## 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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### References:

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