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Stability of functional equations in several variables. (English) Zbl 0907.39025
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).

In 1940 S. Ulam proposed the following problem. Given a metric group $G\left(·,\rho \right)$, a number $\epsilon >0$ and a map $f:G\to G$ which satisfies the inequality $\rho \left(f\left(x·y\right)$, $f\left(x\right)·f\left(y\right)\right)<\epsilon$ 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 $\rho \left(a\left(x\right),f\left(x\right)\right)\le k\epsilon$ for all $x$ in $G$? If the answer is affirmative, the equation $a\left(x·y\right)=a\left(x\right)·a\left(y\right)$ is called stable.

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.

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.

##### MSC:
 39B72 Systems of functional equations and inequalities 39-02 Research monographs (functional equations) 39B05 General theory of functional equations 41-02 Research monographs (approximations and expansions)