INTRODUCTION TO FUNCTIONAL EQUATIONS - MSRI .INTRODUCTION TO FUNCTIONAL EQUATIONS ... (e.g. functional equations, inequalities, ï¬nite and inï¬nite sums, etc.) instead. x

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Text of INTRODUCTION TO FUNCTIONAL EQUATIONS - MSRI .INTRODUCTION TO FUNCTIONAL EQUATIONS ... (e.g....

  • INTRODUCTION TO FUNCTIONAL EQUATIONStheory and problem-solving strategies for mathematical competitions and beyond

    COSTAS EFTHIMIOU

    Department of PhysicsUNIVERSITY OF CENTRAL FLORIDA

    VERSION: 2.00September 12, 2010

  • Cover picture: Domenico Fettis Archimedes Thoughtful, Oil on canvas, 1620. State ArtCollections Dresden.

  • To My Mother who taught me that love is notmeasured by how many things

    one can offer to a child butby how many things

    a parent sacrificesfor the child

  • Contents

    INTRODUCTION ix

    ACKNOWLEDGEMENTS xiii

    I BACKGROUND 1

    1 Functions 3

    1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    1.4 Limits & Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    1.5 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    1.6 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    II BASIC EQUATIONS 33

    2 Functional Relations Primer 35

    2.1 The Notion of Functional Relations . . . . . . . . . . . . . . . . . . . . . . . . 35

    2.2 Beginning Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

    3 Equations for Arithmetic Functions 51

    3.1 The Notion of Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . 51

    3.2 Multiplicative Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

    3.3 Linear Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

    3.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

    4 Equations Reducing to Algebraic Systems 69

    4.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    4.2 Group Theory in Functional Equations . . . . . . . . . . . . . . . . . . . . . . 74

    v

  • vi CONTENTS

    5 Cauchys Equations 81

    5.1 First Cauchy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

    5.2 Second Cauchy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    5.3 Third Cauchy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

    5.4 Fourth Cauchy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

    5.5 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

    6 CauchysNQR-Method 95

    6.1 TheNQR-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

    6.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

    7 Equations for Trigonometric Functions 105

    7.1 Characterization of Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . 105

    7.2 DAlembert-Poisson I Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 113

    7.3 DAlembert-Poisson II Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 115

    7.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

    III GENERALIZATIONS 121

    8 Pexider, Vincze & Wilson Equations 123

    8.1 First Pexider Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

    8.2 Second Pexider Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

    8.3 Third Pexider Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

    8.4 Fourth Pexider Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

    8.5 Vincze Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

    8.6 Wilson Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

    8.7 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

    9 Vector and Matrix Variables 137

    9.1 Cauchy & Pexider Type Equations . . . . . . . . . . . . . . . . . . . . . . . . 138

    9.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

    10 Systems of Equations 145

    10.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

    IV CHANGING THE RULES 151

    11 Less Than Continuity 153

    11.1 Imposing Weaker Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

    11.2 Non-Continuous Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

    11.3 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

  • CONTENTS vii

    12 More Than Continuity 169

    12.1 Differentiable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

    12.2 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

    12.3 Stronger Conditions as a Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

    13 Functional Equations for Polynomials 183

    13.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

    13.2 Symmetric Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

    13.3 More on the Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 188

    13.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

    14 Conditional Functional Equations 205

    14.1 The Notion of Conditional Equations . . . . . . . . . . . . . . . . . . . . . . . 205

    14.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

    15 Functional Inequalities 209

    15.1 Useful Concepts and Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

    15.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

    V EQUATIONS WITH NO PARAMETERS 215

    16 Iterations 217

    16.1 The Need for New Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

    16.2 Iterates, Orbits, Fixed Points, and Cycles . . . . . . . . . . . . . . . . . . . . . 220

    16.3 Fixed Points: Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

    16.4 Cycles: Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

    16.5 From Iterations to Difference Equations . . . . . . . . . . . . . . . . . . . . . 230

    16.6 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

    16.7 A Taste of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

    17 Solving by Invariants & Linearization 247

    17.1 Constructing Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

    17.2 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

    17.3 The Abel and Schroder Equations . . . . . . . . . . . . . . . . . . . . . . . . . 250

    17.4 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

    17.5 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

    18 More on Fixed Points 259

    18.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

  • viii CONTENTS

    VI GETTING ADDITIONAL EXPERIENCE 263

    19 Miscellaneous Problems 265

    19.1 Integral Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 26519.2 Problems Solved by Functional Relations . . . . . . . . . . . . . . . . . . . . 26919.3 Assortment of Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

    20 Unsolved Problems 285

    20.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28520.2 Problems That Can Be Solved Using Functions . . . . . . . . . . . . . . . . . 29120.3 Arithmetic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29220.4 Functional Equations With Parameters . . . . . . . . . . . . . . . . . . . . . . 29720.5 Functional Equations with No Parameters . . . . . . . . . . . . . . . . . . . . 30520.6 Fixed Points and Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30820.7 Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31120.8 Systems of Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . 31420.9 Conditional Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . 31520.10Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31620.11Functional Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32120.12Functional Equations Containing Derivatives . . . . . . . . . . . . . . . . . . 32420.13Functional Relations Containing Integrals . . . . . . . . . . . . . . . . . . . . 325

    VII AUXILIARY MATERIAL 329

    ACRONYMS & ABBREVIATIONS 331

    SET CONVENTIONS 333

    NAMED EQUATIONS 335

    BIBLIOGRAPHY 337

    INDEX 341

  • Introduction

    One of the most beautiful mathematical topics I encountered as a student was the topicof functional equations that is, the topic that deals with the search of functions whichsatisfy given equations, such as

    f (x + y) = f (x) + f (y) .

    This topic is not only remarkable for its beauty but also impressive for the fact that functionalequations arise in all areas of mathematics and, even more, science, engineering, and socialsciences. They appear at all levels of mathema