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Functional Equations with Causal Operators by C. Corduneanu (Stability and Control: Theory, Methods and Applications: Taylor & Francis, STM) Stability and Control: Theory, Methods and Applications is a series of books and monographs on the theory of stability and control Edited by V. Lakshmikantham, Florida Institute of Technology, USA and A.A. Martynyuk, Institute of Mechanics, Kiev, Ukraine

Functional Equations with Causal Operators by C. Corduneanu is intended to provide basic theory of functional equations (including functional differential equations) with causal operators. These equations encompass most of the types of equations which are used in applied science and engineering: ordinary differential equations, integral equations of the Volterra type, equations with delayed argument and integrodifferential equations of the Volterra type.

Functional Equations with Causal Operators presents the connection between equations with causal operators and the classical types of functional equation encountered in much of the literature in this field. The work provides basic theorems of existence and uniqueness of solutions, and properties of solutions or families of solutions. It also describes in detail the fundamentals of linear equations, stability theory and several applications and examples.

Contents:

• Auxiliary concepts
• Existence Theory for Functional Equations with Causal Operators
• Linear and Quasilinear Equations with Causal Operators
• Stability Theory
• Neutral Functional Equations

Readership: Researchers and graduates interested in applied mathematics, functional differential equations and control problems.

This book is dedicated to the investigation of functional or functional differential equations involving causal operators. These operators are also called nonanticipative, or abstract Volterra operators. The term "causal" is prevalent in the engineering literature.

The definition of causal operators is very simple: an operator, acting on a given function space is called causal, if for any pair of function of E, such that x and y coincide on an interval also coincide on that interval. In other words, the values of Vx up to a given point t are determined only by the values taken by the function x on the interval.

The idea of considering such operators appears implicitly in Volterra's work, but a sharp definition and further consideration appear in the paper of L. Tonelli. In this paper, the functional equation

is considered, where the second term in parantheses means the restriction of the function x to the interval. The notation is obviously inspired by Volterra's work, and the above equation reminds us instantly of the Volterra integral equation

Tonelli's paper was dedicated to proving the existence and uniqueness of the solution of the functional equation he has devised by means of causal operators. The equation has been investigated in the space of continuous functions and the hypotheses are formulated in such a way that the compactness of the operator A is assured. This result of Tonelli is, very likely, the first existence result for equations with general causal operators.

The next significant step in developing the theory of functional equations involving cauasal operators was made in 1938 by A.N. Tychonoff. The definition given by Tychonoff to causal operators is as formulated above, and besides the existence of solutions the importance of these types of operators or equations for other fields is emphasized.

In retrospect, it may appear somewhat strange that the concept of a causal (or, as both Tonelli and Tychonoff call it, Volterra) operator did not attract the immediate attention of researchers. A possible explanation may be that at the time this concept was advanced, the relatively new methods of functional analysis did not constitute the main tool of many investigators.

Gradually, the theory of functional equations with causal operators has caught the attention of researchers. From the 1960s we mention the papers by R. Driver, C. Corduneanu, and Z.B. Caljuk. The last-quoted paper deals with functional inequalities with causal operators.

In the 1970s there have been many authors dealing with causal operators/equations. For the first time in book form, these operators are discussed in the book by L. Neustadt [1]. Journal papers have been published by M. Kwapisz and his coworkers, by V.G. Kurbatov, L.A. Zhivotovskii, S. Szufia. A group of researchers from Russia (Perm Technical University), under the leadership of N.V. Azbelev, has started the systematic investigation of linear functional differential equations with causal operators. The activity of this group continues nowadays, with its members in Russia, Israel and other countries. Also in the 1970s, a good deal of research work has been conducted with regard to the equations with infinite delay. The books by J.K. Hale and A.D. Myshkis have also contributed to making the subject of causal operators an attractive topic for researchers.

In the 1980s, the theory of functional or functional differential equations with causal operators made serious steps towards its maturity. A large number of papers were pub­lished during this decade in the USA, in the former Soviet Union, Italy and other countries. Most of the basic problems of this theory, including stability theory, approximation pro­cedures and other aspects, have been considered by many authors. Our list of references provides a large number of items from that period, without any claim to being complete. It is also important to notice that during the 1980s, a large number of engineering papers (system science) were produced. The books by W.J. Rugh and M. Schetzen cover some of these topics. Fundamental results concerning causal general operators have been obtained by Sandberg not necessarily related to the theory of functional equations.

The 1990s were characterized by an increasing interest in the theory of functional equations with causal operators, often related to applications. Several books including results concern­ing functional equations with causal operators have been published, or are in an advanced state of publication: G. Gripenberg, S.O. Londen and O. Staffans, C. Corduneanu, N.V. Azbelev, V.P. Maksimov and L.F. Rakhmatullina, N.V. Azbelev and P.M. Simonov contain chapters or conspicuous sections dedicated to this theory.

The journal literature is currently growing steadily, with at least one periodic publication called Functional Differential Equations being mostly dedicated to this theory. A serial publication with the same title has been published by N.V. Azbelev and his collaborators in Perm. Many other journals include contributions on this subject, authored by researchers from Russia, Ukraine, Israel, the USA, Italy, Ireland, Japan, Georgia, Australia, Poland, Germany and other countries.

