.3.1. Let T =2N0 and t0 = 1. Setting t = 2n , s = 2m , a(t) = 1t , C(t, s) = 1 x (s), Eq. (6.3.18) turns into the following: and G (s, x(s)) = 2s x(2n ) = 2−n −

n−1 

2m−2n−1 x (2m ) , n = 0, 1, ....

s , t2

(6.3.19)

m=0

Letting kˆ = 1 and αˆ = 13 , all conditions of Theorem 6.3.3 hold. Since H (t, x (t)) = G (t, x (t)) − x (t)   1 = − 1 x (t) 2t we can choose k = 1 to make sure that (H2) holds. Since ∫ a(t) − k

t

C(t, s)a(s)Δs = 2−n −

t0

n−1 

2m−2n

m=0 −2n

=2

≥ 0, for n = 0, 1, 2, ...

the condition (6.3.11) holds. Then by Theorem 6.3.5 we know that there  exists a unique nonnegative bounded solution of (6.3.19). Since a(t) = 2−n ∈ L 2 [1, ∞)2N0 one may easily verify by letting α = β = 12 that all conditions of Theorem 6.3.6 are satisfied.  That is,  if x(t) is a nonnegative bounded solution of (6.3.19), then x(t) ∈ L 2 [1, ∞)2N0 .

6.4 Open Problems Open Problem 1. It is clear that Theorem 6.2.3 concerns Volterra integro-dynamic equations when we Δ-differentiate (6.2.7) and obtain

6.4 Open Problems

263

x Δ (t) = f Δ (t) + C(σ(t), t)g(x) +

∫

t

C(t, s)g(x(s))Δs.

(6.4.1)

0

We recommend the short open problem: Generalize (6.4.1) to ∫ t Δ x (t) = h(t) + A(t)g(x) + B(t, s)r(x(s))Δs

(6.4.2)

0

by examining the statement and proof of Theorem 6.2.3 and placing conditions on (6.4.2) so that Theorem 6.2.3 can be stated and proved for (6.4.2).

Chapter 7

Periodic Solutions: The Natural Setup

Summary Among time scales, periodic ones deserve a special interest since they enable researchers to develop a theory for the existence of periodic solutions of dynamic equations on time scales (see for example Bi et al. (Nonlinear Anal 68(5):1226–1245, 2008), Bohner and Warth (Appl Anal 86(8):1007–1015, 2007), Kaufmann and Raffoul (J Math Anal Appl 319(1):315–325, 2006; Electron J Differ Equ 2007(27):12, 2007)). This chapter is devoted to the study of periodic solutions based on the definition of periodicity given by Kaufmann and Raffoul (J Math Anal Appl 319(1):315– 325, 2006) and Atıcı et al. (Dynamic Systems and Applications, vol. 3, pp. 43–48, 2001). In this section, we cover natural setup of periodicity. For the conventional definition of periodic time scale and periodic functions on periodic time scales we refer to Definitions 1.5.1 and 1.5.2, respectively. Based on the Definitions 1.5.1 and 1.5.2, periodicity and existence of periodic solutions of dynamic equations on time scales were studied by various researchers and for first papers on the subject we refer to Adıvar and Raffoul (Ann Mat Pura Appl 188(4):543–559, 2009; Comput Math Appl 58(2):264–272, 2009), Bi et al. (Nonlinear Anal 68(5):1226–1245, 2008), Kaufmann and Raffoul (J Math Anal Appl 319(1):315–325, 2006; Electron J Differ Equ 2007(27):12, 2007), and Liu and Li (Nonlinear Anal 67(5):1457–1463, 2007). We begin by introducing the concept of periodicity and then apply the concept to finite delay neutral nonlinear dynamical equations and to infinite delay Volterra integrodynamic equations of variant forms by appealing to Schaefer fixed point theorem. The a priori bound will be obtained by the aid of Lyapunov-like functionals. Effort will be made to expose the improvement when we set the time scales to be the set of reals or integers. Then we proceed to study the existence of periodic solutions of almost linear Volterra integro-dynamic equations by utilizing Krasnosel’ski˘ı fixed point theorem (see Theorem 1.1.17). We will be examining the relationship between boundedness and the existence of periodic solutions. We will also be giving existence results for almost automorphic solutions of the delayed neutral dynamic system. We will end this chapter by proving existence results by using a fixed point theorem based on the large contraction. © Springer Nature Switzerland AG 2020 M. Adıvar, Y. N. Raffoul, Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales, https://doi.org/10.1007/978-3-030-42117-5_7

265

266

7 Periodic Solutions: The Natural Setup

Among time scales, periodic ones deserve a special interest since they enable researchers to develop a theory for the existence of periodic solutions of dynamic equations on time scales (see for example [41, 52, 99, 100]). This chapter is devoted to the study of periodic solutions based on the definition of periodicity given by Kaufmann and Raffoul [99] and Atıcı et al. [32]. In this section, we cover natural setup of periodicity. For the conventional definition of periodic time scale and periodic functions on periodic time scales we refer to Definitions 1.5.1 and 1.5.2, respectively. Based on the Definitions 1.5.1 and 1.5.2, periodicity and existence of periodic solutions of dynamic equations on time scales were studied by various researchers and for first papers on the subject we refer to [10, 11, 41, 99, 100, 115]. We begin by introducing the concept of periodicity and then apply the concept to finite delay neutral nonlinear dynamical equations and to infinite delay Volterra integrodynamic equations of variant forms by appealing to Schaefer fixed point theorem. The a priori bound will be obtained by the aid of Lyapunov-like functionals. Effort will be made to expose the improvement when we set the time scales to be the set of reals or integers. Then we proceed to study the existence of periodic solutions of almost linear Volterra integro-dynamic equations by utilizing Krasnosel’ski˘ı fixed point theorem (see Theorem 1.1.17). We will be examining the relationship between boundedness and the existence of periodic solutions. We will also be giving existence results for almost automorphic solutions of the delayed neutral dynamic system. We will end this chapter by proving existence results by using a fixed point theorem based on the large contraction. Results of the periodicity parts can be found in [10, 11, 13, 99–101, 144], and [157]. For emphasis we restate, from Chap. 1, the following two definitions that we credit Atıcı et al. [32] and Kaufmann and Raffoul [99] for. Definition 7.0.1. We say that a time scale T is periodic if there exists a p > 0 such that if t ∈ T, then t ± p ∈ T. For T  R, the smallest positive p is called the period of the time scale. Definition 7.0.2. Let T  R be a periodic time scale with period p. We say that the function f : T → R is periodic with period T if there exists a natural number n such that T = np, f (t ± T) = f (t) for all t ∈ T and T is the smallest number such that f (t ± T) = f (t). If T = R, we say that f is periodic with period T > 0 if T is the smallest positive number such that f (t ± T) = f (t) for all t ∈ T. Remark 7.0.1. If T is a periodic time scale with period p, then σ(t + np) = σ(t) + np. Consequently, the graininess function μ satisfies μ(t + np) = σ(t + np) − (t + np) = σ(t) − t = μ(t) and so is a periodic function with period p.

7.1 Neutral Nonlinear Dynamic Equation Let T be a periodic time scale. We use a fixed point theorem due to Krasnosel’ski˘ı (i.e., Theorem 1.1.17) to show that the nonlinear neutral dynamic system with delay

7.1 Neutral Nonlinear Dynamic Equation

267

x Δ (t) = −a(t)x σ (t) + c(t)x Δ (t − k) + q (t, x(t), x(t − k)) , t ∈ T, has a periodic solution. We assume that k is a mixed constant if T = R and is a multiple of the period of T if T  R. Under a slightly more stringent inequality we show that the periodic solution is unique using the contraction mapping principle. All the results of this section can be found in [99]. Let T be a periodic time scale such that 0 ∈ T. We will show the existence of periodic solutions for the nonlinear neutral dynamic system with delay x Δ (t) = −a(t)x σ (t) + c(t)x Δ (t − k) + q (t, x(t), x(t − k)) , t ∈ T.

(7.1.1)

We assume that k = mp if T has period p and k is fixed if T = R. In [120], the authors used Krasnosel’ski˘ı fixed point theorem and showed the existence of a periodic solution of (7.1.1) in the particular case T = R. Also, the existence of unique periodic solution of (7.1.1) was obtained by the aid of the contraction mapping principle. Similar results were obtained concerning (7.1.1) in [120] in the case T = Z. For T = R and assuming x = 0 is a solution of (7.1.1), in [137], Raffoul used the notion of fixed point theory and obtained conditions that guaranteed the asymptotic stability of the zero solution.

7.1.1 Existence of Periodic Solutions Let T > 0, T  k, T ∈ T be fixed and if T  R, T = np for some n ∈ N. Define PT := {φ ∈ C(T, R) : φ(t + T) = φ(t)},

(7.1.2)

where C(T, R) is the space of all real valued continuous functions. Then PT is a Banach space when it is endowed with the supremum norm  x := sup |x(t)|. t ∈[0,T ]T

The next lemma belongs to Kaufmann and Raffoul [99]. Lemma 7.1.1. Let x ∈ PT . Then  x σ  exists and  x σ  =  x. Proof. Since x ∈ PT , then x(0) = x(T) and x σ (0) = x σ (T). For all t ∈ [0, T]T , |x σ (t)| ≤  x. Hence,  x σ  ≤  x. Since x ∈ C(T, R), there exists a t0 ∈ [0, T]T such that  x = |x(t0 )|. If t0 is left scattered, then σ(ρ(t0 )) = t0 . And so,  x σ  ≥ |x σ (ρ(t0 ))| = x(t0 ) =  x. If t0 is dense, then σ(t0 ) = t0 and again  x σ  =  x. Suppose t0 is left dense and right scattered. Note if t0 = 0, then we work with t0 = T. Fix ε > 0 and consider a sequence {τk } such that τk ↑ t0 . Note that σ(τk ) ≤ t0 for all k. By the continuity of x, there exists a K such that for all k > K, |x(t0 ) − x σ (τk )| < ε. This implies that  x − ε ≤  x σ . Since ε > 0 was arbitrary, then  x =  x σ  and the proof is complete.

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7 Periodic Solutions: The Natural Setup

In this subsection, we assume that a is continuous, a(t) > 0 for all t ∈ T and a(t + T) = a(t),

c(t + T) = c(t),

(7.1.3)

where cΔ (t) is continuous. We also require that q(t, x, y) be continuous and periodic in t and Lipschitz continuous in x and y. That is, q(t + T, x, y) = q(t, x, y)

(7.1.4)

and for some positive constants L and E, |q(t, x, y) − q(t, z, w)| ≤ L x − z + E  y − w .

(7.1.5)

Lemma 7.1.2. Suppose (7.1.3)–(7.1.5) hold. If x ∈ PT , then x(t) is a solution of Eq. (7.1.1) if and only if,   −1 x(t) = c(t)x(t − k) + 1 − e a (t, t − T) ∫ t ' ( − r(s)x σ (s − k) + q(s, x(s), x(s − k)) e a (t, s) Δs, ×

(7.1.6)

t−T

where

r(s) = a(s)cσ (s) + cΔ (s).

(7.1.7)

x Δ (t) + a(t)x σ (t) = c(t)x Δ (t − k) + q (t, x(t), x(t − k))

(7.1.8)

Proof. Rewrite (7.1.1) as

and let x ∈ PT be a solution of (7.1.1). Multiply both sides of (7.1.8) by ea (t, 0) and then integrate from t − T to t to obtain, ∫ t ∫ t ' ' (Δ ( ea (s, 0)x(s) Δs = c(s)x Δ (s − k) + q(s, x(s), x(s − k)) ea (s, 0) Δs. t−T

t−T

Consequently, we have ea (t, 0)x(t) − ea (t − T, 0)x(t − T) ∫ t ' ( c(s)x Δ (s − k) + q(s, x(s), x(s − k)) ea (s, 0) Δs. = t−T

Divide both sides of the above equation by ea (t, 0). Since x ∈ PT , we have   x(t) 1 − e a (t, t − T) ∫ t (7.1.9) ' ( c(s)x Δ (s − k) + q(s, x(s), x(s − k)) e a (t, s) Δs, = t−T

where we have used Lemma 1.1.3 to simplify the exponentials.

7.1 Neutral Nonlinear Dynamic Equation

269

Now consider the first term of the integral on the right-hand side of (7.1.9): ∫ t e a (t, s)c(s)x Δ (s − k) Δs. t−T

Integrate by parts to obtain ∫ t e a (t, s)c(s)x Δ (s − k) Δs t−T

= e a (t, t)c(t)x(t − k) − e a (t, t − T)c(t − T)x(t − T − k) ∫ t ' (Δ e a (t, s)c(s) s x σ (s − k) Δs. − t−T

Since c(t) = c(t − T) and x(t) = x(t − T), the above equality reduces to ∫ t e a (t, s)c(s)x Δ (s − k) Δs t−T (7.1.10) ∫ t ' ( ( Δs σ ' e a (t, s)c(s) x (s − k) Δs. = c(t)x(t − k) 1 − e a (t, t − T) − t−T

Substitute the right-hand side of (7.1.10) into (7.1.9) and expand the delta-derivative to get  −1  x(t) = c(t)x(t − k) + 1 − e a (t, t − T) ∫ t ' a(s)e a (t, σ(s))cσ (s)x σ (s − k) − e a (t, s)cΔ (s)x σ (s − k) × t−T ( + e a (t, s)q(s, x(s), x(s − k)) Δs.

(7.1.11)

A final application of Lemma 1.1.3 yields the desired result and the proof is complete. Define the mapping H : PT → PT by  −1  (Hϕ)(t) := c(t)ϕ(t − k) + 1 − e a (t, t − T) ∫ t (7.1.12) ' ( − r(s)ϕσ (s − k) + q(s, ϕ(s), ϕ(s − k)) e a (t, s) Δs. × t−T

To apply Theorem 1.1.17 we need to construct two mappings; one map is a contraction and the other map is compact. We express Eq. (7.1.12) as (Hϕ)(t) = (Bϕ)(t) + (Aϕ)(t), where A, B are given by (Bϕ)(t) := c(t)ϕ(t − k)

(7.1.13)

270

7 Periodic Solutions: The Natural Setup

and  −1  (Aϕ)(t) := 1 − e a (t, t − T) ∫ t ' ( − r(s)ϕσ (s − k) + q(s, ϕ(s), ϕ(s − k)) e a (t, s) Δs ×

(7.1.14)

t−T

and r(s) is defined in (7.1.7) Lemma 7.1.3. Suppose (7.1.3)–(7.1.5) hold. Then A : PT → PT as defined by (7.1.14) is compact. Proof. We first show that A : PT → PT . Evaluate (7.1.14) at t + T.   −1 (Aϕ)(t + T) = 1 − e a (t + T, t) ∫ t+T ' ( − r(s)ϕσ (s − k) + q(s, ϕ(s), ϕ(s − k)) e a (t + T, s) Δs. × t

(7.1.15)

Use Theorem 1.1.9 with u = s − T and the periodicity of c, ϕ, and q to get   −1 (Aϕ)(t + T) = 1 − e a (t + T, t) ∫ t ' ( − r(u + T)ϕσ (u − k) + q(u, ϕ(u), ϕ(u − k)) e a (t + T, u + T) Δu. × t−T

(7.1.16)

Now, since a(t + T) = a(t), c(t + T) = c(t) and μ(t + T) = μ(t), then r(u + T) = a(u + T)cσ (u + T) + cΔ (u + T) = a(u)cσ (u) + cΔ (u) = r(u). That is, r(u) is periodic with period T. Also, from (7.1.3) and Theorem 1.1.9, we have e a (t + T, u + T) = e a (t, u) and e a (t + T, t) = e a (t, t − T). Thus (7.1.16) becomes   −1 (Aϕ)(t + T) = 1 − e a (t, t − T) ∫ t ' ( − r(u)ϕσ (u − k) + q(u, ϕ(u), ϕ(u − k)) e a (t, u) Δu × t−T

= (Aϕ)(t). That is, A : PT → PT . To see that A is continuous, we let ϕ, ψ ∈ PT with ϕ ≤ C and ψ ≤ C and define η := max |(1 − e a (t, t − T))−1 |, t ∈[0,T ]T

β := max |r(t)|, t ∈[0,T ]T

γ :=

max

u ∈[t−T,t]T

e a (t, u).

(7.1.17) Given > 0, take δ = /M such that ϕ − ψ < δ. By making use of Lipschitz inequality (7.1.5) in (7.1.14) we get

7.1 Neutral Nonlinear Dynamic Equation

< < 0, T ∈ T be fixed and if T  R, T = np for some n ∈ N. Let PT be the Banach space of all real valued continuous functions on T (see (7.1.2)). In this section, we assume that a is continuous, a(t) > 0 for all t ∈ T and     a(t + T) = a(t), id − g (t + T) = id − g (t), (7.2.2) where, id is the identity function on T. We also require that Q(t, x) and G(t, x, y) are continuous and periodic in t and Lipschitz continuous in x and y. That is, Q(t + T, x) = Q(t, x), G(t + T, x, y) = G(t, x, y),

(7.2.3)

and there are positive constants E1, E2, E3 such that |Q(t, x) − Q(t, y)| ≤ E1  x − y, for x, y ∈ R,

(7.2.4)

and |G(t, x, y) − G(t, z, w)| ≤ E2  x − z + E3  y − w, for x, y, z, w ∈ R.

(7.2.5)

Lemma 7.2.1. Suppose (7.2.2), (7.2.3) hold. If x ∈ PT , then x is a solution of Eq. (7.2.1) if and only if,     −1 x(t) = Q t, x(t − g(t)) + 1 − e a (t, t − T) ∫ t #    $ − a(s)Q σ s, x(s − g(s)) + G s, x(s), x(s − g(s)) e a (t, s) Δs. × t−T

(7.2.6)

276

7 Periodic Solutions: The Natural Setup

Proof. Let x ∈ PT be a solution of (7.2.1). First we write (7.2.1) as     {x(t) − Q t, x(t − g(t)) }Δ = −a(t){x σ (t) − Q σ t, x(t − g(t)) }     −a(t)Q σ t, x(t − g(t)) + G t, x(t), x(t − g(t)) . Multiply both sides by ea (t, 0) and then integrate from t − T to t to obtain ∫ t '   (Δ ea (s, 0){x(s) − Q s, x(s − g(s)) } Δs t−T ∫ t #    $ − a(s)Q σ s, x(s − g(s)) + G s, x(s), x(s − g(s)) ea (s, 0) Δs. = t−T

Consequently, we have    ea (t, 0) x(t) − a(t)Q t, x(t − g(t))    − ea (t − T, 0) x(t − T) − a(t − T)Q t − T, x(t − T − g(t − T)) ∫ t '    $ − a(s)Q σ s, x(s − g(s)) + G s, x(s), x(s − g(s)) ea (s, 0) Δs. = t−T

After making use of (7.2.2), (7.2.3), and x ∈ PT , we divide both sides of the above equation by ea (t, 0) to obtain  −1    x(t) = Q t, x(t − g(t)) + 1 − e a (t, t − T) ∫ t #    $ − a(s)Q σ s, x(s − g(s)) + G s, x(s), x(s − g(s)) e a (t, s) Δs, × t−T

where we have used Lemma 1.1.3 to simplify the exponentials. Since each step is reversible, the converse follows. This completes the proof. Define the mapping H : PT → PT by  −1    (Hϕ)(t) := Q t, ϕ(t − g(t)) + 1 − e a (t, t − T) ∫ t #    $ − a(s)Q σ s, ϕ(s − g(s)) + G s, ϕ(s), ϕ(s − g(s)) e a (t, s) Δs. × t−T

(7.2.7) To apply Theorem 1.1.17 we need to construct two mappings; one map is a contraction and the other map is compact. We express Eq. (7.2.7) as (Hϕ)(t) = (Bϕ)(t) + (Aϕ)(t), where A, B are given by   (Bϕ)(t) = Q t, ϕ(t − g(t))

(7.2.8)

7.2 Nonlinear Dynamic Equation

277

and  −1  (Aϕ)(t) = 1 − e a (t, t − T) ∫ t '    $ − a(s)Q σ s, ϕ(s − g(s)) + G s, ϕ(s), ϕ(s − g(s)) e a (t, s) Δs. × t−T

(7.2.9)

Lemma 7.2.2. Suppose (7.2.2)–(7.2.5) hold. Then A : PT → PT , as defined by (7.2.9), is compact. Proof. We first show that A : PT → PT . Evaluate (7.2.9) at t + T. ∫ t+T #  −1    − a(s)Q σ s, ϕ(s − g(s)) (Aϕ)(t + T) = 1 − e a (t + T, t) × t (7.2.10)  $ + G s, ϕ(s), ϕ(s − g(s)) e a (t + T, s) Δs. Use Theorem 1.1.9 with u = s − T and conditions (7.2.2)–(7.2.3) to get  −1  (Aϕ)(t + T) = 1 − e a (t + T, t) ∫ t #   − a(u + T)Q σ u − k, ϕ(u − k − g(u − k)) × t−T  $ + G s, ϕ(u − k), ϕ(u − k − g(u − k)) e a (t + T, u + T) Δu. From Lemma 1.1.3 and Theorem 1.1.9, we have e a (t + T, u + T) = e a (t, u) and e a (t + T, t) = e a (t, t − T). Thus (7.2.10) becomes ∫ t #    −1  − a(u)Q σ u, ϕ(u − g(u)) (Aϕ)(t + T) = 1 − e a (t, t − T) × t−T  $ + G u, ϕ(u), ϕ(u − g(u)) e a (t + T, u) Δu = (Aϕ)(t). That is, A : PT → PT . To see that A is continuous, we let ϕ, ψ ∈ PT with ϕ ≤ C and ψ ≤ C and define   −1  η := max  1 − e a (t, t − T) , ρ := max |a(t)|, γ := max e a (t, u). t ∈[0,T ]T

t ∈[0,T ]T

u ∈[t−T,t]T

(7.2.11) Given ε > 0, take δ = ε/M with M = η γ T(ρ E1 + E2 + E3 ) where, E1 , E2 , and E3 are given by (7.2.4) and (7.2.5) such that ϕ − ψ < δ. Using (7.2.9) we get < < < Aϕ − Aψ < ≤ η γ

∫

T

#

$ ρ E1 ϕ − ψ + (E2 + E3 )ϕ − ψ Δu

0

≤ M ϕ − ψ < ε. This proves that A is continuous.

278

7 Periodic Solutions: The Natural Setup

We - .need to show that A is compact. Consider the sequence of periodic functions ϕn ⊂ PT and assume that the sequence is uniformly bounded. Let R > 0 be such that ϕn  ≤ R, for all n ∈ N. In view of (7.2.4) and (7.2.5) we arrive at |Q(t, x)| = |Q(t, x) − Q(t, 0) + Q(t, 0)| ≤ |Q(t, x) − Q(t, 0)| + |Q(t, 0)| ≤ E1  x + α. Similarly, |G(t, x, y)| = |G(t, x, y) − G(t, 0, 0) + G(t, 0, 0)| ≤ |G(t, x, y) − G(t, 0, 0)| + |G(t, 0, 0)| ≤ E2  x + E3  y + β, where α = supt ∈[0,T ]T |Q(t, 0)| and β = supt ∈[0,T ]T |G(t, 0, 0)|. ∫ t #     −1  − a(s)Q σ s, ϕ(s − g(s)) |(Aϕn )(t)| =  1 − e a (t, t − T) × t−T   $  + G s, ϕ(s), ϕ(s − g(s)) e a (t, s) Δs ∫ t         − a(s) Q σ s, ϕ(s − g(s))  + G s, ϕ(s), ϕ(s − g(s))  Δs ≤ ηγ 't−T ( ≤ ηγT ρ(E1 R + α) + (E2 + E3 )R + β ≡ D. Thus the sequence { Aϕn } is uniformly bounded. Now, it can be easily checked that     (Aϕn )Δ (t) = −a(t)(Aϕn )σ (t) − a(t)Q σ t, ϕ(t − g(t)) + G t, ϕ(t), ϕ(t − g(t)) . Consequently, |(Aϕn )Δ (t)| ≤ Dρ + ρ(E1 R + α) + (E2 + E3 )R + β for all n. That is (Aϕn )Δ  ≤ F, for some positive constant F. Thus the sequence { Aϕn } is uniformly bounded and equi-continuous. The Arzel`a–Ascoli theorem implies that there is a subsequence { Aϕnk } which converges uniformly to a continuous T-periodic function ϕ∗ . Thus A is compact. Lemma 7.2.3. Let B be defined by (7.2.8) and E1 ≤ ζ < 1. Then B : PT → PT is a contraction. Proof. Trivially, B : PT → PT . For ϕ, ψ ∈ PT , we have Bϕ − Bψ = sup |Bϕ(t) − Bψ(t)| t ∈[0,T ]T

(7.2.12)

7.2 Nonlinear Dynamic Equation

279

≤ E1 sup |ϕ(t − g(t)) − ψ(t − g(t))| t ∈[0,T ]T

≤ ζ ϕ − ψ. Hence B defines a contraction mapping with contraction constant ζ. In preparation for the next result we define the following set: M = {ϕ ∈ PT : ||ϕ|| ≤ J}.

(7.2.13)

Theorem 7.2.1. Suppose the hypothesis of Lemma 7.2.1 hold. Suppose (7.2.2)– (7.2.5) hold. Let α = supt ∈[0,T ]T |Q(t, 0)| and β = supt ∈[0,T ]T |G(t, 0, 0)|. Let J be a positive constant satisfying the inequality # $ α + E1 J + ηT γ ρ (E1 J + α) + (E2 + E3 )J + β ≤ J. (7.2.14) Then Eq. (7.2.1) has a solution in M. Proof. By Lemma 7.2.2, A is continuous and AM is contained in a compact set. Also, from Lemma 7.2.3, the mapping B is a contraction and it is clear that B : PT → PT . Next, we show that if ϕ, ψ ∈ M, we have || Aϕ + Bψ|| ≤ J. Let ϕ, ψ ∈ M with ||ϕ||, ||ψ|| ≤ J. Then  Aϕ + Bψ

∫

T' ( |a(u)|(α + E1 ||ϕ||) + E2 ||ϕ|| + E3 ||ϕ|| + β Δu ≤ E1 ||ψ|| + α + η γ # 0 $ ≤ α + E1 J + ηT γ ρ(E1 J + α) + (E2 + E3 )J + β ≤ J.

We now see that all the conditions of Krasnosel’ski˘ı’s theorem (i.e., Theorem 1.1.17) are satisfied. Thus there exists a fixed point z in M such that z = Az + Bz. By Lemma 2.1, this fixed point is a solution of (7.2.1). Hence Eq. (7.2.1) has a Tperiodic solution. Theorem 7.2.2. Suppose (7.2.2)–(7.2.5) hold. If E1 + η γ T(ρ E1 + E2 + E3 ) < 1, then Eq. (7.2.1) has a unique T-periodic solution. Proof. Let the mapping H be given by (7.2.7). For ϕ, ψ ∈ PT , we have   Hϕ − Hψ ≤ E1 + η γ T(ρ E1 + E2 + E3 ) ϕ − ψ. This completes the proof by invoking the contraction mapping principle.

280

7 Periodic Solutions: The Natural Setup

It is worth noting that Theorems 7.2.1 and 7.2.2 are not applicable to functions such as   G t, ϕ(t), ϕ(t − g(t)) = f1 (t)ϕ2 (t) + f2 (t)ϕ2 (t − g(t)), where, f1 (t), f2 (t), and g(t) > 0 are continuous and periodic on some applicable time scale. To accommodate such functions, we state the following corollary, in which the functions G and Q are required to satisfy local Lipschitz conditions. We note that conditions (7.2.4) and (7.2.5) require that the functions G and Q be globally Lipschitz. Corollary 7.2.1. Suppose (7.2.2)–(7.2.5) hold and let α and β be the constants defined in Theorem 7.2.1. Let J be a positive constant and the set M be defined as in 7.2.13. Suppose there are positive constants E1∗, E2∗ and E3∗ so that for x, y, z and w ∈ M we have |Q(t, x) − Q(t, y)| ≤ E1∗  x − y, |G(t, x, y) − G(t, z, w)| ≤ E2∗  x − z + E3∗  y − w, and

# $ α + E1∗ J + ηT γ ρ (E1∗ J + α) + (E2∗ + E3∗ )J + β ≤ J.

(7.2.15)

Then Eq. (7.2.1) has a solution in M. If in addition, E1∗ + η γ T(ρ E1∗ + E2∗ + E3∗ ) < 1, then the solution in M is unique. Proof. Let the mapping H be given by (7.2.7). Then the results follow immediately from Theorems 7.2.1 and 7.2.2. Now we give an example. Example 7.2.1. Let T be a periodic time scale. For small positive ε1 and ε2, we consider the perturbed van der Pol equation   Δ  x Δ = −2x σ (t) + ε1 b(t)x 2 (t − g(t)) + ε2 c(t) + x 2 (t) ,

(7.2.16)

where, b(t + T) = b(t),

    id − g (t + T) = id − g (t),

and

c(t + T) = c(t). (7.2.17)

Also, we assume that the functions b, c, and g are continuous with id − g : T → T strictly increasing. So we have a(t) = 2, and

Q(t, x(t − g(t))) = ε1 b(t)x 2 (t − g(t))

  G(t, x(t), x(t − g(t))) = ε2 c(t) + x 2 (t) .

