Analytic solutions of functional equations cheng li

Analytic Solutions of

Functional

Equations

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N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I

World Scientific

Sui Sun Cheng

National Tsing Hua University, R. O. China

Wenrong Li

Binzhou University, P. R. China

Analytic Solutions of

Functional

Equations

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British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center,

Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from

the publisher.

ISBN-13 978-981-279-334-8

ISBN-10 981-279-334-8

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or

mechanical, including photocopying, recording or any information storage and retrieval system now known or to

be invented, without written permission from the Publisher.

Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd.

Published by

World Scientific Publishing Co. Pte. Ltd.

5 Toh Tuck Link, Singapore 596224

USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Printed in Singapore.

ANALYTIC SOLUTIONS OF FUNCTIONAL EQUATIONS

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Preface

Functions are used to describe natural processes and forms. By means of finite or

infinite operations, we may build many types of ‘derived’ functions such as the sum

of two functions, the composition of two functions, the derivative function of a given

function, the power series functions, etc.

Yet a large number of natural processes and forms are not explicitly given by

nature. Instead, they are ‘implicitly defined’ by the laws of nature. Therefore

we have functional equations (or more generally relations) involving our unknown

functions and their derived functions.

When we are given one such functional equation as a mathematical model, it

is important to try to find some or all solutions, since they may be used for pre￾diction, estimation and control, or for suggestion of alternate formulation of the

original physical model. In this book, we are interested in finding solutions that are

‘polynomials of infinite order’, or more precisely, power series functions.

There are many reasons for trying to find such solutions. First of all, it is

sometimes ‘obvious’ from experimental observations that we are facing with natural

processes and forms that can be described by ‘smooth’ functions such as power series

functions. Second, power series functions are basically ‘generated by’ sequences of

numbers, therefore, they can easily be manipulated, either directly, or indirectly

through manipulations of sequences. Indeed, finding power series solutions are not

more complicated than solving recurrence relations or difference equations. Solving

the latter equations may also be difficult, but in most cases, we can ‘calculate’ them

by means of modern digital devices equipped with numerical or symbolic packages!

Third, once formal power series solutions are found, we are left with the convergence

or stability problem. This is a more complicated problem which is not completely

solved. Fortunately, there are now several standard techniques which have been

proven useful.

In this book, basic tools that can be used to handle power series functions and

analytic functions will be given. They are then applied to functional equations in

which derived functions such as the derivatives, iterates and compositions of the un￾known functions are involved. Although there are numerous functional equations in

the literature, our main objective is to show by introductory examples how analytic

v

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vi Analytic Solutions of Functional Equations

solutions can be derived in relatively easy manners.

To accomplish our objective, we keep in mind that this book should be suitable

for the senior and first graduate students as well as anyone who is interested in a

quick introduction to the frontier of related research. Only basic second year ad￾vanced engineering mathematics such as the theory of a complex variable and the

theory of ordinary differential equations are required, and a large body of seem￾ingly unrelated knowledge in the literature is presented in an integrated and unified

manner.

A synopsis of the contents of the various chapters follows.

• The book begins with an elementary example in Calculus for motivation.

Basic definitions, symbols and results are then introduced which will be

used throughout the book.

• In Chapter 2, various types of sequences are introduced. Common opera￾tions among sequences are then presented. In particular, scalar, term by

term, convolution and composition products and their properties are dis￾cussed in detail. Algebraic derivation is also introduced.

• Power series functions are treated as generating functions of sequences and

their relations are fully discussed. Stability properties are discussed and

Cauchy’s majorant method is introduced. The Siegel’s lemma is an impor￾tant tool in deriving majornats.

• In Chapter 4, the basic implicit function theorem for analytic functions is

proved by Newton’s binomial expansion theorem. Schr¨oder and Poincar´e

type implicit functions together with several others are discussed. Applica￾tion of the implicit theorems for finding power series solutions of polynomial

or rational type functional equations are illustrated.

• In Chapter 5 analytic solutions for several classic ordinary differential equa￾tions or systems are derived. The Cauchy-Kowalewski existence theorem

for partial differential equations is treated as an application. Then several

selected functional differential equations are discussed and their analytic

solutions found.