This book evolved during the period 1991-1998, when the author and his former students held a weekly seminar at the University of Texas at Arlington. In particular, Dr Mehran Mandavi and Dr Yizeng Li were active and wrote their PhD theses about functional dif­ferential equations with causal operators. After they graduated and went on to teach at other institutions of higher learning, the cooperation between us continued without interruption. The material contained in this book is in greatest part based on the work we developed separately or jointly. Our list of references is complete with respect to the contributions we have made to this theory, including some applications.

Some of the topics treated in this book have also been the object of some graduate courses the author has taught at the University of Texas of Arlington, starting in the late 1980s until his retirement in 1996. These courses had various titles, and we will mention here those on applied nonlinear analysis, an introduction to control theory, advanced applied differential equations.

One of the aims while teaching such topics was to present a unified treatment of existence theory, as well as other aspects of the theory of functional differential equations, including ordinary differential equations, equations with delayed argument (both finite and infinite), integral equations of Volterra type and integrodifferential equations involving Volterra inte­gral operators. Indeed, the theory of functional differential equations with causal operators has very powerful unifying qualities — a feature we have attempted to illustrate in this book. As a matter of fact, our involvement in the theory of functional equations with causal operators was motivated by the needs of the teaching process, especially for engineering and science graduate students.

Briefly, the structure of the book is as follows: Chapter 1 is an introduction to the concept of functional equations, in general, with particular concern for equations with causal ope­rators; Chapter 2 contains some auxiliary material, mostly pertaining to functional analysis (both linear and nonlinear), as a preparation of the reader for understanding the rest of the material presented in the book; Chapter 3 deals entirely with the existence theory for functional or functional differential equations with causal operators, emphasizing the fact that these equations contain as particular cases several classes of functional equations encountered in the literature; Chapter 4 is dedicated to the theory of linear and quasilinear equations with causal operators, including the global character of existence results and the integral representation of solutions; Chapter 5 contains an introduction to stability theory for equations with causal operators, featuring both the method of "first approximation" and the "comparison method" based on Liapunov functionals and differential inequalities; Chapter 6 is completely dedicated to the theory of neutral functional equations with causal operators, and it is based on the work of the author and of M. Mandavi; Chapter 7 contains some applications of the general theory to problems in optimal control, some generalizations of the existence results by means of Leray—Schauder principle, as well as a review of certain results available in the literature.

In our view, the book is addressed to graduate students in science and engineering, whose scientific horizon should extend beyond the classical theory of ordinary differential equations.

In order to be able to adequately describe phenomena whose evolution is sensibly influ­enced by their past, we need a theory of functional equations with causal operators, with all its basic constituents: existence, uniqueness, approximation, continuous dependence with respect to data, stability, global behavior and relationships with other problems such as con­trol theory, mechanics of materials with memory. It is our hope that this goal is partially fulfilled by this book. There certainly remains much more to be done in the future.

It contains information about the historical development of the material and biographical information of key developers of the theories. It contains many exercises, of varying difficulty. It includes ideas for individual student projects and presentations. What really makes this book different from many other excellent books on the subject are: The choice of topics. The level of the target audience. The ideas offered for student projects (as outlined in Chapter 6). The inclusion of biographical and historical information.

The organization of the book offers flexibility. I like to have my students present material in class. The material that they present ranges in difficulty from "short" exercises, to proofs of standard theorems, to introductions to subjects that lie outside the scope of the main body of such a course.

Chapters 1 through 5 serve as the core of the course. The first two chapters introduce metric spaces, normed spaces, and inner product spaces and their topology. The third chapter is on Lebesgue integration, motivated by probability theory. Aside from the material on probability, the Lebesgue theory offered here is only what is deemed necessary for its use in functional analysis. Fourier analysis in Hilbert space is the subject of the fourth chapter, which draws connections between the first two chapters and the third. The final chapter of this main body of the text introduces the reader to bounded linear operators acting on Banach spaces, Banach algebras, and spectral theory. It is my opinion that every course should end with material that truly challenges the students and leaves them asking more questions than perhaps can be answered. The last three sections of Chapter 5, as well as several sections of Chapter 6, are written with this view in mind. I realize the time constraints placed on such a course. In an effort to abbreviate the course, some material of Chapter 3 can be safely omitted. A good course can include only an outline of Chapter 3, and enough proofs and examples to give a flavor for measure theory.

Chapter 6 consists of seven independent sections. Each time that I have taught this course, I have had the students select topics that they will study individ­ually and teach to the rest of the class. These sections serve as resources for these projects. Each section discusses a topic that is nonstandard in some way. For example, one section gives a proof of the classical Weierstrass approxima­tion theorem and then gives a fairly recent (1980s) proof of Marshall Stone's generalization of Weierstrass's theorem. While there are several proofs of the Stone‑Weierstrass theorem, this is the first that does not depend on the classi­cal result. In another section of this chapter, two arguments are given that no function can be continuous at each rational number and discontinuous at each irrational number. One is the usual Baire category argument; the other is a less well known and more elementary argument due to Volterra. Another section discusses the role of Hilbert spaces in quantum mechanics, with a focus on Heisenberg's uncertainty principle.

Appendices A and B are very short. They contain material that most students will know before they arrive in the course. However, occasionally, a student appears who has never worked with complex numbers, seen De Morgan's Laws, etc. I find it convenient to have this material in the book. I usually spend the first day or two on this material.

The biographies are very popular with my students. I assign each student one of these (or other) "key players" in the development of linear analysis. Then, at a subject‑appropriate time in the course, I have that one student give (orally) a short biography in class. They really enjoy this aspect of the course, and some end up reading (completely due to their own enthusiasm) a book like Constance Reid's Hilber

 

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