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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.
Prasanna K. Sahoo, Department of Mathematics, University of Louisville, Kentucky, USA Palaniappan Kannappan, Department of Pure Mathematics, University of Waterloo, Ontario, Canada
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
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