7.2 Nonlinear Dynamic Equation

281

Then for ϕ, ψ ∈ M we have   −1 η = 1 − e 2 (t, t − T) , ρ = 3 and γ ≤ 1. Let ι = sup |b(t)|, and κ = sup |c(t)|. t ∈[0,T ]T

Then,

t ∈[0,T ]T

α = 0, β = ε2 κ, E1∗ = ε1 ι J, E3∗ = 0, and E2∗ = ε2 J.

Thus, inequality (7.2.15) becomes # $ ε1 ιJ 2 + ηT 3ε1 J 2 + ε2 J 2 + ε2 κ ≤ J which is satisfied for small ε1 and ε2 . Hence, Eq. (7.2.16) has a T-periodic solution, by Corollary 7.2.1. Moreover, if # $ ε1 ιJ + ηγT 3ε1 ιJ + ε2 J < 1 is satisfied for small ε1 and ε2, then Eq. (7.2.16) has a unique T-periodic solution.

7.2.2 Stability Lyapunov functions and functionals have been successfully used to obtain boundedness, stability, and the existence of periodic solutions of differential equations, differential equations with functional delays, and functional differential equations. When studying differential equations with functional delays using Lyapunov functionals, many difficulties arise if the delay is unbounded or if the differential equation in question has unbounded terms. In [137], Raffoul studied, via fixed point theory, the asymptotic stability of the zero solution of the scalar neutral differential equation   (7.2.18) x (t) = −a(t)x(t) + c(t)x (t − g(t)) + q t, x(t), x(t − g(t) , where a(t), b(t), g(t), and q are continuous in their respective arguments. It is clear that Eq. (7.2.1) is more general than (7.2.18). This section is mainly concerned with the asymptotic stability of the zero solution of (7.2.1). We assume that the functions Q and G are continuous, as before. Also, we assume that g(t) is continuous and g(t) ≥ g ∗ > 0, for all t ∈ T such that t ≥ t0 for some t0 ∈ T and that Q(t, 0) = G(t, 0, 0) = 0 and Q and G obey the Lipschitz conditions (7.2.4) and (7.2.5). The techniques used in this section are adapted from the paper [137]. As before, we assume a time scale, T, that is unbounded above and below and that 0 ∈ T. Also, we assume that g : T → R and that id − g : T → T is strictly increasing.

282

7 Periodic Solutions: The Natural Setup

To arrive at the correct mapping, we rewrite (7.2.1) as in the proof of Lemma 7.2.1. Multiply both sides by ea (t, 0) and then integrate from 0 to t to obtain #  $ x(t) = Q(t, x(t − g(t))) + x(0) − Q 0, x(−g(0)) e a (0, t) ∫ t#    $ + − a(u)Q σ u, x(u − g(u)) + G u, x(u), x(u − g(u)) e a (t, u) Δu. 0

(7.2.19) Thus, we see that x is a solution of (7.2.1) if and only if it satisfies (7.2.19). Let ψ : (−∞, 0]T → R be a given Δ-differentiable bounded initial function. We say x(t) := x(t, 0, ψ) is a solution of (7.2.1) if x(t) = ψ(t) for t ≤ 0 and satisfies (7.2.1) for t ≥ 0. We say the zero' solution of (7.2.1) is stable at t0 (if for each ε > 0, there is a δ = δ(ε) > 0 such that ψ : (−∞, t0 ]T → R with ψ < δ implies |x(t, t0, ψ)| < ε. Let Cr d = Cr d (T, R) be the space of all rd-continuous functions from T → R and define the set S by S = {ϕ ∈ Cr d : ϕ(t) = ψ(t) if t ≤ 0, ϕ(t) → 0 as t → ∞, and ϕ is bounded} .   Then, S, || · || is a complete metric space where, || · || is the supremum norm. For the next theorem we impose the following conditions: e a (t, 0) → 0, as t → ∞, there is an α > 0 such that ∫ t# $ |a(u)|E1 + E2 + E3 e a (t, u) Δu ≤ α < 1, t ≥ 0, E1 +

(7.2.20)

(7.2.21)

0

t − g(t) → ∞, as t → ∞,

(7.2.22)

Q(t, 0) → 0, as t → ∞.

(7.2.23)

and Theorem 7.2.3. If (7.2.4), (7.2.5) and (7.2.20)–(7.2.23) hold, then every solution x(t, 0, ψ) of (7.2.1) with small continuous initial function ψ(t), is bounded and goes to zero as t → ∞. Moreover, the zero solution is stable at t0 = 0. Proof. Define the mapping P : S → S by   Pϕ (t) = ψ(t) if t ≤ 0 and



   Pϕ (t) = Q(t, ϕ(t − g(t))) + ψ(0) − Q(0, ψ(−g(0))) e a (t, 0) ∫ t# + − a(u)Q σ (u, ϕ(u − g(u))) 0 $ + G(u, ϕ(u), ϕ(u − g(u))) e a (t, u) Δu, if t ≥ 0.

7.2 Nonlinear Dynamic Equation

283

It is clear that for ϕ ∈ S, Pϕ is continuous. Let ϕ ∈ S with ||ϕ|| ≤ K, for some positive constant K. Let ψ(t) be a small given continuous initial function with ||ψ|| < δ, δ > 0. Then,       ||Pϕ|| ≤ E1 K +  ψ(0) − Q 0, ψ(−g(0)) e a (t, 0) ∫ t# $ + |a(u)|E1 + E2 + E3 e a (t, u) Δu K 0 (7.2.24) ∫ t#  $  |a(u)|E1 + E2 + E3 e a (t, u) Δu K ≤ 1 + E1 δ + E1 K + 0   ≤ 1 + E1 δ + αK, which implies that, ||Pϕ|| ≤ K,  for the right δ. Thus, (7.2.24) implies that (Pϕ)(t) is bounded. Next we show that Pϕ (t) → 0 as t → ∞. The second term on the right   side of Pϕ (t) tends to zero, by condition (7.2.20). Also, the first term on the right side tends to zero, because of (7.2.22), (7.2.23) and the fact that ϕ ∈ S. It is left to show that the integral term goes to zero as t → ∞. Let ε > 0 be given and ϕ ∈ S with ||ϕ|| ≤ K, K > 0. Then, there exists a t1 > 0 so that for t > t1 , |ϕ(t − g(t))| < ε. Due to condition (7.2.20), there exists a t2 > t1 such ε . that for t > t2 implies that e a (t, t1 ) < αK Thus for t > t2, we have  ∫ t#    $   − a(u)Q σ u, ϕ(u − g(u)) + G u, ϕ(u), ϕ(u − g(u)) e a (t, u) Δu  0 ∫ t1 # $ ≤K |a(u)|E1 + E2 + E3 e a (t, u) Δu 0 ∫ t# $ |a(u)|E1 + E2 + E3 e a (t, u) Δu +ε t1

≤ Ke a (t, t1 )

∫ 0

t1 #

$ |a(u)|E1 + E2 + E3 e a (t1, u) Δu + αε

≤ αKe a (t, t1 ) + αε ≤ ε + αε.   Hence, Pϕ (t) → 0 as t → ∞. It remains to show that P is a contraction under the supremum norm. Let ζ, η ∈ S. Then ∫ t#  +  $ ,   |a(u)|E1 + E2 + E3 e a (t, u)Δu ||ζ − η|| (Pζ)(t) − (Pη)(t) ≤ E1 + 0

≤ α||ζ − η||. Thus, by the contraction mapping principle, P has a unique fixed point in S which solves (7.2.1), is bounded and tends to zero as t tends to infinity. The stability of the

284

7 Periodic Solutions: The Natural Setup

zero solution at t0 = 0 follows from the above work by simply replacing K by ε. This completes the proof. Example 4.2. Let 1 1 T = −2Z ∪ 2Z ∪ {0} = {· · · , −2, −1, − , · · · , 0, · · · , , 1, 2, . . . }. 2 2 and let ψ be a continuous initial function, ψ : −2Z ∪ {0} → R with ||ψ|| ≤ δ for small δ > 0. For small ε1 and ε2, we consider the nonlinear neutral dynamic equation ' (Δ   x Δ (t) = −2x σ (t) + ε1 x 2 (t − (t/2)) + ε2 x 2 (t) + x 2 (t − (t/2)) .

(7.2.25)

The function g is given by g(t) = 2t and satisfies g : T → R and id − g : T → T is strictly increasing. Suppose (7.2.26) 0 < 4(2ε1 + ε2 )δ(1 + ε1 δ) < 1. Let S be defined by + , S = ϕ : T → R| ϕ(t) = ψ(t) if t ≤ 0, ϕ(t) → 0 as t → ∞, ϕ ∈ C and ||ϕ|| ≤ K , for some positive constant K satisfying the inequality  1 − 1 − 4(2ε1 + ε2 )δ(1 + ε1 δ) 1 −

2 + γ. μ(t0 )

(7.3.6)

Note that stability of the delay difference equation x(t + 1) = b(t)x(t) − a(t)x(t − g(t)), t ∈ Z has been studied in [95] by making use of Lyapunov’s method. Hereafter, we develop a time scale analog of Lyapunov’s method used in [95, p. 3]. Let the Lyapunov functional V be defined by ∫ t V(t, x(.)) = |x(t)| + γ (7.3.7) |x(s)| Δs. δ(t)

Evidently, the delay function δ : [t0, ∞)T → [δ(t0 ), ∞)T defined in previous section satisfies the assumptions made for the function ν in Theorem 1.1.3. Therefore, Lemma 2.4.2 enables us to arrive at    (7.3.8) V Δ (t, x(.)) = |x(t)| Δ + γ |x(t)| − |x(δ(t))| δΔ (t) . As a consequence of Lemma 1.2.1, Lemma 1.2.2, and (7.3.3)–(7.3.8) we get that V Δ (t, x(.)) ≤ ζ(t) |x(t)| for all t ∈ T, where

 ζ(t) =

(7.3.9)

b(t) + γ if t ∈ T+x 2 − μ(t) − b(t) + γ if t ∈ T−x

and the sets T−x and T−x are defined by (1.2.8). Note that (7.3.4) and (7.3.6) imply ζ(t) < 0 for all t ∈ T.

288

7 Periodic Solutions: The Natural Setup

This along with (7.3.9) shows the stability of zero solution of Eq. (7.3.3) using [90, Theorem 2]. As a consequence of discussion above, we can give the following theorem: Theorem 7.3.1. Let T be a time scale that is unbounded above. If (7.3.4) holds, then zero solution of Eq. (7.3.3) is stable. Observe that for Eq. (7.3.1) the condition (7.3.4) does not hold since b(t) = 0 in (7.3.1). Hence, the use of fixed point theory is justified.

7.3.2 Periodicity Consider Eq. (7.3.1). For the sake of inverting equation (7.3.1) we write it in the form ∫ t Δ Δ x (t) = p(t)x(t) − p(s)x(s)Δs , (7.3.10) δ(t)

where

p(t) = −a(δ−1 (t)).

(7.3.11)

Let T be a periodic time scale (see Definition 1.5.1). Suppose that there is a positive real T such that a(t + T) = a(t) (7.3.12) for all t ∈ T. It follows from (7.3.11) and (7.3.12) that p(t + T) = p(t). In addition to assumptions in (7.3.2), we also assume that the delay function δ : T → T satisfies δ(t ± T) = δ(t) ± T . (7.3.13) Hereafter, in this subsection, we suppose that 1 − e p (t, t − T)  0.

(7.3.14)

We multiply Eq. (7.3.10) by e p (σ(t), t0 ), use the product rule, and integrate the obtained equation from t − T to t to find  ∫ t−T ∫ t x(t) = (1 − e p (t, t − T))−1 e p (t, t − T) p(s)x(s)Δs − p(s)x(s)Δs δ(t−T ) δ(t) ∫ s  ∫ t

p(s)e p (t, s) p(u)x(u)Δu Δs . + t−T

δ(s)

For x ∈ PT define the mapping H by

7.4 Integro-Dynamic Equation

289

 ∫ t−T ∫ t H x(t) = (1 − e p (t, t − T))−1 e p (t, t − T) p(s)x(s)Δs − p(s)x(s)Δs δ(t−T ) δ(t) ∫ s  ∫ t

p(s)e p (t, s) p(r)x(r)Δr Δs . + δ(s)

t−T

It can be easily shown by making use of the substitution u = s + T, (7.3.11), Theorem 1.1.9, and the equalities e p (t + T, s + T) = e p (t, s) , e p (t + T, t) = e p (t, t − T) that (H x)(t + T) = (H x)(t). Let PT be the Banach space of all real valued continuous functions on T (see (7.1.2)). Thus, H : PT → PT . Similar to that of Theorem 2.4.1 we can prove the following theorem. Theorem 7.3.2. Assume (7.3.12)–(7.3.14) hold for all t ∈ T. If there is an α > 0 such that ∫ t ∫ t−T   (1 − e p (t, t − T))−1  |p(s)| Δs + e p (t, t − T) |p(s)| Δs ∫ +

t

t−T

δ(t) s

δ(t)−T

∫ | p(s)| e p (t, s)

δ(s)

|p(r)| ΔrΔs ≤ α < 1,

then Eq. (7.3.10) has a unique T-periodic solution.

7.4 Integro-Dynamic Equation In this section, using topological degree method and Schaefer’s fixed point theorem, we deduce the existence of periodic solutions of nonlinear system of integro-dynamic equations on periodic time scales. Furthermore, we provide several applications to scalar equations, where we develop a time scale analogue of Lyapunov’s direct method and prove an analogue of Sobolev’s inequality on time scales to arrive at a priori bound on all periodic solutions. Therefore, we improve and generalize the corresponding results in continuous case (see [67]). Most of the results of this section can be found in [10] and some are new. Existence of periodic solutions of Volterra-type nonlinear integro-differential and summation equations has been intensively investigated in the literature including [67, 77, 94], and references therein. In recent years, time scale versions of wellknown equations have taken prominent attention (e.g., [26, 27, 29–50, 53, 131– 141, 146]) since the introduction of the new derivative concept by Stefan Hilger. For a comprehensive review of this topic we direct the reader to the monograph [50].

290

7 Periodic Solutions: The Natural Setup

Let T be a periodic time scale with period P including 0. Let T > 0 be fixed and if T  R, T = nP for some n ∈ N (see Definition 1.5.1). This section focuses on nonlinear system of infinite delay integro-dynamic equations of the form ∫ t Δ x (t) = Dx (t) + f (x (t)) + K (t, s) g (x (s)) Δs + p (t) , (7.4.1) −∞

where x Δ (t) is n × 1 column vector determined by Δ-derivative of components of x (t), D is an n×n constant matrix with real entries, f , g : Rn → Rn , p : T → Rn , and K is an n × n matrix valued function with real entries. We shall assume throughout this section that the following assumptions hold: A 1. f and g are continuous functions, and p is T periodic, A.2. the kernel K (t, s) is continuous in (t, s) for s ≤ t, K (t, s) = 0 for s > t, K (t + T, s + T) = K (t, s) for all (t, s) ∈ T × T, and

∫ sup t ∈T

t

−∞

|K (t, s)| Δs < ∞,

(7.4.2)

(7.4.3)

where |.| denotes the matrix norm induced by a norm, also denoted by |.|, in Rn . In this section, we prove an existence theorem for periodic solutions of the system of integro-dynamic equations (7.4.1) whose special cases include the systems of integro-differential equations (T = R) held in [67] and Volterra summation equations (T = Z) treated by [140]. Moreover, we apply our existence theorem to the scalar equations on periodic time scales. Some of our results in this section are new even for some special cases. To differentiate the Lyapunov functionals given in further sections we will employ the next lemma which can be proven similar to [41, Lemma 2.9] by using Theorem 1.1.10. Lemma 7.4.1. Let T be a periodic time scale with period P. Suppose that f : T × T → R satisfies the assumptions of Theorem 1.1.10, then ∫

t

f (t, s)Δs

Δ

∫ = f (σ(t), t) − f (σ(t), t − T) +

t−T

t

f Δ (t, s)Δs,

t−T

where T = n0 P and n0 ∈ N is a positive constant.

7.4.1 Existence of Periodic Solutions We will state and prove our main result in this section. Define PT = {ϕ ∈ C(T, Rk ) : ϕ(t+T) = ϕ(t)} where, C(T, Rk ) is the space of all vector valued continuous functions on T. Then PT is a Banach space with the norm

7.4 Integro-Dynamic Equation

291

/

 |x| 0 = max

i=1,2,...,k

sup |xi (t)| .

t ∈[0,T ]T

Consider Eq. (7.4.1), and corresponding family of systems of equations x Δ (t) = λ [−γI + D] x (t) + γx (t)   ∫ t K (t, s) g (x (s)) Δs + p (t) , + λ f (x (t)) +

(7.4.4)

−∞

where 0 ≤ λ ≤ 1 and γ ∈ R. Lemma 7.4.2. If x ∈ PT , then x is a solution of Eq. (7.4.4) if, and only if, ∫  −1 t  x (t) = λ 1 − eγ (t, t − T) A (s, x (s)) eγ (t, σ (s)) Δs,

(7.4.5)

t−T

where

∫

A (s, x (s)) = {−γI + D} x (s) + f (x (s)) +

s

−∞

K (s, r) g (x (r)) Δr + p (s) .

Proof. Let x ∈ PT be a solution of (7.4.4). Equation (7.4.4) can be expressed as x Δ (t) = λA (t, x (t)) + γx (t) .

(7.4.6)

Multiplying both sides of (7.4.6) by e γ (σ (t) , t0 ) we get x Δ (t) e γ (σ (t) , t0 ) − γx (t) e γ (σ (t) , t0 ) x (t) = λA (t, x (t)) e γ (σ (t) , t0 ) . From (vi) in Lemma 1.1.3 we have '

x (t) e γ (t, t0 )

(Δ

= λA (t, x (t)) e γ (σ (t) , t0 ) ,

Taking integral from t − T to t, we arrive at ∫ x (t) e γ (t, t0 ) − x (t − T) e γ (t − T, t0 ) = λ

t

t−T

A (s, x (s)) e γ (σ (s) , t0 ) Δs.

By x (t) = x (t − T) for x ∈ PT , and e γ (t − T, t0 ) e γ (σ (s) , t0 ) = eγ (t, t − T) , = eγ (t, σ (s)) , e γ (t, t0 ) e γ (t, t0 ) we find (7.4.5). Since each step in the above work is reversible, the proof is complete. It is worth mentioning that in the special case T = R, Lemma 7.4.2 provides a solution different than the one given in [67, Theorem 2.3, (2λ )]. Using periodicity of the kernel K (t, s), we obtain

292

7 Periodic Solutions: The Natural Setup

A (s, x (s)) = A (s + T, x (s + T))

(7.4.7)

for x ∈ PT . Define the operator H by  −1  (H x) (t) = 1 − eγ (t, t − T)

∫

t

t−T

A (s, x (s)) eγ (t, σ (s)) Δs.

(7.4.8)

Then (7.4.5) is equivalent to the equation λH x = x.

(7.4.9)

Moreover, it can be also shown by making use of the substitution u = s + T and the equalities eγ (t + T, σ (s + T)) = eγ (t, σ (s)) , eγ (t + T, t) = eγ (t, t − T) that 

(H x) (t + T) = 1 − eγ (t + T, t)

 −1

∫

t+T

t

 −1  = 1 − eγ (t, t − T)  −1  = 1 − eγ (t, t − T)

∫

t

t−T ∫ t t−T

A (s, x (s)) eγ (t + T, σ (s)) Δs

A (u + T, x (u + T)) eγ (t + T, σ (u + T)) Δu A (u, x (u)) eγ (t, σ (u)) Δu

= (H x) (t) . Thus, H : PT → PT . On the other hand, (7.4.7) and Lemma 7.4.1 imply that Δ ∫ t H Δ x(t) = (1 − eγ (σ(t), σ(t) − T))−1 A(s, x(s))eγ (t, σ(s))Δs t−t Δ ∫ t  + (1 − eγ (t, t − T))−1 A(s, x(s))eγ (t, σ(s))Δs t−t = (1 − eγ (σ(t), σ(t) − T))−1 A(t, x(t))eγ (σ(t), σ(t))  ∫ t A(s, x(s))eγ (t, σ(s))Δs −A(t − T, x(t − T))eγ (σ(t), σ(t − T)) + γ t−t ∫ t γeγ (t, t − T) A(s, x(s))eγ (t, σ(s))Δs + (1 − eγ (t, t − T))(1 − eγ (σ(t), σ(t) − T)) t−t = A(t, x(t)) + γ(1 − eγ (t, t − T))−1 H x(t), where kγ = 1 − eγ (t, t − T) does not depend on t ∈ T. The next result concerns the compactness of the operator H.

(7.4.10)

7.4 Integro-Dynamic Equation

293

Lemma 7.4.3. The operator H : PT → PT , as defined by (7.4.8), is continuous and compact. Proof. Define the set X := {x ∈ PT : |x| 0 ≤ B}. Obviously X is closed and bounded in PT . For φ1, φ2 ∈ X, we obtain ∫ t |Hφ1 (t) − Hφ2 (t)| ≤ E1 E2 | A (s, φ1 (s)) − A (s, φ2 (s))| Δs t−T ∫ t ≤ E1 E2 |−γI + D| |φ1 (s) − φ2 (s)| Δs t−T ∫ t + E1 E2 | f (φ1 (s)) − f (φ2 (s))| Δs t−T ∫ t ∫ s + E1 E2 |K (s, r)| |g (φ1 (r)) − g (φ2 (r))| ΔrΔs, t−T

where

−∞

  −1   E1 = maxt ∈[0,T ]  1 − eγ (t, t − T)  ,   E2 = maxs ∈[t−T,t] eγ (t, σ (s)) .

Since f and g are continuous and φ1, φ2 ∈ X, f and g are uniformly continuous on [−B, B]n . That is, H is continuous in φ. We wish to prove compactness of the operator H. In order to do so, we shall employ the Arzel`a–Ascoli theorem. We need to show that the set W = {Hφn (t) : φn ∈ X } ⊂ Rn is relatively compact and the sequence {Hφn } n∈N is equicontinuous. It follows from the compactness of [−B, B]n in Rn and the continuity of the functions f , g : Rn → Rn that there exists a positive constant C such that | A (s, x (s))| ≤ C for all x ∈ X and s ∈ [0, T]T .

(7.4.11)

Let t ∈ [0, T]T , φn ∈ X for all n ∈ N. Then ∫ t | A (s, φn (s))| Δs ≤ E1 E2TC = N. |Hφn (t)| ≤ E1 E2

(7.4.12)

t−T

Hence, {Hφn} n∈N is uniformly bounded. Finally, we get by (7.4.10) and (7.4.11) and  (7.4.12) that H Δ φn (t) is bounded for φn ∈ X. Thus, {Hφn } n∈N is equicontinuous. Consequently, the Arzel`a–Ascoli theorem yields compactness of the operator H. Now we are in a position that we can state and prove the following existence result. Theorem 7.4.1. If there exists a positive number B such that for any T-periodic solution of (7.4.4), 0 < λ < 1 satisfies |x| 0 ≤ B, then the nonlinear system (7.4.1) has a solution in PT .

294

7 Periodic Solutions: The Natural Setup

Proof. Let H be defined by (7.4.8). Then, by Lemma 7.4.3, T-periodic operator H is continuous and compact. The hypothesis |x| ≤ B rules out part (ii) of Schaefer’s fixed point theorem (Theorem 1.1.16) and thus x = λH x has a solution for λ = 1, which solves Eq. (7.4.1). This concludes the proof.

7.4.2 Applications to Scalar Equations In this subsection, we are concerned with the scalar integro-dynamic equation ∫ t Δ x (t) = ax (t) + f (x (t)) + K (t, s) g (x (s)) Δs + p (t) , t ∈ T (7.4.13) −∞

and the corresponding family of equations xλΔ (t) = [λ (α + a) − α] xλ (t) + λ f (xλ (t)) + λ

∫

t

−∞

K (t, s) g (x (s)) Δs + λp (t)

(7.4.14) for t ∈ T, where T is a periodic time scale with period P and 0 < λ < 1. Besides A.1 and A.2, we also suppose that there exist a function J : R+ → R+ and a positive constant Q such that ∫ ∞ J (u) Δu = Q (7.4.15) |K (t, t − u)| ≤ J (u) with 0

∫

and supt ∈T

∫

t

−∞

∞

|K (u, s)| ΔuΔs < ∞.

(7.4.16)

t

Note that (7.4.15) implies ∫ supt ∈T

∞

|K (u, t)| Δu ≤ Q.

t

We handle Eq. (7.4.13) for the cases a > 0, a < 0, and a = 0, and provide sufficient conditions guaranteeing the existence of periodic solutions. The main steps of the existence proofs can be summarized as follows. First, we differentiate the Lyapunov functionals V1 and V2 , where ∫ t ∫ ∞ V1 (t, xλ (.)) = |xλ (t)| + λ (7.4.17) |K (u, s)| |g (xλ (s))| ΔuΔs −∞

t

and ∫ V2 (t, xλ (.)) = |xλ (t)| − λ

t

−∞

∫ t

∞

|K (u, s)| |g (xλ (s))| ΔuΔs.

(7.4.18)

7.4 Integro-Dynamic Equation

295

Then to obtain the a priori bounds for solutions of Eq. (7.4.14), we use periodicity of the functionals V1 and V2 and time scale analogue of Sobolev’s inequality. Given the a priori bounds and Theorem 7.4.1, the proofs are completed. Different than the analysis of the integro-differential equation for the cases a > 0, a < 0, and a = 0 in [67], finding the estimates for Δ-derivatives of V1 and V2 makes our analysis quite challenging. Next we prove an analogue of Sobolev’s inequality on an arbitrary (not necessarily periodic) time scale. The inequality is essential when proving the existence of a priori bound on all possible periodic solutions of (7.4.14). Corollary 7.4.1 (Sobolev’s Inequality). If x ∈ Cr d , then   |x| 1 + σ(T)  x Δ 1 ≥ T |x| 0 where

∫ |x| 1 =

T

|x (s)| Δs.

