ABSTRACT
In his 1823 manuscript, Abel had considered, among others, the functional equation
f(x+ y) = g(xy) + h(x− y), x, y ∈ R, (24.1) where f, g, h : R→ R. In the same manuscript he gave the differentiable solutions of (24.1). Hilbert suggested in connection with his fifth problem, that, while the theory of differential equations provides elegant and powerful techniques for solving functional equations, the differentiability assumptions are not inherently required (see Acze´l (1989)). Motivated by Hilbert’s suggestion many researchers in functional equations have treated various functional equations without any regularity assumptions. The general solution of (24.1) was given by Acze´l (1989) and also independently by Lajko´ (1994) (see also Lajko´ (1987)) without any regularity assumption. Chung, Ebanks, Ng and Sahoo (1994) determined the general solution of the Abel functional equation for f, g, h : F→ G, where F is a field belonging to a certain class, and G is an abelian group. When the general solution of a functional equation is known without any regularity assumptions, it is possible to obtain the Hyers-Ulam type stability result. In this chapter, we investigate the Hyers-Ulam stability of this Abel functional equation.
