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Aczél, János

The state of the second part of Hilbert’s fifth problem. (English) Zbl 0676.39004

Bull. Am. Math. Soc., New Ser. 20, No. 2, 153-163 (1989).
Hilbert stated the second part of his fifth problem as follows: “In particular the functional equations treated by Abel with so much ingenuity... and other equations occuring in the literature of mathematics, do not directly involve anything which necessitates the requirement of the differentiability of the accompanying functions... In all these cases, then the problem arises: In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumption?”
The author concentrates on Abel’s results and reports where the Hilbert program has been carried through: the differentiability or continuity conditions used by Abel have been successfully replaced by weaker conditions.
Reviewer: M.C.Zdun

Cited in 5 Reviews
Cited in 26 Documents

MSC:

39B62 Functional inequalities, including subadditivity, convexity, etc.
39-02 Research exposition (monographs, survey articles) pertaining to difference and functional equations
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
01A65 Development of contemporary mathematics
39B12 Iteration theory, iterative and composite equations
39B99 Functional equations and inequalities
39B52 Functional equations for functions with more general domains and/or ranges

Keywords:

Hilbert’s fifth problem; measurability; Abel’s functional equation; differentiability; continuity
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References:

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