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Stability of functional equations in random normed spaces. (English) Zbl 1281.46001
Springer Optimization and Its Applications 86. New York, NY: Springer (ISBN 978-1-4614-8476-9/hbk; 978-1-4614-8477-6/ebook). xix, 246 p. (2013).
The essence of the stability of functional equations is “When is it true that a function satisfying a functional equation \({\mathcal E}\) approximately must be close to an exact solution of \({\mathcal E}\)?”. In 1940, S. M. Ulam [Problems in modern mathematics. First published under the title ‘A collection of mathematical problems.’ (Science Editions.) New York: John Wiley and Sons (1964; Zbl 0137.24201)] raised the first stability problem, which was partially solved in the next year for the Cauchy functional equation \(f(x+y)=f(x)+f(y)\) by D. H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222–224 (1941; Zbl 0061.26403)]. T. Aoki [J. Math. Soc. Japan 2, 64–66 (1950; Zbl 0040.35501)] and Th. M. Rassias [Proc. Am. Math. Soc. 72, 297–300 (1978; Zbl 0398.47040)] extended Hyers’ theorem for additive and linear mappings, respectively, when the Cauchy difference is allowed to be unbounded. The result of Rassias has influenced the development of what is now called the Hyers-Ulam-Rassias stability theory. Since then, many mathematicians have investigated the stability of various functional equations and mappings in different settings. The first attempt for establishing the stability of functional equations in “Fuzzy Mathematics” in the spirit of Hyers-Ulam-Rassias was done by A. K. Mirmostafaee and the reviewer [Fuzzy Sets Syst. 159, No. 6, 720–729 (2008; Zbl 1178.46075)] in which they established three versions of the stability of the Cauchy functional equation in a linear space \(X\) together with a fuzzy norm \(N: X\times \mathbb{R} \to [0,1]\), where one may regard \(N(x, t)\) as the truth value of the statement ‘the norm of \(x\) is less than or equal to the real number \(t\)’. This fact was not included in the short literature review of the main theme of the book. Inspired by this paper, many mathematicians have investigated the stability of the Cauchy, Jensen, quadratic, cubic functional equations and their mixed types in other similar settings such as random normed spaces and non-Archimedean random spaces equipped with a triangular norm. In particular, D. MiheŇ£ and V. Radu [J. Math. Anal. Appl. 343, No. 1, 567–572 (2008; Zbl 1139.39040)] initiated the study of the stability of the Cauchy functional equation in random normed spaces by using the fixed point alternative theorem first developed by V. Radu [Fixed Point Theory 4, No. 1, 91–96 (2003; Zbl 1051.39031)].
The book under review is essentially a collection of several recent papers related to the stability of functional equations in the framework of fuzzy and random normed spaces. The first two chapters nicely provide background of the theory of random normed spaces. The literature on the Cauchy, Jensen, quadratic and cubic functional equations is significant. The Cauchy and cubic equations are almost completely investigated in the book. The Jensen equation is not considered. The quadratic equation is only investigated in Chapter 3 for random normed spaces under special triangular norms. In Chapter 8, some notions such as ‘non-Archimedean Lie \(C^*\)-algebras’ are introduced which appear to be artificial. The book could prove to be useful for graduate students who are interested in the Hyers-Ulam-Rassias stability of functional equations.

MSC:
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
39-02 Research exposition (monographs, survey articles) pertaining to difference and functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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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