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Approximate homomorphisms. (English) Zbl 0806.47056
Summary: We present a survey of ideas and results stemming from the following stability problem of S. M. Ulam. Given a group \(G_ 1\), a metric group \(G_ 2\) and \(\varepsilon> 0\), find \(\delta>0\) such that, if \(f: G_ 1\to G_ 2\) satisfies \(d(f(xy),f(x)f(y))\leq \delta\) for all \(x,y\in G_ 1\), then there exists a homomorphism \(g: G_ 1\to G_ 2\) such that \(d(f(x),g(x))\leq \varepsilon\) for all \(x\in G_ 1\). For Banach spaces the problem was solved by D. Hyers (1941) with \(\delta=\varepsilon\) and \(g(x)= \lim_{n\to\infty} f(2^ n x)/2^ n\).
Section 2 deals with the case where \(G_ 1\) is replaced by an Abelian semigroup \(S\) and \(G_ 2\) by a sequentially complete locally convex topological vector space \(E\). The necessity for the commutativity of \(S\) and the sequential completeness of \(E\) are also considered.
The method of invariant means is demonstrated in Section 3 for mappings from a right (left) amenable semigroup into the complex numbers.
In Section 4 we present results by the second author and others, where the Cauchy difference \(Cf(x,y)= f(x+ y)- f(x)- f(y)\) may be unbounded but satisfies a weaker inequality.
Approximately multiplicative maps are discussed in Section 5, including a stability theorem for homomorphisms of rotations of the circle into itself and approximately multiplicative maps between Banach algebras.
Section 6 is devoted to the work of Z. Moszner (1985) on different definitions of stability.
Results by Z. Gajda and R. Ger (1987) on subadditive set valued mappings from an Abelian semigroup \(S\) to a class of subsets of a Banach space \(X\) are dealt with in Section 7. Furthermore, a result by A. Smajdor (1990) on the stability of a functional equation of Pexider type for set valued maps is presented.
Recent works of K. Baron and others on functional congruences, stemming from theorems of J. G. van der Corput (1940), are outlined in Section 8. Section 9 contains remarks and unsolved problems.

MSC:
47J05 Equations involving nonlinear operators (general)
47H04 Set-valued operators
39A10 Additive difference equations
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