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Functional equations and inequalities. Solutions and stability results. (English) Zbl 1371.39019

Series on Concrete and Applicable Mathematics 21. Hackensack, NJ: World Scientific (ISBN 978-981-3147-60-7/hbk; 978-981-3149-97-7/pbk). xvi, 380 p. (2017).
The book is written in a clear manner with a good degree of correctness. The book contains eighteen chapters, as follows: In Chapters 1, 2, 3, the authors present some of well-known functional equations, their historical development and their applications, and present various methods of solving functional equations.
Chapters 4, 5, 6 deal with the general solutions of several types of quadratic and cubic functional equations.
Chapter 7 contains the general solutions of quintic and sextic functional equations.
The method of obtaining the general solution of a variety of mixed-type functional equations is introduced in Chapters 8, 9.
Chapter 10 is concerned with the study of the general solutions of two and three variables functional equations.
In Chapter 11, they discuss the famous problem posed by the mathematician S. M. Ulam concerning the stability of functional equations.
They present some results associated with the Hyers-Ulam stability of functional equations in non-Archimedean spaces, fuzzy normed spaces, quasi-Banch spaces in Chapter 12.
In Chapter 13, they deal with Hyers-Ulam stability of an additive functional inequality, a Cauchy-type additive functional inequality, a Cauchy-Jensen-type functional inequality, and a quadratic functional inequality using the direct and the fixed point methods.
In Chapters 14 and 15, they discuss the generalized Hyers-Ulam stability of a general cubic functional equation in Felbin’s type fuzzy normed spaces and stability of functional equations in \(C^*\)-algebras.
Chapters 16, 17 are concerned with the investigation of the Ulam stability of mixed-type mappings on restricted domains and the Hyers-Ulam stability of related topics on distributions and hyperfunctions. The last chapter contains some exercises and open problems.
This book is a useful source for undergraduate and graduate students and people interested in functional equations.

MSC:

39B05 General theory of functional equations and inequalities
39-02 Research exposition (monographs, survey articles) pertaining to difference and functional equations
39B62 Functional inequalities, including subadditivity, convexity, etc.
39B52 Functional equations for functions with more general domains and/or ranges
39B82 Stability, separation, extension, and related topics for functional equations
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46S40 Fuzzy functional analysis
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