The continuous solution of a functional equation of Abel

Abstract

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. Aczel in 1987, during the 25th International Symposium on Functional Equations in Hamburg-Rissen. In particular Aczel 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: Af(I × I) → ℝ andϕ: I → ℝ that satisfy (1). HereI denotes a real interval containing 0 andAf(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.