caa a22 4500 465824935 CHVBK 20180323112146.0 cr unu---uuuuu 170327e19900201xx s 000 0 eng 10.1007/BF01833940 doi (NATIONALLICENCE)springer-10.1007/BF01833940 Sablik Maciej Institute of Mathematics, Silesian University, ul. Bankowa 14, PL-40 007, Katowice, Poland aut The continuous solution of a functional equation of Abel [Elektronische Daten] [Maciej Sablik] Summary: The functional equationϕ(x) + ϕ(y) = ψ(xf(y) + yf(x)) (1) for the unknown functionsf, ϕ andψ mapping reals into reals appears in the title of N. H. Abel's paper [1] from 1827 and its differentiable solutions are given there. In 1900 D. Hilbert pointed to (1), and to other functional equations considered by Abel, in the second part of his fifth problem. He asked if these equations could be solved without, for instance, assumption of differentiability of given and unknown functions. Hilbert's question was recalled by J. Aczél in 1987, during the 25th International Symposium on Functional Equations in Hamburg-Rissen. In particular Aczél asked for all continuous solutions of (1). An answer to his question is contained in our paper. We determine all continuous functionsf: I → ℝ,ψ: A f (I × I) → ℝ andϕ: I → ℝ that satisfy (1). HereI denotes a real interval containing 0 andA f (x,y) := xf(y) + yf(x), x, y ∈ I. The list contains not only the differentiable solutions, implicitly described by Abel, but also some nondifferentiable ones. Applying some results of C. T. Ng and A. Járai we are able to obtain even a more general result. For instance, the assertion (i.e. the list of solutions) remains unchanged if we replace continuity ofϕ andψ by local boundedness ofϕ orψ∣f(0)I from above or below. Strengthening a bit the assumptions onf we can preserve a large part of the assertion requiring only the measurability of eitherϕ orψ∣f(0)I. Birkhäuser Verlag, 1990 Primary 39B30 nationallicence Secondary 29M99 nationallicence aequationes mathematicae Birkhäuser-Verlag 39/1(1990-02-01), 19-39 0001-9054 39:1<19 1990 39 10 https://doi.org/10.1007/BF01833940 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF01833940 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Sablik Maciej Institute of Mathematics, Silesian University, ul. Bankowa 14, PL-40 007, Katowice, Poland aut NATIONALLICENCE 773 0- aequationes mathematicae Birkhäuser-Verlag 39/1(1990-02-01), 19-39 0001-9054 39:1<19 1990 39 10 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer