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Zbl 0907.39025
Hyers, Donald H.; Isac, George; Rassias, Themistocles M.
Stability of functional equations in several variables. (English)
[B] Progress in Nonlinear Differential Equations and their Applications. 34. Boston, MA: Birkhäuser. vii, 313 p. DM 178.00; öS 1300.00; sFr. 148.00 (1998). ISBN 0-8176-4024-X/hbk

In 1940 S. Ulam proposed the following problem. Given a metric group $G (\cdot, \rho)$, a number $\varepsilon>0$ and a map $f:G\to G$ which satisfies the inequality $\rho (f(x\cdot y)$, $f(x) \cdot f(y))< \varepsilon$ for all $x,y$ in $G$, does there exist an automorphism $a$ of $G$ and a constant $k>0$, depending only on $G$, such that $\rho (a(x), f(x))\le k\varepsilon$ for all $x$ in $G$? If the answer is affirmative, the equation $a(x\cdot y)= a(x) \cdot a(y)$ is called stable.\par {\it D. H. Hyers} [Proc. Nat. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.26403)] gave the first result, and the subject has been developed by an increasing number of mathematicians, particularly during the last two decades.\par The book under review is an exhaustive presentation of the results in the field, not called Hyers-Ulam stability. It contains chapters on approximately additive and linear mappings, stability of the quadratic functional equation, approximately multiplicative mappings, functions with bounded $n$-th differences, approximately convex functions. The book is of interest not only for people working in functional equations but also for all mathematicians interested in functional analysis.
[G.L.Forti (Milano)]

MSC 2000:: *39B72 Functional inequalities involving unknown functions
39-02 Research monographs (functional equations)
39B05 General theory of functional equations
41-02 Research monographs (approximations and expansions)

Keywords: textbook; Hyers-Ulam stability; additive and linear mappings; quadratic functional equation; approximately multiplicative mappings; approximately convex functions

Citations: Zbl 0061.26403

Cited in: Zbl 1221.39038 Zbl 1214.39013 Zbl 1162.39018 Zbl 1186.47039 Zbl 1153.39031 Zbl 1130.39025 Zbl 1106.39026 Zbl 1085.39026 Zbl 1085.47048 Zbl 1096.39029 Zbl 1051.39032 Zbl 0983.39014 Zbl 0983.39015 Zbl 0983.39013 Zbl 0976.39031 Zbl 0976.39030 Zbl 0929.39015 Zbl 0926.39013 Zbl 0918.39009

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