0

Proof. From 1.2.11, we have |x (t)| Δ ≤ |x (t)| Δ for all t ∈ T. Integration by parts (Lemma 1.1.2) yields the following: ∫ t ∫ t ∫ t  Δ   x (s) Δs. σ(s) |x (s)| Δ Δs ≥ t |x (t)| − σ(t) |x (s)| Δs = t |x (t)| − 0

0

0

Taking supremum over the interval [0, T]T , we obtain the desired inequality. In the following theorem, we let φ (x) = − f (x) and show that there is a priori bound for the solution xλ of (7.4.14) for 0 < λ < 1. Then we use Theorem 7.4.1 to infer the existence of periodic solutions of Eq. (7.4.13) whenever a < 0. Theorem 7.4.2. Let a < 0 and x σ (t)φ (x(t)) > 0 for all x  0 and t ∈ T. Suppose that there exist positive constants η, β, and M such that |1 + μ(t)(a − η)| ≤ 1

(7.4.19)

− |φ (x(t))| + Q |g (x(t))| ≤ −β |g(x(t))| + M

(7.4.20)

and for all t ∈ T and x ∈ PT , where Q is given by (7.4.15). Then Eq. (7.4.13) has a solution in PT . Proof. Let xλ ∈ PT be a solution of (7.4.14). Setting α = −a the Eq. (7.4.14) can be rewritten as ∫ t K (t, s) g (xλ (s)) Δs + λp (t) . xλΔ (t) = axλ (t) − λφ (xλ (t)) + λ (7.4.21) −∞

296

7 Periodic Solutions: The Natural Setup

If t ∈ T−x , then t is right scattered, so x x σ < 0 implies

xλσ (t) | xλσ (t) |

= − |xxλλ (t) (t) | , and hence,

xφ (x) < 0 since x σ φ (x) > 0. On the other hand, the condition |1 + μ(t)(a − η)| ≤ 1 2 − a ≤ −η for all t ∈ T−x . It follows from Lemma 1.2.2 and guarantees that − μ(t) (7.4.21) that |xλ (t)| Δ = −

2 xλ (t) Δ x (t) |xλ (t)| − μ(t) |xλ (t)| λ

2 ≤ (− − a) |xλ (t)| − λ |φ (xλ (t))| + λ μ(t)

∫t |K (t, s)| |g (xλ (s))| Δs

−∞

+ |p| 0

(7.4.22) ∫t

≤ −η |xλ (t)| − λ |φ (xλ (t))| + λ

|K (t, s)| |g (xλ (s))| Δs + |p| 0 (7.4.23)

−∞

for all t ∈ T−x . Similarly, if t ∈ T+x , then xφ(x) > 0, and from Lemma 1.2.2 and (7.4.21) we find |xλ (t)| Δ =

xλ (t) Δ x (t) |xλ (t)| λ

∫

≤ a |xλ (t)| − λ |φ (xλ (t))| + λ

t

|K (t, s)| |g (xλ (s))| Δs + |p| 0 .

−∞

(7.4.24)

Combining (7.4.23) and (7.4.24) we get that |xλ (t)| Δ ≤ −η∗ |xλ (t)| − λ |φ (xλ (t))| + λ

∫

t

−∞

|K (t, s)| |g (xλ (s))| Δs + |p| 0 .

for all t ∈ T, where η∗ > 0 is a constant given by η∗ = min {η, −a} .

(7.4.25) (7.4.26)

Define the Lyapunov functional V1 by (7.4.17). It is obvious from (7.4.2) and (ii) in Theorem 1.1.9 that V1 (t + T, xλ (.)) = V1 (t, xλ (.)) , i.e., V1 is periodic for xλ ∈ PT . We get by (7.4.17) and (7.4.25) that ∫ ∞ V1Δ (t, xλ (.)) = |xλ (t)| Δ + λ |g (xλ (t))| |K (u, t)| Δu ∫ −λ

σ(t)

t

−∞ ∗

|K (t, s)| |g (xλ (s))| Δs.

≤ −η |xλ (t)| − λ |φ (xλ (t))| + λQ |g (xλ (t))| + |p| 0 ≤ −η∗ |xλ (t)| − λβ |g (xλ (t))| + L ≤ −η∗ |xλ (t)| + L (7.4.27)

7.4 Integro-Dynamic Equation

297

for all t ∈ T, where L = M + |p| 0 . Hence, we obtain ∫ 0= 0

T

V1Δ (s, xλ (.)) Δs ≤ −η∗

∫

T

|xλ (s)| Δs + T L,

0

which gives a priori bound R = T L/η∗ for |xλ | 1 . The second inequality in (7.4.27) ∫T yields the a priori bound L1 = LT/β for λ 0 |g (xλ (t))| Δt. Similarly, from the first inequality in (7.4.27), we find the a priori bound L2 = QL1 + T |p| 0 for the integral ∫T λ 0 |φ (xλ (t))| Δt. Thus, using (7.4.15) and (7.4.21) we arrive at ∫ t

t+T

 Δ   x (s) Δs ≤ −aR + L2 + λ λ

∫

T

∫

T

∫

−∞ ∞

0

∫

≤ −aR + L2 + λ 0

+ T |p| 0

∫

≤ −aR + L2 + λ 0

u

0 ∞

∫

≤ −aR + L2 + λL1 0

|K (u, s)| |g (xλ (s))| ΔsΔu + T |p| 0 |K (u, u − r)| |g (xλ (u − r))| ΔrΔu ∫

J (r) Δr ∞

0

T

|g (xλ (u − r))| Δu + T |p| 0

J (r) Δr + T |p| 0 = R∗ .

Here, [49, Remark 2.17] allows us to interchange the order of integration. Hence, Corollary 7.4.1 provides a priori bound B = T1 (R + σ (T) R∗ ) for |x| 0 . Consequently, from Theorem 7.4.1, we deduce existence of a periodic solution of Eq. (7.4.13) in PT . Example 7.4.1. Let T = {t = k − q m : k ∈ Z, m ∈ N0 }, where 0 < q < 1. Evidently, μ(t) = q m (1 − q) for t = k − q m . This shows that 0 < μ(t) < 1 − q for all t ∈ T. Choose η = (1 − q)−1 . One may easily verify that the condition (7.4.19) holds if (q−1)−1 < a < 0. If we can guarantee that the functions f and g satisfy the conditions of Theorem 7.4.2, and the kernel K obeys the assumptions given in Section 7.4, then the existence of a solution x ∈ PT of Eq. (7.4.13) follows. Note that the Δ-derivative of a function ϕ ∈ C 1 (T, R) defined on this time scale is given by ϕΔ (t) =

ϕ(k − q m+1 ) − ϕ(k − q m ) for t = k − q m . q m (1 − q)

The following theorem covers the case a > 0. Theorem 7.4.3. Let a > 0 and x f (x) > 0 for x  0. Suppose that there exist positive constants, β and M, such that | f (x)| − Q |g (x)| ≥ β |g(x)| − M, where Q is given by (7.4.15). Then Eq. (7.4.13) has a solution in PT .

298

7 Periodic Solutions: The Natural Setup

Proof. The proof is similar to that of Theorem 7.4.2. We only outline the details. Set α = −a and rewrite Eq. (7.4.14) as in (7.4.21). Lemma 1.2.2 and (1.2.11) imply xλ Δ x for all t ∈ T. (7.4.28) |xλ | Δ ≥ |xλ | λ Then from (7.4.18), (7.4.21), and (7.4.28) we have ∫ V2Δ (t, xλ (.)) = |xλ (t)| Δ − λ |g (xλ (t))| ∫ +λ

∞

σ(t)

t

−∞

|K (u, t)| Δu

|K (t, s)| |g (xλ (s))| Δs

≥ a |xλ (t)| + λ | f (xλ (t))| − λQ |g (xλ (t))| − |p| 0 ≥ a |xλ (t)| + λβ |g (xλ (t))| − M − |p| 0 ≥ a |xλ (t)| − L,

(7.4.29)

where L = M+|p| 0 . That is, there exists an R = T L/a > 0 such that |xλ | 1 ≤ R. There∫T ∫T fore, the a priori bounds for the integrals λ 0 |g (xλ (t))| Δt and λ 0 | f (xλ (t))| Δt can be obtained from the second and first inequalities in (7.4.29), respectively. The proof is completed in a similar way to that of Theorem 7.4.2. In the case a = 0, existence of periodic solutions is guaranteed by the next theorem. Theorem 7.4.4. Assume that a = 0 and that all remaining hypothesis of Theorem 7.4.3. In addition, suppose that there exists θ > 0 such that |g(x)| ≥ θg(|x|) ≥ 0, g(|x|) is convex downward, and g(u) → ∞ as u → ∞. Then Eq. (7.4.13) has a solution in PT . Proof. Let a = 0. Set α = −1 and rewrite (7.4.14) as ∫ t K (t, s) g (x (s)) Δs + λp (t) . xλΔ (t) = (1 − λ) xλ (t) + λ f (xλ (t)) + λ −∞

Consider the Lyapunov functional (7.4.18) to obtain V2Δ (t, xλ (.)) ≥ (1 − λ) |xλ (t)| + λ | f (xλ (t))| − λQ |g (xλ (t))| − |p| 0 ≥ (1 − λ) |xλ (t)| + λβ |g (xλ (t))| − M − |p| 0 . (7.4.30) If 0 < λ ≤ 12 , then from (7.4.30), V2Δ (t, xλ (.)) ≥ 12 |xλ (t)| − L, and so there exists R1 > 0 such that |xλ | 1 ≤ R1 . Let 12 < λ < 1, then we get by (7.4.30) that 1 1 β |g (xλ (t))| − L ≥ βθg (|xλ (t)|) − L. 2 2 Therefore, from Jensen’s inequality (see Theorem 1.1.11), we find V2Δ (t, xλ (.)) ≥

∫ 0= 0

T

V2Δ (t, xλ (t)) Δt

∫ T 1 g (|xλ (t)|) − LT ≥ βθ 2 0  ∫ T  1 1 ≥ βθT g |xλ (t)| Δt − LT . 2 T 0

(7.4.31)

7.4 Integro-Dynamic Equation

299

Since g(u) → ∞ as u → ∞, it follows from (7.4.31) that there exists R2 > 0 such that |xλ | 1 ≤ R2 . Let R = max {R1, R2 }, so |xλ | 1 ≤ R for all 0 < λ < 1. A priori ∫T ∫T bounds for the integrals λ 0 |g (xλ (t))| Δs and λ 0 | f (xλ (t))| Δs follow from the second and first inequalities in (7.4.30), respectively. The remaining part of the proof is same as that of Theorem 7.4.2. Example 7.4.2. In the special case T = R, Eq. (7.4.13) becomes the integro-differential equation ∫ t  x (t) = ax(t) + f (x(t)) + K(t, s)g(x(s))ds + p(t), t ∈ R −∞

for which the existence of periodic solutions has been investigated in [67] under the following conditions: (i) a < 0 and − | f (x)| + Q |g (x)| ≤ −β |g(x)| + M for some β > 0 and M > 0, (ii) a ≥ 0 and | f (x)| − Q |g (x)| ≥ β |g(x)| − M for some β > 0 and M > 0. Since the real line R contains no right scattered points, i.e., μ(t) = 0 for all t ∈ R, we can rule out the condition “|1 + μ (a − η)| ≤ 1” in Theorem 7.4.2. Hence, results of [67] can be obtained from Theorems 7.4.2 to 7.4.4 in the particular case T = R. Henceforth, we provide alternative conditions that guarantee the existence of periodic solutions. Theorem 7.4.5. Suppose that a > 0 and there exists a positive constant β such that β < a and (7.4.32) | f (x)| + Q |g (x)| ≤ β |x| , hold. Then Eq. (7.4.13) has a solution in PT . Proof. Set α = −a and rewrite (7.4.14) as in (7.4.21). As we did in (7.4.29) we obtain V2Δ (t, xλ (.)) ≥ a |xλ (t)| − λ | f (xλ (t))| − λQ |g (xλ (t))| − |p| 0 ≥ (a − β) |xλ (t)| − |p| 0 . Hence, we can find a priori bound for |xλ | 1 . On the other hand, a priori bounds ∫T ∫T for the integrals λ 0 |g (xλ (t))| Δt and λ 0 | f (xλ (t))| Δt can be easily obtained from the condition (7.4.32). The proof is completed as it is done in the proof of Theorem 7.4.2. Theorem 7.4.6. Assume that a < 0. Also, we assume that there exist positive constants β < −a and η > β such that (7.4.32) and |1 + μ(t)(a − η)| ≤ 1 hold for all t ∈ T. Then Eq. (7.4.13) has a solution in PT .

(7.4.33)

300

7 Periodic Solutions: The Natural Setup

Proof. We will proceed with a proof similar to that of Theorem 7.4.2. Set α = −a in (7.4.14) and rewrite it in the form of (7.4.21). Applying similar arguments in (7.4.23) and (7.4.24) we get that ∫ t Δ |K (t, s)| |g (xλ (s))| Δs + λ + |p| 0 , |xλ (t)| ≤ ζ(t) |xλ (t)| + λ | f (xλ (t))| + −∞

where

 ζ(t) =

−η for t ∈ T−x . a for t ∈ T+x

Taking the Lyapunov functional (7.4.17) and the condition (7.4.33) into account we find ∫ ∞ V1Δ (t, xλ (.)) = |xλ (t)| Δ + λ |g (xλ (t))| |K (u, t)| Δu ∫ −λ

σ(t)

t

−∞

|K (t, s)| |g (xλ (s))| Δs.

≤ ζ(t) |xλ (t)| + λ | f (xλ (t))| + λQ |g (xλ (t))| + |p| 0 ≤ ζ ∗ |xλ (t)| + |p| 0 , where ζ ∗ < 0 is given by

ζ ∗ = max {ζ(t) + β} . t ∈T

∫T This gives a priori bound for |xλ | 1 . A priori bounds for the integrals λ 0 |g (xλ (t))| Δt ∫T and λ 0 | f (xλ (t))| Δt can be easily obtained from the condition (7.4.32). The proof is completed in a similar way to that of Theorem 7.4.2. Theorem 7.4.7. Let the periodic time scale T consists only of right scattered points. Assume that a < 0. We also assume that there exist positive constants β and η > β such that (7.4.32) and |1 + μ(t)a| ≥ 1 + ημ(t) (7.4.34) hold for all t ∈ T. Then Eq. (7.4.13) has a solution in PT . Proof. Letting α = −a we rewrite Eq. (7.4.14) as in (7.4.21). From Theorem 1.1.1 (ii), (7.4.21), and (7.4.34) we arrive at  σ   x (t) − |xλ (t)| Δ |xλ (t)| = λ μ(t)    xλ (t) + μ(t)x Δ  − |xλ (t)| = μ(t)   ∫ t |1 + aμ(t)| − 1 ≥ |K (t, s)| |g (xλ (s))| Δs |xλ (t)| − λ | f (xλ (t))| − λ μ(t) −∞ − |p| 0

7.4 Integro-Dynamic Equation

301

∫ ≥ η |xλ (t)| − λ | f (xλ (t))| − λ

t

−∞

|K (t, s)| |g (xλ (s))| Δs − |p| 0 .

It follows from (7.4.32) that V2Δ (t, xλ (.)) ≥ η |xλ (t)| − λ | f (xλ (t))| − λQ |g (xλ (t))| − |p| 0 ≥ (η − β) |xλ (t)| − |p| 0 . The proof is completed as in Theorem 7.4.2. Example 7.4.3. Let T = Z. Then Eq. (7.4.13) turns into the familiar Volterra difference equation x(t + 1) = bx(t) + f (x(t)) +

t−1 

K(t, j)g(x( j)) + p(t),

(7.4.35)

j=−∞

where b = a + 1. In [140], the author assumed that the terms f , g, K, and p obey the same conditions as that of Eq. (7.4.13) and proved the existence of periodic solutions of Eq. (7.4.35) in the following cases: (i) |b| < 1 and there exists a positive constant β such that | f (x)| + Q |g (x)| ≤ β |x| , and |b| − 1 < −β, (ii) |b| > 1 and there exists a positive constant β such that | f (x)| + Q |g (x)| ≤ β |x| , and |b| − 1 > β. Theorems 7.4.5, 7.4.6, and 7.4.7 imply these results in the particular case when T = Z. Remark 7.4.1. The study [140] does not prove the existence of periodic solutions of Eq. (7.4.35) in the case b = 1. Theorem 7.4.4 not only helps us to get over this constraint but also gives more general result since there are many periodic time scales other than R and Z. On the other hand, Theorems 7.4.2 and 7.4.3 assume weaker conditions than the conditions supposed to hold in Theorems 7.4.5 and 7.4.6.

302

7 Periodic Solutions: The Natural Setup

7.5 Schaefer Theorem and Infinite Delay Volterra Integro-Dynamic Equations In this section, using Schaefer’s fixed point theorem (i.e., Theorem 1.1.16), we analyze the existence of periodic solutions of nonlinear system of integro-dynamic equations on periodic time scales. We provide an example in which the a priori bound is established by the aid of constructing a suitable Lyapunov functional and Sobolev’s inequality on time scales which was proved in [10]. Contents of this section are new. Let T be a periodic time scale with period P including 0. Let T > 0 be fixed and if T  R, T = nP for some n ∈ N. This section focuses on infinite delay nonlinear integro-dynamic equations of the form ∫ t K (t − s) x (s) Δs + p (t) , (7.5.1) x Δ (t) = Dx (t) + −∞

where x Δ (t) denotes the Δ-derivative of x (t), D is an n × n constant matrix with real entries, p : T → Rn , and K is an n × n matrix function with real valued entries. After developing the existence of a periodic solution of (7.5.1), we make our results directly available for showing the existence of periodic solutions of the system dynamic equation on time scale x Δ (t) = Dx(t) + p(t), t ∈ T.

(7.5.2)

In order to make use of Schaefer’s fixed point theorem, we first show or assume that all T-periodic solutions of the homotopy equation ∫ t ( ' x Δ (t) = λ Dx (t) + K (t − s) x (s) Δs + p (t) , (7.5.3) −∞

for λ ∈ (0, 1) have a priori bound and then conclude that (7.5.1) has a T-periodic solution. The a priori bound on all T-periodic solutions of (7.5.3) will be established by means of Lyapunov functional. In the previous chapters and in [26] the authors proved general theorems regarding the boundedness of solutions of the functional dynamical system on time scale x Δ (t) = G(t, x(s); 0 ≤ s ≤ t) := G(t, x(.)) and then as an application they constructed Lyapunov functionals and showed boundedness on all solutions of the vector Volterra integro-dynamic equation ∫ t Δ x = Ax(t) + C(t, s)x(s)Δs + g(t), 0

where t ≥ 0, x(t) = φ(t) for 0 ≤ t ≤ t0, φ(t) is a given bounded continuous initial k × 1 vector function.

7.5 Schaefer Theorem and Infinite Delay Volterra Integro-Dynamic Equations

303

7.5.1 Existence of Periodic Solutions Define PT = {ϕ ∈ C(T, Rk ) : ϕ(t + T) = ϕ(t)}, where C(T, Rk ) is the space of all vector valued continuous functions on T. Then PT is a Banach space with the norm ||x|| = sup |x(t)|. t ∈[0,T ]T

If D is a matrix, then |D| means the sum of the absolute values of the elements. Definition 7.5.1. The vector mean η(x) of a vector x ∈ PT is defined by 1 η(x) := T We define * x by

∫

T

x(s)Δs. 0

* x (t) := x(t) − η(x).

Note that * x ∈ PT0 , where PT0 is the subspace of PT whose elements have mean vector ∫T zero. That is, η(φ) = T1 0 φ(s)Δs = 0, for φ ∈ PT0 . We shall assume throughout this section that the following assumptions hold: A 1. ∫p ∈ PT0 . ∞ A 2. 0 |K(u)|Δu < ∞.

∫∞ Note that if p ∈ PT and p  PT0 , (i.e., η(p) = β  0) and if D + 0 K(u)Δu is ∫∞ nonsingular, then the translation y(t) = x(t) + α, where α = [D + 0 K(u)Δu]−1 transforms (7.5.1) into an equation with forcing function having vector mean zero. We note that if ϕ ∈ PT , then ∫ 1 T ϕ(s)Δs|| ≤ 2||ϕ||. || ϕ *|| = ||ϕ − T 0 ∫t Also, if γ(t) = −∞ K(t − s)* ϕ(s)Δs, then ∫

T

∫ γ(t)Δt =

0

T

∫

T

∫

0

∫ =

0

∫ =

t

−∞ ∞

K(t − s)* ϕ(s)ΔsΔt

K(u)* ϕ(t − u)ΔuΔt, (u = t − s) ∫ ∞ T K(u) ϕ *(t − u)ΔtΔu = 0, 0

0

(7.5.4)

0

since the inner integral is 0 for each fixed u. Given a vector ϕ(t) ∈ PT , we define the mapping H : PT → PT by ∫ t (Hϕ)(t) = ((L ϕ *)(s) + p(s))Δs, 0

(7.5.5)

304

7 Periodic Solutions: The Natural Setup

where,

∫ (L ϕ *)(t) = D ϕ *(t) +

t

−∞

K(t − s)* ϕ(s)Δs.

Lemma 7.5.1. Assume A1 hold. If H is defined by (7.5.5), then H : PT → PT . ∫T Proof. Let ϕ ∈ PT and γ(t) = −∞ K(t − s)* ϕ(s)Δs. Then, ∫ (Hϕ)(t + T) − (Hϕ)(t) =

t+T



 Dϕ *(s) + γ(s) + p(s) Δs

0

∫ −

t



 Dϕ *(s) + γ(s) + p(s) Δs

0

∫ =

t+T



 Dϕ *(s) + γ(s) + p(s) Δs = 0,

t

by the fact that p(t) ∈ PT0 and (7.5.4). The following result concerns the compactness of the operator H. Lemma 7.5.2. Assume A1, A2 hold. The operator H : PT → PT , as defined by (7.5.5), is continuous and compact. Proof. Define the set X := {x ∈ PT : ||x|| ≤ B}. Obviously X is closed and bounded in PT . For φ1, φ2 ∈ X, we obtain |(Hϕ1 )(t) − (Hϕ2 )(t)| ≤ |D|T | ϕ *1 (t) − ϕ *2 (t)|

∫ ∫ ∞  t  *2 (t)| + |K(u)|  ϕ *1 (t) − ϕ *2 (t)Δu ≤ |D|T | ϕ *1 (t) − ϕ 0 0 ∫ ∞   ≤ 2 |D|T + T |K(u)| ||ϕ1 − ϕ2 ||. 0

This shows that H is continuous in ϕ. . We wish to prove compactness of the operator H. In order to do so, we shall employ the Arzel`a–Ascoli theorem. We need to show that the set W = {Hφn (t) : φn ∈ X } ⊂ Rn is relatively compact and the sequence {Hφn } n∈N is equicontinuous. Let t ∈ [0, T]T , φn ∈ X for all n ∈ N. Then ∫ ∞   |K(u)|Δu + T ||p||. (7.5.6) |Hφn (t)| ≤ 2||ϕ|| T |D| + T 0

Hence, {Hφn } n∈N is uniformly bounded. Finally, (H Δ φn ) (t) = (Hφn ) (t) and hence by (7.5.6) we have |H Δ φn (t) | is bounded for φn ∈ X. Thus, {Hφn } n∈N is equicontinuous. Consequently, the Arzel`a–Ascoli theorem yields compactness of the operator H. Now we are in a position that we can state and prove our main result.

7.5 Schaefer Theorem and Infinite Delay Volterra Integro-Dynamic Equations

305

Theorem 7.5.1. Assume the hypothesis of Lemmas 7.5.1 and 7.5.2 hold. If there exists a positive number B such that for any T-periodic solution of (7.5.3), 0 < λ < 1 satisfies ||x|| ≤ B, then Eq. (7.5.1) has a solution in PT . Proof. Let H be defined by (7.5.5). By Lemma 7.5.1, the operator H is T-periodic and by Lemma 7.5.2, H is continuous and compact. The hypothesis |x| ≤ B rules out part (ii) of Schaefer’s fixed point theorem and thus x = λH x has a solution for λ = 1, which solves Eq. (7.5.1). This concludes the proof.

7.5.2 Application to Infinite Delay System In the next theorem we establish sufficient conditions that guarantee the existence of a T-periodic solutions vector Volterra integro-dynamic equation with infinite delay ∫ t x Δ (t) = Dx (t) + K (t − s) x (s) Δs + p (t) (7.5.7) −∞

where D and K are k × k matrix p(t), x are k ×1 vector functions where p(t +T) = p(t) for all t ∈ [0, T]T . We accomplish this by showing the homotopy equation ∫ t $ # Δ (7.5.8) x (t) = λ Dx (t) + K (t − s) x (s) Δs + p (t) , λ ∈ (0, 1] −∞

has a priori bound on all its T-periodic solutions and then appeal to Theorem 7.5.1 For what to follow we write p and x for p(t) and x(t), respectively. Theorem 7.5.2. Suppose K T (t − s) = K(t − s). Let I be the k × k identity matrix. Assume there exist positive constants ξ, β1 , and k × k positive definite constant symmetric matrix B such that # $ DT B + BD + μ(t)DT BD ≤ −ξ I, (7.5.9) ∫ − ξ + μ(t)|D Bp| +

t

T

+

∫−∞ ∞

∫ |B||K(t − s)|Δs + μ(t)

σ(t)−s

t

−∞

|DT B||K(t − s)|Δs

|K(u)|Δu ≤ −β1,

(7.5.10)

and 

∫

− 1 + |B| + μ(t) 1 + |D B| + |B|

t

T

Then (7.5.7) has a T-periodic solution.

−∞

 K(t − s)Δs ≤ 0.

(7.5.11)

306

7 Periodic Solutions: The Natural Setup

Proof. Let the matrix B be defined by (7.5.9) and define ∫ t ∫ ∞ T V(t, x) = x Bx + λ |K(u)|Δux 2 (s)Δs. −∞

(7.5.12)

t−s

Here xT x = x 2 = (x12 + x22 + · · · + xk2 ). Using the product rule, we have along the solutions of (7.5.8) that ∫ t V Δ = (x Δ )T Bx + (x σ )T Bx Δ − λ |K(t − s)|x 2 (s)Δs −∞ ∫ ∞ |K(u)|Δux 2 +λ σ(t)−s

Δ T

Δ T

∫

Δ

= (x ) Bx + (x + μ(t)x ) Bx − λ ∫ ∞ |K(u)|Δux 2 +λ

t

−∞

|K(t − s)|x 2 (s)Δs

σ(t)−s

Δ T

Δ

Δ T

Δ

= (x ) Bx + x Bx + μ(t)(x ) Bx − λ ∫ ∞ |K(u)|Δux 2 +λ σ(t)−s

T

∫

∫

t

−∞

|K(t − s)|x 2 (s)Δs

$T K (t − s) x (s) Δs + p (t) Bx −∞ ∫ t $ # T + x Bλ Dx (t) + K (t − s) x (s) Δs + p (t) −∞ ∫ t $T # + μ(t)λ Dx (t) + K (t − s) x (s) Δs + p (t) −∞ ∫ t $ # × B Dx (t) + K (t − s) x (s) Δs + p (t) −∞ ∫ t ∫ ∞ −λ |K(t − s)|x 2 (s)Δs + λ |K(u)|Δux 2 . #

= λ Dx (t) +

t

−∞

σ(t)−s

(7.5.13)

By noting that the right-hand side of (7.5.13) is scalar and by recalling that B is a symmetric matrix, expression (7.5.13) simplifies to ∫ t   V Δ = λxT AT B + BA + μ(t)AT BA x + 2λ|xT ||Bp| + 2λ xT BK(t − s)x(s)Δs −∞ ∫ t ∫ t # + λμ(t) 2xT DT Bp + 2pT B K(t − s)x(s)Δs + 2xT DT B K(t − s)x(s)Δs −∞ −∞ ∫ t ∫ t $ + xT (s)K(t − s)Δs B K(t − s)x(s)Δs + pT B p −∞ −∞ ∫ t ∫ ∞ 2 −λ |K(t − s)|x (s)Δs + λ |K(u)|Δux 2 −∞

σ(t)−s

7.5 Schaefer Theorem and Infinite Delay Volterra Integro-Dynamic Equations

307

∫ t ≤ −λξ x 2 + 2λ|xT ||Bp| + 2λ |xT ||B||K(t − s)||x(s)|Δs −∞ ∫ t #∫ t + λμ(t) |K(t − s)|2|pT B||x(s)|Δs + 2 |xT ||DT B||K(t − s)||x(s)|Δs −∞ −∞ ∫ t ∫ t $ T + x (s)K(t − s) B Δs K(t − s)x(s)Δs + |pT Bp| + 2|xT ||DT Bp| −∞ −∞ ∫ t ∫ ∞ 2 −λ |K(t − s)|x (s)Δs + λ |K(u)|Δux 2 . (7.5.14) −∞

σ(t)−s

Next, we perform some calculations to simplify inequality (7.5.14). Since 2λ|xT ||Bp| = 2λ|x||Bp|, 2λμ(t)|xT ||DT Bp| = 2λμ(t)|xT ||DT Bp| 1/2 |DT Bp| 1/2 ∫ 2λ

−∞

∫ λμ(t)

t

t

≤ λμ(t)(x 2 |DT Bp| + |DT Bp|), ∫ t T |x ||B||K(t − s)||x(s)|Δs ≤ λ |B||K(t − s)|(x 2 + x 2 (s))Δs, −∞

∫

|K(t − s)|2|p B||x(s)|Δs ≤ λμ(t)

t

T

−∞

−∞

|K(t − s)|(|pT B| 2 + x 2 (s))Δs,

and ∫ 2λμ(t)

t

∫ |x ||D B||K(t − s)||x(s)|Δs ≤ λμ(t) T

−∞

t

T

−∞

|DT B||K(t − s)|(x 2 + x 2 (s))Δs,

we have ∫

∫

t

t

x (s)K(t − s)Δs B K(t − s)x(s)Δs −∞ −∞ ∫ t ∫ t T ≤ λμ(t)|B| | x (s)K(t − s)Δs|| K(t − s)x(s)Δs| −∞ −∞ ∫ ∫ t  t 23 23 2 + λμ(t)|B| 2 ≤ λμ(t)|B| xT (s)K(t − s)Δs K(t − s)x(s)Δs −∞ −∞ ∫ t 2 = λμ(t)|B| K(t − s)x(s)Δs −∞ ∫ t 2 1 1 = λμ(t)|B| |K(t − s)| 2 |K(t − s)| 2 |x(s)|Δs −∞ ∫ t ∫ t ≤ λμ(t)|B| |K(t − s)|Δs |K(t − s)|x 2 (s)Δs. λμ(t)

T

−∞

−∞

Substituting the last inequality into (7.5.14) and making use of (7.5.9), (7.5.10), and (7.5.11) we get that

308

7 Periodic Solutions: The Natural Setup

∫ t ' V Δ (t, x) = λ − ξ + μ(t)|DT Bp| + |B||K(t − s)|Δs −∞ ∫ t ∫ ∞ ( + μ(t) |DT B||K(t − s)|Δs + |K(u)|Δu x 2 + 2λ|x||Bp| −∞

'

σ(t)−s

+ λ − 1 + |B| + μ(t)(1 + |D B| ∫ t ∫ ( t + |B| K(t − s)Δs) K(t − s)x(s)Δs T

−∞

−∞

≤ λ(−β1 x 2 + γ|x| + M) ≤ λ(−β2 x 2 − γ1 |x| + Q), for appropriate constants β2, γ1, and Q, ∫ t where γ = 2|Bp|, and M = μ(t)|DT Bp| + μ(t)|pT B| 2 |K(t − s)|Δs. −∞

The Sobolev‘s Inequality (Corollary 7.4.1) enables us to find the a priori bound on all T-periodic solutions. Let V be given by (7.5.12) and let xλ ∈ PT . Then V (t + T, xλ (.)) = V (t, xλ (.)) , i.e., V is periodic for xλ ∈ PT . Thus an integration of V Δ from 0 to T, yields ∫ 0= 0

T

V Δ (s, xλ (.)) Δs ≤ −λγ1

Or, for λ ∈ (0, 1), then

∫

T

|x(s)|Δs ≤

0

∫ 0

T

|xλ (s)| Δs + λMT .

