Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis

Soon-Mo Jung is a highly respected mathematician who was born in 1957 in Seocheon, South Korea. He received his BS in Atomic Nuclear Engineering at the Seoul National University and  received his BS, MS, and PhDs in Mathematics at the University of Stuttgart.Dr. Jung has published approximately 150 research papers in the areas of functional equations, classical analysis, analytical geometry, measure theory, fractals, number theory, and algebra, and has published 5 books. This  present volume will be his first book published with Springer. However, Soon-Mo has contributed papers to several edited volumes and journals such as “Nonlinear Analysis and Variational Problems” (SOIA 35), Acta Mathematica Sinica, Bulletin of the Brazilian Math Society, Proceedings Mathematical Sciences, and Journal of Central South University of Technology.

• -1. Introduction. -2. Additive Cauchy Equation (Behavior of additive functions, Hyers-Ulam stability, Hyers-Ulam-Rassias stability, Stability on a restricted domain, Method of invariant means, Fixed point method, Composite functional congruences, Pexider equation, Remarks). -3. Generalized Additive Cauchy Equations (Functional equation f(ax+by)=af(x)+bf(y), Additive Cauchy equations of general form, Functional equation f(x+y)2=(f(x)+f(y))2). -4. Hosszủ’s Functional Equation (Stability in the sense of Borelli, Hyers-Ulam stability, Generalized Hosszủ’s equation is not stable on the unit interval, Hosszủ’s functional equation of Pexider type). -5. Homogeneous Functional Equation(Homogeneous equation between Banach algebras, Superstability on a restricted domain, Homogeneous equation between vector spaces, Homogeneous equation of Pexide type). -6. Linear Functional Equations (A system for linear functions, Functional equation f(x+cy)=f(x)+cf(y), Stability for other equations).-7. Jensen’s Functional Equation (Hyers-Ulam-Rassias stability, Stability on a restriced domain, Fixed point method, Lobačevskíí;s functional equation). -8. Quadratic Functional Equations (Hyers-Ulam-Rassias stability, Stability on a restricted domain, Fixedpoint method, Quadratic functional equation of other type, Quadratic functional equation of Pexider type). -9. Exponential Functional Equations (Superstability, Stability in the sense of Ger, Stability on a restricted domain, Exponential functional equation of other type). -10. Multiplicative Functional Equations (Superstability, δ-multiplicative functional, Theory of AMNM algebras, Functional equation f(xy)= f(x)y, Functional equation f(x+y)= f(x)f(y)f(1/x+1/y)). -11. Logarithmic Functional Equations (Functional equation f(xy)= yf(x), Superstability of equation f(xy)= yf(x), Functional equation of Heuvers). -12. Trigonometric Functional Equations (Cosinefunctional equation, Sine functional equation, Trigonometric equations with two unknowns, Butler-Rassias functional equation, Remarks). -13. Isometric Functional Equation (Hyers-Ulam stability, Stability on a restricted domain, Fixed point method, Wigner equation). -14. Miscellaneous (Associativity equation, Equation of multiplicative derivation, Gamma functional equation). -Bibliography. -Index.

From the reviews: ...I am more than happy to write my opinion on Soon-Mo Jung's book, as a new addition in this rapidly growing field of mathematics, which will help interested mathematicians and graduate students to understand further this beautiful domain of research... The book contains 14 chapters...concludes with a very useful bibliography of 364 references and an index. It will definitely guide mathematics students to a decisive first step into this abstract, yet intriguing, field of mathematics. The author has succeeded in presenting to both mathematicians and graduate students an invaluable source of essential mathematics. The book will certainly before a standard reference for stability of functional equations in nonlinear analysis. Themistocles M. Rassias (EMS Newsletter, December 2011) "This book is intended to provide an overview of the theory of the stability of functional equations. It is very useful for undergraduate/graduate students and also for anyone interested in Hyers-Ulam stability problems. The book is well written; the style is very accessible and clear." (Pasc Gavruta, Mathematical Reviews, Issue 2012 c) "This interesting book is devoted to an exposition of some new significant results of the Hyers-Ulam-Rassias stability of functional equations, difference equations and related topics in Functional Analysis. ... This book is well written in a concise, clear and readable style. ... The book is a good source for specialists and graduate students working in functional equations." (Mohammad Sal Moslehian, Zentralblatt MATH, Vol. 1221, 2011)