Introduction to Functional Equations

Theory and Problem-solving Strategies for Mathematical Competitions and Beyond
Author: Costas Efthimiou
Publisher: American Mathematical Soc.
ISBN: 0821853147
Category: Mathematics
Page: 363
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Functions and their properties have been part of the rigorous precollege curriculum for decades. And functional equations have been a favorite topic of the leading national and international mathematical competitions. Yet the subject has not received equal attention by authors at an introductory level. The majority of the books on the topic remain unreachable to the curious and intelligent precollege student. The present book is an attempt to eliminate this disparity. The book opens with a review chapter on functions, which collects the relevant foundational information on functions, plus some material potentially new to the reader. The next chapter presents a working definition of functional equations and explains the difficulties in trying to systematize the theory. With each new chapter, the author presents methods for the solution of a particular group of equations. Each chapter is complemented with many solved examples, the majority of which are taken from mathematical competitions and professional journals. The book ends with a chapter of unsolved problems and some other auxiliary material. The book is an invaluable resource for precollege and college students who want to deepen their knowledge of functions and their properties, for teachers and instructors who wish to enrich their curricula, and for any lover of mathematical problem-solving techniques. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.

Lectures on Functional Equations and Their Applications


Author: J. Aczel,Hansjorg Oser
Publisher: Courier Corporation
ISBN: 0486445232
Category: Mathematics
Page: 510
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Numerous detailed proofs highlight this treatment of functional equations. Starting with equations that can be solved by simple substitutions, the book then moves to equations with several unknown functions and methods of reduction to differential and integral equations. Also includes composite equations, equations with several unknown functions of several variables, vector and matrix equations, more. 1966 edition.

Functional Equations and Modelling in Science and Engineering


Author: Enrique Castillo
Publisher: CRC Press
ISBN: 9780824787172
Category: Mathematics
Page: 352
View: 3176

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Provides engineers and applied scientists with some selected results of functional equations and their applications, with the intention of changing the way they think about mathematical modelling. Many of the proofs are simplified or omitted, so as not to bore or confuse engineers. Functional equati

Functional Equations and Inequalities


Author: Themistocles RASSIAS
Publisher: Springer Science & Business Media
ISBN: 9401143412
Category: Mathematics
Page: 336
View: 5473

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This volume provides an extensive study of some of the most important topics of current interest in functional equations and inequalities. Subjects dealt with include: a Pythagorean functional equation, a functional definition of trigonometric functions, the functional equation of the square root spiral, a conditional Cauchy functional equation, an iterative functional equation, the Hille-type functional equation, the polynomial-like iterative functional equation, distribution of zeros and inequalities for zeros of algebraic polynomials, a qualitative study of Lobachevsky's complex functional equation, functional inequalities in special classes of functions, replicativity and function spaces, normal distributions, some difference equations, finite sums, decompositions of functions, harmonic functions, set-valued quasiconvex functions, the problems of expressibility in some extensions of free groups, Aleksandrov problem and mappings which preserve distances, Ulam's problem, stability of some functional equation for generalized trigonometric functions, Hyers-Ulam stability of Hosszú's equation, superstability of a functional equation, and some demand functions in a duopoly market with advertising. Audience: This book will be of interest to mathematicians and graduate students whose work involves real functions, functions of a complex variable, functional analysis, integral transforms, and operational calculus.

Functional Equations in Applied Sciences


Author: Enrique Castillo,Andres Iglesias,Reyes Ruiz-Cobo
Publisher: Elsevier
ISBN: 9780080477916
Category: Mathematics
Page: 408
View: 9846

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The book provides the reader with the different types of functional equations that s/he can find in practice, showing, step by step, how they can be solved. A general methodology for solving functional equations is provided in Chapter 2. The different types of functional equations are described and solved in Chapters 3 to 8. Many examples, coming from different fields, as geometry, science, engineering, economics, probability, statistics, etc, help the reader to change his/her mind in order to state problems as functional equations as an alternative to differential equations, and to state new problems in terms of functional equations or systems. An interesting feature of the book is that it deals with functional networks, a powerful generalization of neural networks that allows solving many practical problems. The second part of the book, Chapters 9 to 13, is devoted to the applications of this important paradigm. The book contains many examples and end of chapter exercises, that facilitates the understanding of the concepts and applications. · A general methodology for solving functional equations is provided in Chapter 2. · It deals with functional networks, a powerful generalization of neural networks. · Many examples, coming from different fields, as geometry, science, engineering, economics, probability, statistics, etc, illustrate the concept of functional equation. · Functional equations are presented as a powerful alternative to differential equations. · The book contains end of chapter exercises.

Functional Equations and Inequalities in Several Variables


Author: Stefan Czerwik
Publisher: World Scientific
ISBN: 9789810248376
Category: Mathematics
Page: 410
View: 4144

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This book outlines the modern theory of functional equations and inequalities in several variables. It consists of three parts. The first is devoted to additive and convex functions defined on linear spaces with semilinear topologies. In the second part, the problems of stability of functional equations in the sense of Ulam-Hyers-Rassias and in some function spaces are considered. In the last part, the functional equations in set-valued functions are dealt with ? for the first time in the mathematical literature. The book contains many fresh results concerning those problems.

On Functions and Functional Equations


Author: Smital
Publisher: CRC Press
ISBN: 9780852744185
Category: Mathematics
Page: 164
View: 1466

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On Functions and Functional Equations introduces the main topics in iteration theory and the theory of functional equations with emphasis on applications in the fields of mathematics, physics, biology, chemistry, and electronics and mechanical engineering. The book contains many classical results as well as important, more recent results. It also includes numerous exercise and some problems that have yet to be resolved. The book is accessible to readers having a secondary level mathematical education.

Iterative Functional Equations


Author: Marek Kuczma,Bogdan Choczewski,Roman Ger
Publisher: Cambridge University Press
ISBN: 9780521355612
Category: Mathematics
Page: 552
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A cohesive and comprehensive account of the modern theory of iterative functional equations. Many of the results included have appeared before only in research literature, making this an essential volume for all those working in functional equations and in such areas as dynamical systems and chaos, to which the theory is closely related. The authors introduce the reader to the theory and then explore the most recent developments and general results. Fundamental notions such as the existence and uniqueness of solutions to the equations are stressed throughout, as are applications of the theory to such areas as branching processes, differential equations, ergodic theory, functional analysis and geometry. Other topics covered include systems of linear and nonlinear equations of finite and infinite ORD various function classes, conjugate and commutable functions, linearization, iterative roots of functions, and special functional equations.

Functional Equations on Groups


Author: Henrik Stetkær
Publisher: World Scientific
ISBN: 9814513148
Category: Mathematics
Page: 396
View: 4106

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This volume provides an accessible and coherent introduction to some of the scientific progress on functional equations on groups in the last two decades. It presents the latest methods of treating the topic and contains new and transparent proofs. Its scope extends from the classical functional equations on the real line to those on groups, in particular, non-abelian groups. This volume presents, in careful detail, a number of illustrative examples like the cosine equation on the Heisenberg group and on the group SL(2, ℝ). Some of the examples are not even seen in existing monographs. Thus, it is an essential source of reference for further investigations. Contents:IntroductionAround the Additive Cauchy EquationThe Multiplicative Cauchy EquationAddition and Subtraction FormulasLevi–Civita's Functional EquationThe Symmetrized Sine Addition FormulaEquations with Symmetric Right Hand SideThe Pre-d'Alembert Functional EquationD'Alembert's Functional EquationD'Alembert's Long Functional EquationWilson's Functional EquationJensen's Functional EquationThe Quadratic Functional EquationK-Spherical FunctionsThe Sine Functional EquationThe Cocycle EquationAppendices:Basic Terminology and ResultsSubstitutes for CommutativityThe Casorati DeterminantRegularityMatrix-Coefficients of RepresentationsThe Small Dimension LemmaGroup Cohomology Readership: Advanced undergraduates, graduates and professional mathematicians interested in harmonic analysis and/or functional equations. Keywords:Functional Equation;Group;Harmonic AnalysisKey Features:Most of the material of the book can be found only in research papers, so it is a good source of referenceThe book is self-contained and provides the necessary background material needed to go further into the subject and to explore the research literatureThe book may be used as a textbook for graduate students and even ambitious undergraduate in mathematics, because it presents the material in an accessible way, originating from a course for students at master's levelExercises at the end of each chapter, some with answers, help to provide more examples to enable the student to grasp the topic betterReviews: “It is an excellent, well composed and self-contained monograph, written in good and clear English. It can serve as a complete and independent introduction to the field of trigonometric functional equations and as an excellent source of suitable references for further study. The scope of solutions considered extends from real functions to those acting between groups, also non-abelian.” Prof Janusz Brzdęk Pedagogical University Kraków, Poland “The book is written as an accessible introduction to trigonometric functional equations for graduate students and working mathematicians. It gives a very readable account of recent research in the area, as well as more than 200 references for further study. It would make an excellent textbook for a graduate course on the topic. This monograph is a valuable contribution to the literature on functional equations.” Zentralblatt MATH