TM := R1 . γ1

An integration of (7.5.8) from 0 to T yields ∫ 0

T

∫ T ∫ T∫ u # $ |x Δ (s)Δs ≤ = λ |D| |x(s)|Δs + K(u − s)x(s)ΔsΔs + T ||p|| 0 0 −∞ ∫ ∞ ∫ T ∫ T $ # |x(s)|Δs + |K(u)|Δu |x(s)|Δs + T ||p|| ≤ λ |D| 0 0 0 ∫ ∞ $∫ T # |K(u)|Δu |x(s)|Δs + λT ||p|| ≤ λ |D| + 0 0 ∫ ∞ # $TM ≤ |D| + |K(u)|Δu + λT ||p|| := R2 . γ1 0

Using Corollary 7.4.1, we obtain T ||x|| ≤

TM + σ(T)R2 = R1 + σ(T)R2 . γ1

Or, ||x|| ≤

1 R1 + σ(T)R2 ), T

7.6 Connection Between Boundedness and Periodicity

309

which is a priori bound on all T-periodic solutions of (7.5.8). This eliminates (ii) of Schaefer Theorem ([149] ) and therefore, (7.5.8) has a T-periodic solution for λ = 1. This completes the proof.

7.6 Connection Between Boundedness and Periodicity Let T be a time scale that is unbounded above and include 0. In this section, we use a direct mapping and then utilize a Krasnosel’ski˘ı fixed point theorem (see Theorem 1.1.17) to show the existence of solutions of the nonlinear functional neutral dynamic system with delay. Then, we consider a special form of the delay neutral equation and use the contraction mapping principle and show the existence of a uniform bound on all solutions and then conclude the existence of a unique periodic solution. Finally, the connection between the boundedness of solutions and the existence of periodic solutions leads us to the extension of Massera’s theorem to functional differential equations on general periodic time scales. Some of the results of this section are published in [101]. Thus, we examine the existence of solutions of the nonlinear functional neutral dynamical equation     (7.6.1) x Δ (t) = f t, x(t), x Δ (t − h(t)) + g t, x(t), x(t − h(t)) ; t, t − h(t) ∈ T, where f , g, and h are continuous, h : T → [0, h0 ]T for some positive constant h0 ∈ T, and f , g : T × R × R → R. In [99], the authors have considered a variation of (7.6.1). They studied the existence of periodic solutions of the neutral dynamical system   x Δ (t) = −a(t)x σ (t) + c(t)x Δ (t − h(t)) + g x(t), x(t − h(t)) , t, t − h(t) ∈ T, (7.6.2) where T is a periodic time scale and a, b, and h are periodic. In [100], the authors showed that   x Δ (t) = −a(t)x σ (t) + (Q(t, x(t), x(t − g(t))))Δ + G t, x(t), x(t − g(t)) , t ∈ T, (7.6.3) has a periodic solution. In both papers, the authors obtained the existence of a periodic solutions using a Krasnosel’ski˘ı fixed point theorem. Moreover, under a slightly more stringent inequality they showed that the periodic solution is unique using the contraction mapping principle. The authors also showed that the zero solution was asymptotically stable using the contraction mapping principle provided that Q(t, 0, 0) = G(t, 0, 0) = 0. In obtaining the existence of a periodic solution of (7.6.2) and (7.6.3) and the stability of the zero solution of (7.6.3), the authors inverted (7.6.2) and (7.6.3) and generated a variation of parameters-like formula. This formula was the sum of two mappings; one mapping was shown to be compact and the other was shown to be a contraction.

310

7 Periodic Solutions: The Natural Setup

We remark that the inversion of either (7.6.2) or (7.6.3) was made possible by the linear term −a(t)x σ (t), a luxury that (7.6.1) does not enjoy. A neutral differential equation is an equation where the immediate growth rate is affected by the past growth rate. This can be observed in the behavior of a stock price or the growth of a child. Also, in the case T = R, neutral equations arise in circuit theory (see [36]) and in the study of drug administration and populations (see [37, 108, 109]). The results of this section extend the results of [59] to time scales. Also, it is worth mentioning that the book [62] contains a wealth of information regarding stability and periodicity using fixed point theory.

7.6.1 Existence For emphasis, we consider     x Δ (t) = f t, x(t), x Δ (t − h(t)) + g t, x(t), x(t − h(t)) ; t, t − h(t) ∈ T,

(7.6.4)

where f , g, and h are continuous, h : T → [0, h0 ]T for some positive constant h0 ∈ T, f , g : T × R × R → R. Since our equation has a delay and the derivative enters nonlinearly on the right side, we must ask for an initial function η ∈ C 1 ([−h0, 0)T, R) whose Δ-derivative at 0 satisfies the expression     ηΔ (0) = f 0, η(0), ηΔ (−h(0)) + g 0, η(0), η(−h(0)) . (7.6.5) In the next definition, we state what we mean by a solution for (7.6.4) in terms of a given initial function. Definition 7.6.1. Let η ∈ C 1 ([−h0, 0)T, R) be a given bounded initial function that satisfies (7.6.5). We say x(t, η) is a solution of (7.6.4) on an interval [−h0, r)T , where r > 0, and r ∈ T, if x(t, η) = η(t) on [−h0, 0]T and satisfies (7.6.4) on [0, r)T . The above definition allows us to continue the solution on [r, r1 )T , for some r1 > r under the requirement of certain conditions. Let Cr d = Cr d ([−h0, r]T, R) be the space of all rd-continuous functions and define the set S by . S = ϕ ∈ Cr d : ϕ(t) = ηΔ (t) on [−h0, 0]T .   Then S, ν is a complete metric space where ν(ϕ, ψ) = ϕ − ψ = sup {|ϕ(t) − ψ(t)|}. t ∈[0,r]T

If ϕ ∈ S, we define  Φ(t) =

η(0) +

η(t), t ∈ [−h0, 0]T ∫t ϕ(s)Δs, t ∈ [0, r]T 0

7.6 Connection Between Boundedness and Periodicity

311

It clear that ΦΔ (t) = ϕ(t) on [0, r]T and Φ ∈ Cr d . Next, we suppose there is an a > 0 and define a subset S ∗ of S by S ∗ = {ϕ ∈ S : |ϕ(t) − ηΔ (0)| ≤ a}, such that there are an α > 0 and β < 1/2 so that for ϕ, ψ ∈ S ∗ we have         f t, Φ(t), ϕ(t − h(t)) − f t, Ψ(t), ψ(t − h(t))    ≤ α|Φ(t) − Ψ(t)| + βϕ(t − h(t)) − ψ(t − h(t)).

(7.6.6)

To be able to use Krasnosel’ski˘ı’s fixed point theorem (i.e., Theorem 1.1.17, we define the two required mappings as follows. For ϕ, ψ ∈ S ∗   (Aϕ)(t) = g t, Φ(t), ϕ(t − h(t)) and

  (Bϕ)(t) = f t, Φ(t), ϕ(t − h(t)) .

Then (7.6.6) implies that   (Bϕ)(t) − (Bψ)(t) ≤ α|Φ(t) − Ψ(t)| + β|ϕ(t − h(t)) − ψ(t − h(t))|.

(7.6.7)

It is obvious from the constructions of sets S and S ∗ that fixed points of S ∗ are solutions of (7.6.4). Theorem 7.6.1. If η satisfies (7.6.5) and (7.6.7), then there is an r > 0 such that the solution x(t, η) of (7.6.4) exists on [0, r)T . Proof. Let a be given as in the set S ∗ . Since the functions f and g are continuous in their respective arguments, then for ϕ ∈ S ∗ , we can find a positive constant L(a) depending on a, such that     (Bϕ)(t) + (Aϕ)(t) ≤ L(a), t ∈ T. Moreover, AS ∗ is equicontinuous. Now, for a fixed a > 0, we claim that there is an r > 0 such that and for all t ∈ [0, r]T ,       f t, Φ(t), ϕ(t − h(t)) − f 0, η(0), ηΔ (−h(0))  ≤ a/2. (7.6.8) To see this         f t, Φ(t), ϕ(t − h(t)) − f 0, η(0), ηΔ (−h(0))    ≤ α|Φ(t) − η(0)| + βϕ(t − h(t)) − ηΔ (−h(0))   ∫ t      ≤ α sup η(0) + ϕ(s) Δs − η(0) + βϕ(t − h(t)) − ηΔ (−h(0)) 0 t ∈[0,r]T    ≤ α t|ϕ(ξ)| + β ϕ(t − h(t)) − ηΔ (−h(0)), ξ ∈ (0, t)T,

312

7 Periodic Solutions: The Natural Setup

where |ϕ(ξ)| = sup |ϕ(ξ)|. ξ ∈(0,t)T

On one hand, suppose h(0) = 0. Then 0 ≤ t − h(t) ≤ t. As a consequence, for ϕ ∈ S ∗ , we have     βϕ(t − h(t)) − ηΔ (−h(0)) = βϕ(t − h(t)) − ηΔ (0) ≤ βa < a/2. On the other hand, if h(0) > 0, then there is an r1 > 0 such that t − h(t) ≤ 0 for t ∈ [0, r1 ]T . Hence, ϕ(t − h(t)) = ηΔ (t − h(t)) for t ∈ [0, r1 ]T . Since ηΔ is continuous, there is an r2 > 0 so that |ϕ(t − h(t)) − ηΔ (0)| < a, for t ∈ [0, r2 ]T . Thus, if we choose r ∗ ∈ (0, min{r1, r2 }), then we can set r = r ∗ , so that for t ∈ [0, r ∗ ]T , we have   ϕ(t − h(t)) − ηΔ (−h(0)) ≤ a. (7.6.9) Due to inequality (7.6.9) and since β < 1/2, we can find a positive number q < a/2 so that   βϕ(t − h(t)) − ηΔ (−h(0)) ≤ q. Moreover, since ϕ ∈ S ∗, we can choose an r, r ∈ (0, r ∗ )T so that for t ∈ [0, r]T , we have   α t|ϕ(ξ)| + βϕ(t − h(t)) − ηΔ (−h(0)) ≤ α t|ϕ(ξ)| + q ≤ a/2, which proves (7.6.8). This shows that A is compact. Finally, we claim that we can make r small enough so that for ϕ ∈ S ∗ , t ∈ [0, r]T , we have   g(t, Φ(t), Φ(t − h(t))) − g(0, η(0), η(−h(0))) ≤ a/2. (7.6.10) The proof of the claim follows from the fact g is uniformly continuous on any bounded set. For 0 ≤ t − h(t), we have        g t, Φ(t), Φ(t − h(t)) − g 0, η(0), η(−h(0))    ≤ |t − 0| + |Φ(t) − η(0)| + Φ(t − h(t)) − η(−h(0))   ∫ t−h(t)     ≤ t + t|ϕ(ξ)| + η(0) + ϕ(s) Δs − η(−h(0))   0 ≤ t + t|ϕ(ξ)| + (t − h(t)|ϕ(ξ)| ≤ r[1 + 2|ϕ(ξ)|], where |ϕ(ξ)| =

sup |ϕ(ξ)|, which can be made arbitrary small. Due to the con-

ξ ∈(0,r)T

tinuity of η, the case t − h(t) < 0 readily follows. This completes the proof of (7.6.10). We now go back to the proof of the theorem. It readily follows from (7.6.7) that for ϕ, ψ ∈ S ∗ , there is a λ < 1 so that the Bϕ − Bψ ≤ λϕ − ψ.

(7.6.11)

7.6 Connection Between Boundedness and Periodicity

313

Next we show that if ϕ, ψ ∈ S ∗ , then Aψ + Bϕ ∈ S ∗ . We remark that (Aψ)(0) + (Bψ)(0) = ηΔ (0), where ηΔ (0) is given by (7.6.5). As a consequence, we have by (7.6.8) and (7.6.10) |(Aψ)(t) + (Bϕ)(t) − ηΔ (0)| = |(Aψ)(t) − (Aψ)(0) + (Bϕ)(t) − (Bψ)(0)|      = (Aψ)(t) − g 0, η(0), η(−h(0)) + (Bϕ)(t) − f 0, η(0), ηΔ (−h(0))      ≤ (Aψ)(t) − g 0, η(0), η(−h(0))     + (Bϕ)(t) − f 0, η(0), ηΔ (−h(0))  ≤ a/2 + a/2 = a. This completes the proof of Aψ + Bϕ ∈ S ∗ . Also, (7.6.11) shows that B is a contraction. Hence all the conditions of Theorem  1.1.17 are satisfied, which imply that there is ϕ ∈ S ∗ , such that ϕ = Aϕ + Bϕ.

7.6.2 Connection Between Boundedness and Periodicity Intuitively, in the study of stability or periodic solutions in dynamical systems one will have to ask for the existence of solutions in the sense that solutions exist for all time or remain bounded. Thus, we may look at boundedness of solutions as a necessary condition before studying stability or attempt to search for a periodic solution. In this section, we examine the relationship between the boundedness of solutions and the existence of a periodic solution of the nonlinear nonautonomous delay dynamical system of the form   (7.6.12) x Δ (t) = −a(t)x σ (t) + b(t)g x(t − r(t)) + q(t), where T is unbounded above and below, and 0 ∈ T. We assume that a, b : T → R are continuous and q : [0, ∞)T → R is continuous. In order for the function x(t − r(t)) to be well defined over T, we assume that r : T → R and that id − r : T → T is strictly increasing. The proof of Lemma 7.6.1 below can be easily deduced from [99], and hence we omit the proof. Lemma 7.6.1. x is a solution of Eq. (7.6.12) if and only if ∫ t x(t) = x(0)e a (t, 0) + b(s)g(x(s − r(s))e a (t, s) Δs 0 ∫ t q(s)e a (t, s) Δs. + 0

314

7 Periodic Solutions: The Natural Setup

Let ψ : (−∞, 0]T → R be a given bounded Δ-differentiable initial function. We say x := x(·, 0, ψ) is a solution of (7.6.12) if x(t) = ψ(t) for t ≤ 0 and satisfies (7.6.12) for t ≥ 0. If ψ : (−∞, 0]T → R , then we set ψ =

sup

s ∈(−∞,0]T

|ψ(s)|.

Definition 7.6.2. Let ψ be as defined above. We say solutions of (7.6.12) are uniformly bounded if for each B1 > 0, there exists B2 > 0 such that [t0 ≥ 0, ψ ≤ B1, t ≤ t0 ] ⇒ |x(·, 0, ψ)| < B2 . For the next theorem we assume the following. There is a positive constant Q so that ∫ t |q(s)|e a (t, s) Δs ≤ Q, (7.6.13) 0

∫

t

a(s) Δs → ∞,

(7.6.14)

|b(s)|e a (t, s) Δs < α,

(7.6.15)

0

there is an α < 1 so that

∫ 0

t

0 ≤ r(t), t − r(t) → ∞ as t → ∞,

(7.6.16)

g(0) = 0 and |g(x) − g(y)| < |x − y|.

(7.6.17)

and if x, y ∈ R, then

Theorem 7.6.2. If (7.6.13)–(7.6.17) hold, then solutions of (7.6.12) are uniformly bounded at t0 = 0. Proof. First by (7.6.14), there is a constant k > 1 so that e a (t, 0) ≤ k. Let B1 be given so that if ψ : (−∞, 0]T → R be a given bounded initial function, 1 +Q . Let ψ ≤ B1 . Define the constant B2 by B2 = k B1−α S = {ϕ ∈ Cr d : ϕ(t) = ψ(t) if t ∈ (−∞, 0]T, ϕ ≤ B2 } .   Then S,  ·  is a complete metric space where  ·  is the supremum norm. For ϕ ∈ S, define the mapping P   Pϕ (t) = ψ(t), t ≤ 0,

7.6 Connection Between Boundedness and Periodicity

315

and 

∫ t  Pϕ (t) = ψ(0)e a (t, 0) + b(s)g(ϕ(s − r(s))e a (t, s) Δs 0 ∫ t q(s)e a (t, s) Δs, t ≥ 0. + 0

It follows from (7.6.17) that |g(x)| = |g(x) − g(0) + g(0)| ≤ |g(x) − g(0)| + |g(0)| ≤ |x|. This implies that |(Pϕ)(t)| ≤ k B1 + αB2 + Q = B2 . Thus, P : S → S. It easy to show, using (7.6.17), that P is a contraction with contraction constant α. Hence there is a unique fixed point in S, which solves (7.6.12).  We end this section by examining the existence of a periodic solution of (7.6.12). Let T be a periodic time scale such that 0 ∈ T. Let T > 0, T ∈ T be fixed and if T  R, T = np for some n ∈ N. Define PT = {ϕ ∈ C(T, R) : ϕ(t + T) = ϕ(t)}, where C(T, R) is the space of all real valued continuous functions on T. Then PT is a Banach space when it is endowed with the supremum norm  x = sup |x(t)|. t ∈[0,T ]T

Here we let the function q : (−∞, ∞)T → R. Since we are searching for a periodic solution, we must ask that a(t + T) = a(t), b(t + T) = b(t), r(t + T) = r(t), and q(t + T) = q(t).

(7.6.18)

Lemma 7.6.2. Suppose (7.6.14)–(7.6.18) hold. If x(t) ∈ PT , then x(t) is a solution of Eq. (7.6.12) if and only if ∫ t ∫ t x(t) = b(s)g(x(s − r(s)))e a (t, s) Δs + q(s)e a (t, s) Δs. (7.6.19) −∞

−∞

Proof. Due to condition (7.6.14) and the fact that p(t) is periodic, condition (7.6.13) is satisfied. Thus, by Theorem 7.6.2, solutions of (7.6.12) are bounded for all t ∈ (−∞, ∞)T . As a consequence, if we multiply both sides of (7.6.12) by ea (s, 0), and then integrate from −∞ to t we obtain (7.6.19). By taking the Δ-derivative on both sides of (7.6.19) we obtain (7.6.12). Theorem 7.6.3. Assume the hypothesis of Lemma 7.6.2. Then (7.6.12) has a unique T-periodic solution.

316

7 Periodic Solutions: The Natural Setup

Proof. For φ ∈ PT , define a mapping H : PT → PT by ∫ t ∫ (Hφ)(t) = b(s)g(φ(s − r(s)))e a (t, s) Δs + −∞

t

−∞

p(s)e a (t, s) Δs.

It is easy to verify that H is periodic and defines a contraction on PT . Thus, H has a unique fixed point in PT by the contraction mapping principle, which solves (7.6.12) by Lemma 7.6.2. We remark that Theorems 7.6.2 and 7.6.3 show a clear connection between boundedness and the existence of a periodic solution. In the case T = R, this result is known as Massera’s theorem, see [122]. Below we state and prove Massera’s theorem for the general case of T being a periodic time scale. We begin with a lemma that will be needed in the proof. Lemma 7.6.3. Let T be a periodic time scale with period T > 0. Let F : T × R → R be continuous. Suppose that F(t + T, x) = F(t, x) and that F(t, x) satisfy a local Lipschitz condition with respect to x. 1. If x(t) is a solution of x Δ = F(t, x), then x(t +T) is also a solution of x Δ = F(t, x). 2. The equation x Δ = F(t, x) has a T-periodic solution if and only if there is a pair (t0, x0 ) with x(t0 + T; t0, x0 ) = x0 where x(t; t0, x0 ) is the unique solution of x Δ = F(t, x), x(t0 ) = x0 . Proof. For part (1), let q(t) = x(t + T). Then, qΔ (t) = x Δ (t + T) = F(t + T, x(t + T)) = F(t, q(t)), and the proof of part 1. is complete. For part 2., first suppose that x(t; t0, x0 ) is T-periodic. Then, x(t0 + T; t0, x0 ) = x(t0 ; t0, x0 ) = x0 . Now suppose that there exists (t0, x0 ) such that x(t0 + T; t0, x0 ) = x0 . From part 1., q(t) ≡ x(t +T; t0, x0 ) is also a solution of x Δ = F(t, x). Since q(t0 ) = x(t0 +T; t0, x0 ) = x0 , then by the uniqueness of solutions of initial value problems, x(t + T; t0, x0 ) = q(t) = x(t; t0, x0 ). This completes the proof of part 2. The next theorem is a generalization of Massera’s Theorem [122] to general time scales. Theorem 7.6.4 (Massera Theorem). Let T be a periodic time scale with period T > 0. Let F : T × R → R be continuous. Suppose that F(t + T, x) = F(t, x) and that F(t, x) satisfies a local Lipschitz condition with respect to x. If the equation x Δ = F(t, x)

(7.6.20)

has a solution bounded in the future, then it has a T-periodic solution. Proof. Let x(t) be the solution of (7.6.20) such that |x(t)| ≤ M for all t ∈ T, t ≥ 0. Define the sequence {xn (t)} by xn (t) = x(t + nT), n = 0, 1, 2, . . . . By Lemma 7.6.3,

7.7 Almost Linear Volterra Integro-Dynamic Equations

317

xn (t) is a solution of (7.6.20) for each n and furthermore, |xn (t)| ≤ M for t ≥ 0. There are two cases to consider. Case 1. Suppose that for some n, xn (0) = xn+1 (0). By uniqueness of solutions for initial value problems we have x(t + nT) = x(t + (n + 1)T) for all t ∈ T. Thus, x(t) is a T-periodic solution of (7.6.20). Case 2. Suppose that xn (0)  xn+1 (0) for all n. We may assume, without loss of generality, that x(0) < x1 (0). By uniqueness, we have x(t) < x1 (t) for all t ∈ T, t ≥ 0. In particular, xn (0) = x(0 + nT) < x1 (0 + nT) = xn+1 (0). Hence, xn (t) < xn+1 (t) for all t ∈ T, t ≥ 0. Thus, {xn (t)} is an increasing sequence bounded above by M. Thus xn (t) → x ∗ (t) for each t ∈ T, t ≥ 0 as n → ∞. Since |F(t, x)| ≤ J for t ∈ T and |x| ≤ M, then |x Δ (t)| ≤ J, t ∈ T. From the Mean Value Theorem (see [50, Corollary 1.68]) we have |xn (t) − xn (s)| ≤ supr ∈[s,t]κ |F Δ (r, xn )| |t − s| ≤ J |t − s| for t, s ∈ T with 0 ≤ s ≤ t, n ≥ 0. Using the Arzel`a–Ascoli Theorem, we get that on any compact subinterval of T there exists a subsequence of {xn (t)} that converges uniformly. We know that the original sequence is monotone, and so, the original sequence is convergent on any compact interval. Since for each n, ∫ t

xn (t) = xn (0) +

F(s, xn (s)) Δs,

0

then the limiting function x ∗ (t) is a solution of (7.6.20). Finally, since x ∗ (T) = lim xn (T) = lim xn+1 (0) = x ∗ (0), then by Lemma 7.6.3, the limiting function is a T-periodic solution of (7.6.20) and the proof is complete.

7.7 Almost Linear Volterra Integro-Dynamic Equations Using Krasnosel’ski˘ı’s fixed point theorem, we deduce the existence of periodic solutions of nonlinear system of integro-dynamic equations on periodic time scales. These equations are studied under a set of assumptions on the functions involved in the equations. The equations will be called almost linear when these assumptions hold. Consider the nonlinear infinite delay Volterra integro-dynamic equation on time scales ∫ t Δ C(t, s)g(x(s))Δs + p(t), t ∈ (−∞, ∞)T . (7.7.1) x (t) = a(t)h(x(t)) + −∞

We assume that the functions h, a, p, and g are continuous and that there exist constants H, G, and positive constants H ∗ , G∗ such that

and

|h(x) − H x| ≤ H ∗,

(7.7.2)

|g(x) − Gx| ≤ G∗ .

(7.7.3)

Equation (7.7.1) will be called almost linear if (7.7.2) and (7.7.3) hold.

318

7 Periodic Solutions: The Natural Setup

This section combines the known continuous and discrete cases and many other time scales that are periodic. Thus the results of this section are new for the continuous and discrete cases. All results of this section can be found in [144]. In [10], the authors used the notion of degree theory in combination with Lyapunov functionals and showed the existence of periodic solutions of system of integrodynamic equations on time scales without the requirement of (7.7.2) and (7.7.3). For more on integro-differential equations, we refer to [84–167].

7.7.1 Periodic Solutions Let T be a periodic time scale with period P (see Definition 1.5.1). Let T > 0 be fixed and if T  R, T = nP for some n ∈ N. In this section we investigate the existence of a periodic solution of (7.7.1) using Krasnosel’ski˘ı’s fixed point theorem (see Theorem 1.1.17). The next lemma is essential to our next result. Its proof can be found in [99]. Lemma 7.7.1. Let x ∈ PT . Then  x σ  exists and  x σ  =  x. In this section we assume that for all (t, s) ∈ T × T, ∫ t sup |C(t, s)| Δs < ∞. t ∈T

(7.7.4)

−∞

We assume a ∈ R + . This implies that e (H a) (t, t − T) < 1. Suppose there exists a constant T > 0 such that for t ∈ T, we have a(t + T) = a(t), p(t + T) = p(t), C(t + T, s + T) = C(t, s).

(7.7.5)

Let M be the complete metric space of continuous T-periodic functions φ : (−∞, ∞)T → (−∞, ∞) with the supremum metric. Then, for any positive constant m the set (7.7.6) PT = { f ∈ M : || f || ≤ m}, is a closed convex subset of M. Lemma 7.7.2. If x ∈ PT , then x is a solution of Eq. (7.7.1) if and only if, ∫ t ' ( x(t) = [1−e (H a) (t, t −T)]−1 Ha(u)x σ (u)+a(u)h(x(u))+ k(u) e (H a) (t, u)Δu, t−T

(7.7.7)

where ∫ k(t) = p(t) +

t

−∞

'

(

C(t, s) g(x(s)) − Gx(s) Δs +

∫

t

−∞

C(t, s)Gx(s)Δs.