• In Chapter 6 analytic solutions for functional equations involving iter￾ates of the unknown functions (or more general composition with other

known functions) are treated. These equations are distinguished by whether

derivatives of the unknown functions are involved. The last section is con￾cerned with the existence of power solutions.

Some of the material in this book is based on classical theory of analytic func￾tions, and some on theory of functional equations. However, a large number of

material is based on recent research works that have been carried out by us and a

number of friends and graduate students during the last ten years.

Our thanks go to J. G. Si, X. P. Wang, T. T. Lu and J. J. Lin for their hard

works and comments. We would also like to remark that without the indirect help

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Preface vii

of many other people, this book would never have appeared.

We tried our best to eliminate any errors. If there are any that have escaped our

attention, your comments will be much appreciated. We have also tried our best

to rewrite all the material that we draw from various sources and cite them in our

notes sections. We beg your pardon if there are still similarities left unattended or

if there are any original sources which we have missed.

Sui Sun Cheng and Wenrong Li

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Contents

Preface v

1. Prologue 1

1.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2. Sequences 11

2.1 Lebesgue Summable Sequences . . . . . . . . . . . . . . . . . . . . 11

2.2 Relatively Summable Sequences . . . . . . . . . . . . . . . . . . . . 18

2.3 Uniformly Summable Sequences . . . . . . . . . . . . . . . . . . . . 21

2.4 Properties of Univariate Sequences . . . . . . . . . . . . . . . . . . 25

2.4.1 Common Sequences . . . . . . . . . . . . . . . . . . . . . . 25

2.4.2 Convolution Products . . . . . . . . . . . . . . . . . . . . . 26

2.4.3 Algebraic Derivatives and Integrals . . . . . . . . . . . . . 32

2.4.4 Composition Products . . . . . . . . . . . . . . . . . . . . 34

2.5 Properties of Bivariate Sequences . . . . . . . . . . . . . . . . . . . 42

2.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3. Power Series Functions 49

3.1 Univariate Power Series Functions . . . . . . . . . . . . . . . . . . 49

3.2 Univariate Analytic Functions . . . . . . . . . . . . . . . . . . . . . 56

3.3 Bivariate Power Series Functions . . . . . . . . . . . . . . . . . . . 63

3.4 Bivariate Analytic Functions . . . . . . . . . . . . . . . . . . . . . 67

3.5 Multivariate Power Series and Analytic Functions . . . . . . . . . . 68

3.6 Matrix Power Series and Analytic Functions . . . . . . . . . . . . . 71

3.7 Majorants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.8 Siegel’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

ix

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x Analytic Solutions of Functional Equations

4. Functional Equations without Differentiation 83

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2 Analytic Implicit Function Theorem . . . . . . . . . . . . . . . . . 86

4.3 Polynomial and Rational Functional Equations . . . . . . . . . . . 90

4.4 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.4.1 Equation I . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.4.2 Equation II . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4.3 Equation III . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.4.4 Equation IV . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.4.5 Equation V . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.4.6 Schr¨oder and Poincar´e Equations . . . . . . . . . . . . . . 110

4.5 Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5. Functional Equations with Differentiation 123

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.3 Neutral Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.4 Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.5 Cauchy-Kowalewski Existence Theorem . . . . . . . . . . . . . . . 139

5.6 Functional Equations with First Order Derivatives . . . . . . . . . 141

5.6.1 Equation I . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.6.2 Equation II . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.6.3 Equation III . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.6.4 Equation IV . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.6.5 Equation V . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.6.6 Equation VI . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.7 Functional Equations with Higher Order Derivatives . . . . . . . . 152

5.7.1 Equation I . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.7.2 Equation II . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.7.3 Equation III . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.7.4 Equation IV . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6. Functional Equations with Iteration 175