Handbook of Functional Equations

Functional Inequalities
Author: Themistocles M. Rassias
Publisher: Springer
ISBN: 1493912461
Category: Mathematics
Page: 555
View: 5663

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As Richard Bellman has so elegantly stated at the Second International Conference on General Inequalities (Oberwolfach, 1978), “There are three reasons for the study of inequalities: practical, theoretical, and aesthetic.” On the aesthetic aspects, he said, “As has been pointed out, beauty is in the eye of the beholder. However, it is generally agreed that certain pieces of music, art, or mathematics are beautiful. There is an elegance to inequalities that makes them very attractive.” The content of the Handbook focuses mainly on both old and recent developments on approximate homomorphisms, on a relation between the Hardy–Hilbert and the Gabriel inequality, generalized Hardy–Hilbert type inequalities on multiple weighted Orlicz spaces, half-discrete Hilbert-type inequalities, on affine mappings, on contractive operators, on multiplicative Ostrowski and trapezoid inequalities, Ostrowski type inequalities for the Riemann–Stieltjes integral, means and related functional inequalities, Weighted Gini means, controlled additive relations, Szasz–Mirakyan operators, extremal problems in polynomials and entire functions, applications of functional equations to Dirichlet problem for doubly connected domains, nonlinear elliptic problems depending on parameters, on strongly convex functions, as well as applications to some new algorithms for solving general equilibrium problems, inequalities for the Fisher’s information measures, financial networks, mathematical models of mechanical fields in media with inclusions and holes.

Functional Equations in Several Variables


Author: J. Aczel,Jean G. Dhombres
Publisher: Cambridge University Press
ISBN: 9780521352765
Category: Mathematics
Page: 462
View: 5971

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Deals with modern theory of functional equations in several variables and their applications to mathematics, information theory, and the natural, behavioral, and social sciences. The authors emphasize applications, although not at the expense of theory, and have kept the prerequisites to a minimum; the reader should be familiar with calculus and some simple structures of algebra and have a basic knowledge of Lebesque integration. For the applications the authors have included references and explained the results used. The book is designed so that the chapters may be read almost independently of each other, enabling a selection of material to be chosen for introductory and advanced courses. Each chapter concludes with exercises and further results, 400 in all, which extend and test the material presented in the text. The history of functional equations is well documented in a final chapter which is complemented by an encyclopedic bibliography of over 1600 items. This volume will be of interest to professionals and graduate students in pure and applied mathematics.

Functional Equations and Inequalities with Applications


Author: Palaniappan Kannappan
Publisher: Springer Science & Business Media
ISBN: 0387894926
Category: Mathematics
Page: 810
View: 3756

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Functional Equations and Inequalities with Applications presents a comprehensive, nearly encyclopedic, study of the classical topic of functional equations. This self-contained monograph explores all aspects of functional equations and their applications to related topics, such as differential equations, integral equations, the Laplace transformation, the calculus of finite differences, and many other basic tools in analysis. Each chapter examines a particular family of equations and gives an in-depth study of its applications as well as examples and exercises to support the material.

Functional Equations — Results and Advances


Author: Zoltan Daroczy,Zsolt Páles
Publisher: Springer Science & Business Media
ISBN: 1475752881
Category: Mathematics
Page: 361
View: 9176

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The theory of functional equations has been developed in a rapid and productive way in the second half of the Twentieth Century. First of all, this is due to the fact that the mathematical applications raised the investigations of newer and newer types of functional equations. At the same time, the self development of this theory was also very fruitful. This can be followed in many monographs that treat and discuss the various methods and approaches. These developments were also essentially influenced by a number jour nals, for instance, by the Publicationes Mathematicae Debrecen (founded in 1953) and by the Aequationes Mathematicae (founded in 1968), be cause these journals published papers from the field of functional equa tions readily and frequently. The latter journal also publishes the yearly report of the International Symposia on Functional Equations and a comprehensive bibliography of the most recent papers. At the same time, there are periodically and traditionally organized conferences in Poland and in Hungary devoted to functional equations and inequali ties. In 2000, the 38th International Symposium on Functional Equations was organized by the Institute of Mathematics and Informatics of the University of Debrecen in Noszvaj, Hungary. The report about this meeting can be found in Aequationes Math. 61 (2001), 281-320.

