Generalized Cauchy difference functional equations

Abstract

This article concerns primarily the study of functional equations of the form f(x) + f(y) − f(x + y) = g(H(x, y)), where H is a given function of two (real or complex) variables, and where f and g are unknown functions. The problem of finding general methods to solve equations of this type was posed at the 37th International Symposium on Functional Equations in 1999 by R. Ger and L. Reich. We present results including the following. Let I be a nonvoid real interval or complex region with I + I ⊂ I. Suppose there exists a particular solution (f0, g0) on I such that f0 is analytic and not affine (or, in the real case, continuously differentiable and strictly convex). If g0 and H are analytic (or, in the real case, continuously differentiable), then all solutions of the equation are of the form \(g(u) = A_{1} \circ g_{0} (u) + c,\;f(x) = A_{1} \circ f_{0} (x) + A_{2} (x) + c,\) where A1, A2 are arbitrary additive functions and c is an arbitrary constant. Analogous results hold for functional equations of the form \(f(x) + f(y) - f \circ \theta ^{{ - 1}} (\theta (x) + \theta (y)) = g(H(x,y))\) with invertible smooth θ.