Publ. Math. Debrecen 60 / 1-2 (2002), 29–46 On a generalized functional equation of Abel By MACIEJ SABLIK (Katowice) Abstract. We present some results concerning the following generalization of a functional equation of Abel ψ (xf (y)+ yg(x)) = ϕ(x)+ ϕ(y). With f = g we get the original Abel’s equation that was mentioned explixitly by D. Hilbert in the second part of his fifth problem. The present generalization implies many applications in the theory of functional equations, particularly those dealing with determination of parametrized subsemigroups. We solve the equation in the class of continuous real functions defined in an interval containing 0. 1. Introduction In the second part of his fifth problem D. Hilbert (cf. [13]) dealt with functional equations, usually investigated only under the assumption of the differentiability of functions involved, and asked the following: In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumption? In particular, Hilbert mentioned explicitly the following equation (A) ψ (xf (y)+ yf (x)) = ϕ(x)+ ϕ(y) which was considered by N. Abel (cf. [1]). Hilbert’s question was re- called by J. Acz´ el during the Twenty-fifth International Symposium on Functional Equations in 1987 (see [2] and [3]). In our papers [15], [17] Mathematics Subject Classification : 39B22. Key words and phrases : functional equation of Abel, continuous solution, Hilbert’s fifth problem.