Superstability of Generalized Multiplicative Functionals

Let X be a set with a binary operation[composite function] such that, for each x,y,z∈X , either(x[composite function]y)[composite function]z=(x[composite function]z)[composite function]y , or z[composite function](x[composite function]y)=x[composite function](z[composite function]y) . We show the su...

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Published in:Journal of inequalities and applications 2009-01, Vol.2009 (1), p.486375-486375
Main Authors: Miura, Takeshi, Takagi, Hiroyuki, Tsukada, Makoto, Sin-Ei Takahasi
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Takagi, Hiroyuki
Tsukada, Makoto
Sin-Ei Takahasi
description Let X be a set with a binary operation[composite function] such that, for each x,y,z∈X , either(x[composite function]y)[composite function]z=(x[composite function]z)[composite function]y , or z[composite function](x[composite function]y)=x[composite function](z[composite function]y) . We show the superstability of the functional equationg(x[composite function]y)=g(x)g(y) . More explicitly, if [straight epsilon]≥0 andf:X[arrow right]... satisfies |f(x[composite function]y)-f(x)f(y)|≤[straight epsilon] for each x,y∈X , then f(x[composite function]y)=f(x)f(y) for allx,y∈X , or |f(x)|≤(1+1+4[straight epsilon])/2 for all x∈X . In the latter case, the constant (1+1+4[straight epsilon])/2 is the best possible.
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Probability
title Superstability of Generalized Multiplicative Functionals
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