In this paper we prove the Hyers-Ulam stability of the following K-quadratic functional equation [...] where E is a real (or complex) vector space. This result was used to demonstrate the Hyers-Ulam stability on a set of Lebesgue measure zero for the same functional equation.
Let \(G\) be a compact group, let \(n \in N\setminus \{0,1\}\) be a fixed element and let \(\sigma\) be a continuous automorphism on \(G\) such that \(\sigma^n=I\). Using the non-abelian Fourier transform, we determine the non-zero continuous solutions \(f:G \to C\) of the functional equation \[ f(xy)+\sum_{k=1}^{n-1}f(\sigma^k(y)x)=nf(x)f(y),\ x,y \in G,\] in terms of unitary characters of \(G\).
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