Perturbations of Cauchy differences

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

$$\begin{aligned} f(x+y)-f(x)-f(y)&= \alpha x y \\ f(x+y)-f(x)-f(y)&= \alpha (x y)\\ f(x+y)-f(x)-f(y)&= \alpha (x)\alpha (y). \end{aligned}$$

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.