(7.7.8)

7.7 Almost Linear Volterra Integro-Dynamic Equations

319

Proof. For convenience we put (7.7.1) in the form x (t) + Ha(t)x σ (t) = Ha(t)x σ (t) + a(t)h(x(t)) + p(t) ∫ t ∫ t ' ( C(t, s) g(x(s)) − Gx(s) Δs + C(t, s)Gx(s)Δs. + −∞

(7.7.9)

−∞

Let ∫ k(t) = p(t) +

t

−∞

'

(

C(t, s) g(x(s)) − Gx(s) Δs +

∫

t

−∞

C(t, s)Gx(s)Δs.

Then we may write (7.7.9) as x (t) + Ha(t)x σ (t) = Ha(t)x σ (t) + a(t)h(x(t)) + k(t).

(7.7.10)

Let x(t) ∈ PT and assume (7.7.5). Multiply both sides of (7.7.10) with e H a (t, 0) and then integrate both sides from t − T to t to obtain e H a (t, 0)x(t) − e H a (t − T, 0)x(t − T) ∫ t ' ( Ha(u)x σ (u) + a(u)h(x(u)) + k(u) e H a (u, 0)Δu. = t−T

Divide both sides of the above equation by e H a (t, 0) and use the fact that x(t − T) = x(t) to obtain, x(t)[1 − e (H a) (t, t − T)] ∫ t ' ( = Ha(u)x σ (u) + a(u)h(x(u)) + k(u) e (H a) (t, u)Δu, t−T

where we have used Lemma 1.1.3 to simplify the exponentials. Since every step is reversible and by using Lemma 7.4.1, the converse holds. Define mappings A and B from PT into M as follows. For φ ∈ PT , +∫ t (Aφ)(t) = [1 − e (H a) (t, t − T)]−1 a(u)[h(φ(u)) + Hφσ (u)]e (H a) (t, u)Δu t−T ∫ t ∫ u , C(u, s)[g(φ(s)) − Gφ(s)] Δs e (H a) (t, u)Δu , + t−T

−∞

and for ψ ∈ PT −1

+∫

t

(Bψ)(t) = [1 − e (H a) (t, t − T)] t−T ∫ t , p(u)e (H a) (t, u)Δu . + t−T

∫

u

−∞

C(u, s)Gψ(s) Δs e (H a) (t, u)Δu

320

7 Periodic Solutions: The Natural Setup

It can be easily verified that both (Aφ)(t) and (Bψ)(t) are T-periodic and continuous in t. Assume ∫ ∫ u   t |C(u, s)|G∗ |ψ(s)| Δs e (H a) (t, u)Δu ≤ α < 1, sup [1 − e (H a) (t, t − T)]−1  t ∈T

t−T

−∞

(7.7.11)

and ∫  + t sup [1 − e (H a) (t, t − T)]−1  |a(u)|H ∗ e (H a) (t, u)Δu t ∈T t−T ∫ t ∫ u , G∗ |C(u, s)| Δs e (H a) (t, u)Δu ≤ β < ∞. + t−T

−∞

Choose the constant m of (7.7.6) satisfying ∫   t sup [1 − e (H a) (t, t − T)]−1  |p(u)|e (H a) (t, u)Δu + αm + β ≤ m. t ∈T

(7.7.12)

(7.7.13)

t−T

Lemma 7.7.3. Assume (7.7.5), (7.7.11), and (7.7.13). Then map B is a contraction from PT into PT . Proof. For φ ∈ PT , ∫ ∫ u   t |(Bφ)(t)| ≤ m[1 − e (H a) (t, t − T)]−1  |C(u, s)|G ds e (H a) (t, u)Δu t−T −∞ ∫  t  |p(u)|e (H a) (t, u)Δu + 1 − e (H a) (t, t − T)]−1  t−T ∫   t |p(u)|e (H a) (t, u)Δu + αm ≤ sup 1 − e (H a) (t, t − T)]−1  t ∈T

t−T

< m.

For φ, ψ ∈ PT , we obtain, using (7.7.11), |(Bφ)(t) − (Bψ)(t)| ≤ α||φ − ψ||. This proves that B is a contraction mapping from PT into PT . Lemma 7.7.4. Assume (7.7.1), (7.7.2), (7.7.4), (7.7.5), (7.7.12), and (7.7.13). Then map A from PT into PT is continuous, and APT is contained in a compact subset of M. Proof. For any φ ∈ PT , it follows from (7.7.12) and (7.7.13) that |(Aφ)(t)| ≤ β ≤ m.

(7.7.14)

So, A maps PT into PT and the set { Aφ} for φ ∈ PT is uniformly bounded. To show that A is a continuous map we let {φn } be any sequence of functions in PT with

7.8 Almost Automorphic Solutions of Delayed Neutral Dynamic Systems on Time Scales

321

||φn − φ|| → 0 as n → ∞. Then one can easily verify that || Aφn − Aφ|| → 0 as n → ∞. This proves that A is a continuous mapping. It is trivial to show that |(Aφ)Δ (t)| is bounded. This would show that the set { Aφ} for φ ∈ PT is equicontinuous by using (7.7.14). Therefore, by the Arzel`a–Ascoli Theorem, APT is contained in a compact subset of M. We are now ready to use the fixed point theorem of Krasnosel’ski˘ı (i.e., Theorem 1.1.17) to show the existence of a continuous T-periodic solution of (7.7.1). Theorem 7.7.1. Suppose assumptions of Lemmas 7.7.3 and 7.7.4 hold. Then (7.7.1) has a continuous T-periodic solution. Proof. φ, ψ ∈ PT , we get   |(Aφ)(t) + (Bψ)(t)| ≤ sup 1 − e (H a) (t, t − T)]−1  t ≥0

∫

t

t−T

|p(u)|e (H a) (t, u)Δu

+ αm + β ≤ m. which proves that Aφ + Bψ ∈ PT . Therefore, by Krasnosel’ski˘ı’s theorem there exists a function x(t) in PT such that x(t) = Ax(t) + Bx(t). This proves that (7.7.1) has a continuous T-periodic solution x(t).

7.8 Almost Automorphic Solutions of Delayed Neutral Dynamic Systems on Time Scales In this section, we study the existence of almost automorphic solutions of delayed neutral dynamic system on periodic time scales (see Definition 1.5.1). We use exponential dichotomy and prove uniqueness of projector of exponential dichotomy to obtain some limited results leading to sufficient conditions for the existence of almost automorphic solutions to neutral system. Unlike the existing literature we prove our existence results without assuming boundedness of the coefficient matrices in the system. Hence, we significantly improve the results in the existing literature. Finally, we also provide an existence result for an almost periodic solutions of the system. All of the results in this section can be found in [20]. The theory of neutral type equations has attracted a prominent attention due to the potential of its application in variety of fields in the natural sciences dealing with models analyzing and controlling real life processes. In particular, investigation of

322

7 Periodic Solutions: The Natural Setup

periodic solutions of neutral dynamic systems has a special importance for scientists interested in biological models of certain type of populations having periodical structures (see [54, 116], and [119]). There is a vast literature on neutral type equations on continuous and discrete domains which focus on the stability, oscillation, and existence results (see [94, 137, 138, 150], and references therein). We may refer the reader to [94, 99, 100, 138, 143], and [147] for studies handling existence of periodic solutions or related topics of neutral dynamic equations on time scales. A more general approach to periodicity notion on time scales has been introduced in [6] and applied to neutral dynamic systems in [19]. Periodicity as we know it may impose strong restrictions on some specific real life models including functions that are not strictly periodic but having values close enough to each other at every different period. Many mathematical models (see, e.g., [127] and [126]) in signal processing require the use of almost periodic function, informally, being a nearly periodic function where any one period is virtually identical to its adjacent periods but not necessarily similar to periods much farther away in time. The theory of almost periodic functions was first introduced by H. Bohr and generalized by A. S. Besitovitch, W. Stepanov, S. Bochner, and J. von Neumann at the beginning of twentieth century. We have the following definitions. A continuous function f : R → R is said to be almost periodic if the following characteristic property holds: D1. For any ε > 0, the set E (ε, f (x)) := {τ ∈ R : | f (x + τ) − f (x)| < ε for all x ∈ R} is relatively dense in the real line R. That is, for any ε > 0, there exists a number l (ε) > 0 such that any interval of length l (ε) contains a number in E (ε, f (x)). Afterwards, S. Bochner showed that almost periodicity is equivalent to the following characteristic property which is also called the normality condition: D2. From any sequence of the form { f (x + hn )} , where hn are real numbers, one can extract a subsequence converging uniformly on the real line (see [40]). Theory of almost automorphic functions was first studied by S. Bochner [42]. It is a property of a function which can be obtained by replacing convergence with uniform convergence in normality definition D2. More explicitly, a continuous - . function f : R → R is said to be almost automorphic if for every sequence hn n∈Z+ of real numbers there exists a subsequence {hn } such that limm→∞ limn→∞ f (t + hn − hm ) = f (t) for each t ∈ R. For more reading on almost automorphic functions, we refer to [75]. Almost automorphic solutions of difference equations have been investigated in [31, 69] and [118]. Additionally, C. Lizama and J. G. Mesquita generalized the notion of almost automorphy in [117] by studying almost automorphic solutions of dynamic equations on time scales that are invariant under translation. In [125] Mishra et al. investigated almost automorphic solutions to functional differential equation

7.8 Almost Automorphic Solutions of Delayed Neutral Dynamic Systems on Time Scales

d (x (t) − F1 (t, x (t − g (t)))) = A (t) x (t) + F2 (t, x (t) , x (t − g (t))) dt

323

(7.8.1)

using the theory of evolution semigroup. Note that almost periodic solutions of Eq. (7.8.1) have also been studied in [1] by means of the theory of evolution semigroup. In this study, we propose the existence results for almost automorphic solutions of the delayed neutral dynamic system x Δ (t) = A(t)x(t) + QΔ (t, x(t − g(t))) + G(t, x(t), x(t − g(t))) by using fixed point theory. The highlights of this section can be summarized as follows: • In this section, we use exponential dichotomy instead of theory of evolution semigroup since the conditions required by theory of evolution are strict and not easy to check (for related discussion see [70]). • We prove uniqueness of projector of exponential dichotomy on periodic time scales (see Theorem 7.8.3). • In [117], the authors obtain the limiting properties of exponential dichotomy by using the product integral on time scales (see [152]). This method requires boundedness of inverse matrices A(t)−1 and (I + μ (t) A(t))−1 as compulsory conditions. We obtain our limit results by using a different technique, including uniqueness of projector of exponential dichotomy, which does not require boundedness of the inverse matrices A(t)−1 and (I + μ (t) A(t))−1 (see Theorem 7.8.4). • Using a different approach we improve the existence results of [117]. Furthermore, our results also extend the results of [69] and [118] in the particular time scale T = Z.

7.8.1 Almost Automorphy Notion on Time Scales This subsection is devoted to the discussion of almost automorphic functions and their properties on time scales. Definition 7.8.1 ([117]). Let X be a (real or complex) Banach space and T be an additively periodic time scale. Define the set T by T := {p ∈ R : t ± p ∈ T, ∀t ∈ T.}

(7.8.2)

Then, an r d-continuous function f : T →X is called almost automorphic in T if for   every sequence αn ∈ T, there exists a subsequence (αn ) such that lim f (t + αn ) = f¯(t)

n→∞

is well defined for each t ∈ T and

324

7 Periodic Solutions: The Natural Setup

lim f¯(t − αn ) = f (t)

n→∞

for every t ∈ T. Remark 7.8.1. In particular cases T = R and T = Z, it is well known that every continuous periodic function is almost periodic and every almost periodic function is almost automorphic. This relationship is preserved on an additively periodic time scale T. Hereafter, we denote by A(X) the set of all almost automorphic functions on an additively periodic time scale. Obviously, A(X) is a Banach space when it is endowed by the norm  f  A(X) = sup  f (t) X , t ∈T

where . X is the norm defined on X. The following result summarizes the main properties of almost automorphic functions on time scales: Theorem 7.8.1 ([117]). Let T be an additively periodic time scale and suppose r d-continuous functions f , g : T →X are almost automorphic. Then (i) f + g is almost automorphic function on time scales, (ii) c f is almost automorphic function on time scales for every scalar c, (iii) For each l ∈ T, the function fl : T →X defined by fl (t) := f (l + t) is almost automorphic on time scales, (iv) The function fˆ : T →X defined by fˆ(t) := f (−t) is almost automorphic on time scales, (v) Every almost automorphic function on a time scale is bounded, that is  f  A(X) < ∞, < < (vi) < f¯< A(X) ≤  f  A(X) , where lim f (t + αn ) = f¯(t) and lim f¯(t − αn ) = f (t).

n→∞

n→∞

The following definition is also necessary for our further analysis. Definition 7.8.2 ([117]). Let X be a (real or complex) Banach space and T be a periodic time scale. Then, an r d-continuous function f : T × X →X  is called almost automorphic for t ∈ T for each x ∈ X, if for every sequence αn ∈ T , there exists a subsequence (αn ) such that lim f (t + αn, x) = f¯(t, x)

n→∞

is well defined for each t ∈ T, x ∈ X and lim f¯(t − αn, x) = f (t, x)

n→∞

for every t ∈ T and x ∈ X.

(7.8.3)

7.8 Almost Automorphic Solutions of Delayed Neutral Dynamic Systems on Time Scales

325

Remark 7.8.2. Almost automorphic functions depending two or more variables have the similar properties of almost automorphic functions of one variable. If f , g : T × X →X are almost automorphic, then f + g and c f (c is a constant) are almost automorphic. Furthermore,  f (., x) A(X) = sup  f (t, x) X < ∞ for each x ∈ X t ∈T

and

< < < f¯(., x)
0 T = {t = k − q m : k ∈ Z, m ∈ N0, q ∈ (0, 1)}

Delay function δ(t) = t − τ, τ ∈ R+ δ(t) = t − kτ, τ ∈ Z+ δ(t) = t − τ, τ ∈ Z+

Throughout this section we suppose that the functions a, h, and G are continuous in their respective domains and that for at least T > 0 a(t + T) = a(t), δ(t + T) = δ(t) + T, G(t, .) = G(t + T, .), t ∈ T.

(7.9.5)

To avoid obtaining the zero solution we also suppose that G(t, 0)  a(t)h(0) for some t ∈ T. Suppose that T is a periodic time scale and that R is an arbitrary time scale that is closed under addition. The space PT (T, R), given by PT (T, R) := {ϕ ∈ C(T, R) : ϕ(t + T) = ϕ(t)}, is a Banach space when it is endowed with the supremum norm  x := sup |x(t)| = sup |x(t)|. t ∈[0,T ]T

t ∈T

For a fixed constant α > 0, define the set Mα by Mα := {φ ∈ PT (T, R) : φ ≤ α} .

(7.9.6)

Then, Mα is a closed, bounded, and convex subset of the Banach space PT (T, R). Hereafter, we use the notation γ = −a and assume that γ ∈ R + .

7.9 Large Contraction and Existence of Periodic Solutions

345

Lemma 7.9.1. If x ∈ PT (T, R), then x is a solution of Eq. (7.9.4) if, and only if, ∫ t+T x (t) = kγ (7.9.7) {a(s)H(x(s)) + G(s, x(δ(s)))} eγ (t + T, σ (s)) Δs, t

 −1  where kγ := 1 − eγ (t + T, t) , γ(t) := −a(t), and H(x(t)) := x(t) − h(x(t)).

(7.9.8)

Proof. Let x ∈ PT (T, R) be a solution of (7.9.4). Equation (7.9.4) can be expressed as x Δ (t) − γ(t)x(t) = a(t)H(x(t)) + G(t, x(δ(t))).

(7.9.9)

Multiplying both sides of (7.9.9) by e γ (σ (t) , t0 ) we get -

. x Δ (t) − γ(t)x (t) e γ (σ (t) , t0 ) = {a(t)H(x(t)) + G(t, x(δ(t)))} e γ (σ (t) , t0 ) .

Since e γ (t, t0 )Δ = −γ(t)e γ (σ (t) , t0 ) we find '

x (t) e γ (t, t0 )

(Δ

= {a(t)H(x(t)) + G(t, x(δ(t)))} e γ (σ(t), t0 ) .

Taking integral from t − T to t, we arrive at ∫ x (t + T) e γ (t + T, t0 ) − x (t) e γ (t, t0 ) =

t+T

{a(s)H(x(s))

t

+G(s, x(δ(s)))} e γ (σ(s), t0 ) Δs. Using x (t + T) = x (t) for x ∈ PT (T, R), and e γ (t, t0 ) e γ (σ (s) , t0 ) = eγ (t + T, t) , = eγ (t + T, σ (s)) , e γ (t + T, t0 ) e γ (t + T, t0 ) we obtain (7.9.7). Since each step in the above work is reversible, the proof is complete. Lemma 7.9.2. If p ∈ R + , then

∫

0 < e p (t, s) ≤ exp



t

p(r)Δr

(7.9.10)

s

for all t ∈ T. Proof. If p ∈ R + , then Log(1 + μ(t)p(t)) = log(1 + μ(t)p(t)) ∈ R. It follows from the definition of the exponential function that e p (t, s) > 0. On the other hand, since we have exp(u) ≥ 1 + u, and therefore, u ≥ log(1 + u) for all u ∈ ( − 1, ∞), we find

346

7 Periodic Solutions: The Natural Setup

∫

t

e p (t, s) = exp s

∫ ≤ exp

t

1 log(1 + μ(r)p(r)) Δr μ(r)  p(r) Δr .



s

This completes the proof. As a consequence of Lemma 7.9.2 we note that for γ ∈ R + , t ∈ [0, T]T , and s ∈ [t, t + T)T , ∫ t+T  eγ (t + T, σ (s)) exp σ(s) γ(r)Δr  ≤  1 − eγ (t + T, t) 1 − eγ (t + T, t) ∫ 2T  exp 0 |γ(r)| Δr   ≤ := K. (7.9.11) 1 − eγ (T, 0) Let the maps A and B be defined by ∫ t+T eγ (t + T, σ (s)) (Aϕ)(t) = Δs, t ∈ T G(s, ϕ(δ(s))) 1 − eγ (t + T, t) t ∫

and

(7.9.12)

eγ (t + T, σ (s)) Δs, t ∈ T, (7.9.13) 1 − eγ (t + T, t) t respectively. It is clear from (7.9.5) that the maps A and B are T periodic. In the proof of next result, we use the time scale version of Lebesgue Dominated Convergence theorem. For a detailed study on Δ-Riemann and Lebesgue integrals on time scales we refer to [51]. (Bψ)(t) =

t+T

a(s)H(ψ(s))

Lemma 7.9.3. Suppose that there exists a positive valued function ξ : T → (0, ∞) which is continuous on [0, T)T such that |G(t, ϕ(δ(t)))| ≤ ξ(t) for all t ∈ T and ϕ ∈ Mα .

(7.9.14)

Then the mapping A, defined by (7.9.12), is continuous on Mα . Proof. To see that A is a continuous mapping, let {ϕi }i ∈N be a sequence of functions in Mα such that ϕi → ϕ as i → ∞. Since (7.9.14) holds, the continuity of G, and the Dominated Convergence theorem yield /  ∫ T     G(s, ϕi (δ(s))) − G(s, ϕ(δ(s))) Δs lim sup  Aϕi (t) − Aϕ(t) ≤ K lim i→∞

i→∞

t ∈[0,T ]T

∫ ≤K 0

= 0,

T

0

  lim G(s, ϕ j (δ(s))) − G(s, ϕ(δ(s))) Δs

i→∞

7.9 Large Contraction and Existence of Periodic Solutions

347

where K is defined as in (7.9.11). This shows the continuity of the mapping A. The proof is complete. One may illustrate with the following example what kind of functions ξ, satisfying (7.9.14), can be chosen to show the continuity of A. Example 7.9.1. If we assume that G(t, x) satisfies a Lipschitz in x, i.e., there is a positive constant k such that |G(t, z) − G(t, w)| ≤ k z − w, for z, w ∈ PT ,

(7.9.15)

then for ϕ ∈ Mα , |G(t, ϕ(δ(t)))| = |G(t, ϕ(δ(t))) − G(t, 0) + G(t, 0)| ≤ |G(t, ϕ(δ(t))) − G(t, 0)| + |G(t, 0)| ≤ kα + |G(t, 0)|. In this case we may choose ξ as ξ(t) = kα + |G(t, 0)| .

(7.9.16)

Another possible ξ satisfying (7.9.14) is the following: ξ(t) = |g(t)| + |p(t)| |y(t)| n ,

(7.9.17)

where n = 1, 2, ..., g and p are continuous functions on T, and y ∈ Mα . Remark 7.9.1. Condition (7.9.15) is strong since it requires the function G to be globally Lipschitz. A lesser condition is (7.9.14) in which ξ can be directly chosen as in (7.9.16) or (7.9.17). In the next two results we assume that for all t ∈ T and ψ ∈ Mα ∫ t+T eγ (t + T, σ (s)) Δs ≤ α, J := {|a(s)||H(ψ(s))| + ξ(s)} 1 − eγ (t + T, t) t

(7.9.18)

where ξ is defined by (7.9.14). Lemma 7.9.4. In addition to the assumptions of Lemma 7.9.3, suppose also that (7.9.18) holds. Then A is continuous in ϕ ∈ Mα and maps Mα into a compact subset of Mα . Proof. Let ϕ ∈ Mα . Continuity of A in ϕ ∈ Mα follows from Lemma 7.9.3. Now, by (7.9.12), (7.9.14), and (7.9.18) we have |(Aϕ)(t)| < α. Thus, Aϕ ∈ Mα . Let ϕi ∈ Mα, i = 1, 2, .... Then from the above discussion we conclude that || Aϕ j || ≤ α.

348

7 Periodic Solutions: The Natural Setup

This shows A(Mα ) is uniformly bounded. It is left to show that A(Mα ) is equicontinuous. Since ξ is continuous and T-periodic, by (7.9.14) and differentiation of (7.9.12) with respect to t ∈ T (for the differentiation rule see [10, Lemma 1]) we arrive at   (Aϕ j )Δ (t) = |G(t, ϕ j (δ(t))) − a(t)(Aϕi )(t)| ≤ ξ(t) + |a(t)| |(Aϕi )(t)| ≤ ξ(t) + ||a|| || Aϕi || ≤ L, for t ∈ [0, T]T , where L is a positive constant. Thus, the estimation on |(Aϕi )Δ (t)| and (7.9.3) imply that A(Mα ) is equicontinuous. Then Arzel`a–Ascoli theorem yields compactness of the mapping A. The proof is complete. From closedness of the time scale R under addition we get that (Aφ + Bϕ)(t) ∈ R for all t ∈ T and φ, ϕ ∈ Mα .

(7.9.19)

Theorem 7.9.4. Suppose all assumptions of Lemma 7.9.4 hold. If B is a large contraction on Mα , then (7.9.4) has a periodic solution in Mα . Proof. Let A and B be defined by (7.9.12) and (7.9.13), respectively. By Lemma 7.9.4, the mapping A is compact and continuous. Then using (7.9.18), (7.9.19), and the periodicity of A and B, we have Aϕ + Bψ : Mα → Mα for ϕ, ψ ∈ Mα . Hence an application of Theorem 7.9.1 implies the existence of a periodic solution in Mα . This completes the proof. The next result gives a relationship between the mappings H and B in the sense of large contraction. Lemma 7.9.5. Let a be a positive valued function. If H is large contraction on Mα , then so is the mapping B. Proof. If H is large contraction on Mα , then for x, y ∈ Mα , with x  y, we have ||H x − H y|| ≤ ||x − y||. Since γ = −a ∈ R + and a is positive valued, γ(t) < 0 for all t ∈ T. Thus, it follows from the equality ' (Δ a(s)eγ (t + T, σ (s)) = eγ (t + T, s) s , where Δs indicates the delta derivative with respect to s, and Remark 4.1.1 that ∫

eγ (t + T, σ (s)) a(s)|H(x)(s) − H(y)(s)|Δs 1 − eγ (t, t − T) t ∫ t+T ||x − y|| ≤ a(s)eγ (t + T, σ (s)) Δs 1 − eγ (t + T, t) t

|Bx(t) − By(t)| ≤

t+T

= ||x − y||.

7.9 Large Contraction and Existence of Periodic Solutions

349

Taking the supremum over the set [0, T]T , we get that Bx − By ≤ ||x − y||. One may also show in a similar way that Bx − By ≤ δ||x − y|| holds if we know the existence of a 0 < δ < 1 such that for all ε > 0 [x, y ∈ Mα, ||x − y|| ≥ ε] ⇒ ||H x − H y|| ≤ δ||x − y||. The proof is complete. From Theorem 7.9.4 and Lemma 7.9.5, we deduce the following result. Corollary 7.9.1. In addition to the assumptions of Theorem 7.9.4, suppose also that a is a positive valued function. If H is a large contraction on Mα , then (7.9.4) has a periodic solution in Mα .

7.9.2 Classification of Large Contractions Corollary 7.9.1 shows that having a large contraction on a class of periodic functions plays a substantial role in proving existence of periodic solutions. We deduce by the next theorem that H.1. h : R → R is continuous on Uα and differentiable on Uακ , H.2. h is strictly increasing on Uα , H.3. sup h (t) ≤ 1 t ∈(−α,α)

are the conditions implying that the mapping H in (7.9.8) is a large contraction on the set Mα . Thus we have the following general theorem that classifies the functions that are large contraction. Theorem 7.9.5. Let h : R → R be a function satisfying (H.1–H.3). Then the mapping H in (7.9.8) is a large contraction on the set Mα . Proof. Let φ, ϕ ∈ Mα with φ  ϕ. Then φ(t)  ϕ(t) for some t ∈ R. Let us denote this set by D(φ, ϕ), i.e., D(φ, ϕ) = {t ∈ R : φ(t)  ϕ(t)} . For all t ∈ D(φ, ϕ), we have |Hφ(t) − Hϕ(t)| = |φ(t) − h(φ(t)) − ϕ(t) + h(ϕ(t))|     h(φ(t)) − h(ϕ(t))  = |φ(t) − ϕ(t)| 1 − . φ(t) − ϕ(t)

(7.9.20)

350

7 Periodic Solutions: The Natural Setup

Since h is a strictly increasing function we have h(φ(t)) − h(ϕ(t)) > 0 for all t ∈ D(φ, ϕ). φ(t) − ϕ(t)

(7.9.21)

For each fixed t ∈ D(φ, ϕ) define the interval Ut ⊂ [−α, α] by  (ϕ(t), φ(t)) if φ(t) > ϕ(t) Ut = . (φ(t), ϕ(t)) if φ(t) < ϕ(t) Mean Value Theorem implies that for each fixed t ∈ D(φ, ϕ) there exists a real number ct ∈ Ut such that h(φ(t)) − h(ϕ(t)) = h (ct ). φ(t) − ϕ(t) By (H.2–H.3) we have 0≤

inf

u ∈(−α,α)

h (u) ≤ inf h (u) ≤ h (ct ) ≤ sup h (u) ≤ u ∈Ut

u ∈Ut

sup

u ∈(−α,α)

h (u) ≤ 1. (7.9.22)

Hence, by (7.9.20)–(7.9.22) we obtain   |Hφ(t) − Hϕ(t)| ≤ 1 −

  inf h (u) |φ(t) − ϕ(t)| . u ∈(−α,α) 

(7.9.23)

for all t ∈ D(φ, ϕ). This implies a large contraction in the supremum norm. To see this, choose a fixed ε ∈ (0, 1) and assume that φ and ϕ are two functions in Mα satisfying ε ≤ sup |φ(t) − ϕ(t)| = φ − ϕ . t ∈D(φ,ϕ)

If |φ(t) − ϕ(t)| ≤

ε 2

for some t ∈ D(φ, ϕ), then we get by (7.9.22) and (7.9.23) that

1 (7.9.24) φ − ϕ . 2   Since h is continuous and strictly increasing, the function h u + ε2 − h(u) attains ε its minimum on the closed and bounded interval [−α, α]. Thus, if 2 < |φ(t) − ϕ(t)| for some t ∈ D(φ, ϕ), then by (H.2) and (H.3) we conclude that |H(φ(t)) − H(ϕ(t))| ≤ |φ(t) − ϕ(t)| ≤

1≥ where

h(φ(t)) − h(ϕ(t)) > λ, φ(t) − ϕ(t)

1 min {h(u + ε/2) − h(u) : u ∈ [−α, α]} > 0. 2α Hence, (7.9.20) implies λ :=

7.9 Large Contraction and Existence of Periodic Solutions

|Hφ(t) − Hϕ(t)| ≤ (1 − λ) φ(t) − ϕ(t) .

