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Exponential functional equation on spheres. (English) Zbl 1205.39032

Let \(X\) be a real, at least 2-dimensional, normed space and let \(Y\) be a uniquely 2-divisible Abelian semigroup with a neutral element. For \(f_1,f_2,f_3: X\to Y\) the conditional functional equation
\[ f_1(x+y)=f_2(x)f_3(y)\qquad \text{whenever}\;\|x\|=\|y\| \]
is considered. For this equation both the solutions are given and, for \(Y=\mathbb{K}\), the stability is proved. Namely, assuming that the quotient \(f_1(x+y)/f_2(x)+f_3(y)\) is close to 1 whenever \(\|x\|=\|y\|\), there exist \(g_1,g_2,g_3\) satisfying the considered exponential equation on spheres and such that \(f_i/g_i\) are close to 1 on \(X\).

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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