## Introduction to Functional Equations

Published:
Author(s):

Hardback
\$104.95
ISBN 9781439841112
Cat# K11911
eBook
ISBN 9781439841167
Cat# KE11901

### Features

• Provides self-guided tutorials and detailed proofs
• Includes many functional equations not covered in other books
• Presents theoretical discussions on the real numbers to help readers understand abstract concepts in the simplest setting
• Ends each chapter with material on more abstract settings as well as the latest developments of the main equations

### Summary

Introduction to Functional Equations grew out of a set of class notes from an introductory graduate level course at the University of Louisville. This introductory text communicates an elementary exposition of valued functional equations where the unknown functions take on real or complex values.

In order to make the presentation as manageable as possible for students from a variety of disciplines, the book chooses not to focus on functional equations where the unknown functions take on values on algebraic structures such as groups, rings, or fields. However, each chapter includes sections highlighting various developments of the main equations treated in that chapter. For advanced students, the book introduces functional equations in abstract domains like semigroups, groups, and Banach spaces.

Functional equations covered include:

• Cauchy Functional Equations and Applications
• The Jensen Functional Equation
• Pexider's Functional Equation
• D'Alembert Functional Equation
• Trigonometric Functional Equations
• Pompeiu Functional Equation
• Hosszu Functional Equation
• Davison Functional Equation
• Abel Functional Equation
• Mean Value Type Functional Equations
• Functional Equations for Distance Measures

The innovation of solving functional equations lies in finding the right tricks for a particular equation. Accessible and rooted in current theory, methods, and research, this book sharpens mathematical competency and prepares students of mathematics and engineering for further work in advanced functional equations.

Introduction
Functional Equations
Solution of Additive Cauchy Functional Equation
Discontinuous Solution of Additive Cauchy Equation
Other Criteria for Linearity
Additive Functions on the Complex Plane
Concluding Remarks
Exercises

Remaining Cauchy Functional Equations
Introduction
Solution of Exponential Cauchy Equation
Solution of Logarithmic Cauchy Equation
Solution of Multiplicative Cauchy Equation
Concluding Remarks
Exercises

Cauchy Equations in Several Variables
Introduction
Additive Cauchy Equations in Several Variables
Multiplicative Cauchy Equations in Several Variables
Other Two Cauchy Equations in Several Variables
Concluding Remarks
Exercises

Extension of the Cauchy Functional Equations
Introduction
Concluding Remarks
Exercises

Applications of Cauchy Functional Equations
Introduction
Area of Rectangles
Definition of Logarithm
Simple and Compound Interests
Characterization of Geometric Distribution
Characterization of Discrete Normal Distribution
Characterization of Normal Distribution
Concluding Remarks

More Applications of Functional Equations
Introduction
Sum of Powers of Integers
Sum of Powers of Numbers on Arithmetic Progression
Number of Possible Pairs Among n Things
Cardinality of a Power Set
Sum of Some Finite Series
Concluding Remarks

The Jensen Functional Equation
Introduction
Convex Function
The Jensen Functional Equation
A Related Functional Equation
Concluding Remarks
Exercises

Pexider's Functional Equations
Introduction
Pexider's Equations
Pexiderization of the Jensen Functional Equation
A Related Equation
Concluding Remarks
Exercises

Introduction
Continuous Solution of Quadratic Functional Equation
Contents xvii
Concluding Remarks
Exercises

D'Alembert Functional Equation
Introduction
Continuous Solution of d'Alembert Equation
General Solution of d'Alembert Equation
A Characterization of Cosine Functions
Concluding Remarks
Exercises

Trigonometric Functional Equations
Introduction
Solution of a Cosine-Sine Functional Equation
Solution of a Sine-Cosine Functional Equation
Solution of a Sine Functional Equation
Solution of a Sine Functional Inequality
An Elementary Functional Equation
Concluding Remarks
Exercises

Pompeiu Functional Equation
Introduction
General Solution Pompeiu Functional Equation
A Generalized Pompeiu Functional Equation
Pexiderized Pompeiu Functional Equation
Concluding Remarks
Exercises

Hosszu Functional Equation
Introduction
Hosszu Functional Equation
A Generalization of Hosszu Equation
Concluding Remarks
Exercises

Davison Functional Equation
Introduction
Continuous Solution of Davison Functional Equation
General Solution of Davison Functional Equation
Concluding Remarks
Exercises

Abel Functional Equation
Introduction
General Solution of Abel Functional Equation
Concluding Remarks
Exercises

Mean Value Type Functional Equations
Introduction
The Mean Value Theorem
A Mean Value Type Functional Equation
Generalizations of Mean Value Type Equation
Concluding Remarks
Exercises

Functional Equations for Distance Measures
Introduction
Solution of two functional equations
Some Auxiliary Results
Solution of a generalized functional equation
Concluding Remarks
Exercises

Introduction
Cauchy Sequence and Geometric Series
Hyers Theorem
Generalizations of Hyers Theorem
Concluding Remarks
Exercises

Stability of Exponential Cauchy Equations
Introduction
Stability of Exponential Equation
Ger Type Stability of Exponential Equation
Concluding Remarks
Exercises
Stability of d'Alembert and Sine Equations
Introduction
Stability of d'Alembert Equation
Stability of Sine Equation
Concluding Remarks
Exercises

Introduction
Stability of a Functional Equation of Drygas
Concluding Remarks
Exercises

Stability of Davison Functional Equation
Introduction
Stability of Davison Functional Equation
Generalized Stability of Davison Equation
Concluding Remarks
Exercises

Stability of Hosszu Functional Equation
Introduction
Stability of Hossz_u Functional Equation
Stability of Pexiderized Hossz_u Functional Equation
Concluding Remarks
Exercises

Stability of Abel Functional Equation
Introduction
Stability Theorem
Concluding Remarks
Exercises
Bibliography
Index

### Author Bio(s)

Prasanna K. Sahoo, Department of Mathematics, University of Louisville, Kentucky, USA

Palaniappan Kannappan, Department of Pure Mathematics, University of Waterloo, Ontario, Canada

### Editorial Reviews

The book includes several interesting and fundamental techniques for solving functional equations in real or complex realms. There exist many useful exercises as well as well-organized concluding remarks in each chapter. … This book is written in a clear and readable style. It is useful for researchers and students working in functional equations and their stability.
—Mohammad Sal Moslehian, Mathematical Reviews, Issue 2012b