##
**Iterative functional equations.**
*(English)*
Zbl 0703.39005

Encyclopedia of Mathematics and Its Applications, 32. Cambridge etc.: Cambridge University Press. xiv, 552 p. £55.00/hbk; $ 99.50/hbk (1990).

Functional equations have a long and distinguished history but only the latter part of this century brought a large progress in this branch of mathematics. The first books giving a cohesive and extensive treatment of the subject were published in the 1960s: one by J. Aczél [Lectures of functional equations and their applications (1966; Zbl 0139.093)] and the other by the first author [Functional equations in a single variable (1968; Zbl 0196.164)]. In these monographs there have been laid down the foundations of theories of two considerably different (both in methods and in the kind of results) topics: functional equations in several variables and functional equations in a single variable, respectively. “Iterative functional equations”, the title of the book just reviewed, is simply another name of those in a single variable.

There are some types of equations often referred to as functional equations which are not dealt with in this book. This regards almost all equations in which infinitesimal operations are performed on the unknown functions, among others differential and differential-difference equations, integral equations as well as those occuring in the theory of dynamic programming. Also difference equations and recurrences are not separately discussed (however, some results contained in the book can be applied to finite differences). All these equations form independent branches having their special methods and problems.

The subject matter of this monograph is similar to that of the first author’s previous book. However, because of a great number of new results published in the last two decades, it was practically impossible to give a complete presentation of the modern theory of iterative functional equations. So the authors decided to exclude some classical topics as construction of general solutions or some iteration problems strictly connected with functional equations (continuous iteration groups and semigroups).

The main stress has been laid on fundamental questions of existence and uniqueness of solutions. Following the reasoning of the authors the reader can get to know some iterative and fixed-point methods characteristic for functional equations in a single variable. A specific leading motive of the whole treatise is the problem of linearization and special equations connected with it: Schröder’s and Abel’s equations. The reader is systematically being acquainted with their crucial role in many aspects of the theory of functional equations as well as of iteration theory. Other topics covered include among others linear and nonlinear equations in various function classes, conjugate and commuting functions, equations containing superpositions of the unknown function (especially those connected with iterative roots and invariant curves) and linear functional inequalities. At the end of each chapter one can find a special section called “Notes” in which some supplementary results and facts are briefly discussed. Many pieces of background information and references are given in “Comments” terminating most of sections.

A pretty large number of applications of the theory to differential equations, probability theory, ergodic theory and geometry makes this book interesting not only for all those working in functional equations but also for specialists from other areas of mathematics. Since functional equations in a single variable are closely related to iterations, most of all it may be of great interest for those concerned with discrete dynamical systems. Some parts of the material are organized in such a way that the reader can observe how questions coming from applications lead to problems in functional equations (cf., for instance, Chapter 2: Linear equations and branching processes). In other situations some interesting and often unexpected applications crown a purely theoretical part.

The authors provide a readable, though rigorous, book avoiding advanced methods throughout. The clarity and freshness of style make that the reader can find a great pleasure and a privilege satisfaction in studying the theory presented.

Reviewer’s remarks: The reader should be warned of the annoying error made in Section 4.3. A devoted to the Siegel set (unfortunately the same can be found in the former book by the first author): after Definition 4.3.1 (p. 156) the formula \(``\log | s^ n-1| =O(\log n)\), \(n\to \infty ''\) should be replaced by \(``\log | s^ n-1|^{- 1}=O(\log n)\), \(n\to \infty ''.\)

There are also some comments misleading the reader a little. Among others it seems that “Comments” at the end of Section 5.6 (p. 216) do not emphasize enough the contribution of J. Matkowski to the proof of Theorem 5.6.1 who answered two fundamental questions related to this result and unsolved for some years: superfluity of the assumption (5.6.10) and uniqueness of \(C^ r\) solutions of the nonlinear equation.