6.1 Equations without Derivatives . . . . . . . . . . . . . . . . . . . . 175

6.1.1 Babbage Type Equations . . . . . . . . . . . . . . . . . . . 176

6.1.2 Equations Involving Several Iterates . . . . . . . . . . . . . 182

6.1.3 Equations of Invariant Curves . . . . . . . . . . . . . . . . 190

6.2 Equations with First Order Derivatives . . . . . . . . . . . . . . . 197

6.2.1 Equation I . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

6.2.2 Equation II . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

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Contents xi

6.2.3 Equation III . . . . . . . . . . . . . . . . . . . . . . . . . . 206

6.2.4 Equation IV . . . . . . . . . . . . . . . . . . . . . . . . . . 212

6.2.5 First Order Neutral Equation . . . . . . . . . . . . . . . . 214

6.3 Equations with Second Order Derivatives . . . . . . . . . . . . . . 222

6.3.1 Equation I . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

6.3.2 Equation II . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

6.3.3 Equation III . . . . . . . . . . . . . . . . . . . . . . . . . . 235

6.3.4 Equation IV . . . . . . . . . . . . . . . . . . . . . . . . . . 240

6.4 Equations with Higher Order Derivatives . . . . . . . . . . . . . . 244

6.4.1 Equation I . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

6.4.2 Equation II . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

6.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Appendix A Univariate Sequences and Properties 259

A.1 Common Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 259

A.2 Sums and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

A.3 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

A.4 Algebraic Derivatives and Integrals . . . . . . . . . . . . . . . . . . 261

A.5 Tranformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

A.6 Limiting Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 263

A.7 Operational Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

A.8 Knowledge Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

A.9 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

A.10 Operations for Analytic Functions . . . . . . . . . . . . . . . . . . 267

Bibliography 271

Index 283

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Chapter 1

Prologue

1.1 An Example

As an elementary but motivating example, let y(t) be the cash at hand of a corpo￾ration at time t ≥ 0. Suppose the corporation invests its cash into a project which

guarantees a positive interest rate r so that

dy

dt = ry, t ≥ 0. (1.1)

What is the cash at hand of the corporation at any time t > 0 given that y(0) = 1?

One way to solve this problem in elementary analysis is to assume that y = y(t)

is a “power series function” of the form

y(t) = a0 + a1t + a2t

2 + a3t

3 + · · · ,

then we have

a0 = y(0) = 1.

By formally operating the power series y(t) term by term, we further have

y

0

(t) = a1 + 2a2t + 3a3t

2 + · · · ,

and

ry(t) = ra0 + ra1t + ra2t

2 + · · · .

In view of (1.1), we see that

a1 + 2a2t + 3a3t

2 + · · · ≡ ra0 + ra1t + ra2t

2 + · · · .

By comparing coefficients on both sides, we may proceed formally and write

a1 = r, 2a2 = ra1, 3a3 = ra2, ...,

This yields

a1 = r, a2 =

r

2

2

, a3 =

r

3

3 · 2

, ..., an =

r

n

n!

, ...,

so that

y(t) = 1 + rt +

r

2

2! t

2 +

r

3

3! t

3 + · · · , (1.2)

1

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2 Analytic Solutions of Functional Equations

which is a “formal power series function”.

In order that the formal solution (1.2) is a true solution, we need either to

show that y(t) is meaningful on [0, ∞) and that the operations employed above are

legitimate, or, we may show that y(t) is equal to some previously known function

and show that this function satisfies (1.1) and y(0) = 1 directly. If these can be

done, then a power series solution exists and is given by (1.2).

Such solutions often reveal important quantitative as well as qualitative infor￾mation which can help us understand the complex behavior of the physical systems

represented by these equations.

In this book, we intend to provide some elementary properties of power series

functions and its applications to finding solutions of equations involving unknown

functions and/or their associated functions such as their iterates and derivatives.

1.2 Basic Definitions

Basic concepts from real and complex analysis and the theory of linear algebra will

be assumed in this book. For the sake of completeness, we will, however, briefly

go through some of these concepts and their related information. We will also

introduce here some common notations and conventions which will be used in this

book.

First of all, sums and products of a set of numbers are common. However, empty

sums or products may be encountered. In such cases, we will adopt the convention

that an empty sum is taken to be zero, while an empty product will be taken as

one.

