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Zbl 0989.39009
Baker, John A.
Distributional methods for functional equations.
(English)
[J] Aequationes Math. 62, No.1-2, 136-142 (2001). ISSN 0001-9054; ISSN 1420-8903/e

The author studies two functional equations, due to Z. Daróczy and K. Heuvers respectively, viz. $$f\left({x+y\over 2}\right)+ f \left( {2xy\over x+y}\right) =f(x)+f(y),\ x,y\in I,\tag D$$ where $I$ is an open subinterval of $]0,\infty[$, and $$f(x+y)- f(x)-f(y)= f\left({1\over x}+{1 \over y}\right), \quad x>0,\ y>0.\tag H$$ He carefully describes the tools from the theory of distributions (composition of distributions with submersions and elliptic regularity for ordinary differential equations) that he needs for these and similar functional equations. Using ordinary differential equations that arise from differentiating (D) and (H) twice, he then proves by simple arguments that the distributional solutions of (D) and (H) are $f(x)=\alpha \log x+ \beta$ and $f(x)= \alpha\log x$, respectively, where $\alpha,\beta\in C$ are arbitrary.
[Henrik Stetkaer (Aarhus)]
MSC 2000:
*39B22 Functional equations for real functions
46F10 Operations with distributions (generalized functions)

Keywords: functional equations; distributional solutions

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