It happens also that some papers are misquoted. For instance Note 7.9.8 suggests that Baron-Jarczyk [1] and Dankiewicz [2] contain results which not differ essentially from those reported on in Section 7.4, whereas actually these papers present quite different methods and theorems uncomparable with those given in the book. Similarly, it seems that the references (excluding Dhombres [3]) given in connection with equation (11.9.14) misinform about the problems solved in the papers quoted.

There are some types of equations often referred to as functional equations which are not dealt with in this book. This regards almost all equations in which infinitesimal operations are performed on the unknown functions, among others differential and differential-difference equations, integral equations as well as those occuring in the theory of dynamic programming. Also difference equations and recurrences are not separately discussed (however, some results contained in the book can be applied to finite differences). All these equations form independent branches having their special methods and problems.

The subject matter of this monograph is similar to that of the first author’s previous book. However, because of a great number of new results published in the last two decades, it was practically impossible to give a complete presentation of the modern theory of iterative functional equations. So the authors decided to exclude some classical topics as construction of general solutions or some iteration problems strictly connected with functional equations (continuous iteration groups and semigroups).

The main stress has been laid on fundamental questions of existence and uniqueness of solutions. Following the reasoning of the authors the reader can get to know some iterative and fixed-point methods characteristic for functional equations in a single variable. A specific leading motive of the whole treatise is the problem of linearization and special equations connected with it: Schröder’s and Abel’s equations. The reader is systematically being acquainted with their crucial role in many aspects of the theory of functional equations as well as of iteration theory. Other topics covered include among others linear and nonlinear equations in various function classes, conjugate and commuting functions, equations containing superpositions of the unknown function (especially those connected with iterative roots and invariant curves) and linear functional inequalities. At the end of each chapter one can find a special section called “Notes” in which some supplementary results and facts are briefly discussed. Many pieces of background information and references are given in “Comments” terminating most of sections.

A pretty large number of applications of the theory to differential equations, probability theory, ergodic theory and geometry makes this book interesting not only for all those working in functional equations but also for specialists from other areas of mathematics. Since functional equations in a single variable are closely related to iterations, most of all it may be of great interest for those concerned with discrete dynamical systems. Some parts of the material are organized in such a way that the reader can observe how questions coming from applications lead to problems in functional equations (cf., for instance, Chapter 2: Linear equations and branching processes). In other situations some interesting and often unexpected applications crown a purely theoretical part.

The authors provide a readable, though rigorous, book avoiding advanced methods throughout. The clarity and freshness of style make that the reader can find a great pleasure and a privilege satisfaction in studying the theory presented.

Reviewer’s remarks: The reader should be warned of the annoying error made in Section 4.3. A devoted to the Siegel set (unfortunately the same can be found in the former book by the first author): after Definition 4.3.1 (p. 156) the formula \(``\log | s^ n-1| =O(\log n)\), \(n\to \infty ''\) should be replaced by \(``\log | s^ n-1|^{- 1}=O(\log n)\), \(n\to \infty ''.\)

There are also some comments misleading the reader a little. Among others it seems that “Comments” at the end of Section 5.6 (p. 216) do not emphasize enough the contribution of J. Matkowski to the proof of Theorem 5.6.1 who answered two fundamental questions related to this result and unsolved for some years: superfluity of the assumption (5.6.10) and uniqueness of \(C^ r\) solutions of the nonlinear equation.

It happens also that some papers are misquoted. For instance Note 7.9.8 suggests that Baron-Jarczyk [1] and Dankiewicz [2] contain results which not differ essentially from those reported on in Section 7.4, whereas actually these papers present quite different methods and theorems uncomparable with those given in the book. Similarly, it seems that the references (excluding Dhombres [3]) given in connection with equation (11.9.14) misinform about the problems solved in the papers quoted.

Reviewer: W.Jarczyk

### MSC:

39B12 | Iteration theory, iterative and composite equations |

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |

39B22 | Functional equations for real functions |

39B52 | Functional equations for functions with more general domains and/or ranges |

39B72 | Systems of functional equations and inequalities |