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The stability problem: New results and counterexamples. (English) Zbl 0595.39010

In 1940 S. Ulam investigated the stability of the Cauchy functional equation \(f(x+y)=f(x)+f(y)\). Definition: If G and H are metric groups and \(\epsilon >0\) then a function \(f: G\to H\) is called an \(\epsilon\)- homomorphism if for any x, y in G it follows that \(d(f(x+y),f(x),f(y))\leq \epsilon\). Ulam’s problem: Ulam asked about the existence of homomorphisms \(h: G\to H\), corresponding to \(\epsilon\)-homomorphisms \(f: G\to H\), such that d(f(x),h(x))\(\leq \epsilon\) for any x in G.
If such an h exists then f is called a perturbation of h. In this paper the author employs methods from his 1979 paper [Proc. R. Soc. Edinb., Sect. A 84, 279-281 (1979; Zbl 0439.39004)] to show the existence of homomorphisms h for \(0<\epsilon <1/6\) when G is \(I_ n\), the cyclic group of order n, and H is either G or I, the rotations of the circle. He also shows that in general the results can not be extended to \(\epsilon\geq 1/4\). He then shows the non-existence of the homomorphisms for the group Z of integers and for the finite Abelian groups. Finally, he presents a table of computer generated results for maps from \(I_ n\) to \(I_ n\) for \(n<13\).
Reviewer: K.Heuvers

MSC:

39B99 Functional equations and inequalities
39B72 Systems of functional equations and inequalities
39B52 Functional equations for functions with more general domains and/or ranges
20K01 Finite abelian groups
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups

Citations:

Zbl 0439.39004
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References:

[1] CenzerD., ?The Stability Problem for Transformations of the Circle?, Proc. Royal Soc. Edinburgh 84A, 279-281 (1979).
[2] De laHarpeP. and KaroubiM., ?Representations approchees d’un groupe dans une algebre de Banach?, Manuscripta Math. 22, 293-310 (1977). · Zbl 0371.22007 · doi:10.1007/BF01172669
[3] HyersD. H., ?On the Stability of the Linear Functional Equation?, Proc. Nat. Acad. Sci. U.S.A. 27, 222-224 (1941). · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[4] ShapiroH. N., ?Note on a Problem in Number Theory?, Bull. Amer. Math. Soc. 54, 890-893 (1948). · Zbl 0032.26202 · doi:10.1090/S0002-9904-1948-09090-5
[5] Ulam, S., Sets, Numbers and Universes, MIT Press, 1974. · Zbl 0558.00017
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