On a generalization of a functional equation associated with Simpson’s rule

Abstract

The general solution of the functional equation f(x) - g(x) = (x - y)[h(sx + ty) + ф(х) + ψ(у)] for all x,y ε R (the set of reals) with s and t being a pńori known parameters is determined without any regularity assumptions (differentiability, continuity, measurability, etc.) imposed on the real functions f, g, h, ф and ψ. The motivation for studying this equation came from Simpson’s rule for evaluating definite integrals. Special cases of this equation include functioned equations studied by Aczél [1] and Haruki [2].