351

(7.9.25)

Consequently, combining (7.9.24) and (7.9.25) we obtain |Hφ(t) − Hϕ(t)| ≤ δ φ − ϕ , 

where δ = max

 1 , 1 − λ < 1. 2

The proof is complete. Example 7.9.2. For a fixed real number α > 0 and a fixed integer k ∈ Z+ define the function h : R → R by h(s) =

s2k+1 , for s ∈ R. (2k + 1)(α + 1)2k

(7.9.26)

Then the mapping H defined by (7.9.8) is a large contraction on the set Mα where Mα is given by (7.9.6). Example 7.9.3. For a fixed α > 0 and a fixed k ∈ Z+ define the function h : R → R by h(u) = u2k+1, u ∈ R. Then the mapping H defined by (7.9.8) is a large contraction on the set

where

Mν = {φ ∈ P(T, R) : φ ≤ ν} ,

(7.9.27)

ν := (2k + 1)−1/2k .

(7.9.28)

Theorem 7.9.6. Let T be a periodic time scale. For a fixed k ∈ Z+ set G(u, x(δ(u))) = b(u)x 2k+1 (δ(t)) + c(u) and h(u) := u2k+1 for u ∈ R, and define the mappings A and B as in (7.9.12) and (7.9.13), respectively. If a is positive valued T periodic function and c ∈ PT (T, R) is a function which is not equally zero, then ∫ t+T 1 ξ(s)eγ (t + T, σ (s)) Δs ≤ ν, 2kν 2k+1 + (7.9.29) 1 − eγ (t + T, t) t implies the existence of nonzero periodic solution x ∈ Mν of the equations x Δ (t) = −a(t)x 2k+1 (t) + b(t)x 2k+1 (δ(t)) + c(t).

352

7 Periodic Solutions: The Natural Setup

Proof. First, it follows from Theorem 7.9.5 that the mapping H given by (7.9.8) is a large contraction on Mν . Also for x ∈ Mν, we have |x(t)| 2k+1 ≤ ν 2k+1, and therefore, G(u, x(δ(u))) ≤ ν 2k+1 |b(u)| + |c(u)| := ξ(t),

(7.9.30)

i.e., (7.9.14) holds. Using the standard techniques of calculus one may verify that   |H(x(t))| =  x(t) − x 2k+1 (t) ≤ 2k(2k + 1)−(2k+1)/2k = 2kν 2n+1 Since a(t) > 0 and for all x ∈ Mα , we get by (7.9.29) that 2kν 2k+1 J≤ 1 − eγ (t + T, t) +

1 1 − eγ (t + T, t)

= 2kν 2k+1 +

∫ ∫

t+T

t

t

t+T

a(s)eγ (t + T, σ (s)) Δs ξ(s)eγ (t + T, σ (s)) Δs

1 1 − eγ (t + T, t)

∫ t

t+T

ξ(s)eγ (t + T, σ (s)) Δs ≤ ν.

Thus, (7.9.18) is satisfied. The proof is completed by making use of Corollary 7.9.1. Note that in [18] it has been shown that ∫ t+T   ∫ t +T 4(5−5/4 ) + η 5−5/4 |b(u)| + |c(u)| e− u a(s)ds du ≤ 5−1/4

(7.9.31)

t

is the condition guaranteeing that the totally nonlinear delay differential equation x (t) = −a(t)x(t)5 + b(t)x(t − r(t))5 + c(t)

(7.9.32)

has a T-periodic solution in M1/ √4 5 , where a(t) > 0 for all t ∈ R,   ∫T   η := (1 − e− 0 a(s)ds )−1  , and c( 0) ∈ PT (R, R). Observe that Theorem 7.9.6 covers this result.

7.10 Open Problem Extend the idea of large contraction in Sect. 7.9 to prove the existence of periodic solutions of the almost linear Volterra integral equations in Sect. 7.7.

Chapter 8

Periodicity Using Shift Periodic Operators

Summary In Chap. 7, we had to require the time scale to be additive in order to show the existence of periodic solutions. We devote this chapter to the study of periodic solutions of functional delay dynamical systems. The periodic delays are expressed in terms of shift operators (see Definition 1.4.4) that were developed by Adıvar (Electron J Qual Theory Differ Equ 2010(7):1–22, 2010). We begin the chapter with the discussion of periodicity in shift operators over different time scales and provide examples. We proceed to unifying Floquet theory for homogeneous and nonhomogeneous dynamical systems. Then we consider neutral dynamical systems using Floquet theory and Krasnosel’ski˘ı fixed point theorem. We move to the existence of almost automorphic solutions of delayed neutral dynamic system using exponential dichotomy and Krasnosel’ski˘ı fixed point theorem. This chapter should serve as the foundation and guidance for future research on periodicity using the well-defined shift operators, by the authors. Part of this chapter is new and the rest can be found in Adıvar (Math Slovaca 63(4):817–828, 2013), Adıvar and Koyuncuoğlu (Appl Math Comput 273:1208–1233, 2016; Appl Math Comput 242:328–339, 2014), and Koyuncuoğlu (q-Floquet Theory and Its Extensions to Time Scales Periodic in Shifts. Thesis (Ph.D.), 2016). In Chap. 7, we had to require the time scale to be additive in order to show the existence of periodic solutions. We devote this chapter to the study of periodic solutions of functional delay dynamical systems. The periodic delays are expressed in terms of shift operators (see Definition 1.4.4) that were developed by Adıvar in [4]. We begin the chapter with the discussion of periodicity in shift operators over different time scales and provide examples. We proceed to unifying Floquet theory for homogeneous and nonhomogeneous dynamical systems. Then we consider neutral dynamical systems using Floquet theory and Krasnosel’ski˘ı fixed point theorem. We move to the existence of almost automorphic solutions of delayed neutral dynamic system using exponential dichotomy and Krasnosel’ski˘ı fixed point theorem. This chapter should serve as the foundation and guidance for future research on periodicity

© Springer Nature Switzerland AG 2020 M. Adıvar, Y. N. Raffoul, Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales, https://doi.org/10.1007/978-3-030-42117-5_8

353

354

8 Periodicity Using Shift Periodic Operators

using the well-defined shift operators, by the authors. Part of this chapter is new and the rest can be found in [6, 9, 19], and [107].

8.1 Periodicity in Shifts Shift operators are introduced in Sect. 1.4.2. In this section we present some fundamental results on periodicity in shifts given in Definition 1.5.3. We first start with the following observation. Lemma 8.1.1. Let T be a time scale that is periodic in shifts with the period P.   −1 = δ−P defined by δ−P (t) := Then, δ+P is an invertible mapping with the inverse δ+P δ− (P, t). Proof. By P.4 of Definition 1.4.4 the mapping δ+P : T∗ → T∗ defined by δ+P (t) = δ+ (P, t) is surjective. On the other hand, we know by P.1 of Definition 1.4.4 that shift operators δ± are strictly increasing in their second arguments. That is, the mapping δ+P (t) := δ+ (P, t) is injective. This along with P.4 of Definition 1.4.4 shows that the operator δ+P is invertible with the inverse δ−P . In the next two results, we suppose that T is a periodic time scale in shifts δ± with period P and show that the operators δ±P : T∗ → T∗ are commutative with the forward jump operator σ : T → T. That is,     δ±P ◦ σ (t) = σ ◦ δ±P (t) for all t ∈ T∗ . (8.1.1) Lemma 8.1.2. The mapping δ+T : T∗ → T∗ preserves the structure of the points in T∗ . That is, t)) = δ+ (P, 4 t). σ(4 t) = 4 t implies σ(δ+ (P, 4 σ(4 t) > 4 t implies σ(δ+ (P, 4 t) > δ+ (P, 4 t). Proof. By definition we have σ(t) ≥ t for all t ∈ T∗ . Thus, by P.1 of Definition 1.4.4 δ+ (P, σ(t)) ≥ δ+ (P, t). Since σ(δ+ (P, t)) is the smallest element satisfying σ(δ+ (P, t)) ≥ δ+ (P, t), we get

δ+ (P, σ(t)) ≥ σ(δ+ (P, t)) for all t ∈ T∗ .

If σ(4 t) = 4 t, then (8.1.2) implies δ+ (P, 4 t) = δ+ (P, σ(4 t)) ≥ σ(δ+ (P, 4 t)).

(8.1.2)

8.1 Periodicity in Shifts

That is,

355

t) = σ(δ+ (P, 4 t)) provided σ(4 t) = 4 t. δ+ (P, 4

If σ(4 t) > 4 t, then by definition of σ we have (4 t, σ(4 t))T∗ = (4 t, σ(4 t))T∗ = 

(8.1.3)

and by P.1 of Definition 1.4.4 δ+ (P, σ(4 t)) > δ+ (P, 4 t). Suppose contrary that δ+ (P, 4 t) is right dense, i.e., σ(δ+ (P, 4 t)) = δ+ (P, 4 t). This along with (8.1.2) implies t), δ+ (P, σ(4 t)))T∗  . (δ+ (P, 4 Pick one element s ∈ (δ+ (P, 4 t), δ+ (P, σ(4 t)))T∗ . Since δ+ (P, t) is strictly increasing in t and invertible there should be an element t ∈ (4 t, σ(4 t))T∗ such that δ+ (P, t) = s. t) must be right scattered, i.e., σ(δ+ (P, 4 t)) > This contradicts (8.1.3). Hence, δ+ (P, 4 t). The proof is complete. δ+ (P, 4 Corollary 8.1.1. We have

Thus,

δ+ (P, σ(t)) = σ(δ+ (P, t)) for all t ∈ T∗ .

(8.1.4)

δ− (P, σ(t)) = σ(δ− (P, t)) for all t ∈ T∗ .

(8.1.5)

Proof. The equality (8.1.4) can be obtained as we did in the proof of preceding lemma. By (8.1.4) we have δ+ (P, σ(s)) = σ(δ+ (P, s)) for all s ∈ T∗ . Substituting s = δ− (P, t) we obtain δ+ (P, σ(δ− (P, t))) = σ(δ+ (P, δ− (P, t))) = σ(t). This and (iii) of Lemma 1.4.6 imply σ(δ− (P, t)) = δ− (P, σ(t)) for all t ∈ T∗ . The proof is complete. Observe that (8.1.4) along with (8.1.5) yields (8.1.1). Definition 8.1.1 (Periodic Function in Shifts δ± ). Let T be a time scale that is periodic in shifts δ± with the period P. We say that a real valued function f defined on T∗ is periodic in shifts δ± if there exists a T ∈ [P, ∞)T∗ such that (T, t) ∈ D± and f (δ±T (t)) = f (t) for all t ∈ T∗,

(8.1.6)

356

8 Periodicity Using Shift Periodic Operators

where δ±T (t) := δ± (T, t). The smallest number T ∈ [P, ∞)T∗ such that (8.1.6) holds is called the period of f . Example 8.1.1. By Definition 1.5.3 we know that the set of reals R is periodic in shifts δ± defined by (1.4.21–1.4.22) associated with the initial point t0 = 1. The function   ln |t| π , t ∈ R∗ := R − {0} f (t) = sin ln (1/2) is periodic in shifts δ± defined by (1.4.21)–(1.4.22) with the period T = 4 since   ±1  f  t4  if t ≥ 0 f (δ± (T, t)) = f t/4±1 if t < 0   ln |t| ± 2 ln (1/2) π = sin ln (1/2)   ln |t| = sin π ± 2π ln (1/2)   ln |t| = sin π ln (1/2) = f (t) for all t ∈ R∗ (see Fig. 8.1).

1.0

0.5

0.005

0.010

-0.5

-1.0

Fig. 8.1: Graph of f (t) = sin



ln |t | ln(1/2) π



8.1 Periodicity in Shifts

357

Example 8.1.2. The time scale qZ = {q n : n ∈ Z and q > 1} ∪ {0} is periodic in shifts δ± (P, t) = P±1 t with the period P = q. The function f defined by ln t ln f (t) = (−1) q , t ∈ qZ

(8.1.7) ∗

is periodic in shifts δ± with the period T = q2 since δ+ (q2, t) ∈ qZ = qZ and ln t ln t ±2 f δ± (q , t) = (−1) ln q = (−1) ln q = f (t) 



2

for all t ∈ qZ . However, f is not periodic in the sense of Definition 7.0.2 since there exists no positive number positive number T so that f (t ± T) = f (t) holds. In the following, we introduce Δ-periodic function in shifts. Definition 8.1.2 (Δ-Periodic Function in Shifts δ± ). Let T be a time scale that is periodic in shifts δ± with period P. We say that a real valued function f defined on T∗ is Δ-periodic in shifts δ± if there exists a T ∈ [P, ∞)T∗ such that (T, t) ∈ D± for all t ∈ T∗, the shifts δ±T are Δ − differentiable with rd-continuous derivatives, and

f (δ±T (t))δ±ΔT (t) = f (t) T∗ ,

(8.1.8) (8.1.9) (8.1.10)

for all t ∈ where := δ± (T, t). The smallest number T ∈ [P, ∞)T∗ such that (8.1.8)–(8.1.10) hold is called the period of f . δ±T (t)

Notice that Definition 8.1.1 and Definition 8.1.2 give the classic periodicity definition (i.e., Definition 7.0.2) on time scales whenever δ±T (t) = t ± T are the shifts satisfying the assumptions of Definitions 8.1.1 and 8.1.2. Example 8.1.3. The real valued function g(t) = 1/t defined on 2Z = {2n : n ∈ Z} is Δ-periodic in shifts δ± (T, t) = T ±1 t with the period T = 2 since f (δ± (2, t)) δ±Δ (2, t) =

1 ±1 1 2 = = f (t). t 2±1 t

Theorem 1.1.9 is essential for the proof of next theorem. Theorem 8.1.1. Let T be a time scale that is periodic in shifts δ± with period P ∈ (t0, ∞)T∗ and f a Δ-periodic function in shifts δ± with the period T ∈ [P, ∞)T∗ . Suppose that f ∈ Cr d (T), then ∫ t0

t

∫ f (s)Δs =

δ±T (t)

δ±T (t0 )

f (s)Δs.

358

8 Periodicity Using Shift Periodic Operators

Proof. Substituting v(s) = δ+T (s) and g(s) = f (δ+T (s)) in Theorem 1.1.9 and taking (8.1.10) into account we have ∫

∫

δ+T (t)

f (s)Δs =

δ+T (t0 )

ν(t)

ν(t0 ) t

∫ =

g(ν −1 (s))Δs

g(s)ν Δ (s)Δs

t0

∫ =

t

t0

∫ =

t

f (δ+T (s))δ+ΔT (s)Δs f (s)Δs.

t0

The equality

∫

δ−T (t)

δ−T (t0 )

∫ f (s)Δs =

t

f (s)Δs

t0

can be obtained similarly. The proof is complete.

8.1.1 Periodicity in Quantum Calculus In this part of the book we show the connection between periodicity in quantum calculus and periodicity in shifts over the particular time scale qN0 := {q n : n = 0, 1, 2...}, where q > 1 is a constant. In the existing literature, there are two different periodicity definitions for the functions defined on the time scale qN0 . To the best of our knowledge, the periodicity notion on qN0 has been first introduced by Bohner and Chieochan in [45]. They defined a P-periodic function on qN0 as follows: Definition 8.1.3 ([45]). Let P ∈ N. A function f : qN0 → R is said to be P-periodic if   (8.1.11) f (t) = q P f q P t for all t ∈ qN0 . Afterwards, Adıvar [6] (see also [9]) introduced a more general periodicity notion on time scales that are not necessarily additively periodic (see Definition 8.1.1). In particular case T = qN0 , Definition 8.1.1 leads to the following definition: Definition 8.1.4 ([6]). Let P ∈ N. A function f : qN0 → R is said to be P-periodic if   f q P t = f (t) for all t ∈ qN0 . In accordance with the periodicity notion on the continuous domain R, Definition 8.1.4 regards a periodic function to be the one repeating its values after a certain number of steps on qN0 . On the other hand, Definition 8.1.3 resembles the periodicity on R in geometric meaning. That is, periodicity in Definition 8.1.3 is based on the

8.1 Periodicity in Shifts

359

equality of areas lying below the graph of the function at each period. For example, ln t the function h(t) = (−1) ln q on qN0 is a 2-periodic function according to Definition 8.1.4, since h(tq2 ) = h(t) holds. On the other hand, the function g(t) = 1/t is 1-periodic with respect to Definition 8.1.3 since it satisfies qg(qt) = g(t). Hereafter, we show that the periodicity notions introduced in Definitions 8.1.3 and 8.1.4 are closely related. This connection helps the construction of relationship between coefficients of equations whose solutions are periodic with respect to Definitions 8.1.3 and 8.1.4, respectively. 8.1.1.1 Connection Between Two q-Periodicity Notions on q N0 In this part, we propose some results to discuss the connection between two qperiodicity notions. Now, we can list the following observations establishing a connection between two periodicity definitions. Proposition 8.1.1. Let f : qN0 → R. Then f is periodic with respect to Definition 8.1.3 if and only if f˜ (t) = t f (t) is periodic with respect to Definition 8.1.4 with the same period. Proposition 8.1.2. The function x is a P-periodic solution of the following first order q-difference equation Dq x(t) + a(t)x σ (t) = f (t, t x (t)) , t ∈ qN0 ,

(8.1.12)

with respect to Definition 8.1.3 if and only if x(t) ˜ := t x(t) is a P-periodic solution of the first order q-difference equation Dq x(t) ˜ + a(t) ˜ x˜ σ (t) = f˜ (t, x˜ (t)) , t ∈ qN0 , where a(t) ˜ :=

(8.1.13)

ta (t) − 1 , qt

and f˜ (t, x˜ (t)) = t f (t, t x (t)), with respect to Definition 8.1.4. Here, Dq f represents the q-derivative of f defined by 

 f (qt) − f (t) , t ∈ qN0 . Dq f (t) = (q − 1)t

Proof. Let x be a solution of (8.1.12). Then tDq x(t) + x σ (t) − x σ (t) + ta(t)x σ (t) = t f (t, t x (t)) , implying Dq (t x(t)) +

ta(t) − 1 qt x σ (t) = t f (t, t x(t)). qt

(q-derivative)

360

8 Periodicity Using Shift Periodic Operators

Therefore, x˜ solves (8.1.13). The proof that x solves (8.1.12) is similar and hence we omit. Proposition 8.1.1 implies that x is P-periodic with respect to Definition 8.1.13 if and only if x˜ is P-periodic with respect to Definition 8.1.12. Suppose that a : qN0 → R is a function with (1 + (q − 1)ta(t))  0 for all t ∈ qN0 . Based upon the function a define the functions ea (q n, q m ) :=

n−1 

(1 + (q − 1)q k a(q k ))

k=m

and

e a (q n, q m ) := ea (q n, q m )−1 .

Multiplying both sides of Eqs. (8.1.12) and (8.1.13) with ea (t, 1) and ea˜ (t, 1) , respectively, we obtain the following integral equations: 

∫q



Pt

  e a q P t, s f (s, sx (s)) dq s,

x(t) = q P e a q P t, t x (t) + q P

(8.1.14)

t

and





∫q

Pt

x(t) ˜ = e a˜ q P t, t x˜ (t) +

  e a˜ q P t, s f (s, x˜ (s)) dq s,

(8.1.15)

t

for t ∈ qN0 , where the q-integral is defined by ∫

qn

qm

n−1 

f (s)dq s := (q − 1)

q k f (q k ).

(q-integral)

k=m

Next, the generalizations of (8.1.14) and (8.1.15) have the form of q-Volterra integral equations as follows: ∫q

Pt

x(t) = g (t, t x (t)) +

C (t, s) f (s, sx (s)) dq s,

(8.1.16)

t

and

∫q x(t) ˜ = g˜ (t, x˜ (t)) +

Pt

C˜ (t, s) f˜ (s, x˜ (s)) dq s,

(8.1.17)

t

where g, g, ˜ f , f˜ : qN0 × R → R are continuous in their second variable and C, C˜ : N N 0 0 q × q → R, t C˜ (t, s) = C (t, s) , s f˜ (t, x˜ (t)) = t f (t, t x (t)) ,

(8.1.18) (8.1.19)

8.1 Periodicity in Shifts

361

and g˜ (t, x˜ (t)) = tg (t, t x (t)) .

(8.1.20)

Similar to the Proposition 8.1.2, we can establish a linkage between two periodicity notions in terms of periodic solutions of the integral equations (8.1.16) and (8.1.17). Proposition 8.1.3. Assume that C, f , g and x satisfy   q P C q P t, q P s = C(t, s),    q P f q P t, q P t x q P t = f (t, t x (t)) ,    q P g q P t, q P t x q P t = g (t, t x (t)) .

(8.1.21) (8.1.22) (8.1.23)

Then x(t) is a periodic solution of (8.1.16) with respect to Definition 8.1.3 if and only if x˜ (t) = t x (t) is a periodic solution of (8.1.17) with respect to Definition 8.1.4. Proof. Assume (8.1.21–8.1.23) hold and let x(t) solve (8.1.16) and be P-periodic with respect to Definition 8.1.3. Let us multiply both sides of (8.1.16) by t, i.e., ∫q

Pt

t x(t) = tg (t, t x (t)) +

tC (t, s) f (s, sx (s)) dq s, t

or

∫q

Pt

t x(t) = tg (t, t x (t)) +

t C (t, s) s f (s, sx (s)) dq s. s

t

By employing (8.1.18–8.1.20), we get ∫q x(t) ˜ = g˜ (t, x˜ (t)) +

Pt

C˜ (t, s) f˜ (s, x˜ (s)) dq s.

(8.1.24)

t

Notice that x(t) ˜ is a P-periodic solution of (8.1.24) with respect to Definition 8.1.4. To show this, consider 







x(q ˜ P t) = q P t x q P t = g˜ q P t, x˜ q P t



q ∫

2P t

+

  C˜ q P t, s f˜ (s, x˜ (s)) dq s

qPt

   ∫      = g˜ q P t, x˜ q P t + C˜ q P t, q P s f˜ q P s, x˜ q P s dq s qPt





t

= q tg q t, q t x q t P

∫q + t

Pt

P

P

P



   qPt  P P  P C q t, q s q s f q P s, q P sx q P s dq s. s

362

8 Periodicity Using Shift Periodic Operators

Using (8.1.21–8.1.23) we get ∫q

Pt

t C (t, s) s f (s, sx (s)) dq s s

x(q ˜ P t) = tg (t, t x (t)) + t

∫q = g˜ (t, x˜ (t)) +

Pt

C˜ (t, s) f˜ (s, x˜ (s)) dq s. t

The proof of the necessity part can be done by following a similar procedure used in the sufficiency part, hence, we omit it. Remark 8.1.1. Above given results reveal the basic linkage between two periodicity definitions [45] and [9]. In particular, Propositions 8.1.2 and 8.1.3 show the relationship between q-difference equations having periodic solutions with respect to Definitions 8.1.4 and 8.1.3. This provides a procedure for the rearrangement of an existence result based on Definition 8.1.4 to obtain an existence result based on Definition 8.1.3, and vice versa.

8.2 Floquet Theory on Time Scales Periodic in Shifts δ± Floquet theory is an important tool to study periodic solutions and stability theory of dynamic systems. Indeed, this is a century-old theory which is introduced by Gaston Floquet in 1883 in order to analyze the solutions of systems of linear differential equations with periodic coefficients (see [82]). Afterwards, this theory is extended to difference equations/systems, integral equations, integro-differential equations, and partial differential equations (see [21, 22, 30, 35, 110], and [105]). The time scale variant of Floquet theory was first studied by Ahlbrandt and Ridenhour [24], and this study basically focuses on Floquet’s theorem on mixed domains and Putzer representations of matrix logarithms. Adamec [3] criticizes the approach in [24] and stated his concern about suitability of using real exponential function instead of time scale exponential function. Afterwards, DaCunha’s work made a significant contribution to Floquet theory on time scales. Lyapunov stability and unified Floquet theory regarding Lyapunov transformations are first handled by DaCunha in 2004 in his Ph.D. thesis [71]. The study in [71] not only improves the results of [24] but also extends the study of Floquet theory on general time scales. In [71], DaCunha unified Floquet theory for nonautonomous linear dynamic systems based on Lyapunov transformations by using matrix exponential on time scales (see [50, Section 5]). Afterwards, DaCunha and Davis improved the results of [71] in [73]. Note that in [73] and [71] the authors study Floquet theory only for the additive periodic time scales. However, additive periodicity assumption is a strong restriction for the class of time scales on which dynamic equations with periodic solutions can be constructed. For instance, the time scale

8.2 Floquet Theory on Time Scales Periodic in Shifts δ±

363

qZ := {q n : n ∈ Z} ∪ {0}, q > 1 is not additive periodic. Since q-difference equations are the dynamic equations constructed on the time scale qZ , there is no way to investigate periodic solutions of q-difference equations by means of additive periodicity arguments. A q-difference equation is an equation including a q-derivative Dq , given by Dq ( f ) (t) =

f (qt) − f (t) , t ∈ qZ , (q − 1) t

of its unknown function. Notice that the q-derivative Dq ( f ) of a function f turns into ordinary derivative f  if we let q → 1. As an alternative to difference equations the q-difference equations are used for the discretization of differential equations. Hence, one may intuitively deduce that if periodicity is possible for the solutions of differential equations, then it should be possible to study the periodicity of solutions of q-difference equations. In this section, Floquet theory is reconstructed on more general domains including both additively and nonadditively periodic time scales. All the results in this section can be found in [9] and [107]. Hereafter, T is supposed to be a T-periodic time scale in shifts δ± and that the shift operators δ± are Δ-differentiable with r d-continuous derivatives. For brevity, the term “periodic in shifts” is used to mean periodicity in shifts δ± . Throughout the chapter, the notation δ±T (t) is employed to indicate the shifts δ± (T, t). Furthermore, the notation δ±(n) (T, t), n ∈ N, represents the n-times composition of shifts, δ±T , by itself, namely, δ±(n) (T, t) := δ±T ◦ δ±T ◦ ... ◦ δ±T (t) . ABCD n−times

Observe that, the period of a function f does not have to be equal to period of the time scale on which f is determined. However, for simplicity of the results the period of time scale T is assumed to be equal to period of all the functions defined on T. The following definition plays a key role in the following analysis: Definition 8.2.1 ([73, Definition 2.1]). A Lyapunov transformation is an invertible matrix L (t) ∈ Cr1d (T, Rn×n ) satisfying L (t) ≤ ρ and |det L (t)| ≥ η for all t ∈ T where ρ and η are arbitrary positive reals.

8.2.1 Floquet Theory Based on New Periodicity Concept: Homogeneous Case Consider the following regressive nonautonomous linear dynamic system x Δ (t) = A (t) x (t) , x (t0 ) = x0,

(8.2.1)

364

8 Periodicity Using Shift Periodic Operators

where A : T∗ → Rn×n is Δ-periodic in shifts with period T. Observe that if the time scale is additively periodic, then δ±Δ (T, t) = 1 and Δ-periodicity in shifts becomes the same as the periodicity in shifts. Hence, the homogeneous system focused in this section is more general than that of [71] and [73]. In [71], the solution of the system (8.2.1) (for an arbitrary matrix A) is expressed by the equality x (t) = Φ A(t, t0 )x0 , where Φ A(t, t0 ), called the transition matrix for the system (8.2.1), is given by ∫t

∫t

Φ A(t, t0 ) = I +

A(τ1 )Δτ1 + t0

A(τ1 ) t0

∫t +

∫τ1

∫τ1 A(τ1 )

t0

A(τ2 )Δτ2 Δτ1 + . . . t0

∫τi−1 A(τ2 ) . . . A(τi )Δτi . . . Δτ1 + . . . .

t0

(8.2.2)

t0

As mentioned in [73] the matrix exponential e A(t, t0 ) is not always identical to Φ A(t, t0 ) since A (t) e A (t, t0 ) = e A (t, t0 ) A (t) is always true but the equality A (t) Φ A (t, t0 ) = Φ A (t, t0 ) A (t) is not. One has e A(t, t0 ) ≡ Φ A(t, t0 ) only if the matrix A satisfies ∫t

∫t A(τ)Δτ =

A(t) s

A(τ)Δτ A(t). s

In preparation for the next result, the following set + , P (t0 ) := δ+(k) (T, t0 ) , k = 0, 1, 2, . . .

