Maciej Sablik

The continuous solution of a functional equation of Abel

SummaryThe 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. Abels 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. Hilberts 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: 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.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.

Final Part of the Answer to a Hilbert’s Question

We present new results concerning the following functional equation of Abel

On some local topological semigroups

\psi \left( {xf\left( y \right) + yf\left( x \right)} \right) = \phi \left( x \right) + \phi \left( y \right)

Taylor's theorem and functional equations

D. Hilbert in the second part of his fifth problem asked whether it can be solved without differentiability assumption on the unknown functions ψ, f and ϕ. We gave earlier (cf. [9] and [10]) a positive answer assuming however that 0 is either in the domain or the range of f. Now we solve the equation in the remaining case and thus complete the answer to Hilbert’s question.

On a functional equation in actuarial mathematics

In the present paper we consider a system of equations (1), (2) introduced in [2] in connection with the ancient Greek problem of duplicating the cube. We prove also some results in the case of a restricted domain, namely, if y = 2x:.1

A Different Version of Flett's Mean Value Theorem and an Associated Functional Equation

SummaryWe determine all continuous functionsf, defined on a real intervalI with 0∈ I, taking values in ℝ and such that the operationAf:I × I → ℝ given by

The law of large numbers and a functional equation

A_f (x,y) = xf(y) + yf(x)