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Generalizations of a “folk-theorem” on simple functional equations in a single variable. (English) Zbl 0725.39003
The “folk theorem” the authors start with is the following “Theorem 1. The general solution of (1) \(f(2x)=2f(x)\) on \({\mathbb{R}}_+=]0,\infty [\) is given by \(f(x)=xp(\ln x/\ln 2)\), where p is an arbitrary periodic function with period 1. The solution is continuous, n-times differentiable, \(C^ n\) (n\(\leq \infty)\) or analytic on \({\mathbb{R}}_+\) if p has the respective properties on \({\mathbb{R}}.\)
If f is defined at \(x=0\) and (1) is satisfied then \(f(0)=0\). If f is defined on [0,a[ and bounded on \([2^{-m},2^{-m+1}[\) for one m then f is continuous at 0. However, every solution, differentiable from the right at 0 (and defined on a right neighbourhood of 0, where it satisfies (1)) is of the form \(f(x)=cx.''\)
In the rest of the paper the authors (cf. the abstract) “... present similar and further results concerning general, \(C^ n\) (n\(\leq \infty)\), analytic, locally monotonic or \(\gamma\)-th order convex solutions of the somewhat more general equation \(f(kx)=k^{\gamma}f(x)\) (k\(\neq 1\) a positive, \(\gamma\) a real constant), which seems to be of importance in meteorology...”.

MSC:
39B22 Functional equations for real functions
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References:
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