(8.2.3)

and the function Θ (t) :=

m(t) 

  δ− δ+(j−1) (T, t0 ) , δ+(j) (T, t0 ) + G (t) ,

(8.2.4)

j=1

where

+ , m (t) := min k ∈ N : δ+(k) (T, t0 ) ≥ t 

and G (t) := are defined.

  if t ∈ P (t0 ) −δ− t, δ+(m(t)) (T, t0 ) if t  P (t0 )

(8.2.5)

0

(8.2.6)

8.2 Floquet Theory on Time Scales Periodic in Shifts δ±

365

Remark 8.2.1. For an additive periodic time scale one always has Θ (t) = t − t0 . In order to construct the matrix R which is a solution of the matrix exponential equation, the definition of real power of a matrix is essential. Definition 8.2.2 (Real Power of a Matrix [73, Definition A.5]). Given an n × n k nonsingular matrix M with elementary divisors {(λ − λi )mi }i=1 and any r ∈ R, the real power of the matrix M is given by r

M :=

k  i=1

Pi (M) λir

⎡m   j⎤ ⎢ i −1 Γ (r + 1) M − λi I ⎥⎥ ⎢ ⎢ ⎥, λi ⎢ j=0 j!Γ (r − j + 1) ⎥ ⎣ ⎦

(8.2.7)

where Pi (λ) := ai (λ) bi (λ) ,   bi (λ) := Π kji λ − λ j , j=1

 ai (λ) 1 = , p (λ) i=1 (λ − λi )mi k

and p (λ) is the characteristic polynomial of M. It is known by Proposition A.3 in [73] that M s+r = M s M r for any r, s ∈ R

Theorem 8.2.1. For a nonsingular, n×n constant matrix M a solution R : T → Cn×n of matrix exponential equation   eR δ+T (t0 ) , t0 = M can be given by M T [Θ(σ(t))−Θ(s)] − I , σ (t) − s 1

R (t) = lim s→t

(8.2.8)

where I is the n × n identity matrix and Θ is as in (8.2.4). Proof. Let’s construct the matrix exponential function eR (t, t0 ) as follows: eR (t, t0 ) := M T Θ(t) for t ≥ t0 , 1

(8.2.9)

where Θ is given by (8.2.4) and real power of a nonsingular matrix M is given by Definition 8.2.2. To show that the function eR (t, t0 ) constructed in (8.2.9) is the matrix exponential we first observe that eR (t0, t0 ) = M T Θ(t0 ) = I, 1

366

8 Periodicity Using Shift Periodic Operators

where we use (8.2.9) along with Θ (t0 ) = G (t0 ) = 0. Second, differentiating (8.2.9) yields eΔR (t, t0 ) = R (t) eR (t, t0 ) . To see this, first suppose that t is right-scattered. Then, eΔR (t, t0 ) =

eR (σ (t) , t0 ) − eR (t, t0 ) σ (t) − t M T Θ(σ(t)) − M T Θ(t) σ (t) − t 1

=

1

M T [Θ(σ(t))−Θ(t)] − I 1 Θ(t) MT = σ (t) − t = R (t) eR (t, t0 ) . 1

If t is right dense, then σ (t) = t. Setting s = t + h in (8.2.4) and using (8.2.9) one can obtain eΔR (t, t0 ) = lim

h→0

eR (t + h, t0 ) − eR (t, t0 ) h M T Θ(t+h) − M T Θ(t) h 1

= lim

h→0

1

M T [Θ(t+h)−Θ(t)] − I 1 Θ(t) MT h→0 h = R (t) eR (t, t0 ) . 1

= lim

In any case, we have eΔR (t, t0 ) = R (t) eR (t, t0 ). Finally, we obtain     Θ δ+T (t0 ) = δ− t0, δ+T (t0 ) = δ+T (t0 ) = T, and therefore,

  1 T eR δ+T (t0 ) , t0 = M T Θ(δ+ (t0 )) = M.

Corollary 8.2.1. The eigenvectors of the matrices R (t) and M are same. Proof. For any eigenpairs {λi, vi }, i = 1, 2, ..., n of M, using Mvi = λi vi one can write that 1 1 [Θ(σ(t))−Θ(s)] lim M T [Θ(σ(t))−Θ(s)] vi = lim λiT vi . s→t

s→t

This implies 1 T

λ R(t)vi = lim  i s→t 

[Θ(σ(t))−Θ(s)]

σ(t) − s

− 1  vi . 

(8.2.10)

8.2 Floquet Theory on Time Scales Periodic in Shifts δ±

!

Substituting γi (t) = lims→t

1 [Θ(σ(t ))−Θ(s)]

λiT

σ(t)−s

367

"

−1

into (8.2.10) one can conclude that

n . R(t) has the eigenpairs {γi (t), vi }i=1

Lemma 8.2.1. Let T be a time scale and P ∈ R (T∗, Rn×n ) be a Δ-periodic matrix valued function in shifts with period T, i.e.   P (t) = P δ±T (t) δ±ΔT (t) . Then the solution of the dynamic matrix initial value problem Y Δ (t) = P (t) Y (t) , Y (t0 ) = Y0,

(8.2.11)

is unique up to a period T in shifts. That is   Φ P (t, t0 ) = Φ P δ+T (t) , δ+T (t0 )

(8.2.12)

for all t ∈ T∗ . Proof. By [50], the unique solution to (8.2.11) is Y (t) = Φ P (t, t0 )Y0 . Observe that Y Δ (t) = ΦΔP (t, t0 )Y0 = P(t)Φ P (t, t0 )Y0 and Y (t0 ) = Φ P (t0, t0 )Y0 = Y0 . To verify (8.2.12) we first need to show that Φ P (δ+ (T, t), δ+ (T, t0 ))Y0 is also solution for (8.2.11). Since the shift operator δ+ is strictly increasing, the chain rule [50, Theorem 1.93] yields [Φ P (δ+ (T, t), δ+ (T, t0 ))Y0 ]Δ = P (δ± (T, t)) δ+Δ (T, t)Φ P (δ+ (T, t), δ+ (T, t0 ))Y0 = P(t)Φ P (δ+ (T, t), δ+ (T, t0 ))Y0 . On the other hand, we have Φ P (δ+ (T, t), δ+ (T, t0 ))t=t0 Y0 = Φ P (δ+ (T, t0 ), δ+ (T, t0 ))Y0 = Y0 . This means Φ P (δ+ (T, t), δ+ (T, t0 ))Y0 solves (8.2.11). From the uniqueness of solution of (8.2.11), we get (8.2.12). Corollary 8.2.2. Let T be a time scale and P ∈ R (T∗, Rn×n ) be a Δ-periodic matrix valued function in shifts. Then   e P (t, t0 ) = e P δ+T (t) , δ+T (t0 ) . (8.2.13) Theorem 8.2.2 (Floquet Decomposition). Let A be a matrix valued function that is Δ-periodic in shifts with period T. The transition matrix for A can be given in the form

368

8 Periodicity Using Shift Periodic Operators

Φ A (t, τ) = L (t) eR (t, τ) L −1 (τ) , for all t, τ ∈ T∗,

(8.2.14)

where R : T → R is Δ-periodic function in shifts and L (t) ∈ Cr1d (T∗, Rn×n ) is periodic in shifts with the period T. n×n

Proof. Let the matrix function   R is defined as in Theorem 8.2.1 with a constant n × n matrix M := Φ A δ+T (t0 ) , t0 . Then,     eR δ+T (t0 ) , t0 = Φ A δ+T (t0 ) , t0 , and the Lyapunov transformation L (t) is defined as L (t) := Φ A (t, t0 ) e−1 R (t, t0 )

(8.2.15)

(see Definition 8.2.1). Obviously, L (t) ∈ Cr1d (T∗, Rn×n ) and L is invertible. The equality (8.2.16) Φ A (t, t0 ) = L (t) eR (t, t0 ) along with (8.2.15) implies −1 Φ A (t0, t) = e−1 (t) R (t, t0 ) L

= eR (t0, t) L −1 (t) .

(8.2.17)

Combining (8.2.16) and (8.2.17), one can obtain (8.2.14). The periodicity in shifts of L is shown by using (8.2.12)–(8.2.13) as follows:       T δ , t L δ+T (t) = Φ A δ+T (t) , t0 e−1 (t) 0 + R       T T = Φ A δ+ (t) , δ+ (t0 ) Φ A δ+T (t0 ) , t0 eR t0, δ+T (t)         = Φ A δ+T (t) , δ+T (t0 ) Φ A δ+T (t0 ) , t0 eR t0, δ+T (t0 ) eR δ+T (t0 ) , δ+T (t)     = Φ A δ+T (t) , δ+T (t0 ) eR δ+T (t0 ) , δ+T (t)     T T = Φ A δ+T (t) , δ+T (t0 ) e−1 δ , δ (t) (t ) 0 + + R = Φ A (t, t0 ) e−1 R (t, t0 ) = L (t) . The following result is the extension of [73, Theorem 3.7] and it can be proved similarly. Theorem 8.2.3. Assume that the transition matrix Φ A of the unified Floquet system (8.2.1) has the decomposition of the form Φ A (t, t0 ) = L (t) eR (t, t0 ). Then, x (t) = Φ A (t, t0 ) x0 is a solution of (8.2.1) if and only if the linear dynamic equation z Δ (t) = R (t) z (t) , z (t0 ) = x0 . has a solution of the form z (t) = L −1 (t) x (t).

8.2 Floquet Theory on Time Scales Periodic in Shifts δ±

369

Next, the necessary and sufficient condition for the existence of solution of Floquet system (8.2.1) which is periodic in shifts is given. Theorem 8.2.4. The solution of the unified Floquet system (8.2.1) has a T-periodic solution in shifts with an initial state x (t0 ) = x0  0 if and only if at least one of the eigenvalues of     eR δ+T (t0 ) , t0 = Φ A δ+T (t0 ) , t0 is 1. Proof. Let x (t) be a solution of the periodic system (8.2.1) which is T-periodic in shifts corresponding with the nonzero initial state x (t0 ) = x0 . Then according to Theorem 8.2.2, the Floquet decomposition of x can be written as x (t) = Φ A (t, t0 ) x0 = L (t) eR (t, t0 ) L −1 (t0 ) x0, which also yields       x δ+T (t) = L δ+T (t) eR δ+T (t) , t0 L −1 (t0 ) x0 . By T-periodicity of x and L in shifts, one can obtain   eR (t, t0 ) L −1 (t0 ) x0 = eR δ+T (t) , t0 L −1 (t0 ) x0, and therefore,     eR (t, t0 ) L −1 (t0 ) x0 = eR δ+T (t) , δ+T (t0 ) eR δ+T (t0 ) , t0 L −1 (t0 ) x0 .   Since eR δ+T (t) , δ+T (t0 ) = eR (t, t0 ) the last equality implies   eR (t, t0 ) L −1 (t0 ) x0 = eR (t, t0 ) eR δ+T (t0 ) , t0 L −1 (t0 ) x0   L −1 (t0 ) x0 = eR δ+T (t0 ) , t0 L −1 (t0 ) x0 .   Hence, the matrix eR δ+T (t0 ) , t0 has an eigenvector L −1 (t0 ) x0 corresponding to the eigenvalue 1.   Conversely, assume that 1 is an eigenvalue of eR δ+T (t0 ) , t0 with corresponding  eigenvector z0. This means z0 is real valued and nonzero. Using eR (t, t0 ) = eR δ+T (t) , δ+T (t0 ) , one can arrive at the following equality:     z δ+T (t) = eR δ+T (t) , t0 z0     = eR δ+T (t) , δ+T (t0 ) eR δ+T (t0 ) , t0 z0   = eR δ+T (t) , δ+T (t0 ) z0

and thus

370

8 Periodicity Using Shift Periodic Operators

= eR (t, t0 ) z0 = z (t) , which shows that z (t) = eR (t, t0 ) z0 is T-periodic in shifts. Applying the Floquet decomposition and setting x0 := L (t0 ) z0 , the nontrivial solution x of (8.2.1) is obtained as follows: x (t) = Φ A (t, t0 ) x0 = L (t) eR (t, t0 ) L −1 (t0 ) x0 = L (t) eR (t, t0 ) z0 = L (t) z (t) , which is T-periodic in shifts since L and z are T-periodic in shifts. This completes the proof. ' ±k ( ±k ∪ Example 8.2.1. Suppose that T = ∪∞ {0}. Then, T is 3-periodic in k=0 3 , 2.3 shifts δ± (s, t) = s±1 t. Setting A (t) = 1t I2×2 , one may get A (δ± (3, t)) δ±Δ (3, t) = A (3t) 3 = A (t) which shows that A is Δ-periodic in shifts with the period 3. Consider the system 1 0 x Δ (t) = t 1 x (t) , x(1) = x0 0 t whose transition matrix is given by Φ A (t, 1) = Then



0 e1/t (t, 1) . 0 e1/t (t, 1)

   0 e (3, 1) . Φ A δ+3 (1) , 1 = Φ A (3, 1) = 1/t 0 e1/t (3, 1)

As in Theorem 8.2.1, one can write that  0 e (3, 1) eR (3, 1) = Φ A (3, 1) = 1/t = M. 0 e1/t (3, 1) On the other hand, by (8.2.8) and (8.2.9) one can obtain eR (t, 1) = M 3 Θ(t)  1 3m(t)−3m(t ) /t ] if t  P (1) M 3[ , = m(t) M if t ∈ P (1) 1

and M 3 [Θ(σ(t))−Θ(s)] − I R (t) = lim s→t σ (t) − s    1 3 2 Θ 3 [ ( 2 t )−Θ(t)] − I if σ (t) > t t M = , '1 (Δ Θ Log if σ = t (t) [M] (t) 3 1

8.2 Floquet Theory on Time Scales Periodic in Shifts δ±

371

where P (t) and m (t) are defined by (8.2.3) and (8.2.5), respectively. Then the matrix function L (t) which is 3-periodic in shifts is given as L (t) = Φ A (t, 1) e−1 R (t, 1)   e1/t (t, 1) e1/t (3, 1) 0 0 = 0 e1/t (t, 1) 0 e1/t (3, 1)

− 31 Θ(t)

.

Example 8.2.2. Consider the time scale T = R that is periodic in shifts δ± (s, t) = s±1 t associated with the initial point t0 = 1. Let us define the matrix function A (t) : T∗ → Rn×n as follows:   ⎤ ⎡1 ln |t | 0 ⎥ ⎢ t sin π ln 2 ⎥.  A (t) = ⎢⎢ ln |t | ⎥ 1 0 ⎢ t sin π ln 2 ⎥⎦ ⎣ Then A(t) is Δ-periodic in shifts with the period 4. The following system   ⎤ ⎡1 ln t 0 sin π ⎥ ⎢ t ln 2 Δ ⎥ x (t) , x(1) = x0 ⎢   x (t) = ⎢ 1 ln t ⎥ sin π 0 ⎢ t ln 2 ⎥⎦ ⎣ has the transition matrix

 Φ A (t, 1) =

where u(t) =

1 t

0 eu(t) (t, 1) , 0 eu(t) (t, 1)

  ln t sin π ln 2 . Moreover, 

Φ A δ+4 (1) , 1



 = Φ A (4, 1) =

10 = M. 01

Thus, R(t) is 2 × 2 zero matrix, and hence, eR (t, 1) = I. Finally, the matrix function L(t) which is 4-periodic in shifts is obtained as follows: L(t) = Φ A (t, 1) e−1 R (t, 1) = Φ A (t, 1) .

8.2.2 Floquet Theory Based on New Periodicity Concept: Nonhomogeneous Case Now we turn our attention to the nonhomogeneous regressive time varying linear dynamic initial value problem x Δ (t) = A (t) x (t) + F (t) , x (t0 ) = x0,

(8.2.18)

372

8 Periodicity Using Shift Periodic Operators

where A : T∗ → Rn×n , F ∈ R (T∗, Rn ). Hereafter, both A and F are supposed to be Δ-periodic in shifts with the period T. Lemma 8.2.2. A solution x (t) of (8.2.18) is T-periodic in shifts if and only if   x δ+T (t0 ) = x (t0 ) for all t ∈ T∗ . Proof. Suppose that x (t) is T-periodic in shifts. Let us define z (t) as   z (t) = x δ+T (t) − x (t) .

(8.2.19)

Obviously z (t0 ) = 0. Moreover, if we take delta derivative of both sides of (8.2.19), we have the following: $Δ  #  z Δ (t) = x δ+T (t) − x (t)   = x Δ δ+T (t) − x Δ (t)   Δ = x Δ δ+T (t) δ+T (t) − x Δ (t)     Δ  Δ  = A δ+T (t) x δ+T (t) δ+T (t) + F δ+T (t) δ+T (t) − A (t) x (t) − F (t) . Since A and F are both Δ-periodic in shifts with the period T, we have   z Δ (t) = A (t) x δ+T (t) + F (t) − A (t) x (t) − F (t) #  $  = A (t) x δ+T (t) − x (t) = A (t) z (t) .   By uniqueness of solutions, we can conclude that z (t) ≡ 0 and that x δ+T (t) = x (t) for all t ∈ T∗ . Theorem 8.2.5. The solution of (8.2.18) is T-periodic in shifts δ± for any initial point t0, and corresponding initial state x (t0 ) = x0 if and only if the T-periodic homogeneous initial value problem z Δ (t) = A (t) z (t) , z (t0 ) = z0,

(8.2.20)

has no T-periodic solution in shifts for any nonzero initial state z (t0 ) = z0 . Proof. In [5], the following representation for the solution of (8.2.18) is given: x (t) = X (t) X

−1

∫t (τ) x0 +

X (t) X −1 (σ (s)) F (s) Δs,

τ

where X (t) is a fundamental matrix solution of the homogenous system (8.2.1) with respect to initial condition x (τ) = x0 . As it is done in [5], x (t) can be expressed as

8.2 Floquet Theory on Time Scales Periodic in Shifts δ±

∫

t

x (t) = Φ A (t, t0 ) x0 +

373

Φ A (t, σ (s)) F (s) Δs.

t0

  By the previous lemma x (t) is T-periodic in shifts if and only if x δ+T (t0 ) = x0 or equivalently #



I − Φ A δ+T (t0 ) , t0

$

T δ∫ + (t0 )

  Φ A δ+T (t0 ) , σ (s) F (s) Δs.

x0 =

(8.2.21)

t0

By guidance of Theorem 8.2.4, it should be shown that  (8.2.18) has a solution with respect to initial condition x (t0 ) = x0 if and only if eR δ+T (t0 ) , t0 has no eigenvalues equal to 1.    Let eR δ+T (η) , η = Φ A δ+T (η) , η , for some η ∈ T∗ , has no eigenvalues equal to 1. That is,  # $ det I − Φ A δ+T (η) , η  0. Invertibility and periodicity of Φ A imply    $  # 0  det Φ A δ+T (t0 ) , δ+T (η) I − Φ A δ+T (η) , η Φ A (η, t0 )   $  # = det Φ A δ+T (t0 ) , δ+T (η) Φ A (η, t0 ) − Φ A δ+T (t0 ) , t0 .

(8.2.22)

'  ( By periodicity of Φ A, the invertibility of I − Φ A δ+T (t0 ) , t0 is equivalent to (8.2.22) for any t0 ∈ T∗ . Thus, (8.2.21) has a solution #

x0 = I − Φ A



δ (t )  $ −1 ∫+ 0   T δ+ (t0 ) , t0 Φ A δ+T (t0 ) , σ (s) F (s) Δs T

t0

for any t0 ∈ T∗ and for any Δ-periodic function F in shifts with period T. Suppose that (8.2.21) has a solution for every t0 ∈ T∗ and every Δ-periodic function F in shifts with period T. Let us define the set P− (t) as , + P− (t) = k ∈ Z : δ−(k) (T, t) .   It is clear that, P− (t) = P− δ+T (t) . Additionally, let the function ξ be defined by 

ξ (t) :=



δ+ΔT (s)

s ∈P− (t)∩[t0,t)



= δ+ΔT (δ− (T, t))

 −1

 −1

  −1   −1   −   × δ+ΔT δ−(2) (T, t) × . . . × δ+ΔT δ−(m (t)) (T, t) ,

. where m− (t) = max k ∈ Z : δ−(k) (T, t) ≥ t0 . By definition of ξ, one can write

374

8 Periodicity Using Shift Periodic Operators

  ξ δ+T (t) =



 s ∈P− ( δ+T (t))∩[t0,δ+T (t))





=

s ∈P− (t)∩[t0,δ+T (t))

  −1 = δ+ΔT (t) 

= δ+ΔT (t)

 −1

δ+ΔT (s)

δ+ΔT (s)

 −1





 −1

δ+ΔT (s)

 −1

s ∈P− (t)∩[t0,t)

ξ (t) ,

which shows that ξ is Δ-periodic in shifts with period T. For an arbitrary t0 and corresponding F0, define a regressive and Δ-periodic function F in shifts as follows:    ' (8.2.23) F (t) := Φ A σ (t) , δ+T (t0 ) ξ (t) F0, t ∈ t0, δ+T (t0 ) ∩ T. Then, T δ∫ + (t0 )



T δ∫ + (t0 )



Φ A δ+T (t0 ) , σ (s) F (s) Δs = F0

ξ (s) Δs.

t0

(8.2.24)

t0

Thus, (8.2.21) can be rewritten as follows: #



I − Φ A δ+T (t0 ) , t0

$

T δ∫ + (t0 )

x0 =

ξ (s) Δs.

(8.2.25)

t0

For any F which is defined in (8.2.23), and hence for any corresponding F0 , (8.2.25) has a solution for x0 by assumption. Thus,  $ # det I − Φ A δ+T (t0 ) , t0  0.     Consequently, eR δ+T (t0 ) , t0 = Φ A δ+T (t0 ) , t0 has no eigenvalue 1. Then, we can conclude by Theorem 8.2.4 that (8.2.20) has no periodic solution in shifts.

8.2.3 Floquet Multipliers and Floquet Exponents of Unified Floquet Systems This part of the thesis is devoted to Floquet multipliers and Floquet exponents of systems periodic with respect to new periodicity perception on time scales. Let Φ A (t, t0 ) be the transition matrix and Φ (t) the fundamental matrix at t = τ of the system (8.2.1). Then any fundamental matrix Ψ (t) can be represented as follows:

8.2 Floquet Theory on Time Scales Periodic in Shifts δ±

Ψ (t) = Φ (t) Ψ (τ) or Ψ (t) = Φ A (t, t0 ) Ψ (t0 ) .

375

(8.2.26)

Additionally, for a nonzero initial vector x0 ∈ Rn , the monodromy operator M : Rn → Rn is defined by     (8.2.27) M (x0 ) := Φ A δ+T (t0 ) , t0 x0 = Ψ δ+T (t0 ) Ψ−1 (t0 ) x0. In parallel to preceding chapter, the eigenvalues of monodromy operator M are called Floquet (characteristic) multipliers of the system (8.2.1). Similar to [73, Theorem 5.2 (i)], the following remark can be given: Remark 8.2.2. The monodromy operator of the linear system (8.2.1) is invertible, and consequently, every Floquet (characteristic) multiplier is nonzero. Theorem 8.2.6. The monodromy operator M corresponding to different fundamental matrices of the system (8.2.1) is unique. Proof. Suppose that M1 and M2 are the monodromy operators corresponding to fundamental matrices Ψ1 (t) and Ψ2 (t) , respectively. One can write the monodromy operator M2 (x0 ) corresponding to Ψ2 (t) as   M2 (x0 ) = Ψ2 δ+T (t0 ) Ψ2−1 (t0 ) x0 . Using (8.2.26) yields   M2 (x0 ) = Ψ2 δ+T (t0 ) Ψ2−1 (t0 ) x0   = Ψ1 δ+T (t0 ) Ψ2 (τ) Ψ2−1 (τ) Ψ1−1 (t0 ) x0   = Ψ1 δ+T (t0 ) Ψ1−1 (t0 ) x0 = M1 (x0 ) . By using Theorem 8.2.2, (8.2.26), and (8.2.27), one can obtain Φ A (t, t0 ) = Ψ1 (t) Ψ1−1 (t0 ) = L (t) eR (t, t0 ) L −1 (t0 ) and

    M (x0 ) = Φ A δ+T (t0 ) , t0 x0 = Ψ1 δ+T (t0 ) Ψ1−1 (t0 ) x0 .

(8.2.28)

(8.2.29)

The following equation is obtained by combining (8.2.28) and (8.2.29)           Φ A δ+T (t0 ) , t0 = Ψ1 δ+T (t0 ) Ψ1−1 (t0 ) = L δ+T (t0 ) eR δ+T (t0 ) , t0 L −1 δ+T (t0 ) . By using the periodicity in shifts of L, the following equality     Φ A δ+T (t0 ) , t0 = L (t0 ) eR δ+T (t0 ) , t0 L −1 (t0 )

(8.2.30)

376

8 Periodicity Using Shift Periodic Operators

is obtained. Hence, the Floquet multipliers  of the unified Floquet system (8.2.1) are the eigenvalues of the matrix eR δ+T (t0 ) , t0 . Definition 8.2.3. The Floquet exponent of the system (8.2.1) is the function γ (t) satisfying the equation   eγ δ+T (t0 ) , t0 = λ, where λ is the Floquet multiplier of the system. The next result can be proven similar to [73, Theorem 5.3]. Theorem 8.2.7. Let R (t) be a matrix function as in Theorem 8.2.1, with eigenvalues γ1 (t) , . . . , γn (t) repeated according to multiplicities. Then γ1k (t) , . . . , γnk (t) are the eigenvalues of Rk (t) and eigenvalues of eR are eγ1 , . . . , eγn . Theorem 8.2.8. The Floquet exponent γ of (8.2.1) with corresponding Floquet mul◦ tiplier λ is not unique. That is, γ (t) ⊕ ı δ T 2πk is also a Floquet exponent for (8.2.1) + (t0 )−t0 for all k ∈ Z. Proof. For all k ∈ Z, we have     eγ ⊕◦ı 2π k δ+T (t0 ) , t0 = eγ δ+T (t0 ) , t0 e◦ı T ( t )−t δ+ 0 0



 2π k T ( t )−t δ+ 0 0



= eγ δ+T (t0 ) , t0 exp    

= eγ δ+T (t0 ) , t0 exp    

= eγ δ+T (t0 ) , t0 exp  





δ+T (t0 ) , t0



  ◦ log 1 + μ (τ) ı δ T 2πk (t )−t

T δ∫ + (t0 )

+

t0

0

 Δτ 

0

μ (τ)   log exp i δ2πkμ(τ) T (t )−t

T δ∫ + (t0 )

+

0

μ (τ)

t0





0

 Δτ  

T δ∫ + (t0 )

t0

 i2πk  Δτ δ+T (t0 ) − t0  

= eγ δ+T (t0 ) , t0 ei2πk   = eγ δ+T (t0 ) , t0 , which gives the desired result. Lemma 8.2.3. Let T be a time scale that is p-periodic in shifts δ± associated with p δ (t)−t the initial point t0 and k ∈ Z. If δ p+(t )−t ∈ Z, then the functions e◦ı 2π k and +

e ◦ı

2π k p δ+ ( t0 )−t0

are p periodic in shifts.

