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Distributional methods for functional equations. (English) Zbl 0989.39009
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.

39B22Functional equations for real functions
46F10Operations with distributions (generalized functions)
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