The union of two sets A and B will be denoted by A ∪ B or A + B, their

intersection by A∩B or A·B, their difference by A\B, and their Cartesian product by

A×B. The notations A2

, A3

, ..., stand for the Cartesian products A×A, A×A×A, ...,

respectively. It is also natural to set A1 = A. The number of elements in a set Ω

will be denoted by |Ω| .

The set of real numbers will be denoted by R, the set of all complex numbers

by C, the set of integers by Z, the set of positive integers by Z

+, and the set of

nonnegative integers by N. We will also use F to denote either R or C.

It is often convenient to extend the real number system by the addition of

two elements, ∞ (which may also be written as +∞) and −∞. This enlarged set

[−∞, ∞] is called the set of extended real numbers. In addition to the usual oper￾ations involving the real numbers, we will also require −∞ < x < ∞, x + ∞ = ∞,

x −∞ = −∞ and x/∞ = 0 for x ∈ R; x ·∞ = ∞ and x · −∞ = −∞ for x > 0; and

∞ + ∞ = ∞, − ∞ − ∞ = −∞, ∞ · (±∞) = ±∞, − ∞ · (±∞) = ∓∞, 0 · ∞ = 0.

In the sequel, the equation

1

u

= v

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Prologue 3

will be met where v ∈ [0, ∞]. The solution u will be taken as ∞ if v = 0 and as 0 if

v = ∞.

The imaginary number √

−1 in C will be denoted by i. The symbols 0! and 0

0

will be taken as 1. Given a complex number z and an integer n, the n-th power of

z is defined by z

0 = 1, z

n+1 = z

nz if n ≥ 0 and z

−n = (z

−1

)

n if z 6= 0 and n > 0.

Recall also that for any complex number z = x + iy where x, y ∈ R, its real

part is R(z) = x, its imaginary part is I(z) = y, its conjugate is z

∗ = x − iy and

its modulus or absolute value is |z| =

￾

x

2 + y

2

1/2

. We have |z + w| ≤ |z| + |w| ,

|zw| = |z| |w| and (zw)

∗ = z

∗w

∗

for any z, w ∈ C.

Given a nonzero z = x + iy ∈ C, if we let θ be the angle measured from the

positive x-axis to the line segment joining the origin and the point (x, y), then we

see that

z = |z|(cos θ + isin θ).

We define an argument of the nonzero z to be any angle t ∈ R (which may or may

not lie inside [0, 2π)) for which

z = |z|(cost + isin t),

and we write arg z = t. A concrete choice of arg z is made by defining arg0

z to be

that number t0, called the principal argument, in the range (−π, π] such that

z = |z|(cost0 + isin t0).

We may then write

arg0

(zw) = arg0

z + arg0 w (mod 2π).

It is also easy to show that for any z 6= 0, given any positive integer n, there

are exactly n distinct complex numbers z0, z1, ..., zn−1 such that z

n

i = z for each

i = 0, 1, ..., n − 1. The numbers z0, z1, ..., zn−1 are called the n-th roots of z. The

geometric picture of the n-th roots is very simple: they lie on the circle centered

at the origin of radius |z|

1/n and are equally spaced on this circle with one of the

roots having polar angle 1

n

arg0

z.

Given a real or complex number α, and any real or complex valued functions f

and g, we define −f, αf, f · g, and f + g by (−f)(z) = −f(z), (αf)(z) = αf(z),

(f · g)(z) = f(z)g(z) and (f + g)(z) = f(z) + g(z) as usual, while |f| is defined by

|f|(z) = |f(z)| . If no confusion is caused, the product f · g is also denoted by fg.

The zeroth power of a function, denoted by f

0

, is defined by f

0

(z) = 1, while

the n-th power, denoted by f

n, is defined by f

n(z) = (f(z))n.

The composition of f and g is denoted by f ◦ g. The iterates of f are formally

defined by f

[0](z) = z, f

[1](z) = f(z), f

[2](z) = f(f(z)), ..., and f

[n]

is called the

n-th iterate of f. Note that f

[n] may not be defined if the range of f

[n−1] does not

lie inside the domain of f.

The n-th derivative of a function is defined by

f

0

(z) = f

(1)(z) = limw→0

f(z + w) − f(z)