Introduction to Functional Equations
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|Format: ||Hardback, 464 pages|
|Other Information: ||15 black & white illustrations|
|Published In: ||United States, 15 February 2011|
Functional equations form a modern branch of mathematics. This book provides an elementary yet comprehensive introduction to the field of functional equations and stabilities. Concentrating on functional equations that are real or complex, the authors present many fundamental techniques for solving these functional equations. Topics covered in the text include Cauchy equations, additive functions, functional equations for distance measures, and Pexider's functional equations. Each chapter points to various developments in abstract domains, such as semigroups, groups, or Banach spaces, and includes exercises for both self-study and classroom use.
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 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
About the Author
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
Chapman & Hall/CRC|
15+ years |