Perturbations of Cauchy differences

TL;DR

This paper studies functional equations related to perturbations of Cauchy differences, characterizing solutions involving additive and exponential functions, and extends previous bilinearity results with explicit solution representations.

Contribution

It extends prior work on Cauchy difference bilinearity by providing new characterizations and explicit solutions for perturbed equations under various assumptions.

Findings

Solutions often reduce to additive or exponential functions

Explicit representations for Levi-Civita type equations are provided

Conditions for existence of real-valued solutions are identified

Abstract

This paper investigates functional equations arising from perturbations of Cauchy differences. We study equations of the form \[ f(x+y)-f(x)-f(y)=B(x,y) \quad \text{or} \quad f(xy)-f(x)f(y) = B(x,y) \] where $B$ is a biadditive mapping, and also more general cases where the inhomogeneity depends on unknown functions \begin{align*} f(x+y)-f(x)-f(y)&= \alpha x y \\[2.5mm] f(x+y)-f(x)-f(y)&= \alpha (x y)\\[2.5mm] f(x+y)-f(x)-f(y)&= \alpha(x)\alpha(y). \end{align*} Our results extend previous work on the bilinearity of the Cauchy exponential difference by Alzer and Matkowski. We characterize solutions under various structural and regularity assumptions, including additive and exponential Cauchy differences, and show that solutions often reduce to additive functions, exponential polynomials, or combinations thereof. For Levi-Civita type equations, we provide explicit representations of solutions in terms of additive and exponential components. Furthermore, we determine conditions under which real-valued solutions exist and describe their forms. The paper concludes with open problems concerning generalized equations that cannot be solved by the methods presented here, suggesting directions for future research.