Mean Value Theorems and Functional Equations


Author: Prasanna Sahoo,Thomas Riedel
Publisher: World Scientific
ISBN: 9789810235444
Category: Mathematics
Page: 245
View: 477

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This book takes a comprehensive look at mean value theorems and their connection with functional equations. Besides the traditional Lagrange and Cauchy mean value theorems, it covers the Pompeiu and the Flett mean value theorems as well as extension to higher dimensions and the complex plane. Furthermore the reader is introduced to the field of functional equations through equations that arise in connection with the many mean value theorems discussed.

Functional Equations


Author: David Leigh-Lancaster
Publisher: Aust Council for Ed Research
ISBN: 0864314922
Category: Education
Page: 114
View: 8136

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Functional equations provides mathematics teachers with an introduction to elementary aspects of functional equations. These equations are linked to function in various topics of the senior secondary mathematics curriculum including transformations, identities difference equations and mathematical modelling.

Linear Functional Equations. Operator Approach


Author: Anatolij Antonevich
Publisher: Springer Science & Business Media
ISBN: 9783764329310
Category: Mathematics
Page: 183
View: 5333

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In this book we shall study linear functional equations of the form m bu(x) == Lak(X)U(Qk(X)) = f(x), (1) k=l where U is an unknown function from a given space F(X) of functions on a set X, Qk: X -+ X are given mappings, ak and f are given functions. Our approach is based on the investigation of the operators given by the left-hand side of equa tion (1). In what follows such operators will be called functional operators. We will pay special attention to the spectral properties of functional operators, first of all, to invertibility and the Noether property. Since the set X, the space F(X), the mappings Qk and the coefficients ak are arbitrary, the class of operators of the form (1) is very rich and some of its individ ual representatives are related with problems arising in various areas of mathemat ics and its applications. In addition to the classical theory of functional equations, among such areas one can indicate the theory of functional-differential equations with deviating argument, the theory of nonlocal problems for partial differential equations, the theory of boundary value problems for the equation of a vibrating string and equations of mixed type, a number of problems of the general theory of operator algebras and the theory of dynamical systems, the spectral theory of au tomorphisms of Banach algebras, and other problems.

Functional Equations with Causal Operators


Author: C. Corduneanu
Publisher: CRC Press
ISBN: 9780203166376
Category: Mathematics
Page: 184
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Functional equations encompass most of the equations used in applied science and engineering: ordinary differential equations, integral equations of the Volterra type, equations with delayed argument, and integro-differential equations of the Volterra type. The basic theory of functional equations includes functional differential equations with cau

Volterra Integral and Functional Equations


Author: G. Gripenberg,S. O. Londen,O. Staffans
Publisher: Cambridge University Press
ISBN: 9780521372893
Category: Mathematics
Page: 701
View: 8883

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This book looks at the theories of Volterra integral and functional equations.

Functional Equations, Inequalities and Applications


Author: Themistocles RASSIAS
Publisher: Springer Science & Business Media
ISBN: 940170225X
Category: Mathematics
Page: 224
View: 2640

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Functional Equations, Inequalities and Applications provides an extensive study of several important equations and inequalities, useful in a number of problems in mathematical analysis. Subjects dealt with include the generalized Cauchy functional equation, the Ulam stability theory in the geometry of partial differential equations, stability of a quadratic functional equation in Banach modules, functional equations and mean value theorems, isometric mappings, functional inequalities of iterative type, related to a Cauchy functional equation, the median principle for inequalities and applications, Hadamard and Dragomir-Agarwal inequalities, the Euler formulae and convex functions and approximate algebra homomorphisms. Also included are applications to some problems of pure and applied mathematics. This book will be of particular interest to mathematicians and graduate students whose work involves functional equations, inequalities and applications.