0

0

p δ+ ( t0 )−t0

8.2 Floquet Theory on Time Scales Periodic in Shifts δ± p

Proof. If

δ+ (t)−t p δ+ (t0 )−t0

377

∈ Z, then p

e◦ı

2π k p δ+ ( t0 )−t0

δ+ (t)

∫   p  i2πk δ+ (t) , t0 = exp  Δτ  p δ+ (t0 ) − t0  t0  p δ + (t) ∫ ∫t

 i2πk i2πk

   = exp  Δτ  exp  Δτ  p p δ+ (t0 ) − t0 δ (t0 ) − t0 t0 +   t    ∫t p δ (t) − t i2πk

 Δτ  = exp i2πk p+ exp  p δ+ (t0 ) − t0 δ+ (t0 ) − t0 t0  ∫t i2πk

 Δτ  = e◦ı 2π k (t, t0 ) = exp  p p δ (t0 ) − t0 δ+ ( t0 )−t0 t0 + 

which proves the periodicity of e◦ı

2π k p δ+ ( t0 )−t0

proven by using the periodicity of e◦ı

. The periodicity of e ◦ı

2π k p δ+ ( t0 )−t0

2π k p δ+ ( t0 )−t0

can be

and the identity e α = 1/eα .

p

δ (t)−t

Remark 8.2.3. Notice that the condition δ p+(t )−t ∈ Z holds not only for all additive 0 + 0 periodic time scales but also for the many time scales that are periodic in shifts. ' ±k ( ±(k+1) ∪ For example the time scales 2Z and ∪∞ {0} are periodic in shifts k=0 3 , 2.3 δ± (s, t) = s±1 t associated with the initial point t0 = 1 and the condition is always satisfied for p = 2 and p = 3, respectively.

p

δ+ (t)−t p δ+ (t0 )−t0

∈Z

Theorem 8.2.9. If the unified Floquet system (8.2.1) has a Floquet exponent γ (t), then the corresponding transition matrix Φ A can be decomposed as Φ A (t, t0 ) = L (t) eR (t, t0 ) , where γ (t) is an eigenvalue of R (t) . Proof. Consider the Floquet decomposition Φ A (t, t0 ) = * L (t) eR* (t, t0 ) and let γ be a Floquet exponent of (8.2.1) with corresponding Floquet   multiplier λ. Moreover, there is an eigenvalue γ˜ (t) of R˜ (t) so that eγ˜ δ+T (t0 ), t0 = λ, where γ˜ (t) can be defined as 2πk ◦ γ˜ (t) := γ (t) ⊕ ı T δ+ (t0 ) − t0 by Theorem 8.2.8. Setting * (t) ◦ı R (t) := R

2πk I δ+T (t0 ) − t0

378

8 Periodicity Using Shift Periodic Operators

and L (t) := L˜ (t) e◦ı

2π k T ( t )−t δ+ 0 0

I

(t, t0 ) ,

then one can write * (t) := R (t) ⊕ ◦ı R

δ+T

2πk I, (t0 ) − t0

and hence, L (t) eR (t, t0 ) = L˜ (t) e◦ı

2π k T ( t )−t δ+ 0 0

I

(t, t0 ) eR (t, t0 ) = L˜ (t) e◦ı

2π k T ( t )−t δ+ 0 0

I ⊕R

(t, t0 )

= L˜ (t) eR˜ (t, t0 ) . This shows that Φ A (t, t0 ) = L (t) eR (t, t0 ) is an alternative Floquet decomposition where γ (t) is an eigenvalue of R (t) . Theorem 8.2.10. Let γ(t) be a Floquet exponent of the system (8.2.1) and λ be the corresponding Floquet multiplier. Then, the unified Floquet system (8.2.1) has a nontrivial solution of the form x (t) = eγ (t, t0 ) κ (t) satisfying

(8.2.31)

  x δ+T (t) = λx (t) ,

where κ is a T-periodic function in shifts. Proof. Let Φ A (t, t0 ) be the transition matrix of (8.2.1) and Φ A (t, t0 ) = L (t) eR (t, t0 ) is Floquet decomposition such that γ (t) is an eigenvalue of R (t). There exists a nonzero vector u  0 such that R (t) u = γ (t) u, and therefore, eR (t, t0 ) u = eγ (t, t0 ) u. Then, the solution x (t) := Φ A (t, t0 ) u can be represented as follows: x (t) = L (t) eR (t, t0 ) u = eγ (t, t0 ) L (t) u. Setting κ (t) = L (t) u, the last equality implies (8.2.31). Thus, the first part of the theorem is proven. The second part is proven by the following equality.       x δ+T (t) = eγ δ+T (t) , t0 q δ+T (t)     = eγ δ+T (t) , δ+T (t0 ) eγ δ+T (t0 ) , t0 q (t)   = eγ δ+T (t0 ) , t0 eγ (t, t0 ) L (t) u   = eγ δ+T (t0 ) , t0 x (t) = λx (t) . The next result can be proven in a similar way to [73, Theorem 5.3].

8.2 Floquet Theory on Time Scales Periodic in Shifts δ±

379

Theorem 8.2.11. Let R(t) be a matrix function, with eigenvalues γ1 (t), . . . , γn (t) repeated according to multiplicities. Then γ1k (t), . . . , γnk (t) are the eigenvalues of Rk (t) and eigenvalues of eR are eγ1 , . . . , eγn . The next result shows that two solutions of (8.2.1) according to two distinct Floquet multipliers are linearly independent. Theorem 8.2.12. Let λ1 and λ2 be the characteristic multipliers of the system (8.2.1) and γ1 and γ2 are Floquet exponents such that eγi (δ+T (t0 ), t0 ) = λi , i = 1, 2. If λ1  λ2 , then there exists T-periodic functions κ1 and κ2 in shifts such that xi (t) = eγi (t, t0 )κi (t), i = 1, 2 are linearly independent solutions of the system (8.2.1). Proof. Let Φ A (t, t0 ) = L (t) eR (t, t0 ) and γ1 (t) be an eigenvalue of R (t) corre sponding to nonzero eigenvector v1 . Since λ2 is an eigenvalue of Φ A δ+T (t0 ) , t0 , by Theorem 8.2.11 there is an eigenvalue γ (t) of R (t) satisfying     eγ δ+T (t0 ) , t0 = λ2 = eγ2 δ+T (t0 ) , t0 . ◦

. Furthermore, λ1  λ2 Hence, for some k ∈ Z we have γ2 (t) = γ (t) ⊕ ı δ T 2πk + (t0 )−t0 implies that γ(t)  γ1 (t). If v2 is a nonzero eigenvector of R (t) corresponding to eigenvalue γ(t), then the eigenvectors v1 and v2 are linearly independent. Similar to the related part in the proof of Theorem 8.2.10, one can state the solutions of the system (8.2.1) as follows: x1 (t) = eγ1 (t, t0 ) L (t) v1

(8.2.32)

and x2 (t) = eγ (t, t0 ) L (t) v2 . Since x1 (t0 ) = L(t0 )v1 and x2 (t0 ) = L(t0 )v2 , the solutions x1 (t) and x2 (t) are linearly independent. Moreover, the solution x2 can be rewritten in the following form: x2 (t) = eγ2 (t, t0 ) eγ γ2 (t, t0 )L(t)ν2 = eγ2 (t, t0 ) e ◦ı Letting κ1 (t) = L (t) v1 and κ2 (t) = e ◦ı respectively, we complete the proof.

2π k T ( t )−t δ+ 0 0

2π k T ( t )−t δ+ 0 0

(t, t0 ) L(t)ν2 .

(8.2.33)

(t, t0 ) L(t)ν2 in (8.2.32) and (8.2.33),

380

8 Periodicity Using Shift Periodic Operators

8.3 Stability Properties of Unified Floquet Systems In this section, the unified Floquet theory established in previous sections is employed to investigate the stability characteristics of the regressive periodic system x Δ (t) = A (t) x (t) , x (t0 ) = x0 .

(8.3.1)

By Theorem 8.2.1, the matrix R in the Floquet decomposition of Φ A is given by   1 [Θ(σ(t))−Θ(s)] Φ A δ+T (t0 ) , t0 T −I . R (t) = lim s→t σ (t) − s

(8.3.2)

Also, Theorem 8.2.3 concludes that the solution z(t) of the regressive system z Δ (t) = R (t) z (t) , z (t0 ) = x0

(8.3.3)

can be expressed in terms of the solution x(t) of the system (8.3.1) as follows: z(t) = L −1 (t)x(t), where L(t) is the Lyapunov transformation given by (8.2.15). In preparation for the main result, the following definitions and results which can be found in [71] and [73] are presented. Definition 8.3.1 (Stability). The unified Floquet system (8.3.1) is uniformly stable if there exists a constant α > 0 such that the following inequality  x(t) ≤ α  x(t0 ) , t ≥ t0 holds for any initial state and corresponding solution. Theorem 8.3.1. Let Φ A be the transition matrix of the system (8.3.1). Then, (8.3.1) is uniformly stable if and only if there exists a α > 0 such that the inequality Φ A(t, t0 ) ≤ α, t ≥ t0 satisfied. Definition 8.3.2 (Exponential Stability). The unified Floquet system (8.3.1) is uniformly exponentially stable if there exist α, β > 0 such that the inequality  x(t) ≤  x(t0 ) αe β (t, t0 ), t ≥ t0 holds for any initial state and corresponding solution. Moreover, necessary and sufficient conditions for exponential stability can be stated as the following: Theorem 8.3.2. The system (8.3.1) is uniformly exponentially stable if and only if there exist α, β > 0 such that the inequality Φ A(t, t0 ) ≤ αe β (t, t0 ), t ≥ t0 satisfied for the transition matrix Φ A.

8.3 Stability Properties of Unified Floquet Systems

381

Definition 8.3.3 (Asymptotical Stability). In addition to uniform stability condition, if for any given c > 0 there exists a K > 0 such that the inequality  x(t) ≤ c  x(t0 ) , t ≥ t0 + K holds, then the unified Floquet system (8.3.1) is uniformly asymptotically stable. Definition 8.3.4 ([133]; See Also [73, Definition 7.1]). The scalar function γ : T∗ → C is uniformly regressive if there exists a constant θ > 0 such that 0 < θ −1 ≤ |1 + μ (t) γ (t)| , for all t ∈ Tκ . Lemma 8.3.1. Each eigenvalue of the matrix R (t) in (8.3.3) is uniformly regressive. Proof. Define Λ(t, s) by Λ(t, s) := Θ (σ (t)) − Θ (s) .

(8.3.4)

As it is done in Corollary 8.2.1, let 1

Λ(t,s)

T − 1

λ γi (t) = lim  i  , i = 1, 2, ..., k s→t σ (t) − s  

be any of the k ≤ n distinct eigenvalues of R (t). Now, there are two cases: 1. If |λi | ≥ 1, then   1     Λ(t,s)   1 Λ(t,s)  λiT − 1 T    > 1.  = lim λ |1 + μ (t) γi (t)| = lim 1 + μ (s)  s→t  σ (t) − s  s→t  i 2. If 0 ≤ |λi | < 1, then   1     Λ(t,s)   1 Λ(t,s)  λiT − 1 T    ≥ |λi | .  = lim λ |1 + μ (t) γi (t)| = lim 1 + μ (s)  s→t  σ (t) − s  s→t  i Setting θ −1 := min{1, |λ1 | , . . . , |λk |}, then one can obtain 0 < θ −1 < |1 + μ (t) γi (t)| . Definition 8.3.5 ([73, Definition 7.3]). A matrix function H(t) is said to admit a dynamic eigenvector w (t), where w (t) is a Δ-differentiable nonzero vector function, if the following equality is satisfied w Δ (t) = H (t) w (t) − ξ (t) w σ (t) , t ∈ Tk

(8.3.5)

for corresponding dynamic eigenvalue ξ (t). Then the pair {ξ (t) , w (t)} is called a dynamic eigenpair. Additionally, the mode vector χ for a function H(t) is given by

382

8 Periodicity Using Shift Periodic Operators

χi := eξi (t, t0 ) wi (t) ,

(8.3.6)

where {ξi (t) , wi (t)} is associated dynamic eigenpair. The following results can be proven in a similar way to [73, Lemma 7.4, Theorem 7.5]: Lemma 8.3.2. Any regressive matrix function H has n dynamic eigenpairs with linearly independent eigenvectors. Moreover, if the dynamic eigenvectors form the columns of a matrix function W (t), then W satisfies the matrix dynamic eigenvalue problem W Δ (t) = H (t) W (t)−W σ (t) Ξ (t) , where Ξ (t) := diag [ξ1 (t) , . . . , ξn (t)] . (8.3.7) where Ξ (t) := diag [ξ1 (t) , . . . , ξn (t)] . Theorem 8.3.3. Solutions to the uniformly regressive (but not necessarily periodic) time varying linear dynamic system (8.3.1) are: 1. stable if and only if there exists a γ > 0 such that every mode vector χi (t) of A (t) satisfies  χi (t) ≤ γ < ∞, t > t0 , for all 1 ≤ i ≤ n; 2. asymptotically stable if and only if, in addition to (1),  χi (t) → 0, t > t0 , for all 1 ≤ i ≤ n, 3. exponentially stable if and only if there exists γ, λ > 0 with −λ ∈ R + (T, R) such that  χi (t) ≤ γeλ (t, t0 ), t > t0 , for all 1 ≤ i ≤ n. Definition 8.3.6. For each k ∈ N0 the mappings hk : T∗ × T∗ → R+ , recursively defined by ∫t  h0 (t, t0 ) :≡ 1, hk+1 (t, t0 ) =

lim

s→τ t0

 Λ(τ, s) hk (τ, t0 )Δτ for n ∈ N0, σ(τ) − s

(8.3.8)

are called monomials, where Λ(t, s) is given by (8.3.4). Remark 8.3.1. For an additive periodic time scale we always have Θ (t) = t − t0 , and hence, Λ(t, s) = σ(t) − s. Lemma 8.3.3. Let T be a time scale which is unbounded above and γ (t) be an eigenvalue of R(t). If there exists a constant H ≥ t0 such that      −1 Λ(t, s) (8.3.9) Reμ γ (t) > 0 − lim inf s→t σ(t) − s t ∈[H,∞)T holds, then lim hk (t, t0 ) eγ (t, t0 ) = 0, k ∈ N0 .

t→∞

(8.3.10)

8.3 Stability Properties of Unified Floquet Systems

383

Proof. It suffices to show that limt→∞ hk (t, t0 ) eReμ γ(t) (t, t0 ) = 0 (see [89, Theorem 7.4]). Let us proceed by mathematical induction. For k = 0, it is known that h0 (t, t0 ) = 1 and by [133], we have lim eReμ γi (t) (t, t0 ) = 0 for t0 ∈ T.

t→∞

Suppose that it is true for a fixed k ∈ N and consider the (k + 1)th step. lim hk+1 (t, t0 ) eReμ γ(t) (t, t0 )

t→∞

⎡∫ t ⎤   ⎢ ⎥ Λ(τ, s) = lim ⎢⎢ lim hk (τ, t0 ) Δτ ⎥⎥ e Reμ γ(t) (t, t0 )−1 t→∞ ⎢ s→τ σ(τ) − s ⎥ ⎣t0 ⎦    e Λ(t, s) Reμ γ(t) (σ(t), t0 ) = lim lim hk (t, t0 ) t→∞ s→t σ(t) − s − Reμ γ(t)   ⎡ ⎤ Λ(t,s) ⎢ lims→t σ(t)−s hk (t, t0 ) eReμ γ(t) (σ(t), t0 ) ⎥ ⎢ ⎥ = lim ⎢ ⎥, t→∞ ⎢ ⎥ − Reμ γ(t) ⎢ ⎥ ⎣ ⎦

(8.3.11)

where (8.3.9) is used together with [50, Theorem 1.120] and [53, Theorem 3.4] to obtain the second equality. The last term in (8.3.11) can be written as   ⎡ ⎤ Λ(t,s) ⎢ lims→t σ(t)−s hk (t, t0 ) eReμ γ(t) (σ(t), t0 ) ⎥ ⎢ ⎥ lim ⎢ ⎥ t→∞ ⎢ ⎥ − Reμ γ(t) ⎢ ⎥ ⎣ ⎦ ⎤ ⎡  ⎢ 1 + μ(t) Reμ γ(t) hk (t, t0 ) eRe γ(t) (t, t0 ) ⎥ μ ⎥ ⎢ = lim ⎢ ⎥    −1  t→∞ ⎢ ⎥ Λ(t,s) ⎥ ⎢ − lims→t σ(t)−s Reμ (γ (t)) ⎦ ⎣ ⎤ ⎡   ⎥ ⎢ ⎢ 1 + μ(t) Reμ γ(t) hk (t, t0 ) eReμ γ(t) (t, t0 ) ⎥ ⎥. ⎢   (8.3.12) ≤ lim ⎢    −1 ⎥ t→∞ ⎢ Λ(t,s) ⎥ inf Re − lim (γ (t)) ⎥ ⎢ t ∈[H,∞)T s→t σ(t)−s μ ⎦ ⎣ Now, one may use (8.2.4) and (8.3.4) to get the inequality  ! 1 "  λ T Λ(t,s) − 1   1 + μ(t) Reμ γ(t) = 1 + μ(t) lim  ≤ max {1, |λ|} s→t  σ (t) − s  which along with (8.3.12) implies lim hk+1 (t, t0 ) eReμ γ(t) (t, t0 ) = 0

t→∞

as desired.

384

8 Periodicity Using Shift Periodic Operators

n Theorem 8.3.4. Let {γi (t)}i=1 be the set of conventional eigenvalues of the matrix n be the set of corresponding linearly independent R(t) given in (8.3.2) and {wi (t)}i=1 n is a set dynamic eigenvectors as defined by Lemma 8.3.2. Then, {γi (t) , wi (t)}i=1 of dynamic eigenpairs of R(t) with the property that for each 1 ≤ i ≤ n there are positive constants Di > 0 such that

wi (t) ≤ Di

m i −1 

hk (t, t0 ) ,

(8.3.13)

k=0

holds where hk (t, t0 ), k = 0, 1, ..., mi − 1, are the monomials defined as in (8.3.8) and mi is the dimension of the Jordan block which contains the i th eigenvalue, for all 1 ≤ i ≤ n. n Proof. By Lemma 8.3.2, it is obvious that, {γi (t) , wi (t)}i=1 is the set of eigenpairs of R (t). First, there exists an appropriate n × n constant, nonsingular matrix S which   transforms Φ A δ+T (t0 ) , t0 to its Jordan canonical form given by   J := S −1 Φ A δ+T (t0 ) , t0 S

⎡ Jm1 (λ1 ) ⎤ ⎢ ⎥ ⎢ ⎥ Jm2 (λ2 ) ⎢ ⎥ =⎢ , (8.3.14) ⎥ .. ⎢ ⎥ . ⎢ ⎥ ⎢ Jmd (λd ) ⎥⎦ n×n ⎣   d mi = n, λi are the eigenvalues of Φ A δ+T (t0 ) , t0 . By utilizing where d ≤ n, i=1 the above determined matrix S, define the following: K (t) := S −1 R (t) S

  1 Λ(t,s) Φ A δ+T (t0 ) , t0 T − I = S  lim S s→t σ (t) − s   1   Λ(t,s) S −1 Φ A δ+T (t0 ) , t0 T S−I . = lim s→t σ (t) − s −1

This along with [73, Theorem A.6] yields J T Λ(t,s) − I . σ (t) − s 1

K (t) = lim s→t

Note that, K (t) has the block diagonal form K (t) = diag [K1 (t) , . . . , Kd (t)]

8.3 Stability Properties of Unified Floquet Systems

385

in which each Ki (t) given by Ki (t) := lim Ki (t) s→t

⎡ ⎢ ⎢ 1 ⎢ λ T Λ(t, s) −1 ⎢ i ⎢ σ(t)−s ⎢ ⎢ ⎢ := lim ⎢⎢ s→t ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1 1 T Λ(t, s)−1 T Λ(t,s)λi

(σ(t)−s)2! 1 Λ(t, s)

λiT −1 σ(t)−s

...

... .. .

n−2 

 1 Λ(t, s)−n+1  [ T1 Λ(t,s)−k ]λiT   k=0  (n−1)!(σ(t)−s) n−3 

 1 Λ(t, s)−n+2  [ T1 Λ(t,s)−k ]λiT   k=0  (n−2)!(σ(t)−s)

.. .

1 Λ(t, s) λiT −1

σ(t)−s

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ . ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ mi ×mi

It should be mentioned that, since R (t) and K (t) are similar, they have the same conventional eigenvalues 1

[Λ(t,s)]

T − 1

λ γi (t) = lim  i  , i = 1, 2, ..., n, s→t σ (t) − s   with corresponding multiplicities. Moreover, if we set the dynamic eigenvalues of K(t) to be same as conventional eigenvalues γi (t), then the corresponding dynamic n of K (t) can be given by ui (t) = S −1 wi (t). eigenvectors {ui (t)}i=1 n is a set of dynamic This claim can be proven by showing that {γi (t) , ui (t)}i=1 eigenpairs of K (t) . By Definition 8.3.5, one can write that

uiΔ (t) = S −1 wiΔ (t) = S −1 R (t) wi (t) − S −1 γi (t) wiσ (t) = K (t) S −1 wi (t) − γi (t) S −1 wiσ (t)

= K (t) ui (t) − γi (t) uiσ (t) ,

(8.3.15)

for all 1 ≤ i ≤ n and this proves our claim. Now, it should be shown that ui (t) n is the set of dynamic eigenpairs of K (t) , it satisfies (8.3.13). Since {γi (t) , ui (t)}i=1 satisfies (8.3.15) for all 1 ≤ i ≤ n. By choosing the i th block of K (t) with dimension mi × mi , we can construct the following linear dynamic system: v Δ (t) = K˜ i (t) v(t)

386

8 Periodicity Using Shift Periodic Operators

⎡ ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢ ⎢ = lim ⎢⎢ s→t ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1 T Λ(t,s) (σ(t)−s)λi

0

( T1 Λ(t,s))( T1 Λ(t,s)−1) (σ(t)−s)λi 2!

1 T Λ(t,s) (σ(t)−s)λi

...

..

0

..

n−2 

  [ T1 Λ(t,s)−k ]   k=0  (n−1)!(σ(t)−s)λin−1 n−3 

  [ T1 Λ(t,s)−k ]   k=0  (n−2)!(σ(t)−s)λin−2

.. .

. .

1 T Λ(t,s) (σ(t)−s)λi

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ v(t), (8.3.16) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

where K˜ i (t) (t) := Ki (t) γi (t) I. There are mi linearly independent solutions of (8.3.16). Let us denote these solutions by vi, j (t), where i corresponds to the i th block i−1  ms , with m0 = 0. Then, matrix Ki (t) and j = 1, . . . , mi . For 1 ≤ i ≤ d, define li = s=0

the form of an arbitrary n × 1 column vector uli +j for i ≤ j ≤ m can be given as uli +j = [ 0, . . . , 0 , vi,T j (t), 0, . . . , 0 ]1×n . ABCD ABCD ABCD m1 +...+mi−1

mi

(8.3.17)

mi+1,...,m d

Considering all the vector solutions of (8.3.15), the solution of the n × n matrix dynamic equation U Δ (t) = K (t) U (t) − U σ (t) Γ (t) , where Γ (t) := diag [γ1 (t) , . . . , γn (t)] , can be written as # U (t) := u1, . . . , um1 , . . . , u( i−1 mk ), . . . , u( i mk ), . . . , u( d ⎡ ⎡ v1,1 v1,2 ⎢⎢ ⎢⎢ ⎢⎢ v1,1 ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ ⎢ = ⎢⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

, un k=1 mk )−1

k=1

k=1

. . . v1,m1 ⎤⎥ ⎥ .. . v1,m1 −1 ⎥⎥ .. ⎥⎥ .. . . ⎥ v1,1 ⎥⎦ m

1 ×m1

..

.

⎡ vd,1 vd,2 ⎢ ⎢ ⎢ vd,1 ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

... .. . .. .

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ . ⎥ ⎥ vd,md ⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ vd,md −1 ⎥⎥ ⎥ ⎥ .. ⎥ ⎥ ⎥ . ⎥ ⎥ vd,1 ⎥⎦ m ×m ⎥ d d ⎦ n×n

The mi linearly independent solutions of (8.3.16) have the form ' (T vi,1 (t) := vi,mi (t) , 0, . . . , 0 mi ×1 ,

$

8.3 Stability Properties of Unified Floquet Systems

387

' (T vi,2 (t) := vi,mi −1 (t) , vi,mi (t) , 0, . . . , 0 mi ×1 , .. .

' (T vi,mi (t) := vi,1 (t) , vi,2 (t) , . . . , vi,mi −1 (t) , vi,mi (t) mi ×1 . Then, we have the dynamic equations Δ vi,m (t) = 0, i Δ vi,m (t) = lim i −1 s→t

Δ (t) = lim vi,m i −2

[Λ(t, s)] vi,mi (t) , T(σ(t) − s)λi " ! 1  [ T1 Λ(t, s) − k] k=0

2(σ(t) −

s→t

.. .

vi,mi (t) + lim

s)λi2

s→t

Λ(t, s) vi,mi −1 (t) , T(σ(t) − s)λi

" ! m −2 i  1 [ T Λ(t, s) − k]

Δ (t) = lim vi,1 s→t

+ lim s→t

k=0

(mi − 1)!(σ(t) − s)λimi −1 " ! m −3 i  1 [ T Λ(t, s) − k] k=0

(mi − 2)!(σ(t) − s)λ mi −2 " ! 1  1 [ T Λ(t, s) − k]

. . . + lim

k=0

2(σ(t) −

s→t

vi,mi (t)

vi,mi −1 (t) +

vi,3 (t) + lim

s)λi2

s→t

Λ(t, s) vi,2 (t) . T(σ(t) − s)λi

Moreover, we have the following solutions: ∫t vi,mi (t) = 1, vi,mi −1 (t) = t0

! ∫t vi,mi −2 (t) =

lim

s→τ t0

∫t +

lim

s→τ t0

Λ(τ, s) vi,mi (τ) Δτ, s→τ T(σ(τ) − s)λi lim

1  [ T1 Λ(τ, s) − k] k=0

2(σ(τ) − s)λi2

" vi,mi (τ) Δτ

Λ(τ, s) vi,mi −1 (τ) Δτ, T(σ(τ) − s)λi

388

8 Periodicity Using Shift Periodic Operators

.. .

! m −2 i  ∫t

vi,1 (t) =

1 T Λ(τ, s)

− k]

k=0

lim

(mi − 1)!(σ(τ) − s)λimi −1 " ! m −3 i  1 T Λ(τ, s) − k]

lim

(mi − 2)!(σ(τ) − s)λimi −2

s→τ t0

∫t +

"

k=0

s→τ t0

∫t ...+

lim

s→τ t0

vi,mi (τ) Δτ

vi,mi −1 (τ)Δτ+

Λ(τ, s) vi,2 (τ) Δτ. T(σ(τ) − s)λi

Then we can show that each vi, j is bounded. There exist constants Bi, j , i = 1, . . . , d and j = 1, . . . , mi, such that   vi,m (t) = 1 ≤ Bi,m h0 (t, t0 ) = Bi,m , i i i  vi,m

  i −1 (t) ≤

   ∫t    Λ(τ, s)   lim s→τ T(σ(τ) − s)λi vi,mi (τ) Δτ t0

1 ≤ T |λi |

∫t t0



 Λ(τ, s) lim h0 (τ, t0 ) Δτ s→τ σ(τ) − s

h1 (t, t0 ) ≤ ≤ Bi,mi −1 h1 (t, t0 ) , T |λi |  "  ! 1     1 [ T Λ(τ, s) − k]  ∫ t     k=0   vi,m −2 (t) ≤  lim  vi,mi (τ) Δτ i 2 s→τ  2(σ(τ) − s)λi  t0      

∫t +

lim

s→τ t0

 Λ(τ, s) vi,mi −1 (τ) Δτ. T(σ(τ) − s)λi

Since 0 ≤ Θ (σ (τ)) − Θ (s) ≤ T as s → τ, we get

  1   Λ(τ, s) − k  ≤ k as s → τ for k = 1, 2, .... T 

8.3 Stability Properties of Unified Floquet Systems

389

Then  vi,m

  i −2 (t) ≤

1 2T λi2

1 + 2 2 T λi =



 Λ(τ, s) lim h0 (τ, t0 ) Δτ s→τ σ(τ) − s

∫t t0



∫t lim

s→τ t0

 Λ(τ, s) h1 (τ, t0 ) Δτ σ(τ) − s

h1 (t, t0 ) h2 (t, t0 ) + 2T λi2 T 2 λi2

≤ Bi,mi −2

2 

h j (t, t0 )

j=1

.. . m i −1    vi,1  ≤ Bi,1 h j (t, t0 ) . j=1

-

.

Setting βi := max j=1,...,mi Bi, j for each 1 ≤ i ≤ d, we obtain m i −1  < <