# Introduction to Functional Equations

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## Book Description

**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
- Quadratic 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.

## Table of Contents

**Additive Cauchy Functional Equation **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

Extension of Additive Functions

Concluding Remarks

Exercises

**Applications of Cauchy Functional Equations**

Introduction

Area of Rectangles

Definition of Logarithm

Simple and Compound Interests

Radioactive Disintegration

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

**Quadratic Functional Equation**

Introduction

Biadditive Functions

Continuous Solution of Quadratic Functional Equation

A Representation of Quadratic Functions

Contents xvii

Pexiderization of Quadratic Equation

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

**Stability of Additive Cauchy Equation**

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

**Stability of Quadratic Functional Equations**

Introduction

Stability of the Quadratic Equation

Stability of Generalized Quadratic Equation

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

**Introduction**

Stability of Hosszu Functional Equation

Stability of Hosszu Functional Equation

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(s)

### Biography

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

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

## 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