466 Pages 13 B/W Illustrations
    by Chapman & Hall

    466 Pages 13 B/W Illustrations
    by Chapman & Hall

    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.

    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

    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

    Biography

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

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

    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